Chapter 28 Game Theory

Three fundamental elements to describe a game: Players, (pure) strategies or actions, payoffs.

Color Matching B b r 1, -1 -1, 1 b A r -1, 1 1, -1

Payoff matrices for Two-person games. Simultaneous(-move) games. Finite games: Both the numbers of players and of alternative pure strategies are finite

The Prisoner’s Dilemma B Confess Deny Confess A Deny -3*, -3* 0, -5 -5, 0 -1, -1

Method of iterated elimination of strictly dominated strategies. Dominant strategies, and dominated strategies. Method of iterated elimination of strictly dominated strategies.

The Prisoner’s Dilemma shows also that a Nash equilibrium does not necessarily lead to a Pareto efficient outcome. Two-win games.

A pair of strategies is a Nash equilibrium if A’s choice is optimal given B’s choice, and vice versa. Nash is a situation, or a strategy combination of no incentive to deviate unilaterally.

Battle of Sexes Girl Soccer Ballet 2*, 1* 0, 0 Soccer Boy Ballet 2*, 1* 0, 0 Soccer Boy Ballet -1, -1 1*, 2*

Method of underlining relatively advantageous strategies. Double underlining gives Nash. There can be no, one, and multiple (pure) Nash equilibria.

Price Struggle Pepsi L H L Coke H 3*, 3* 6, 1 1, 6 5, 5

How if there is no Nash of pure strategies? Mixed strategies (by probability). Method of response functions.

Color Matching again B b r q 1-q 1, -1 -1, 1 b p A r 1-p -1, 1 1, -1

With EUA = 1 pq + (-1)p (1-q) + (-1) (1-p) q + 1 (1-p) (1-q) = pq - p + pq - q + pq + 1 - p - q + pq = 4 pq - 2 p - 2 q + 1 = 2 p (2 q - 1) + (1 - 2 q ) , we have 1 if q > 1/2 , p = [0, 1] if q = 1/2 , 0 if q < 1/2 .

Similarly, 0 if p > 1/2 , q = [0, 1] if p = 1/2 , 1 if p < 1/2 . q 1 N 1 p

Method of response functions: The intersections of response functions give Nash equilibria ((p*, q*) = (1/2, 1/2) in example) Nash Theorem: There is always a ( maybe “mixed”) Nash equilibrium for any finite game

Sequential games. Games in extensive form versus in normal form.

Battle of Sexes again Ballet ( 1, 2) Girl ( -1, -1) Ballet Soccer Boy ( 0, 0) Soccer ( 2, 1) Soccer

Strategies as Plans of Actions. Boy’s strategies: Ballet, and Soccer. Girl’s Strategies: Ballet strategy; Soccer strategy; Strategy to follow; and Strategy to oppose.