Battle of the sexes (1) | Opera | Football | | Opera | 2, 1 | 0, 0 | | Football | 0, 0 | 1, 2 | | Battle of the sexes (2) | Opera | Football | | Opera | 2, 1 | 0, 0 | | Football | -1, -1 | 1, 2 | | The Battle of the Sexes is a two player game used in game theory. Imagine a couple, Kelly and Chris. Kelly would most of all like to go to the football game. Chris 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? Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns. ...
The the payoff matrix labeled "Battle of the sexes (1)" is an example of Battle of the Sexes, where Chris chooses a row and Kelly chooses a column. It has been suggested that this article or section be merged with normal form game. ...
This representation does not account for the additional harm that might come from going to different locations and going to the wrong one. 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. The game of chicken (also referred to as playing chicken) is a game in which two players engage in an activity that will result in serious damage unless one of them backs down. ...
Equilibrium Analysis
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 equilibria in mixed strategies, where Kelly and Chris go to their preferred event more often than the other. For the payoffs listed above, each player attends their preferred event with probability 2/3. A pure strategy is a term used to refer to strategies in Game theory. ...
In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, whereby the players reach an outcome to mutual advantage. ...
A mixed strategy is used in game theory economics to describe a strategy comprising possible moves and a probability distribution which corresponds to how frequently each move is chosen. ...
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 equilibria (when it exists) is inefficient. The players will miscoordinate with probability 5/9, leaving each player with an expected return of 2/3 (less than the return one would receive from constantly going to one's less favored event). Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns. ...
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 Kelly and Chris 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 Kelly nor Chris have any incentives to alter their proposed actions this will 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 of Kelly and Chris are exactly equal.
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