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Game Theory-Battle of the Sexes

In the battle of sexes game, there is a game with variable pay offs instead of specific numerical values. In this game, it is assumed that a couple decide to go for an entertainment while both forget to communicate about the event where they want to go. The husband and the wife each prefer to go to a different event. At the same time, they prefer to go together. Hence, they have to find a solution about their trip simultaneously but without communication about the event. This is the generalized form of the battle of sexes game.

This game has two pure strategy equilibrium. The preference of each player will be different strategy equilibrium. Moreover, they prefer equilibrium to non-equilibrium outcomes. Thus, both equilibria are Pareto optimal. There is the existence of a mixed strategy equilibrium also.

The general form of the battle of sexes is given as below.

Player 2

 L R Player 1 U a,w b,x D c,y d,z

There are the following relations, which are assumed to expect

d>a>b; d>a>c

w>z>y; w>z>x

Here a two-player coordination game is the case. In the payoff matrix, wife chooses a row and the husband chooses a column. Thus, the payoff to the wife is denoted by the first number in each cell and the second number in each cell shows the payoff to the husband.

The two pure strategy Nash equilibria are unfair since there is chance of one player frequently playing better than the other. The mixed strategy also can be considered as inefficient due to the chance of being mis coordinated. Correlated equilibrium is a possible resolution for this. The decision can be to correlate their strategies in the game depending on the choice's outcome in case of chance of access to a commonly observed randomizing device. Thus, it is sure that achievement of perfect coordination is there. Moreover, the expected payoffs to the participants will be always equal prior to the game. In this game, there can be option of burning money where a player can destroy some of her utility. Then the strategies will definitely change based on this option. Then each player will have four strategies. If this option is not selected, it means that the player expects better outcome than choosing the burning money option. This is also a means of information conveyance to the other player regarding his choice of action.

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