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Every nontrivial simple cooperative game that is a convex cooperative game is a unanimity game. If two coalitions win, the supermodular inequality forces their intersection to win as well. Intersecting the finite nonempty family of winning coalitions gives a winning set $K\ne \varnothing$. Every winning coalition contains $K$; conversely monotonicity makes every superset of $K$ win. Thus the game is $u_K$. Conversely, every unanimity game is a supermodular set function, as a direct check of the winning and losing cases shows.

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]

Use the usual empty-coalition normalization $v(\varnothing )=0$ for a transferable utility game. Write $P_i=\{1,\dots ,i-1\}$, so the proposed payoff is the marginal contributionThe vector is the marginal contribution vector for the natural player order.

First, convex cooperative games have increasing marginal contributions. If $A\subseteq B$ and $i\notin B$, apply the defining supermodular set function inequality to $A\cup \{i\}$ and $B$:Rearranging gives

The vector is efficient by a telescoping sum:Now fix any coalition $T$. For each $i\in T$, its earlier members are $T\cap P_i\subseteq P_i$. Increasing marginal contributions therefore giveSum over the members of $T$ in increasing order. The right-hand side is another telescoping sum, now along the growing coalition $T$:Efficiency and these inequalities for every coalition are exactly the defining requirements of the core of a cooperative game. Thus $x$ belongs to the core. The same proof works for any player ordering, not only the natural order.

The normalization is essential to the literal statement: the printed part does not explicitly state it. Without $v(\varnothing )=0$, take the constant characteristic function of a coalitional game $v(T)=1$. It satisfies the displayed convex cooperative game inequality with equality, but every proposed $x_i$ is zero, so efficiency fails when $N$ is nonempty. Under the standard normalized definition, the proof above establishes the intended result.