In game theory, 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 r ...

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 w ...

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. 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 Professional boxing, or prizefighting, is regulated, sanctioned boxing. Professional boxing bouts are fought for a purse that is divided between the boxers as determined by contract. Most professional bouts are supervised by a regulatory autho ...

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, normal-form representations are not graphical ''per se'', but rather represent the game by way of a matrix. While this approach can be of greater use in identifyin ...

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 the players each have a payoff of 2 because they at least get to attend their favored events.

Equilibrium analysis

This game has two

pure strategy In game theory, a player's strategy is any of the options which they choose in a setting where the outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game ...

Nash equilibria In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...

, 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 player's strategy is any of the options which they choose in a setting where the outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game ...

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 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 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 game theory, a correlated equilibrium is a solution concept that is more general than the well known Nash equilibrium. It was first discussed by mathematician Robert Aumann in 1974. The idea is that each player chooses their action according ...

. 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.

Burning money

Interesting strategic changes can take place in this game if one allows one player the option of " burning money" – that is, allowing that player to destroy some of their utility. Consider the battle of the sexes here (called ''Unburned''). Before making his decision, the man (the row player) can, in view of the woman (the column player), choose to set fire to 2 payoff points, leading to the ''Burned'' subgame payoffs pictured to the right. The man and woman then simultaneously and independently choose ''Prize Fight'' or ''Ballet''. If one iteratively deletes weakly dominated strategies then one arrives at a unique solution where the man does not burn the money and both players choose ''Prize Fight''. The odd thing about this result is that by simply having the opportunity to burn money (but not actually using it), the man is able to secure his favored equilibrium. The reasoning that results in this conclusion is known as forward induction and is somewhat controversial.For a detailed explanation, se

p8 Section 4.5. Loosely, if the woman sees the man burn the money, she would think that he is doing so only because he is convinced it would make her choose ''Prize Fight''. She should also think, however, that if he could so convince her but he does not burn the money, he must think that she is going to choose ''Prize Fight'' anyway.

Notes

References

* Fudenberg, D. and Jean Tirole, Tirole, J. (1991) ''Game theory'', MIT Press. (see Chapter 1, section 2.4) * Kelsey, D. and S. le Roux (2015): An Experimental Study on the Effect of Ambiguity in a Coordination Game, Theory and Decision.

External links

GameTheory.net

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