Mixed Strategies in Game Theory
Coordination games are a type of game in which players have a strong incentive to choose the same strategy as their opponent. In these games, there are typically two pure strategy Nash equilibria, meaning that there are two strategies that are best responses to each other. However, when players are uncertain about which strategy their opponent will choose, mixed strategies may be used to find a Nash equilibrium.
One classic example of a coordination game is the Battle of the Sexes. In this game, a couple must decide whether to go to a football game or an opera. The husband prefers the football game, but would rather go to the opera than go alone. The wife prefers the opera, but would rather go to the football game than go alone. In this case, there are two pure strategy Nash equilibria: (football, football) and (opera, opera). However, if the husband believes there is a 50% chance the wife will choose the football game and a 50% chance she will choose the opera, he may choose a mixed strategy of 50% football and 50% opera. If the wife believes there is a 50% chance the husband will choose the football game and a 50% chance he will choose the opera, she may also choose a mixed strategy of 50% football and 50% opera. This results in a mixed strategy Nash equilibrium of (50% football, 50% opera) for both players.
Another example of a coordination game is the Stag Hunt. In this game, two hunters must decide whether to hunt a stag or a hare. Hunting a stag requires cooperation, as both hunters must work together to catch it. Hunting a hare can be done alone. If both hunters choose to hunt a hare, they each receive 1 unit of payoff. If they both choose to hunt a stag, they each receive 2 units of payoff. However, if one hunter chooses to hunt a hare while the other chooses to hunt a stag, the hare hunter gets no payoff and the stag hunter only receives 1 unit of payoff. In this case, there are two pure strategy Nash equilibria: (stag, stag) and (hare, hare). However, if the hunters believe there is a 50% chance the other hunter will choose stag and a 50% chance they will choose hare, they may choose a mixed strategy of 50% stag and 50% hare. This results in a mixed strategy Nash equilibrium of (50% stag, 50% hare) for both players.
Overall, mixed strategies can be useful in coordination games when players are uncertain about their opponent's choices. By using a mixed strategy, players can find a Nash equilibrium even when there are no pure strategy equilibria.
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