Mixed Strategies in Game Theory
In game theory, the Battle of the Sexes is a classic example of a coordination game where two players have to decide on a common activity. The game gets its name from the following scenario: A couple wants to spend an evening together, but they disagree on what to do. The husband would prefer to watch a football game, while the wife would rather go to the ballet. Both would prefer to be together rather than alone, but they have different preferences. There are two possible outcomes: watching the football game or going to the ballet. The goal is to coordinate on the same activity. The payoffs for each player are as follows:
A pure strategy equilibrium in this game is when both players choose the same activity.
However, there are also mixed strategy equilibria where each player randomly chooses an activity with a certain probability. For example, if each player chooses ballet with a probability of 2/3 and football with a probability of 1/3, then the expected payoff for each player is 5/3. This is a Nash equilibrium because neither player can increase their payoff by unilaterally changing their strategy.
Another example of mixed strategies in the Battle of the Sexes game is when each player chooses ballet with a probability of 3/4 and football with a probability of 1/4. In this case, the expected payoff is 5/4 for each player. This is also a Nash equilibrium because neither player can increase their payoff by unilaterally changing their strategy.
Overall, the Battle of the Sexes game is an interesting example of how players can coordinate on a common activity even when they have different preferences. Mixed strategies provide a way to achieve coordination without being stuck in a pure strategy equilibrium.
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