Minimax is a rule for decision-making under uncertainty: choose the action that makes your worst possible outcome as good as possible, i.e., minimize the maximum loss.
Minimax in decision theory
Suppose you choose an action (a) and the state of the world is (s). If your loss is (L(a,s)), the minimax decision rule is:
[ a^* \in \arg\min_a \max_s L(a,s). ]
It is “pessimistic” in the sense that it evaluates each action by its worst-case scenario.
Minimax in zero-sum games
In a two-player zero-sum game (one player’s gain is the other’s loss), minimax is part of the equilibrium logic.
Let (u(a,b)) be player 1’s payoff when player 1 chooses (a) and player 2 chooses (b). Player 1’s maximin value is:
[ \max_a \min_b u(a,b), ]
and player 2’s minimax value (from player 1’s perspective) is:
[ \min_b \max_a u(a,b). ]
The minimax theorem (von Neumann) says that in finite zero-sum games, when players are allowed to use mixed strategies, these two values coincide. That is why mixed-strategy Nash equilibria exist and can be computed by minimax reasoning.
Practical example
In games like matching pennies, there is no best pure strategy: any predictable move can be exploited. The minimax/mixed-strategy solution randomizes in a way that prevents the opponent from earning a systematic advantage.
Related Terms
- Maximin
- Game Theory
- Zero-Sum Game
- Mixed Strategy
- Saddle Point
- Nash Equilibrium
- Subgame Perfect Equilibrium