Convexity criterion for an induced subgraph game ($v\text{ is supermodular}⟺w_{ij}\ge 0\text{ for all }i\ne j$)

A loop-free induced subgraph game is a convex cooperative game exactly when all edge weights are nonnegative. Necessity follows by testing the supermodular inequality on two distinct singleton coalitions. For sufficiency,Only edges connecting the two set differences survive cancellation. Loop contributions cancel as well, so the same off-diagonal condition remains sufficient and necessary when self-loops are added.