Battle of the sexes (game theory)
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In game theory, battle of the sexes (BoS) is a two-player coordination game, however the game also involves elements of conflict. The traditional method of BOS was to use male and female players, however some authors refer to the game as Bach or Stravinsky and designate the players simply as Player 1 and Player 2, rather than assigning sex.
Imagine that Player 1 and Player 2 agreed to meet this evening, but cannot recall if they will be attending a Bach concert or a Stravinsky concert (and the fact that they forgot is common knowledge). Player 1 would prefer to go to the Stravinsky concert. Player 2 would rather go to the Bach concert. Both would prefer to go to the same place rather than different ones. If they cannot communicate, where should they go?
A 2x2 matrix is used to convey the choices of the players, and a numeric representation is given to their preferred choices. Player 1 is depicted in the rows whereas player 2 is the columns. The payoff matrix labeled "Bach or Stravinsky (1)" is an example of Battle of the Sexes, where Player 1 chooses a row and Player 2 chooses a column. In each cell, the first number represents the payoff to the Player 1 and the second number represents the payoff to Player 2.
This 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. Player 1 goes to the Bach concert while Player 2 goes to the Stravinsky concert, satisfying neither). To account for this, the game would be represented in "Bach or Stravinsky (2)".
In the Bach or Stravinsky game depicted above, the Players must decide which option they would like to choose, keeping in mind they would prefer to be together rather than apart. Both Players can determine that there are two options which are strategically dominated (see strategic dominance), which would both give them a payoff of 0. The other two options left would be the decisions the players should make, as they will both receive utility for either choice. As the BOS is a simultaneous game, the Players know that one of them will have to decide to choose the others dominant strategy. This is where the elements of conflict come into play. As their preferred activities differ from one another, they will experience the conflict, however must involve communication and coordination to decide so as to not have coordination failure and achieve 0 payoff. Although a simultaneous game, the players have the power of disclosing their strategy. If Player 1 initially states that they are going to the Bach concert, it is now up to Player 2 to either agree and go along, as it is in their best interest to achieve some utility because they are together, or be separated. This could work vice versa as well. If Player 1 discloses their strategy, it would be in the best interest of Player 2 to agree and go to the Bach concert to gain utility and be together for the evening. However, without any form of communication, studies  have shown that there will be frequent coordination failures, with up to 52% of failures each time this game is played. Therefore, there is a need for some form of communication between players to achieve a positive result.
This game has two pure strategy Nash equilibria, one where both go to the Bach concert, and another where both go to the Stravinsky concert. There is also a mixed strategies Nash equilibrium in both games, where the players go to their preferred event more often than the other. For the payoffs listed in the first game, each player attends their preferred event with probability 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 (when it exists) is inefficient. The players will miscoordinate with probability 13/25, leaving each player with an expected return of 6/5 (less than the return one would receive from constantly going to one's less favored event).
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 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 Bach in the event of heads and Stravinsky 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 – that would 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 for the players are exactly equal.
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 version of Bach or Stravinsky pictured here (called Unburned). Before making the decision, Player 1 (the row player) can, in view of Player 2 (the column player), choose to set fire to 2 points making the game Burned pictured to the right. This results in a game with four strategies for each player. The row player can choose to burn or not burn the money and also choose to play Stravinsky or Bach. The column player observes whether or not the row player burns and then chooses either to play Stravinsky or Bach.
If one iteratively deletes weakly dominated strategies then one arrives at a unique solution where Player 1 does not burn the money and plays Stravinsky and where Player 2 plays Stravinsky. The odd thing about this result is that by simply having the opportunity to burn money (but not actually using it), Player 1 is able to secure their favored equilibrium. The reasoning that results in this conclusion is known as forward induction and is somewhat controversial. In brief, by choosing not to burn money, the player is indicating they expect an outcome that is better than any of the outcomes available in the "burned" version, and this conveys information to the other party about which branch they will take.
- 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.
- Osborne, Rubinstein (1994). A course in game theory. The MIT Press.
- Alonso-Sanz, Ramón (2012). The Spatialized, Continuous-Valued Battle of the Sexes. Dynamic games and applications, 2012-06, Vol.2 (2), p.177-194
- Luce, R.D. and Raiffa, H. (1957) Games and Decisions: An Introduction and Critical Survey, Wiley & Sons. (see Chapter 5, section 3).
- Friedman, J.W. (1994), Introduction and overview, in Friedman, J.W. (ed.), Problems of Coordination in Economic Activity, Boston: Kluwer Academic Publishers, pp. 3–15.
- Lau, Sau-Him Paul ; Mui, Vai-Lam. (2008-09). Using Turn Taking to Mitigate Coordination and Conflict Problems in the Repeated Battle of the Sexes Game. Theory and decision, 2008-09, Vol.65 (2), p.153-183
- For a detailed explanation, see  Archived 2012-10-15 at the Wayback Machine p8 Section 4.5.