Mixedstrategy

Mixedstrategy, in game theory, refers to a strategy in which a player randomizes over available actions by assigning probabilities to each action. This allows a player to keep opponents uncertain and to optimize expected outcomes when no single action is best.

Formally, in a finite strategic game with players N = {1,...,n}, each player i has a finite set S_i of pure strategies. A mixed strategy for player i is a probability distribution p_i over S_i. A mixed-strategy profile p = (p_1,...,p_n) assigns to every player a distribution. If players independently sample actions according to their p_i, the probability of a pure-action profile s = (s_1,...,s_n) is ∏_i p_i(s_i). The expected payoff to player i is u_i(p) = ∑_{s∈S} (∏_i p_i(s_i)) u_i(s), where u_i(s) is i’s payoff from the action profile s.

Key concepts include the support of a mixed strategy (the set of actions given positive probability) and the view that a pure strategy is a degenerate mixed strategy placing probability 1 on a single action.

A mixed-strategy Nash equilibrium is a strategy profile in which no player can improve their expected payoff by unilaterally changing their own mixed strategy. Existence is guaranteed for every finite game: at least one Nash equilibrium exists in mixed strategies (Nash, 1950).

Example: In rock–paper–scissors, the equilibrium is for each player to randomize uniformly among the three moves, giving each action probability 1/3 and equal expected payoffs against any opponent strategy.

Applications include modeling strategic uncertainty, facilitating analysis when pure equilibria do not exist, and underpinning computational methods for finding equilibria.