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# \$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)\$

Let $$f(x,y)$$ be a real function of the variables $$x,y$$ (which can be real vectors). Under what conditions do we have the following equality:

$$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$$

For example, this equality is true if $$f(x,y) = xy$$ and $$x,y$$ are real scalars.

Note that this is not the same as Von Neumann’s minimax theorem (https://en.wikipedia.org/wiki/Minimax_theorem), because here the role of the variables is exchanged (e.g., $$x$$ is minimized on the left-hand side, but it is maximized on the right-hand side).

Though I do not know if convexity/concavity of $$f(x,y)$$ with respect to either arguments plays a role here (like it does for Von Neumann’s minimax), I am using the `convex`-related tags here since that’s the context where I’ve seen related questions. Similarly I am tagging `game-theory`, though I’m not sure it’s directly applicable.

I also expect that the a condition for this equality to be true is that the saddle-points of both sides of the equation be attained at points where the gradient of $$f$$ vanishes (see my answer below).