Mixed Strategies and Repeated Games Game Theory 2 Mixed Strategies and Repeated Games

Mixed Strategies Mixed Strategies: Players make random choices among two or more options based on sets of chosen probabilities Pure Strategies: Players make specific choices among two or more options Some games may have no Nash Equilibria in Pure Strategies Every game has at least one Nash Equilibrium when mixed strategies are allowed Examples Matching Pennies The Battle of the Sexes

Matching Pennies Player B Heads Tails Player A 1, -1 -1, 1

Mixed Strategies: Matching Pennies Heads ½, Tails ½ ¼ probability of getting any outcome Expected payoff is zero for each player Any other probabilities give at least one player a worse payoff Example: Heads ¾ , Tails ¼ Probabilities Heads, Heads 9/16 Tails, Tails 1/16 Heads, Tails 6/16 Payoffs Player A: + ¼ Player B: - ¼

Battle of the Sexes Jim Wrestling Opera Joan 2, 1 0, 0 1, 2

Mixed Strategies: Battle of the Sexes Nash Equilibrium Jim: Wrestling 1/3 , Opera 2/3 Joan: Wrestling 2/3 , Opera 1/3 Expected Payoffs: 2/3 Proof Say Jim chooses W 1/3, O 2/3 as above Joan chooses W with probability p, O with (1-p) Both choose O: 2/3(1-p) Both Choose W: 1/3 p Joan’s expected payoff: 2(1/3 p) + 1(2/3)(1-p) = (2/3)p + 2/3 – (2/3) p = 2/3 Independent of p: Joan cannot improve her payoff

Battle of the Sexes: Examples Suppose Jim chooses W ½ , O ½ Joan chooses W p, O (1-p) Joan’s Payoff: 2(1/2)p + 1(1/2)(1-p) = ½ p +1/2 Joan would choose p = 1 to max her payoff When Joan chooses p = 1, Jim is better off with W 1, O 0 Suppose Jim chooses W ¼ , O ¾ Joan’s payoff: 2(1/4)p + 1(3/4)(1-p) = ½ p + ¾ - ¾ p = ¾ - ¼ p Joan chooses p = 0 to max her payoff Jim is better off then to choose W = 0, O = 1 Pure Strategies preferred to all other mixed strategies

Why Mixed Strategies? Mixed Strategy expected payoff is zero in Matching Pennies In Battle of the Sexes, Pure Strategy equilibria yield higher payoffs Possibilities Equilibria when pure strategies fail (poker, for instance) Players are risk lovers Often not realistic Would firms really expect a rival to randomly Set prices Choose advertising strategies Enter or exit markets

Repeated Games Low Price High Price 10,10 100, -50 -50, 100 50, 50 Consider Repeating A Prisoners’ Dilemma Game Firm Low Price 2 High Price Firm 1 10,10 100, -50 -50, 100 50, 50

Infinitely Repeated Games Strategies depend on present values of payoffs Suppose the discount rate is 10%. The payoff for any player is Π = π1 + π2/(1.1) + π3/(1.1)2 + π4/(1.1)3….. Note: 1 + d + d2 + d3 +…. = 1/(1- d); d + d2 + d3 +…. = d/(1- d) => 1 + 1/(1.1) + 1/(1.1)2 + 1/(1.1)3… = 1/[1- (1/1.1)] = 11 Payoffs for infinitely repeated pricing game Collusion forever: 50(11) = 550 Nash forever: 10(11) = 110 Cheat today, Nash after: 100 + 10(10) = 200 Defect from Nash, collude after: -50 + 50(10) = 450

The Tit for Tat Strategy A player plays the same strategy played by its rival in the previous period Collusion (cooperation) is the Nash Equilibrium when one player plays tit-for-tat with certainty Collusion (cooperation) can be the Nash Equilibrium even if there is uncertainty about a player playing tit-for-tat Players may also cycle through Collude, Cheat, Defect, Collude outcomes when one player plays tit-for-tat Applies even for some (sufficiently low) probability that the game will end in the next period

Finitely Repeated Games When a game has a known last period Cheating always pays in the last period Both players know this, such that whoever cheats first gets a higher payoff The game “unravels” from the end backwards to the beginning Consequently, the Low Price equilibrium is the Nash Equilibrium If the end of the game is probabilistic (not known with certainty), High Price (cooperation) can be a Nash Equilibrium.