IDC Revision

Mixed Strategy

Also known as · randomised strategy · mixed-strategy Nash equilibrium

A mixed strategy assigns a probability distribution over pure strategies: player ii plays strategy siks_i^k with probability pkp_k, where ∑kpk=1\sum_k p_k = 1. A mixed-strategy Nash Equilibrium requires every player to be indifferent between all pure strategies they play with positive probability — otherwise they would deviate entirely to the strictly better one:

EUi(sik)=EUi(sij)for all k,j played with positive probability.EU_i(s_i^k) = EU_i(s_i^j) \quad \text{for all } k, j \text{ played with positive probability}.
type: payoff-matrix
rows: Rock,Paper,Scissors
cols: Rock,Paper,Scissors
payoffs: 0,0;-1,1;1,-1|1,-1;0,0;-1,1|-1,1;1,-1;0,0

Rock-Paper-Scissors is the canonical example: best responses cycle around the loop and no pure-strategy NE exists, so the unique equilibrium is to randomise uniformly over the three actions.

When to use

Solve for a mixed-strategy NE whenever a game has no pure-strategy Nash Equilibrium (e.g. Rock-Paper-Scissors, zero-sum games), or when you are asked for all equilibria of a coordination game (Battle of the Sexes has two pure NE plus one mixed NE). The trick is to make the opponent indifferent: solve EUj(strategy A)=EUj(strategy B)EU_j(\text{strategy A}) = EU_j(\text{strategy B}) for your own mixing probability.

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