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, they must have the cooperation of their 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. 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, they must have the cooperation of their 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 pure Nash equilibrium is when both players choose to defect. 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 and another that is payoff dominant. The payoff matrix in Figure 1 illustrates a generic stag hunt, where a > b ≥ d > c {displaystyle a>bgeq 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. 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. 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.