= Solution
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 $\pi$, then
$$
\phi_i(v,N)=\frac1{n!}\sum_\pi\bigl(v(P_i^\pi\cup\{i\})-v(P_i^\pi)\bigr)
=\sum_{S\subseteq N\setminus\{i\}}\frac{|S|!(n-|S|-1)!}{n!}\bigl(v(S\cup\{i\})-v(S)\bigr).
$$
For 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> give
$$
v(P_i^\pi\cup\{i\})-v(P_i^\pi)
\geq v((P_i^\pi\cap T)\cup\{i\})-v(P_i^\pi\cap T).
$$
Averaging proves <Shapley population monotonicity in a convex game>:
$$
\boxed{\phi_i(v,N)\geq\phi_i(v,T)\qquad(i\in T).}
$$
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 gives
$$
\sum_{i\in T}\phi_i(v,N)\geq\sum_{i\in T}\phi_i(v,T)=v(T).
$$
Together with efficiency for $N$, these are exactly the <core of a cooperative game> inequalities. Hence \b[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 $n\geq1$ workers by $1,\ldots,n$. 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 $\binom{n-1}{k}$ predecessor choices, is
$$
\binom{n-1}{k}\frac{(k+1)!(n-k-1)!}{(n+1)!}
=\frac{k+1}{n(n+1)}.
$$
The fair wage is thus
$$
\boxed{w=\frac1{n(n+1)}\sum_{k=0}^{n-1}(k+1)\bigl(p(k+1)-p(k)\bigr)
=\frac{np(n)-\sum_{k=0}^{n-1}p(k)}{n(n+1)}.}
$$
This is <Shapley wages in an entrepreneur-worker game>. The entrepreneur's payoff, by efficiency, is
$$
\boxed{x_0=p(n)-nw=\frac1{n+1}\sum_{k=0}^{n}p(k).}
$$
It 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
$$
\boxed{w=\frac\alpha2,\qquad x_0=\frac{\alpha n}{2}.}
$$
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 \b[the complete Shapley allocation lies in the core], including the entrepreneur's payoff.
One can also verify the <core> inequalities directly. Nonentrepreneur <coalitions> receive $kw\geq0$, their value. For a <coalition> containing the entrepreneur and $k<n$ workers, it suffices that
$$
w\leq\frac{p(n)-p(k)}{n-k}.
$$
Since $k+1\leq n+1$ in the wage sum, $w\leq\frac1n\sum_{j=0}^{n-1}d_j$. 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 $x_0+kw=p(n)-(n-k)w\geq p(k)$; the grand <coalition> has equality by efficiency. This confirms the core conclusion without assuming $p(0)=0$.
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