Battle of the sexes (game theory)
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In game theory, the battle of the sexes is a two-player coordination game that also involves elements of conflict. The game was introduced in 1957 by R. Duncan Luce and Howard Raiffa in their classic book, Games and Decisions.[1] 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.[2] 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. 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
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
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
Notes
- ^ Luce, R.D. and Raiffa, H. (1957) Games and Decisions: An Introduction and Critical Survey, Wiley & Sons (see Chapter 5, section 3).
- ^ Osborne, Martin and Ariel Rubinstein (1994). A Course in Game Theory. The MIT Press.
References
- Fudenberg, D. and 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