payoffvector

In game theory, a payoff vector is an n-dimensional vector that records the payoffs received by each of the players for a particular outcome or strategy profile in a normal-form game. If there are n players, each player i has a payoff function u_i: S -> R, where S is the set of strategy profiles. For a profile s = (s1, ..., sn), the payoff vector is u(s) = (u_1(s), ..., u_n(s)). Payoffs are typically real numbers and may represent utilities, money, or other measures of value.

Example: in a two-player game with strategy sets S1 and S2, the profile (s1, s2) yields a payoff vector (u1(s1,s2), u2(s1,s2)).

Uses: payoff vectors allow comparison of outcomes, assessment of efficiency, and formulation of solution concepts. A central notion is the Nash equilibrium: a strategy profile s* where no player can increase their own payoff by deviating unilaterally. Payoff vectors also facilitate Pareto comparisons: one vector Pareto-dominates another if it weakly improves all components and strictly improves at least one.

Variations: payoffs may be expected values in stochastic or repeated games, leading to an expected payoff vector E[u(s)]. In cooperative games, a payoff vector can describe how the value of the grand coalition is distributed among players (e.g., as given by the Shapley value or the core). Notation commonly uses u_i(s) for each component or π_i(s).