[1 1 .] We found two pure strategy Nash Equilibria for the Battle of the Sexes G

Question

[1 1 .] We found two pure strategy Nash Equilibria for the Battle of the Sexes Game with Miguel and Vanesa There also exists a mixed strategy Nash Equilibrium. Let p be the probability that Vanesa picks Football and q be the probability that Miguel picks football. A. Find the best response correspondences for each player Hint: Follow the matching pennies example done in class. B. Graph the the best response correspondences. Let p be on the horizontal axis and q be on the vertical axis C. Find the mixed strategy NE: (p*,q*)

Explanation / Answer

We have seen the two Nash equilibria in pure strategies: both watch mud wrestling and both go to the opera. Equilibrium in mixed strategies has Joan choosing wrestling with probability 2/3, where Jim chooses wrestling with probability 1/3

Since these are mixed strategies of random selection we should not expect Jim and Joan to use these mixed strategies, unless they are risk loving or irrational. Since mixed stratgeies are usually not chosen. This seems to be working when we compare their payofffs. The expected payoff from mixed strategies to Joan is 2/3*2 + 1/3*0 = 4/3 and to Jim is 1/3*1 + 2/3*0 = 1/3. If they agree to either of the two choices, each of them will have a payoff of at least 1 as in the pure strategies, while the expected payoff is less for Jim in this case.

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