mathematics//game theory//zero-sum game

A zero-sum game is a game in which the players' payoffs add up to zero in every outcome, so that whatever one gains the other loses; it is the model of pure opposition used for worst-case design: an intruder against a perimeter, a jammer against a radio link, a disturbance against a controller. A two-player zero-sum game needs only one payoff matrix \(G\), the first player's; the second player's is \(-G\), and each cell is a single number that one side pushes up and the other down.

A zero-sum game is a game in which the players' payoffs add up to zero in every outcome, so that whatever one gains the other loses; it is the model of pure opposition used for worst-case design: an intruder against a perimeter, a jammer against a radio link, a disturbance against a controller. A two-player zero-sum game needs only one payoff matrix GGG, the first player's; the second player's is −G-G−G, and each cell is a single number that one side pushes up and the other down.

Pure opposition has a strong consequence. Since the two sides want exactly opposite things, there is nothing to negotiate and no outcome both prefer, and the game has a single value: what the first player can guarantee and the second can hold it to. The minimax theorem says that with mixed strategies this value is the same whichever player is made to reveal its strategy first, and a linear program computes it. In the inspection game the matrix holds the intruder's gain (0 when it meets the inspector, the gate's value otherwise) and the value is 2/32/32/3.

Modelling an opponent as zero-sum is pessimistic by construction: it assumes the other side wants exactly what hurts you most. That is the right assumption against a designed attack and a costly one against a competitor that has aims of its own.

Control engineers play zero-sum games without calling them that. H-infinity control chooses the controller against the worst disturbance of bounded energy, and robust optimization chooses the decision against the worst parameter in a set. Pursuit and evasion, an interceptor against a drone that dodges, is a zero-sum game in continuous dynamics, a differential game of the family the game theory note describes.

Zero-sum describes the payoffs, not the scarcity of what is fought over. Two fleets sharing airspace both lose in a collision, and two firms in a market can both profit; such games have outcomes better for both, the minimax guarantee no longer predicts play, and the solution concept becomes the Nash equilibrium.

A useful check before modelling a threat as zero-sum is to ask whether the adversary pays for its own actions. A jammer that drains its battery, or an intruder who risks arrest, has costs the matrix must hold, and with them the game stops being zero-sum.