Zero-Sum

In game theory, zero-sum describes a situation in which a participant's gain or loss is exactly balanced by the losses or gains of the other participant(s). It is so named because when the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero. Chess and Go are examples of a zero-sum game: it is impossible for both players to win. Zero-sum can be thought of more generally as constant sum where the benefits and losses to all players sum to the same value. Cutting a cake is zero- or constant-sum because taking a larger piece reduces the amount of cake available for others. In contrast, non-zero-sum describes a situation in which the interacting parties' aggregate gains and losses is either less than or more than zero.

Situations where participants can all gain or suffer together, such as a country with an excess of bananas trading with another country for their excess of apples, where both benefit from the transaction, are referred to as non-zero-sum. Other non-zero-sum games are games in which the sum of gains and losses by the players are always more or less than what they began with. For example, a game of poker, disregarding the house's rake, played in a casino is a zero-sum game unless the pleasure of gambling or the cost of operating a casino is taken into account, making it a non-zero-sum game.

The concept was first developed in game theory and consequently zero-sum situations are often called zero-sum games though this does not imply that the concept, or game theory itself, applies only to what are commonly referred to as games. In pure strategies, each outcome is Pareto optimal (generally, any game where all strategies are Pareto optimal is called a conflict game). Nash equilibria of two-player zero-sum games are exactly pairs of minimax strategies.

In 1944 John von Neumann and Oskar Morgenstern proved that any zero-sum game involving n players is in fact a generalized form of a zero-sum game for two players, and that any non-zero-sum game for n players can be reduced to a zero-sum game for n + 1 players; the (n + 1) player representing the global profit or loss. This suggests that the zero-sum game for two players forms the essential core of mathematical game theory.