Stag hunt

In game theory, the stag hunt is a game that 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 has been taken to be a useful analogy for social cooperation, such as international agreements on climate change.

The stag hunt differs from the Prisoner's Dilemma in that there are two pure strategy Nash equilibria: when both players cooperate and both players defect. In the Prisoner's Dilemma, in contrast, despite the fact that both players cooperating is Pareto efficient, the only Nash equilibrium is when both players choose to defect.

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 symmetric stag hunt

	Stag	Hare
Stag	2, 2	0, 1
Hare	1, 0	1, 1
Fig. 2: Stag hunt example

Formal definition

Formally, a stag hunt is a game with two pure strategy Nash equilibria—one that is risk dominant and another that is payoff dominant. The payoff matrix in Figure 1 illustrates a generic stag hunt, where $a>b\ge d>c$ . Often, games with a similar structure but without a risk dominant Nash equilibrium are called assurance game. 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.

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 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.

	Cooperate	Defect
Cooperate	2, 2	0, 3
Defect	3, 0	1, 1
Fig. 3: Prisoner's dilemma example

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. The payoff matrix would need adjusting if players who defect against cooperators might be punished for their defection. For instance, if 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.

Examples of the stag hunt

The original stag hunt dilemma is as follows: a group of hunters have tracked a large stag, and found it to follow a certain path. If all the hunters work together, they can kill the stag and all eat. If they are discovered, or do not cooperate, the stag will flee, and all will go hungry.

The hunters hide and wait along a path. An hour goes by, with no sign of the stag. Two, three, four hours pass, with no trace. A day passes. The stag may not pass every day, but the hunters are reasonably certain that it will come. However, a hare is seen by all hunters moving along the path.

If a hunter leaps out and kills the hare, he will eat. However, it results in the trap laid for the stag to be wasted, and the others will starve. There is no certainty that the stag will arrive; the hare is present. The dilemma is that if one hunter waits, he risks one of his fellows killing the hare for himself, sacrificing everyone else. This makes the risk twofold; risk the stag never coming, or risk another man taking the kill.

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.

Several animal behaviors have been described as stag hunts. One is 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, but success depends on the cooperation of many individual protozoa. Another example is the hunting practices of orcas (known as carousel feeding). Orcas cooperatively corral large schools of fish to the surface and stun them by hitting them with their tails. Since this requires that the fish have no way to escape, it requires the cooperation of many orcas.

Author James Cambias describes a solution to the game as the basis for an extraterrestrial civilization in his 2014 science fiction book A Darkling Sea.

References

Skyrms, Brian. (2004) The Stag Hunt and the Evolution of Social Structure. Cambridge: Cambridge University Press.

External links

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

Topics in game theory

Definitions	Normal-form game Extensive-form game Escalation of commitment Graphical game Cooperative game Succinct game Information set Hierarchy of beliefs Preference

Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian-Nash Perfect Bayesian Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium

Strategies	Dominant strategies Pure strategy Mixed strategy Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy

Classes of games	Symmetric game Perfect information Simultaneous game Sequential game Repeated game Signaling game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global games Strictly determined game

Games	Prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Rock-paper-scissors Pirate game Dictator game Public goods game Blotto games War of attrition El Farol Bar problem Fair division Fair cake-cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Screening game Prisoners and hats puzzle Trust game Princess and monster game Monty Hall problem Rendezvous problem

Theorems	Minimax theorem Nash's theorem Purification theorem Folk theorem Revelation principle Arrow's impossibility theorem

Key figures	Albert W. Tucker Amos Tversky Ariel Rubinstein Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin Jean Tirole Jean-François Mertens John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Thomas Schelling William Vickrey

See also	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition List of game theorists List of games in game theory No-win situation Topological game Tragedy of the commons Tyranny of small decisions

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