Mixed Strategy (Indifference)
Probability that makes an opponent indifferent between their options.
Formula
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Example
In matching pennies the equilibrium is to play 50/50.
How it works
In a two-action zero-sum or 2x2 strategic setting, this formula gives the mixing probability that makes the other player exactly indifferent between their two choices. It comes from setting the expected payoff from one of the opponent’s options equal to the expected payoff from the other, then solving for the probability p. When the opponent cannot improve by switching actions, they have no incentive to deviate, which is the core idea behind a mixed-strategy equilibrium.
When to use it
Use this when analyzing a 2x2 game and you need the probability that randomizes between actions so the opposing player is indifferent. It is especially useful in competitive economics, auctions, security games, and simplified strategic models where best responses depend on expected payoffs.
Worked example, step by step
Suppose Player 1 is choosing between two actions, and Player 2’s payoff matrix entries are: If Player 1 plays action A: Player 2 gets a = 8 if they choose Left, and c = 2 if they choose Right If Player 1 plays action B: Player 2 gets b = 5 if they choose Left, and d = 11 if they choose Right We want the probability p that Player 1 plays action A so Player 2 is indifferent between Left and Right. Use the formula: p = (d - c) / ((a - b) + (d - c)) Substitute values: p = (11 - 2) / ((8 - 5) + (11 - 2)) Compute: p = 9 / (3 + 9) = 9 / 12 = 0.75 Final answer: Player 1 should play action A with probability 0.75 (and action B with probability 0.25) to make Player 2 indifferent.
Common mistakes to avoid
- Using the wrong player's payoffs when forming a, b, c, d
- Forgetting that p must be a probability between 0 and 1, which may fail if the game parameters do not support an interior mixed strategy
- Mixing up which action p refers to and substituting the complement 1 - p without checking the payoff setup
Did You Know?
John Nash proved every finite game has at least one equilibrium — the result that won him a Nobel Prize.
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