Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question

1 Answer 1

up vote 3 down vote accepted
+50

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.

share|improve this answer
    
@Greinecker, good answer. Some reference? Book or an article? Thanks. –  Elias Oct 26 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 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. –  Elias Oct 27 at 14:32

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.