Battle of the sexes

This article covers the battle of the sexes as a game in game theory. For more meanings, see Gender Warfare or Battle of the Sexes .

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The battle of the sexes (Engl. Battle of the sexes ) is a problem from the game theory and represents the coordination games with distribution conflict. Two players want to spend the evening together, but forget to agree on the location. Either a soccer game or a concert is possible . Both players have to decide independently of each other. The football game is preferred by the man, the concert by the woman . The game can also be represented in extensive form .

Bimatrix representation

Balance in pure strategies

The payout bimatrix for symmetrical two-player game is as follows: ${\displaystyle N=\{1,2\}}$

Man Woman	Soccer	concert
Soccer	(3, 1)	(0, 0)
concert	(0, 0)	(1, 3)

The man's “ payoff ” comes first, the woman second. If the woman goes to the football stadium, the man's best answer would be to go there too. The reverse is also true, so the top left cell is a Nash equilibrium . The same applies to the concert hall. So there are two Nash equilibria in pure strategies . ${\displaystyle NGG=\{({\text{Fussball}},{\text{Fussball}}),({\text{Konzert}},{\text{Konzert}})\}}$

The problem with this game is that there are no dominant strategies . If the two players choose their favorite alternative at the same time (woman chooses strategy concert, man chooses strategy football), there will be no meeting at all, which is not ideal for both. In this case, they would rather go to the place that the other prefers - the main thing is that they are together. But if both think that way and want to meet the other, they don't meet again.

Balance in mixed strategies

Since every finite game has a Nash equilibrium (possibly one in mixed strategies), one way out of the problem described above is for the players to randomly decide (randomize) which location they will go to tonight. There is a balance in mixed strategies for this . A Von Neumann Morgenstern utility function is established . For the benefit of the man arises

${\displaystyle {\begin{aligned}U_{m}&=3\cdot F_{m}\cdot F_{f}+0\cdot F_{m}\cdot (1-F_{f})+0\cdot (1-F_{m})\cdot F_{f}+1\cdot (1-F_{m})\cdot (1-F_{f})\\&=3\cdot F_{m}\cdot F_{f}+1-F_{f}-F_{m}+F_{m}\cdot F_{f}\\&=1+4\cdot F_{m}\cdot F_{f}-F_{m}-F_{f}\end{aligned}}}$

and for the benefit of women

${\displaystyle {\begin{aligned}U_{f}&=1\cdot F_{m}\cdot F_{f}+0\cdot F_{m}\cdot (1-F_{f})+0\cdot (1-F_{m})\cdot F_{f}+3\cdot (1-F_{m})\cdot (1-F_{f})\\&=F_{m}\cdot F_{f}+3-3\cdot F_{f}-3\cdot F_{m}+3\cdot F_{m}\cdot F_{f}\\&=3+4\cdot F_{m}\cdot F_{f}-3\cdot F_{f}-3\cdot F_{m}\end{aligned}}}$

Here is the probability that the woman will go to soccer and the probability that the man will go to soccer. If one player plays the randomized strategy corresponding to the Nash equilibrium in mixed strategies, the other player is indifferent between the pure strategies that he plays in this Nash equilibrium with a positive probability, i.e. H. each of these pure strategies brings him the same expected utility. This can be used to calculate the Nash equilibrium. The following must then apply: ${\displaystyle F_{f}}$ ${\displaystyle F_{m}}$

${\displaystyle U_{m}=1+4\cdot 1\cdot F_{f}-1-F_{f}=1+4\cdot 0\cdot F_{f}-0-F_{f}}$

${\displaystyle U_{f}=3+4\cdot F_{m}\cdot 1-3\cdot 1-3\cdot F_{m}=3+4\cdot F_{m}\cdot 0-3\cdot 0-3\cdot F_{m}}$

It follows from the first equation and from the second . It follows that both should go to their partner's favorite place in 25% of all cases. The game of battle of the sexes is usually chosen to start with mixed strategies because it is comparatively easy to calculate. The percentages for mixed strategies, e.g. B. in tennis or in penalty shootouts (see Dixit / Nalebuff), where there are also no dominant strategies, but the repetition rate is correspondingly high. In the case of a non-symmetrical evaluation, other results, albeit basically similar, result. ${\displaystyle F_{f}=0{,}25}$ ${\displaystyle F_{m}=0{,}75}$

literature

• Manfred J. Holler and Gerhard Illing: Introduction to game theory . 6th edition. Springer, 2006, ISBN 3-540-27880-X , pp. 11-12 and 88-90 , doi : 10.1007 / 3-540-29948-3 .