Stag hunt

In game theory, the stag hunt (also referred to as the assurance game, trust dilemma or common interest game) describes a situation or game where participants would be better off cooperating to achieve a more ambitious goal (hunting a stag together, which succeeds), but can choose a safer option (hunting a hare on one's own) that protects them from a bad outcome (hunting a stag alone while the other hunts a hare, which fails). This sets up a stylized conflict between safety and social cooperation based on the mathematical payoffs of each option.

In the most common account of this dilemma, two hunters must decide separately, and without the other knowing, whether to hunt a stag or a hare. However, both hunters know the only way to successfully hunt a stag is with the other's help. One hunter can catch a hare alone with less effort and less time, but it is worth far less than a stag and has much less meat. But both hunters would be better off if both choose the more ambitious and more rewarding goal of getting the stag, giving up some autonomy in exchange for the other hunter's cooperation and added might.

This situation is often seen as a useful analogy for many kinds of social cooperation, such as international agreements on climate change.

The stag hunt problem originated with philosopher Jean-Jacques Rousseau in his Discourse on Inequality, although the most common mathematized formulation differs from the original presentation.

Formally, a stag hunt is a game with two pure strategy Nash equilibria, that is, stable attractors where an individual player can't improve their position with a different strategy if the other players' strategy remains constant—one that is risk dominant (hunting hares individually) and another that is payoff dominant (hunting a stag together) . The payoff matrix in Figure 1 illustrates a generic stag hunt, where ${\displaystyle a>b\geq d>c}$.

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

In addition to the pure strategy Nash equilibria, the stag hunt has one mixed strategy Nash equilibrium, that is, one in which the players choose either option with some probability. 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.

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).