Minimax Theorems

We suppose that X and Y are nonempty sets and f : X × Y → R. A minimax theorem is a theorem that asserts that, under certain conditions , inf Y sup X f = sup X inf Y f, that is to say, inf y∈Y sup x∈X f (x, y) = sup x∈X inf y∈Y f (x, y). The purpose of this article is to give the reader the flavor of the different kind of mini-max theorems, and of the techniques that have been used to prove them. This is a very large area, and it would be impossible to touch on all the work that has been done in it in the space that we have at our disposal. The choice that we have made is to give the historical roots of the subject, and then go directly to the most recent results. The reader who is interested in a more complete narrative can refer to the 1974 survey article [35] by E. In his investigation of games of strategy , J.von Neumann realized that, even though a two– person zero–sum game did not necessarily have a solution in pure strategies, it did have to have one in mixed strategies. Here is a statement of that seminal result ([32], translated into English in [33]): Theorem 1 (1928). Let A be an m×n matrix, and X and Y be the sets of nonnegative row and column vectors with unit sum. Then min y∈Y max x∈X xAy = max x∈X min y∈Y xAy. Despite the fact that the statement of this result is quite elementary, the proof was quite sophisticated, and depended on an extremely ingenious induction argument. Nine years later in [34], von Neumann showed that the bilinear character of Theorem 1 was not needed when he extended it as follows, using Brouwer's fixed– point theorem: