 Solution

A loop-free induced subgraph game is a convex cooperative game if and only if every edge weight is nonnegative. Necessity follows by applying the supermodular inequality to $S=\{i\}$ and $T=\{j\}$. Since both singleton values and the empty value are zero,

Conversely, direct cancellation of common edge contributions gives, for any coalitions $S,T$,Every summand is nonnegative, proving that $v$ is a supermodular set function and hence convexity. Equivalently, a player's marginal contribution to $S$ is ${\Delta }_i(S)={\sum }_{j\in S}{w}_{ij}$, which cannot decrease as $S$ grows. This establishes the convexity criterion for an induced subgraph game directly.^[ref]