Payoff

Payoff is the amount received or the cost incurred as a result of a decision, action, or random event. In economics, finance, game theory, and related fields, the payoff is often a numerical representation of value, which may be monetary, utility-based, or otherwise meaningful to a participant. A payoff can be realized at contract maturity, exercise, or upon the occurrence of a specified event.

In game theory, a payoff represents the utility or value a player obtains from a particular outcome of the game. Payoffs are usually presented in matrices or as functions of the players’ strategies. When payoffs are cardinal, they convey the strength of preferences; when they are ordinal, only the ranking of outcomes matters. Analyses such as Nash equilibrium compare payoffs to determine best responses.

In finance and derivatives, the payoff of a contract is the amount paid at expiration or upon exercise, as a function of the underlying asset’s price or other variables. For example, the payoff of a long call option is max(S − K, 0), and the payoff of a long put option is max(K − S, 0). Other instruments have linear or piecewise payoffs, and valuation often uses the expected payoff discounted to present value.

In insurance and actuarial science, the payoff is the benefit paid under a policy, typically dependent on events such as claims or mortality outcomes. Payoff concepts also appear in risk management, where discounted or risk-adjusted payoffs are analyzed for pricing and decision making.

Payoff is distinct from profit: net payoff may subtract costs, premiums, or fees. In stochastic settings, the payoff is modeled as a random variable whose properties inform pricing, hedging, and strategy selection.