Mixed Strategies in Game theory

14/03/2023 1 By indiafreenotes

Mixed strategies are a key concept in game theory that refers to a situation where a player chooses to play more than one strategy with a certain probability distribution. In other words, instead of playing a single strategy every time, the player selects different strategies randomly based on a probability distribution.

Mixed strategies are used in game theory to find the equilibrium of a game when no pure strategy Nash equilibrium exists. A Nash equilibrium is a set of strategies where no player can improve their payoff by changing their strategy unilaterally. In some games, however, there may not be a Nash equilibrium in pure strategies, which means that each player has a dominant strategy to play. In such cases, mixed strategies are used to find an equilibrium.

To find a mixed strategy equilibrium, players assign a probability distribution over the available strategies, which must satisfy certain conditions. The probabilities should sum up to one, and each strategy should have a non-negative probability. The expected payoff for each player under this probability distribution is then calculated, and the Nash equilibrium is found when no player has an incentive to change their strategy.

Mixed strategies are used in many different types of games, such as the famous Prisoner’s Dilemma and Battle of the Sexes. They provide a useful tool for analyzing games where players have incomplete information or where there are multiple equilibria.

Mixed strategies Types with examples

There are two main types of mixed strategies that are commonly used in game theory: symmetric and asymmetric mixed strategies.

  1. Symmetric Mixed Strategies: In symmetric mixed strategies, all players use the same probability distribution over the strategies. In other words, players have the same strategy set and they randomize over those strategies in the same way. This is often used in games where players have identical strategies and payoffs.

Example: The classic example of a game that uses symmetric mixed strategies is the Matching Pennies game. In this game, two players each have a penny and choose to show either the heads or tails side. The payoff depends on whether the two pennies match or not. Each player randomizes over their choices with equal probability, so the probability of matching is 1/2.

  1. Asymmetric Mixed Strategies: In asymmetric mixed strategies, players use different probability distributions over their strategies. This is often used in games where players have different strategies and payoffs.

Example: Consider the game of Rock-Paper-Scissors. In this game, each player can choose to play either rock, paper, or scissors. Each choice wins against one choice and loses against another, and ties with itself. If both players play pure strategies, there is no Nash equilibrium. However, if each player randomly chooses each option with equal probability, then there is a mixed strategy Nash equilibrium.

In asymmetric mixed strategies, players use different probability distributions over their strategies, so each player has a different optimal mix of strategies. For example, if Player 1 chooses rock, paper, and scissors with probabilities 1/2, 1/3, and 1/6, respectively, then Player 2’s optimal mix of strategies is to choose rock, paper, and scissors with probabilities 1/3, 1/3, and 1/3, respectively.