Outcome (game theory)

In game theory, an outcome refers to a specific state of the game resulting from the choices of all players. It represents the culmination of all players' strategies and determines the payoffs each player receives. An outcome is often defined as a complete specification of all relevant aspects of the game at its conclusion, including actions taken, final resource allocations, and payoffs realized.

An outcome is distinct from a strategy profile, which is the set of strategies chosen by each player. A strategy profile leads to an outcome. Given a strategy profile and the rules of the game, the outcome can be determined. In games with randomness or incomplete information, the relationship between strategy profiles and outcomes may be probabilistic.

The value of an outcome for a player is typically represented by a payoff function that assigns a numerical value (utility) to each outcome for each player. This payoff represents the desirability of that outcome to the player. Players are generally assumed to act in a way that maximizes their expected payoff.

Outcomes can be classified in various ways. For example, an outcome can be:

• Efficient (Pareto Optimal): An outcome where it is impossible to make any player better off without making another player worse off.
• Inefficient (Pareto Suboptimal): An outcome where it is possible to make at least one player better off without making any player worse off.
• Equilibrium: An outcome that is stable, in the sense that no player has an incentive to unilaterally deviate from their chosen strategy, given the strategies of the other players. A Nash Equilibrium is a common type of equilibrium outcome.

Understanding the possible outcomes of a game, and the strategies that lead to them, is central to game-theoretic analysis. Analyzing the payoffs associated with each outcome allows for predictions of rational player behavior and the identification of stable solutions.