Chain-to-Chain Competition Under Demand Uncertainty

Abstract

In this paper, we aim to study the structure choice of supply chains under competitive environment with uncertain demand. We consider two competing supply chains, each of which chooses to either vertically integrate or decentralize with coordinating contracts. We first analyze firms’ strategic behavior under given supply chain structures: two integrated chains (II), two decentralized chains (DD), and a mixed structure with one decentralized chain and one integrated chain. We then compare different supply chain structures and examine the equilibrium structure choice. We find that the equilibrium structure depends on the product characteristics. For substitutable products, DD is the equilibrium supply chain structure choice, whereas for complementary products, II is the equilibrium structure. Furthermore, a high demand uncertainty strengthens these equilibrium choices.

Appendix: Proofs

Proof of Theorem 3.1

The expected excess inventory at retailer (i, j) can be expressed as

$$\begin{aligned} \mathrm{E}[(q_{ij}-D_{ij}({\varvec{q}}))^+]&= \int _0^{q_{ij}} \mathsf{P}[D_{ij}({\varvec{q}}) \leqslant u] \mathrm{d}u \nonumber \\&=\int _0^{q_{ij}} F_{i}\Big (u-\alpha _i q_{ij} + \beta _i q_{i,-j}+ \frac{1}{2}\gamma _{-i} (q_{-i,1}+q_{-i,2})\Big )\, \mathrm{d}u \nonumber \\&= \int _0^{F_i^{-1}(S_{ij})} F_i(u) \mathrm{d}u. \end{aligned}$$

(A.1)

The industry’s profit function in (3.1) can be written as

$$\begin{aligned}&\sum _{i,j} \bigg [(p_i-c_i)q_{ij} - (p_i-v_i) \int _0^{q_{ij}} F_{i}\Big (u-\alpha _i q_{ij} + \beta _i q_{i,-j}+ \frac{1}{2}\gamma _{-i} (q_{-i,1}+q_{-i,2})\Big ) \mathrm{d}u \bigg ]. \end{aligned}$$

The discussion after (3.1) reveals the concavity of the objective function. Hence, the optimal inventory levels are given by the first-order conditions:

$$\begin{aligned} p_i-c_i&= (p_i-v_i) \big ((1-\alpha _i)S_{ij} + \beta _i S_{i,-j}\big )\\&\quad +\frac{1}{2}(p_{-i}-v_{-i})\gamma _i (S_{-i,j}+S_{-i,-j}), \quad i,j=1,2, \end{aligned}$$

where we use the service level defined in (2.3). This system of equations determines the optimal service levels in (3.2), which in turn determine the optimal inventory in (3.3) by using the relation in (2.5). The optimal inventory and service level and the expression in (A.1) lead to the profit in (3.4).

Next, we prove the monotonicity properties. The optimal service level in (3.2) can be written as

$$\begin{aligned} S_i^*&= \frac{(1-\alpha _{-i}+\beta _{-i}) (p_i-c_i) - \gamma _i(p_{-i}-c_{-i})}{(p_i-v_i) ((1-\alpha _1+\beta _1)(1-\alpha _2+\beta _2)-\gamma _1\gamma _2) }\\&= \frac{(1-\alpha _{-i}+\beta _{-i}) (p_i-c_i) - \gamma _i(p_{-i}-c_{-i})}{(p_i-v_i) \eta }. \end{aligned}$$

We can derive

$$\begin{aligned} \frac{\partial S_{i}^*}{\partial (\alpha _i-\beta _i)}&\geqslant 0, \\ \frac{\partial S_{-i}^*}{\partial (\alpha _i-\beta _i)}&= \frac{(p_{-i}-c_{-i})\gamma _1\gamma _2 -(1-\alpha _{-i}+\beta _{-i})\gamma _{-i}(p_i-c_i) }{(p_{-i}-v_{-i})\eta ^2}\ =\ \frac{-\gamma _{-i}(p_i-v_i)\gamma _i}{(p_{-i}-v_{-i})\eta }. \end{aligned}$$

Thus $S_{-i}^*$ decreases (increases) in $\alpha _i-\beta _i$ if $\gamma >0$ $(<0)$. Consequently, $\gamma _{-i} F^{-1}_{-i}(S_{-i}^*)$ decreases in $\alpha _i-\beta _i$.

Therefore, in Eq. (3.3), when $\alpha _i-\beta _i$ increases, the denominator decreases while the numerator increases. Thus $q_i^*$ increases in $\alpha _i-\beta _i$.

To examine the relation between $q^*_i$ and $\alpha _{-i}-\beta _{-i}$, we derive

$$\begin{aligned}&\quad \frac{\partial q^*_i}{\partial (\alpha _{-i}-\beta _{-i})} \nonumber \\&=-\frac{\gamma _{-i}}{\eta }\bigg [ q_{-i}^* -\frac{1-\alpha _{-i}+\beta _{-i}}{\gamma _{-i}} \frac{\partial F_i^{-1}}{\partial S_i} \frac{\partial S_i^*}{\partial (\alpha _{-i}-\beta _{-i})}\nonumber \\&\quad + \frac{\partial F_{-i}^{-1}}{\partial S_{-i}} \frac{\partial S_{-i}}{\partial (\alpha _{-i}-\beta _{-i})} \bigg ]. \end{aligned}$$

(A.2)

All three terms in the brackets are positive, and thus, the sign of the above derivative only depends on the sign of $\gamma$ in front of the brackets.

In equilibrium,

$$\begin{aligned} D_{ij}({\varvec{q}}^*)&= \xi _{ij} + (\alpha _i- \beta _i) q^*_i - \gamma _{-i} q^*_{-i}. \end{aligned}$$

Suppose $\alpha _i-\beta _i$ increases or $\gamma _i$ decreases, and suppose the industry keeps the inventory levels ${\varvec{q}}^*$ unchanged. Then, the above relation suggests that demand $D_{ij}$ will increase, and the industry’s profit increases. Therefore, if the industry optimally chooses inventory levels in response to the changes in $\alpha _i-\beta _i$ and $\gamma _i$, $\pi ^*$ will increase further.

Proof of Theorem 3.2

The problem in (3.5) can be written as

$$\begin{aligned}&\max _{q_{i1},q_{i2}}\ \sum _j \bigg [ (p_i-c_i)q_{ij} - (p_i-v_i)\nonumber \\&\quad \int _0^{q_{ij}} F_{i}\Big (u-\alpha _i q_{ij} + \beta _i q_{i,-j}+ \frac{1}{2}\gamma _{-i} (q_{-i,1}+q_{-i,2})\Big ) \hbox {d}u \bigg ]. \end{aligned}$$

The first-order condition is

$$\begin{aligned} p_i-c_i&= (p_i-v_i) \big ((1-\alpha _i)S_{ij} + \beta _i S_{i,-j} \big ), \qquad j=1,2. \end{aligned}$$

Solving the this system of equations, we obtain the optimal service level in (3.6).

Using (2.5), the equilibrium inventory levels are given by (3.7). The expression for chain i’s profit in (3.8) can be derived in the same way as in the proof of Theorem 3.1.

We next prove the monotonicity. If $\alpha _i-\beta _i$ increases, then $S_i^{\mathrm{II}}$ in (3.6) increases, which also increases $F_i^{-1}(S_i^{\mathrm{II}})$ in (3.7). At the same time, a higher $\alpha _i-\beta _i$ reduces $\eta$. Hence, $q_i^{\mathrm{II}}$ increases in $\alpha _i-\beta _i$.

Differentiating $q_i^{\mathrm{II}}$ with respect to $\alpha _{-i}-\beta _{-i}$ and after some algebraic manipulations, we obtain

$$\begin{aligned} \frac{\partial q_i^{\mathrm{II}}}{\partial (\alpha _{-i}-\beta _{-i})}&= - \frac{\gamma _{-i}}{\eta } \bigg [ \frac{\partial F_{-i}^{-1}}{\partial S_{-i}} \frac{\partial S^{\mathrm{II}}_{-i}}{\partial (\alpha _{-i}-\beta _{-i})} + q_{-i}^{\mathrm{II}}\bigg ]. \end{aligned}$$

(A.3)

All terms in the square bracket are positive, and therefore, if $\gamma >0$ $(\gamma <0)$, $q_i^{\mathrm{II}}$ decreases (increases) in $\alpha _{-i}-\beta _{-i}$.

From the profit expression in (3.8), we see that $\alpha _{-i}-\beta _{-i}$ affects $\pi _i^{\mathrm{II}}$ only through $q_i^{\mathrm{II}}$. Thus, if $\gamma >0$ $(\gamma <0)$, $\pi _i^{\mathrm{II}}$ decreases (increases) in $\alpha _{-i}-\beta _{-i}$.

Finally, we prove the monotonicity of $\pi _i^{\mathrm{II}}$ in $\alpha _i-\beta _i$. Let $\alpha _i-\beta _i<\widetilde{\alpha }_i-\widetilde{\beta }_i$. Under $\alpha _i-\beta _i$, using (2.3), the equilibrium satisfies

$$\begin{aligned} F_i\big (\big (1-\alpha _i+\beta _i\big ) q_i^{\mathrm{II}}+ \gamma _{-i} q_{-i}^{\mathrm{II}}\big ) = S^{\mathrm{II}}_i. \end{aligned}$$

Under $\widetilde{\alpha }_i-\widetilde{\beta }_i$, denote the new equilibrium inventory as $\widetilde{q}_1^{\;{\mathrm{II}}}$ and $\widetilde{q}_2^{\;{\mathrm{II}}}$. Suppose chain $-i$ choose the equilibrium inventory $\widetilde{q}_{-i}^{\;{\mathrm{II}}}$, but chain i takes a suboptimal strategy $q_i^s$ to maintain its original service level under $\alpha _i-\beta _i$. That is,

$$\begin{aligned} F_i\big ((1-\widetilde{\alpha }_i+\widetilde{\beta }_i)q_i^s + \gamma _{-i}\widetilde{q}_{-i}^{\;{\mathrm{II}}}\big ) = S^{\mathrm{II}}_i. \end{aligned}$$

As we show by (A.3), if $\gamma >0$ $(\gamma <0)$, $q_{-i}^{\mathrm{II}}$ decreases (increases) in $\alpha _i-\beta _i$. In either case, $\gamma q_{-i}^{\mathrm{II}}$ decreases in $\alpha _i-\beta _i$. Hence, $\gamma _{-i} q_{-i}^{\mathrm{II}}> \gamma _{-i} \widetilde{q}_{-i}^{\;{\mathrm{II}}}$. Together with $1-\alpha _i+\beta _i>1-\widetilde{\alpha }_i+\widetilde{\beta }_i>0$, we have

$$\begin{aligned} q_i^s > q_i^{\mathrm{II}}. \end{aligned}$$

Under this suboptimal inventory level $q_i^s$, the profit of chain i is

$$\begin{aligned} \pi _i&= 2(p_i-c_i)q_i^s - 2(p_i-v_i)\int _0^{F_i^{-1}(S^{\mathrm{II}}_i)}F_i(u)\mathrm{d}u \\&> 2(p_i-c_i)q_i^{\mathrm{II}}- 2(p_i-v_i)\int _0^{F_i^{-1}(S^{\mathrm{II}}_i)}F_i(u)\mathrm{d}u \ = \ \pi _i^{\mathrm{II}}. \end{aligned}$$

Since a suboptimal inventory for chain i yields a higher profit, the optimal profit $\pi _i^{\mathrm{II}}$ in the equilibrium must be even higher. This proves $\pi _i^{\mathrm{II}}$ increases in $\alpha _i-\beta _i$.

Proof of Theorem 3.3

To show the quasi-concavity of the objective function in (3.11), it suffices to show that the first-order derivative crosses zero value from above only once. The first-order derivative with respect to $S^c_i$ is

$$\begin{aligned} \bigg [\frac{2(p_i-c_i)(1-\alpha _{-i}+\beta _{-i})}{\eta } - 2(p_i-v_i) S^c_i \bigg ] \frac{\partial F^{-1}_i(S^c_i)}{\partial S^c_i}. \end{aligned}$$

(A.4)

Because $F_i^{-1}(y)$ is assumed to be strictly increasing in y for $y\in (0,1)$, we have ${\partial F^{-1}_i}/{\partial S^c_i} > 0$. The term in the brackets is strictly decreasing in $S^c_i$. Hence, as $S^c_i$ increases, the derivative in (A.4) crosses zero value from above only once. Thus, the objective function in (3.11) is quasi-concave.

Service level $S_i^{\mathrm{DD}}$ in (3.13) clearly increases in $\alpha _1-\beta _1$ and $\alpha _2-\beta _2$.

If $\gamma <0$, then $q_i^{\mathrm{DD}}$ in (3.14) increases in $\alpha _i - \beta _i$, because as $\alpha _i - \beta _i$ increases, $\eta$ decreases, $F^{-1}_i(S_i^{\mathrm{DD}})$ increases, and $- \gamma _{-i} F^{-1}_{-i}(S_{-i}^{\mathrm{DD}})$ increases.

Similar to (A.2), we can derive

$$\begin{aligned} \frac{\partial q_i^{\mathrm{DD}}}{\partial (\alpha _{-i}-\beta _{-i})}&= -\frac{\gamma _{-i}}{\eta }\left[ q_{-i}^{\mathrm{DD}}+ \frac{\partial F_{-i}^{-1}}{\partial S_{-i}} \frac{\partial S_{-i}}{\partial (\alpha _{-i}-\beta _{-i})} \right] \\&\quad +\frac{1-\alpha _{-i}+\beta _{-i}}{\eta } \frac{\partial F_i^{-1}}{\partial S_i} \frac{\partial S_i^{\mathrm{DD}}}{\partial (\alpha _{-i}-\beta _{-i})}. \end{aligned}$$

Because $\gamma <0$ and $S_i^{\mathrm{DD}}$ increases in $\alpha _{-i}-\beta _{-i}$, the above derivative is positive, and thus $q_i^{\mathrm{DD}}$ increases in $\alpha _{-i}-\beta _{-i}$.

Finally, we prove the monotonicity of $\pi _i^{\mathrm{DD}}$ in $\alpha _i-\beta _i$ using the same method as in the proof of the monotonicity of $\pi _i^{\mathrm{II}}$ in $\alpha _i-\beta _i$ in Theorem 3.2. Let $\alpha _i-\beta _i<\widetilde{\alpha }_i-\widetilde{\beta }_i$. Under $\alpha _i-\beta _i$, the equilibrium satisfies $F_i\big ((1-\alpha _i+\beta _i) q_i^{\mathrm{DD}}+ \gamma _{-i} q_{-i}^{\mathrm{DD}}\big ) = S^{\mathrm{DD}}_i.$ Under $\widetilde{\alpha }_i-\widetilde{\beta }_i$, suppose chain $-i$ choose the equilibrium inventory $\widetilde{q}_{-i}^{\;{\mathrm{DD}}}$, but chain i takes a suboptimal strategy $q_i^s$ to maintain its original service level: $F_i\big ((1-\widetilde{\alpha }_i+\widetilde{\beta }_i)q_i^s + \gamma _{-i}\widetilde{q}_{-i}^{\;{\mathrm{DD}}}\big ) = S^{\mathrm{DD}}_i.$ As we have shown, if $\gamma <0$, $q_{-i}^{\mathrm{DD}}$ increases in $\alpha _i-\beta _i$. Hence, $\gamma _{-i} q_{-i}^{\mathrm{DD}}> \gamma _{-i} \widetilde{q}_{-i}^{\;{\mathrm{DD}}}$. Together with $1-\alpha _i+\beta _i>1-\widetilde{\alpha }_i+\widetilde{\beta }_i>0$, we have $q_i^s > q_i^{\mathrm{DD}}.$ Under this suboptimal inventory level $q_i^s$, the profit of chain i is

$$\begin{aligned} \pi _i&= 2(p_i-c_i)q_i^s - 2(p_i-v_i)\int _0^{F_i^{-1}(S^{\mathrm{DD}}_i)}F_i(u)\mathrm{d}u \\&> 2(p_i-c_i)q_i^{\mathrm{DD}}- 2(p_i-v_i)\int _0^{F_i^{-1}(S^{\mathrm{DD}}_i)}F_i(u)\mathrm{d}u \ = \ \pi _i^{\mathrm{DD}}. \end{aligned}$$

Since a suboptimal inventory for chain i yields a higher profit, the optimal profit $\pi _i^{\mathrm{DD}}$ in the equilibrium must be even higher. This proves $\pi _i^{\mathrm{DD}}$ increases in $\alpha _i-\beta _i$ when $\gamma <0$.

The monotonicity of $\pi _i^{\mathrm{DD}}$ in $\alpha _{-i}-\beta _{-i}$ can be shown using the same approach as above.

Proof of Theorem 3.4

The game dynamics is described and the equilibrium is proved in the paper. Here, we show the monotonicity results.

The effects of $\alpha _2-\beta _2$ on the equilibrium follow the same lines of the proof as in Theorem 3.2.

The effects of $\alpha _1-\beta _1$ on the equilibrium follow the same lines of the proof as in Theorem 3.3.

Proof of Corollary 4.1

Comparing $S_i^{\mathrm{II}}$ in (3.6) and $S_i^{\mathrm{DD}}$ in (3.13), and noting $\gamma _1\gamma _2>0$, we have the relation $S_i^{\mathrm{II}}< S_i^{\mathrm{DD}}$.

When $\gamma <0$, comparing $S_i^{\mathrm{DD}}$ in (3.13) and $S_i^*$ in (3.2), we obtain $S_i^{\mathrm{DD}}< S_i^*$.

When $\gamma >0$, we have

$$\begin{aligned} S_i^*-S_i^{\mathrm{II}}&= \frac{p_i-c_i-\frac{\gamma _i(p_{-i}-c_{-i})}{1-\alpha _{-i}+\beta _{-i}}}{(p_i-v_i)\big (1-\alpha _i+\beta _i-\frac{\gamma _1\gamma _2}{1-\alpha _{-i}+\beta _{-i}}\big )} - \frac{p_i-c_i}{(p_i-v_i)(1-\alpha _i+\beta _i)}\\&= \frac{(p_i-c_i)(1-\alpha _{-i}+\beta _{-i})-\gamma _i(p_{-i}-c_{-i}) }{(p_i-v_i)\eta }\\&\quad - \frac{ (p_i-c_i)(1-\alpha _{-i}+\beta _{-i})-\gamma _1\gamma _2 \frac{(p_i-c_i)}{(1-\alpha _i+\beta _i)} }{(p_i-v_i)\eta }\\&= -\frac{\gamma _i\Big [(p_{-i}-c_{-i})-\frac{\gamma _{-i}(p_i-c_i)}{1-\alpha _i+\beta _i}\Big ]}{(p_i-v_i)\eta }\\&= -\frac{\gamma _i S_{-i}^* }{1-\alpha _i+\beta _i} \ < \ 0. \end{aligned}$$

This proves $S_i^* < S_i^{\mathrm{II}}$ when $\gamma >0$.

Proof of Corollary 4.3

The equilibrium inventory levels in (3.7), (3.14), and (3.22) from Theorems 3.2–3.4 are listed below:

$$\begin{aligned} q_i^{\mathrm{II}}&= \frac{1}{\eta } \Big [ (1-\alpha _{-i}+\beta _{-i})F^{-1}_i\big (S^{\mathrm{II}}_i\big ) - \gamma _{-i} F^{-1}_{-i}\big (S^{\mathrm{II}}_{-i}\big ) \Big ], \\ q_i^{\mathrm{DD}}&= \frac{1}{\eta } \Big [(1-\alpha _{-i}+\beta _{-i})F^{-1}_i\big (S_i^{\mathrm{DD}}\big ) - \gamma _{-i} F^{-1}_{-i}\big (S_{-i}^{\mathrm{DD}}\big )\Big ],\\ q_1^{\mathrm{DI}}&= \frac{1}{\eta } \Big [(1-\alpha _2+\beta _2)F^{-1}_1\big (S_1^{\mathrm{II}}\big ) - \gamma _2 F^{-1}_2\big (S_2^{\mathrm{DD}}\big )\Big ],\\ q_2^{\mathrm{DI}}&= \frac{1}{\eta } \Big [(1-\alpha _1+\beta _1)F^{-1}_2\big (S_2^{\mathrm{DD}}\big ) - \gamma _1 F^{-1}_1\big (S_1^{\mathrm{II}}\big )\Big ]. \end{aligned}$$

When $\gamma \ne 0$, we have $S_i^{\mathrm{DD}}> S_i^{\mathrm{II}}$, $i=1,2$, which immediately leads to the order of $q_i^{\mathrm{II}}$, $q_i^{\mathrm{DI}}$, and $q_i^{\mathrm{DD}}$ in the theorem.

Next, we prove the order of profits. Theorems 3.2–3.4 imply that

$$\begin{aligned} \pi _1^{\mathrm{II}}- \pi _1^{\mathrm{DI}}&= 2(p_1-c_1)(q_1^{\mathrm{II}}-q_1^{\mathrm{DI}}),\\ \pi _2^{\mathrm{DD}}- \pi _2^{\mathrm{DI}}&= 2(p_2-c_2)(q_2^{\mathrm{DD}}-q_2^{\mathrm{DI}}). \end{aligned}$$

Thus, the order of $\pi _1^{\mathrm{II}}$ and $\pi _1^{\mathrm{DI}}$ is the same as that of $q_1^{\mathrm{II}}$ and $q_1^{\mathrm{DI}}$, and the order of $\pi _2^{\mathrm{DD}}$ and $\pi _2^{\mathrm{DI}}$ is the same as that of $q_2^{\mathrm{DD}}$ and $q_2^{\mathrm{DI}}$.

We next derive the order of $\pi _2^{\mathrm{DI}}$ and $\pi _2^{\mathrm{II}}$. Define an auxiliary function,

$$\begin{aligned} \psi (x)&\overset{{\mathrm{def}}}{=} 2(p_2-c_2)\frac{-\gamma _1F^{-1}_1\big (S_1^{\mathrm{II}}\big ) + (1-\alpha _1+\beta _1) x}{\eta } - 2(p_2-v_2)\int _0^x F_2(u)\mathrm{d}u. \end{aligned}$$

We have

$$\begin{aligned} \frac{\mathrm{d}\psi (x)}{\mathrm{d}x}&= \frac{2(p_2-c_2)(1-\alpha _1+\beta _1)}{\eta } - 2(p_2-v_2) F_2(x), \end{aligned}$$

(A.5)

$$\begin{aligned} \frac{\mathrm{d}\psi (F_2^{-1}(S_2^{\mathrm{DD}}))}{\mathrm{d}x}&= \frac{2(p_2-c_2)(1-\alpha _1+\beta _1)}{(1-\alpha _1+\beta _1)(1-\alpha _2+\beta _2)-\gamma _1\gamma _2} - \frac{2(p_2-c_2)}{(1-\alpha _2+\beta _2)-\frac{\gamma _1\gamma _2}{1-\alpha _1+\beta _1}}\nonumber \\&= 0. \end{aligned}$$

(A.6)

Equation (A.5) implies that $\mathrm{d}\psi (x)/\mathrm{d}x$ decreases in x, and it decreases to zero value when $x=F_2^{-1}(S_2^{\mathrm{DD}})$ in view of (A.6). Hence, $\mathrm{d}\psi (x)/\mathrm{d}x \geqslant 0$ for all $x\in [F_2^{-1}(S^{\mathrm{II}}_2), F_2^{-1}(S^{\mathrm{DD}}_2)]$, and therefore, $\psi (x)$ increases in x, which leads to

$$\begin{aligned} \pi _2^{\mathrm{II}}\ =\ \psi \big (F_2^{-1}\big (S^{\mathrm{II}}_2\big )\big )\ \leqslant \ \psi \big (F_2^{-1}\big (S^{\mathrm{DD}}_2\big )\big )\ =\ \pi _2^{\mathrm{DI}}. \end{aligned}$$

The order of $\pi _1^{\mathrm{DI}}$ and $\pi _1^{\mathrm{DD}}$ can be derived in the same logic with an auxiliary function:

$$\begin{aligned} \phi (x)&\overset{{\mathrm{def}}}{=} 2(p_1-c_1)\frac{(1-\alpha _2+\beta _2) x -\gamma _2F^{-1}_2\big (S^D_2\big )}{\eta } - 2(p_1-v_1)\int _0^x F_1(u)\mathrm{d}u. \end{aligned}$$

We can show that $\phi (x)$ increases in x for $\in \big [F_1^{-1}\big (S_1^{\mathrm{II}}\big ), F_1^{-1}\big (S_1^{\mathrm{DD}}\big )\big ]$, and therefore,

$$\begin{aligned} \pi _1^{\mathrm{DI}}\ =\ \phi \big (F_1^{-1}\big (S_1^{\mathrm{II}}\big )\big )\ \leqslant \ \phi \big (F_1^{-1}\big (S_1^{\mathrm{DD}}\big )\big )\ =\ \pi _1^{\mathrm{DD}}. \end{aligned}$$

Proof of Lemma 4.6

(a) For any $x \leqslant \widehat{x}$, if $G(x)=0$, then $F(x) \geqslant G(x)$ is already satisfied. If $0 < G(x) \leqslant G(\widehat{x})=\widehat{y} < 1$, then by definition, we have $F^{-1}(\widehat{y})-F^{-1}(G(x)) \geqslant G^{-1}(\widehat{y}) - G^{-1}(G(x))$, or equivalently $\widehat{x} - F^{-1}(G(x)) \geqslant \widehat{x} - x$, and thus $F^{-1}(G(x)) \leqslant x$ or $G(x) \leqslant F(x)$. Similarly, we can prove $F(x) \leqslant G(x)$ for all $x \geqslant \widehat{x}$.

For $y \leqslant \widehat{y}$, by definition, $F^{-1}(\widehat{y})-F^{-1}(y) \geqslant G^{-1}(\widehat{y}) - G^{-1}(y)$, which implies $F^{-1}(y) \leqslant G^{-1}(y)$. Similarly, for $y \geqslant \widehat{y}$, $F^{-1}(y) \geqslant G^{-1}(y)$.

(b) If there does not exist $\widehat{x}$ such that $F(\widehat{x}) = G(\widehat{x}) \in (0,1)$, then we have either $F(x) \geqslant G(x)$ or $G(x) \geqslant F(x)$ for all $x\in [0,\infty )$. Since $\mathrm{E}[X] \geqslant \mathrm{E}[Y]$, only the later case stands. Thus $\int _a^\infty (G(x)-F(x))\mathrm{d}x \geqslant 0$, for all $a \geqslant 0$. Hence $X \geqslant _v Y$.

If there exists $\widehat{x}$, such that $F(\widehat{x})=G(\widehat{x})\in (0,1)$, then part (a) applies. We have $G(x)-F(x) \leqslant 0$ for $x \leqslant \widehat{x}$ and $G(x)-F(x) \geqslant 0$ for $x \geqslant \widehat{x}$. Thus the function

$$\begin{aligned} s(a) \overset{{\mathrm{def}}}{=} \int _{-a}^{\infty } G(x)-F(x)\mathrm{d}x,\quad a\in (-\infty ,0) \end{aligned}$$

is increasing and then decreasing in a. $\mathrm{E}[X] \geqslant \mathrm{E}[Y]$ implies that $s(0)=\int _0^\infty G(x)-F(x) \mathrm{d}x \geqslant 0$. Together with $s(-\infty ) = 0$, we have $s(a) \geqslant 0$ for all $a\in (-\infty ,0)$. This implies $X \geqslant _v Y$.

(c) Integrating by parts, we have $\int _0^{F^{-1}(x)}F(u)\mathrm{d}u = xF^{-1}(x) - \int _0^x F^{-1}(u)\mathrm{d}u$. Thus,

$$\begin{aligned}&\quad \int _0^{F^{-1}(x)}F(u)\mathrm{d}u - \int _0^{G^{-1}(x)}G(u)\mathrm{d}u \\&= \int _0^x \Big [(F^{-1}(x)-F^{-1}(u))-(G^{-1}(x)-G^{-1}(u))\Big ]\mathrm{d}u \ \geqslant \ 0, \end{aligned}$$

where the last inequality is due to the fact that $X \geqslant _d Y$ so that the integrand is always non-negative. Similarly, we can prove

$$\begin{aligned} \displaystyle \int ^\infty _{F^{-1}(y)}(1-F(u))\mathrm{d}u \geqslant \int ^\infty _{G^{-1}(y)}(1-G(u))\mathrm{d}u. \end{aligned}$$

Proof of Theorem 4.7

We only need to prove $g_i(y,F_i) \leqslant g_i(y,G_i)$, for all $y\in (0,1)$. Since $\mathrm{E}[\xi _{ij}^F] = \mathrm{E}[\xi _{ij}^G]$, there must exist $\widehat{x}$ such that $F(\widehat{x})=G(\widehat{x}) = \widehat{y}$.

From Lemma 4.6 (a) and (c), when $y \leqslant \widehat{y}$, $F_i^{-1}(y) \leqslant G_i^{-1}(y)$ and $\int _0^{F_i^{-1}(y)} F_i(u)\mathrm{d}u \geqslant \int _0^{G_i^{-1}(y)}G_i(u)\mathrm{d}u$. These together imply that

$$\begin{aligned} g_i(y,F_i) \leqslant g_i(y,G_i)\quad \text{ for } y \leqslant \widehat{y}. \end{aligned}$$

We note that $\mathrm{E}[\xi _{ij}^F] = \mathrm{E}[\xi _{ij}^G]$ implies that

$$\begin{aligned} \int _0^{\infty }(1-F_i(u))\mathrm{d}u&= \int _0^{\infty }(1-G_i(u))\mathrm{d}u, \end{aligned}$$

or equivalently,

$$\begin{aligned}&\qquad g_i(y, F_i) + (1-\gamma _i)F_i^{-1}(y)+\int _{F_i^{-1}(y)}^{\infty }(1-F_i(u))\mathrm{d}u\\&=g_i(y, G_i) + (1-\gamma _i)G_i^{-1}(y)+\int _{G_i^{-1}(y)}^{\infty }(1-G_i(u))\mathrm{d}u. \end{aligned}$$

When $y \geqslant \widehat{y}$, from Lemmas 4.6 (a) and 4.6 (c), we have $F^{-1}(y) \geqslant G^{-1}(y)$ and $\int _{F^{-1}(y)}^\infty (1-F_i(u))\mathrm{d}u \geqslant \int _{G_i^{-1}(y)}^\infty (1-G_i(u))\mathrm{d}u$. Thus $g_i(y,F_i) \leqslant g_i(y,G_i)$ for $y \geqslant \widehat{y}$.

Together, we have $g_i(y,F_i) \leqslant g_i(y,G_i)$ for all $y\in (0,1)$. Hence,

$$\begin{aligned} \pi ^\mathrm{X}(F_1,F_2) \ \leqslant \ \pi ^\mathrm{X}(G_1,G_2). \end{aligned}$$

To prove the second part regarding to the individual supply chain profit, we express supply chain’s profit as

$$\begin{aligned} \pi ^\mathrm{X}_i(F_1,F_2)&= 2(p_i-v_i) \widetilde{g}_i\big (S_i^\mathrm{X}, F_i\big ) -\frac{2(p_i-c_i)\gamma _{-i}}{\eta }F^{-1}_{-i}\big (S_{-i}^\mathrm{X}\big ), \end{aligned}$$

where $\widetilde{g}_i\big (S_i^\mathrm{X}, F_i\big ) = S_i^{\mathrm{DD}}F_i^{-1}\big (S^\mathrm{X}_i\big )-\int _0^{F_i^{-1}\big (S^\mathrm{X}_i\big )}F_i(u)\mathrm{d}u$ is similar to $g_i(y, F_i)$ and is also decreasing when F becomes more dispersed while keeping mean constant. Hence the result.

If $F_{-i}$ becomes more dispersed, the direction of the change of $\pi _i^\mathrm{X}$ is unclear. It depends on how the more dispersed distribution is shaped. In particular, it depends on whether $\widehat{x} \leqslant S_{-i}^\mathrm{X}$.

Proof of Theorem 4.8

$$\begin{aligned} \left| \pi ^{\mathrm{X}_1}(F_1,F_2) - \pi ^{\mathrm{X}_2}(F_1,F_2) \right|&= \left| \sum _i\displaystyle 2(p_i-v_i) \int _{S_i^{\mathrm{X}_2}}^{S_i^{\mathrm{X}_1}} \frac{\mathrm{d}g_i(y;F_i)}{\mathrm{d}y} \mathrm{d}y \right| \\&= \sum _i\displaystyle 2(p_i-v_i) \int _{S_i^{\mathrm{X}_2}}^{S_i^{\mathrm{X}_1}} \left| \frac{\mathrm{d}g_i(y;F_i)}{\mathrm{d}y} \right| \mathrm{d}y \\&= \sum _i\displaystyle 2(p_i-v_i) \int _{S_i^{\mathrm{X}_2}}^{S_i^{\mathrm{X}_1}} \left| \gamma _i-y\right| \frac{\mathrm{d} F_i^{-1}(y)}{\mathrm{d}y} \mathrm{d}y \\&\geqslant \sum _i\displaystyle 2(p_i-v_i) \int _{S_i^{\mathrm{X}_2}}^{S_i^{\mathrm{X}_1}} \left| \gamma _i-y\right| \frac{\mathrm{d} G_i^{-1}(y)}{\mathrm{d}y} \mathrm{d}y \\&= \left| \pi ^{\mathrm{X}_1}(G_1,G_2) - \pi ^{\mathrm{X}_2}(G_1,G_2)\right| , \end{aligned}$$

where the second equality is due to the fact that $\displaystyle \int _{S_i^{\mathrm{X}_2}}^{S_i^{\mathrm{X}_1}} \frac{\mathrm{d}g_i(y;F_i)}{\mathrm{d}y} \mathrm{d}y$ are of the same signs for $i=1,2$, and the inequality follows from (4.5).