Shapley population monotonicity in a convex game (${\varphi }_i(v,N)\ge {\varphi }_i(v,T)(i\in T\subseteq N)$)

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.