# Risk Analysis of a Two-Level Supply Chain Subject to Misplaced Inventory
## 3 Decision Policies of the Supply Chain with Information Asymmetry
### The Benefits of Information Sharing  
Even though information sharing is beneficial to the centralized supply chain, the supplier may have no incentive to induce the retailer to share information about inventory errors. By Proposition 1, we quantify the benefits of information sharing for the supply chain participants.  
**Proposition 1**: _Compared with the case of information asymmetry, information sharing on inventory errors leads to the following results:_  
\begin{tabular}{l l} (1) & _the supplier lowers her wholesale price, i.e.,_ \(w_{2}^{*}\leq w_{1}^{*}\)_;_ \\ (2) & _the retailer orders more, i.e.,_ \(Q_{2}^{*}\geq Q_{1}^{*}\)_;_ \\ (3) & _the retailer always benefits from information sharing, i.e.,_ \(\pi_{2}^{\eta}\geq\pi_{1}^{\eta}\)_;_ \\ (4) & _the supplier benefits from information sharing if and only if:_ \\ \end{tabular}  
\[E\leq\left\{\begin{array}{cl}\frac{\eta D_{\max}\beta^{2}(p-s)}{4(1-\beta) ^{2}},&\mbox{if }\beta\leq\frac{p-c}{2(p-s)}\\ \frac{\eta D_{\max}A_{1}}{4(1-\beta)^{2}(p-s)},&\mbox{if }\frac{p-c}{2(p-s)}\leq\beta\leq \frac{p-c}{p-s}\\ 0,&\mbox{otherwise}\end{array}\right.. \tag{11}\]  
_where \(A_{1}=(p-s)(1-\beta)+s-c\)._  
**Proof.** From Equations (A3) and (10), we obtain:  
\[Q_{2}^{*}-Q_{1}^{*}=\left\{\begin{array}{ll}\frac{\eta D_{\max}(p-s)\beta}{2(1- \beta)^{2}(p-s)},&\text{if }\beta\leq\frac{p-c}{2(p-s)}\\ \frac{\eta D_{\max}((p-s)(1-\beta)+s-c)}{2(1-\beta)^{2}(p-s)},&\text{if }\frac{p-c}{2(p-s)}\leq \beta\leq\frac{p-c}{p-s}\\ 0,&\text{otherwise}\end{array}\right.. \tag{12}\]  
It is obvious that \(Q_{2}^{*}-Q_{1}^{*}\geq 0\), indicating that the retailer orders more after information sharing. Moreover, we observe that the retailer has a high tendency to stop ordering (i.e., \(\beta>(p-c)/(2(p-s))\)) when he has private information about inventory errors, whereas he can bear a higher level of inventory inaccuracy (i.e., \(\beta>(p-c)/(p-s)\)) after sharing information to the supplier.  
Moreover, Equation (9) shows that \(w_{2}\) decreases in \(\beta\), where \(\beta=0\) indicates the situation of no inventory errors. Therefore, we infer that the supplier charges a lower wholesale price when she has public information about inventory errors. Since the wholesale price is decreased and the order quantity is increased after information sharing, the performance of the risk-averse retailer is improved from information sharing. Meanwhile, it is straightforward to obtain \(\pi_{2}^{\eta}-\pi_{1}^{\eta}=(p-s)(1-\beta)^{2}(Q_{2}^{*2}-Q_{1}^{*2})/(2\eta D _{\max})\geq 0\) from Equation (12), which means that the retailer always benefits from information sharing.  
Even though the supplier receives more orders from the retailer, she may not benefit from information sharing due to its cost, \(E\). We define \(\Delta\pi_{IS}^{S}\) as the supplier's benefit from information sharing, where \(\Delta\pi_{IS}^{S}=\pi_{2}^{S}-\pi_{1}^{S}\). From \(\Delta\pi_{IS}^{S}\geq 0\), we derive the conditions under which the supplier can benefit from information sharing. \(\square\)  
Obviously, the supplier cannot benefit from information sharing when the retailer has serious inventory errors (i.e., \(\beta>(p-c)/(p-s)\)). In this situation, the retailer will stop placing orders and thereby the supplier will avoid the additional cost of information sharing. When the inventory errors are not serious (\(\beta<(p-c)/(p-s)\)), the supplier is motivated by a higher order quantity and is willing to induce the retailer to share information, as long as the benefits exceeds the costs. Since the supplier benefits less from information sharing when the magnitude of inventory errors becomes smaller, supply chain participants may not share information when there are few inventory errors.