In game theory, mixed strategy equilibria occur when players choose actions randomly, using a probability distribution to determine which action to take. Unlike pure strategy equilibria, mixed strategy equilibria may involve players choosing different actions with positive probabilities. In this lesson, we will explore how to interpret mixed strategy equilibria and what they tell us about the game.

Interpreting Mixed Strategy Equilibria

When analyzing mixed strategy equilibria, we look at the expected payoffs of each player given their probability distribution. A mixed strategy equilibrium is considered “stable” if no player can improve their expected payoff by changing their strategy. In other words, each player is choosing the best response to the other player's strategy, given their own probability distribution.

One of the main challenges in interpreting mixed strategy equilibria is that they can be counterintuitive. For example, in the game of matching pennies, where each player chooses heads or tails, the mixed strategy equilibrium involves each player choosing heads and tails with equal probability. This may seem strange, as it suggests that players are deliberately trying to be unpredictable. However, when we analyze the expected payoffs of each player, we see that this strategy is actually the best response to the other player's strategy.

Another important concept when interpreting mixed strategy equilibria is the concept of support. The support of a mixed strategy is the set of actions that a player chooses with positive probability. In a mixed strategy equilibrium, each player's support must be non-empty, as otherwise they would be playing a pure strategy. Additionally, there may be multiple mixed strategy equilibria in a game, each with a different support.

Conclusion

Mixed strategy equilibria can provide insight into the behavior of players in a game, but they may be difficult to interpret at first glance. By analyzing the expected payoffs of each player and the support of their strategies, we can better understand the logic behind these equilibria.

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