Minimax Theorem

In any finite two-player zero-sum game, max-min and min-max values coincide in mixed strategies.

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

Definition

Von Neumann (1928): $\max_{x\in\Delta^m}\min_{y\in\Delta^n} x^\top A y=\min_{y\in\Delta^n}\max_{x\in\Delta^m} x^\top A y=v(A).$

The common value $v$ is the value of the game; any saddle-point $(x^*,y^*)$ attains it.

Intuition

Rational worst-case planning yields the same outcome for both sides.

It guarantees an optimal defensive strategy for each player in pure conflict.

Worked example

Matching Pennies has value $0$ with optimal mixes $(1/2,1/2)$ for both.

A $3\times 3$ RPS matrix has value $0$ with $(1/3,1/3,1/3)$ optimal.

The math

Proof by LP duality: the row-player's LP $\max v\text{ s.t. }A^\top x\ge v\mathbf{1},\mathbf{1}^\top x=1$ is dual to the column-player's, so strong duality gives the minimax equality.

Generalizations include Sion's minimax theorem for quasi-concave/convex payoffs on compact convex sets.

Where it is used

Foundation of LP duality, robust optimization, statistical decision theory, and adversarial training of generative models.

Fictitious play converges to the minimax value in zero-sum games.

More in Game theory

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