Minimax Value
Best worst-case payoff a player can guarantee.
Formula
Variables
Example
Choose the move whose worst outcome is least bad.
How it works
The minimax value is the payoff you can secure by first choosing the strategy that makes your worst possible outcome as good as possible. For each strategy a, you look at the opponent’s best response b against it, take the minimum payoff you might receive, and then choose the strategy whose minimum is largest. This works because in adversarial settings you assume the other player will exploit your weakness, so the relevant score is the payoff you can still guarantee even under the most unfavorable response.
When to use it
Use this in zero-sum or adversarial games, robust decision-making, security planning, and any setting where an opponent or uncertainty source is assumed to act against you. It is especially useful when you want a conservative strategy that guarantees the best guaranteed outcome rather than the best average outcome.
Worked example, step by step
Suppose you have three strategies A1, A2, and A3, and the opponent has two responses B1 and B2. Payoffs u(a,b): A1: u(A1,B1)=4, u(A1,B2)=1 A2: u(A2,B1)=3, u(A2,B2)=2 A3: u(A3,B1)=5, u(A3,B2)=-1 Step 1: Find the worst-case payoff for each strategy. For A1: min(4, 1) = 1 For A2: min(3, 2) = 2 For A3: min(5, -1) = -1 Step 2: Choose the strategy with the largest of these minimum values. max(1, 2, -1) = 2 So the minimax value is v = 2, achieved by strategy A2.
Common mistakes to avoid
- Using the opponent’s best outcome instead of your worst-case outcome
- Taking the maximum first and then the minimum, which changes the meaning of the decision
- Confusing the minimax value with the actual payoff that will occur, rather than the guaranteed lower bound
Did You Know?
Von Neumann's 1928 minimax theorem proved every two-player zero-sum game has a well-defined optimal value.
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