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Is there any relationship between the minimax theorems $$ \mbox{Regularly hypothesis on } f:U\times V\to\mathbb{R} \implies \min_{u}\max_{v}f(u,v)=\max_{v}\min_{u}f(u,v) $$ and fixed point theorems $$ \mbox{Regularly hypothesis on } F:X\to X \implies F(x)=x, \mbox{ for some } x\in X ? $$ More precisely, there is some kind teorem minimax implies some fixed point theorem? Or fixed point theorem implies some type minimax theorem?

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up vote 3 down vote accepted

One direction is given historically in game theory. Nash's result on the existence of Nash equilibria, essentially a fixed point theorem, implies the minimax theorem of von Neumann. Here is the coarse structure of the argument. We want to rewrite $$f:U\times V\to\mathbb{R} \implies \min_{u}\max_{v}f(u,v)=\max_{v}\min_{u}f(u,v)$$ as the solution to a fixed point problem. For this we have to generalize the idea of a fixed point to correspondences, or set-valued mappings. If $\phi:S\to 2^S$ maps points in $S$ to subsets of $S$, we say that $s$ is a fixed point of $\phi$ if $s\in\phi(s)$. One can often prove fixed point theorems for correspondences from fixed point theorems for functions by using selection or approximation theorems.

So we define a correspondence $F:U\times V\to 2^{U\times V}$ by letting $$F(u,v)=\{u'\in U:u'\text{ minimizes } f(\cdot,v)\}\times\{v'\in V:v'\text{ maximizes } f(u,\cdot)\}.$$ Now the fixed points of $F$ are exactly the solutions to the minimax problem.

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@Greinecker, good answer. Some reference? Book or an article? Thanks. – MathOverview Oct 26 '14 at 14:32
@Elias For the finite dimensional case, there is the nice book Fixed Point Theorems with Applications to Economics and Game Theory by Kim Border. – Michael Greinecker Oct 27 '14 at 12:14
@Grinecker, thanks! I went back to take interest in this topic again. I'm interesting in writing a paper review. Something very simple. But first I want to check if what I had in mind was not already done before. – MathOverview Oct 27 '14 at 14:32

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