Convex cooperative game ($v(S\cup T)+v(S\cap T)\ge v(S)+v(T)$)

A transferable utility game is convex when its coalition-value function is supermodular. Equivalently a player's marginal contribution cannot decrease when the preceding coalition grows. The implication follows by applying the displayed inequality to $A\cup \{i\}$ and $B$ with $A\subseteq B$, $i\notin B$; the reverse implication follows by adding successive marginal inequalities. This property concerns coalition values, not geometrical convexity of a strategy space.

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.

Choose a uniform random ordering of $N$ and restrict it to $T$. The restriction is uniform, and each player's predecessor set in $T$ is contained in their predecessor set in $N$. Increasing marginal contributions in a convex cooperative game make the full-population contribution at least the restricted contribution. Averaging proves the displayed Shapley value inequality. Summing over $i\in T$ and using efficiency in the restricted game proves every core inequality.

Increasing marginal contributions imply $m_i^{\pi }\ge v((S\cap P_i)\cup \{i\})-v(S\cap P_i)$ for $i\in S$. Summing telescopes to ${\sum }_{i\in S}{m}_i^{\pi }\ge v(S)$, while efficiency follows by telescoping over the full ordering. Hence every marginal contribution vector is in the core of a cooperative game. The core is a convex set, so their average, the Shapley value, is in it too. This gives an elementary proof of nonemptiness and stability for a convex cooperative game.