The impact of ex-post information sharing on a two-echelon supply chain with horizontal competition and capacity constraint

Abstract

We consider a two-echelon supply chain consisting of one capacity-constrained supplier and two Cournot-competitive retailers. The incumbent retailer has private demand information and takes strategic information sharing to maximize its expected profit. The supplier carries out capacity allocation strategy when the capacity cannot satisfy both retailers’ total order quantity. We model three scenarios of information sharing: (1) no member is informed; (2) only the entrant retailer is informed; and (3) both the supplier and entrant retailer are informed. We characterize the conditions under which information sharing may benefit or hurt the supply chain and different members and meaningful information sharing is achieved via “cheap talk”. Compared with no information sharing, when only the entrant retailer is informed, the incumbent retailer always has incentive to deflate its demand information to make the entrant retailer order less. Both retailers cannot achieve meaningful information sharing. When the supplier and entrant retailer both are informed, the incumbent retailer faces a trade-off between the desire to receive more products and the fear of intense competition and high wholesale price. The meaningful information sharing is achieved and all members are benefitted. The numerical examples verify theoretical results.

Appendix

Proof of Proposition 1

When $A = A_{H}$, the actual total orders are $\frac{1}{2}A_{H} - \frac{1}{6}\overline{A} - \frac{1}{3}c >$ $\frac{1}{3}\left( {\overline{A} - c} \right)$. If $K$ is greater than the total orders, each retailer receives its initial order quantity. However, if $K$ is less than the total orders, then the incumbent retailer receives $\alpha K$ from the supplier, and the entrant retailer receives $\left( {1 - \alpha } \right)K$. When $A = A_{L}$, the actual total orders are $\frac{1}{2}A_{L} - \frac{1}{6}\overline{A} - \frac{1}{3}c$. If $K$ is greater than $\frac{1}{3}\left( {\overline{A} - c} \right)$, the supplier satisfies the total orders. If $K$ falls into $\left[ {\frac{1}{2}A_{L} - \frac{1}{6}\overline{A} \frac{1}{3}c, \frac{1}{3}\left( {\overline{A} - c} \right)} \right)$, the supplier anticipates that the total orders cannot be satisfied and sets the wholesale price $w^{{NS^{*} }} = \overline{A} - \frac{3}{2}K$. However, the actual total orders are less than $K$. The supplier does not need to allocate $K$. If $K$ is less than $\frac{1}{2}A_{L} - \frac{1}{6}\overline{A} - \frac{1}{3}c$, the supplier cannot satisfy the total orders and needs to allocate $K$. □

Proof of Proposition 2

Under horizontal information sharing, the supplier maintains the same wholesale price and allocation strategy as that under the scenario of no information sharing. The incentive compatibility condition of the incumbent retailer to achieve truthful information sharing is as follows:

$$ \pi_{i}^{{HS^{*} }} \left( {A_{i} ,m_{i} } \right) \ge \pi_{i}^{{HS^{*} }} \left( {A_{i} ,m_{j} } \right),\quad {\text{for}}\;i,j \in \left\{ {H,L} \right\}\;{\text{and}}\;j \ne i. $$

(A1)

When $R_{i}$ observes $A_{L}$, he will not send a $m_{H}$ to $R_{e}$. This result is because $m_{H}$ increases the competition. Hence, $R_{i}$ always truthfully sends a $m_{L}$ to $R_{e}$. When $R_{i}$ observes $A_{H}$, if $R_{i}$ truthfully shares a $m_{H}$, $R_{i}$ receives $Q_{i}^{{HS^{*} }} \left( {A_{H} ,m_{H} } \right) = \left\{ {\begin{array}{*{20}l} {\frac{1}{3}A_{H} - \frac{1}{6}\overline{A} - \frac{1}{6}c,} \hfill & {{\text{if}}\; K \ge \frac{2}{3}A_{H} - \frac{1}{3}\overline{A} - \frac{1}{3}c} \hfill \\ {\alpha K,} \hfill & {{\text{if}}\; K < \frac{2}{3}A_{H} - \frac{1}{3}\overline{A} - \frac{1}{3}c} \hfill \\ \end{array} } \right.$. If $R_{i}$ shares a $m_{L}$, $R_{i}$ receives $Q_{i}^{{HS^{*} }} \left( {A_{H} ,m_{L} } \right) = \left\{ {\begin{array}{*{20}l} {\frac{1}{2}A_{H} - \frac{1}{6}A_{L} - \frac{1}{6}\overline{A} - \frac{1}{6}c,} \hfill & {{\text{if}}\; K \ge \frac{1}{2}A_{H} + \frac{1}{6}A_{L} - \frac{1}{3}\overline{A} - \frac{1}{3}c} \hfill \\ {\alpha K,} \hfill & {{\text{if}}\; K < \frac{1}{2}A_{H} + \frac{1}{6}A_{L} - \frac{1}{3}\overline{A} - \frac{1}{3}c} \hfill \\ \end{array} } \right.$ when $0 < r < \frac{3}{4}$. When $\frac{3}{4} \le r < 1$, $R_{i}$ receives $Q_{i}^{{HS^{*} }} \left( {A_{H} ,m_{L} } \right) = \left\{ {\begin{array}{*{20}l} {\frac{1}{2}A_{H} - \frac{1}{6}A_{L} - \frac{1}{6}\overline{A} - \frac{1}{6}c,} \hfill & {{\text{if}}\; K \ge \frac{1}{3}\left( {\overline{A} - c} \right)} \hfill \\ {\frac{1}{2}A_{H} - \frac{1}{6}A_{L} - \frac{1}{3}\overline{A} + \frac{1}{2}K,} \hfill & {{\text{if}}\; \frac{1}{2}A_{H} + \frac{1}{6}A_{L} - \frac{1}{3}\overline{A} - \frac{1}{3}c \le K < \frac{1}{3}\left( {\overline{A} - c} \right)} \hfill \\ {\alpha K.} \hfill & {{\text{if}} \;K < \frac{1}{2}A_{H} + \frac{1}{6}A_{L} - \frac{1}{3}\overline{A} - \frac{1}{3}c} \hfill \\ \end{array} } \right..$

Further, $Q_{i}^{{HS^{*} }} \left( {A_{H} ,m_{H} } \right) \le Q_{i}^{{HS^{*} }} \left( {A_{H} ,m_{L} } \right)$ for any $K$. If $K \ge \frac{1}{2}A_{H} + \frac{1}{6}A_{L} - \frac{1}{3}\overline{A} - \frac{1}{3}c,$ $R_{i}$’s expected profit increases with the quantity of received products because the sent message does not alter the wholesale price. $R_{i}$ has an incentive to induce the entrant retailer to order less. $R_{i}$, therefore, always sends $m_{L}$ to $R_{e}$. Truthful information sharing cannot be achieved.

If $K < \frac{1}{2}A_{H} + \frac{1}{6}A_{L} - \frac{1}{3}\overline{A} - \frac{1}{3}c$, the supplier cannot satisfy the total orders and needs to allocate $K$. Regardless of which message is sent to $R_{e}$, $R_{i}$ receives $\alpha K$ from the supplier. $R_{i}$ would truthfully share his private information with $R_{e}$.□

Proof of Proposition 3

Under the truthful information sharing in Proposition 2, $R_{i}$ always receives $\alpha K$ and $R_{e}$ receives $\left( {1 - \alpha } \right)K$. The received product quantity of each retailer is the same as that under the scenario of no information sharing. Because the supplier makes the same wholesale price as that in the scenario of no information sharing, truthful information sharing does not alter the expected profits of both retailers. □

Proof of Proposition 4

• (i) If $R_{i}$ sends a $m_{H}$ to both $R_{e}$ and $S$, $R_{i}$’s order quantity is $q_{i}^{{PS^{*} }} \left( {A_{i} ,m_{H} ,w\left( {m_{H} } \right)} \right) = \frac{1}{2}A_{i} - \frac{1}{6}m_{H} - \frac{1}{3}w\left( {m_{H} } \right)$ and $R_{e}$’s order quantity is $q_{e}^{{PS^{*} }} \left( {m_{H} ,w\left( {m_{H} } \right)} \right) = \frac{1}{3}\left( {m_{H} - w\left( {m_{H} } \right)} \right)$. The supplier sets $w^{{PS^{*} }} \left( {m_{H} } \right) = \left\{ {\begin{array}{ll} {\frac{1}{2}\left( {m_{H} + c} \right),} & {\hbox{if} \; K \ge \frac{1}{3}\left( {m_{H} - c} \right)} \\ {m_{H} - \frac{3}{2}K,} & {\hbox{if} \; K < \frac{1}{3}\left( {m_{H} - c} \right)} \\ \end{array} } \right.$. When $A = A_{H}$, $R_{i}$ receives $Q_{i}^{{PS^{*} }} \left( {A_{H} ,m_{H} ,w\left( {m_{H} } \right)} \right) = \left\{ {\begin{array}{ll} {\frac{1}{6}\left( {A_{H} - c} \right),} & {\hbox{if} \; K \ge \frac{1}{3}\left( {A_{H} - c} \right)} \\ {\left( {\alpha + \beta } \right)K.} & {\hbox{if} \; K < \frac{1}{3}\left( {A_{H} - c} \right)} \\ \end{array} } \right.$. When $A = A_{L}$, $R_{i}$ receives $Q_{i}^{{PS^{*} }} \left( {A_{L} ,m_{H} ,w\left( {m_{H} } \right)} \right) = \left\{ {\begin{array}{*{20}l} {\frac{1}{2}A_{L} - \frac{1}{3}A_{H} - \frac{1}{6}c,} \hfill & {{\text{if}}\; K \ge \frac{1}{3}\left( {A_{H} - c} \right)} \hfill \\ {\frac{1}{2}\left( {A_{L} - A_{H} + K} \right),} \hfill & {{\text{if}}\; \frac{1}{2}A_{L} - \frac{1}{6}A_{H} - \frac{1}{3}c \le K < \frac{1}{3}\left( {A_{H} - c} \right)} \hfill \\ {\left( {\alpha + \beta } \right)K.} \hfill & {{\text{if}}\; K < \frac{1}{2}A_{L} - \frac{1}{6}A_{H} - \frac{1}{3}c} \hfill \\ \end{array} } \right..$

• (ii) If $R_{i}$ sends a $m_{L}$ to both $R_{e}$ and $S$, $R_{i}$’s order quantity is $q_{i}^{{PS^{*} }} \left( {A_{i} ,m_{L} ,w\left( {m_{H} } \right)} \right) = \frac{1}{2}A_{i} - \frac{1}{6}m_{L} - \frac{1}{3}w\left( {m_{L} } \right)$ and $R_{e}$’s order quantity is $q_{e}^{{PS^{*} }} \left( {m_{L} ,w\left( {m_{L} } \right)} \right) = \frac{1}{3}\left( {m_{L} - w\left( {m_{L} } \right)} \right)$. The supplier sets $w^{{PS^{*} }} \left( {m_{L} } \right) = \left\{ {\begin{array}{ll} {\frac{1}{2}\left( {m_{L} + c} \right),} & {\hbox{if} \; K \ge \frac{1}{3}\left( {m_{L} - c} \right)} \\ {m_{L} - \frac{3}{2}K,} & {\hbox{if} \; K < \frac{1}{3}\left( {m_{L} - c} \right)} \\ \end{array} } \right.$. When $A = A_{H}$, $R_{i}$ receives $Q_{i}^{{PS^{*} }} \left( {A_{H} ,m_{L} ,w\left( {m_{L} } \right)} \right) = \left\{ {\begin{array}{ll} {\frac{1}{2}A_{H} - \frac{1}{3}A_{L} - \frac{1}{6}c,} & {\hbox{if} \; K \ge \frac{1}{2}A_{H} - \frac{1}{6}A_{L} - \frac{1}{3}c} \\ {\alpha K.} & {\hbox{if} \; K < \frac{1}{2}A_{H} - \frac{1}{6}A_{L} - \frac{1}{3}c} \\ \end{array} } \right.$. When $A = A_{L}$, $R_{i}$ receives $Q_{i}^{{PS^{*} }} \left( {A_{L} ,m_{L} ,w\left( {m_{L} } \right)} \right) = \left\{ {\begin{array}{ll} {\frac{1}{6}\left( {A_{L} - c} \right),} & {\hbox{if} \; K \ge \frac{1}{3}\left( {A_{L} - c} \right)} \\ {\alpha K.} & {\hbox{if} \; K < \frac{1}{3}\left( {A_{L} - c} \right)} \\ \end{array} } \right.$.

$R_{i}$’s expected profit $\pi_{i}^{{PS^{*} }} \left( {A_{i} ,m_{j} ,w\left( {m_{j} } \right)} \right)$ is listed in Table 3. To achieve truthful information sharing, the following incentive compatibility constraints conditions must be satisfied:

$$ \left\{ {\begin{array}{ll} {\pi_{i} \left( {A_{H} ,m_{H} ,w\left( {m_{H} } \right)} \right) \ge \pi_{i} \left( {A_{H} ,m_{L} ,w\left( {m_{L} } \right)} \right) \left( {IC_{HL} } \right)} \\ {\pi_{i} \left( {A_{L} ,m_{L} ,w\left( {m_{L} } \right)} \right) \ge \pi_{i} \left( {A_{L} ,m_{H} ,w\left( {m_{H} } \right)} \right) \left( {IC_{LH} } \right)} \\ \end{array} .} \right. $$

(A2)

Table 3 The expected profit of $R_{i}$ upon observing $A_{i}$ and publicly announcing $m_{j}$

• (a) If $K \ge \frac{1}{2}A_{H} - \frac{1}{6}A_{L} - \frac{1}{3}c$, then $\left( {IC_{HL} } \right)$ is $\frac{1}{36}\left( {A_{H} - c} \right)^{2} \ge \left( {\frac{1}{2}A_{H} - \frac{1}{3}A_{L} - \frac{1}{6}c} \right)^{2} .$ Simplifying the above inequation obtains $A_{L} \ge A_{H}$, which contradicts the assumption. $\left( {IC_{HL} } \right)$ cannot be satisfied.

• (b) If $\frac{1}{3}\left( {A_{H} - c} \right) \le K < \frac{1}{2}A_{H} - \frac{1}{6}A_{L} - \frac{1}{3}c$, then (A2) can be written as follows:

  $$ \begin{aligned} & \frac{1}{36}\left( {A_{H} - c} \right)^{2} \ge \frac{1}{2}\alpha K\left( {2A_{H} - A_{L} - 2K - c} \right);\quad (IC_{HL} ) \\ & \frac{1}{36}\left( {A_{L} - c} \right)^{2} \ge \left( {\frac{1}{2}A_{L} - \frac{1}{3}A_{H} - \frac{1}{6}c} \right)^{2} .\quad (IC_{LH} ) \\ \end{aligned} $$

  Simplifying the above inequations, the conditions can be reduced to $\beta \ge 1 - \frac{{\left( {A_{H} - c} \right)^{2} }}{{9K\left( {2A_{H} - A_{L} - 2K - c} \right)}}$ and $A_{H} \ge A_{L}$. Define $\theta = \frac{{A_{H} - c}}{{A_{L} - c}}$. To ensure that $Q_{i}^{{PS^{*} }}$ and $Q_{e}^{{PS^{*} }}$ under all information sharing scenarios are positive, assume that $1 < \theta < \frac{3}{2}$. According to $\frac{1}{3}\left( {A_{H} - c} \right) \le K < \frac{1}{2}A_{H} - \frac{1}{6}A_{L} - \frac{1}{3}c$, we have $\frac{1}{2} < \frac{{\left( {A_{H} - c} \right)^{2} }}{{9K\left( {2A_{H} - A_{L} - 2K - c} \right)}} < 1$. If $1 - \frac{{\left( {A_{H} - c} \right)^{2} }}{{9K\left( {2A_{H} - A_{L} - 2K - c} \right)}} \le \beta < 1$, then ($IC_{HL}$) and ($IC_{LH}$) can be satisfied. A meaningful information sharing can be achieved.

• (c) If $\frac{1}{3}\left( {A_{L} - c} \right) \le K < \frac{1}{3}\left( {A_{H} - c} \right)$, then (A2) can be written as follows:

  $$ \begin{aligned} & \frac{1}{2}K^{2} \left( {\alpha + \beta } \right) \ge \frac{1}{2}\alpha K\left( {2A_{H} - A_{L} - 2K - c} \right);\quad (IC_{HL} ) \\ & \frac{1}{36}\left( {A_{L} - c} \right)^{2} \ge \frac{1}{4}\left[ {K - \left( {A_{H} - A_{L} } \right)} \right]^{2} .\quad (IC_{LH} ) \\ \end{aligned} $$

  Simplifying the condition ($IC_{HL}$) obtains $\beta \ge \frac{{2A_{H} - A_{L} - 3K - c}}{{2A_{H} - A_{L} - K - c}}$. The condition ($IC_{LH}$) is reduced to $K \le A_{H} - \frac{2}{3}A_{L} - \frac{1}{3}c$. This inequation can be satisfied since $K < \frac{1}{3}\left( {A_{H} - c} \right)$. Consequently, meaningful information sharing can be achieved if $\frac{{2A_{H} - A_{L} - 3K - c}}{{2A_{H} - A_{L} - K - c}} \le \beta < 1$.

• (d) If $\frac{1}{2}A_{L} - \frac{1}{6}A_{H} - \frac{1}{3}c \le K < \frac{1}{3}\left( {A_{L} - c} \right)$, then (A2) can be denoted by the following:

  $$ \begin{aligned} & \frac{1}{2}K^{2} \left( {\alpha + \beta } \right) \ge \frac{1}{2}\alpha K\left( {A_{H} - A_{L} + K} \right);\quad (IC_{HL} ) \\ & \frac{1}{2}\alpha K^{2} \ge \frac{1}{2}\left( {\alpha + \beta } \right)K\left[ {K - 2\left( {A_{H} - A_{L} } \right)} \right].\quad (IC_{LH} ) \\ \end{aligned} $$

  The condition ($IC_{HL}$) is reduced to $\beta \ge \frac{{A_{H} - A_{L} }}{{K + A_{H} - A_{L} }}$, and the condition ($IC_{LH}$) is reduced to $\beta \le 1 - \left( {\frac{{K - \left( {A_{H} - A_{L} } \right)}}{K}} \right)^{2}$. Obviously, $1 - \left( {\frac{{K - \left( {A_{H} - A_{L} } \right)}}{K}} \right)^{2} > \frac{{A_{H} - A_{L} }}{{K + A_{H} - A_{L} }}$. A meaningful information sharing can be achieved if $\frac{{A_{H} - A_{L} }}{{K + A_{H} - A_{L} }} \le \beta \le 1 - \left( {\frac{{K - \left( {A_{H} - A_{L} } \right)}}{K}} \right)^{2}$.

• (e) If $K < \frac{1}{2}A_{L} - \frac{1}{6}A_{H} - \frac{1}{3}c$, the condition ($IC_{LH}$) turns to $\frac{1}{2}\alpha K^{2} \ge \frac{1}{2}\left( {\alpha + \beta } \right)K\left[ {K - 2\left( {A_{H} - A_{L} } \right)} \right]$, which is reduced to $\beta \le \frac{{A_{H} - A_{L} }}{{K - \left( {A_{H} - A_{L} } \right)}}$. Since $K < \frac{1}{2}A_{L} - \frac{1}{6}A_{H} + \frac{1}{3}c$, then $0 < \frac{{A_{H} - A_{L} }}{{K + \left( {A_{H} - A_{L} } \right)}} < \frac{{A_{H} - A_{L} }}{{K - \left( {A_{H} - A_{L} } \right)}} < 1$. A meaningful information sharing is achieved if $\frac{{A_{H} - A_{L} }}{{K + A_{H} - A_{L} }} \le \beta < min\left\{ {\frac{{A_{H} - A_{L} }}{{K - \left( {A_{H} - A_{L} } \right)}},1} \right\}$. □

Proof of Proposition 5

Based on the wholesale price and the received products in Proposition 1, the expected profit of each member under the scenario of no information sharing is evaluated. Four possible scenarios are considered: (i) $\frac{1}{2}A_{H} - \frac{1}{6}\overline{A} - \frac{1}{3}c \le K < \frac{1}{2}A_{H} - \frac{1}{6}A_{L} - \frac{1}{3}c$; (ii) $\frac{1}{3}\left( {A_{H} - c} \right) \le K < \frac{1}{2}A_{H} - \frac{1}{6}\overline{A} - \frac{1}{3}c$; (iii) $\frac{1}{3}\left( {\overline{A} - c} \right) \le K < \frac{1}{3}\left( {A_{H} - c} \right)$ and (iv) $K < \frac{1}{3}\left( {\overline{A} - c} \right)$.

• (a) First, in scenario (i), under the scenario of no information sharing, $R_{i}$’s expected profit is $\left( {\frac{1}{2}A_{H} - \frac{1}{3}\overline{A} - \frac{1}{6}c} \right)^{2}$. This outcome is obviously more than $\frac{1}{36}\left( {A_{H} - c} \right)^{2}$, which is the expected profit under the scenario of information sharing. Information sharing hurts $R_{i}$. In scenario (ii), $R_{i}$’s expected profit under the scenario of no information sharing is $\frac{1}{2}\alpha K\left( {2A_{H} - \overline{A} - 2K - c} \right)$. Since $1 - \frac{{\left( {A_{H} - c} \right)^{2} }}{{9K\left( {2A_{H} - A_{L} - 2K - c} \right)}} \le \beta < 1$, then $\alpha \le \frac{{\left( {A_{H} - c} \right)^{2} }}{{18K\left( {2A_{H} - A_{L} - 2K - c} \right)}} < \frac{{\left( {A_{H} - c} \right)^{2} }}{{18K\left( {2A_{H} - \overline{A} - 2K - c} \right)}}$. Further, $\frac{1}{2}\alpha K\left( {2A_{H} - \overline{A} - 2K - c} \right) < \frac{1}{2}K \cdot \frac{{\left( {A_{H} - c} \right)^{2} \left( {2A_{H} - \overline{A} - 2K - c} \right)}}{{18K\left( {2A_{H} - \overline{A} - 2K - c} \right)}} = \frac{1}{36}\left( {A_{H} - c} \right)^{2}$. Information sharing benefits $R_{i}$. In scenario (iii), under the scenario of meaningful information sharing, $R_{i}$’s expected profit changes to $\frac{1}{2}\left( {\alpha + \beta } \right)K^{2}$. Then, $\frac{1}{2}\left( {\alpha + \beta } \right)K^{2} - \frac{1}{2}\alpha K\left( {2A_{H} - \overline{A} - 2K - c} \right) = \frac{1}{2}K\left[ {K - \alpha \left( {2A_{H} - \overline{A} - c - K} \right)} \right] > \frac{1}{2}K\left[ {K - \alpha \left( {2A_{H} - A_{L} - c - K} \right)} \right]$. Since $\frac{{2A_{H} - A_{L} - 3K - c}}{{2A_{H} - A_{L} - K - c}} \le \beta < 1$, then $0 < \alpha \le \frac{K}{{2A_{H} - A_{L} - c - K}}$ and $K - \alpha \left( {2A_{H} - A_{L} - c - K} \right) \ge 0$. Information sharing also benefits $R_{i}$. In scenario (iv), $R_{i}$’s expected profit under the scenario of no information sharing turns to $\frac{1}{2}\alpha K\left( {2A_{H} - 2\overline{A} + K} \right)$. Since $0 < \alpha = \frac{1 - \beta }{2} \le \frac{K}{{2A_{H} - A_{L} - c - K}}$, then $\frac{1}{2}\left( {\alpha + \beta } \right)K^{2} - \frac{1}{2}\alpha K\left( {2A_{H} - 2\overline{A} + K} \right) = \frac{1}{2}K\left[ {K - 2\alpha \left( {K + A_{H} - \overline{A} } \right)} \right] \ge \frac{1}{2}K\left[ {K - \frac{2K}{{2A_{H} - A_{L} - c - K}}\left( {K + A_{H} - \overline{A} } \right)} \right] = \frac{{2\overline{A} - A_{L} - 3K - c}}{{2\left( {2A_{H} - A_{L} - K - c} \right)}}K^{2} > \frac{{\overline{A} - A_{L} }}{{2\left( {2A_{H} - A_{L} - K - c} \right)}}K^{2} > 0$. Information sharing benefits $R_{i}$.

• (b) Second, in scenario (i), under the scenario of no information sharing, $R_{e}$’s expected profit is $\frac{1}{6}\left( {\overline{A} - c} \right)\left( {\frac{1}{2}A_{H} - \frac{1}{3}\overline{A} - \frac{1}{6}c} \right)$. This outcome is more than the expected profit $\frac{1}{36}\left( {A_{H} - c} \right)^{2}$ under the scenario of information sharing, since $1 < \theta = \frac{{A_{H} - c}}{{A_{L} - c}} < \frac{3}{2}$. Information sharing hurts $R_{e}$. In scenario (ii), $R_{e}$’s expected profit is $\frac{1}{2}\left( {1 - \alpha } \right)K\left( {2A_{H} - \overline{A} - 2K - c} \right)$ under the scenario of no information sharing. $\pi_{e}^{{PS^{*} }} - \pi_{e}^{{NS^{*} }} = \frac{1}{36}\left( {A_{H} - c} \right)^{2} - \frac{1}{2}\left( {1 - \alpha } \right)K\left( {2A_{H} - \overline{A} - 2K - c} \right) = \frac{1}{36}\left( {A_{H} - c} \right)^{2} + \left( {1 - \alpha } \right)\left[ {K^{2} - K\left( {A_{H} - \frac{1}{2}\overline{A} - \frac{1}{2}c} \right)} \right]$. Since $\frac{1}{3}\left( {A_{H} - c} \right) \le K < \frac{1}{2}A_{H} - \frac{1}{6}\overline{A} - \frac{1}{3}c$, then $K^{2} - K\left( {A_{H} - \frac{1}{2}\overline{A} - \frac{1}{2}c} \right) < 0$. Hence, $\pi_{e}^{{PS^{*} }} - \pi_{e}^{{NS^{*} }} < \frac{1}{36}\left( {A_{H} - c} \right)^{2} + \left( {1 - \frac{{\left( {A_{H} - c} \right)^{2} }}{{18K\left( {2A_{H} - \overline{A} - 2K - c} \right)}}} \right)\left[ {K^{2} - K\left( {A_{H} - \frac{1}{2}\overline{A} - \frac{1}{2}c} \right)} \right]$ since $0 < \alpha < \frac{{\left( {A_{H} - c} \right)^{2} }}{{18K\left( {2A_{H} - \overline{A} - 2K - c} \right)}}$. Denote the formula $\frac{1}{36}\left( {A_{H} - c} \right)^{2} + \left( {1 - \frac{{\left( {A_{H} - c} \right)^{2} }}{{18K\left( {2A_{H} - \overline{A} - 2K - c} \right)}}} \right)\left[ {K^{2} - K\left( {A_{H} - \frac{1}{2}\overline{A} - \frac{1}{2}c} \right)} \right]$ as $f\left( K \right)$. Then, $\mathop {\hbox{max} }\limits_{{\frac{1}{3}\left( {A_{H} - c} \right) \le K < \frac{1}{2}A_{H} - \frac{1}{6}\overline{A} - \frac{1}{3}c}} f\left( K \right) < f\left( {\frac{1}{2}A_{H} - \frac{1}{6}\overline{A} - \frac{1}{3}c} \right) = - \frac{1}{36}\left[ {7\left( {A_{H} - c} \right)^{2} - 9\left( {A_{H} - c} \right)\left( {\overline{A} - c} \right) + 2\left( {\overline{A} - c} \right)^{2} } \right] = - \frac{1}{36}\left[ {\left( {A_{H} - c} \right) - \left( {\overline{A} - c} \right)} \right]\left[ {7\left( {A_{H} - c} \right) - 2\left( {\overline{A} - c} \right)} \right] < 0$. Information sharing also hurts $R_{e}$. In scenario (iii), $R_{e}$’s expected profit under the scenario of information sharing changes to $\frac{1}{2}\left( {1 - \alpha - \beta } \right)K^{2}$. Since $\alpha \le \frac{K}{{2A_{H} - A_{L} - c - K}} < \frac{1}{2}$, then $\pi_{e}^{{PS^{*} }} - \pi_{e}^{{NS^{*} }} = \frac{1}{2}K\left[ {K - \left( {1 - \alpha } \right)\left( {2A_{H} - \overline{A} - c} \right)} \right] < \frac{1}{2}K\left[ {K - \frac{1}{2}\left( {2A_{H} - \overline{A} - c} \right)} \right] = - \frac{1}{4}K\left[ {2A_{H} - \overline{A} - 2K - c} \right] < 0.$ Information sharing hurts $R_{e}$. In scenario (iv), $R_{e}$’s expected profit under the scenario of no information sharing changes to $\frac{1}{2}\left( {1 - \alpha } \right)K\left( {2A_{H} - 2\overline{A} + K} \right)$. $\pi_{e}^{{PS^{*} }} - \pi_{e}^{{NS^{*} }} = \frac{1}{2}\left( {1 - \alpha - \beta } \right)K^{2} - \frac{1}{2}\left( {1 - \alpha } \right)K\left( {2A_{H} - 2\overline{A} + K} \right) = - \frac{1}{2}\beta K^{2} - \left( {1 - \alpha } \right)K\left( {A_{H} - \overline{A} } \right) < 0$. Information sharing still hurts $R_{e}$.

• (c) Third, in scenario (i), under the scenario of no information sharing, $S$’s expected profit is $\frac{1}{2}\left( {\overline{A} - c} \right)\left[ {\frac{1}{2}\left( {A_{H} - c} \right) - \frac{1}{6}\left( {\overline{A} - c} \right)} \right]$. This outcome is less than the expected profit under the scenario of information sharing, which is $\frac{1}{6}\left( {A_{H} - c} \right)^{2}$. Information sharing benefits $S$. In scenario (ii), $S$’s expected profit changes to $\frac{1}{2}\left( {\overline{A} - c} \right)K$ under the scenario of no information sharing. Since $\frac{1}{3}\left( {A_{H} - c} \right) \le K < \frac{1}{2}A_{H} - \frac{1}{6}\overline{A} - \frac{1}{3}c$, then $\pi_{S}^{{PS^{*} }} - \pi_{S}^{{NS^{*} }} = \frac{1}{6}\left( {A_{H} - c} \right)^{2} - \frac{1}{2}\left( {\overline{A} - c} \right)K > \frac{1}{6}\left( {A_{H} - c} \right)^{2} - \frac{1}{2}\left( {\overline{A} - c} \right) \cdot \left( {\frac{1}{2}A_{H} - \frac{1}{6}\overline{A} - \frac{1}{3}c} \right) = \frac{1}{12}\left( {A_{H} - \overline{A} } \right)\left[ {2\left( {A_{H} - c} \right) - \left( {\overline{A} - c} \right)} \right] > 0$. Information sharing also benefits $S$. In scenario (iii), information sharing benefits $S$ if $K\left( {A_{H} - \frac{3}{2}K - c} \right) > \frac{1}{2}\left( {\overline{A} - c} \right)K$ holds. It is satisfied since $K \le \frac{1}{3}\left( {A_{H} - c} \right)$ and $A_{H} > \overline{A}$. In scenario (iv), $S$’s expected profit is $K\left( {\overline{A} - \frac{3}{2}K - c} \right)$ under the scenario of no information sharing. The outcome is obviously less than the expected profit under the scenario of information sharing, which is $K\left( {A_{H} - \frac{3}{2}K - c} \right)$. □

Proof of Proposition 6

Similar to Proposition 5, given the demand state $A = A_{H}$, the expected profit of the supply chain is $\Pi = \left( {p - c} \right)Q.$ Because $\frac{{d\Pi }}{dQ} = \left( {p - c} \right) + \frac{\partial p\left( Q \right)}{\partial Q}Q = A_{H} - c - 2Q$, it is increasing with the sale quantity $Q$ when $Q < \frac{1}{2}\left( {A_{H} - c} \right)$. In scenarios (i) and (ii), information sharing raises downstream competition and leads to a higher wholesale price. As a result, the total sale quantity decreases, which hurts the supply chain. In scenarios (iii) and (iv), the total sale quality is locked by the capacity $K$. Information sharing has an effect on the expected profit of the supply chain. □

Proof of Proposition 7

When $R_{i}$ observes $A_{L}$, four possible scenarios are considered: (i) $\frac{1}{3}\left( {\overline{A} - c} \right) \le K < \frac{1}{2}A_{H} - \frac{1}{6}A_{L} - \frac{1}{3}c$; (ii) $\frac{1}{3}\left( {A_{L} - c} \right) \le K < \frac{1}{3}\left( {\overline{A} - c} \right)$; (iii) $\frac{1}{2}A_{L} - \frac{1}{6}\overline{A} - \frac{1}{3}c \le K < \frac{1}{3}\left( {A_{L} - c} \right)$; and (iv) $K < \frac{1}{2}A_{L} - \frac{1}{6}\overline{A} - \frac{1}{3}c$.

• (a) In scenario (i), $R_{i}$’s expected profit under the scenario of no information sharing is $\left( {\frac{1}{2}A_{L} - \frac{1}{3}\overline{A} - \frac{1}{6}c} \right)^{2}$. This outcome is clearly less than $\frac{1}{36}\left( {A_{L} - c} \right)^{2}$, which is the expected profit under the scenario of information sharing. In scenario (ii), $R_{i}$’s expected profit changes to $\left( {\frac{1}{2}A_{L} - \frac{1}{2}\overline{A} + \frac{1}{2}K} \right)^{2}$ under the scenario of no information sharing. $\pi_{iL}^{{PS^{*} }} - \pi_{iL}^{{NS^{*} }} = \frac{1}{36}\left( {A_{L} - c} \right)^{2} - \left( {\frac{1}{2}A_{L} - \frac{1}{2}\overline{A} + \frac{1}{2}K} \right)^{2} = \left[ {\frac{1}{2}\left( {\overline{A} - c} \right) - \frac{1}{3}\left( {A_{L} - c} \right) - \frac{1}{2}K} \right]\left[ {\frac{2}{3}\left( {A_{L} - c} \right) - \frac{1}{2}\left( {\overline{A} - c} \right) + \frac{1}{2}K} \right]$. Since $K < \frac{1}{3}\left( {\overline{A} - c} \right)$, then $\pi_{iL}^{{PS^{*} }} - \pi_{iL}^{{NS^{*} }} > \left[ {\frac{1}{2}\left( {\overline{A} - c} \right) - \frac{1}{3}\left( {A_{L} - c} \right) - \frac{1}{6}\left( {\overline{A} - c} \right)} \right]\left[ {\frac{2}{3}\left( {A_{L} - c} \right) - \frac{1}{2}\left( {\overline{A} - c} \right) + \frac{1}{2}K} \right] = \frac{1}{3}\left( {\overline{A} - A_{L} } \right)\left[ {\frac{2}{3}\left( {A_{L} - c} \right) - \frac{1}{2}\left( {\overline{A} - c} \right) + \frac{1}{2}K} \right] > 0$. Information sharing benefits $R_{i}$. In scenario (iii), $R_{i}$’s expected profit changes to $\frac{1}{2}\alpha K^{2}$ under the scenario of information sharing. This outcome is more than $\left( {\frac{1}{2}A_{L} - \frac{1}{2}\overline{A} + \frac{1}{2}K} \right)^{2}$ if and only if $\frac{{A_{H} - A_{L} }}{{K + A_{H} - A_{L} }} \le \beta \le 1 - \left( {\frac{{K - \left( {\overline{A} - A_{L} } \right)}}{K}} \right)^{2}$. In scenario (iv), $R_{i}$’s expected profit changes to $\frac{1}{2}\alpha K\left( {2A_{L} - 2\overline{A} + K} \right)$ under the scenario of no information sharing. This outcome is obviously less than the expected profit under the scenario of information sharing, which is $\frac{1}{2}\alpha K^{2}$.

• (b) In scenario (i), $R_{e}$’s expected profit under the scenario of no information sharing is $\frac{1}{6}\left( {\overline{A} - c} \right)\left( {\frac{1}{2}A_{L} - \frac{1}{3}\overline{A} - \frac{1}{6}c} \right)$. This outcome is less than the expected profit under the scenario of information sharing which is $\frac{1}{36}\left( {A_{L} - c} \right)^{2}$. In scenario (ii), under the scenario of no sharing information, $R_{e}$’s expected profit changes to $\frac{1}{2}K\left( {\frac{1}{2}A_{L} - \frac{1}{2}\overline{A} + \frac{1}{2}K} \right)$. $\pi_{e}^{{PS^{*} }} - \pi_{e}^{{NS^{*} }} = - \frac{1}{4}K^{2} + \frac{1}{4}\left( {\overline{A} - A_{L} } \right)K + \frac{1}{36}\left( {A_{L} - c} \right)^{2}$. Since $\frac{1}{3}\left( {A_{L} - c} \right) \le K < \frac{1}{3}\left( {\overline{A} - c} \right)$, then $\mathop {\hbox{min} }\limits_{K} \left( {\pi_{e}^{{PS^{*} }} - \pi_{e}^{{NS^{*} }} } \right) = - \frac{1}{4}\left( {\frac{1}{3}\left( {\overline{A} - c} \right)} \right)^{2} + \frac{1}{4}\left( {\overline{A} - A_{L} } \right)\left( {\frac{1}{3}\left( {\overline{A} - c} \right)} \right) + \frac{1}{36}\left( {A_{L} - c} \right)^{2} = \frac{1}{36}\left( {\overline{A} - A_{L} } \right)\left[ {2\left( {\overline{A} - c} \right) - \left( {A_{L} - c} \right)} \right] > 0$. In scenario (iii), under the scenario of information sharing, $R_{e}$’s profit is $\frac{1}{2}\left( {1 - \alpha } \right)K^{2}$. The outcome is higher than the expected profit $\frac{1}{2}K\left( {\frac{1}{2}A_{L} - \frac{1}{2}\overline{A} + \frac{1}{2}K} \right)$. Information sharing benefits $R_{e}$. In scenario (iv), $R_{e}$’s expected profit under the scenario of no information sharing is $\frac{1}{2}\left( {1 - \alpha } \right)K\left( {2A_{L} - 2\overline{A} + K} \right)$. This outcome is less than $\frac{1}{2}\left( {1 - \alpha } \right)K^{2}$ under the scenario of information sharing. Information sharing also benefits $R_{e}$.

• (c) In scenario (i), $S$’s expected profit is $\frac{1}{2}\left( {\overline{A} - c} \right)\left[ {\frac{1}{2}\left( {A_{L} - c} \right) - \frac{1}{6}\left( {\overline{A} - c} \right)} \right]$ under the scenario of no information sharing and $\frac{1}{6}\left( {A_{L} - c} \right)^{2}$ under the scenario of information sharing. Information sharing hurts $S$ since $1 < \theta = \frac{{A_{H} - c}}{{A_{L} - c}} < \frac{3}{2}$. In scenario (ii), $S$’s expected profit under the scenario of no information sharing changes to $\left( {\overline{A} - \frac{3}{2}K - c} \right)\left( {\frac{1}{2}A_{L} - \frac{1}{2}\overline{A} + K} \right)$. $\pi_{S}^{{PS^{*} }} - \pi_{S}^{{NS^{*} }} = \frac{3}{2}K^{2} - K\left( {\frac{7}{4}\overline{A} - \frac{3}{4}A_{L} - c} \right) + \frac{1}{6}\left( {A_{L} - c} \right)^{2} - \frac{1}{2}\left( {\overline{A} - c} \right)\left( {A_{L} - c} \right) + \frac{1}{2}\left( {\overline{A} - c} \right)^{2}$. Because $\frac{1}{3}\left( {A_{L} - c} \right) \le K < \frac{1}{3}\left( {\overline{A} - c} \right)$, $\pi_{S}^{{PS^{*} }} - \pi_{S}^{{NS^{*} }}$ is decreasing in $K$. Then $\mathop {\hbox{max} }\limits_{K} \left( {\pi_{S}^{{PS^{*} }} - \pi_{S}^{{NS^{*} }} } \right) = \left. {\left( {\pi_{S}^{{PS^{*} }} - \pi_{S}^{{NS^{*} }} } \right)} \right|_{{K = \frac{1}{3}\left( {A_{L} - c} \right)}} < 0.$ Information sharing hurts $S$. In scenario (iii), $S$’s expected profit under the scenario of information sharing changes to $K\left( {A_{L} - \frac{3}{2}K - c} \right)$. $\pi_{S}^{{PS^{*} }} - \pi_{S}^{{NS^{*} }} = K\left( {A_{L} - \frac{3}{2}K - c} \right) - \left( {\overline{A} - \frac{3}{2}K - c} \right)\left( {\frac{1}{2}A_{L} - \frac{1}{2}\overline{A} + K} \right) = \left( {\overline{A} - A_{L} } \right)\left[ {\frac{1}{2}\left( {\overline{A} - c} \right) - \frac{7}{4}K} \right]$. Information sharing benefits $S$ if and only if $\frac{1}{2}A_{L} - \frac{1}{6}\overline{A} - \frac{1}{3}c \le K < \frac{2}{7}\left( {\overline{A} - c} \right)$. In scenario (iv), $S$’s expected profit under the scenario of no information sharing turns to $K\left( {\overline{A} - \frac{3}{2}K - c} \right)$. Information sharing hurts $S$.

• (d) Similar to Proposition 6, the expected profit of the supply chain is increasing in the sale quantity $Q$ when $Q < \frac{1}{2}\left( {A_{L} - c} \right)$. Truthfully sharing a low demand message reduces the competition and leads a low wholesale price. The retailers’ total orders increase. In scenarios (i) and (ii), the total sale quantity under the scenario of no information sharing are $\frac{1}{2}A_{L} - \frac{1}{6}\overline{A} - \frac{1}{3}c$ and $\frac{1}{2}A_{L} - \frac{1}{2}\overline{A} + K$ respectively, which are less than the sale quantity $\frac{1}{3}\left( {A_{L} - c} \right)$ under the scenario of information sharing. Information sharing benefits the supply chain. In scenario (iii), the total sale quantity is $K$ under the scenario of information sharing. This outcome is more than $\frac{1}{2}A_{L} - \frac{1}{2}\overline{A} + K$ under the scenario of no information sharing. Information sharing also benefits the supply chain. In scenario (iv), the total sale quantity is locked by the capacity $K$ under the scenarios of both public and no information sharing. Information sharing does not affect the expected profit of the supply chain. □

Proof of Proposition 8

We discuss $S$’s ex-ante expected profit under the scenarios of public and no information sharing. Considering the following five possible scenarios: (i) If $\frac{1}{3}\left( {A_{H} - c} \right) \le K < \frac{1}{2}A_{H} - \frac{1}{6}A_{L} - \frac{1}{3}c$, $S$’s ex-ante expected profit under the scenario of information sharing is $\frac{1}{6}\left[ {r\left( {A_{H} - c} \right)^{2} + \left( {1 - r} \right)\left( {A_{L} - c} \right)^{2} } \right]$. The outcome is higher than that under the scenario of no information sharing which is $\frac{1}{6}\left( {\overline{A} - c} \right)^{2}$; (ii) If $\frac{1}{3}\left( {\overline{A} - c} \right) \le K < \frac{1}{3}\left( {A_{H} - c} \right)$, $S$’s ex-ante expected profit is $E\left[ {\pi_{S}^{{PS^{*} }} } \right] = r\left( {A_{H} - \frac{3}{2}K - c} \right)K + \frac{1}{6}\left( {1 - r} \right)\left( {A_{L} - c} \right)^{2}$ under the scenario of information sharing and $E\left[ {\pi_{S}^{{NS^{*} }} } \right] = r\frac{1}{2}\left( {\overline{A} - c} \right)K + \left( {1 - r} \right)\frac{1}{2}\left( {\overline{A} - c} \right)\left( {\frac{1}{2}A_{L} - \frac{1}{6}\overline{A} - \frac{1}{3}c} \right)$ under the scenario of no information sharing. $E\left[ {\pi_{S}^{{PS^{*} }} } \right] - E\left[ {\pi_{S}^{{NS^{*} }} } \right] = r\left[ { - \frac{3}{2}K^{2} + K\left[ {\left( {A_{H} - c} \right) - \frac{1}{2}\left( {\overline{A} - c} \right)} \right]} \right] + \left( {1 - r} \right)\left[ {\frac{1}{6}\left( {A_{L} - c} \right)^{2} - \frac{1}{4}\left( {\overline{A} - c} \right)\left( {A_{L} - c} \right) + \frac{1}{12}\left( {\overline{A} - c} \right)^{2} } \right]$. Since $\frac{1}{3}\left( {\overline{A} - c} \right) \le K < \frac{1}{3}\left( {A_{H} - c} \right)$, then $\mathop {\hbox{min} }\limits_{K} \left( {E\left[ {\pi_{S}^{{PS^{*} }} } \right] - E\left[ {\pi_{S}^{{NS^{*} }} } \right]} \right) > \left. {\left( {E\left[ {\pi_{S}^{{PS^{*} }} } \right] - E\left[ {\pi_{S}^{{NS^{*} }} } \right]} \right)} \right|_{{K = \frac{1}{3}\left( {A_{H} - c} \right)}}$. It can be reduced to $\left[ {\frac{1}{6}\left[ {r\left( {A_{L} - c} \right)^{2} + \left( {1 - r} \right)\left( {A_{L} - c} \right)^{2} - \left( {\overline{A} - c} \right)^{2} } \right]} \right] + \frac{1}{12}\left( {1 - r} \right)\left( {\overline{A} - c} \right)\left( {\overline{A} - A_{L} } \right) > 0$; (iii) If $\frac{1}{3}\left( {A_{L} - c} \right) \le K < \frac{1}{3}\left( {\overline{A} - c} \right)$, $S$’s ex-ante expected profit under the scenario of no information sharing changes to $E\left[ {\pi_{S}^{{NS^{*} }} } \right] = r\left( {\overline{A} - \frac{3}{2}K - c} \right)K + \left( {1 - r} \right)\left( {\overline{A} - \frac{3}{2}K - c} \right)\left( {\frac{1}{2}A_{L} - \frac{1}{2}\overline{A} + K} \right)$. Then $E\left[ {\pi_{S}^{{PS^{*} }} } \right] - E\left[ {\pi_{S}^{{NS^{*} }} } \right] = \left( {1 - r} \right)\left[ {\frac{3}{2}K^{2} - K\left[ {\frac{1}{4}\left( {A_{L} - c} \right) + \frac{3}{4}\left( {\overline{A} - c} \right)} \right]} \right] + \frac{1}{6}\left( {1 - r} \right)\left[ {\left( {A_{L} - c} \right)^{2} - 3\left( {A_{L} - c} \right)\left( {\overline{A} - c} \right) + 3\left( {\overline{A} - c} \right)^{2} } \right]$. Since $\frac{1}{3}\left( {A_{L} - c} \right) \le K < \frac{1}{3}\left( {\overline{A} - c} \right)$, $\mathop {\hbox{min} }\limits_{K} \left( {E\left[ {\pi_{S}^{{PS^{*} }} } \right] - E\left[ {\pi_{S}^{{NS^{*} }} } \right]} \right) = \left. {\left( {E\left[ {\pi_{S}^{{PS^{*} }} } \right] - E\left[ {\pi_{S}^{{NS^{*} }} } \right]} \right)} \right|_{{K = \frac{1}{4}\left( {\overline{A} - c} \right) + \frac{1}{12}\left( {A_{L} - c} \right)}} = \frac{1}{96}\left( {1 - r} \right)\left( {A_{L} - \overline{A} } \right)\left[ {15\left( {A_{L} - c} \right) - 39\left( {\overline{A} - c} \right)} \right] > 0$; (iv) If $\frac{1}{2}A_{L} - \frac{1}{6}\overline{A} - \frac{1}{3}c \le K < \frac{1}{3}\left( {A_{L} - c} \right)$, $S$’s ex-ante expected profit under the scenario of information sharing changes to $r\left( {A_{H} - \frac{3}{2}K - c} \right)K + \left( {1 - r} \right)\left( {A_{L} - \frac{3}{2}K - c} \right)K$, which is equal to $\left( {\overline{A} - \frac{3}{2}K - c} \right)K$. This outcome is obviously more than that under the scenario of no information sharing which is $r\left( {\overline{A} - \frac{3}{2}K - c} \right)K + \left( {1 - r} \right)\left( {\overline{A} - \frac{3}{2}K - c} \right)\left( {\frac{1}{2}A_{L} - \frac{1}{2}\overline{A} + K} \right)$; (v) If $K < \frac{1}{2}A_{L} - \frac{1}{6}\overline{A} - \frac{1}{3}c$, $S$’s ex-ante expected profit is $\left( {\overline{A} - \frac{3}{2}K - c} \right)K$ under the scenarios of both public and no information sharing. As a result, information sharing always benefits $S$. □

Proof of Proposition 9

It is obvious from Propositions 4 and 8.□