Stag Hunt Game

Stag Hunt is a famous game in game theory that involves two players. In this game, both players have a choice to hunt a stag or a hare. Hunting a stag requires cooperation between the players since they must coordinate to hunt the stag successfully. On the other hand, hunting a hare can be done by a player alone. If both the players hunt the stag, they get a large reward, but if they both hunt the hare, they get a small reward. If one player hunts the hare while the other hunts the stag, the one who hunts the hare gets nothing, and the one who hunts the stag gets a small reward.

Mixed Strategies

Mixed strategies can be used in Stag Hunt games to obtain a higher expected payoff. In a Stag Hunt game, there are two pure strategies: hunting the stag and hunting the hare. If both players choose the same pure strategy, they get a payoff based on that strategy's outcome. However, if one player chooses one pure strategy while the other chooses another, then they get a lower payoff. By using mixed strategies, players can randomize their choices between the two pure strategies. This reduces the predictability of the opponent's choice and can lead to higher payoffs.

For example, suppose both players use a mixed strategy in a Stag Hunt game. Player 1 chooses to hunt the stag with a probability of p and the hare with a probability of (1-p). Player 2 chooses to hunt the stag with a probability of q and the hare with a probability of (1-q). The expected payoff for Player 1 can be calculated as follows: if both players hunt the stag, the payoff is 5; if both players hunt the hare, the payoff is 1; if Player 1 hunts the stag while Player 2 hunts the hare, the payoff is 0; and if Player 1 hunts the hare while Player 2 hunts the stag, the payoff is 2. The expected payoff for Player 1 is then given by:

E(P1) = 5pq + 1(1-p)(1-q) + 0p(1-q) + 2(1-p)q

This expression can be simplified to:

E(P1) = 5pq + 1 - p - q

Similarly, the expected payoff for Player 2 can be calculated. By finding the values of p and q that maximize E(P1) and E(P2), we can obtain the mixed strategy equilibrium for the game.

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