Two players secretly pick heads or tails. Is there a strategy that beats pure guessing?
Two players each show a coin as heads or tails at the same moment. If the coins match, Player A wins both; if they differ, Player B wins both. Since each player wants to predict the other while staying unpredictable themselves, is there a clever strategy that does better than choosing randomly?
Reveal the answer
No — the only stable strategy for both players is a pure 50/50 random choice. Any pattern more predictable than a coin flip, however subtle, can eventually be detected and exploited by the opponent. It's game theory's simplest worked example of a mixed-strategy Nash equilibrium in a zero-sum game.
— Wikipedia contributors, Matching pennies — Encyclopedia entry; classic game theory example