Payoffs

Payoffs refer to the value an actor receives as a result of a decision, action, or event. In economics and game theory, a payoff is a numerical outcome used to represent preferences, often called the payoff or the payoff function. Payoffs can be monetary or non-monetary and may reflect costs, revenues, profits, utilities, or other measures of value. In decision problems, the payoff of a strategy profile is the vector of outcomes for all players.

In game theory, each player has a payoff function mapping strategy profiles to real numbers. The goal is to choose strategies that maximize one's payoff, given others' choices. Payoffs can be common knowledge; games can be zero-sum, where one player's gain equals another's loss, or general-sum, where all players may gain or lose.

In probability and decision theory, the expected payoff is the probability-weighted average of possible payoffs, used to compare options under uncertainty. Risk attitudes may transform payoffs into utilities, affecting decision rules.

In finance and economics, payoff often refers to the cash or value received at a future date from a financial instrument, conditional on states of the world. For derivatives, payoffs are determined by the instrument's payoff function, such as max(S - K, 0) for a call option or max(K - S, 0) for a put option. The concept is central to pricing, hedging, and risk management.

Payoffs help formalize choices in strategic settings and financial planning, bridging observable outcomes and underlying preferences or risk tolerances.