Variant of Parthasarathy's minimax theorem

Question

Does there exist a variant of Parthasarathy's minimax theorem [1] that relaxes the assumption that the spaces $X$ and $Y$ are $[0,1]$?

[1] https://en.wikipedia.org/wiki/Parthasarathy%27s_theorem

Answer 1

Let $f\colon X\times Y\to\mathbb R$, where $X$ and $Y$ are any sets. Suppose that the function $f$ is generalized concave-convex in the sense that for any $x_0,x_1$ in $X$, any $y_0,y_1$ in $Y$, and any $t\in[0,1]$ there exist $x_t\in X$ and $y_t\in Y$ such that for all $x\in X$ and $y\in Y$ $$f(x_t,y)\ge(1-t)f(x_0,y)+tf(x_1,y)$$ and $$f(x,y_t)\le(1-t)f(x,y_0)+tf(x,y_1).$$

A necessary and sufficient condition for $$\sup_{x\in X}\,\inf_{\rule{0pt}{6.6pt}y\in Y}\,f(x,y) =\inf_{\rule{0pt}{7pt}y\in Y}\,\sup_{x\in X}\,f(x,y)$$ was given in this paper; you may also want to see the corresponding zbMATH review.

In Parthasarathy's minimax theorem, the "payoff" function $f$ is affine in each of its two arguments and hence concave-convex, and hence generalized concave-convex.

Answer 2

Going over the proof of Parthasarathy (1970), I didn't see where he uses the assumption that X and Y are the unit intervals, and not more general sets. Read his proof and find the exact conditions on X and Y that are required for his approach.