Mixed Strategies in Game Theory
In game theory, a mixed strategy is a probability distribution over pure strategies. While pure strategies are represented by a single action, mixed strategies represent a combination of actions that are played with certain probabilities. In this sense, mixed strategies are a way to introduce randomness into a game.
In a two-player game, a mixed strategy can be represented as a pair of probability distributions: one for each player. For example, consider the game of Rock-Paper-Scissors. A mixed strategy for player 1 could be (1/3, 1/3, 1/3), which means that player 1 plays Rock, Paper, and Scissors with equal probability. Similarly, a mixed strategy for player 2 could be (1/3, 1/3, 1/3).
To find a mixed strategy equilibrium, we need to find a pair of mixed strategies (one for each player) such that neither player can improve their expected payoff by unilaterally changing their strategy. This is known as the Nash equilibrium.
One way to find a mixed strategy equilibrium is to use the concept of expected payoff. The expected payoff of a mixed strategy is the sum of the payoffs of each pure strategy multiplied by its probability. For example, if player 1 plays (1/3, 1/3, 1/3) and player 2 plays (1/3, 1/3, 1/3), the expected payoff of player 1 for playing Rock is (1/3)(0) + (1/3)(-1) + (1/3)(1) = 0. Similarly, the expected payoff of player 2 for playing Rock is (1/3)(1) + (1/3)(0) + (1/3)(-1) = 0. If we calculate the expected payoffs for all pure strategies and find that no player can increase their expected payoff by changing their strategy, we have found a mixed strategy equilibrium.
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