A notation mistake in MinMaximizer?

Question

I've been reading the game theory from this source. In there clearly mentioned that:

But I think it's not true. Isn't it true to write it as follows:

$$\min_{x_2\in X_2} (x^*_1)^TAx_2 = \min_{x_2\in X_2} \max_{x_1\in X_1}(x_1)^TAx_2$$

Thanks

Answer 1

If you think the second equality is true, then so is the first. This follows from the minimax theorem. In particular, we have that

$$ \max_{x_1 \in X_1} \min_{x_2 \in X_2} \, x_1^T A x_2 = \min_{x_2 \in X_2} \max_{x_1 \in X_1} \, x_1^T A x_2, $$

as long as $X_1$ and $X_2$ are compact and convex. These are standard assumptions in game theory.