Suppose that $\mathcal{X} = \{x_{1},x_{2}\}$ and $\mathcal{Y}=\{y_{1},y_{2}\}$ are two sets, each containing two real numbers.
Now suppose I have some function $f: \mathcal{X} \times \mathcal{Y} \to \mathbb{R}$, and I want to solve the minimax problem: \begin{align}\min_{x \in \mathcal{X}} \max_{y \in \mathcal{Y}} f(x,y).\end{align} What is the proper notation for the value of $y^* \in \mathcal{Y}$ that obtains in the inner maximum? Would it be proper to write: $$ y^* = \min_{x \in \mathcal{X}} \arg\max_{y \in \mathcal{Y}} f(x,y)?$$
Let: $$y^{*}(x) = \arg \max_{y \in \mathcal{Y}} f(x,y)$$ Furthermore, let: $$x^{*} = \arg\min_{x \in \mathcal{X}} f(x,y^{*}(x))$$ Then the value of $y$ that obtains in the inner maximum can be written as $y^{*}(x^{*})$.