I am solving two-stage optimization problems in the form of $$\max_{x \in X}\min_{y \in Y} f(x,y),$$ where $f(x,y)$ is the solution of a mixed integer linear program (MIP). As the constraints of the MIP link the $x$ and $y$ elements, it is clear that we cannot apply the classic minimax theorem to interchange the minimization and the maximization. (Actually, I am solving a trilevel optimization problem and interchanging the minimization and the maximization will be very helpful).
We all know that the minimax theorem states that $$\max_{x \in X}\min_{y \in Y} f(x,y) \leq \min_{y \in Y}\max_{x \in X} f(x,y)$$ for any function $f$ mapping the $X,Y$ space to a real set (assuming the max and min can be attained).
My question is the following: In the above sketched case, this equation will not be strict (i.e., it won't hold with equality). However, are there any particular useful results on how large this gap might be, both worst-case as well as numerically? If it is not too big (in general), it might suffice as a bound on the larger (and not relevant for this question) trilevel optimization problem.