About a simplified version of Sion's minimax theorem.

Question

The Sion's minimax theorem is stated as:

Theorem minimax of sion. Let $X$ be a compact convex subset of a linear topological space and $Y$ a convex subset of a linear topological space. Let $f$ be a real-valued function on $X \times Y$ such that

• $f(x, \cdot)$ is upper semicontinuous and quasi-concave on $Y$ for each $x \in X$.
• $f( \cdot, y)$is lower semicontinuous and quasi-convex on $X$ for each $y \in Y$.

Then $$\inf_{x \in X}\sup_{y \in Y}f(x,y) = \sup_{y \in Y}\inf_{x \in X}f(x,y)$$

The proof of this theorem ( since its context is of linear topological spaces and your stament uses semi continuity, quasi convexity and quasi concavity) is very intricate. For the applications I'm looking for, a minimax theorem with a context in m-dimentional Euclidean space.

I believe in the old Chinese proverb: "You do not have to use a cannon to kill a fly." But, on the other hand, Von Neumann's minimax theorem does not meet the requirements of the text I am writing.

Thus, for my purposes it is necessary that I prove the following theorem of the minimax type:

Let $X\subset \mathbb{R}^n$ and $Y\subset \mathbb{R}^m$ be a compact and convex. Let a real-valued function on $f:X \times Y \to\mathbb{R}$ such that

• $f(x, \cdot)$ is continuous and concave on $Y$ for each $x \in X$,

• $f( \cdot, y)$ is continuous and convex on $X$ for each $y \in Y$.

Then there exists a saddle point $(x_0,y_0)$, i.e. $$ f(x_0,y)\leq f(x_0,y_0) \leq f(x,y_0), $$ for all $(x,y)\in X\times Y$ and $$ \max_{y\in Y}\min_{x\in X} f(x,y)=\min_{x\in X}\max_{y\in Y} f(x,y) $$

I am sure that the version of the minimax theorem I enunciated has already been proven somewhere.

My question is the one that follows.

Is there any reference that states and proves the minimax theorem that I wrote above? If this bibliographic reference is not possible, is it possible to reproduce here a proof of this theorem?

Answer 1

A proof of Sion's theorem in the context of Euclidean spaces can be found as 16.9 Theorem 16.9. in

Border, Kim C. Fixed point theorems with applications to economics and game theory. Cambridge university press, 1989.

A not necessarily easier, but, for game theorists, potentially more natural, proof would consist in proving that the zero-sum game has a Nash equilibrium via the Kakutani fixed-point theorem (or the finite-dimensional version of Glicksberg's theorem) and then verifying as in the finite game case that a Nash equilibrium gives you a minimax solution.