(8th edition) (the pearson series in economics) robert pindyck, daniel rubinfeld microecon 522

CHAPTER 13 • Game Theory and Competitive Strategy 497 TABLE 13.7 THE BATTLE OF THE SEXES Jim Joan Wrestling Opera Wrestling 2, 0, Opera 0, 1, One reason to consider mixed strategies is that some games (such as “Matching Pennies”) not have any Nash equilibria in pure strategies It can be shown, however, that once we allow for mixed strategies, every game has at least one Nash equilibrium.7 Mixed strategies, therefore, provide solutions to games when pure strategies fail Of course, whether solutions involving mixed strategies are reasonable will depend on the particular game and players Mixed strategies are likely to be very reasonable for “Matching Pennies,” poker, and other such games A firm, on the other hand, might not find it reasonable to believe that its competitor will set its price randomly THE BATTLE OF THE SEXES Some games have Nash equilibria both in pure strategies and in mixed strategies An example is “The Battle of the Sexes,” a game that you might find familiar It goes like this Jim and Joan would like to spend Saturday night together but have different tastes in entertainment Jim would like to go to the opera, but Joan prefers mud wrestling As the payoff matrix in Table 13.7 shows, Jim would most prefer to go to the opera with Joan, but prefers watching mud wrestling with Joan to going to the opera alone, and similarly for Joan First, note that there are two Nash equilibria in pure strategies for this game— the one in which Jim and Joan both watch mud wrestling, and the one in which they both go to the opera Joan, of course, would prefer the first of these outcomes and Jim the second, but both outcomes are equilibria—neither Jim nor Joan would want to change his or her decision, given the decision of the other This game also has an equilibrium in mixed strategies: Joan chooses wrestling with probability 2/3 and opera with probability 1/3, and Jim chooses wrestling with probability 1/3 and opera with probability 2/3 You can check that if Joan uses this strategy, Joan cannot better with any other strategy, and vice versa.8 The outcome is random, and Jim and Joan will each have an expected payoff of 2/3 Should we expect Jim and Joan to use these mixed strategies? Unless they’re very risk loving or in some other way a strange couple, probably not By agreeing to either form of entertainment, each will have a payoff of at least 1, which exceeds the expected payoff of 2/3 from randomizing In this game More precisely, every game with a finite number of players and a finite number of actions has at least one Nash equilibrium For a proof, see David M Kreps, A Course in Microeconomic Theory (Princeton, NJ: Princeton University Press, 1990), p 409 Suppose Joan randomizes, letting p be the probability of wrestling and (1 - p) the probability of opera Because Jim is using probabilities of 1/3 for wrestling and 2/3 for opera, the probability that both will choose wrestling is (1/3)p, and the probability that both will choose opera is (2/3)(1 - p) Thus, Joan’s expected payoff is 2(1/3)p + 1(2/3)(1 - p) = (2/3)p + 2/3 - (2/3)p = 2/3 This payoff is independent of p, so Joan cannot better in terms of expected payoff no matter what she chooses