I am not familiarly with Game Theory but came across the question of how to compute $v :=\sup_{\mathbf{p}} \inf_{\mathbf{q}} \sum_{i,j=1}^n p_i q_j a_{ij}$ subject to $\sum p_i = 1$, $p_i \geq 0 \ \forall i$ and likewise for $\mathbf{q}$. I've learned that this is one of the fundamental objects of study of Game Theory, but from the references on the subject available to me, very little is said about how to compute this in general, aside from the fact that it can be understood as the following linear program:
$\max \sum q_j$ subject to $\sum_j a_{ij} q_j \leq 1$ for all $i$ and $q_j \geq 0$ for all $j$.
Let $\mathbf{q}^*$ be an optimal solution of the above program, then $v$ defined above is equal to $\frac{1}{\sum q*_j}$.
I would like to know if this is the "gold standard" method at this level of generality. Thank you.