Inventory centralization with risk-averse newsvendors

Abstract

In the paper, we study the inventory centralization problem with risk-averse newsvendors using cooperative game theory. Pareto-optimality and collective rationality conditions are adopted to characterize the stable profit allocations among coalition members. Pareto-optimal profit allocation rules are derived for exponential and power utilities while the necessary and sufficient conditions are shown for stable profit allocations. We also show that the game model has a nonempty core when the exponential and power utilities are adopted to describe the risk-averse behavior.

Appendix: Proofs of theorems and lemmas

Proof of Lemma1

Since

$$\begin{aligned} v\big (\varPi (S, q)\big )=v\big (p\min \{q,D_{S}\}-cq\big ), \end{aligned}$$

we have

$$\begin{aligned} E\big [v\big (\varPi (S, q)\big )\big ]=\int _{0}^{q}v\big (px-cq\big )f_{S}(x)dx+v\big ((p-c)q\big )\cdot (1-F_{S}(q)). \end{aligned}$$

Let $\text {I}({q\le D_{S}})$ equal 1 if $q\le D_{S}$ and 0 otherwise.

First, we see that

$$\begin{aligned} \frac{d E\big [v\big (\varPi (S, q)\big )\big ]}{d q}= & {} \int _{0}^{q}(-c)v'\big (px-cq\big )f_{S}(x)dx\\&+\int _{q}^{+\infty } (p-c)v'\big ((p-c)q\big )f_{S}(x)dx\\= & {} -c\big [\int _{0}^{q}v'\big (px-cq\big )f_{S}(x)dx+\int _{q}^{+\infty }v'\big ((p-c)q\big )f_{S}(x)dx\big ]\\&+\int _{q}^{+\infty }pv'\big ((p-c)q\big )f_{S}(x)dx\\= & {} -c\int _{0}^{+\infty }v'\big (p\min \{x,q\}-cq\big )f_{S}(x)dx\\&+\int _{0}^{+\infty }p\text {I}(q\le x)v'\big (p\min \{q,x\}-cq\big )f_{S}(x)dx\\= & {} -cE\big [v'(p\min \{q, D_{S}\}-cq)\big ]\\&+ \,\,E\big [p\text {I}({q\le D_{S}})v'\big (p\min \{q, D_{S}\}-cq\big )\big ]\\= & {} E\big [v'(p\min \{q, D_{S}\}-cq)\cdot \big (p\text {I}({q\le D_{S}})-c\big )\big ]. \end{aligned}$$

Then, we have

$$\begin{aligned} \frac{d^{2} E\big [v\big (\varPi (S, q)\big )\big ]}{d q^{2}}= & {} c^2\int _{0}^{q}v''\big (px-cq\big )f_{S}(x)dx\\&+\int _{q}^{+\infty }(-c)(p-c)v''\big ((p-c)q\big )f_{S}(x)dx\\&+\int _{q}^{+\infty }p(p-c)v''\big ((p-c)q\big )f_{S}(x)dx-pv'\big ((p-c)q\big )f_{S}(q)\\= & {} c^2 \int _{0}^{+\infty }v''\big (p\min \{x,q\}-cq\big )f_{S}(x)dx\\&+\int _{0}^{+\infty }p^2 \text {I}({q\le x})v''\big (p\min \{x,q\}-cq\big )f_{S}(x)dx\\&- \,\,2\int _{0}^{+\infty }cp\text {I}({q\le x})v''\big (p\min \{x,q\}-cq\big )f_{S}(x)dx\\&- \,\,pv'\big ((p-c)q\big )f_{S}(q)\\= & {} E\big [v''(\varPi (S, q))\cdot \big (p\text {I}({q\le D_{S}})-c\big )^{2}\big ]-pv'\big ((p-c)q\big )f_{S}(q). \end{aligned}$$

Since v is strictly increasing and strictly concave, we have $v'(\cdot )> 0$ and $v''(\cdot )< 0$ for $q\ge 0$. Hence $\frac{d^{2} E[v(\varPi (S, q))]}{d q^{2}}< 0$ for $q\ge 0$, which means that $E[v(\varPi (S, q))]$ is a strictly concave function of q. $\square$

Proof of Theorem 1

Suppose $(X_{i})_{i\in S}$ is a Pareto-optimal risk sharing scheme of S. Then there exists an order $q\ge 0$ such that $\sum _{i\in S} X_{i}=\varPi (S, q)$. From Sect. 3.1, there exist $a_{i}>0$, $i\in S$ with $\sum _{i\in S} a_{i}=1$ such that $(X_{i})_{i\in S}$ is the solution of (4) and (5). From Raiffa (1970), for the exponential utilities (6), we have

$$\begin{aligned} X_{i}=\frac{1/\alpha _{i}}{\sum _{j\in S}1/\alpha _{j}} \varPi (S, q)-\frac{1/\alpha _{i}}{\sum _{j\in S}1/\alpha _{j}}\sum _{j\in S} \frac{1}{\alpha _{j}}\ln \frac{a_{i}}{a_{j}}, \quad \textit{for}~ i\in S. \end{aligned}$$

From (6), when newsvendor $i\in S$ has a profit $X_{i}$, his maximum expected utility is reached at

$$\begin{aligned} \arg \max _{q\ge 0} E\Bigg [\frac{1}{\alpha _{i}}\Bigg (1-\exp \Bigg (-\frac{1}{\sum _{j\in S}1/\alpha _{j}}\varPi (S, q)-\frac{1}{\sum _{j\in S}1/\alpha _{j}}\sum _{j\in S} \frac{1}{\alpha _{j}}\ln \frac{a_{i}}{a_{j}}\Bigg )\Bigg )\Bigg ]. \end{aligned}$$

Since

$$\begin{aligned}&E\left[ \frac{1}{\alpha _{i}}\left( 1-\exp (-\frac{1}{\sum _{j\in S}1/\alpha _{j}}\varPi (S, q)-\frac{1}{\sum _{j\in S}1/\alpha _{j}}\sum _{j\in S} \frac{1}{\alpha _{j}}\ln \frac{a_{i}}{a_{j}})\right) \right] \\&\quad =\frac{1}{\alpha _{i}}-\frac{1}{\alpha _{i}}\cdot E\left[ \exp (-\frac{1}{\sum _{j\in S}1/\alpha _{j}}\varPi (S, q))\right] \cdot \exp \Bigg (-\frac{1}{\sum _{j\in S}1/\alpha _{j}}\sum _{j\in S} \frac{1}{\alpha _{j}}\ln \frac{a_{i}}{a_{j}}\Bigg ), \end{aligned}$$

we have

$$\begin{aligned}&\arg \max _{q\ge 0} E\Bigg [\frac{1}{\alpha _{i}}\Bigg (1-\exp \Bigg (-\frac{1}{\sum _{j\in S}1/\alpha _{j}}\varPi (S, q)-\frac{1}{\sum _{j\in S}1/\alpha _{j}}\sum _{j\in S} \frac{1}{\alpha _{j}}\ln \frac{a_{i}}{a_{j}}\Bigg )\Bigg )\Bigg ]\\= & {} \arg \max _{q\ge 0} \Bigg ( -\exp \Bigg (-\frac{1}{\sum _{j\in S}1/\alpha _{j}}\varPi (S, q)\Bigg ) \Bigg )\\= & {} \arg \max _{q\ge 0} E\Bigg [\Bigg (\sum _{j\in S}\frac{1}{\alpha _{j}}\Bigg )\Bigg (1-\exp \Bigg (-\frac{1}{\sum _{j\in S}1/\alpha _{j}}\varPi (S, q)\Bigg )\Bigg )\Bigg ]. \end{aligned}$$

In Lemma 1, let $v(x)=(\sum _{j\in S}\frac{1}{\alpha _{j}})(1-\exp (-\frac{1}{\sum _{j\in S}1/\alpha _{j}}x))$, for $x\in \mathbb {R}$. Then we see that $E\big [(\sum _{j\in S}\frac{1}{\alpha _{j}})\big (1-\exp (-\frac{1}{\sum _{j\in S}1/\alpha _{j}}\varPi (S, q))\big )\big ]$ is a strictly concave function of $q\ge 0$ and, consequently,

$$\begin{aligned} \arg \max _{q\ge 0} E\Bigg [\Bigg (\sum _{j\in S}\frac{1}{\alpha _{j}}\Bigg )\Bigg (1-\exp \Bigg (-\frac{1}{\sum _{j\in S}1/\alpha _{j}}\varPi (S, q)\Bigg )\Bigg )\Bigg ] \end{aligned}$$

is unique.

Therefore, every newsvendor $i\in S$ obtains his maximum expected utility at the same order $q_{S}^{\star }$ given by (7). Since $(X_{i})_{i\in S}$ is a Pareto-optimal risk sharing scheme of coalition S, we have $q=q_{S}^{\star }$ and

$$\begin{aligned} X_{i}=\frac{1/\alpha _{i}}{\sum _{j\in S}1/\alpha _{j}} \varPi (S, q_{S}^{\star })-\frac{1/\alpha _{i}}{\sum _{j\in S}1/\alpha _{j}}\sum _{j\in S} \frac{1}{\alpha _{j}}\ln \frac{a_{i}}{a_{j}}, \quad \textit{for}~i\in S. \end{aligned}$$

For each $i\in S$, let $z_{i}=-\frac{1/\alpha _{i}}{\sum _{j\in S}1/\alpha _{j}}\sum _{j\in S} \frac{1}{\alpha _{j}}\ln \frac{a_{i}}{a_{j}}$. Then

$$\begin{aligned} \sum _{i\in S} z_{i}= & {} \sum _{i\in S} \Bigg (-\frac{1/\alpha _{i}}{\sum _{j\in S}1/\alpha _{j}}\sum _{j\in S} \frac{1}{\alpha _{j}}\ln \frac{a_{i}}{a_{j}}\Bigg )\\= & {} \sum _{i\in S} \Bigg (-\frac{1/\alpha _{i}}{\sum _{j\in S}1/\alpha _{j}}\sum _{j\in S} \frac{1}{\alpha _{j}}(\ln a_{i}-\ln a_{j})\Bigg )\\= & {} -\sum _{i\in S}\frac{1}{\alpha _{i}}\ln a_{i}+\sum _{j\in S}\frac{1}{\alpha _{j}}\ln a_{j}=0. \end{aligned}$$

Hence $(X_{i})_{i\in S}$ is an element of (8).

Suppose $(X_{i})_{i\in S}$ is an element of (8), i.e., there exist $z_{i}\in \mathbb {R}$, $i\in S$ with $\sum _{i\in S} z_{i}=0$ such that

$$\begin{aligned} X_{i}=\frac{1/\alpha _{i}}{\sum _{j\in S}1/\alpha _{j}} \varPi (S, q_{S}^{\star })+z_{i}, \quad \textit{for}~i\in S. \end{aligned}$$

From the matrix theory (Bernstein 2005), there exist $a_{i}>0$, $i\in S$ with $\sum _{i\in S} a_{i}=1$ such that $z_{i}=-\frac{1/\alpha _{i}}{\sum _{j\in S}1/\alpha _{j}}\sum _{j\in S} \frac{1}{\alpha _{j}}\ln \frac{a_{i}}{a_{j}}$, for each $i\in S$. From Sect. 3.1, corresponding to $a_{i}$, $i\in S$, there is a Pareto-optimal risk sharing scheme $(X_{i}')_{i\in S}$, where

$$\begin{aligned} X_{i}'=\frac{1/\alpha _{i}}{\sum _{j\in S}1/\alpha _{j}} \varPi (S, q)-\frac{1/\alpha _{i}}{\sum _{j\in S}1/\alpha _{j}}\sum _{j\in S} \frac{1}{\alpha _{j}}\ln \frac{a_{i}}{a_{j}}, \end{aligned}$$

for $i\in S$ and some $q\ge 0$. It is not difficult to verify that $q=q_{S}^{\star }$. Hence $X_{i}=X_{i}'$, for $i\in S$. Hence $(X_{i})_{i\in S}$ is a Pareto-optimal risk sharing scheme of S.

Consequently, the set of Pareto-optimal risk sharing schemes of S is given by (8). Moreover, we can see that the Pareto-optimal order of S is given by (7). $\square$

Proof of Lemma 2

For newsvendor $i\in N$, let

$$\begin{aligned} T_{i}(X_{i})=-\frac{\ln \big (1-\alpha _{i}U_{i}(X_{i})\big )}{\alpha _{i}}, \end{aligned}$$

where $X_{i}$ is the profit of i. Note that for two possible profits of newsvendor i, say, $X_{i}$ and $X_{i}'$, we know $T_{i}(X_{i})\ge T_{i}(X_{i}')$ if and only if $U_{i}(X_{i})\ge U_{i}(X_{i}')$.

Let $(X_{i})_{i\in N}$ be a Pareto-optimal risk sharing scheme of N and $t_{i}=T_{i}(X_{i})$, $i\in N$. Then, by Theorem 1, there exist $z_{i}\in \mathbb {R}$, $i\in N$, with $\sum _{i\in N}z_{i}=0$ such that

$$\begin{aligned} X_{i}=-\frac{1}{\alpha _{i}} \ln E\big [\exp (-\alpha _{N}\varPi (N, q_{N}^{\star }))\big ]+z_{i},\quad i\in N. \end{aligned}$$

Therefore,

$$\begin{aligned} \sum \limits _{i\in N} T_{i}(X_{i})=-\frac{1}{\alpha _{N}}\ln E\big [\exp (-\alpha _{N}\varPi (N, q_{N}^{\star }))\big ]. \end{aligned}$$

Suppose that (10) is not true, i.e., that $\sum \limits _{i\in S} t_{i}< -\frac{1}{\alpha _{S}}\ln E[\exp (-\alpha _{S} \varPi (S, q_{S}^{\star }))]$ holds for some $\emptyset \ne S\subseteq N$. Then, from Lemma 4 of (Baton and Lemaire, 1981), there exist $t_{i}'\in \mathbb {R}$, $i\in S$, such that $t_{i}<t_{i}'$ and

$$\begin{aligned} \sum \limits _{i\in S}t_{i}'= -\frac{1}{\alpha _{S}}\ln E[\exp (-\alpha _{S} \varPi (S, q_{S}^{\star }))]. \end{aligned}$$

For $i\in S$, we define

$$\begin{aligned} z_{i}'=t_{i}'+\frac{1}{\alpha _{i}} \ln E\big [\exp (-\alpha _{S}\varPi (S, q_{S}^{\star }))\big ] \end{aligned}$$

and

$$\begin{aligned} X_{i}'=\frac{\alpha _{S}}{\alpha _{i}} \varPi (S, q_{S}^{\star })+z_{i}'. \end{aligned}$$

Then, we have $T_{i}(X_{i}')=t_{i}'>t_{i}=T_{i}(X_{i})$ for $i\in S$. Consequently, $U_{i}(X_{i}')>U_{i}(X_{i})$ for $i\in S$. This means $(X_{i})_{i\in N}$ is not Pareto-optimal, which leads to a contradiction. This completes the proof of sufficiency.

Now, let $t_{i}$, $i\in N$, be real numbers that satisfy (9) and (10). For $i\in N$, define

$$\begin{aligned} X_{i}=\frac{\alpha _{N}}{\alpha _{i}}\varPi (N, q_{N}^{\star })+z_{i} \end{aligned}$$

and

$$\begin{aligned} z_{i}=t_{i}+\frac{1}{\alpha _{i}} \ln E\big [\exp (-\alpha _{N}\varPi (N, q_{N}^{\star }))\big ]. \end{aligned}$$

Then, $(X_{i})_{i\in N}$ is a Pareto-optimal risk sharing scheme of N. Suppose that $(X_{i})_{i\in N}$ is not stable, then there is a coalition $S\subseteq N$ such that $(X_{i}')_{i\in S}$ is a Pareto-optimal risk sharing scheme of S that dominates $(X_{i})_{i\in N}$, i.e., $U_{i}(X_{i}')>U_{i}(X_{i})$ for $i\in S$. Consequently, $T_{i}(X_{i}')>T_{i}(X_{i})$ for $i\in S$ and $\sum \limits _{i\in S} T_{i}(X_{i}')>\sum \limits _{i\in S} T_{i}(X_{i})$. Since $(X_{i}')_{i\in S}$ is a Pareto-optimal risk sharing scheme of S, we have

$$\begin{aligned} \sum \limits _{i\in S} T_{i}(X_{i}')=-\frac{1}{\alpha _{S}}\ln E[\exp (\alpha _{S} \varPi (S, q_{S}^{\star }))]. \end{aligned}$$

By calculation, we obtain $T_{i}(X_{i})=t_{i}$ for $i\in S$. Therefore,

$$\begin{aligned} -\frac{1}{\alpha _{S}}\ln E[\exp (-\alpha _{S} \varPi (S, q_{S}^{\star }))]>\sum \limits _{i\in S} t_{i}, \end{aligned}$$

which leads to a contradiction. Therefore, $(X_{i})_{i\in N}$ must be stable. This completes the proof of necessity. $\square$

Proof of Theorem 2

From Lemma 2, the core is nonempty if and only if the following system has a feasible solution:

$$\begin{aligned} {\left\{ \begin{array}{ll} \sum \limits _{i\in N} t_{i}=-\frac{1}{\alpha _{N}}\ln E\big [\exp (-\alpha _{N}\varPi (N, q_{N}^{\star }))\big ],\\ \sum \limits _{i\in S} t_{i}\ge -\frac{1}{\alpha _{S}} \ln E\big [\exp (-\alpha _{S}\varPi (S, q_{S}^{\star }))\big ],\quad \forall ~\emptyset \ne S\subseteq N, \end{array}\right. } \end{aligned}$$

(17)

where $t_{i}\in \mathbb {R}$, for $i\in N$. From the Bondareva–Shapley Theorem (Bondareva 1963; Shapley 1967) , (17) has a feasible solution if and only if

$$\begin{aligned} \sum \limits _{\emptyset \ne S\subseteq N} \lambda (S)\Big (-\frac{1}{\alpha _{S}}\ln E\big [\exp (-\alpha _{S}\varPi (S, q_{S}^{\star }))\big ]\Big )\le -\frac{1}{\alpha _{N}}\ln E\big [\exp (-\alpha _{N}\varPi (N, q_{N}^{\star }))\big ].\nonumber \\ \end{aligned}$$

(18)

for all possible functions $\lambda {:}\,2^{N}\setminus \emptyset \rightarrow \mathbb {R}_{+}$ with $\sum \limits _{\begin{array}{c} S: i\in S, \emptyset \ne S\subseteq N \end{array}}\lambda (S)=1$, for $i\in N$, where $\mathbb {R}_{+}$ is the set of nonnegative real numbers.

To verify (18), we need the inequality

$$\begin{aligned} \sum \limits _{i=1}^{m}t_{i}\ln E \big [\exp (Z_{i})\big ]\ge \ln E \Bigg [\exp \Bigg (\sum \limits _{i=1}^{m}t_{i}Z_{i}\Bigg )\Bigg ], \end{aligned}$$

where $t_{i}>0$, $\sum \limits _{i=1}^{m}t_{i}=1$, m is a positive integer, and $Z_{i}$ is a random variable for each $1\le i\le m$. Note that

$$\begin{aligned} \sum \limits _{i=1}^{m}t_{i}\ln \big (E \big [\exp (Z_{i})\big ]\big )= & {} \ln \big (\varPi _{i=1}^{m}(E \big [\exp (Z_{i})\big ])^{t_{i}}\big )\\= & {} \ln \big (\varPi _{i=1}^{m} E \big [\exp (\frac{1}{t_{i}}t_{i}Z_{i})\big ]^{t_{i}}\big )\\\ge & {} \ln \big (E \big [\varPi _{i=1}^{m} \exp (t_{i}Z_{i}) \big ] \big )~~(\textit{H}\ddot{o}{} \textit{lder Inequality})\\= & {} \ln \big (E \big [ \exp (\sum \limits _{i=1}^{m}t_{i}Z_{i} ) \big ] \big ) \end{aligned}$$

Consequently,

$$\begin{aligned}&\sum \limits _{\emptyset \ne S\subseteq N}\lambda (S)\Big (-\frac{1}{\alpha _{S}}\ln E\big [\exp (-\alpha _{S}\varPi (S, q_{S}^{\star }))\big ]\Big )\\&\quad =\sum \limits _{\emptyset \ne S\subseteq N}\lambda (S)\sum \limits _{i\in S}\Big (-\frac{1}{\alpha _{i}}\Big ) \ln E\big [\exp (-\alpha _{S}\varPi (S, q_{S}^{\star }))\big ]\\&\quad =\sum \limits _{i\in N}\sum \limits _{\emptyset \ne S\subseteq N: i\in S} \lambda _{S}\Big (-\frac{1}{\alpha _{i}}\Big ) \ln E\big [\exp (-\alpha _{S}\varPi (S, q_{S}^{\star }))\big ]\\&\quad =\sum \limits _{i\in N}\Big (-\frac{1}{\alpha _{i}}\Big )\sum \limits _{\emptyset \ne S\subseteq N: i\in S} \lambda _{S} \ln E\Bigg [\exp \Bigg (-\alpha _{S}\varPi (S, q_{S}^{\star })\Bigg )\Bigg ]\\&\quad \le \sum \limits _{i\in N}\Big (-\frac{1}{\alpha _{i}}\Big ) \cdot \ln E\Bigg [\exp \Bigg (\sum \limits _{\emptyset \ne S\subseteq N: i\in S}-\alpha _{S}\varPi (S, q_{S}^{\star })\lambda _{S}\Bigg )\Bigg ]\\&\quad =-\frac{1}{\alpha _{N}}\sum \limits _{i\in N} \frac{\alpha _{N}}{\alpha _{i}} \ln E\Bigg [\exp \Bigg (-\sum \limits _{\emptyset \ne S\subseteq N: i\in S} \alpha _{S}\lambda _{S}\varPi (S, q_{S}^{\star })\Bigg )\Bigg ]\\&\quad \le -\frac{1}{\alpha _{N}}\ln E\Bigg [\exp \Bigg (-\sum \limits _{i\in N} \frac{\alpha _{N}}{\alpha _{i}}\cdot \sum \limits _{\emptyset \ne S\subseteq N: i\in S} \alpha _{S}\lambda _{S}\varPi (S, q_{S}^{\star })\Bigg )\Bigg ]\\&\quad =-\frac{1}{\alpha _{N}} \ln E\Bigg [\exp \Bigg (-\alpha _{N}\sum \limits _{\emptyset \ne S\subseteq N}\sum \limits _{i\in S}\lambda _{S}\varPi (S, q_{S}^{\star })\Bigg )\Bigg ]\\&\quad \le -\frac{1}{\alpha _{N}} \ln E\big [\exp (-\alpha _{N}\varPi (N, q_{N}^{\star }))\big ]. \end{aligned}$$

This finishes the proof. $\square$

Proof of Theorem 3

Suppose $(X_{i})_{i\in S}$ is a Pareto-optimal risk sharing scheme of S. Then there exists an order $q\ge 0$ such that $\sum _{i\in S} X_{i}=\varPi (S, q)$. From Sect. 3.1, there exist $a_{i}>0$, $i\in S$ with $\sum _{i\in S} a_{i}=1$ such that $(X_{i})_{i\in S}$ is the solution of (4) and (5). From Gerber and Pafum (1998), for the power utilities (11), we have

$$\begin{aligned} X_{i}=\frac{\frac{s_{i}}{a_{i}^{1/b}}}{\sum _{j\in S}\frac{s_{j}}{a_{j}^{1/b}}} \varPi (S, q)+s_{i}-\frac{\frac{s_{i}}{a_{i}^{1/b}}}{\sum _{j\in S}\frac{s_{j}}{a_{j}^{1/b}}}\sum _{j=1}^{n}s_{j}, \quad \textit{~for~}i\in S. \end{aligned}$$

From (11), when newsvendor $i\in S$ has a profit $X_{i}$, his maximum expected utility is reached at

$$\begin{aligned} \arg \max _{q\ge 0} -E\big [\sum _{j\in S}s_{j}-\varPi (S, q)\big ]^{b+1}. \end{aligned}$$

Since we do not assume a specific distribution for the demand of each newsvendor, there is no explicit expression for $\arg \max \limits _{q\ge 0} -E\big [\sum _{j\in S}s_{j}-\varPi (S, q)\big ]^{b+1}$. In Lemma 1, let $v(x)=-(\sum _{i\in S}s_{i}-x)^{b+1}$, for $x<\sum _{i\in S}s_{i}$. Then we can see that $-E\big [\sum _{i\in S}s_{i}-\varPi (S, q)\big ]^{b+1}$ is a strictly concave function of q, for $q\ge 0$. Therefore,

$$\begin{aligned} \arg \max _{q\ge 0}-E\big [\sum _{j\in S}s_{j}-\varPi (S, q)\big ]^{b+1} \end{aligned}$$

is unique.

Moreover,

$$\begin{aligned} \arg \max _{q\ge 0}-E\Bigg [\sum _{j\in S}s_{j}-\varPi (S, q)\Bigg ]^{b+1}= & {} \arg \max _{q\ge 0}E\Bigg [-\Bigg (\sum _{j\in S}s_{j}-\varPi (S, q)\Bigg )^{b+1}\Bigg ]\\= & {} \arg \max \limits _{q\ge 0} E\Bigg [{s_{i}^{b+1}-\Bigg (\sum _{i\in S}s_{i}-\varPi (S, q)\Bigg )^{b+1}}\Bigg ]\\= & {} \arg \max \limits _{q\ge 0} E\Bigg [\frac{s_{i}^{b+1}-\big (\sum _{i\in S}s_{i}-\varPi (S, q)\big )^{b+1}}{(b+1)s_{i}^{b}}\Bigg ]. \end{aligned}$$

Therefore, every newsvendor $i\in S$ obtains his maximum expected utility at the same order $q_{S}^{\star \star }$ given by (12). Since $(X_{i})_{i\in S}$ is a Pareto-optimal risk sharing scheme of coalition S, we have $q=q^{\star \star }$ and

$$\begin{aligned} X_{i}=\frac{\frac{s_{i}}{a_{i}^{1/b}}}{\sum _{j\in S}\frac{s_{j}}{a_{j}^{1/b}}} \varPi (S, q_{S}^{\star \star })+s_{i}-\frac{\frac{s_{i}}{a_{i}^{1/b}}}{\sum _{j\in S}\frac{s_{j}}{a_{j}^{1/b}}}\sum _{j=1}^{n}s_{j}, \quad \textit{~for~}i\in S. \end{aligned}$$

For each $i\in S$, let $z_{i}=\frac{\frac{s_{i}}{a_{i}^{1/b}}}{\sum _{j\in S}\frac{s_{j}}{a_{j}^{1/b}}}$, for $i\in S$. It is obvious that $\sum _{i\in S} z_{i}=1$. Hence $(X_{i})_{i\in S}$ is an element of (13).

Suppose $(X_{i})_{i\in S}$ is an element of (13), i.e., there exist $z_{i}>0$, $i\in S$, with $\sum _{i\in S} z_{i}=1$ such that

$$\begin{aligned} X_{i}=z_{i}\varPi (S, q_{S}^{\star \star })+s_{i}-z_{i}\sum _{j\in S}s_{j}, \quad ~\textit{for}~i\in S. \end{aligned}$$

It is easy to verify that there exist $a_{i}>0$, $i\in S$, with $\sum _{i\in S} a_{i}=1$ such that $z_{i}=\frac{\frac{s_{i}}{a_{i}^{1/b}}}{\sum _{j\in S}\frac{s_{j}}{a_{j}^{1/b}}}$, for $i\in S$. From Sect. 3.1, corresponding to $a_{i}>0$, $i\in S$, there is a Pareto-optimal risk sharing scheme $(X_{i}')_{i\in S}$, where

$$\begin{aligned} X_{i}'=\frac{\frac{s_{i}}{a_{i}^{1/b}}}{\sum _{j\in S}\frac{s_{j}}{a_{j}^{1/b}}} \varPi (S, q)+s_{i}-\frac{\frac{s_{i}}{a_{i}^{1/b}}}{\sum _{j\in S}\frac{s_{j}}{a_{j}^{1/b}}}\sum _{j=1}^{n}s_{j}, \end{aligned}$$

for $i\in S$ and some $q\ge 0$. We can verify that $q=q_{S}^{\star \star }$. Hence $X_{i}=X_{i}'$, for $i\in S$, and $(X_{i})_{i\in S}$ is a Pareto-optimal risk sharing scheme of S.

Consequently, the set of Pareto-optimal risk sharing schemes of S is given by (13). Moreover, the Pareto-optimal order of S is given by (12). $\square$

Proof of Lemma 3

For newsvendor $i\in N$, let

$$\begin{aligned} T_{i}(X_{i})=-\big (s_{i}^{b+1}-(b+1)s_{i}^{b}U(X_{i})\big )^{\frac{1}{b+1}}, \end{aligned}$$

where $X_{i}$ is the profit of i. Note that for two possible profits of newsvendor i, $X_{i}$ and $X_{i}'$, we know $T_{i}(X_{i})\ge T_{i}(X_{i}')$ if and only if $U_{i}(X_{i})\ge U_{i}(X_{i}')$.

Let $(X_{i})_{i\in N}$ be a Pareto-optimal risk sharing scheme in N and let $t_{i}=T_{i}(X_{i})$, for $i\in N$. Then, by Theorem 3, there exist $z_{i}>0$, $i\in N$, with $\sum _{i\in N}z_{i}=1$ such that

$$\begin{aligned} X_{i}=z_{i}\varPi (N, q_{N}^{\star \star })+s_{i}-z_{i}\sum _{j\in N}s_{j},\quad i\in N. \end{aligned}$$

Consequently, we have

$$\begin{aligned} \sum \limits _{i\in N} T_{i}(X_{i})=-\Bigg (E\Bigg [\sum _{i\in N}s_{i}-\varPi (N, q_{N}^{\star \star })\Bigg ]^{b+1}\Bigg )^{\frac{1}{b+1}}. \end{aligned}$$

Suppose (15) is not true, i.e., that $\sum \nolimits _{i\in S} t_{i}< -\big (E\big [\sum _{i\in S}s_{i}-\varPi (S, q_{S}^{\star \star })\big ]^{b+1}\big )^{\frac{1}{b+1}}$ holds for some $\emptyset \ne S\subseteq N$. Then, from Lemma 4 of (Baton and Lemaire, 1981), there exist $t_{i}'\in \mathbb {R}$, $i\in S$, such that $t_{i}<t_{i}'$ for $i\in S$ and

$$\begin{aligned} \sum \limits _{i\in S}t_{i}'= -\Bigg (E\Bigg [\sum _{i\in S}s_{i}-\varPi (S, q_{S}^{\star \star })\Bigg ]^{b+1}\Bigg )^{\frac{1}{b+1}}. \end{aligned}$$

For $i\in S$, define

$$\begin{aligned} z_{i}'=-\frac{t_{i}'}{\big (E\big [\sum _{i\in N}s_{i}-\varPi (N, q_{N}^{\star \star })\big ]^{b+1}\big )^{\frac{1}{b+1}}} \end{aligned}$$

and

$$\begin{aligned} X_{i}'=z_{i}'\varPi (S,q_{S}^{\star \star })+s_{i}-z_{i}'\sum _{j\in S}s_{j}. \end{aligned}$$

Therefore, $T_{i}(X_{i}')=t_{i}'>t_{i}=T_{i}(X_{i})$, for $i\in S$. Consequently, $U_{i}(X_{i}')>U_{i}(X_{i})$, for $i\in S$, and $(X_{i})_{i\in N}$ is not a Pareto-optimal risk sharing scheme, which leads to a contradiction. This finishes the proof of sufficiency.

Let $t_{i}$, $i\in N$, be real numbers that satisfy (14) and (15). For $i\in N$, define

$$\begin{aligned} z_{i}=-\frac{t_{i}}{\big (E\big [\sum _{i\in N}s_{i}-\varPi (N, q_{N}^{\star \star })\big ]^{b+1}\big )^{\frac{1}{b+1}}} \end{aligned}$$

and

$$\begin{aligned} X_{i}=z_{i}\varPi (N,q_{N}^{\star \star })+s_{i}-z_{i}\sum _{j\in N}s_{j}. \end{aligned}$$

Then, $(X_{i})_{i\in N}$ is a Pareto-optimal risk sharing scheme. Suppose that $(X_{i})_{i\in N}$ is not stable, then there is a coalition $S\subseteq N$ such that $(X_{i}')_{i\in S}$ is a Pareto-optimal risk sharing scheme of S and $(X_{i}')_{i\in S}$ dominates $(X_{i})_{i\in N}$, i.e., $U_{i}(X_{i}')>U_{i}(X_{i})$, for $i\in S$. Consequently, $T_{i}(X_{i}')>T_{i}(X_{i})$, for $i\in S$, and $\sum \limits _{i\in S} T_{i}(X_{i}')>\sum \limits _{i\in S} T_{i}(X_{i})$. Since $(X_{i}')_{i\in S}$ is a Pareto-optimal risk sharing scheme in S, we have

$$\begin{aligned} \sum \limits _{i\in S} T_{i}(X_{i}')=-\Bigg (E\Bigg [\sum _{i\in S}s_{i}-\varPi (S, q_{S}^{\star \star })\Bigg ]^{b+1}\Bigg )^{\frac{1}{b+1}}. \end{aligned}$$

By calculation, we have $T_{i}(X_{i})=t_{i}$, for $i\in S$. Therefore,

$$\begin{aligned} -\Bigg (E\Bigg [\sum _{i\in S}s_{i}-\varPi (S, q_{S}^{\star \star })\Bigg ]^{b+1}\Bigg )^{\frac{1}{b+1}}>\sum \limits _{i\in S} t_{i}, \end{aligned}$$

which leads to a contradiction. Hence $(X_{i})_{i\in N}$ must be stable. This finishes the proof of necessity. $\square$

Proof of Theorem 3

From Lemma 3, $\mathcal {C}\big (N, \varPi (\cdot , \cdot , \cdot ), \varvec{u}(\cdot )\big )\ne \emptyset$ if and only if the following system has a solution:

$$\begin{aligned} {\left\{ \begin{array}{ll} \sum \limits _{i\in N} t_{i}=-\Bigg (E\Bigg [\sum _{i\in N}s_{i}-\varPi (S, q_{N}^{\star \star })\Bigg ]^{b+1}\Bigg )^{\frac{1}{b+1}},\\ \sum \limits _{i\in S} t_{i}\ge -\Bigg (E\Bigg [\sum _{i\in S}s_{i}-\varPi (N, q_{S}^{\star \star })\Bigg ]^{b+1}\Bigg )^{\frac{1}{b+1}},\quad \forall ~\emptyset \ne S\subseteq N, \end{array}\right. } \end{aligned}$$

(19)

where $t_{i}<0$ $i\in N$, are variables.

From the Bondareva–Shapley Theorem, system (19) has a solution if and only if

$$\begin{aligned} \sum \limits _{\emptyset \ne S\subseteq N} \lambda (S) \Bigg (E\Bigg [\sum _{i\in S}s_{i}-\varPi (S, q_{S}^{\star \star })\Bigg ]^{b+1}\Bigg )^{\frac{1}{b+1}}{\ge } \Bigg (E\Bigg [\sum _{i\in N}s_{i}-\varPi (N, q_{N}^{\star \star })\Bigg ]^{b+1}\Bigg )^{\frac{1}{b+1}},\qquad \end{aligned}$$

(20)

for all possible functions $\lambda {:}\,2^{N}\setminus \emptyset \rightarrow \mathbb {R}_{+}$ with $\sum \nolimits _{\begin{array}{c} S: i\in S, \emptyset \ne S\subseteq N \end{array}}\lambda (S)=1$, for $i\in N$.

Using the Minkowski Inequality, we have

$$\begin{aligned}&\sum \limits _{\emptyset \ne S\subseteq N} \lambda (S) \Bigg (E\Bigg [\sum _{i\in S}s_{i}-\varPi (S, q_{S}^{\star \star })\Bigg ]^{b+1}\Bigg )^{\frac{1}{b+1}}\\&\quad =\sum \limits _{\emptyset \ne S\subseteq N} \Bigg (E\Bigg [\lambda (S)\sum _{i\in S}s_{i}-\lambda (S)\varPi (N, q_{N}^{\star \star })\Bigg ]^{b+1}\Bigg )^{\frac{1}{b+1}}\\&\quad \ge \Bigg (E\sum \limits _{\emptyset \ne S\subseteq N} \Bigg [\lambda (S)\sum _{i\in S}s_{i}-\lambda (S)\varPi (S, q_{S}^{\star \star })\Bigg ]^{b+1}\Bigg )^{\frac{1}{b+1}}\\&\quad =\Bigg (E\Bigg [\sum _{i\in N}s_{i}-\sum \limits _{\emptyset \ne S\subseteq N}\lambda (S)\varPi (S, q_{S}^{\star \star })\Bigg ]^{b+1}\Bigg )^{\frac{1}{b+1}}\\&\quad \ge \Bigg (E\Bigg [\sum _{i\in N}s_{i}-\varPi (N, \sum \limits _{\emptyset \ne S\subseteq N}\lambda (S)q_{S}^{\star \star })\Bigg ]^{b+1}\Bigg )^{\frac{1}{b+1}}\\&\quad \ge \Bigg (E\Bigg [\sum _{i\in N}s_{i}-\varPi (N, q_{N}^{\star \star })\Bigg ]^{b+1}\Bigg )^{\frac{1}{b+1}}. \end{aligned}$$

Thus inequality (20) is satisfied and the proof is complete. $\square$