Matching Pennies

Matching pennies is the simplest game that demonstrates why mixed strategies are necessary. Two players simultaneously choose heads or tails; one wins if the choices match, the other wins if they differ. The game is a zero-sum game with no pure-strategy equilibrium — any predictable choice can be exploited by the opponent.

The unique Nash equilibrium requires both players to randomize 50-50, making their choices statistically unpredictable. This is not a description of human behavior but a normative prediction: a player who deviates from 50-50 randomization can be systematically exploited. The game illustrates that rationality sometimes requires deliberate randomness, not because randomness is inherently valuable but because unpredictability is a strategic resource. Matching pennies is the formal ancestor of all applications of mixed strategies, from auction design to sports strategy to military deterrence.