Mixed Strategy

A probability distribution over a player's pure strategies.

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

A mixed strategy for player $i$ is $\sigma_i\in\Delta(S_i)$ with $\sigma_i(s_i)\ge 0$ and $\sum_{s_i}\sigma_i(s_i)=1$ ; expected payoff is $u_i(\sigma)=\sum_s\prod_j\sigma_j(s_j)\,u_i(s)$ .

A mixed Nash equilibrium requires each player to be indifferent among all pure strategies in their support.

Intuition

Randomization is optimal when predictability is exploitable — it keeps opponents guessing.

A mixing player must be indifferent among the pure strategies in the support, otherwise they'd collapse to a pure best response.

Worked example

In Matching Pennies, each player mixes $(1/2,1/2)$ giving expected payoff $0$ .

Rock-Paper-Scissors has a unique symmetric mixed equilibrium $(1/3,1/3,1/3)$ with expected payoff $0$ .

The math

Nash's existence theorem guarantees a mixed equilibrium in every finite game. Harsanyi (1973) showed mixing is the limit of slightly perturbed pure-strategy play (purification).

Support characterization: if $\mathrm{supp}(\sigma_i^*)=T_i$ then $u_i(s_i,\sigma_{-i}^*)$ is constant on $T_i$ and $\ge$ its value off $T_i$ .

Where it is used

Used in security games, poker bluffing, penalty-kick analysis, and randomized algorithms.

Essential for analyzing games lacking pure equilibria.

More in Game theory

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