ZeroSum

Zero-sum, in game theory, refers to a competitive situation where the total payoff to all participants remains constant across outcomes. In the common two-player formulation, one player's gain is exactly the other player's loss; the sum of the payoffs is zero for every outcome. This concept is often contrasted with non-zero-sum or constant-sum games.

Mathematically, in a two-player game with payoff functions u1(a, b) and u2(a, b), the zero-sum condition requires u1(a, b) + u2(a, b) = 0 for all strategy pairs (a, b). Players may randomize their choices via mixed strategies to maximize expected payoff. The two-player, finite zero-sum setting is characterized by the minimax theorem: there exists a value v and optimal strategies for both players such that each player can guarantee at least v, and the opponent cannot push the payoff above v. Computation often uses linear programming or regret-minimization techniques.

Examples include standard matrix games like Matching Pennies and Rock-Paper-Scissors, where a win yields +1, a loss -1, and a tie 0. In such games, equilibrium strategies are often mixed rather than pure, reflecting the need to keep opponents uncertain.

In broader contexts, zero-sum describes any competitive interaction where gains and losses across participants cancel each other out, including various stochastic or extensive-form representations. It is contrasted with non-zero-sum games, where participants’ interests may partially align, leading to outcomes in which the total payoff is not fixed.

See also: non-zero-sum game, constant-sum game, minimax, Nash equilibrium.