Solution
Matching Pennies
Show the problem again
Two players simultaneously show a penny heads or tails. If the pennies match, Player 1 wins $1 from Player 2; if they differ, Player 2 wins $1 from Player 1. In the unique Nash equilibrium, what is the expected value of the game to each player?
Worked solution
The value is 0. No pure equilibrium exists; one player always wants to deviate. In mixed strategies, each player must make the other indifferent: playing heads with probability p makes the opponent indifferent only when p = 1/2. Both mixing 50/50 is the unique equilibrium, with expected payoff 0 to each.
Source: Standard zero-sum game of the game-theory literature, dating to von Neumann's minimax work. Statement written for AxiomIQ.