Stag hunt

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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.

An example of the payoff matrix for the stag hunt is pictured in Figure 2.

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

	Stag	Hare
Stag	4, 4	0, 3
Hare	3, 0	3, 3
Fig. 2: Stag hunt example

1 Formal definition
2 The stag hunt and social cooperation
3 Examples of the stag hunt
4 References
5 External links

[edit] Formal definition

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 $a>b\ge d>c$ . 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 satisify 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.

[edit] The stag hunt and social cooperation

"Nature and Appearance of Deer" taken from "Livre du Roy Modus," created in the 14th Century

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 2003).

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

[edit] Examples of the stag hunt

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 bacteria. 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.

[edit] References

Skyrms, Brian. (2003) The stag hunt and Evolution of Social Structure Cambridge: Cambridge University Press.

[edit] External links

The stag hunt at GameTheory.net
The stag hunt (pdf) by Brian Skyrms

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Equilibrium concepts	Nash equilibrium · Subgame perfection · Bayes-Nash · Trembling hand · Correlated equilibrium · Sequential equilibrium · Quasi-perfect equilibrium · Evolutionarily stable strategy
Strategies	Dominant strategies · Mixed strategy · Grim trigger · Tit for Tat
Classes of games	Symmetric game · Perfect information · Dynamic game · Repeated game · Signaling game · Cheap talk · Zero-sum game · Mechanism design
Games	Prisoner's dilemma · Coordination game · Chicken · Battle of the sexes · Stag hunt · Matching pennies · Ultimatum game · Minority game · Rock, Paper, Scissors · Pirate game · Dictator game
Theorems	Minimax theorem · Purification theorems · Folk theorem · Revelation principle
Related topics	Mathematics · Economics · Behavioral economics · Evolutionary game theory · Population genetics · Behavioral ecology · Adaptive dynamics · List of game theorists