Mixed-Strategy Nash Equilibrium

Randomize over best responses; indifference fixes the probabilities.

The idea

A mixed strategy for a player is a probability distribution over that player's actions; a player using one randomizes among them. Some games have no stable choice of one action per player — whatever one player fixes, another gains by switching — and mixed strategies are what restore stability.

A profile of mixed strategies, one per player, is a mixed-strategy Nash equilibrium when no player can raise their own expected payoff by switching to any other mixed strategy while the other players' mixes stay fixed.

Equilibrium forces an indifference condition. Suppose a player's equilibrium mix puts positive probability on two actions. Given the opponents' mixes, each action has a definite expected payoff; if one were larger, the player could gain by moving all of that probability onto it, and the profile would not be an equilibrium. So every action a player mixes over must yield the same expected payoff.

A player's own probabilities never appear in their own expected payoffs — the opponents' probabilities determine those. Solving one player's indifference condition therefore pins down the opponent's mixing probabilities, not that player's own: each player mixes exactly so as to keep the other willing to randomize.

The figure plots this condition: against the opponent's mixing probability $q$, each of a player's pure actions traces a straight expected-payoff line, and only at the crossing $q^{*}$ do the two actions tie — the one mix that leaves the player willing to randomize.

Ways to work on it

• Walkthrough. Find an equilibrium in randomized strategies by making each player indifferent.
• Practice. Solve an indifference condition for the opponent's mixing probability.
• Hardest. Find the mixed equilibrium of an asymmetric coordination game.

Not sure where to start? Take the ten-question placement test.