# Minimax Theorems V.S. Fixed Point Theorems.

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 theorem minimax implies some fixed point theorem? Or fixed point theorem implies some type minimax theorem?

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.