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 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.
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