Théorème du minimax
The Minimax Theorem, formulated by mathematician John von Neumann in the 1920s, is a crucial concept in théorie des jeux, particularly in the analysis of zero-sum games. In a zero-sum game, one player’s gain is exactly balanced by the losses of another player, creating a situation of perfect competition.
The theorem states that for every finite two-player zero-sum game, there exists a strategy for both players that minimizes their maximum losses (hence ‘minimax’). In other words, each player can determine a strategy that maximizes their minimum payoff, ensuring they are safeguarded against the worst-case scenario.
Mathematically, if Player A aims to maximize their minimum payoff, and Player B aims to minimize Player A’s maximum payoff, the Minimax Theorem assures that both players can find a mixed strategy equilibrium. This equilibrium occurs when both players choose their strategies in such a way that neither can improve their outcome by unilaterally changing their strategy.
The Minimax Theorem has wide applications beyond traditional games, influencing fields such as economics, théorie de la décision, and intelligence artificielle, particularly in algorithm design for competitive environments like chess and poker. It helps in developing strategies that can effectively counter opponents’ moves while optimizing the player’s own outcomes.