Strategy profile

In game theory, player's strategy is any of the options he or she can choose in a setting where the outcome depends not only on his own actions but on the action of others. A player's strategy will determine the action the player will take at any stage of the game.

The strategy concept is sometimes (wrongly) confused with that of a move. A move is an action taken by a player at some point during the play of a game (e.g., in chess, moving white's Bishop a2 to b3). A strategy on the other hand is a complete algorithm for playing the game, telling a player what to do for every possible situation throughout the game.

A strategy profile (sometimes called a strategy combination) is a set of strategies for all players which fully specifies all actions in a game. A strategy profile must include one and only one strategy for every player.

A player's strategy set defines what strategies are available for them to play.

A player has a finite strategy set if they have a number of discrete strategies available to them. For instance, in a single game of rock-paper-scissors, each player has the finite strategy set {rock, paper, scissors}.

A strategy set is infinite otherwise. For instance, an auction with mandated bid increments may have an infinite number of discrete strategies in the strategy set {$10, $20, $30, ...}. Alternatively, the cake cutting game has a bounded continuum of strategies in the strategy set {Cut anywhere between zero percent and 100 percent of the cake}.

In a dynamic game, the strategy set consists of the possible rules a player could give to a robot or agent on how to play the game. For instance, in the ultimatum game, the strategy set for the second player would consist of every possible rule for which offers to accept and which to reject.

In a Bayesian game, the strategy set is similar to that in a dynamic game. It consists of rules for what action to take for any possible private information.

In applied game theory, the definition of the strategy sets is an important part of the art of making a game simultaneously solvable and meaningful. The game theorist can use knowledge of the overall problem to limit the strategy spaces, and ease the solution.

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