Game Theory and Social Psychology

Transcription

1 Game Theory and Social Psychology cf. Osborne, ch 4.8 Kitty Genovese: attacked in NY in front of 38 witnesses no one intervened or called the police Why not? \Indierence to one's neighbour and his troubles is a conditioned reex of life in NY as it is in other big cities" (Rosenthal, 1964)

2 Experiments in social psychology: lone witness to a problem is very likely to help As group size (n) increases, decline in probability that at least one person helps Explanations in social psychology: a) diusion of responsibility { larger n, lower the psychological cost of not helping b) audience inhibition { larger n; greater the embarassment suered by the helper in the event that help is inappropriate. c) social inuence { a person infers that if others are not helping, then its not appropriate to help.

3 Explanations assume that the benet from helping is decreasing in n Is there an explanation where benets and costs constant, but prob. that no one helps rises with n?

4 Suppose each of n witnesses prefers that a crime is reported than not gets utility v from it being reported personal cost c if she makes the report, v > c If only one person, she will report the crime for sure Each has to choose between fr; Ng Pure strategy equilibrium where exactly one witness i reports and no one else does. i gets v c from R (given that no one else does)

5 if she deviates and chooses N; will get 0 < v c any other player gets v from R (given that i is choosing R) woud get v c from choosing R: No pure strategy equilibrium where more than one person reports. How to coordinate? How will we realize that i is the person who will report? Symmetric mixed strategy equilibrium. Each person reports with probability p:

6 Pr(no one reports) = (1 p) n u(n; p; n) = Pr(at least one other person reports) v = f1 (1 p) n 1 gv: u(r; p; n) = v c: In a mixed strategy equilibrium, u(n; p; n) = u(r; p; n):

7 Pr(at least one other person reports) v = v c RHS is independent of n; so LHS must also be same for all values of n: Pr(at least one other person reports) = v v c Pr(no one reports) = Pr(no other person reports) (1 p) So if p is decreasing in n; Pr(no one reports) increases as n increases.

8 To determine p; v c = [1 (1 p) n 1 ]v 1 p = n 1c v = c v 1 n 1 p is decreasing in n; so Pr(no one reports) increases as n increases. Example c=v = 1 2 ; 1 p = n 11 2

9 if n = 2; p = 0:5: Pr (no report)= 1 4 : n = 3; p = 0:29; Pr (no report)=(0:707) 3 = 0:353 n = 9; p = 0:08; = Pr(no report) = 0:917 9 = 0:46 Not surprising that each individual reports less frequently when n increases But prob of no report also rises.

10 Games in extensive form Set of players N = f1; 2; :::; Ng set of actions A set of nodes (or histories) X x 0 is the initial node or empty history any other x 2 X = (a 1 ; a 2 ; ::; a k ) results from this sequence of actions a h 2 A

11 A(x) is the set of actions available at x A(x 0 ) is the set of actions that nature chooses between at x 0 is a prob. distribution on A(x 0 ) nature moves only once, at x 0 Set of terminal nodes E; where A(z) = ; for z 2 E any terminal node describes a complete play of the game : Xn(E [ fx 0 g)! I tells us which player chooses at any node x

12 X i is the set of decision nodes for player i I i is a partition of X i ;player i's infomation partition. i.e. I i is a collection of player i's information sets, x; x 0 2 I i (x) ) A(x) = A(x 0 ) u i : E! R is i's vnm utility function =< I; A; X; E; ; ; I; (u i ) i2i > is nite if A and X are nite.

13 A pure strategy for player i is a function s i : I i! A satisfying s i (I(x)) 2 A(x) S i is the set of pure strategies for i If we x a pure strategy for each player, this in conjunction with nature's inital choice, determines a probability distribution over E gives rise to expected payo for each player. (S i ; u i ) i2i is the strategic form of We can therefore analyze by analysing its strategic form

14 Games of perfect information is a game of perfect information if I (x) is a singleton set for every x 2 X Can be solved by backwards induction. a node x is penultimate if every action in A(x) results in a node belonging to E at any penultimate node x; let (x) select an action that maximizes his payo Let u x be the resulting payo vector Delete all the branches following x; and assign payo vector u x to x

15 This gives rise to new game tree 1 Repeat this procedure until every node has been assigned an action. Results in backward induction pure strategy prole. Theorem If s is a backward induction strategy prole of a game of perfect information, then s is a Nash equilibrium of : Suppose s is not a NE, so that there exists s 0 i such that u i (s 0 i ; s i) > u i (s) Then there must be a choice of nature a 1 and two end nodes e and e 0 where e is induced by s and e 0 is induced by (s 0 i ; s i) following a 1 such that u i (e 0 ) > u i (e): The set S of decison nodes following a 1 : i can do better is non-empty.

16 Let x 2 S be such that there is no element of S that strictly follows it:. x belongs to player i at any node following x; i cannot improve on s i If i follows that if i takes an action a 0 at x; and follows s i thereafter, he can increase his payo. But s i prescribes the payo maximizing action at x; given that s is played thereafter. Contradiction.

17 One step deviation principle for BI strategies: if a nite number of deviations is protable, then there must be one deviation that is protable. Pure strategy NE exist in games of perfect information.

18 Games of imperfect information: subgame perfect equilibrium a node x denes a subgame of if I(x) is a singleton set and whenever a decision node y weakly follows x; and z 2 I(y); then z weakly follows x: x is the subgame dened by node x a pure strategy prole s in induces a pure strategy prole in x A pure strategy prole s is a subgame perfect equilibrium of in every subgame of ; if s induces a NE Backwards induction strategies are subgame perfect equilibria

19 Mixed strategy in extensive form games: randomization over the pure strategies in the strategic form Behavior strategies: local randomization Local strategy at each information set { randomization over the actions available. behavior strategies: local strategies for each information set A game has perfect recall if a player does not forget what he already knows. Let node x; y belong to a player and suppose that y results from (x; a 1 ; a 2 ; ::; a k )

20 Then if w and y belong to the same information set, w must result from (z; a 1 ; a 0 2 ; ::; a0 h ) where z 2 I(x) We will assume perfect recall Kuhn's theorem: In a game with perfect recall any mixed strategy has an equivalent behavior strategy and vice versa. More convenient to use behavior strategies

21 Thm: A subgame perfect equilibrium exists in any nite extensive form game. Consider any subgame which does not contain a proper subgame. Nash equilibrium exists in this subgame satises perfect recall, and hence behavior strategies dened. Replace this subgame with its NE payo vector Continue this process...ends in nitely many steps. local strategies dened at each step. Behavior strategy for i : local strategies so dened.