N-player prisoner's dilemma - Revision history

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Xkorj58: /* Practice */

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‎Practice

@@ Line 16: / Line 16: @@
 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>
-[[File:N-player_graph_1.png]]
+[[File:N-player_graph_1.png|center]]
 =Parameters=
@@ Line 111: / Line 111: @@
 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>
-[[File:Tragedyoc.jpg|500px]]
+[[File:Tragedyoc.jpg|500px|center]]

@@ Line 96: / Line 96: @@
 We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix.
-[[File:N-player_graph_2.png|thumb||<ref name="Szilagyi"></ref>]]
+[[File:N-player_graph_2.png|thumb|center||<ref name="Szilagyi"></ref>]]
@@ Line 102: / Line 102: @@
 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>
-[[File:N-player_graph_3.png|thumb||<ref name="Szilagyi"></ref>]]
+[[File:N-player_graph_3.png|thumb|center||<ref name="Szilagyi"></ref>]]
 =Examples=

@@ Line 35: / Line 35: @@
 ''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.
-Firstly, distinction he used for his work model.
+Firstly, distinction he used for his work model<ref name="Szilagyi"></ref>.
 * 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.
@@ Line 96: / Line 96: @@
 We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix.
-[[File:N-player_graph_2.png||<ref name="Szilagyi"></ref>]]
+[[File:N-player_graph_2.png|thumb||<ref name="Szilagyi"></ref>]]
@@ Line 102: / Line 102: @@
 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>
-[[File:N-player_graph_3.png||<ref name="Szilagyi"></ref>]]
+[[File:N-player_graph_3.png|thumb||<ref name="Szilagyi"></ref>]]
 =Examples=

@@ Line 89: / Line 89: @@
 |}
-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">
+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>
 =Models=
@@ Line 96: / Line 96: @@
 We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix.
-[[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>]]
+[[File:N-player_graph_2.png||<ref name="Szilagyi"></ref>]]
 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.
-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>
+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>
-[[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>]]
+[[File:N-player_graph_3.png||<ref name="Szilagyi"></ref>]]
 =Examples=

@@ Line 33: / Line 33: @@
 ''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.
-''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.
+''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.
 Firstly, distinction he used for his work model.
@@ Line 89: / Line 89: @@
 |}
-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>
+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">
 =Models=

@@ Line 14: / Line 14: @@
 * The dominant strategies intersect at a deficient equilibrium point
-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>
+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>
 [[File:N-player_graph_1.png]]
@@ Line 89: / Line 89: @@
 |}
-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.
+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>
 =Models=
@@ Line 96: / Line 96: @@
 We will use something like the first graph. At first, we will infer 2-player graph out of 2-player matrix.
-[[File:N-player_graph_2.png]]
+[[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>]]
 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.
-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:
+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>
-[[File:N-player_graph_3.png]]
+[[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>]]
 =Examples=
-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.
+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>
@@ Line 114: / Line 114: @@
-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.
+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> .
@@ Line 121: / Line 121: @@
 [[File:Zimbardo.jpg|thumb|150px|right||P.Zimbardo]] [[File:Milgram.jpg|thumb|150px|left||S. Milgram]]
-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.
+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>
-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.
+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>
 =Practice=
@@ Line 165: / Line 165: @@
 '''''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.''
 =References=

@@ Line 1: / Line 1: @@
 =Introduction=
-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.
+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.
 =Problem definition=
-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.
+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].
 '''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.

@@ Line 128: / Line 128: @@
 Try to solve this N-player game:
 players are waiting to get checked into a flight.
-The consumers utility is based on when they get checked in:
+*6 players are waiting to get checked into a flight.
 {| class="wikitable"
@@ Line 156: / Line 157: @@
-The airline will check people randomly if no one is in line, but those in line will be checked-in first
+*The airline will check people randomly if no one is in line, but those in line will be checked-in first
-Players incur a 2 units loss of utility by waiting in line.
+*Players incur a 2 units loss of utility by waiting in line.
-What is the equilibrium number of people standing in line?
+*What is the equilibrium number of people standing in line?
 '''''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.''