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The concept of the bystander effect, a social psychological phenomenon where individuals are less likely to offer help in an emergency situation when others are present. The document uses the example of Kitty Genovese and the 38 witnesses who did not intervene during her attack in New York City. the reasons behind this phenomenon, including diffusion of responsibility, audience inhibition, and social influence. The document also discusses the mathematical modeling of this situation using game theory, specifically the Prisoner's Dilemma, to understand the equilibrium strategies and the probability of no one reporting a crime as the group size increases.
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Game Theory and Social Psychology
cf. Osborne, ch 4. Kitty Genovese: attacked in NY in front of 38 witnesses no one intervened or called the police Why not? \IndiNY as it is in other big cities" (Rosenthal, 1964) erence to one's neighbour and his troubles is a conditioned re ex of life in
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) di usion of responsibility { larger n, lower the psychological cost of not helping b) audience inhibition { largerin the event that help is inappropriate. n; greater the embarassment su ered by the helper
c) social in uence { a person infers that if others are not helping, then its notappropriate to help.
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; N g Pure strategy equilibrium where exactly one witnessdoes. i reports and no one else
i gets v c from R (given that no one else does)
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:
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.
To determine p;
v c = [1 (1 p)n ^1 ]v
1 p = n v^ c^1 =^ ^ vc^ ^ n ^11
p is decreasing in n; so Pr(no one reports) increases as n increases.
Example
c=v =^12 ; 1 p = n ^121
Games in extensive form
Set of players N = f 1 ; 2 ; :::; N g 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 ; ::; ak) results from this sequence of actions ah 2 A
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
A pure strategy for player i is a function si : Ii! A satisfying si(I(x)) 2 A(x) Si is the set of pure strategies for i If wechoice, determines a probability distribution over x a pure strategy for each player, this in conjunction with nature's inital E
gives rise to expected payo for each player. (Si; ui)i 2 I is the strategic form of We can therefore analyze by analysing its strategic form
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 ux be the resulting payo vector Delete all the branches following x; and assign payo vector ux to x
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 si Ifincrease his payo i follows that if .i takes an action a^0 at x; and follows si thereafter, he can
Butthereafter. si prescribes the payo maximizing action at x; given that s is played
Contradiction.
One step deviation principle for BI strategies: if apro table, then there must be one deviation that is pro table. nite number of deviations is
Pure strategy NE exist in games of perfect information.
Mixed strategy in extensive form games: randomization over the pure strategiesin 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 ; ::; ak
Then if(z; a w and y belong to the same information set, w must result from 1 ; a^02 ; ::; a^0 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 behaviorstrategy and vice versa.
More convenient to use behavior strategies
