Consider two arbitrary matrices $A, B \in \mathbb{R}^{N \times M}$. Then the feasible payoff region for a two-player normal form game $(A,B)$ would be $$\{(p^TAq, p^TBq) \vert\ p \in \Delta^{N-1}, q \in \Delta^{M-1}\} \subset \mathbb{R}^2$$, where $\Delta^{N-1}$ is the N-1 probability simplex. The feasible payoff region is simply the set of (expected) payoffs which can be achieved by some pair of strategies $p,q$ for a given game. I've seen such a region plotted for a few games in a game theory textbook, but it's unclear to me how one would go about drawing this region in general, or what sort of properties, if any, we could prove about these regions.
In the $N=M=2$ case, a brute force sampling of $p$ and $q$ is sufficient to visualize the regions -- here for 16 randomly generated games:
From this, all I could say is that the regions can be nonconvex, appear to be composed of line-segments or asymptotic curves, and don't appear to have more than 4 distinct "elements".
So my questions are
Is there an alternative description of the class of shapes which can arise in the case of $N=M=2$? What about for general $N, M$? Given that the regions are not convex, could we even find a game with a payoff region which has a "hole" in it?
Is there an efficient way to draw the feasible region, or even just test if some point $(x,y)$ is inside the region for any given game with arbitrary (but reasonably small) $N, M$?