Mixed strategy ( $σ_{i} \in Δ (S_{i})$ )

A mixed strategy is a probability distribution over a player's pure strategies. For a finite game with

m_{i}

such strategies, the possible mixed strategies form the probability simplex

Δ (S_{i}) = {σ_{i} \in R^{m_{i}} : σ_{i} (a) \geq 0, \sum_{a \in S_{i}} σ_{i} (a) = 1} .

A pure strategy is included as a mixed strategy placing all probability on one choice. The support of a mixed strategy consists of the choices with positive probability. In a sequential game, sampling a complete pure strategy once is the definition; it is not automatically equivalent to fresh randomization at every move.

In a bimatrix game, write

A

for the row player's payoff matrix and

B

for the column player's, and write

x, y

for their probability vectors. Independent randomization gives expected payoffs

U_{1} (x, y) = x^{T} A y, U_{2} (x, y) = x^{T} B y .

For example, in matching pennies, choosing heads with probabilities

p = 3/4

and

q = 1/4

gives joint probabilities

3/16, 9/16, 1/16, 3/16

for

(H, H), (H, T), (T, H), (T, T)

. The row player's expected payoff is

(3 - 9 - 1 + 3) /16 = - 1/4

, and the column player's is

1/4

.

Randomization can prevent an opponent exploiting a predictable choice. It does not mean every choice is equally good against every opponent: only best responses belong in the support of a mixed strategy at a Nash equilibrium. A shared random signal can instead produce a lottery over joint action profiles whose joint probabilities do not factor into independent mixed strategies.

Table of contents

Fully mixed strategy ( $σ_{i} (a) > 0 for every a \in A_{i}$ )

A mixed strategy assigning strictly positive probability to every available pure strategy. Its support of a mixed strategy is the entire action set. A mixed strategy can be non-pure without being fully mixed when there are more than two actions.

Support of a mixed strategy ( $supp (p) = {i : p_{i} > 0}$ )

The support of a mixed strategy consists of the pure strategies used with positive probability. At a Nash equilibrium, every supported pure strategy is a best response, while unused strategies may give no higher payoff.

// Target: computer-science.bigb

Strategy support ( $supp (s) = {i : s_{i} > 0}$ )

The support of a finite mixed strategy consists of the pure strategies assigned positive probability. At a Nash equilibrium, every action in the support is a best response. In a game satisfying nondegeneracy of a bimatrix game, the two equilibrium supports have equal size.

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