Indifference

Use equal-payoff conditions to identify candidate mixtures, then check both players.

12 minutes · advanced · Original Poker4Quant lesson

What you will learn

Derive each player’s mixing probability from the other player’s indifference condition.

Your mixture changes the opponent’s incentives

In an interior mixed equilibrium of a finite game, every action a player uses with positive probability must tie in expected payoff against the opponent’s strategy. Otherwise shifting weight toward a better action would improve value.

To find the row player’s mixing probability p, equalize the column player’s action values. To find column probability q, equalize the row player’s values. Solving for your own indifference determines the opponent’s mixture, a direction that is easy to reverse.

Equalities are candidates, not a universal certificate

A solution must lie between zero and one, and unused actions must not offer a profitable deviation. In a two-by-two game where both probabilities are interior and both players’ two actions tie, these checks establish mutual best responses.

If an equality yields a probability outside the unit interval, do not clip it to a boundary and call it equilibrium. Reconsider which actions could be supported, and check pure-strategy best responses directly.

Worked example

A zero-sum row-payoff table is A=[3,−1], B=[0,2]. Derive an interior equilibrium.

Row values against L-probability q are 4q−1 and 2−2q. Equate them: q=1/2.
Column L gives row value 3p; column R gives 2−3p. The minimizing column player is indifferent when p=1/3.
Both probabilities are interior. At p=1/3, either column gives row value 1.
At q=1/2, either row also gives value 1.

The candidate is mutual best response: row chooses A one-third, column chooses L one-half, and row’s expected value is 1 under the specified zero-sum payoffs.

Common mistakes

Solving your own indifference and labeling the result your mixing frequency.
Accepting negative or above-one probabilities.
Ignoring an unused action that might improve payoff.

Keep these ideas

The opponent’s indifference determines your mixture.
Validate probability bounds and all available deviations.
A solved equality is part of an equilibrium argument, not its replacement.