Zero sum game

In game theory and economic theory, a zero-sum game is a mathematical representation of a situation in which each participant’s gain or loss of utility is exactly balanced by the losses or gains of the utility of the other participants.

If the total gains of the participants are added up and the total losses are subtracted, they will sum to zero. Thus, cutting a cake, where taking a larger piece reduces the amount of cake available for others as much as it increases the amount available for that taker, is a zero-sum game if all participants value each unit of cake equally.

In contrast, non-zero-sum describes a situation in which the interacting parties’ aggregate gains and losses can be less than or more than zero. A zero-sum game is also called a strictly competitive game while non-zero-sum games can be either competitive or non-competitive. Zero-sum games are most often solved with the minimax theorem which is closely related to linear programming duality, or with Nash equilibrium.

Many people have a cognitive bias towards seeing situations as zero-sum, known as zero-sum bias.

Definition

The zero-sum property (if one gains, another loses) means that any result of a zero-sum situation is Pareto optimal. Generally, any game where all strategies are Pareto optimal is called a conflict game.

Zero-sum games are a specific example of constant sum games where the sum of each outcome is always zero. Such games are distributive, not integrative; the pie cannot be enlarged by good negotiation.

Situations where participants can all gain or suffer together are referred to as non-zero-sum. Thus, a country with an excess of bananas trading with another country for their excess of apples, where both benefit from the transaction, is in a non-zero-sum situation.

Other non-zero-sum games are games in which the sum of gains and losses by the players are sometimes more or less than what they began with.

The idea of Pareto optimal payoff in a zero-sum game gives rise to a generalized relative selfish rationality standard, the punishing-the-opponent standard, where both players always seek to minimize the opponent’s payoff at a favorable cost to himself rather to prefer more than less.

The punishing-the-opponent standard can be used in both zero-sum games (e.g. warfare game, chess) and non-zero-sum games (e.g. pooling selection games)

Examples

Zero-sum games are found in game theory, but are less common than non-zero sum games.

Poker and gambling are popular examples of zero-sum games since the sum of the amounts won by some players equals the combined losses of the others.

In game theory, the game of “Matching Pennies” is often cited as an example of a zero-sum game. The game involves two players – let’s call them A and B – simultaneously placing a penny on the table; the payoff depends on whether the pennies match or not. If both pennies are heads or tails, Player A wins and keeps Player B’s penny; if they do not match, Player B wins and keeps Player A’s penny.

This is a zero-sum game because one player’s gain is the other’s loss.

Zero-sum games are essentially bets. In the financial markets, for instance, speculators essentially place bets on the future prices of certain commodities. Thus, if you disagree with the consensus that wheat prices are going to fall, you might buy a futures contract.

If your prediction is right and wheat prices increase, you could make money by selling the futures contract before it expires.