Minimax theorem

In the mathematical area of game theory and of convex optimization, a minimax theorem is a theorem that claims that under certain conditions on the sets

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 theory.

Von Neumann is quoted as saying "As far as I can see, there could be no theory of games ... without that theorem ...

I thought there was nothing worth publishing until the Minimax Theorem was proved".

[3] Since then, several generalizations and alternative versions of von Neumann's original theorem have appeared in the literature.

[4][5] Von Neumann's original theorem[2] was motivated by game theory and applies to the case where Under these assumptions, von Neumann proved that In the context of two-player zero-sum games, the sets

correspond to the strategy sets of the first and second player, respectively, which consist of lotteries over their actions (so-called mixed strategies), and their payoffs are defined by the payoff matrix

Von Neumann's minimax theorem can be generalized to domains that are compact and convex, and to functions that are concave in their first argument and convex in their second argument (known as concave-convex functions).

is a continuous function that is concave-convex, i.e. Then we have that Sion's minimax theorem is a generalization of von Neumann's minimax theorem due to Maurice Sion,[6] relaxing the requirement that It states:[6][7] Let

be a convex subset of a linear topological space and let

be a compact convex subset of a linear topological space.

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