The stag hunt differs from the Prisoner's Dilemma in that there are two Nash equilibria: when both players cooperate and both players defect. In the Prisoners Dilemma, however, 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.
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, 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.
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.