Mixed Strategy

A mixed strategy is a probability distribution over pure strategies in a game, allowing a player to introduce deliberate randomness into their choice. Unlike a pure strategy — which specifies a single action with certainty — a mixed strategy makes the player's move unpredictable to opponents, which can be advantageous in games where predictability is exploitable. John Nash proved that every finite game has at least one equilibrium in mixed strategies, even when no pure-strategy equilibrium exists.

The classic example is matching pennies: two players simultaneously choose heads or tails, with one player winning if the choices match and the other winning if they differ. No pure strategy equilibrium exists — any predictable choice can be exploited — but the mixed strategy of randomizing 50-50 is a stable equilibrium. The concept extends to games with many actions and players, and it underlies applications from auction theory to military strategy, where unpredictability itself is a resource.