game theory Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed ...

, the battle of the sexes is a two-player

coordination game A coordination game is a type of simultaneous game found in game theory. It describes the situation where a player will earn a higher payoff when they select the same course of action as another player. The game is not one of pure conflict, which ...

that also involves elements of conflict. The game was introduced in 1957 by R. Duncan Luce and

Howard Raiffa Howard Raiffa ( ; January 24, 1924 – July 8, 2016) was an American academic who was the Frank P. Ramsey Professor (Emeritus) of Managerial Economics, a joint chair held by the Business School and Harvard Kennedy School at Harvard University. He ...

in their classic book, ''Games and Decisions''. Some authors prefer to avoid assigning sexes to the players and instead use Players 1 and 2, and some refer to the game as Bach or Stravinsky, using two concerts as the two events.Osborne, Martin and Ariel Rubinstein (1994). ''A Course in Game Theory.'' The MIT Press. The game description here follows Luce and Raiffa's original story. Imagine that a man and a woman hope to meet this evening, but have a choice between two events to attend: a prize fight and a

ballet Ballet () is a type of performance dance that originated during the Italian Renaissance in the fifteenth century and later developed into a concert dance form in France and Russia. It has since become a widespread and highly technical form of ...

. The man would prefer to go to prize fight. The woman would prefer the ballet. Both would prefer to go to the same event rather than different ones. If they cannot communicate, where should they go? The

payoff matrix In game theory, normal form is a description of a ''game''. Unlike extensive-form game, extensive form, normal-form representations are not Graph (discrete mathematics), graphical ''per se'', but rather represent the game by way of a matrix (mathe ...

labeled "Battle of the Sexes (1)" shows the payoffs when the man chooses a row and the woman chooses a column. In each cell, the first number represents the man's payoff and the second number the woman's. This standard representation does not account for the additional harm that might come from not only going to different locations, but going to the wrong one as well (e.g. the man goes to the ballet while the woman goes to the prize fight, satisfying neither). To account for this, the game would be represented in "Battle of the Sexes (2)", where in the top right box, the players each have a payoff of 1 because they at least get to attend their favored events.

Equilibrium analysis

This game has two

pure strategy In game theory, a move, action, or play is any one of the options which a player can choose in a setting where the optimal outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the actio ...

Nash equilibria In game theory, the Nash equilibrium is the most commonly used solution concept for non-cooperative games. A Nash equilibrium is a situation where no player could gain by changing their own strategy (holding all other players' strategies fixed) ...

, one where both players go to the prize fight, and another where both go to the ballet. There is also a

mixed strategy In game theory, a move, action, or play is any one of the options which a player can choose in a setting where the optimal outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the actio ...

Nash equilibrium, in which the players randomize using specific probabilities. For the payoffs listed in Battle of the Sexes (1), in the mixed strategy equilibrium the man goes to the prize fight with probability 3/5 and the woman to the ballet with probability 3/5, so they end up together at the prize fight with probability 6/25 = (3/5)(2/5) and together at the ballet with probability 6/25 = (2/5)(3/5). This presents an interesting case for

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 is inefficient: the players will miscoordinate with probability 13/25, leaving each player with an expected return of 6/5 (less than the payoff of 2 from each's less favored pure strategy equilibrium). It remains unclear how expectations would form that would result in a particular equilibrium being played out. 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 players could flip a coin before choosing their strategies, they might agree to correlate their strategies based on the coin flip by, say, choosing ballet in the event of heads and prize fight in the event of tails. Notice that once the results of the coin flip are revealed neither player has any incentives to alter their proposed actions if they believe the other will not. The result is that perfect coordination is always achieved and, prior to the coin flip, the expected payoffs for the players are exactly equal. It remains true, however, that even if there is a correlating device, the

in which the players ignore it will remain; correlated equilibria require both the existence of a correlating device and the expectation that both players will use it to make their decision.

Notes

References

* Fudenberg, D. and Tirole, J. (1991) ''Game theory'', MIT Press. (see Chapter 1, section 2.4) *

External links

GameTheory.net

Cooperative Solution with Nash Function
by Elmer G. Wiens {{DEFAULTSORT:Battle Of The Sexes (Game Theory) Non-cooperative games