Game theory

Zero-Sum Game

A game in which one player's gain exactly offsets the others' losses.

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

Definition

A two-player game is zero-sum if u1(s)+u2(s)=0u_1(s)+u_2(s)=0 for all s∈S1×S2s\in S_1\times S_2; it is constant-sum if u1(s)+u2(s)=cu_1(s)+u_2(s)=c.

Equivalently it is described by a single payoff matrix A∈Rm×nA\in\mathbb{R}^{m\times n} with row player maximizing x⊤Ayx^\top A y and column player minimizing.

Intuition

Pure conflict — whatever one side gains the other loses.

Optimal play is max-min / min-max: maximize your guaranteed payoff against a worst-case opponent.

Worked example

Chess, Go, and Matching Pennies are zero-sum (draws counted as split).

A rock-paper-scissors tournament with {−1,0,+1}\{-1,0,+1\} payoffs is zero-sum.

The math

Von Neumann's Minimax Theorem: max⁡xmin⁡yx⊤Ay=min⁡ymax⁡xx⊤Ay\max_x\min_y x^\top A y = \min_y\max_x x^\top A y in mixed strategies, giving a unique value vv.

The equilibrium can be computed by linear programming in polynomial time.

Where it is used

Underpins adversarial ML (GANs, robust optimization), pursuit-evasion, and security resource allocation.

LP duality between the two players' optimization problems is foundational.

Go deeper

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Assembled from the ReLU.chat curated knowledge base. These explanations are concise on purpose; check the sources for anything important.