Minimax Theorem
Minimizing the maximum loss
The minimax theorem states that in zero-sum games, one player should attempt to minimize their opponent's maximum payoff while their opponent attempts to maximize their own minimum payoff. This leads to a Nash equilibrium in mixed strategies, where both players use a mixture of pure strategies to achieve the best possible outcome.
- In zero-sum games, one player's gain is the other's loss.
- Players should attempt to minimize their opponent's maximum payoff.
- The minimum of the maximum payoffs equals the maximum of the minimum payoffs.
- Identify the game as a zero-sum gameDetermine if the game is a zero-sum game, where one player's gain is the other's loss.Pro tipConsider the payoffs and outcomes of the game to determine if it is zero-sum.WarningNot all games are zero-sum, so be careful in applying the minimax theorem.
- Determine the pure strategies available to each playerIdentify the pure strategies available to each player in the game.Pro tipConsider the possible actions and outcomes for each player.WarningBe sure to include all possible pure strategies.
- Calculate the payoffs for each pure strategy combinationCalculate the payoffs for each combination of pure strategies.Pro tipUse the payoffs to determine the best mixture of pure strategies.WarningBe careful in calculating the payoffs, as small errors can lead to incorrect conclusions.
- Find the Nash equilibrium in mixed strategiesUse the payoffs to find the Nash equilibrium in mixed strategies, where both players use a mixture of pure strategies to achieve the best possible outcome.Pro tipConsider the minimax theorem and the principles of game theory to find the Nash equilibrium.WarningThe Nash equilibrium may not always be unique or easy to find.
In soccer penalty kicks, the kicker and goalie engage in a zero-sum game, where the kicker tries to score and the goalie tries to prevent the score.
In Rock-Paper-Scissors, two players engage in a zero-sum game, where each player tries to win by choosing the correct move.
The minimax theorem was first proposed by John von Neumann and later elaborated by him and Oscar Morgenstern in their book 'Theory of Games and Economic Behavior'.