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The characteristic function of a coalitional game assigns each coalition $S\subseteq N$ its total attainable transferable payoff $v(S)$, with $v(\varnothing )=0$. In a transferable utility game the members may divide that total among themselves. This game-theoretic characteristic function is not the characteristic function of a probability distribution.

The Shapley value assigns to player $i$ the average marginal contribution that player makes over all uniformly random orderings of the players. If $P_i^{\pi }$ is the predecessor set in ordering $\pi$, thenFor any ordering, summing all contributions telescopes to $v(N)$, so the Shapley value is efficient. It treats interchangeable players equally, pays a null player zero, and is additive in the characteristic function. These properties give its standard fairness axioms and uniquely characterize the Shapley value. The random-order interpretation explains why every position and predecessor coalition receives a principled weight rather than privileging a particular bargaining order.

For a convex cooperative game, couple a uniform ordering of $N$ with its restriction to $T$. The restricted ordering is uniform on $T$. For $i\in T$, the restricted predecessor set is $P_i^{\pi }\cap T\subseteq P_i^{\pi }$, so increasing marginal contributions giveAveraging proves Shapley population monotonicity in a convex game:The comparison is meaningful for players who participate in the restricted game; no restricted value is defined for a player outside $T$. Summing over $i\in T$ and applying efficiency in that restricted game givesTogether with efficiency for $N$, these are exactly the core of a cooperative game inequalities. Hence the Shapley value belongs to the core of a convex game. This proves the requested stability, not just individual rationality.

Now label the entrepreneur by $0$ and the $n\ge 1$ workers by $1,\dots ,n$. A coalition without the entrepreneur has value zero; one containing the entrepreneur and $k$ workers has value $p(k)$. Workers are interchangeable, so their Shapley values agree. Fix one worker. Their contribution is zero if the entrepreneur appears after them. If the entrepreneur and exactly $k$ of the other $n-1$ workers appear before them, it is $p(k+1)-p(k)$. For a fixed choice of those workers, the $k+1$ predecessors can be ordered in $(k+1)!$ ways and the remaining $n-k-1$ successors in $(n-k-1)!$ ways. Therefore the probability of this event, summing over the $\left(\genfrac{}{}{0ex}{}{n-1}{k}\right)$ predecessor choices, isThe fair wage is thusThis is Shapley wages in an entrepreneur-worker game. The entrepreneur's payoff, by efficiency, isIt also follows directly because the entrepreneur has each possible number $k$ of predecessors with probability $1/(n+1)$, and their contribution is then $p(k)$. When $p(k)=\alpha k$, the differences are all $\alpha$ and ${\sum }_{k=0}^{n-1}(k+1)=n(n+1)/2$, so

Finally suppose $p$ is convex and nondecreasing on nonnegative integers, so $d_k=p(k+1)-p(k)$ is nonnegative and nondecreasing. We verify that the entire coalitional game is convex. The entrepreneur's contribution to a set of $k$ workers is $p(k)$, which increases with that set. A worker's contribution is zero when the entrepreneur is absent and is $d_k$ when the entrepreneur and $k$ other workers are present. Enlarging a predecessor set either leaves both contributions zero, introduces the entrepreneur and a nonnegative contribution, or increases $k$ and hence $d_k$. Every player's marginal contributions therefore increase, giving a convex cooperative game. The result already proved yields the complete Shapley allocation lies in the core, including the entrepreneur's payoff.

One can also verify the core inequalities directly. Nonentrepreneur coalitions receive $kw\ge 0$, their value. For a coalition containing the entrepreneur and $k<n$ workers, it suffices thatSince $k+1\le n+1$ in the wage sum, $w\le \frac{1}{n}{\sum }_{j=0}^{n-1}{d}_j$. Nondecreasing $d_j$ makes this full average no greater than the average of the final $n-k$ differences, which is the displayed right side. Thus $x_0+kw=p(n)-(n-k)w\ge p(k)$; the grand coalition has equality by efficiency. This confirms the core conclusion without assuming $p(0)=0$.