In the mathematical area of game theory, a minimax theorem is a theorem providing conditions that guarantee that the max–min inequality is also an equality. The first theorem in this sense is von Neumann's minimax theorem about zero-sum games published in 1928,[1] which was 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".[2]
Since then, several generalizations and alternative versions of von Neumann's original theorem have appeared in the literature.[3][4]
Formally, von Neumann's minimax theorem states:
Let and be compact convex sets. If is a continuous function that is concave-convex, i.e.
Then we have that