Mixed strategy

A mixed strategy is a probability distribution over a player's available pure strategies. It lets a player randomize actions, with each probability specifying how often an action is chosen.

Pure strategy: A complete plan that specifies one action for every possible decision a player may face. A mixed strategy assigns probabilities to these nonrandom plans.
Matching pennies: A two-player zero-sum game in which one player wins when coin choices match and the other wins when they differ. Its equilibrium requires both players to randomize equally between heads and tails.
Émile Borel: A French mathematician whose work helped establish the mathematical study of games and strategies. Borel explored randomized strategies in games before a general existence theorem was known.
Normal-form game: A representation of a game listing each player's strategies and the payoffs for every strategy profile. A mixed strategy assigns probabilities to a player's strategies in this representation.
Correlated equilibrium: A probability distribution over strategy profiles where no player benefits from deviating after receiving a private recommendation. It allows players' recommendations to be correlated, unlike independent mixed-strategy choices.
Expected utility: The probability-weighted average utility of outcomes under uncertainty. Players compare mixed strategies by calculating their expected payoffs.
Rock paper scissors: A simultaneous game in which rock beats scissors, scissors beats paper, and paper beats rock. Uniform randomization prevents an opponent from exploiting a predictable choice.
John von Neumann: A Hungarian-American mathematician who established foundational results in game theory and mathematical economics. His minimax theorem showed that optimal play in finite zero-sum games can require randomization.
Probability distribution: A function or rule assigning probabilities to possible outcomes of a random variable. A mixed strategy is specifically a distribution over a player's pure strategies.
Evolutionarily stable strategy: A strategy that resists invasion by rare mutant strategies in a population game. It concerns resistance to population invasion, not merely best responses in a one-shot game.

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