I have a function $f$ which is linear in the first argument $x \in \mathcal{X}$, and quadratic in the second argument $y \in \mathcal{Y}$. The set $\mathcal{X}$ is countable, and the set $\mathcal{Y}$ is finite. I want to know under what conditions is the following equality true. The sets are bounded.
\begin{equation*} \min_{x \in \mathcal{X}} \max_{y \in \mathcal{Y}} f(x, y) \stackrel{?}= \max_{y \in \mathcal{Y}} \min_{x \in \mathcal{X}} f(x, y) \end{equation*}
