Normal form - Revision history

Jirihradil at 15:12, 24 January 2013

2013-01-24T15:12:35Z

Jirihradil: /* Example 2 */

2013-01-24T15:10:36Z

‎Example 2

Jirihradil: /* Exercises */

2013-01-24T15:09:40Z

‎Exercises

Jirihradil: /* Matching pennies */

2013-01-24T15:08:35Z

‎Matching pennies

Jirihradil: /* The battle of sexes */

2013-01-24T15:08:21Z

‎The battle of sexes

Jirihradil: /* The prisoner's dilemma */

2013-01-24T15:08:09Z

‎The prisoner's dilemma

Jirihradil: /* The prisoner's dilemma */

2013-01-24T15:07:43Z

‎The prisoner's dilemma

Jirihradil: /* The prisoner's dilemma */

2013-01-24T15:07:05Z

‎The prisoner's dilemma

Jirihradil: /* The prisoner's dilemma */

2013-01-24T15:06:39Z

‎The prisoner's dilemma

Jirihradil: /* Terminology */

2013-01-24T15:05:14Z

‎Terminology

← Older revision		Revision as of 15:12, 24 January 2013
Line 1:		Line 1:
−	~~= This page is in progress. When I finish it, I'll remove this notice =~~
−
−
	= Normal form =		= Normal form =
	== What is a Normal form? ==		== What is a Normal form? ==

@@ Line 265: / Line 265: @@
 | cooperate
 | 10, 10
-| 0, 20
+| 20, 0
 |-
 | defect
-| 20, 0
+| 0, 20
 | 1, 1
 |}

@@ Line 236: / Line 236: @@
 == Exercises ==
-Find all Nash equilibria for all of these games.
+Find all Nash equilibria for following games.
 === Example 1 ===

@@ Line 215: / Line 215: @@
 === Matching pennies ===
-The last example (but not the last in general, of course) is “the Matching pennies game”. 2 players have a penny with 2 sides (head and tail). Both of them will flip a penny secretly and then reveal the result. If both pennies have heads or tails, player 1 will take both pennies. If they differ (the first penny has head and the second has tail or vice versa), player 2 will take both pennies.
+The last example (but not the last in general, of course) is „the Matching pennies game”. 2 players have a penny with 2 sides (head and tail). Both of them will flip a penny secretly and then reveal the result. If both pennies have heads or tails, player 1 will take both pennies. If they differ (the first penny has head and the second has tail or vice versa), player 2 will take both pennies.
 The game can be described in the following form:

@@ Line 194: / Line 194: @@
 === The battle of sexes ===
-Another example of using normal form is “the battle of sexes”. It is a two-player coordination game where one of them would like to do something and the another one would like to do something different. A common example is a battle between man and woman where man would like to watch a sport match but woman would like to watch an opera. Both players prefer to stay together and in this case they have to because they have only one TV.
+Another example of using normal form is „the battle of sexes”. It is a two-player coordination game where one of them would like to do something and the another one would like to do something different. A common example is a battle between man and woman where man would like to watch a sport match but woman would like to watch an opera. Both players prefer to stay together and in this case they have to because they have only one TV.
 The normal form matrix is described like a table:

@@ Line 191: / Line 191: @@
 |}
-The best strategy is "to betray" for both prisoners and it's the only pure Nash Equilibrium.
+The best strategy is „to betray" for both prisoners and it's the only pure Nash Equilibrium.
 === The battle of sexes ===

@@ Line 175: / Line 175: @@
 In the worst case I spend 2 years in jail. But if I'm lucky I will get 0 years. The problem can be solved by the Nash equilibrium.
-The matrix is in a normal form, so we can say that „prisoner's dilemma problem was put into the normal form and then the Nash Equilibrium is used to find a solution of the problem.
+The matrix is in a normal form, so we can say that „prisoner's dilemma problem was put into a normal form" and then the Nash Equilibrium is used to find a solution of the problem.
 {| class="wikitable"

@@ Line 171: / Line 171: @@
 A good example for using normal form is a description of a famous Prisoner's dilemma problem. 2 prisoners are in a jail. They can't cooperate together. Both of them would like to avoid imprisonment as much as possible. Each prisoner has a dilemma: should I keep my mouth shut (stay silent) or should I betray the other prisoner? From their point of the view, they're not sure because they don't know how the second prisoner will decide.
-If I betray and the second prison will stay silent I can go out without any punishment. But he will get 3 years (I don't mind about him). If I stay silent and the second one betrays he'll be free but I get 3 years. If both of us keep silent we'll get 1 year. But if we betray, each of us will be punished by 2 years.
+If I betray and the second prison will stay silent I can go out without any punishment. But he will get 3 years (I don't mind about him). If I stay silent and the second one betrays he'll be free but I'll get 3 years. If both of us keep silent we'll get 1 year. But if we betray, each of us will be punished by 2 years.
 In the worst case I spend 2 years in jail. But if I'm lucky I will get 0 years. The problem can be solved by the Nash equilibrium.

@@ Line 169: / Line 169: @@
 == Examples ==
 === The prisoner's dilemma ===
-A good example for using normal form is a description of a famous Prisoner's dilemma problem. 2 prisoners are in a jail. They can't cooperate together. Both of them would like to avoid imprisonment as much as possible. Each prisoner has a dilemma: should I keep my mouth shut (stay silent) or should I betray the other prisoner? From their point of the view, I'm not sure because I don't know how the second prisoner will decide.
+A good example for using normal form is a description of a famous Prisoner's dilemma problem. 2 prisoners are in a jail. They can't cooperate together. Both of them would like to avoid imprisonment as much as possible. Each prisoner has a dilemma: should I keep my mouth shut (stay silent) or should I betray the other prisoner? From their point of the view, they're not sure because they don't know how the second prisoner will decide.
 If I betray and the second prison will stay silent I can go out without any punishment. But he will get 3 years (I don't mind about him). If I stay silent and the second one betrays he'll be free but I get 3 years. If both of us keep silent we'll get 1 year. But if we betray, each of us will be punished by 2 years.

@@ Line 158: / Line 158: @@
 * '''Normal form''' – representation of a game, a description. If you take a look at the normal form then you can explain game for every player in every stage (round) of the game and you know what each player earns (payoffs).
 * '''Strategic form''' – just another term for a Normal form. A synonym.
-* '''Simultaneous game''' – a game in which each player decides without a knowledge of other player's decisions. So it is same as they move simultaneously. In simultaneous game, no sequence can be described. If we would like to describe a sequence between moves then we should to move to sequential games.
+* '''Simultaneous game''' – a game in which each player decides without a knowledge of other player's decisions. So it's the same as they move simultaneously. In a simultaneous game, no sequence can be described. If we would like to describe a sequence between moves then we should to move to sequential games.
 * '''Sequential game''' – a game where we use movement descriptions. For example, Adam will choose „Forward“, then Bob will choose „Left“, then Adam „Forward“, and so on.
-* '''Strategy profile''' – a part of a strategy. One step in strategy which generates a payoff.
+* '''Strategy profile''' – a part of a strategy. One step in a strategy which generates a payoff.
-* '''Strategy''' – each player of the game has a plan – how he will play (behave) in every stage (round) of the game. The set of all actions is called a strategy.
+* '''Strategy''' – each player of the game has a plan – how he's going to play (behave) in every stage (round) of the game. The set of all actions is called a strategy.
 * '''Strategy space''' – set of all possible strategies in the game for a player
-* '''Payoff''' – for each player of a game, this is what a player receives, what he's going to get from the game. Payoff is the reason why player plays the game. Often the payoff is shown as number which represents a utility or an outcome. So we can say that payoff is a utility function. A payoff determines the meaning of a game.
+* '''Payoff''' – for each player of a game, this is what a player receives, what he's going to get from the game. A payoff is the reason why player plays the game. Often a payoff is shown as a number which represents a utility or an outcome. So we can say that payoff is a utility function. A payoff determines the meaning of a game.
-* '''Nash equilibrium''' – a strategy<nowiki> profile with the property that no single player can obtain a higher payoff by deviating unilaterally from this profile [6]. </nowiki>Simple said, nobody can get more at the expense of someone's else.
+* '''Nash equilibrium''' – a strategy profile with the property that no single player can obtain a higher payoff by deviating unilaterally from this profile [[#link6|[6]]]. Simple said, nobody can get more at the expense of someone's else.
 * '''Finite game''' – a game which stops at some time or has limited number of rounds. You can think these games from the beginning to the end.