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Matching Pennies

A classic zero-sum game with no pure strategy Nash equilibrium. You win if the coins match, your opponent wins if they don't. Both players must randomize to achieve equilibrium.

Payoff Matrix (You are Row Player)

H T
H +1, -1 -1, +1
T -1, +1 +1, -1

Mixed Strategy Equilibrium: 50% H, 50% T

Game Statistics

Rounds Played 0
Your Wins 0
Opponent Wins 0
Your Score 0
Your H% -
Opponent H% -

About Matching Pennies

The Game: Two players simultaneously choose Heads or Tails. If the choices match, you win. If they don't match, your opponent wins.

Zero-Sum: Your gain is exactly your opponent's loss. The sum of payoffs is always zero.

No Pure Strategy Equilibrium: If your opponent thinks you'll play H, they should play T. But then you should play T too. But then they should play H... There's no stable pure strategy.

Mixed Strategy Nash Equilibrium: Both players randomize 50-50 between H and T. This makes the opponent indifferent to their choice, and neither player can gain by deviating.

Von Neumann's Minimax Theorem: This game demonstrates that in zero-sum games, players should randomize to guarantee the best worst-case outcome.

Real Applications: Penalty kicks in soccer, tennis serves, poker bluffing, military strategy - any situation where unpredictability is advantageous.

Try it: Can you beat the AI? Or will both of you converge to 50-50 randomization?