http://www.simulace.info/api.php?action=feedcontributions&user=Xkorj58&feedformat=atomSimulace.info - User contributions [en]2024-03-28T13:50:33ZUser contributionsMediaWiki 1.31.1http://www.simulace.info/index.php?title=Vacuum_cleaner&diff=17529Vacuum cleaner2019-02-03T17:49:20Z<p>Xkorj58: </p>
<hr />
<div>=Introduction=<br />
Goal of this Netlogo simulation is to imitate AI of self-moving vacuum cleaner. It's aim is to clean whole room from dust by moving around it and sucking mess out of carpet or ground in general. Nowadays AI in these vacuum cleaners differs. Easiest one is just to randomly move around the room and in enough time room will be cleaned. I'm trying out to figure out few algorithms to make vacuuming faster and therefore more effective.<br />
<br />
=Problem definition=<br />
At first I created few rooms, each with different distribuition of furniture and items in it. Making it either harder to go through or make some cleaners to get stuck. Rooms also have different amount of dust in it, which can make vacuuming longer. <br />
Algorithms should optimize time needed for getting rid of dust in room. Measure will be ticks in Netlogo.<br />
<br />
=== Software ===<br />
Simulation was created in Netlogo 6.0.4.<br />
<br />
=Agents=<br />
===Turtles===<br />
In this simulation there is only one type of turtle agent - vacuum cleaner.<br />
It has one variable (not used in all of used algorithms) called "next-to-wall"<br />
Vacuum cleaner is always sprouted somewhere in the room - not inside of a piece of furniture, heading north.<br />
===Patches===<br />
Patch agents are either brown - furniture and items on ground, black - floor tiles with dust on them, or white - cleared floor tiles.<br />
<br />
=Model Settings=<br />
World of this simulation doesn't wrap - rooms are closed. It's 32x32 patches big and it's origin corner is bottom left one.<br />
<br />
=Interface controls=<br />
[[File:Kitchen.JPG|thumb|right|100px|Kitchen]]<br />
[[File:Living_room.JPG |thumb|right|100px|Living room]]<br />
[[File:Kids_room.JPG |thumb|right|100px|Kids room]]<br />
[[File:Training.JPG |thumb|right|100px|Training]]<br />
[[File:Empty.JPG |thumb|right|100px|Empty]]<br />
===Setup===<br />
Prepares world. At first it clears previous world. After that sets up new world. It creates chosen room and furniture. Then it creates cleaner/s somewhere on the ground. It also resets tick counter.<br />
===Go===<br />
This makes cleaner to vacuum dust underneath it. Then it moves it by chosen algorithmic path and vacuums again - making it as usefull as it can be. In the end it ticks once, for ticks to be counted as measurement of effectivity of cleaning algorithm.<br />
===Room===<br />
I created 3 layouts of a room. Each one with different furniture, making it either harder to go trough or easier.<br />
===== Kitchen =====<br />
Kitchen corner with desk. Big table in the middle of the room and 8 chairs - 4 legs each.<br />
<br />
https://imgur.com/a/8sme1jO<br />
===== Living room =====<br />
Living room has a sofa, TV, armchair, teatable, and 3 libraries. This one would be the easiest one to go through, due to not many small items in it and all furniture being quite symetrical. <br />
<br />
https://imgur.com/a/QT31Abp<br />
<br />
===== Kids room =====<br />
Probably the hardest one to go through. It has big bed, desk with chair, one long locker, one small locker and 3 toys. One of the toys is not symetrical with dead end. <br />
<br />
https://imgur.com/a/3AJw7XX<br />
<br />
===== Training =====<br />
Easy room, made for practicing running around walls. Walls having right hand and left hand corners. <br />
<br />
https://imgur.com/a/5OuKsL9<br />
<br />
===== Empty =====<br />
Empty room without any furniture.<br />
<br />
https://imgur.com/a/zVZB3yK<br />
<br />
===Robots===<br />
<br />
Number of vacuum cleaning robots.<br />
<br />
===Path===<br />
<br />
===== Random =====<br />
<br />
If possible, moves forward and randomly turns in some direction.<br />
<br />
===== Circles with walls =====<br />
At first it tries to find a wall it circles around, making the circle bigger and bigger until it reaches wall (or furniture). When it reaches wall it goes around it and cleans around walls. When it hits cleaned spot in front of itself starts going around again making it go around more times. It also counts stuck points - to see if its moving too long on cleaned area or is hitting wall too often. If it does it randomly spins, heading other directions. This perk makes it sometimes little bit longer to finish cleaning, but makes the clearer never to get stuck.<br />
[[File:Circles.JPG |thumb|right|100px|Circles with walls]]<br />
https://imgur.com/a/GoFvLDA<br />
<br />
===== Vision =====<br />
This algorithm makes vacuum cleaner look for dust around itself. If it finds it, heads the cleaner towards the dust patch. If it doesn't it chooses it's direction randomly.<br />
<br />
===== Circles walls with vision=====<br />
It is a combination of previous two algorithms. It makes cleaner circle around until it finds wall. But instead of heading randomly, if getting stuck, it looks around itself and if finds dust it heads its direction<br />
<br />
=Results=<br />
I tried with BeahviourSpace plugin to go with each algortihm thorugh each room 100 times. Needless to say, that each one finishes the room, just amount of ticks differ.<br />
<br />
Averages:<br />
Random : 8160,338 ticks<br />
Circles with walls : 4437.41 ticks<br />
Vision : 10930.662 ticks<br />
Circles walls with vision: 4366.222 ticks<br />
<br />
I found it quite weird that random got better average score than vision. So i tried to make vision higher. When having vision 2 patches around itself suddenly creates signifacantly better results. <br />
<br />
Vision 2 : 3263.134<br />
Vision 3 : 2595.164<br />
<br />
Result then would be - if vacuum cleaner can see bigger spaces around itself it is best to make him move by that. If it can't see far enough - circling around itself and than around walls would be the best strategy then.<br />
<br />
[[Media:Results_Vacuum_xkorj58.zip]]<br />
[[File:Results Vacuum xkorj58 v.2.zip]]<br />
<br />
http://www.uschovna.cz/zasilka/JDAESA63KE6L8YYA-427/ - I couldn't upload to simulace.info, so i uploaded on this link - will be there until 2.2.2019<br />
<br />
=Conclusion=<br />
Best way to clean your room with AI vacuum cleaner is have its algorithm see around itself. But the area has to be big, if cleaner can't see that far away, best algorithm to use is circling. Nowadays but, there are new cleaners that can scan room before they start cleanup, so they could optimize it's path and be way more effective.<br />
<br />
[[Media:Xkorj58-vacuum.nlogo ]]<br />
<br />
http://www.uschovna.cz/zasilka/JDL5P55VGKJKHYBZ-P8A/<br />
I couldn't upload anything to simulace.info, so here is simulation itself. All images and result are also on external websites. Results and simulation will be available until 2.2.2019 [[User:Xkorj58|Xkorj58]] ([[User talk:Xkorj58|talk]]) 15:31, 19 January 2019 (CET)</div>Xkorj58http://www.simulace.info/index.php?title=File:Results_Vacuum_xkorj58_v.2.zip&diff=17528File:Results Vacuum xkorj58 v.2.zip2019-02-03T17:48:04Z<p>Xkorj58: Version 2 of Results</p>
<hr />
<div>== Summary ==<br />
Version 2 of Results</div>Xkorj58http://www.simulace.info/index.php?title=Vacuum_cleaner&diff=17445Vacuum cleaner2019-01-24T21:22:28Z<p>Xkorj58: </p>
<hr />
<div>=Introduction=<br />
Goal of this Netlogo simulation is to imitate AI of self-moving vacuum cleaner. It's aim is to clean whole room from dust by moving around it and sucking mess out of carpet or ground in general. Nowadays AI in these vacuum cleaners differs. Easiest one is just to randomly move around the room and in enough time room will be cleaned. I'm trying out to figure out few algorithms to make vacuuming faster and therefore more effective.<br />
<br />
=Problem definition=<br />
At first I created few rooms, each with different distribuition of furniture and items in it. Making it either harder to go through or make some cleaners to get stuck. Rooms also have different amount of dust in it, which can make vacuuming longer. <br />
Algorithms should optimize time needed for getting rid of dust in room. Measure will be ticks in Netlogo.<br />
<br />
=== Software ===<br />
Simulation was created in Netlogo 6.0.4.<br />
<br />
=Agents=<br />
===Turtles===<br />
In this simulation there is only one type of turtle agent - vacuum cleaner.<br />
It has one variable (not used in all of used algorithms) called "next-to-wall"<br />
Vacuum cleaner is always sprouted somewhere in the room - not inside of a piece of furniture, heading north.<br />
===Patches===<br />
Patch agents are either brown - furniture and items on ground, black - floor tiles with dust on them, or white - cleared floor tiles.<br />
<br />
=Model Settings=<br />
World of this simulation doesn't wrap - rooms are closed. It's 32x32 patches big and it's origin corner is bottom left one.<br />
<br />
=Interface controls=<br />
[[File:Kitchen.JPG|thumb|right|100px|Kitchen]]<br />
[[File:Living_room.JPG |thumb|right|100px|Living room]]<br />
[[File:Kids_room.JPG |thumb|right|100px|Kids room]]<br />
[[File:Training.JPG |thumb|right|100px|Training]]<br />
[[File:Empty.JPG |thumb|right|100px|Empty]]<br />
===Setup===<br />
Prepares world. At first it clears previous world. After that sets up new world. It creates chosen room and furniture. Then it creates cleaner/s somewhere on the ground. It also resets tick counter.<br />
===Go===<br />
This makes cleaner to vacuum dust underneath it. Then it moves it by chosen algorithmic path and vacuums again - making it as usefull as it can be. In the end it ticks once, for ticks to be counted as measurement of effectivity of cleaning algorithm.<br />
===Room===<br />
I created 3 layouts of a room. Each one with different furniture, making it either harder to go trough or easier.<br />
===== Kitchen =====<br />
Kitchen corner with desk. Big table in the middle of the room and 8 chairs - 4 legs each.<br />
<br />
https://imgur.com/a/8sme1jO<br />
===== Living room =====<br />
Living room has a sofa, TV, armchair, teatable, and 3 libraries. This one would be the easiest one to go through, due to not many small items in it and all furniture being quite symetrical. <br />
<br />
https://imgur.com/a/QT31Abp<br />
<br />
===== Kids room =====<br />
Probably the hardest one to go through. It has big bed, desk with chair, one long locker, one small locker and 3 toys. One of the toys is not symetrical with dead end. <br />
<br />
https://imgur.com/a/3AJw7XX<br />
<br />
===== Training =====<br />
Easy room, made for practicing running around walls. Walls having right hand and left hand corners. <br />
<br />
https://imgur.com/a/5OuKsL9<br />
<br />
===== Empty =====<br />
Empty room without any furniture.<br />
<br />
https://imgur.com/a/zVZB3yK<br />
<br />
===Robots===<br />
<br />
Number of vacuum cleaning robots.<br />
<br />
===Path===<br />
<br />
===== Random =====<br />
<br />
If possible, moves forward and randomly turns in some direction.<br />
<br />
===== Circles with walls =====<br />
At first it tries to find a wall it circles around, making the circle bigger and bigger until it reaches wall (or furniture). When it reaches wall it goes around it and cleans around walls. When it hits cleaned spot in front of itself starts going around again making it go around more times. It also counts stuck points - to see if its moving too long on cleaned area or is hitting wall too often. If it does it randomly spins, heading other directions. This perk makes it sometimes little bit longer to finish cleaning, but makes the clearer never to get stuck.<br />
[[File:Circles.JPG |thumb|right|100px|Circles with walls]]<br />
https://imgur.com/a/GoFvLDA<br />
<br />
===== Vision =====<br />
This algorithm makes vacuum cleaner look for dust around itself. If it finds it, heads the cleaner towards the dust patch. If it doesn't it chooses it's direction randomly.<br />
<br />
===== Circles walls with vision=====<br />
It is a combination of previous two algorithms. It makes cleaner circle around until it finds wall. But instead of heading randomly, if getting stuck, it looks around itself and if finds dust it heads its direction<br />
<br />
=Results=<br />
I tried with BeahviourSpace plugin to go with each algortihm thorugh each room 100 times. Needless to say, that each one finishes the room, just amount of ticks differ.<br />
<br />
Averages:<br />
Random : 8160,338 ticks<br />
Circles with walls : 4437.41 ticks<br />
Vision : 10930.662 ticks<br />
Circles walls with vision: 4366.222 ticks<br />
<br />
I found it quite weird that random got better average score than vision. So i tried to make vision higher. When having vision 2 patches around itself suddenly creates signifacantly better results. <br />
<br />
Vision 2 : 3263.134<br />
Vision 3 : 2595.164<br />
<br />
Result then would be - if vacuum cleaner can see bigger spaces around itself it is best to make him move by that. If it can't see far enough - circling around itself and than around walls would be the best strategy then.<br />
<br />
[[Media:Results_Vacuum_xkorj58.zip]]<br />
<br />
http://www.uschovna.cz/zasilka/JDAESA63KE6L8YYA-427/ - I couldn't upload to simulace.info, so i uploaded on this link - will be there until 2.2.2019<br />
<br />
=Conclusion=<br />
Best way to clean your room with AI vacuum cleaner is have its algorithm see around itself. But the area has to be big, if cleaner can't see that far away, best algorithm to use is circling. Nowadays but, there are new cleaners that can scan room before they start cleanup, so they could optimize it's path and be way more effective.<br />
<br />
[[Media:Xkorj58-vacuum.nlogo ]]<br />
<br />
http://www.uschovna.cz/zasilka/JDL5P55VGKJKHYBZ-P8A/<br />
I couldn't upload anything to simulace.info, so here is simulation itself. All images and result are also on external websites. Results and simulation will be available until 2.2.2019 [[User:Xkorj58|Xkorj58]] ([[User talk:Xkorj58|talk]]) 15:31, 19 January 2019 (CET)</div>Xkorj58http://www.simulace.info/index.php?title=Vacuum_cleaner&diff=17444Vacuum cleaner2019-01-24T21:21:12Z<p>Xkorj58: </p>
<hr />
<div>=Introduction=<br />
Goal of this Netlogo simulation is to imitate AI of self-moving vacuum cleaner. It's aim is to clean whole room from dust by moving around it and sucking mess out of carpet or ground in general. Nowadays AI in these vacuum cleaners differs. Easiest one is just to randomly move around the room and in enough time room will be cleaned. I'm trying out to figure out few algorithms to make vacuuming faster and therefore more effective.<br />
<br />
=Problem definition=<br />
At first I created few rooms, each with different distribuition of furniture and items in it. Making it either harder to go through or make some cleaners to get stuck. Rooms also have different amount of dust in it, which can make vacuuming longer. <br />
Algorithms should optimize time needed for getting rid of dust in room. Measure will be ticks in Netlogo.<br />
<br />
=== Software ===<br />
Simulation was created in Netlogo 6.0.4.<br />
<br />
=Agents=<br />
===Turtles===<br />
In this simulation there is only one type of turtle agent - vacuum cleaner.<br />
It has one variable (not used in all of used algorithms) called "next-to-wall"<br />
Vacuum cleaner is always sprouted somewhere in the room - not inside of a piece of furniture, heading north.<br />
===Patches===<br />
Patch agents are either brown - furniture and items on ground, black - floor tiles with dust on them, or white - cleared floor tiles.<br />
<br />
=Model Settings=<br />
World of this simulation doesn't wrap - rooms are closed. It's 32x32 patches big and it's origin corner is bottom left one.<br />
<br />
=Interface controls=<br />
<br />
===Setup===<br />
Prepares world. At first it clears previous world. After that sets up new world. It creates chosen room and furniture. Then it creates cleaner/s somewhere on the ground. It also resets tick counter.<br />
===Go===<br />
This makes cleaner to vacuum dust underneath it. Then it moves it by chosen algorithmic path and vacuums again - making it as usefull as it can be. In the end it ticks once, for ticks to be counted as measurement of effectivity of cleaning algorithm.<br />
===Room===<br />
I created 3 layouts of a room. Each one with different furniture, making it either harder to go trough or easier.<br />
===== Kitchen =====<br />
Kitchen corner with desk. Big table in the middle of the room and 8 chairs - 4 legs each.<br />
[[File:Kitchen.JPG|thumb|right|100px|Kitchen]]<br />
https://imgur.com/a/8sme1jO<br />
===== Living room =====<br />
Living room has a sofa, TV, armchair, teatable, and 3 libraries. This one would be the easiest one to go through, due to not many small items in it and all furniture being quite symetrical. <br />
[[File:Living_room.JPG |thumb|right|100px|Living room]]<br />
https://imgur.com/a/QT31Abp<br />
<br />
===== Kids room =====<br />
Probably the hardest one to go through. It has big bed, desk with chair, one long locker, one small locker and 3 toys. One of the toys is not symetrical with dead end. <br />
[[File:Kids_room.JPG |thumb|right|100px|Kids room]]<br />
https://imgur.com/a/3AJw7XX<br />
<br />
===== Training =====<br />
Easy room, made for practicing running around walls. Walls having right hand and left hand corners. <br />
[[File:Training.JPG |thumb|right|100px|Training]]<br />
https://imgur.com/a/5OuKsL9<br />
<br />
===== Empty =====<br />
Empty room without any furniture.<br />
[[File:Empty.JPG |thumb|right|100px|Empty]]<br />
https://imgur.com/a/zVZB3yK<br />
<br />
===Robots===<br />
<br />
Number of vacuum cleaning robots.<br />
<br />
===Path===<br />
<br />
===== Random =====<br />
<br />
If possible, moves forward and randomly turns in some direction.<br />
<br />
===== Circles with walls =====<br />
At first it tries to find a wall it circles around, making the circle bigger and bigger until it reaches wall (or furniture). When it reaches wall it goes around it and cleans around walls. When it hits cleaned spot in front of itself starts going around again making it go around more times. It also counts stuck points - to see if its moving too long on cleaned area or is hitting wall too often. If it does it randomly spins, heading other directions. This perk makes it sometimes little bit longer to finish cleaning, but makes the clearer never to get stuck.<br />
[[File:Circles.JPG |thumb|left|100px|Circles with walls]]<br />
https://imgur.com/a/GoFvLDA<br />
<br />
===== Vision =====<br />
This algorithm makes vacuum cleaner look for dust around itself. If it finds it, heads the cleaner towards the dust patch. If it doesn't it chooses it's direction randomly.<br />
<br />
===== Circles walls with vision=====<br />
It is a combination of previous two algorithms. It makes cleaner circle around until it finds wall. But instead of heading randomly, if getting stuck, it looks around itself and if finds dust it heads its direction<br />
<br />
=Results=<br />
I tried with BeahviourSpace plugin to go with each algortihm thorugh each room 100 times. Needless to say, that each one finishes the room, just amount of ticks differ.<br />
<br />
Averages:<br />
Random : 8160,338 ticks<br />
Circles with walls : 4437.41 ticks<br />
Vision : 10930.662 ticks<br />
Circles walls with vision: 4366.222 ticks<br />
<br />
I found it quite weird that random got better average score than vision. So i tried to make vision higher. When having vision 2 patches around itself suddenly creates signifacantly better results. <br />
<br />
Vision 2 : 3263.134<br />
Vision 3 : 2595.164<br />
<br />
Result then would be - if vacuum cleaner can see bigger spaces around itself it is best to make him move by that. If it can't see far enough - circling around itself and than around walls would be the best strategy then.<br />
<br />
[[Media:Results_Vacuum_xkorj58.zip]]<br />
<br />
http://www.uschovna.cz/zasilka/JDAESA63KE6L8YYA-427/ - I couldn't upload to simulace.info, so i uploaded on this link - will be there until 2.2.2019<br />
<br />
=Conclusion=<br />
Best way to clean your room with AI vacuum cleaner is have its algorithm see around itself. But the area has to be big, if cleaner can't see that far away, best algorithm to use is circling. Nowadays but, there are new cleaners that can scan room before they start cleanup, so they could optimize it's path and be way more effective.<br />
<br />
[[Media:Xkorj58-vacuum.nlogo ]]<br />
<br />
http://www.uschovna.cz/zasilka/JDL5P55VGKJKHYBZ-P8A/<br />
I couldn't upload anything to simulace.info, so here is simulation itself. All images and result are also on external websites. Results and simulation will be available until 2.2.2019 [[User:Xkorj58|Xkorj58]] ([[User talk:Xkorj58|talk]]) 15:31, 19 January 2019 (CET)</div>Xkorj58http://www.simulace.info/index.php?title=Vacuum_cleaner&diff=17443Vacuum cleaner2019-01-24T21:20:47Z<p>Xkorj58: </p>
<hr />
<div>=Introduction=<br />
Goal of this Netlogo simulation is to imitate AI of self-moving vacuum cleaner. It's aim is to clean whole room from dust by moving around it and sucking mess out of carpet or ground in general. Nowadays AI in these vacuum cleaners differs. Easiest one is just to randomly move around the room and in enough time room will be cleaned. I'm trying out to figure out few algorithms to make vacuuming faster and therefore more effective.<br />
<br />
=Problem definition=<br />
At first I created few rooms, each with different distribuition of furniture and items in it. Making it either harder to go through or make some cleaners to get stuck. Rooms also have different amount of dust in it, which can make vacuuming longer. <br />
Algorithms should optimize time needed for getting rid of dust in room. Measure will be ticks in Netlogo.<br />
<br />
=== Software ===<br />
Simulation was created in Netlogo 6.0.4.<br />
<br />
=Agents=<br />
===Turtles===<br />
In this simulation there is only one type of turtle agent - vacuum cleaner.<br />
It has one variable (not used in all of used algorithms) called "next-to-wall"<br />
Vacuum cleaner is always sprouted somewhere in the room - not inside of a piece of furniture, heading north.<br />
===Patches===<br />
Patch agents are either brown - furniture and items on ground, black - floor tiles with dust on them, or white - cleared floor tiles.<br />
<br />
=Model Settings=<br />
World of this simulation doesn't wrap - rooms are closed. It's 32x32 patches big and it's origin corner is bottom left one.<br />
<br />
=Interface controls=<br />
<br />
===Setup===<br />
Prepares world. At first it clears previous world. After that sets up new world. It creates chosen room and furniture. Then it creates cleaner/s somewhere on the ground. It also resets tick counter.<br />
===Go===<br />
This makes cleaner to vacuum dust underneath it. Then it moves it by chosen algorithmic path and vacuums again - making it as usefull as it can be. In the end it ticks once, for ticks to be counted as measurement of effectivity of cleaning algorithm.<br />
===Room===<br />
I created 3 layouts of a room. Each one with different furniture, making it either harder to go trough or easier.<br />
===== Kitchen =====<br />
Kitchen corner with desk. Big table in the middle of the room and 8 chairs - 4 legs each.<br />
[[File:Kitchen.JPG|thumb|right|150px|Kitchen]]<br />
https://imgur.com/a/8sme1jO<br />
===== Living room =====<br />
Living room has a sofa, TV, armchair, teatable, and 3 libraries. This one would be the easiest one to go through, due to not many small items in it and all furniture being quite symetrical. <br />
[[File:Living_room.JPG |thumb|right|100px|Living room]]<br />
https://imgur.com/a/QT31Abp<br />
<br />
===== Kids room =====<br />
Probably the hardest one to go through. It has big bed, desk with chair, one long locker, one small locker and 3 toys. One of the toys is not symetrical with dead end. <br />
[[File:Kids_room.JPG |thumb|right|100px|Kids room]]<br />
https://imgur.com/a/3AJw7XX<br />
<br />
===== Training =====<br />
Easy room, made for practicing running around walls. Walls having right hand and left hand corners. <br />
[[File:Training.JPG |thumb|right|100px|Training]]<br />
https://imgur.com/a/5OuKsL9<br />
<br />
===== Empty =====<br />
Empty room without any furniture.<br />
[[File:Empty.JPG |thumb|right|100px|Empty]]<br />
https://imgur.com/a/zVZB3yK<br />
<br />
===Robots===<br />
<br />
Number of vacuum cleaning robots.<br />
<br />
===Path===<br />
<br />
===== Random =====<br />
<br />
If possible, moves forward and randomly turns in some direction.<br />
<br />
===== Circles with walls =====<br />
At first it tries to find a wall it circles around, making the circle bigger and bigger until it reaches wall (or furniture). When it reaches wall it goes around it and cleans around walls. When it hits cleaned spot in front of itself starts going around again making it go around more times. It also counts stuck points - to see if its moving too long on cleaned area or is hitting wall too often. If it does it randomly spins, heading other directions. This perk makes it sometimes little bit longer to finish cleaning, but makes the clearer never to get stuck.<br />
[[File:Circles.JPG |thumb|left|100px|Circles with walls]]<br />
https://imgur.com/a/GoFvLDA<br />
<br />
===== Vision =====<br />
This algorithm makes vacuum cleaner look for dust around itself. If it finds it, heads the cleaner towards the dust patch. If it doesn't it chooses it's direction randomly.<br />
<br />
===== Circles walls with vision=====<br />
It is a combination of previous two algorithms. It makes cleaner circle around until it finds wall. But instead of heading randomly, if getting stuck, it looks around itself and if finds dust it heads its direction<br />
<br />
=Results=<br />
I tried with BeahviourSpace plugin to go with each algortihm thorugh each room 100 times. Needless to say, that each one finishes the room, just amount of ticks differ.<br />
<br />
Averages:<br />
Random : 8160,338 ticks<br />
Circles with walls : 4437.41 ticks<br />
Vision : 10930.662 ticks<br />
Circles walls with vision: 4366.222 ticks<br />
<br />
I found it quite weird that random got better average score than vision. So i tried to make vision higher. When having vision 2 patches around itself suddenly creates signifacantly better results. <br />
<br />
Vision 2 : 3263.134<br />
Vision 3 : 2595.164<br />
<br />
Result then would be - if vacuum cleaner can see bigger spaces around itself it is best to make him move by that. If it can't see far enough - circling around itself and than around walls would be the best strategy then.<br />
<br />
[[Media:Results_Vacuum_xkorj58.zip]]<br />
<br />
http://www.uschovna.cz/zasilka/JDAESA63KE6L8YYA-427/ - I couldn't upload to simulace.info, so i uploaded on this link - will be there until 2.2.2019<br />
<br />
=Conclusion=<br />
Best way to clean your room with AI vacuum cleaner is have its algorithm see around itself. But the area has to be big, if cleaner can't see that far away, best algorithm to use is circling. Nowadays but, there are new cleaners that can scan room before they start cleanup, so they could optimize it's path and be way more effective.<br />
<br />
[[Media:Xkorj58-vacuum.nlogo ]]<br />
<br />
http://www.uschovna.cz/zasilka/JDL5P55VGKJKHYBZ-P8A/<br />
I couldn't upload anything to simulace.info, so here is simulation itself. All images and result are also on external websites. Results and simulation will be available until 2.2.2019 [[User:Xkorj58|Xkorj58]] ([[User talk:Xkorj58|talk]]) 15:31, 19 January 2019 (CET)</div>Xkorj58http://www.simulace.info/index.php?title=Vacuum_cleaner&diff=17442Vacuum cleaner2019-01-24T21:19:49Z<p>Xkorj58: </p>
<hr />
<div>=Introduction=<br />
Goal of this Netlogo simulation is to imitate AI of self-moving vacuum cleaner. It's aim is to clean whole room from dust by moving around it and sucking mess out of carpet or ground in general. Nowadays AI in these vacuum cleaners differs. Easiest one is just to randomly move around the room and in enough time room will be cleaned. I'm trying out to figure out few algorithms to make vacuuming faster and therefore more effective.<br />
<br />
=Problem definition=<br />
At first I created few rooms, each with different distribuition of furniture and items in it. Making it either harder to go through or make some cleaners to get stuck. Rooms also have different amount of dust in it, which can make vacuuming longer. <br />
Algorithms should optimize time needed for getting rid of dust in room. Measure will be ticks in Netlogo.<br />
<br />
=== Software ===<br />
Simulation was created in Netlogo 6.0.4.<br />
<br />
=Agents=<br />
===Turtles===<br />
In this simulation there is only one type of turtle agent - vacuum cleaner.<br />
It has one variable (not used in all of used algorithms) called "next-to-wall"<br />
Vacuum cleaner is always sprouted somewhere in the room - not inside of a piece of furniture, heading north.<br />
===Patches===<br />
Patch agents are either brown - furniture and items on ground, black - floor tiles with dust on them, or white - cleared floor tiles.<br />
<br />
=Model Settings=<br />
World of this simulation doesn't wrap - rooms are closed. It's 32x32 patches big and it's origin corner is bottom left one.<br />
<br />
=Interface controls=<br />
<br />
===Setup===<br />
Prepares world. At first it clears previous world. After that sets up new world. It creates chosen room and furniture. Then it creates cleaner/s somewhere on the ground. It also resets tick counter.<br />
===Go===<br />
This makes cleaner to vacuum dust underneath it. Then it moves it by chosen algorithmic path and vacuums again - making it as usefull as it can be. In the end it ticks once, for ticks to be counted as measurement of effectivity of cleaning algorithm.<br />
===Room===<br />
I created 3 layouts of a room. Each one with different furniture, making it either harder to go trough or easier.<br />
===== Kitchen =====<br />
Kitchen corner with desk. Big table in the middle of the room and 8 chairs - 4 legs each.<br />
[[File:Kitchen.JPG|thumb|right|150px|Kitchen]]<br />
https://imgur.com/a/8sme1jO<br />
===== Living room =====<br />
Living room has a sofa, TV, armchair, teatable, and 3 libraries. This one would be the easiest one to go through, due to not many small items in it and all furniture being quite symetrical. <br />
[[File:Living_room.JPG |thumb|right|150px|Living room]]<br />
https://imgur.com/a/QT31Abp<br />
<br />
===== Kids room =====<br />
Probably the hardest one to go through. It has big bed, desk with chair, one long locker, one small locker and 3 toys. One of the toys is not symetrical with dead end. <br />
[[File:Kids_room.JPG |thumb|right|150px|Kids room]]<br />
https://imgur.com/a/3AJw7XX<br />
<br />
===== Training =====<br />
Easy room, made for practicing running around walls. Walls having right hand and left hand corners. <br />
[[File:Training.JPG |thumb|right|150px|Training]]<br />
https://imgur.com/a/5OuKsL9<br />
<br />
===== Empty =====<br />
Empty room without any furniture.<br />
[[File:Empty.JPG |thumb|right|150px|Empty]]<br />
https://imgur.com/a/zVZB3yK<br />
<br />
===Robots===<br />
<br />
Number of vacuum cleaning robots.<br />
<br />
===Path===<br />
<br />
===== Random =====<br />
<br />
If possible, moves forward and randomly turns in some direction.<br />
<br />
===== Circles with walls =====<br />
At first it tries to find a wall it circles around, making the circle bigger and bigger until it reaches wall (or furniture). When it reaches wall it goes around it and cleans around walls. When it hits cleaned spot in front of itself starts going around again making it go around more times. It also counts stuck points - to see if its moving too long on cleaned area or is hitting wall too often. If it does it randomly spins, heading other directions. This perk makes it sometimes little bit longer to finish cleaning, but makes the clearer never to get stuck.<br />
[[File:Circles.JPG |thumb|right|150px|Circles with walls]]<br />
https://imgur.com/a/GoFvLDA<br />
<br />
===== Vision =====<br />
This algorithm makes vacuum cleaner look for dust around itself. If it finds it, heads the cleaner towards the dust patch. If it doesn't it chooses it's direction randomly.<br />
<br />
===== Circles walls with vision=====<br />
It is a combination of previous two algorithms. It makes cleaner circle around until it finds wall. But instead of heading randomly, if getting stuck, it looks around itself and if finds dust it heads its direction<br />
<br />
=Results=<br />
I tried with BeahviourSpace plugin to go with each algortihm thorugh each room 100 times. Needless to say, that each one finishes the room, just amount of ticks differ.<br />
<br />
Averages:<br />
Random : 8160,338 ticks<br />
Circles with walls : 4437.41 ticks<br />
Vision : 10930.662 ticks<br />
Circles walls with vision: 4366.222 ticks<br />
<br />
I found it quite weird that random got better average score than vision. So i tried to make vision higher. When having vision 2 patches around itself suddenly creates signifacantly better results. <br />
<br />
Vision 2 : 3263.134<br />
Vision 3 : 2595.164<br />
<br />
Result then would be - if vacuum cleaner can see bigger spaces around itself it is best to make him move by that. If it can't see far enough - circling around itself and than around walls would be the best strategy then.<br />
<br />
[[Media:Results_Vacuum_xkorj58.zip]]<br />
<br />
http://www.uschovna.cz/zasilka/JDAESA63KE6L8YYA-427/ - I couldn't upload to simulace.info, so i uploaded on this link - will be there until 2.2.2019<br />
<br />
=Conclusion=<br />
Best way to clean your room with AI vacuum cleaner is have its algorithm see around itself. But the area has to be big, if cleaner can't see that far away, best algorithm to use is circling. Nowadays but, there are new cleaners that can scan room before they start cleanup, so they could optimize it's path and be way more effective.<br />
<br />
[[Media:Xkorj58-vacuum.nlogo ]]<br />
<br />
http://www.uschovna.cz/zasilka/JDL5P55VGKJKHYBZ-P8A/<br />
I couldn't upload anything to simulace.info, so here is simulation itself. All images and result are also on external websites. Results and simulation will be available until 2.2.2019 [[User:Xkorj58|Xkorj58]] ([[User talk:Xkorj58|talk]]) 15:31, 19 January 2019 (CET)</div>Xkorj58http://www.simulace.info/index.php?title=Vacuum_cleaner&diff=17441Vacuum cleaner2019-01-24T21:19:17Z<p>Xkorj58: </p>
<hr />
<div>=Introduction=<br />
Goal of this Netlogo simulation is to imitate AI of self-moving vacuum cleaner. It's aim is to clean whole room from dust by moving around it and sucking mess out of carpet or ground in general. Nowadays AI in these vacuum cleaners differs. Easiest one is just to randomly move around the room and in enough time room will be cleaned. I'm trying out to figure out few algorithms to make vacuuming faster and therefore more effective.<br />
<br />
=Problem definition=<br />
At first I created few rooms, each with different distribuition of furniture and items in it. Making it either harder to go through or make some cleaners to get stuck. Rooms also have different amount of dust in it, which can make vacuuming longer. <br />
Algorithms should optimize time needed for getting rid of dust in room. Measure will be ticks in Netlogo.<br />
<br />
=== Software ===<br />
Simulation was created in Netlogo 6.0.4.<br />
<br />
=Agents=<br />
===Turtles===<br />
In this simulation there is only one type of turtle agent - vacuum cleaner.<br />
It has one variable (not used in all of used algorithms) called "next-to-wall"<br />
Vacuum cleaner is always sprouted somewhere in the room - not inside of a piece of furniture, heading north.<br />
===Patches===<br />
Patch agents are either brown - furniture and items on ground, black - floor tiles with dust on them, or white - cleared floor tiles.<br />
<br />
=Model Settings=<br />
World of this simulation doesn't wrap - rooms are closed. It's 32x32 patches big and it's origin corner is bottom left one.<br />
<br />
=Interface controls=<br />
<br />
===Setup===<br />
Prepares world. At first it clears previous world. After that sets up new world. It creates chosen room and furniture. Then it creates cleaner/s somewhere on the ground. It also resets tick counter.<br />
===Go===<br />
This makes cleaner to vacuum dust underneath it. Then it moves it by chosen algorithmic path and vacuums again - making it as usefull as it can be. In the end it ticks once, for ticks to be counted as measurement of effectivity of cleaning algorithm.<br />
===Room===<br />
I created 3 layouts of a room. Each one with different furniture, making it either harder to go trough or easier.<br />
===== Kitchen =====<br />
Kitchen corner with desk. Big table in the middle of the room and 8 chairs - 4 legs each.<br />
[[File:Kitchen.JPG|thumb|right|150px|Kitchen]]<br />
https://imgur.com/a/8sme1jO<br />
===== Living room =====<br />
Living room has a sofa, TV, armchair, teatable, and 3 libraries. This one would be the easiest one to go through, due to not many small items in it and all furniture being quite symetrical. <br />
[[File:Living_room.JPG |thumb|right|150px|Living room]]<br />
https://imgur.com/a/QT31Abp<br />
<br />
===== Kids room =====<br />
Probably the hardest one to go through. It has big bed, desk with chair, one long locker, one small locker and 3 toys. One of the toys is not symetrical with dead end. <br />
[[File:Kids_room.JPG |thumb|right|150px|Kids room]]<br />
https://imgur.com/a/3AJw7XX<br />
<br />
===== Training =====<br />
Easy room, made for practicing running around walls. Walls having right hand and left hand corners. <br />
[[File:Training.JPG |thumb|right|150px|Training]]<br />
https://imgur.com/a/5OuKsL9<br />
<br />
===== Empty =====<br />
Empty room without any furniture.<br />
[[File:Empty.JPG |thumb|right|150px|Empty]]<br />
https://imgur.com/a/zVZB3yK<br />
<br />
===Robots===<br />
<br />
Number of vacuum cleaning robots.<br />
<br />
===Path===<br />
<br />
===== Random =====<br />
<br />
If possible, moves forward and randomly turns in some direction.<br />
<br />
===== Circles with walls =====<br />
At first it tries to find a wall it circles around, making the circle bigger and bigger until it reaches wall (or furniture). When it reaches wall it goes around it and cleans around walls. When it hits cleaned spot in front of itself starts going around again making it go around more times. It also counts stuck points - to see if its moving too long on cleaned area or is hitting wall too often. If it does it randomly spins, heading other directions. This perk makes it sometimes little bit longer to finish cleaning, but makes the clearer never to get stuck.<br />
[[File:Circles.JPG |thumb|right|150px|Circles with walls]]<br />
https://imgur.com/a/GoFvLDA<br />
<br />
===== Vision =====<br />
This algorithm makes vacuum cleaner look for dust around itself. If it finds it, heads the cleaner towards the dust patch. If it doesn't it chooses it's direction randomly.<br />
<br />
===== Circles walls with vision=====<br />
It is a combination of previous two algorithms. It makes cleaner circle around until it finds wall. But instead of heading randomly, if getting stuck, it looks around itself and if finds dust it heads its direction<br />
<br />
=Results=<br />
I tried with BeahviourSpace plugin to go with each algortihm thorugh each room 100 times. Needless to say, that each one finishes the room, just amount of ticks differ.<br />
<br />
Averages:<br />
Random : 8160,338 ticks<br />
Circles with walls : 4437.41 ticks<br />
Vision : 10930.662 ticks<br />
Circles walls with vision: 4366.222 ticks<br />
<br />
I found it quite weird that random got better average score than vision. So i tried to make vision higher. When having vision 2 patches around itself suddenly creates signifacantly better results. <br />
<br />
Vision 2 : 3263.134<br />
Vision 3 : 2595.164<br />
<br />
Result then would be - if vacuum cleaner can see bigger spaces around itself it is best to make him move by that. If it can't see far enough - circling around itself and than around walls would be the best strategy then.<br />
<br />
[[Media:Results_Vacuum_xkorj58.zip]]<br />
http://www.uschovna.cz/zasilka/JDAESA63KE6L8YYA-427/ - I couldn't upload to simulace.info, so i uploaded on this link - will be there until 2.2.2019<br />
<br />
=Conclusion=<br />
Best way to clean your room with AI vacuum cleaner is have its algorithm see around itself. But the area has to be big, if cleaner can't see that far away, best algorithm to use is circling. Nowadays but, there are new cleaners that can scan room before they start cleanup, so they could optimize it's path and be way more effective.<br />
<br />
[[Media:Xkorj58-vacuum.nlogo ]]<br />
<br />
http://www.uschovna.cz/zasilka/JDL5P55VGKJKHYBZ-P8A/<br />
I couldn't upload anything to simulace.info, so here is simulation itself. All images and result are also on external websites. Results and simulation will be available until 2.2.2019 [[User:Xkorj58|Xkorj58]] ([[User talk:Xkorj58|talk]]) 15:31, 19 January 2019 (CET)</div>Xkorj58http://www.simulace.info/index.php?title=File:Xkorj58-vacuum.nlogo&diff=17440File:Xkorj58-vacuum.nlogo2019-01-24T21:19:06Z<p>Xkorj58: </p>
<hr />
<div></div>Xkorj58http://www.simulace.info/index.php?title=File:Results_Vacuum_xkorj58.zip&diff=17439File:Results Vacuum xkorj58.zip2019-01-24T21:17:57Z<p>Xkorj58: </p>
<hr />
<div></div>Xkorj58http://www.simulace.info/index.php?title=Vacuum_cleaner&diff=17438Vacuum cleaner2019-01-24T21:16:58Z<p>Xkorj58: </p>
<hr />
<div>=Introduction=<br />
Goal of this Netlogo simulation is to imitate AI of self-moving vacuum cleaner. It's aim is to clean whole room from dust by moving around it and sucking mess out of carpet or ground in general. Nowadays AI in these vacuum cleaners differs. Easiest one is just to randomly move around the room and in enough time room will be cleaned. I'm trying out to figure out few algorithms to make vacuuming faster and therefore more effective.<br />
<br />
=Problem definition=<br />
At first I created few rooms, each with different distribuition of furniture and items in it. Making it either harder to go through or make some cleaners to get stuck. Rooms also have different amount of dust in it, which can make vacuuming longer. <br />
Algorithms should optimize time needed for getting rid of dust in room. Measure will be ticks in Netlogo.<br />
<br />
=== Software ===<br />
Simulation was created in Netlogo 6.0.4.<br />
<br />
=Agents=<br />
===Turtles===<br />
In this simulation there is only one type of turtle agent - vacuum cleaner.<br />
It has one variable (not used in all of used algorithms) called "next-to-wall"<br />
Vacuum cleaner is always sprouted somewhere in the room - not inside of a piece of furniture, heading north.<br />
===Patches===<br />
Patch agents are either brown - furniture and items on ground, black - floor tiles with dust on them, or white - cleared floor tiles.<br />
<br />
=Model Settings=<br />
World of this simulation doesn't wrap - rooms are closed. It's 32x32 patches big and it's origin corner is bottom left one.<br />
<br />
=Interface controls=<br />
<br />
===Setup===<br />
Prepares world. At first it clears previous world. After that sets up new world. It creates chosen room and furniture. Then it creates cleaner/s somewhere on the ground. It also resets tick counter.<br />
===Go===<br />
This makes cleaner to vacuum dust underneath it. Then it moves it by chosen algorithmic path and vacuums again - making it as usefull as it can be. In the end it ticks once, for ticks to be counted as measurement of effectivity of cleaning algorithm.<br />
===Room===<br />
I created 3 layouts of a room. Each one with different furniture, making it either harder to go trough or easier.<br />
===== Kitchen =====<br />
Kitchen corner with desk. Big table in the middle of the room and 8 chairs - 4 legs each.<br />
[[File:Kitchen.JPG|thumb|right|100px|Kitchen]]<br />
https://imgur.com/a/8sme1jO<br />
===== Living room =====<br />
Living room has a sofa, TV, armchair, teatable, and 3 libraries. This one would be the easiest one to go through, due to not many small items in it and all furniture being quite symetrical. <br />
[[File:Living_room.JPG |thumb|right||Living room]]<br />
https://imgur.com/a/QT31Abp<br />
<br />
===== Kids room =====<br />
Probably the hardest one to go through. It has big bed, desk with chair, one long locker, one small locker and 3 toys. One of the toys is not symetrical with dead end. <br />
[[File:Kids_room.JPG |thumb|right||Kids room]]<br />
https://imgur.com/a/3AJw7XX<br />
<br />
===== Training =====<br />
Easy room, made for practicing running around walls. Walls having right hand and left hand corners. <br />
[[File:Training.JPG |thumb|right||Training]]<br />
https://imgur.com/a/5OuKsL9<br />
<br />
===== Empty =====<br />
Empty room without any furniture.<br />
[[File:Empty.JPG |thumb|right||Empty]]<br />
https://imgur.com/a/zVZB3yK<br />
<br />
===Robots===<br />
<br />
Number of vacuum cleaning robots.<br />
<br />
===Path===<br />
<br />
===== Random =====<br />
<br />
If possible, moves forward and randomly turns in some direction.<br />
<br />
===== Circles with walls =====<br />
At first it tries to find a wall it circles around, making the circle bigger and bigger until it reaches wall (or furniture). When it reaches wall it goes around it and cleans around walls. When it hits cleaned spot in front of itself starts going around again making it go around more times. It also counts stuck points - to see if its moving too long on cleaned area or is hitting wall too often. If it does it randomly spins, heading other directions. This perk makes it sometimes little bit longer to finish cleaning, but makes the clearer never to get stuck.<br />
[[File:Circles.JPG |thumb|right||Circles with walls]]<br />
https://imgur.com/a/GoFvLDA<br />
<br />
===== Vision =====<br />
This algorithm makes vacuum cleaner look for dust around itself. If it finds it, heads the cleaner towards the dust patch. If it doesn't it chooses it's direction randomly.<br />
<br />
===== Circles walls with vision=====<br />
It is a combination of previous two algorithms. It makes cleaner circle around until it finds wall. But instead of heading randomly, if getting stuck, it looks around itself and if finds dust it heads its direction<br />
<br />
=Results=<br />
I tried with BeahviourSpace plugin to go with each algortihm thorugh each room 100 times. Needless to say, that each one finishes the room, just amount of ticks differ.<br />
<br />
Averages:<br />
Random : 8160,338 ticks<br />
Circles with walls : 4437.41 ticks<br />
Vision : 10930.662 ticks<br />
Circles walls with vision: 4366.222 ticks<br />
<br />
I found it quite weird that random got better average score than vision. So i tried to make vision higher. When having vision 2 patches around itself suddenly creates signifacantly better results. <br />
<br />
Vision 2 : 3263.134<br />
Vision 3 : 2595.164<br />
<br />
Result then would be - if vacuum cleaner can see bigger spaces around itself it is best to make him move by that. If it can't see far enough - circling around itself and than around walls would be the best strategy then.<br />
<br />
<br />
http://www.uschovna.cz/zasilka/JDAESA63KE6L8YYA-427/ - I couldn't upload to simulace.info, so i uploaded on this link - will be there until 2.2.2019<br />
<br />
=Conclusion=<br />
Best way to clean your room with AI vacuum cleaner is have its algorithm see around itself. But the area has to be big, if cleaner can't see that far away, best algorithm to use is circling. Nowadays but, there are new cleaners that can scan room before they start cleanup, so they could optimize it's path and be way more effective.<br />
<br />
http://www.uschovna.cz/zasilka/JDL5P55VGKJKHYBZ-P8A/<br />
I couldn't upload anything to simulace.info, so here is simulation itself. All images and result are also on external websites. Results and simulation will be available until 2.2.2019 [[User:Xkorj58|Xkorj58]] ([[User talk:Xkorj58|talk]]) 15:31, 19 January 2019 (CET)</div>Xkorj58http://www.simulace.info/index.php?title=Vacuum_cleaner&diff=17437Vacuum cleaner2019-01-24T21:15:55Z<p>Xkorj58: </p>
<hr />
<div>=Introduction=<br />
Goal of this Netlogo simulation is to imitate AI of self-moving vacuum cleaner. It's aim is to clean whole room from dust by moving around it and sucking mess out of carpet or ground in general. Nowadays AI in these vacuum cleaners differs. Easiest one is just to randomly move around the room and in enough time room will be cleaned. I'm trying out to figure out few algorithms to make vacuuming faster and therefore more effective.<br />
<br />
=Problem definition=<br />
At first I created few rooms, each with different distribuition of furniture and items in it. Making it either harder to go through or make some cleaners to get stuck. Rooms also have different amount of dust in it, which can make vacuuming longer. <br />
Algorithms should optimize time needed for getting rid of dust in room. Measure will be ticks in Netlogo.<br />
<br />
=== Software ===<br />
Simulation was created in Netlogo 6.0.4.<br />
<br />
=Agents=<br />
===Turtles===<br />
In this simulation there is only one type of turtle agent - vacuum cleaner.<br />
It has one variable (not used in all of used algorithms) called "next-to-wall"<br />
Vacuum cleaner is always sprouted somewhere in the room - not inside of a piece of furniture, heading north.<br />
===Patches===<br />
Patch agents are either brown - furniture and items on ground, black - floor tiles with dust on them, or white - cleared floor tiles.<br />
<br />
=Model Settings=<br />
World of this simulation doesn't wrap - rooms are closed. It's 32x32 patches big and it's origin corner is bottom left one.<br />
<br />
=Interface controls=<br />
<br />
===Setup===<br />
Prepares world. At first it clears previous world. After that sets up new world. It creates chosen room and furniture. Then it creates cleaner/s somewhere on the ground. It also resets tick counter.<br />
===Go===<br />
This makes cleaner to vacuum dust underneath it. Then it moves it by chosen algorithmic path and vacuums again - making it as usefull as it can be. In the end it ticks once, for ticks to be counted as measurement of effectivity of cleaning algorithm.<br />
===Room===<br />
I created 3 layouts of a room. Each one with different furniture, making it either harder to go trough or easier.<br />
===== Kitchen =====<br />
Kitchen corner with desk. Big table in the middle of the room and 8 chairs - 4 legs each.<br />
[[File:Kitchen.JPG|thumb|right||Kitchen]]<br />
https://imgur.com/a/8sme1jO<br />
===== Living room =====<br />
Living room has a sofa, TV, armchair, teatable, and 3 libraries. This one would be the easiest one to go through, due to not many small items in it and all furniture being quite symetrical. <br />
[[File:Living_room.JPG |thumb|right||Living room]]<br />
https://imgur.com/a/QT31Abp<br />
<br />
===== Kids room =====<br />
Probably the hardest one to go through. It has big bed, desk with chair, one long locker, one small locker and 3 toys. One of the toys is not symetrical with dead end. <br />
[[File:Kids_room.JPG |thumb|right||Kids room]]<br />
https://imgur.com/a/3AJw7XX<br />
<br />
===== Training =====<br />
Easy room, made for practicing running around walls. Walls having right hand and left hand corners. <br />
[[File:Training.JPG |thumb|right||Training]]<br />
https://imgur.com/a/5OuKsL9<br />
<br />
===== Empty =====<br />
Empty room without any furniture.<br />
[[File:Empty.JPG |thumb|right||Empty]]<br />
https://imgur.com/a/zVZB3yK<br />
<br />
===Robots===<br />
<br />
Number of vacuum cleaning robots.<br />
<br />
===Path===<br />
<br />
===== Random =====<br />
<br />
If possible, moves forward and randomly turns in some direction.<br />
<br />
===== Circles with walls =====<br />
At first it tries to find a wall it circles around, making the circle bigger and bigger until it reaches wall (or furniture). When it reaches wall it goes around it and cleans around walls. When it hits cleaned spot in front of itself starts going around again making it go around more times. It also counts stuck points - to see if its moving too long on cleaned area or is hitting wall too often. If it does it randomly spins, heading other directions. This perk makes it sometimes little bit longer to finish cleaning, but makes the clearer never to get stuck.<br />
[[File:Circles.JPG |thumb|right||Circles with walls]]<br />
https://imgur.com/a/GoFvLDA<br />
<br />
===== Vision =====<br />
This algorithm makes vacuum cleaner look for dust around itself. If it finds it, heads the cleaner towards the dust patch. If it doesn't it chooses it's direction randomly.<br />
<br />
===== Circles walls with vision=====<br />
It is a combination of previous two algorithms. It makes cleaner circle around until it finds wall. But instead of heading randomly, if getting stuck, it looks around itself and if finds dust it heads its direction<br />
<br />
=Results=<br />
I tried with BeahviourSpace plugin to go with each algortihm thorugh each room 100 times. Needless to say, that each one finishes the room, just amount of ticks differ.<br />
<br />
Averages:<br />
Random : 8160,338 ticks<br />
Circles with walls : 4437.41 ticks<br />
Vision : 10930.662 ticks<br />
Circles walls with vision: 4366.222 ticks<br />
<br />
I found it quite weird that random got better average score than vision. So i tried to make vision higher. When having vision 2 patches around itself suddenly creates signifacantly better results. <br />
<br />
Vision 2 : 3263.134<br />
Vision 3 : 2595.164<br />
<br />
Result then would be - if vacuum cleaner can see bigger spaces around itself it is best to make him move by that. If it can't see far enough - circling around itself and than around walls would be the best strategy then.<br />
<br />
<br />
http://www.uschovna.cz/zasilka/JDAESA63KE6L8YYA-427/ - I couldn't upload to simulace.info, so i uploaded on this link - will be there until 2.2.2019<br />
<br />
=Conclusion=<br />
Best way to clean your room with AI vacuum cleaner is have its algorithm see around itself. But the area has to be big, if cleaner can't see that far away, best algorithm to use is circling. Nowadays but, there are new cleaners that can scan room before they start cleanup, so they could optimize it's path and be way more effective.<br />
<br />
http://www.uschovna.cz/zasilka/JDL5P55VGKJKHYBZ-P8A/<br />
I couldn't upload anything to simulace.info, so here is simulation itself. All images and result are also on external websites. Results and simulation will be available until 2.2.2019 [[User:Xkorj58|Xkorj58]] ([[User talk:Xkorj58|talk]]) 15:31, 19 January 2019 (CET)</div>Xkorj58http://www.simulace.info/index.php?title=File:Circles.JPG&diff=17436File:Circles.JPG2019-01-24T21:14:48Z<p>Xkorj58: </p>
<hr />
<div></div>Xkorj58http://www.simulace.info/index.php?title=File:Empty.JPG&diff=17435File:Empty.JPG2019-01-24T21:13:46Z<p>Xkorj58: </p>
<hr />
<div></div>Xkorj58http://www.simulace.info/index.php?title=File:Training.JPG&diff=17434File:Training.JPG2019-01-24T21:13:35Z<p>Xkorj58: </p>
<hr />
<div></div>Xkorj58http://www.simulace.info/index.php?title=File:Kids_room.JPG&diff=17433File:Kids room.JPG2019-01-24T21:12:29Z<p>Xkorj58: </p>
<hr />
<div></div>Xkorj58http://www.simulace.info/index.php?title=File:Living_room.JPG&diff=17432File:Living room.JPG2019-01-24T21:11:33Z<p>Xkorj58: </p>
<hr />
<div></div>Xkorj58http://www.simulace.info/index.php?title=Vacuum_cleaner&diff=17431Vacuum cleaner2019-01-24T21:10:40Z<p>Xkorj58: </p>
<hr />
<div>=Introduction=<br />
Goal of this Netlogo simulation is to imitate AI of self-moving vacuum cleaner. It's aim is to clean whole room from dust by moving around it and sucking mess out of carpet or ground in general. Nowadays AI in these vacuum cleaners differs. Easiest one is just to randomly move around the room and in enough time room will be cleaned. I'm trying out to figure out few algorithms to make vacuuming faster and therefore more effective.<br />
<br />
=Problem definition=<br />
At first I created few rooms, each with different distribuition of furniture and items in it. Making it either harder to go through or make some cleaners to get stuck. Rooms also have different amount of dust in it, which can make vacuuming longer. <br />
Algorithms should optimize time needed for getting rid of dust in room. Measure will be ticks in Netlogo.<br />
<br />
=== Software ===<br />
Simulation was created in Netlogo 6.0.4.<br />
<br />
=Agents=<br />
===Turtles===<br />
In this simulation there is only one type of turtle agent - vacuum cleaner.<br />
It has one variable (not used in all of used algorithms) called "next-to-wall"<br />
Vacuum cleaner is always sprouted somewhere in the room - not inside of a piece of furniture, heading north.<br />
===Patches===<br />
Patch agents are either brown - furniture and items on ground, black - floor tiles with dust on them, or white - cleared floor tiles.<br />
<br />
=Model Settings=<br />
World of this simulation doesn't wrap - rooms are closed. It's 32x32 patches big and it's origin corner is bottom left one.<br />
<br />
=Interface controls=<br />
<br />
===Setup===<br />
Prepares world. At first it clears previous world. After that sets up new world. It creates chosen room and furniture. Then it creates cleaner/s somewhere on the ground. It also resets tick counter.<br />
===Go===<br />
This makes cleaner to vacuum dust underneath it. Then it moves it by chosen algorithmic path and vacuums again - making it as usefull as it can be. In the end it ticks once, for ticks to be counted as measurement of effectivity of cleaning algorithm.<br />
===Room===<br />
I created 3 layouts of a room. Each one with different furniture, making it either harder to go trough or easier.<br />
===== Kitchen =====<br />
Kitchen corner with desk. Big table in the middle of the room and 8 chairs - 4 legs each.<br />
[[File:Kitchen.JPG|thumb|left||Kitchen]]<br />
https://imgur.com/a/8sme1jO<br />
===== Living room =====<br />
Living room has a sofa, TV, armchair, teatable, and 3 libraries. This one would be the easiest one to go through, due to not many small items in it and all furniture being quite symetrical. https://imgur.com/a/QT31Abp<br />
<br />
===== Kids room =====<br />
Probably the hardest one to go through. It has big bed, desk with chair, one long locker, one small locker and 3 toys. One of the toys is not symetrical with dead end. <br />
https://imgur.com/a/3AJw7XX<br />
<br />
===== Training =====<br />
Easy room, made for practicing running around walls. Walls having right hand and left hand corners. <br />
https://imgur.com/a/5OuKsL9<br />
<br />
===== Empty =====<br />
Empty room without any furniture.<br />
https://imgur.com/a/zVZB3yK<br />
<br />
===Robots===<br />
<br />
Number of vacuum cleaning robots.<br />
<br />
===Path===<br />
<br />
===== Random =====<br />
<br />
If possible, moves forward and randomly turns in some direction.<br />
<br />
===== Circles with walls =====<br />
At first it tries to find a wall it circles around, making the circle bigger and bigger until it reaches wall (or furniture). When it reaches wall it goes around it and cleans around walls. When it hits cleaned spot in front of itself starts going around again making it go around more times. It also counts stuck points - to see if its moving too long on cleaned area or is hitting wall too often. If it does it randomly spins, heading other directions. This perk makes it sometimes little bit longer to finish cleaning, but makes the clearer never to get stuck.<br />
https://imgur.com/a/GoFvLDA<br />
<br />
===== Vision =====<br />
This algorithm makes vacuum cleaner look for dust around itself. If it finds it, heads the cleaner towards the dust patch. If it doesn't it chooses it's direction randomly.<br />
<br />
===== Circles walls with vision=====<br />
It is a combination of previous two algorithms. It makes cleaner circle around until it finds wall. But instead of heading randomly, if getting stuck, it looks around itself and if finds dust it heads its direction<br />
<br />
=Results=<br />
I tried with BeahviourSpace plugin to go with each algortihm thorugh each room 100 times. Needless to say, that each one finishes the room, just amount of ticks differ.<br />
<br />
Averages:<br />
Random : 8160,338 ticks<br />
Circles with walls : 4437.41 ticks<br />
Vision : 10930.662 ticks<br />
Circles walls with vision: 4366.222 ticks<br />
<br />
I found it quite weird that random got better average score than vision. So i tried to make vision higher. When having vision 2 patches around itself suddenly creates signifacantly better results. <br />
<br />
Vision 2 : 3263.134<br />
Vision 3 : 2595.164<br />
<br />
Result then would be - if vacuum cleaner can see bigger spaces around itself it is best to make him move by that. If it can't see far enough - circling around itself and than around walls would be the best strategy then.<br />
<br />
http://www.uschovna.cz/zasilka/JDAESA63KE6L8YYA-427/ - I couldn't upload to simulace.info, so i uploaded on this link - will be there until 2.2.2019<br />
<br />
=Conclusion=<br />
Best way to clean your room with AI vacuum cleaner is have its algorithm see around itself. But the area has to be big, if cleaner can't see that far away, best algorithm to use is circling. Nowadays but, there are new cleaners that can scan room before they start cleanup, so they could optimize it's path and be way more effective.<br />
<br />
http://www.uschovna.cz/zasilka/JDL5P55VGKJKHYBZ-P8A/<br />
I couldn't upload anything to simulace.info, so here is simulation itself. All images and result are also on external websites. Results and simulation will be available until 2.2.2019 [[User:Xkorj58|Xkorj58]] ([[User talk:Xkorj58|talk]]) 15:31, 19 January 2019 (CET)</div>Xkorj58http://www.simulace.info/index.php?title=Vacuum_cleaner&diff=17430Vacuum cleaner2019-01-24T21:10:09Z<p>Xkorj58: </p>
<hr />
<div>=Introduction=<br />
Goal of this Netlogo simulation is to imitate AI of self-moving vacuum cleaner. It's aim is to clean whole room from dust by moving around it and sucking mess out of carpet or ground in general. Nowadays AI in these vacuum cleaners differs. Easiest one is just to randomly move around the room and in enough time room will be cleaned. I'm trying out to figure out few algorithms to make vacuuming faster and therefore more effective.<br />
<br />
=Problem definition=<br />
At first I created few rooms, each with different distribuition of furniture and items in it. Making it either harder to go through or make some cleaners to get stuck. Rooms also have different amount of dust in it, which can make vacuuming longer. <br />
Algorithms should optimize time needed for getting rid of dust in room. Measure will be ticks in Netlogo.<br />
<br />
=== Software ===<br />
Simulation was created in Netlogo 6.0.4.<br />
<br />
=Agents=<br />
===Turtles===<br />
In this simulation there is only one type of turtle agent - vacuum cleaner.<br />
It has one variable (not used in all of used algorithms) called "next-to-wall"<br />
Vacuum cleaner is always sprouted somewhere in the room - not inside of a piece of furniture, heading north.<br />
===Patches===<br />
Patch agents are either brown - furniture and items on ground, black - floor tiles with dust on them, or white - cleared floor tiles.<br />
<br />
=Model Settings=<br />
World of this simulation doesn't wrap - rooms are closed. It's 32x32 patches big and it's origin corner is bottom left one.<br />
<br />
=Interface controls=<br />
<br />
===Setup===<br />
Prepares world. At first it clears previous world. After that sets up new world. It creates chosen room and furniture. Then it creates cleaner/s somewhere on the ground. It also resets tick counter.<br />
===Go===<br />
This makes cleaner to vacuum dust underneath it. Then it moves it by chosen algorithmic path and vacuums again - making it as usefull as it can be. In the end it ticks once, for ticks to be counted as measurement of effectivity of cleaning algorithm.<br />
===Room===<br />
I created 3 layouts of a room. Each one with different furniture, making it either harder to go trough or easier.<br />
===== Kitchen =====<br />
Kitchen corner with desk. Big table in the middle of the room and 8 chairs - 4 legs each.<br />
[[File:Kitchen.jpg|thumb|left|Kitchen]]<br />
https://imgur.com/a/8sme1jO<br />
===== Living room =====<br />
Living room has a sofa, TV, armchair, teatable, and 3 libraries. This one would be the easiest one to go through, due to not many small items in it and all furniture being quite symetrical. https://imgur.com/a/QT31Abp<br />
<br />
===== Kids room =====<br />
Probably the hardest one to go through. It has big bed, desk with chair, one long locker, one small locker and 3 toys. One of the toys is not symetrical with dead end. <br />
https://imgur.com/a/3AJw7XX<br />
<br />
===== Training =====<br />
Easy room, made for practicing running around walls. Walls having right hand and left hand corners. <br />
https://imgur.com/a/5OuKsL9<br />
<br />
===== Empty =====<br />
Empty room without any furniture.<br />
https://imgur.com/a/zVZB3yK<br />
<br />
===Robots===<br />
<br />
Number of vacuum cleaning robots.<br />
<br />
===Path===<br />
<br />
===== Random =====<br />
<br />
If possible, moves forward and randomly turns in some direction.<br />
<br />
===== Circles with walls =====<br />
At first it tries to find a wall it circles around, making the circle bigger and bigger until it reaches wall (or furniture). When it reaches wall it goes around it and cleans around walls. When it hits cleaned spot in front of itself starts going around again making it go around more times. It also counts stuck points - to see if its moving too long on cleaned area or is hitting wall too often. If it does it randomly spins, heading other directions. This perk makes it sometimes little bit longer to finish cleaning, but makes the clearer never to get stuck.<br />
https://imgur.com/a/GoFvLDA<br />
<br />
===== Vision =====<br />
This algorithm makes vacuum cleaner look for dust around itself. If it finds it, heads the cleaner towards the dust patch. If it doesn't it chooses it's direction randomly.<br />
<br />
===== Circles walls with vision=====<br />
It is a combination of previous two algorithms. It makes cleaner circle around until it finds wall. But instead of heading randomly, if getting stuck, it looks around itself and if finds dust it heads its direction<br />
<br />
=Results=<br />
I tried with BeahviourSpace plugin to go with each algortihm thorugh each room 100 times. Needless to say, that each one finishes the room, just amount of ticks differ.<br />
<br />
Averages:<br />
Random : 8160,338 ticks<br />
Circles with walls : 4437.41 ticks<br />
Vision : 10930.662 ticks<br />
Circles walls with vision: 4366.222 ticks<br />
<br />
I found it quite weird that random got better average score than vision. So i tried to make vision higher. When having vision 2 patches around itself suddenly creates signifacantly better results. <br />
<br />
Vision 2 : 3263.134<br />
Vision 3 : 2595.164<br />
<br />
Result then would be - if vacuum cleaner can see bigger spaces around itself it is best to make him move by that. If it can't see far enough - circling around itself and than around walls would be the best strategy then.<br />
<br />
http://www.uschovna.cz/zasilka/JDAESA63KE6L8YYA-427/ - I couldn't upload to simulace.info, so i uploaded on this link - will be there until 2.2.2019<br />
<br />
=Conclusion=<br />
Best way to clean your room with AI vacuum cleaner is have its algorithm see around itself. But the area has to be big, if cleaner can't see that far away, best algorithm to use is circling. Nowadays but, there are new cleaners that can scan room before they start cleanup, so they could optimize it's path and be way more effective.<br />
<br />
http://www.uschovna.cz/zasilka/JDL5P55VGKJKHYBZ-P8A/<br />
I couldn't upload anything to simulace.info, so here is simulation itself. All images and result are also on external websites. Results and simulation will be available until 2.2.2019 [[User:Xkorj58|Xkorj58]] ([[User talk:Xkorj58|talk]]) 15:31, 19 January 2019 (CET)</div>Xkorj58http://www.simulace.info/index.php?title=File:Kitchen.JPG&diff=17429File:Kitchen.JPG2019-01-24T21:09:25Z<p>Xkorj58: </p>
<hr />
<div></div>Xkorj58http://www.simulace.info/index.php?title=WS_2018/2019&diff=17428WS 2018/20192019-01-24T21:05:57Z<p>Xkorj58: </p>
<hr />
<div>Semestral papers from winter term 2018/2019. Please, put here links to the pages with your paper. First you need to have your [[Assignments WS 2018/2019|assignment approved]]<br />
<br />
==Simulations==<br />
<br />
--[[User:xvegm00|xvegm00]] [[User:Xvegm00|Xvegm00]] ([[User talk:Xvegm00|talk]]) 22:13, 8 January 2019 (CET) [[Simulation of semi-intelligent algae]]<br />
<br />
-- Jan Doležálek [[User:Dolj04|Dolj04]] ([[User talk:Dolj04|talk]]) 16:50, 18 January 2019 (CET) [[Optimal size of HDD for virtual Digitization server]]<br />
<br />
-- Jiří Korčák [[User:Xkorj58|Xkorj58]] ([[User talk:Xkorj58|talk]]) 11:09, 19 January 2019 (CET) [[Vacuum cleaner]]<br />
<br />
-- Jan Mandík [[User:Manj01|Manj01]] ([[User talk:Manj01|talk]]) 14:46, 19 January 2019 (CET) [[Ticket Solving Process at a Small IT dev Company]] <br />
<br />
-- [[User:Martin svejda|Martin svejda]] ([[User talk:Martin svejda|talk]]) 18:43, 19 January 2019 (CET) [[evacuation from burning building]]<br />
<br />
-- [[User:Xlazl00|Xlazl00]] ([[User talk:Xlazl00|talk]]) 12:11, 20 January 2019 (CET) [[Medieval Battle Simulation]]<br />
<br />
-- [[User:Qnesa01|Qnesa01]] ([User talk:Qnesa01|talk]]) 16:19, 20 January 2019 (CET) [[Argentinska Intersection]]<br />
<br />
-- Jan Pippal (xpipj04) [[User:Janpippal|Janpippal]] 16:41, 20 January 2019 (CET) [[You are what you eat]]<br />
<br />
-- [[User:Kadj02|Kadj02]] ([[User talk:Kadj02|talk]]) 23:19, 20 January 2019 (CET) [[Slime mold]]<br />
<br />
-- [[User:Xkaij00|Xkaij00]] ([[User talk:Xkaij00|talk]]) 01:38, 21 January 2019 (CET) [[Simulation of north korea migration]]<br />
<br />
-- Tomáš Smysl [[User:Xsmyt00|Xsmyt00]] ([[User talk:Xsmyt00|talk]]) 01:19, 24 January 2019 (CET) [[Cafe simulation]]<br />
<br />
-- Marina Lushnikova [[User:Xlusm05|Xlusm05]] ([[User talk:Xlusm05|talk]]) 19:45, 24 January 2019 (CET) [[Simulation of a public ice rink]]<br />
<br />
==Papers==<br />
-- [[User:Martin svejda|Martin svejda]] ([[User talk:Martin svejda|talk]]) 20:43, 12 January 2019 (CET) [https://en.wikipedia.org/wiki/Data_flow_diagram Complete redo of DFD wikipedia]~<br />
<br />
-- [[User:Xvegm00|Xvegm00]] ([[User talk:Xvegm00|talk]]) 10:44, 17 January 2019 (CET) [[http://www.simulace.info/index.php/Multi-agent_systems Multi-agent systems]]<br />
<br />
-- Jan Pippal (xpipj04) [[User:Janpippal|Janpippal]] 4:48, 20 January 2019 (CET) [https://en.wikipedia.org/wiki/Draft:MMABP MMABP in English]<br />
<br />
-- [[User:Qnesa01|Qnesa01]] ([User talk:Qnesa01|talk]]) 17:19, 20 January 2019 (CET) [[Limits to Growth_ver2]] <br />
<br />
-- Tomáš Smysl (xsmyt00) [[User:Xsmyt00|Xsmyt00]] ([[User talk:Xsmyt00|talk]]) 22:36, 20 January 2019 (CET) [[https://en.wikipedia.org/wiki/ArchiMate ArchiMate wiki]] Note: I had some issues with the Wikipedia image upload - they did not approve my images. [[User:Xsmyt00|Xsmyt00]] ([[User talk:Xsmyt00|talk]]) 13:53, 23 January 2019 (CET) EDIT: Solved.<br />
<br />
-- Jan Doležálek [[User:Dolj04|Dolj04]] ([[User talk:Dolj04|talk]]) 11:09, 21 January 2019 (CET) [[http://www.simulace.info/index.php/Variance_reduction Variance reduction]]<br />
<br />
-- Jan Mandík [[User:Manj01|Manj01]] ([[User talk:Manj01|talk]]) 21:52, 21 January 2019 (CET) [[Vickrey%27s_auction]] <br />
<br />
-- [[User:Kadj02|Kadj02]] ([[User talk:Kadj02|talk]]) 22:21, 23 January 2019 (CET) [[Serious Gaming - textbook text]]<br />
<br />
-- [[User:Xkaij00|Xkaij00]] ([[User talk:Xkaij00|talk]]) 23:14, 23 January 2019 (CET) [https://en.wikipedia.org/wiki/Database_normalization Database normalization (Wikipedia)] - introduced step by step normalization in examples - my username on Wikipedia is "Honzikec"<br />
<br />
-- [[User:Xlazl00|Xlazl00]] ([[User talk:Xlazl00|talk]]) 18:30, 24 January 2019 (CET) [[Leverage point]]<br />
<br />
-- Marina Lushnikova [[User:Xlusm05|Xlusm05]] ([[User talk:Xlusm05|talk]]) 19:29, 24 January 2019 (CET) [[Multiplayer_games]]<br />
<br />
--Jiří Korčák [[User:Xkorj58|Xkorj58]] [[User:Xkorj58|Xkorj58]] ([[User talk:Xkorj58|talk]]) 22:05, 24 January 2019 (CET) [[N-player_prisoner%27s_dilemma]]</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17427N-player prisoner's dilemma2019-01-24T21:02:27Z<p>Xkorj58: </p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with [http://www.simulace.info/index.php/Prisoner%27s_dilemma Prisoner’s dilemma]. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from [http://www.simulace.info/index.php/Game_theory Game theory]. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or [http://www.simulace.info/index.php/Multiplayer_games Multiplayer games].<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true [http://www.simulace.info/index.php/Nash_equilibrium/cs Nash equilibrium]. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <ref name="Prisoner's Dilemma threshold"> Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria</ref><ref name="Bach,LA,Helvik"> Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</ref><br />
<br />
[[File:N-player_graph_1.png|center]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
<br />
''Information'' – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi<ref name="Szilagyi">Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.</ref> described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model<ref name="Szilagyi"></ref>. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<ref name="Szilagyi"></ref><br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png|thumb|center||<ref name="Szilagyi"></ref>]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<ref name="Szilagyi"></ref><br />
<br />
[[File:N-player_graph_3.png|thumb|center||<ref name="Szilagyi"></ref>]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – [http://www.simulace.info/index.php/Tragedy_of_the_commons Tragedy of the commons] . It is inspired by real situation that happened in 18th century in England.<ref name="Morgenstern"> Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.</ref><br />
<br />
<br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc. <ref name="G. Hardin"> G. Hardin, “The Tragedy of the Commons,” Science, 162 (1968) 1243– 1248.</ref><ref>N-Person Prisoner's Dilemma. Stanford.edu [online]. [cit. 2017-06-09]. Dostupné z: https://cs.stanford.edu/people/eroberts/courses/soco/projects/1998-99/game-theory/npd.html</ref><br />
<br />
[[File:Tragedyoc.jpg|500px|center]]<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<ref name="Glance"> Glance, Natalie S.; Huberman, Bernardo A. (March 1994). "The dynamics of social dilemmas". Scientific American</ref> .<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
[[File:Zimbardo.jpg|thumb|150px|right||P.Zimbardo]] [[File:Milgram.jpg|thumb|150px|left||S. Milgram]]<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<ref name="Poundstone">Poundstone, William (1993). Prisoner's Dilemma: John von Neumann, Game Theory, and the Puzzle of the Bomb. New York: Anchor Books ISBN 0-385-41580-X.</ref><br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making. <ref name="Harvey">Harvey, Jerry B. (1988). The Abilene Paradox and Other Meditations on Management. Lexington, Mass: Lexington Books. ISBN 0-669-19179-5</ref><br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
<br />
*6 players are waiting to get checked into a flight.<br />
*The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
*The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
*Players incur a 2 units loss of utility by waiting in line.<br />
*What is the equilibrium number of people standing in line?<br />
<br />
<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<ref name="Rousu"> Matthew Rousu 2014, Game Theory N person games, available at https://www.youtube.com/watch?v=GxCWuktSckQ [22.1.2019]</ref><br />
<br />
<br />
=References=<br />
</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17426N-player prisoner's dilemma2019-01-24T21:01:10Z<p>Xkorj58: </p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with [http://www.simulace.info/index.php/Prisoner%27s_dilemma Prisoner’s dilemma]. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from [http://www.simulace.info/index.php/Game_theory Game theory]. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or [http://www.simulace.info/index.php/Multiplayer_games Multiplayer games].<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true [http://www.simulace.info/index.php/Nash_equilibrium/cs Nash equilibrium]. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <ref name="Prisoner's Dilemma threshold"> Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria</ref><ref name="Bach,LA,Helvik"> Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</ref><br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
<br />
''Information'' – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi<ref name="Szilagyi">Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.</ref> described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model<ref name="Szilagyi"></ref>. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<ref name="Szilagyi"></ref><br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png|thumb|center||<ref name="Szilagyi"></ref>]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<ref name="Szilagyi"></ref><br />
<br />
[[File:N-player_graph_3.png|thumb|center||<ref name="Szilagyi"></ref>]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – [http://www.simulace.info/index.php/Tragedy_of_the_commons Tragedy of the commons] . It is inspired by real situation that happened in 18th century in England.<ref name="Morgenstern"> Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.</ref><br />
<br />
<br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc. <ref name="G. Hardin"> G. Hardin, “The Tragedy of the Commons,” Science, 162 (1968) 1243– 1248.</ref><ref>N-Person Prisoner's Dilemma. Stanford.edu [online]. [cit. 2017-06-09]. Dostupné z: https://cs.stanford.edu/people/eroberts/courses/soco/projects/1998-99/game-theory/npd.html</ref><br />
<br />
[[File:Tragedyoc.jpg|500px]]<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<ref name="Glance"> Glance, Natalie S.; Huberman, Bernardo A. (March 1994). "The dynamics of social dilemmas". Scientific American</ref> .<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
[[File:Zimbardo.jpg|thumb|150px|right||P.Zimbardo]] [[File:Milgram.jpg|thumb|150px|left||S. Milgram]]<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<ref name="Poundstone">Poundstone, William (1993). Prisoner's Dilemma: John von Neumann, Game Theory, and the Puzzle of the Bomb. New York: Anchor Books ISBN 0-385-41580-X.</ref><br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making. <ref name="Harvey">Harvey, Jerry B. (1988). The Abilene Paradox and Other Meditations on Management. Lexington, Mass: Lexington Books. ISBN 0-669-19179-5</ref><br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
<br />
*6 players are waiting to get checked into a flight.<br />
*The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
*The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
*Players incur a 2 units loss of utility by waiting in line.<br />
*What is the equilibrium number of people standing in line?<br />
<br />
<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<ref name="Rousu"> Matthew Rousu 2014, Game Theory N person games, available at https://www.youtube.com/watch?v=GxCWuktSckQ [22.1.2019]</ref><br />
<br />
<br />
=References=<br />
</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17425N-player prisoner's dilemma2019-01-24T21:00:34Z<p>Xkorj58: </p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with [http://www.simulace.info/index.php/Prisoner%27s_dilemma Prisoner’s dilemma]. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from [http://www.simulace.info/index.php/Game_theory Game theory]. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or [http://www.simulace.info/index.php/Multiplayer_games Multiplayer games].<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true [http://www.simulace.info/index.php/Nash_equilibrium/cs Nash equilibrium]. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <ref name="Prisoner's Dilemma threshold"> Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria</ref><ref name="Bach,LA,Helvik"> Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</ref><br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
<br />
''Information'' – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi<ref name="Szilagyi">Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.</ref> described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model<ref name="Szilagyi"></ref>. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<ref name="Szilagyi"></ref><br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png|thumb||<ref name="Szilagyi"></ref>]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<ref name="Szilagyi"></ref><br />
<br />
[[File:N-player_graph_3.png|thumb||<ref name="Szilagyi"></ref>]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – [http://www.simulace.info/index.php/Tragedy_of_the_commons Tragedy of the commons] . It is inspired by real situation that happened in 18th century in England.<ref name="Morgenstern"> Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.</ref><br />
<br />
<br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc. <ref name="G. Hardin"> G. Hardin, “The Tragedy of the Commons,” Science, 162 (1968) 1243– 1248.</ref><ref>N-Person Prisoner's Dilemma. Stanford.edu [online]. [cit. 2017-06-09]. Dostupné z: https://cs.stanford.edu/people/eroberts/courses/soco/projects/1998-99/game-theory/npd.html</ref><br />
<br />
[[File:Tragedyoc.jpg|500px]]<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<ref name="Glance"> Glance, Natalie S.; Huberman, Bernardo A. (March 1994). "The dynamics of social dilemmas". Scientific American</ref> .<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
[[File:Zimbardo.jpg|thumb|150px|right||P.Zimbardo]] [[File:Milgram.jpg|thumb|150px|left||S. Milgram]]<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<ref name="Poundstone">Poundstone, William (1993). Prisoner's Dilemma: John von Neumann, Game Theory, and the Puzzle of the Bomb. New York: Anchor Books ISBN 0-385-41580-X.</ref><br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making. <ref name="Harvey">Harvey, Jerry B. (1988). The Abilene Paradox and Other Meditations on Management. Lexington, Mass: Lexington Books. ISBN 0-669-19179-5</ref><br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
<br />
*6 players are waiting to get checked into a flight.<br />
*The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
*The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
*Players incur a 2 units loss of utility by waiting in line.<br />
*What is the equilibrium number of people standing in line?<br />
<br />
<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<ref name="Rousu"> Matthew Rousu 2014, Game Theory N person games, available at https://www.youtube.com/watch?v=GxCWuktSckQ [22.1.2019]</ref><br />
<br />
<br />
=References=<br />
</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17424N-player prisoner's dilemma2019-01-24T20:59:46Z<p>Xkorj58: </p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with [http://www.simulace.info/index.php/Prisoner%27s_dilemma Prisoner’s dilemma]. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from [http://www.simulace.info/index.php/Game_theory Game theory]. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or [http://www.simulace.info/index.php/Multiplayer_games Multiplayer games].<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true [http://www.simulace.info/index.php/Nash_equilibrium/cs Nash equilibrium]. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <ref name="Prisoner's Dilemma threshold"> Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria</ref><ref name="Bach,LA,Helvik"> Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</ref><br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
<br />
''Information'' – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi<ref name="Szilagyi">Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.</ref> described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<ref name="Szilagyi"></ref><br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png||<ref name="Szilagyi"></ref>]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<ref name="Szilagyi"></ref><br />
<br />
[[File:N-player_graph_3.png||<ref name="Szilagyi"></ref>]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – [http://www.simulace.info/index.php/Tragedy_of_the_commons Tragedy of the commons] . It is inspired by real situation that happened in 18th century in England.<ref name="Morgenstern"> Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.</ref><br />
<br />
<br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc. <ref name="G. Hardin"> G. Hardin, “The Tragedy of the Commons,” Science, 162 (1968) 1243– 1248.</ref><ref>N-Person Prisoner's Dilemma. Stanford.edu [online]. [cit. 2017-06-09]. Dostupné z: https://cs.stanford.edu/people/eroberts/courses/soco/projects/1998-99/game-theory/npd.html</ref><br />
<br />
[[File:Tragedyoc.jpg|500px]]<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<ref name="Glance"> Glance, Natalie S.; Huberman, Bernardo A. (March 1994). "The dynamics of social dilemmas". Scientific American</ref> .<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
[[File:Zimbardo.jpg|thumb|150px|right||P.Zimbardo]] [[File:Milgram.jpg|thumb|150px|left||S. Milgram]]<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<ref name="Poundstone">Poundstone, William (1993). Prisoner's Dilemma: John von Neumann, Game Theory, and the Puzzle of the Bomb. New York: Anchor Books ISBN 0-385-41580-X.</ref><br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making. <ref name="Harvey">Harvey, Jerry B. (1988). The Abilene Paradox and Other Meditations on Management. Lexington, Mass: Lexington Books. ISBN 0-669-19179-5</ref><br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
<br />
*6 players are waiting to get checked into a flight.<br />
*The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
*The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
*Players incur a 2 units loss of utility by waiting in line.<br />
*What is the equilibrium number of people standing in line?<br />
<br />
<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<ref name="Rousu"> Matthew Rousu 2014, Game Theory N person games, available at https://www.youtube.com/watch?v=GxCWuktSckQ [22.1.2019]</ref><br />
<br />
<br />
=References=<br />
</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17423N-player prisoner's dilemma2019-01-24T20:56:58Z<p>Xkorj58: </p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with [http://www.simulace.info/index.php/Prisoner%27s_dilemma Prisoner’s dilemma]. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from [http://www.simulace.info/index.php/Game_theory Game theory]. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or [http://www.simulace.info/index.php/Multiplayer_games Multiplayer games].<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true [http://www.simulace.info/index.php/Nash_equilibrium/cs Nash equilibrium]. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <ref name="Prisoner's Dilemma threshold"> Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria</ref><ref name="Bach,LA,Helvik"> Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</ref><br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
<br />
''Information'' – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi<ref name="Szilagyi">Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.</ref> described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<ref name="Szilagyi"><br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png||<ref>Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.</ref>]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<ref>Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.</ref><br />
<br />
[[File:N-player_graph_3.png||<ref>Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.</ref>]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – [http://www.simulace.info/index.php/Tragedy_of_the_commons Tragedy of the commons] . It is inspired by real situation that happened in 18th century in England.<ref name="Morgenstern"> Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.</ref><br />
<br />
<br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc. <ref name="G. Hardin"> G. Hardin, “The Tragedy of the Commons,” Science, 162 (1968) 1243– 1248.</ref><ref>N-Person Prisoner's Dilemma. Stanford.edu [online]. [cit. 2017-06-09]. Dostupné z: https://cs.stanford.edu/people/eroberts/courses/soco/projects/1998-99/game-theory/npd.html</ref><br />
<br />
[[File:Tragedyoc.jpg|500px]]<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<ref name="Glance"> Glance, Natalie S.; Huberman, Bernardo A. (March 1994). "The dynamics of social dilemmas". Scientific American</ref> .<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
[[File:Zimbardo.jpg|thumb|150px|right||P.Zimbardo]] [[File:Milgram.jpg|thumb|150px|left||S. Milgram]]<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<ref name="Poundstone">Poundstone, William (1993). Prisoner's Dilemma: John von Neumann, Game Theory, and the Puzzle of the Bomb. New York: Anchor Books ISBN 0-385-41580-X.</ref><br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making. <ref name="Harvey">Harvey, Jerry B. (1988). The Abilene Paradox and Other Meditations on Management. Lexington, Mass: Lexington Books. ISBN 0-669-19179-5</ref><br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
<br />
*6 players are waiting to get checked into a flight.<br />
*The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
*The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
*Players incur a 2 units loss of utility by waiting in line.<br />
*What is the equilibrium number of people standing in line?<br />
<br />
<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<ref name="Rousu"> Matthew Rousu 2014, Game Theory N person games, available at https://www.youtube.com/watch?v=GxCWuktSckQ [22.1.2019]</ref><br />
<br />
<br />
=References=<br />
</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17422N-player prisoner's dilemma2019-01-24T20:55:14Z<p>Xkorj58: </p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with [http://www.simulace.info/index.php/Prisoner%27s_dilemma Prisoner’s dilemma]. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from [http://www.simulace.info/index.php/Game_theory Game theory]. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or [http://www.simulace.info/index.php/Multiplayer_games Multiplayer games].<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true [http://www.simulace.info/index.php/Nash_equilibrium/cs Nash equilibrium]. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <ref name="Prisoner's Dilemma threshold"> Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria</ref><ref name="Bach,LA,Helvik"> Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</ref><br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
<br />
''Information'' – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi<ref>Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.</ref> described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<ref>Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.</ref><br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png||<ref>Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.</ref>]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<ref>Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.</ref><br />
<br />
[[File:N-player_graph_3.png||<ref>Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.</ref>]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – [http://www.simulace.info/index.php/Tragedy_of_the_commons Tragedy of the commons] . It is inspired by real situation that happened in 18th century in England.<ref name="Morgenstern"> Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.</ref><br />
<br />
<br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc. <ref name="G. Hardin"> G. Hardin, “The Tragedy of the Commons,” Science, 162 (1968) 1243– 1248.</ref><ref>N-Person Prisoner's Dilemma. Stanford.edu [online]. [cit. 2017-06-09]. Dostupné z: https://cs.stanford.edu/people/eroberts/courses/soco/projects/1998-99/game-theory/npd.html</ref><br />
<br />
[[File:Tragedyoc.jpg|500px]]<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<ref name="Glance"> Glance, Natalie S.; Huberman, Bernardo A. (March 1994). "The dynamics of social dilemmas". Scientific American</ref> .<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
[[File:Zimbardo.jpg|thumb|150px|right||P.Zimbardo]] [[File:Milgram.jpg|thumb|150px|left||S. Milgram]]<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<ref name="Poundstone">Poundstone, William (1993). Prisoner's Dilemma: John von Neumann, Game Theory, and the Puzzle of the Bomb. New York: Anchor Books ISBN 0-385-41580-X.</ref><br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making. <ref name="Harvey">Harvey, Jerry B. (1988). The Abilene Paradox and Other Meditations on Management. Lexington, Mass: Lexington Books. ISBN 0-669-19179-5</ref><br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
<br />
*6 players are waiting to get checked into a flight.<br />
*The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
*The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
*Players incur a 2 units loss of utility by waiting in line.<br />
*What is the equilibrium number of people standing in line?<br />
<br />
<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<ref name="Rousu"> Matthew Rousu 2014, Game Theory N person games, available at https://www.youtube.com/watch?v=GxCWuktSckQ [22.1.2019]</ref><br />
<br />
<br />
=References=<br />
</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17421N-player prisoner's dilemma2019-01-24T20:44:39Z<p>Xkorj58: </p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with [http://www.simulace.info/index.php/Prisoner%27s_dilemma Prisoner’s dilemma]. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from [http://www.simulace.info/index.php/Game_theory Game theory]. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or [http://www.simulace.info/index.php/Multiplayer_games Multiplayer games].<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true [http://www.simulace.info/index.php/Nash_equilibrium/cs Nash equilibrium]. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <ref name="Prisoner's Dilemma threshold"> Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria</ref><br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
<br />
''Information'' – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi<ref>Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.</ref> described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<br />
<br />
[[File:N-player_graph_3.png]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – [http://www.simulace.info/index.php/Tragedy_of_the_commons Tragedy of the commons] . It is inspired by real situation that happened in 18th century in England. <br />
<br />
<br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc. <ref name="G. Hardin"> G. Hardin, “The Tragedy of the Commons,” Science, 162 (1968) 1243– 1248.</ref><ref>N-Person Prisoner's Dilemma. Stanford.edu [online]. [cit. 2017-06-09]. Dostupné z: https://cs.stanford.edu/people/eroberts/courses/soco/projects/1998-99/game-theory/npd.html</ref><br />
<br />
[[File:Tragedyoc.jpg|500px]]<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
[[File:Zimbardo.jpg|thumb|150px|right||P.Zimbardo]] [[File:Milgram.jpg|thumb|150px|left||S. Milgram]]<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
<br />
*6 players are waiting to get checked into a flight.<br />
*The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
*The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
*Players incur a 2 units loss of utility by waiting in line.<br />
*What is the equilibrium number of people standing in line?<br />
<br />
<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<br />
=References=<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17420N-player prisoner's dilemma2019-01-24T20:39:31Z<p>Xkorj58: </p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with [http://www.simulace.info/index.php/Prisoner%27s_dilemma Prisoner’s dilemma]. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from [http://www.simulace.info/index.php/Game_theory Game theory]. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or [http://www.simulace.info/index.php/Multiplayer_games Multiplayer games].<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true [http://www.simulace.info/index.php/Nash_equilibrium/cs Nash equilibrium]. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
Information – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<br />
<br />
[[File:N-player_graph_3.png]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – [http://www.simulace.info/index.php/Tragedy_of_the_commons Tragedy of the commons] . It is inspired by real situation that happened in 18th century in England. <br />
<br />
<br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc.<br />
<br />
[[File:Tragedyoc.jpg|500px]]<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
[[File:Zimbardo.jpg|thumb|150px|right||P.Zimbardo]] [[File:Milgram.jpg|thumb|150px|left||S. Milgram]]<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
<br />
*6 players are waiting to get checked into a flight.<br />
*The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
*The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
*Players incur a 2 units loss of utility by waiting in line.<br />
*What is the equilibrium number of people standing in line?<br />
<br />
<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<br />
=References=<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17419N-player prisoner's dilemma2019-01-24T20:36:28Z<p>Xkorj58: </p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with [http://www.simulace.info/index.php/Prisoner%27s_dilemma Prisoner’s dilemma]. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from [http://www.simulace.info/index.php/Game_theory Game theory]. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or [http://www.simulace.info/index.php/Multiplayer_games Multiplayer games].<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true Nash equilibrium. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
Information – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<br />
<br />
[[File:N-player_graph_3.png]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – tragedy of the commons. It is inspired by real situation that happened in 18th century in England. <br />
<br />
<br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc.<br />
<br />
[[File:Tragedyoc.jpg|500px]]<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
[[File:Zimbardo.jpg|thumb|150px|right||P.Zimbardo]] [[File:Milgram.jpg|thumb|150px|left||S. Milgram]]<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
<br />
*6 players are waiting to get checked into a flight.<br />
*The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
*The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
*Players incur a 2 units loss of utility by waiting in line.<br />
*What is the equilibrium number of people standing in line?<br />
<br />
<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<br />
=References=<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17418N-player prisoner's dilemma2019-01-24T18:51:06Z<p>Xkorj58: /* Practice */</p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with Prisoner’s dilemma. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from Game theory. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or multiplayer games.<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true Nash equilibrium. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
Information – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<br />
<br />
[[File:N-player_graph_3.png]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – tragedy of the commons. It is inspired by real situation that happened in 18th century in England. <br />
<br />
<br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc.<br />
<br />
[[File:Tragedyoc.jpg|500px]]<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
[[File:Zimbardo.jpg|thumb|150px|right||P.Zimbardo]] [[File:Milgram.jpg|thumb|150px|left||S. Milgram]]<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
<br />
*6 players are waiting to get checked into a flight.<br />
*The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
*The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
*Players incur a 2 units loss of utility by waiting in line.<br />
*What is the equilibrium number of people standing in line?<br />
<br />
<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<br />
=References=<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17417N-player prisoner's dilemma2019-01-24T18:50:05Z<p>Xkorj58: /* Examples */</p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with Prisoner’s dilemma. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from Game theory. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or multiplayer games.<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true Nash equilibrium. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
Information – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<br />
<br />
[[File:N-player_graph_3.png]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – tragedy of the commons. It is inspired by real situation that happened in 18th century in England. <br />
<br />
<br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc.<br />
<br />
[[File:Tragedyoc.jpg|500px]]<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
[[File:Zimbardo.jpg|thumb|150px|right||P.Zimbardo]] [[File:Milgram.jpg|thumb|150px|left||S. Milgram]]<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
6 players are waiting to get checked into a flight.<br />
The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
Players incur a 2 units loss of utility by waiting in line.<br />
What is the equilibrium number of people standing in line?<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<br />
=References=<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17416N-player prisoner's dilemma2019-01-24T18:47:47Z<p>Xkorj58: /* Examples */</p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with Prisoner’s dilemma. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from Game theory. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or multiplayer games.<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true Nash equilibrium. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
Information – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<br />
<br />
[[File:N-player_graph_3.png]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – tragedy of the commons. It is inspired by real situation that happened in 18th century in England. <br />
<br />
<br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc.<br />
<br />
[[File:Tragedyoc.jpg|500px]]<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
[[File:Zimbardo.jpg|150px|right|]] [[File:Zimbardo.jpg|150px|left|]]<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
6 players are waiting to get checked into a flight.<br />
The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
Players incur a 2 units loss of utility by waiting in line.<br />
What is the equilibrium number of people standing in line?<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<br />
=References=<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58http://www.simulace.info/index.php?title=File:Milgram.jpg&diff=17415File:Milgram.jpg2019-01-24T18:47:23Z<p>Xkorj58: </p>
<hr />
<div></div>Xkorj58http://www.simulace.info/index.php?title=File:Zimbardo.jpg&diff=17414File:Zimbardo.jpg2019-01-24T18:45:44Z<p>Xkorj58: </p>
<hr />
<div></div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17411N-player prisoner's dilemma2019-01-24T18:43:42Z<p>Xkorj58: /* Examples */</p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with Prisoner’s dilemma. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from Game theory. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or multiplayer games.<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true Nash equilibrium. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
Information – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<br />
<br />
[[File:N-player_graph_3.png]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – tragedy of the commons. It is inspired by real situation that happened in 18th century in England. <br />
<br />
<br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc.<br />
<br />
[[File:Tragedyoc.jpg|500px]]<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
6 players are waiting to get checked into a flight.<br />
The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
Players incur a 2 units loss of utility by waiting in line.<br />
What is the equilibrium number of people standing in line?<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<br />
=References=<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17410N-player prisoner's dilemma2019-01-24T18:42:58Z<p>Xkorj58: /* Examples */</p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with Prisoner’s dilemma. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from Game theory. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or multiplayer games.<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true Nash equilibrium. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
Information – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<br />
<br />
[[File:N-player_graph_3.png]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – tragedy of the commons. It is inspired by real situation that happened in 18th century in England. <br />
<br />
<br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc.<br />
<br />
[[File:Tragedyoc.jpg|200px]]<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
6 players are waiting to get checked into a flight.<br />
The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
Players incur a 2 units loss of utility by waiting in line.<br />
What is the equilibrium number of people standing in line?<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<br />
=References=<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17409N-player prisoner's dilemma2019-01-24T18:41:42Z<p>Xkorj58: /* Examples */</p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with Prisoner’s dilemma. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from Game theory. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or multiplayer games.<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true Nash equilibrium. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
Information – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<br />
<br />
[[File:N-player_graph_3.png]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – tragedy of the commons. It is inspired by real situation that happened in 18th century in England. <br />
<br />
<br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc.<br />
<br />
[[File:Tragedyoc.jpg]]<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
6 players are waiting to get checked into a flight.<br />
The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
Players incur a 2 units loss of utility by waiting in line.<br />
What is the equilibrium number of people standing in line?<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<br />
=References=<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17408N-player prisoner's dilemma2019-01-24T18:40:59Z<p>Xkorj58: /* Examples */</p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with Prisoner’s dilemma. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from Game theory. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or multiplayer games.<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true Nash equilibrium. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
Information – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<br />
<br />
[[File:N-player_graph_3.png]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – tragedy of the commons. It is inspired by real situation that happened in 18th century in England. <br />
<br />
<br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc.<br />
<br />
[[File:Tragedyoc.jpg/320px-Tragedyoc.jpg]]<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
6 players are waiting to get checked into a flight.<br />
The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
Players incur a 2 units loss of utility by waiting in line.<br />
What is the equilibrium number of people standing in line?<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<br />
=References=<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17407N-player prisoner's dilemma2019-01-24T18:40:39Z<p>Xkorj58: /* Examples */</p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with Prisoner’s dilemma. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from Game theory. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or multiplayer games.<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true Nash equilibrium. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
Information – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<br />
<br />
[[File:N-player_graph_3.png]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – tragedy of the commons. It is inspired by real situation that happened in 18th century in England. <br />
<br />
<br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc.<br />
<br />
[[File:320px-Tragedyoc.jpg]]<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
6 players are waiting to get checked into a flight.<br />
The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
Players incur a 2 units loss of utility by waiting in line.<br />
What is the equilibrium number of people standing in line?<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<br />
=References=<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58http://www.simulace.info/index.php?title=File:Tragedyoc.jpg&diff=17406File:Tragedyoc.jpg2019-01-24T18:39:50Z<p>Xkorj58: </p>
<hr />
<div></div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17405N-player prisoner's dilemma2019-01-24T18:38:18Z<p>Xkorj58: /* Models */</p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with Prisoner’s dilemma. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from Game theory. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or multiplayer games.<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true Nash equilibrium. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
Information – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
[[File:N-player_graph_2.png]]<br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here:<br />
<br />
[[File:N-player_graph_3.png]]<br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – tragedy of the commons. It is inspired by real situation that happened in 18th century in England. <br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc.<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
<br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
6 players are waiting to get checked into a flight.<br />
The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
Players incur a 2 units loss of utility by waiting in line.<br />
What is the equilibrium number of people standing in line?<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<br />
=References=<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58http://www.simulace.info/index.php?title=File:N-player_graph_3.png&diff=17404File:N-player graph 3.png2019-01-24T18:38:03Z<p>Xkorj58: </p>
<hr />
<div></div>Xkorj58http://www.simulace.info/index.php?title=File:N-player_graph_2.png&diff=17403File:N-player graph 2.png2019-01-24T18:36:43Z<p>Xkorj58: </p>
<hr />
<div></div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17402N-player prisoner's dilemma2019-01-24T18:35:12Z<p>Xkorj58: </p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with Prisoner’s dilemma. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from Game theory. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or multiplayer games.<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true Nash equilibrium. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <br />
<br />
[[File:N-player_graph_1.png]]<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
Information – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here: <br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – tragedy of the commons. It is inspired by real situation that happened in 18th century in England. <br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc.<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
<br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
6 players are waiting to get checked into a flight.<br />
The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
Players incur a 2 units loss of utility by waiting in line.<br />
What is the equilibrium number of people standing in line?<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<br />
=References=<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58http://www.simulace.info/index.php?title=File:N-player_graph_1.png&diff=17401File:N-player graph 1.png2019-01-24T18:34:25Z<p>Xkorj58: </p>
<hr />
<div></div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17400N-player prisoner's dilemma2019-01-24T18:31:55Z<p>Xkorj58: </p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with Prisoner’s dilemma. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from Game theory. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or multiplayer games.<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true Nash equilibrium. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <br />
<br />
<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
Information – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
| Adventurous (Antisocial)<br />
| Self-Confident (Narcissistic)<br />
|-<br />
| Aggressive (Sadistic)<br />
| Self-Sacrificing (Self-Defeating)<br />
|-<br />
| Conscientious (Obsessive-Compulsive)<br />
| Sensitive (Avoidant)<br />
|-<br />
| Devoted (Dependent)<br />
| Solitary (Schizoid)<br />
|-<br />
| Dramatic (Histrionic)<br />
| Vigilant (Paranoid)<br />
|-<br />
| Idiosyncratic (Schizotypal)<br />
| Exuberant (Cyclothymic)<br />
|-<br />
| Leisurely (Passive-Aggressive)<br />
| Serious (Depressive)<br />
|-<br />
| Mercurial (Borderline)<br />
| Inventive (Compensatory Narcissistic)<br />
|-<br />
|}<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
<br />
=Models=<br />
<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here: <br />
<br />
=Examples=<br />
<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – tragedy of the commons. It is inspired by real situation that happened in 18th century in England. <br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc.<br />
<br />
<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
<br />
<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
<br />
<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
<br />
<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
<br />
<br />
=Practice=<br />
<br />
Try to solve this N-player game:<br />
6 players are waiting to get checked into a flight.<br />
The consumers utility is based on when they get checked in:<br />
<br />
{| class="wikitable"<br />
|-<br />
! Order served<br />
! Utility<br />
|-<br />
| 1<br />
| 20<br />
|-<br />
| 2<br />
| 17<br />
|-<br />
| 3<br />
| 14<br />
|-<br />
| 4<br />
| 11<br />
|-<br />
| 5<br />
| 8<br />
|-<br />
| 6<br />
| 5<br />
|}<br />
<br />
<br />
The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
Players incur a 2 units loss of utility by waiting in line.<br />
What is the equilibrium number of people standing in line?<br />
<br />
'''''Results:''''' ''4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.''<br />
<br />
=References=<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17398N-player prisoner's dilemma2019-01-24T18:18:14Z<p>Xkorj58: </p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with Prisoner’s dilemma. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from Game theory. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or multiplayer games.<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
* Each player can either cooperate or defect<br />
* Defecting is usually dominant strategy for each player<br />
* The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true Nash equilibrium. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <br />
<br />
<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
Information – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
<br />
''Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. <br />
<br />
Firstly, distinction he used for his work model. <br />
<br />
* Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
* Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
* Conformist – agent will act the same way as majority of other agents<br />
* Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
* Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
<br />
* Openness<br />
* Conscientiousness<br />
* Extroversion<br />
* Agreeableness<br />
* Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
<br />
* trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
* forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
* repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
* trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
Adventurous (Antisocial) Self-Confident (Narcissistic)<br />
Aggressive (Sadistic) Self-Sacrificing (Self-Defeating)<br />
Conscientious (Obsessive-Compulsive) Sensitive (Avoidant)<br />
Devoted (Dependent) Solitary (Schizoid)<br />
Dramatic (Histrionic) Vigilant (Paranoid)<br />
Idiosyncratic (Schizotypal) Exuberant (Cyclothymic)<br />
Leisurely (Passive-Aggressive) Serious (Depressive)<br />
Mercurial (Borderline) Inventive (Compensatory Narcissistic)<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
<br />
=Models=<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here: <br />
<br />
=Examples=<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – tragedy of the commons. It is inspired by real situation that happened in 18th century in England. <br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc.<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
=Practice=<br />
Try to solve this N-player game:<br />
6 players are waiting to get checked into a flight.<br />
The consumers utility is based on when they get checked in:<br />
Order served Utility<br />
1 20<br />
2 17<br />
3 14<br />
4 11<br />
5 8<br />
6 5<br />
The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
Players incur a 2 units loss of utility by waiting in line.<br />
What is the equilibrium number of people standing in line?<br />
4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.<br />
<br />
=References=<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17397N-player prisoner's dilemma2019-01-24T18:14:11Z<p>Xkorj58: </p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with Prisoner’s dilemma. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
<br />
N-player prisoner’s dilemma is a multiplayer game from Game theory. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or multiplayer games.<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
- Each player can either cooperate or defect<br />
- Defecting is usually dominant strategy for each player<br />
- The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true Nash equilibrium. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <br />
<br />
<br />
<br />
=Parameters=<br />
<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
Information – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
<br />
''Time'' – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
<br />
''Communication'' – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
<br />
''Space'' – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
<br />
''Participation'' – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
''<br />
Personalities of agents'' – last mentioned parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. Firstly, distinction he used for his work model. <br />
<br />
-Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
-Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
-Conformist – agent will act the same way as majority of other agents<br />
-Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
-Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1), P=0.5 acts randomly, agent is totally unpredictable.<br />
<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
-Openness<br />
-Conscientiousness<br />
-Extroversion<br />
-Agreeableness<br />
-Neuroticism<br />
<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
-trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
-forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
-repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
-trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
<br />
Adventurous (Antisocial) Self-Confident (Narcissistic)<br />
Aggressive (Sadistic) Self-Sacrificing (Self-Defeating)<br />
Conscientious (Obsessive-Compulsive) Sensitive (Avoidant)<br />
Devoted (Dependent) Solitary (Schizoid)<br />
Dramatic (Histrionic) Vigilant (Paranoid)<br />
Idiosyncratic (Schizotypal) Exuberant (Cyclothymic)<br />
Leisurely (Passive-Aggressive) Serious (Depressive)<br />
Mercurial (Borderline) Inventive (Compensatory Narcissistic)<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
<br />
=Models=<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here: <br />
<br />
=Examples=<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – tragedy of the commons. It is inspired by real situation that happened in 18th century in England. <br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc.<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
=Practice=<br />
Try to solve this N-player game:<br />
6 players are waiting to get checked into a flight.<br />
The consumers utility is based on when they get checked in:<br />
Order served Utility<br />
1 20<br />
2 17<br />
3 14<br />
4 11<br />
5 8<br />
6 5<br />
The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
Players incur a 2 units loss of utility by waiting in line.<br />
What is the equilibrium number of people standing in line?<br />
4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.<br />
<br />
=References=<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17395N-player prisoner's dilemma2019-01-24T18:09:29Z<p>Xkorj58: </p>
<hr />
<div><br />
=Introduction=<br />
You have already become familiar with Prisoner’s dilemma. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
<br />
=Problem definition=<br />
N-player prisoner’s dilemma is a multiplayer game from Game theory. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or multiplayer games.<br />
<br />
'''Interesting fact:''' In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
<br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
- Each player can either cooperate or defect<br />
- Defecting is usually dominant strategy for each player<br />
- The dominant strategies intersect at a deficient equilibrium point<br />
<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true Nash equilibrium. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <br />
<br />
<br />
<br />
=Parameters=<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
Information – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
Time – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
Communication – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
Space – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
Participation – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
Personalities of agents – last mention parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. Firstly, distinction he used for his work model. <br />
- Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
- Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
- Conformist – agent will act the same way as majority of other agents<br />
- Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
- Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1)<br />
- P=0.5 acts randomly, agent is totally unpredictable.<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
- Openness<br />
- Conscientiousness<br />
- Extroversion<br />
- Agreeableness<br />
- Neuroticism<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
- trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
- forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
- repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
- trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
Adventurous (Antisocial) Self-Confident (Narcissistic)<br />
Aggressive (Sadistic) Self-Sacrificing (Self-Defeating)<br />
Conscientious (Obsessive-Compulsive) Sensitive (Avoidant)<br />
Devoted (Dependent) Solitary (Schizoid)<br />
Dramatic (Histrionic) Vigilant (Paranoid)<br />
Idiosyncratic (Schizotypal) Exuberant (Cyclothymic)<br />
Leisurely (Passive-Aggressive) Serious (Depressive)<br />
Mercurial (Borderline) Inventive (Compensatory Narcissistic)<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
Models<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here: <br />
<br />
Examples<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – tragedy of the commons. It is inspired by real situation that happened in 18th century in England. <br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc.<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
Practice <br />
Try to solve this N-player game:<br />
6 players are waiting to get checked into a flight.<br />
The consumers utility is based on when they get checked in:<br />
Order served Utility<br />
1 20<br />
2 17<br />
3 14<br />
4 11<br />
5 8<br />
6 5<br />
The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
Players incur a 2 units loss of utility by waiting in line.<br />
What is the equilibrium number of people standing in line?<br />
4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.<br />
<br />
<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58http://www.simulace.info/index.php?title=N-player_prisoner%27s_dilemma&diff=17394N-player prisoner's dilemma2019-01-24T18:07:08Z<p>Xkorj58: Created page with "N-player prisoner’s dilemma Introduction You have already become familiar with Prisoner’s dilemma. But now imagine a different situation. What if there are three prisoners..."</p>
<hr />
<div>N-player prisoner’s dilemma<br />
Introduction<br />
You have already become familiar with Prisoner’s dilemma. But now imagine a different situation. What if there are three prisoners. And even worse, what if there is a lot of them. How will they behave? What strategy will they use. They can cooperate or they have to make fools out of the other. Sometimes though, the environment is shared and not cooperating can have devastating consequences. Imagine a field, a pasture for cattle. There are three farmers who can send their cows on this field. But as there are more and more cows on it, it becomes devastated. Each farmer sends their cows until the field is totally ruined. That is a classical dilemma of this topic. In this chapter you will find out more details about how n-player prisoner’s dilemma works and how is it solved.<br />
-Problem definition-<br />
N-player prisoner’s dilemma is a multiplayer game from Game theory. It is a expansion of classical Prisoner’s dilemma, because its lack of applicability. In real world problems, there usually are more than two players or two sides of an argument. Some of these problems can be turned to two-player games, but there are some you cannot. These situations are called N-player games or multiplayer games. <br />
Interesting fact: In group of 8 players there are 28 one-to-one relations. There are 40.320 potential relationships (8!, groups of 2,3,4,5,…). But in group of 50 players there already are 1225 one-to-one relationships. <br />
Multiplayer games like N-player prisoner’s dilemma have 3 defining properties, which are similar to the characteristics of regular 2-player prisoner’s dilemma.<br />
- Each player can either cooperate or defect<br />
- Defecting is usually dominant strategy for each player<br />
- The dominant strategies intersect at a deficient equilibrium point<br />
What it really means is, that all players want to make their utility higher, therefore they usually incline to defecting strategy. Regardless of what the other players do, each one receives a higher payoff for defecting behavior than for cooperating behavior. But because of limits of the environment equilibrium is usually deficient. We don’t find true Nash equilibrium. All players receive a lower payoff if all defect than if all cooperate. The dominance of the non-cooperative strategy over cooperation, regardless of the overall share of cooperating entities, shows the mutual position of the curves of the functions of cooperation and non-cooperation in the graph. The specific form of function curve depends on the concrete problem’s definition. <br />
<br />
<br />
<br />
Parameters<br />
Beside of defecting strategy, we can find more parameters describing situation and decision making of players. (In game theory in generally we call them agents, as it could be individuals, groups, companies, artificial agents or imaginative organization structures).<br />
Information – agents can either know how others make their decisions, what are their biases and limits, or not know. They can also have or not have information about environment (e.g. limits of pasture)<br />
Time – Decisions agents make can be proceeded simultaneously or in turns. Also, one’s turn can take longer than others, leading to overlaying of turns (one can make two changes in their strategy while the other will only make one change). The game itself can be played only as one interaction or iteration, or as repeated game. This repetition could be certain number, or it could be infinite. If game is infinite again information about it will be very important for decision making<br />
Communication – If agents can communicate among themselves, they can form fellowships and coalitions for improving their results.<br />
Goals – With postmodern emancipation we can assume that some agents come into a game with different goals. Therefore, their rational behavior towards their personal goal could be different from others. <br />
Space – In todays world of fast travel of information, space doesn’t make too much of a difference. But information is not the only thing traveling through space. We can find more spatial problems and definitions than just that. These afterwards create limits for players and their decisions. Often when space is bigger than agents in it, they can move in it, having influence on ever changing environment<br />
Participation – Huge difference for agents is not having to participate. It can refuse to participate in whole game or in some of iterations. For example, in chess you can’t refuse to move, on the other hand, Kutuzov beat Napoleon by avoiding battles.<br />
Personalities of agents – last mention parameter will be personality of agent. Everyone is different and can decide differently then others when faced the same situation. We will introduce personalities as Szilagyi described them. He presents 4 areas of distinction. Need to say, that “personalities” as we understand them here, will be just decision approaches, that can be designated as the most usual ones. Firstly, distinction he used for his work model. <br />
- Pavlovian personality – means that agent, internally, changes probability of repeating the same decision by amount of reward or penalty coming from this decision. How fast he learns from previous wins or loses is called “learning rate”. This personality is derived from experiments of I.P. Pavlov on dogs and Thorndike’s Law of effect.<br />
- Greedy – agent will act same way as any of his friend or foes, who got the best result<br />
- Conformist – agent will act the same way as majority of other agents<br />
- Accountant – agent with this personality will count average payoff or utility of previous decisions and decide by that<br />
- Statistically “predictable” – has a constant parameter “p” – cooperation. Short-time rational agent always defect (p=0), Altruistic agents ignore short term interest and cooperate (p=1)<br />
- P=0.5 acts randomly, agent is totally unpredictable.<br />
Secondly, Big Five approach to personalities, Szilagyi used this distinction in his standing ovation study<br />
- Openness<br />
- Conscientiousness<br />
- Extroversion<br />
- Agreeableness<br />
- Neuroticism<br />
Third, approach that distinguishes what kind of action will agent perform after result.<br />
- trustworthiness (probability of choosing cooperation after reward for cooperation by all members)<br />
- forgiveness (probability of choosing cooperation after “sucker’s payoff” – when cooperating while the opponent/s was defecting)<br />
- repentance (probability of choosing cooperation after “temptation payoff” – when defecting while other one/s cooperating)<br />
- trust (probability of choosing cooperation after receiving punishment for defecting by all members)<br />
Last approach probably the most particular one, personalities defined by 16 mental disorders. These disorders are extremes of regular people’s personalities, therefore combination of them puts together agent’s personality. Traits and disorders are shown in following table. <br />
Adventurous (Antisocial) Self-Confident (Narcissistic)<br />
Aggressive (Sadistic) Self-Sacrificing (Self-Defeating)<br />
Conscientious (Obsessive-Compulsive) Sensitive (Avoidant)<br />
Devoted (Dependent) Solitary (Schizoid)<br />
Dramatic (Histrionic) Vigilant (Paranoid)<br />
Idiosyncratic (Schizotypal) Exuberant (Cyclothymic)<br />
Leisurely (Passive-Aggressive) Serious (Depressive)<br />
Mercurial (Borderline) Inventive (Compensatory Narcissistic)<br />
<br />
All these parameters can be found in regular two player prisoner’s dilemma, but in N-player versions, especially personalities have significant value, as there is more players and more types of situation and more interactions are about to arrive.<br />
Models<br />
As it is very difficult to create a matrix like one in two player game, we need to choose another approach to modelling. In three-player game you would need 3x3x3 matrix, in game of 4 players 4x4x4x4 and so on. It would be almost impossible to make even like 10-player game. Therefore, we need to replace matrix approach. <br />
We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix. <br />
<br />
The letters describe something we already mentioned in previous part. “P” – means punishment to an agent when all agents defect. “R” – means reward to an agent when all agents cooperate. “T” – is temptation’s payoff, it is a utility/reward for agent when defecting while others cooperate and last letter “S” – sucker’s payoff, utility/reward or punishment for cooperating while others defect.<br />
In this graph we can easily replace axis x – second player’s choice with ratio of cooperating agents in the rest of the group. This way we can apply graph on any size of a group, even again on two-player game. Last adding to the graph will be connecting point P with point T – creating “defectors’ payoff function” (D(X)), and also connecting letters S and R – creating “cooperators’ function” (C(X)). These function can have various shapes, don’t have to be linear, and can also go below zero. Look on this graph here: <br />
<br />
Examples<br />
Probably the best-known example of n-player dilemma is, already mentioned, problem with cow pasture – tragedy of the commons. It is inspired by real situation that happened in 18th century in England. <br />
Imagine situation, six farmers share common field, a pasture. Each farmer takes on the pasture one cow every day. If this cow is well fed it will weight 500kg. The pasture can hold up to six cows, if there’s more it will be overgrazed. So, if more cows are added, they will be starving, and the weight of each animal will be lowered by 50 kg. Being a farmer, you want to add more cows, (900kg – 2x450kg > 500kg). Everyone of these farmers are adding more cows, making it dominant strategy. In the end, adding more cows will totally deteriorate and no cows can live there. This problematic dilemma describes situation of merely all environmental issues in todays world. What could governments do about it? They can either privatize the field – sell it to one owner, so he can add as many cows as he wants at his own risk, or they can setup regulations (how many cows can each farmer have). This way cheaters are punished by government and not the environment. There is a certain conspiracy about this problem, because it was published in time of beginnings of privatizations of properties in US and it was taken as a campaign for it. More about this dilemma here. There are many similar examples like water usage, waste dumping etc.<br />
Another example of n-player dilemma is called Unscrupulous diner’s dilemma. Imagine you go on a dinner with 3 friends, before ordering your food you decide that you split the bill equally. However, there are two types of burgers in the menu, regular one (utility of 1) and XXL one (utility of 2). Prices of those burgers though are not adequate to the utility. It is 1$ for regular one and 3$ for XXL one. My dominant strategy would be to eat as much as I can and pay less. So, I will order XXL burger and let others pay for me. Therefore, total amount paid will be 6$ (3+1+1+1) and I will pay just 1.5$, but I will receive 2 utility / you get 2 times better burger but only paid 1.5-time the money. However, everyone of your friends think the same way, each one of them wants to have twice as much burger at just a little bit more price or at least, nobody wants to pay for the douchebag ordering the big burger while being stuck on the regular one. So, in the end everybody orders XXL burger – bill is 12$ and everybody pay 3-times more but only get twice better burger.<br />
Problems like these are sometimes called “freeriding” or “freeriding problems”. One member of the group uses it to gain utility, but if everybody in the group would acts the same way, everyone would lose more than gain. N-player prisoner’s dilemma is in general sociocultural issue. Individual rationality leads to collective irrationality. It presents modern battle between individualism and collectivism. On the other hand, there is also few examples, showing otherwise.<br />
First one is called Volunteer’s dilemma game. It models for example meerkat behavior when being on guard. One member of a group chooses between making a little sacrifice in name of bigger good of the whole group or just wait until somebody else does the sacrifice instead. Mentioned meerkats take on themselves roles of sentries even though when they warn about incoming danger, they face immediate thread of dying because of being discovered first while calling a warning. This dilemma is often connected with Bystander effect and other psychological effects discovered in 1960s and 70s by Milgram and Zimbardo.<br />
Imagine you and your family want to go for a dinner. The restaurant is quite far away, and you would rather stay at home and have dinner there instead of that long drive. But you think that don’t want to be the bad guy who ruins dinner for everybody, so you go. Now imagine, that every member of the family thinks just the same way. And with lack of communication you all end up going to the restaurant, even though nobody truly wanted to go there. This is called Abilane Paradox, because of example with restaurant in town of Abilane. It’s quite the opposite we see in previous dilemmas but, it fills up the overview about decision-making.<br />
Practice <br />
Try to solve this N-player game:<br />
6 players are waiting to get checked into a flight.<br />
The consumers utility is based on when they get checked in:<br />
Order served Utility<br />
1 20<br />
2 17<br />
3 14<br />
4 11<br />
5 8<br />
6 5<br />
The airline will check people randomly if no one is in line, but those in line will be checked-in first<br />
Players incur a 2 units loss of utility by waiting in line.<br />
What is the equilibrium number of people standing in line?<br />
4 people standing in line - The 5th guy gets 6 Utils guaranteed by waiting, but a 50/50 of getting 8 or 5 (average 6.5) Utils if he takes a seat. The -3 for getting picked last vs the -2 for waiting in line for 5th is more than made up for by the chance he gets picked 5th anyway for +2. The 4th guy isnt so lucky. The average of being picked 4/5/6 is only 8, while standing in line guarantees 9.<br />
<br />
<br />
<br />
Hardin, G. The tragedy of the commons. Science 1968, 162, 1243–1248. 2. Von Neumann, J.; <br />
Szilagyi, Miklos N. "An investigation of N-Person prisoners' dilemmas." Complex Systems 14.2 (2003): 155-174.<br />
Morgenstern, O. Theory of Games and Economic Behavior; Wiley: New York, NY, USA, 1944. 3. Maynard Smith, J. Evolution and the Theory of Games; Cambridge University Press: Cambridge, UK, 1982.<br />
Cooperation in n - player Prisoner’s Dilemma threshold game Boza, G. (1,4) , Könnyű, B. (1) and Számadó, Sz. (2,3) 1-Department of Plant Taxonomy and Ecology, Eötvös Loránd University 2-HAS Research Group of Ecology and Theoretical Biology, Eötvös Loránd University 3-Collegium Budapest, Institute for Advanced Study Budapest, Hungary 4-IIASA, International Institute for Applied Systems Analysis Laxenburg, Austria<br />
Bach, LA, Helvik, T & Christiansen, FB 2006, 'The evolution of n-player cooperation—threshold games and ESS bifurcations' Journal of Theoretical Biology, bind 238, s. 426-434.</div>Xkorj58