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It is always true that the left-hand side is at most the right-hand side (max–min inequality) but equality only holds under certain conditions identified by minimax theorems. The first theorem in this sense is von Neumann's minimax theorem about two-player zero-sum games published in 1928, [2] which is considered the starting point of game ...
Game theory is the study of mathematical models ... Émile Borel proved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix is ...
A zero-sum game is also called a strictly competitive game, while non-zero-sum games can be either competitive or non-competitive. Zero-sum games are most often solved with the minimax theorem which is closely related to linear programming duality, [5] or with Nash equilibrium. Prisoner's Dilemma is a classic non-zero-sum game. [6]
By the minimax theorem of John von Neumann, there exists a game value , and mixed strategies for each player, such that the players can guarantee expected value or better by playing those strategies, and such that the optimal pure strategy against either mixed strategy produces expected value exactly .
Constant sum: A game is a constant sum game if the sum of the payoffs to every player are the same for every single set of strategies. In these games, one player gains if and only if another player loses. A constant sum game can be converted into a zero sum game by subtracting a fixed value from all payoffs, leaving their relative order unchanged.
In game theory, folk theorems are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games (Friedman 1971). [1] The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game.
In game theory, the Nash equilibrium is the most commonly-used solution concept for non-cooperative games.A Nash equilibrium is a situation where no player could gain by changing their own strategy (holding all other players' strategies fixed). [1]
In the mathematical theory of games, in particular the study of zero-sum continuous games, not every game has a minimax value. This is the expected value to one of the players when both play a perfect strategy (which is to choose from a particular PDF). This article gives an example of a zero-sum game that has no value. It is due to Sion and ...