(8th edition) (the pearson series in economics) robert pindyck, daniel rubinfeld microecon 522
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CHAPTER 13 • Game Theory and Competitive Strategy 497
TABLE 13.7
THE BATTLE OF THE SEXES
Jim
Joan
Wrestling
Opera
Wrestling
2, 1
0, 0
Opera
0, 0
1, 2
One reason to consider mixed strategies is that some games (such as
“Matching Pennies”) do 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 do 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
7
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 do better in terms of expected payoff no matter what
she chooses.
8
