Battle of the sexes & stag hunt

Posted by: lizwang on: October 13, 2008

• In: liking
• Leave a Comment

The Battle of the Sexes is a two player coordination game used in game theory. Imagine a couple. The husband would most of all like to go to the football game. The wife 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?

Opera Football Opera 3,2 0,0 Football 0,0 2,3 Battle of the Sexes 1

Opera Football Opera 3,2 1,1 Football 0,0 2,3 Battle of the Sexes 2

The payoff matrix labeled "Battle of the sexes (1)" is an example of Battle of the Sexes, where the husband chooses a row and the wife chooses a column.

This representation does not account for the additional harm that might come from going to different locations and going to the wrong one.(ie he goes to the opera while she goes to the football game, satisfying neither) 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.

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 equilibrium in mixed strategies, where the players go to their preferred event more often than the other. For the payoffs listed above, 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 couple 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 the husband nor wife 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 for the players are exactly equal.

 

In game theory, the stag hunt is a game which describes a conflict between safety and social cooperation. Other names for it or its variants include "assurance game", "coordination game", and "trust dilemma". Jean-Jacques Rousseau described a situation in which two individuals go out on a hunt. Each can individually choose to hunt a stag or hunt a hare. Each player must choose an action without knowing the choice of the other. If an individual hunts a stag, he must have the cooperation of his partner in order to succeed. An individual can get a hare by himself, but a hare is worth less than a stag. This is taken to be an important analogy for social cooperation.

Stag Hare Stag A, a C, b Hare B, c D, d Fig. 1: Generic stag hunt

Stag Hare Stag 4, 4 1, 3 Hare 3, 1 3, 3 Fig. 2: Stag hunt example

The stag hunt differs from the Prisoner’s Dilemma in that the greatest potential payoff is both players cooperating, whereas in the prisoners Dilemma, the greatest payoff is in one player cooperating, and the other defecting.

An example of the payoff matrix for the stag hunt is pictured in Figure 2.
Formally, a stag hunt is a game with two pure strategy Nash equilibria – one that is risk dominant another that is payoff dominant. The payoff matrix in Figure 1 illustrates a stag hunt, where . Often, games with a similar structure but without a risk dominant Nash equilibrium are called stag hunts. For instance if a=2, b=1, c=0, and d=1. While (Hare, Hare) remains a Nash equilibrium, it is no longer risk dominant. Nonetheless many would call this game a stag hunt.

In addition to the pure strategy Nash equilibria there is one mixed strategy Nash equilibrium. This equilibrium depends on the payoffs, but the risk dominance condition places a bound on the mixed strategy Nash equilibrium. No payoffs (that satisfy the above conditions including risk dominance) can generate a mixed strategy equilibrium where Stag is played with a probability higher than one half. The best response correspondences are pictured here.

Although most authors focus on the prisoner’s dilemma as the game that best represents the problem of social cooperation, some authors believe that the stag hunt represents an equally (or more) interesting context in which to study cooperation and its problems (for an overview see Skyrms 2004).

There is a substantial relationship between the stag hunt and the prisoner’s dilemma. In biology many circumstances that have been described as prisoner’s dilemma might also be interpreted as a stag hunt, depending on how fitness is calculated. It is also the case that some human interactions that seem like prisoner’s dilemmas may in fact be stag hunts. For example, suppose we have a prisoner’s dilemma as pictured in Figure 3.

But occasionally players who defect against cooperators are punished for their defection. For instance, suppose that the expected punishment is -2, then the imposition of this punishment turns the above prisoner’s dilemma into the stag hunt given at the introduction.

	Cooperate	Defect
Cooperate	4, 4	0, 5
Defect	5, 0	3, 3
Fig. 3: Prisoner’s dilemma example

In addition to the example suggested by Rousseau, David Hume provides a series of examples that are stag hunts. One example addresses two individuals who must row a boat. If both choose to row they can successfully move the boat. However if one doesn’t, the other wastes his effort. Hume’s second example involves two neighbors wishing to drain a meadow. If they both work to drain it they will be successful, but if either fails to do his part the meadow will not be drained.

There are several animal behaviors that have been described as stag hunts.

For example, the coordination of slime molds. In times of stress, individual unicellular protists will aggregate to form one large body. Here if they all act together they can successfully reproduce, however the success depends on the cooperation of many individual protozoa. Also, the hunting practices of orca (known as carousel feeding) are an example of a stag hunt. Here orcas cooperatively corral large schools of fish to the surface and stun them by hitting them with their tails. Since this requires that fish not have any mechanism for escape, it requires the cooperation of many orcas.