For square matrix $P$, define

$$V(P) = \sup_x  \inf_y x^T P y^T$$

where $x$ and $y$ lie on the unit $n-1$-simplex.

($P$ is a payoff matrix for a symmetric game, $x$ and $y$ are mixed strategies, and $V$ is the value of the game.)

It is known that determining $V(P)$ is NP-hard.  However, is there some way to bound $V(P)$ by differentiable functions of the elements of $P$?  If I am not mistaken, $V$ is monotonic in each element of $P$.