Minimax Theorems V.S. Fixed Point Theorems.

Solution 1:

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.