Matching Pennies

A zero-sum $2\times 2$ game with no pure equilibrium and a unique uniform mixed equilibrium.

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

Payoff matrix $\begin{pmatrix}(1,-1)&(-1,1)\\(-1,1)&(1,-1)\end{pmatrix}$ where player 1 wins if the pennies match.

Unique Nash equilibrium: both mix $(1/2,1/2)$ with value $0$ .

Intuition

One player wins when the pennies match; the other when they mismatch.

Any deterministic strategy is exploitable — so both must randomize.

Worked example

Soccer penalty kicks between kicker and goalie approximate a matching-pennies structure; empirical studies find near-equilibrium randomization (Chiappori, Levitt, Groseclose).

Any $2\times 2$ game with cyclic best responses is strategically equivalent.

The math

Canonical example of a game with no pure Nash equilibrium; demonstrates the necessity of mixed strategies.

Optimal randomization is derived via the minimax theorem.

Where it is used

Penalty kicks, serve direction in tennis, randomized audits, and cybersecurity patrolling.

Base case for teaching mixed-strategy equilibrium.

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