Spillfunksjon

Spillfunksjon, or payoff function, is the Norwegian term used in game theory to describe how outcomes yield numerical payoffs to players given a profile of actions. For a game with n players, each player i selects a strategy s_i from a set S_i. The joint strategy profile s = (s_1,...,s_n) belongs to S = S_1 × ... × S_n. The spillfunksjon for player i is a mapping u_i: S → R, assigning a real number that represents i's payoff or utility from the outcome. Collectively, (u_1,...,u_n) describes all players' payoffs.

Payoff functions can be cardinal, encoding the intensity of preferences, or merely ordinal, preserving order of outcomes. They can be deterministic, or incorporate uncertainty by taking expectations with respect to random events or types, yielding expected payoffs E[u_i(s, ω)]. In dynamic or stochastic games, payoffs may depend on the history or time discounting.

A payoff function is used to analyze strategic behavior. A strategy profile s* is a Nash equilibrium if, for every player i, u_i(s_i*, s_-i*) ≥ u_i(s_i, s_-i*) for all s_i ∈ S_i; no player can gain by unilaterally deviating.

Common examples include the Prisoner's Dilemma and coordination games, where payoff functions specify the rewards for different action profiles. Spillfunksjoner are central to computations of equilibria, comparative statics, and mechanism design, and are a standard formal tool in microeconomics, political science, and beyond.