= Shapley wages in an entrepreneur-worker game
{c}
{title2=$w=\frac{1}{n(n+1)}\sum_{k=0}^{n-1}(k+1)(p(k+1)-p(k))$}
An entrepreneur is indispensable and a <coalition> containing the entrepreneur and $k$ of $n$ workers earns $p(k)$; a <coalition> without the entrepreneur earns zero. A worker's <marginal contribution> is nonzero only when the entrepreneur precedes them. The <probability> that the entrepreneur and exactly $k$ other workers precede that worker is $(k+1)/(n(n+1))$, giving the displayed common <Shapley value> wage. The entrepreneur receives $p(n)-nw$. Convex nondecreasing $p$ makes all <marginal contributions> increase with predecessor sets, so the whole <Shapley value> allocation belongs to the <core>.
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