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For other uses see Battle of the sexes.

The Battle of the Sexes is a two player coordination game used in game theory. Imagine a couple. The husband would most of all like to go to the football game. The wife would like to go to the opera. Both would prefer to go to the same place rather than different ones. If they cannot communicate, where should they go?

The payoff matrix labeled "Battle of the sexes (1)" is an example of Battle of the Sexes, where the husband chooses a row and the wife chooses a column.

This representation does not account for the additional harm that might come from going to different locations and going to the wrong one.(ie he goes to the opera while she goes to the football game, satisfying neither) In order to account for this, the game is sometimes represented as in "Battle of the sexes (2)".

This second representation bears some similarity to the Game of chicken.

This game has two pure strategy Nash equilibria, one where both go to the opera and another where both go to the football game. For the first game, there is also a Nash equilibrium in mixed strategies, where the players go to their preferred event more often than the other. For the payoffs listed above, each player attends their preferred event with probability 3/5.

This presents an interesting case for game theory since each of the Nash equilibria is deficient in some way. The two pure strategy Nash equilibria are unfair, one player consistently does better than the other. The mixed strategy Nash equilibrium (when it exists) is inefficient. The players will miscoordinate with probability 13/25, leaving each player with an expected return of 6/5 (less than the return one would receive from constantly going to one's less favored event).

One possible resolution of the difficulty involves the use of a correlated equilibrium. In its simplest form, if the players of the game have access to a commonly observed randomizing device, then they might decide to correlate their strategies in the game based on the outcome of the device. For example, if the couple could flip a coin before choosing their strategies, they might agree to correlate their strategies based on the coin flip by, say, choosing football in the event of heads and opera in the event of tails. Notice that once the results of the coin flip are revealed neither the husband nor wife have any incentives to alter their proposed actions - that would result in miscoordination and a lower payoff than simply adhering to the agreed upon strategies. The result is that perfect coordination is always achieved and, prior to the coin flip, the expected payoffs for the players are exactly equal.

• Luce, R.D. and Raiffa, H. (1957) Games and Decisions: An Introduction and Critical Survey, Wiley & Sons. (see Chapter 5, section 3).
• Fudenberg, D. and Tirole, J. (1991) Game theory, MIT Press. (see Chapter 1, section 2.4)

• GameTheory.net
• Cooperative Solution with Nash Function by Elmer G. Wiens