Minimaxregel

The minimaxregel, or minimax rule, is a decision rule used under uncertainty in decision and game theory. It directs choosing an action that minimizes the worst-case loss across all possible states of nature or opponent responses. In other words, the rule selects the action with the smallest maximum loss, adopting a pessimistic or robust approach to uncertainty.

Formal description can be stated as follows. Let A be the set of available actions and S the set of possible states of nature. Let L(a, s) denote the loss incurred when action a is chosen and state s occurs. The worst-case loss for action a is max_{s in S} L(a, s). The minimax rule chooses a* = argmin_{a in A} max_{s in S} L(a, s). In payoff terms, if P(a, s) is the payoff to a player, the corresponding rule often uses a maximum of potential losses for the opponent or a minimum of potential gains for the decision-maker, depending on the formulation.

In game theory, the minimax principle is central to zero-sum games. The value of the game is v = min_{a} max_{b} u(a, b) for the player facing the opponent’s choices, and by the minimax theorem this equals max_{b} min_{a} u(a, b) under suitable conditions. The minimax solution may require mixed strategies in finite games.

Applications include robust decision-making under uncertainty, economics, and artificial intelligence, especially in adversarial settings and game-playing algorithms where worst-case reasoning is appropriate. Limitations include excessive conservatism if the worst-case scenario is unlikely, and computational complexity in large problems or non-linear payoff structures.