minimaxteoreem

Minimaxteoreem (or the Minimax Theorem) is a fundamental result in game theory, introduced by John von Neumann in 1928. It pertains to two-player, zero-sum games, where one player's gain is exactly the other player's loss. The theorem establishes that in such games, there exists a strategy for each player that guarantees a specific outcome regardless of the opponent's moves.

The core idea of the minimax theorem is formalized in the equality of the players' minimax and maximin values. Specifically, it states that the maximum expected payoff a player can guarantee by choosing the best possible strategy (maximin) is equal to the minimum expected payoff the opponent can enforce against the player’s worst-case strategy (minimax). Mathematically, this is expressed as:

supₓ inf_y 𝐸[x, y] = inf_y supₓ 𝐸[x, y]

where 𝐸[x, y] represents the expected payoff when Player 1 adopts strategy x and Player 2 adopts strategy y.

The theorem has broad implications, providing a foundation for equilibrium analysis in strategic interactions and underlining the concept of saddle points in continuous and discrete games. It assures that in finite, zero-sum games, optimal mixed strategies exist for both players, leading to a stable solution concept known as the Nash equilibrium in such contexts.

The minimax teoreem not only influenced the development of game theory but also found applications in optimization, economics, and decision sciences, shaping methods for solving complex strategic problems.