= Shapley population monotonicity in a convex game
{c}
{title2=$\phi_i(v,N)\geq\phi_i(v,T)\quad(i\in T\subseteq N)$}
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.
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