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doi: 10.3934/jimo.2021207
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## Information sharing when competing manufacturers adopt asymmetric channel in an e-tailer

 1 School of Economics and Business Administration, Chongqing University, Chongqing, China 2 School of Economics and Management, Chongqing University of Arts and Sciences, Chongqing, China

*Corresponding author: Yong Wang

Received  October 2020 Revised  July 2021 Early access December 2021

Fund Project: This work was supported by the National Natural Science Foundation of China (grant number 71672015)

This paper investigates the incentive for information sharing when competing manufacturers sell substitute products through the marketplace channel and the reseller channel respectively. Our analysis shows that the e-tailer's incentive to share information strongly depends on the platform fee, competition intensity, and different information sharing scenarios. If competition intensity is small, or competition intensity is large and the platform fee is enough large, the e-tailer has incentive to alone share information with the manufacturer who is from the marketplace channel; if competition intensity is moderate and the platform fee is small, or competition intensity is large but the platform fee is moderate, it has incentive to share information with both manufacturers; if competition intensity is large but the platform fee is small, it has no incentive to share information. The results also indicate that the double marginalization effect of information sharing is a promoting factor to share information under linear cost, which is different from previous literature. Additionally, we find that the main qualitative insights from the base model are robust even if one monopolist manufacturer employs both channels. And we also compare the incentive of information sharing under asymmetric channel with that under symmetric channel.

Citation: Guoqiang Shi, Yong Wang, Dejian Xia, Yanfei Zhao. Information sharing when competing manufacturers adopt asymmetric channel in an e-tailer. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021207
##### References:
 [1] A. Y. K. Chua, How Web 2.0 supports customer relationship management in Amazon, International J. Electronic Customer Relationship Management, 5 (2011), 288-304.  doi: 10.1504/ijecrm.2011.044693. [2] R. N. Clarke, Collusion and the incentives for information sharing, The Bell Journal of Economics, 14 (1983), 383-394.  doi: 10.2307/3003640. [3] A. T. Coughlan and B. Wernerfelt, On credible delegation by oligopolists: A discussion of distribution channel management, Management Science, 35 (1989), 226-239.  doi: 10.1287/mnsc.35.2.226. [4] S. E. Fawcett, C. Wallin, C. Allred, A. M. Fawcett and G. M. Magnan, Information technology as an enabler of supply chain collaboration: A dynamic-capabilities perspective, J. Supply Chain Management, 47 (2011), 38-59.  doi: 10.1111/j.1745-493x.2010.03213.x. [5] E. Gal-Or, Information sharing in oligopoly, Econometrica, 53 (1985), 329-343.  doi: 10.2307/1911239. [6] A. Y. Ha, Q. Tian and S. Tong, Information sharing in competing supply chains with production cost reduction, Manufacturing & Service Operations Management, 19 (2017), 246-262.  doi: 10.1287/msom.2016.0607. [7] A. Y. Ha, S. Tong and H. Zhang, Sharing demand information in competing supply chains with production diseconomies, Management Science, 57 (2011), 566-581.  doi: 10.1287/mnsc.1100.1295. [8] L. Hao and M. Fan, An analysis of pricing models in the electronic book market, MIS Quarterly, 38 (2014), 1017-1032.  doi: 10.25300/misq/2014/38.4.04. [9] S. Huang, X. Guan and Y.-J. Chen, Retailer information sharing with supplier encroachment, Production and Operations Management, 27 (2018), 1133-1147.  doi: 10.1111/poms.12860. [10] A. Jain and M. Sohoni, Should firms conceal information when dealing with common suppliers?, Naval Res. Logist., 62 (2015), 1-15.  doi: 10.1002/nav.21609. [11] K. Jerath and Z. J. Zhang, Store within a store, J. Marketing Research, 47 (2010), 748-763.  doi: 10.1509/jmkr.47.4.748. [12] B. Jiang, K. Jerath and K. Srinivasan, Firm strategies in the "mid tail" of platform-based retailing, Marketing Science, 30 (2011), 757-775.  doi: 10.1287/mksc.1110.0656. [13] B. Jiang, L. Tian, Y. Xu and F. Zhang, To share or not to share: Demand forecast sharing in a distribution channel, Marketing Science, 35 (2016), 800-809.  doi: 10.1287/mksc.2016.0981. [14] L. Jiang and Z. Hao, Incentive-driven information dissemination in two-tier supply chains, Manufacturing & Service Operations Management, 18 (2016), 393-413.  doi: 10.1287/msom.2016.0575. [15] H. L. Lee and S. Whang, Information sharing in a supply chain, International Journal of Manufacturing Technology and Management, 1 (2000), 79-93.  doi: 10.1504/ijmtm.2000.001329. [16] L. Li, Cournot oligopoly with information sharing, The RAND Journal of Economics, 16 (1985), 521-536. [17] L. Li, Information sharing in a supply chain with horizontal competition, Management Science, 48 (2002), 1196-1212.  doi: 10.1287/mnsc.48.9.1196.177. [18] L. Li and H. Zhang, Confidentiality and information sharing in supply chain coordination, Management Science, 54 (2008), 1467-1481.  doi: 10.1287/mnsc.1070.0851. [19] T. Li and H. Zhang, Information sharing in a supply chain with a make-to-stock manufacturer, Omega, 50 (2015), 115-125.  doi: 10.1016/j.omega.2014.08.001. [20] Z. Li, S. M. Gilbert and G. Lai, Supplier encroachment under asymmetric information, Management Science, 60 (2014), 449-462.  doi: 10.1287/mnsc.2013.1780. [21] W. Shang, A. Y. Ha and S. Tong, Information sharing in a supply chain with a common retailer, Management Science, 62 (2016), 245-263.  doi: 10.1287/mnsc.2014.2127. [22] X. Sun, W. Tang, J. Zhang and J. Chen, The impact of quantity-based cost decline on supplier encroachment, Transportation Research Part E: Logistics and Transportation Review, 147 (2021), 102245.  doi: 10.1016/j.tre.2021.102245. [23] Y. Tan and J. E. Carrillo, Strategic analysis of the agency model for digital goods, Production and Operations Management, 26 (2017), 724-741.  doi: 10.1111/poms.12595. [24] L. Tian, A. J. Vakharia, Y. R. Tan and Y. Xu, Marketplace, reseller, or hybrid: Strategic analysis of an emerging e-commerce model, Production and Operations Management, 27 (2018), 1595-1610.  doi: 10.1111/poms.12885. [25] X. Vives, Duopoly information equilibrium: Cournot and Bertrand, J. Econom. Theory, 34 (1984), 71-94.  doi: 10.1016/0022-0531(84)90162-5. [26] Y. Wang, W. Tang and R. Zhao, Information sharing and information concealment in the presence of a dominant retailer, Computers & Industrial Engineering, 121 (2018), 36-50.  doi: 10.1016/j.cie.2018.04.039. [27] J. Wei, J. Lu and J. Zhao, Interactions of competing manufacturers' leader-follower relationship and sales format on online platforms, European J. Oper. Res., 280 (2020), 508-522.  doi: 10.1016/j.ejor.2019.07.048. [28] L. Wei, J. Zhang and G. Zhu, Incentive of retailer information sharing on manufacturer volume flexibility choice, Omega, 100 (2021), 102210.  doi: 10.1016/j.omega.2020.102210. [29] Y. Yan, R. Zhao and Z. Liu, Strategic introduction of the marketplace channel under spillovers from online to offline sales, European J. Oper. Res., 267 (2018), 65-77.  doi: 10.1016/j.ejor.2017.11.011. [30] Y. Zennyo, Strategic contracting and hybrid use of agency and wholesale contracts in e-commerce platforms, European J. Oper. Res., 281 (2020), 231-239.  doi: 10.1016/j.ejor.2019.08.026. [31] H. Zhang, Vertical information exchange in a supply chain with duopoly retailers, Production and Operations Management, 11 (2002), 531-546.  doi: 10.1111/j.1937-5956.2002.tb00476.x. [32] J. Zhang and J. Chen, Information sharing in a make-to-stock supply chain, J. Ind. Manag. Optim., 10 (2014), 1169-1189.  doi: 10.3934/jimo.2014.10.1169. [33] J. Zhang, S. Li, S. Zhang and R. Dai, Manufacturer encroachment with quality decision under asymmetric demand information, European J. Oper. Res., 273 (2019), 217-236.  doi: 10.1016/j.ejor.2018.08.002. [34] Q. Zhang, W. Tang, G. Zaccour and J. Zhang, Should a manufacturer give up pricing power in a vertical information-sharing channel?, European J. Oper. Res., 276 (2019), 910-928.  doi: 10.1016/j.ejor.2019.01.054. [35] S. Zhang and J. Zhang, Agency selling or reselling: E-tailer information sharing with supplier offline entry, European J. Oper. Res., 280 (2020), 134-151.  doi: 10.1016/j.ejor.2019.07.003.

show all references

##### References:
 [1] A. Y. K. Chua, How Web 2.0 supports customer relationship management in Amazon, International J. Electronic Customer Relationship Management, 5 (2011), 288-304.  doi: 10.1504/ijecrm.2011.044693. [2] R. N. Clarke, Collusion and the incentives for information sharing, The Bell Journal of Economics, 14 (1983), 383-394.  doi: 10.2307/3003640. [3] A. T. Coughlan and B. Wernerfelt, On credible delegation by oligopolists: A discussion of distribution channel management, Management Science, 35 (1989), 226-239.  doi: 10.1287/mnsc.35.2.226. [4] S. E. Fawcett, C. Wallin, C. Allred, A. M. Fawcett and G. M. Magnan, Information technology as an enabler of supply chain collaboration: A dynamic-capabilities perspective, J. Supply Chain Management, 47 (2011), 38-59.  doi: 10.1111/j.1745-493x.2010.03213.x. [5] E. Gal-Or, Information sharing in oligopoly, Econometrica, 53 (1985), 329-343.  doi: 10.2307/1911239. [6] A. Y. Ha, Q. Tian and S. Tong, Information sharing in competing supply chains with production cost reduction, Manufacturing & Service Operations Management, 19 (2017), 246-262.  doi: 10.1287/msom.2016.0607. [7] A. Y. Ha, S. Tong and H. Zhang, Sharing demand information in competing supply chains with production diseconomies, Management Science, 57 (2011), 566-581.  doi: 10.1287/mnsc.1100.1295. [8] L. Hao and M. Fan, An analysis of pricing models in the electronic book market, MIS Quarterly, 38 (2014), 1017-1032.  doi: 10.25300/misq/2014/38.4.04. [9] S. Huang, X. Guan and Y.-J. Chen, Retailer information sharing with supplier encroachment, Production and Operations Management, 27 (2018), 1133-1147.  doi: 10.1111/poms.12860. [10] A. Jain and M. Sohoni, Should firms conceal information when dealing with common suppliers?, Naval Res. Logist., 62 (2015), 1-15.  doi: 10.1002/nav.21609. [11] K. Jerath and Z. J. Zhang, Store within a store, J. Marketing Research, 47 (2010), 748-763.  doi: 10.1509/jmkr.47.4.748. [12] B. Jiang, K. Jerath and K. Srinivasan, Firm strategies in the "mid tail" of platform-based retailing, Marketing Science, 30 (2011), 757-775.  doi: 10.1287/mksc.1110.0656. [13] B. Jiang, L. Tian, Y. Xu and F. Zhang, To share or not to share: Demand forecast sharing in a distribution channel, Marketing Science, 35 (2016), 800-809.  doi: 10.1287/mksc.2016.0981. [14] L. Jiang and Z. Hao, Incentive-driven information dissemination in two-tier supply chains, Manufacturing & Service Operations Management, 18 (2016), 393-413.  doi: 10.1287/msom.2016.0575. [15] H. L. Lee and S. Whang, Information sharing in a supply chain, International Journal of Manufacturing Technology and Management, 1 (2000), 79-93.  doi: 10.1504/ijmtm.2000.001329. [16] L. Li, Cournot oligopoly with information sharing, The RAND Journal of Economics, 16 (1985), 521-536. [17] L. Li, Information sharing in a supply chain with horizontal competition, Management Science, 48 (2002), 1196-1212.  doi: 10.1287/mnsc.48.9.1196.177. [18] L. Li and H. Zhang, Confidentiality and information sharing in supply chain coordination, Management Science, 54 (2008), 1467-1481.  doi: 10.1287/mnsc.1070.0851. [19] T. Li and H. Zhang, Information sharing in a supply chain with a make-to-stock manufacturer, Omega, 50 (2015), 115-125.  doi: 10.1016/j.omega.2014.08.001. [20] Z. Li, S. M. Gilbert and G. Lai, Supplier encroachment under asymmetric information, Management Science, 60 (2014), 449-462.  doi: 10.1287/mnsc.2013.1780. [21] W. Shang, A. Y. Ha and S. Tong, Information sharing in a supply chain with a common retailer, Management Science, 62 (2016), 245-263.  doi: 10.1287/mnsc.2014.2127. [22] X. Sun, W. Tang, J. Zhang and J. Chen, The impact of quantity-based cost decline on supplier encroachment, Transportation Research Part E: Logistics and Transportation Review, 147 (2021), 102245.  doi: 10.1016/j.tre.2021.102245. [23] Y. Tan and J. E. Carrillo, Strategic analysis of the agency model for digital goods, Production and Operations Management, 26 (2017), 724-741.  doi: 10.1111/poms.12595. [24] L. Tian, A. J. Vakharia, Y. R. Tan and Y. Xu, Marketplace, reseller, or hybrid: Strategic analysis of an emerging e-commerce model, Production and Operations Management, 27 (2018), 1595-1610.  doi: 10.1111/poms.12885. [25] X. Vives, Duopoly information equilibrium: Cournot and Bertrand, J. Econom. Theory, 34 (1984), 71-94.  doi: 10.1016/0022-0531(84)90162-5. [26] Y. Wang, W. Tang and R. Zhao, Information sharing and information concealment in the presence of a dominant retailer, Computers & Industrial Engineering, 121 (2018), 36-50.  doi: 10.1016/j.cie.2018.04.039. [27] J. Wei, J. Lu and J. Zhao, Interactions of competing manufacturers' leader-follower relationship and sales format on online platforms, European J. Oper. Res., 280 (2020), 508-522.  doi: 10.1016/j.ejor.2019.07.048. [28] L. Wei, J. Zhang and G. Zhu, Incentive of retailer information sharing on manufacturer volume flexibility choice, Omega, 100 (2021), 102210.  doi: 10.1016/j.omega.2020.102210. [29] Y. Yan, R. Zhao and Z. Liu, Strategic introduction of the marketplace channel under spillovers from online to offline sales, European J. Oper. Res., 267 (2018), 65-77.  doi: 10.1016/j.ejor.2017.11.011. [30] Y. Zennyo, Strategic contracting and hybrid use of agency and wholesale contracts in e-commerce platforms, European J. Oper. Res., 281 (2020), 231-239.  doi: 10.1016/j.ejor.2019.08.026. [31] H. Zhang, Vertical information exchange in a supply chain with duopoly retailers, Production and Operations Management, 11 (2002), 531-546.  doi: 10.1111/j.1937-5956.2002.tb00476.x. [32] J. Zhang and J. Chen, Information sharing in a make-to-stock supply chain, J. Ind. Manag. Optim., 10 (2014), 1169-1189.  doi: 10.3934/jimo.2014.10.1169. [33] J. Zhang, S. Li, S. Zhang and R. Dai, Manufacturer encroachment with quality decision under asymmetric demand information, European J. Oper. Res., 273 (2019), 217-236.  doi: 10.1016/j.ejor.2018.08.002. [34] Q. Zhang, W. Tang, G. Zaccour and J. Zhang, Should a manufacturer give up pricing power in a vertical information-sharing channel?, European J. Oper. Res., 276 (2019), 910-928.  doi: 10.1016/j.ejor.2019.01.054. [35] S. Zhang and J. Zhang, Agency selling or reselling: E-tailer information sharing with supplier offline entry, European J. Oper. Res., 280 (2020), 134-151.  doi: 10.1016/j.ejor.2019.07.003.
Supply chain structure. Note: The dashed line indicates that the retailing quantity in the marketplace channel is decided by manufacturer 2
The impacts of information sharing with only manufacturer 2 on the e-tailer
The impacts of information sharing on the supply chain payoffs. (a) Comparing $\left( N, I \right)$ with $\left( N, N \right)$; (b) Comparing $\left( I, I \right)$ with $\left( N, N \right)$; (c) Comparing $\left( I, I \right)$ with $\left( N, I \right)$
Equilibrium information sharing decisions. Note: The solid lines indicate the thresholds of Equilibrium information sharing decisions. The dashed line in $\left( N, I \right)$ area represents the threshold in which $\pi _{e}^{NN} = \pi _{e}^{NI}$ and the dashed line in $\left( I, I \right)$ area represents the threshold in which ${{V}^{NI}} = {{V}^{NN}}$
Information sharing in a monopolist model. (a) Information sharing voluntarily; (b) Equilibrium information sharing decisions
${{f}_{1}}\left( \beta , \gamma \right)>0$ when $\beta \in \left( 0, 1 \right)$ and $\gamma \in \left( 0, 1 \right)$
Thresholds $\beta ^{\left( 3 \right)}$ and $\beta ^{\left( 4 \right)}$
Threshold $\beta ^{\left( 5 \right)}$
Threshold $\beta ^{\left( 6 \right)}$
Operation equilibrium
 $\left( {{x}_{m1}}, {{x}_{m2}} \right)$ Equilibrium outcome Ex-ante profit $\left( N, N \right)$ ${{w}^{NN}}=A\alpha$ $\pi _{e}^{NN}={{\bar{\pi }}_{e}}+\frac{1}{4}\delta$ $q_{1}^{NN}=B\alpha +\frac{1}{2}E\left[ \theta |Y \right]$ $\pi _{m1}^{NN}={{\bar{\pi }}_{m1}}$ $q_{2}^{NN}=C\alpha$ $\pi _{m2}^{NN}={{\bar{\pi }}_{m2}}$ $\left( I, N \right)$ ${{w}^{IN}}=A\alpha +\frac{4-\left( 1+\beta \right){{\gamma }^{2}}}{4-\left( 2+2\beta \right)\gamma }AE\left[ \theta |Y \right]$ $\pi _{e}^{IN}={{\bar{\pi }}_{e}}+\frac{4-\left( {{\gamma }^{2}}-8\gamma \right)\beta }{64}\delta$ $q_{1}^{IN}=B\alpha +\frac{1}{4}E\left[ \theta |Y \right]$ $\pi _{m1}^{IN}={{\bar{\pi }}_{m1}}+\frac{4-\left( 1+\beta \right){{\gamma }^{2}}}{32}\delta$ $q_{2}^{IN}=C\alpha$ $\pi _{m2}^{IN}={{\bar{\pi }}_{m2}}$ $\left( N, I \right)$ ${{w}^{NI}}=A\alpha$ $\pi _{e}^{NI}={{\bar{\pi }}_{e}}+\frac{\left( \beta +{{\beta }^{2}} \right){{\gamma }^{3}}+\left( 1-{{\beta }^{2}}-\beta \right){{\gamma }^{2}}-4\left( 1+\beta \right)\gamma +4\beta +4}{{{\left[ 4-\left( 1+\beta \right){{\gamma }^{2}} \right]}^{2}}}\delta$ $q_{1}^{NI}=B\alpha +2BE\left[ \theta |Y \right]$ $\pi _{m1}^{NI}={{\bar{\pi }}_{m1}}$ $q_{2}^{NI}=C\alpha +\frac{\left( 8-4\gamma \right)E\left[ \theta |Y \right]}{16-\left( 4+4\beta \right){{\gamma }^{2}}}$ $\pi _{m2}^{NI}={{\bar{\pi }}_{m2}}+\frac{{{\left( 2-\gamma \right)}^{2}}\left( 1-\beta \right)}{{{\left[ 4-\left( 1+\beta \right){{\gamma }^{2}} \right]}^{2}}}\delta$ $\left( I, I \right)$ ${{w}^{II}}=A\alpha +AE\left[ \theta |Y \right]$ $\pi _{e}^{II}={{\bar{\pi }}_{e}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $q_{1}^{II}=B\alpha +BE\left[ \theta |Y \right]$ $\pi _{m1}^{II}={{\bar{\pi }}_{m1}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $q_{2}^{II}=C\alpha +CE\left[ \theta |Y \right]$ $\pi _{m2}^{II}={{\bar{\pi }}_{m2}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ Note: $A=\frac{2-\gamma -\beta \gamma }{4}$, $B=\frac{2-\left( 1+\beta \right)\gamma }{8-\left( 2+2\beta \right){{\gamma }^{2}}}$, $C=\frac{8-\left( 1+\beta \right){{\gamma }^{2}}-2\gamma }{2\left[ 8-\left( 2+2\beta \right){{\gamma }^{2}} \right]}$, ${{\bar{\pi }}_{e}}=\frac{3\beta \left( 1+{{\beta }^{2}} \right){{\gamma }^{4}}+4\beta \left( 1+\beta \right){{\gamma }^{3}}+\left( 4-28{{\beta }^{2}}-28\beta \right){{\gamma }^{2}}-16\left( 1+\beta \right)\gamma +64\beta +16}{16{{\left[ 4-\left( 1+\beta \right){{\gamma }^{2}} \right]}^{2}}}{{\alpha }^{2}}$, ${{\bar{\pi }}_{m1}}=\frac{{{\left[ 2-\left( 1+\beta \right)\gamma \right]}^{2}}{{\alpha }^{2}}}{32-\left( 8+8\beta \right){{\gamma }^{2}}}$, and ${{\bar{\pi }}_{m2}}=\frac{\left( 1-\beta \right){{\left[ 8-\left( 1+\beta \right){{\gamma }^{2}}-2\gamma \right]}^{2}}}{16{{\left[ 4-\left( 1+\beta \right){{\gamma }^{2}} \right]}^{2}}}{{\alpha }^{2}}$. We let ${{\bar{\pi }}_{e}}$, ${{\bar{\pi }}_{m1}}$ and ${{\bar{\pi }}_{m2}}$ be the deterministic payoffs (in the absence of uncertainty). Let $\delta =E\left[ {{\left( E\left[ \theta |Y \right] \right)}^{2}} \right]$ for simplicity.
 $\left( {{x}_{m1}}, {{x}_{m2}} \right)$ Equilibrium outcome Ex-ante profit $\left( N, N \right)$ ${{w}^{NN}}=A\alpha$ $\pi _{e}^{NN}={{\bar{\pi }}_{e}}+\frac{1}{4}\delta$ $q_{1}^{NN}=B\alpha +\frac{1}{2}E\left[ \theta |Y \right]$ $\pi _{m1}^{NN}={{\bar{\pi }}_{m1}}$ $q_{2}^{NN}=C\alpha$ $\pi _{m2}^{NN}={{\bar{\pi }}_{m2}}$ $\left( I, N \right)$ ${{w}^{IN}}=A\alpha +\frac{4-\left( 1+\beta \right){{\gamma }^{2}}}{4-\left( 2+2\beta \right)\gamma }AE\left[ \theta |Y \right]$ $\pi _{e}^{IN}={{\bar{\pi }}_{e}}+\frac{4-\left( {{\gamma }^{2}}-8\gamma \right)\beta }{64}\delta$ $q_{1}^{IN}=B\alpha +\frac{1}{4}E\left[ \theta |Y \right]$ $\pi _{m1}^{IN}={{\bar{\pi }}_{m1}}+\frac{4-\left( 1+\beta \right){{\gamma }^{2}}}{32}\delta$ $q_{2}^{IN}=C\alpha$ $\pi _{m2}^{IN}={{\bar{\pi }}_{m2}}$ $\left( N, I \right)$ ${{w}^{NI}}=A\alpha$ $\pi _{e}^{NI}={{\bar{\pi }}_{e}}+\frac{\left( \beta +{{\beta }^{2}} \right){{\gamma }^{3}}+\left( 1-{{\beta }^{2}}-\beta \right){{\gamma }^{2}}-4\left( 1+\beta \right)\gamma +4\beta +4}{{{\left[ 4-\left( 1+\beta \right){{\gamma }^{2}} \right]}^{2}}}\delta$ $q_{1}^{NI}=B\alpha +2BE\left[ \theta |Y \right]$ $\pi _{m1}^{NI}={{\bar{\pi }}_{m1}}$ $q_{2}^{NI}=C\alpha +\frac{\left( 8-4\gamma \right)E\left[ \theta |Y \right]}{16-\left( 4+4\beta \right){{\gamma }^{2}}}$ $\pi _{m2}^{NI}={{\bar{\pi }}_{m2}}+\frac{{{\left( 2-\gamma \right)}^{2}}\left( 1-\beta \right)}{{{\left[ 4-\left( 1+\beta \right){{\gamma }^{2}} \right]}^{2}}}\delta$ $\left( I, I \right)$ ${{w}^{II}}=A\alpha +AE\left[ \theta |Y \right]$ $\pi _{e}^{II}={{\bar{\pi }}_{e}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $q_{1}^{II}=B\alpha +BE\left[ \theta |Y \right]$ $\pi _{m1}^{II}={{\bar{\pi }}_{m1}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $q_{2}^{II}=C\alpha +CE\left[ \theta |Y \right]$ $\pi _{m2}^{II}={{\bar{\pi }}_{m2}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ Note: $A=\frac{2-\gamma -\beta \gamma }{4}$, $B=\frac{2-\left( 1+\beta \right)\gamma }{8-\left( 2+2\beta \right){{\gamma }^{2}}}$, $C=\frac{8-\left( 1+\beta \right){{\gamma }^{2}}-2\gamma }{2\left[ 8-\left( 2+2\beta \right){{\gamma }^{2}} \right]}$, ${{\bar{\pi }}_{e}}=\frac{3\beta \left( 1+{{\beta }^{2}} \right){{\gamma }^{4}}+4\beta \left( 1+\beta \right){{\gamma }^{3}}+\left( 4-28{{\beta }^{2}}-28\beta \right){{\gamma }^{2}}-16\left( 1+\beta \right)\gamma +64\beta +16}{16{{\left[ 4-\left( 1+\beta \right){{\gamma }^{2}} \right]}^{2}}}{{\alpha }^{2}}$, ${{\bar{\pi }}_{m1}}=\frac{{{\left[ 2-\left( 1+\beta \right)\gamma \right]}^{2}}{{\alpha }^{2}}}{32-\left( 8+8\beta \right){{\gamma }^{2}}}$, and ${{\bar{\pi }}_{m2}}=\frac{\left( 1-\beta \right){{\left[ 8-\left( 1+\beta \right){{\gamma }^{2}}-2\gamma \right]}^{2}}}{16{{\left[ 4-\left( 1+\beta \right){{\gamma }^{2}} \right]}^{2}}}{{\alpha }^{2}}$. We let ${{\bar{\pi }}_{e}}$, ${{\bar{\pi }}_{m1}}$ and ${{\bar{\pi }}_{m2}}$ be the deterministic payoffs (in the absence of uncertainty). Let $\delta =E\left[ {{\left( E\left[ \theta |Y \right] \right)}^{2}} \right]$ for simplicity.
Operation equilibrium
 ${{x}_{m}}$ Equilibrium outcome Ex-ante profit $N$ ${{w}^{N}}=D\alpha$ $\pi _{e}^{N}={{\bar{\pi }}_{e}}+\frac{1}{4}\delta$ $q_{1}^{N}=F\alpha +\frac{1}{2}E\left[ \theta |Y \right]$ $\pi _{m}^{N}={{\bar{\pi }}_{m}}$ $q_{2}^{N}=G\alpha$ $I$ ${{w}^{I}}=D\alpha +DE\left[ \theta |Y \right]$ $\pi _{e}^{I}={{\bar{\pi }}_{e}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $q_{1}^{I}=F\alpha +FE\left[ \theta |Y \right]$ $\pi _{m}^{I}={{\bar{\pi }}_{m}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $q_{2}^{I}=G\alpha +GE\left[ \theta |Y \right]$ Note: $D \ = \ \frac{8+{{\left( 1+\beta \right)}^{2}}{{\gamma }^{3}}-4{{\gamma }^{2}}-8\beta \gamma }{16-\left( 6+2\beta \right){{\gamma }^{2}}}$, $F=\frac{2\left( 1-\gamma \right)}{8-\left( 3+\beta \right){{\gamma }^{2}}}$, $G=\frac{8-\left( 1+\beta \right){{\gamma }^{2}}-2\gamma }{2\left[ 8-\left( 3+\beta \right){{\gamma }^{2}} \right]}$, ${{\bar{\pi }}_{e}}= {\small \frac{\left( 5\beta +6{{\beta }^{2}}+{{\beta }^{3}} \right){{\gamma }^{4}}+8\beta {{\gamma }^{3}}+\left( 16-16{{\beta }^{2}}-52\beta \right){{\gamma }^{2}}-32\gamma +64\beta +16}{4{{\left[ 8-\left( 3+\beta \right){{\gamma }^{2}} \right]}^{2}}}{{\alpha }^{2}}}$, and {\small ${{\bar{\pi }}_{m}}=\frac{{{\left( 1+\beta \right)}^{2}}{{\gamma }^{2}}-8\gamma -8\beta +12}{32-\left( 12+4\beta \right){{\gamma }^{2}}}{{\alpha }^{2}}$. We let ${{\bar{\pi }}_{e}}$ and ${{\bar{\pi }}_{m}}$ be the deterministic payoffs (in the absence of uncertainty). Let $\delta =E\left[ {{\left( E\left[ \theta |Y \right] \right)}^{2}} \right]$ for simplicity.
 ${{x}_{m}}$ Equilibrium outcome Ex-ante profit $N$ ${{w}^{N}}=D\alpha$ $\pi _{e}^{N}={{\bar{\pi }}_{e}}+\frac{1}{4}\delta$ $q_{1}^{N}=F\alpha +\frac{1}{2}E\left[ \theta |Y \right]$ $\pi _{m}^{N}={{\bar{\pi }}_{m}}$ $q_{2}^{N}=G\alpha$ $I$ ${{w}^{I}}=D\alpha +DE\left[ \theta |Y \right]$ $\pi _{e}^{I}={{\bar{\pi }}_{e}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $q_{1}^{I}=F\alpha +FE\left[ \theta |Y \right]$ $\pi _{m}^{I}={{\bar{\pi }}_{m}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $q_{2}^{I}=G\alpha +GE\left[ \theta |Y \right]$ Note: $D \ = \ \frac{8+{{\left( 1+\beta \right)}^{2}}{{\gamma }^{3}}-4{{\gamma }^{2}}-8\beta \gamma }{16-\left( 6+2\beta \right){{\gamma }^{2}}}$, $F=\frac{2\left( 1-\gamma \right)}{8-\left( 3+\beta \right){{\gamma }^{2}}}$, $G=\frac{8-\left( 1+\beta \right){{\gamma }^{2}}-2\gamma }{2\left[ 8-\left( 3+\beta \right){{\gamma }^{2}} \right]}$, ${{\bar{\pi }}_{e}}= {\small \frac{\left( 5\beta +6{{\beta }^{2}}+{{\beta }^{3}} \right){{\gamma }^{4}}+8\beta {{\gamma }^{3}}+\left( 16-16{{\beta }^{2}}-52\beta \right){{\gamma }^{2}}-32\gamma +64\beta +16}{4{{\left[ 8-\left( 3+\beta \right){{\gamma }^{2}} \right]}^{2}}}{{\alpha }^{2}}}$, and {\small ${{\bar{\pi }}_{m}}=\frac{{{\left( 1+\beta \right)}^{2}}{{\gamma }^{2}}-8\gamma -8\beta +12}{32-\left( 12+4\beta \right){{\gamma }^{2}}}{{\alpha }^{2}}$. We let ${{\bar{\pi }}_{e}}$ and ${{\bar{\pi }}_{m}}$ be the deterministic payoffs (in the absence of uncertainty). Let $\delta =E\left[ {{\left( E\left[ \theta |Y \right] \right)}^{2}} \right]$ for simplicity.
Operation equilibrium under the pure reseller channel
 $\left( {{x}_{m1}}, {{x}_{m2}} \right)$ Equilibrium outcome Ex-ante profit $\left( N, N \right)$ $w_{1}^{NN}=H\alpha$ $\pi _{e}^{NN}={{\bar{\pi }}_{e}}+\frac{1}{2\left( 1+\gamma \right)}\delta$ $w_{2}^{NN}=H\alpha$ $\pi _{m1}^{NN}={{\bar{\pi }}_{m1}}$ $q_{1}^{NN}=J\alpha +\frac{E\left[ \theta |Y \right]}{2\left( 1+\gamma \right)}$ $\pi _{m2}^{NN}={{\bar{\pi }}_{m2}}$ $q_{2}^{NN}=J\alpha +\frac{E\left[ \theta |Y \right]}{2\left( 1+\gamma \right)}$ $\left( N, I \right)$ $w_{1}^{NI}=H\alpha$ $\pi _{e}^{NI}={{\bar{\pi }}_{e}}+\frac{5+3\gamma }{16\left( 1+\gamma \right)}\delta$ $w_{2}^{NI}=H\alpha +\frac{1-\gamma }{2}E\left[ \theta |Y \right]$ $\pi _{m1}^{NI}={{\bar{\pi }}_{m1}}$ $q_{1}^{NI}=J\alpha +\frac{\left( 2+\gamma \right)E\left[ \theta |Y \right]}{4\left( 1+\gamma \right)}$ $\pi _{m2}^{NI}={{\bar{\pi }}_{m2}}+\frac{1-\gamma }{8\left( 1+\gamma \right)}\delta$ $q_{2}^{NI}=J\alpha +\frac{E\left[ \theta |Y \right]}{4\left( 1+\gamma \right)}$ $\left( I, I \right)$ $w_{1}^{II}=H\alpha +HE\left[ \theta |Y \right]$ $\pi _{e}^{II}={{\bar{\pi }}_{e}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $w_{2}^{II}=H\alpha +HE\left[ \theta |Y \right]$ $\pi _{m1}^{II}={{\bar{\pi }}_{m1}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $q_{1}^{II}=J\alpha +JE\left[ \theta |Y \right]$ $\pi _{m2}^{II}={{\bar{\pi }}_{m2}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $q_{2}^{II}=J\alpha +JE\left[ \theta |Y \right]$ Note: $H=\frac{1-\gamma }{2-\gamma }$, $J=\frac{1}{2\left( 2-\gamma \right)\left( 1+\gamma \right)}$, and ${{\bar{\pi }}_{e}}={{\bar{\pi }}_{m1}}={{\bar{\pi }}_{m2}}=\frac{1-\gamma }{2{{\left( 2-\gamma \right)}^{2}}\left( 1+\gamma \right)}{{\alpha }^{2}}$. We let ${{\bar{\pi }}_{e}}$, ${{\bar{\pi }}_{m1}}$ and ${{\bar{\pi }}_{m2}}$ be the deterministic payoffs (in the absence of uncertainty). Let $\delta =E\left[ {{\left( E\left[ \theta |Y \right] \right)}^{2}} \right]$ for simplicity.
 $\left( {{x}_{m1}}, {{x}_{m2}} \right)$ Equilibrium outcome Ex-ante profit $\left( N, N \right)$ $w_{1}^{NN}=H\alpha$ $\pi _{e}^{NN}={{\bar{\pi }}_{e}}+\frac{1}{2\left( 1+\gamma \right)}\delta$ $w_{2}^{NN}=H\alpha$ $\pi _{m1}^{NN}={{\bar{\pi }}_{m1}}$ $q_{1}^{NN}=J\alpha +\frac{E\left[ \theta |Y \right]}{2\left( 1+\gamma \right)}$ $\pi _{m2}^{NN}={{\bar{\pi }}_{m2}}$ $q_{2}^{NN}=J\alpha +\frac{E\left[ \theta |Y \right]}{2\left( 1+\gamma \right)}$ $\left( N, I \right)$ $w_{1}^{NI}=H\alpha$ $\pi _{e}^{NI}={{\bar{\pi }}_{e}}+\frac{5+3\gamma }{16\left( 1+\gamma \right)}\delta$ $w_{2}^{NI}=H\alpha +\frac{1-\gamma }{2}E\left[ \theta |Y \right]$ $\pi _{m1}^{NI}={{\bar{\pi }}_{m1}}$ $q_{1}^{NI}=J\alpha +\frac{\left( 2+\gamma \right)E\left[ \theta |Y \right]}{4\left( 1+\gamma \right)}$ $\pi _{m2}^{NI}={{\bar{\pi }}_{m2}}+\frac{1-\gamma }{8\left( 1+\gamma \right)}\delta$ $q_{2}^{NI}=J\alpha +\frac{E\left[ \theta |Y \right]}{4\left( 1+\gamma \right)}$ $\left( I, I \right)$ $w_{1}^{II}=H\alpha +HE\left[ \theta |Y \right]$ $\pi _{e}^{II}={{\bar{\pi }}_{e}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $w_{2}^{II}=H\alpha +HE\left[ \theta |Y \right]$ $\pi _{m1}^{II}={{\bar{\pi }}_{m1}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $q_{1}^{II}=J\alpha +JE\left[ \theta |Y \right]$ $\pi _{m2}^{II}={{\bar{\pi }}_{m2}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $q_{2}^{II}=J\alpha +JE\left[ \theta |Y \right]$ Note: $H=\frac{1-\gamma }{2-\gamma }$, $J=\frac{1}{2\left( 2-\gamma \right)\left( 1+\gamma \right)}$, and ${{\bar{\pi }}_{e}}={{\bar{\pi }}_{m1}}={{\bar{\pi }}_{m2}}=\frac{1-\gamma }{2{{\left( 2-\gamma \right)}^{2}}\left( 1+\gamma \right)}{{\alpha }^{2}}$. We let ${{\bar{\pi }}_{e}}$, ${{\bar{\pi }}_{m1}}$ and ${{\bar{\pi }}_{m2}}$ be the deterministic payoffs (in the absence of uncertainty). Let $\delta =E\left[ {{\left( E\left[ \theta |Y \right] \right)}^{2}} \right]$ for simplicity.
Operation equilibrium
 $\left( {{x}_{m1}}, {{x}_{m2}} \right)$ Equilibrium outcome Ex-ante profit $\left( N, N \right)$ $q_{1}^{NN}=K\alpha$ $\pi _{e}^{NN}={{\bar{\pi }}_{e}}$ $q_{2}^{NN}=K\alpha$ $\pi _{m1}^{NN}={{\bar{\pi }}_{m1}}$ $\pi _{m2}^{NN}={{\bar{\pi }}_{m2}}$ $\left( N, I \right)$ $q_{1}^{NI}=K\alpha$ $\pi _{e}^{NI}={{\bar{\pi }}_{e}}+\frac{\beta }{4}\delta$ $q_{2}^{NI}=K\alpha +\frac{E\left[ \theta |Y \right]}{2}$ $\pi _{m1}^{NI}={{\bar{\pi }}_{m1}}$ $\pi _{m2}^{NI}={{\bar{\pi }}_{m2}}+\frac{1-\beta }{4}\delta$ $\left( I, I \right)$ $q_{1}^{II}=K\alpha +KE\left[ \theta |Y \right]$ $\pi _{e}^{II}={{\bar{\pi }}_{e}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $q_{2}^{II}=K\alpha +KE\left[ \theta |Y \right]$ $\pi _{m1}^{II}={{\bar{\pi }}_{m1}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $\pi _{m2}^{II}={{\bar{\pi }}_{m2}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ Note: $K=\frac{1}{2+\gamma }$, ${{\bar{\pi }}_{e}}=\frac{2\beta }{{{\left( 2+\gamma \right)}^{2}}}{{\alpha }^{2}}$, and ${{\bar{\pi }}_{m1}}={{\bar{\pi }}_{m2}}=\frac{\left( 1-\beta \right)}{{{\left( 2+\gamma \right)}^{2}}}{{\alpha }^{2}}$. We let ${{\bar{\pi }}_{e}}$, ${{\bar{\pi }}_{m1}}$ and ${{\bar{\pi }}_{m2}}$ be the deterministic payoffs (in the absence of uncertainty). Let $\delta = E\left[ {{\left( E\left[ \theta |Y \right] \right)}^{2}} \right]$ for simplicity.
 $\left( {{x}_{m1}}, {{x}_{m2}} \right)$ Equilibrium outcome Ex-ante profit $\left( N, N \right)$ $q_{1}^{NN}=K\alpha$ $\pi _{e}^{NN}={{\bar{\pi }}_{e}}$ $q_{2}^{NN}=K\alpha$ $\pi _{m1}^{NN}={{\bar{\pi }}_{m1}}$ $\pi _{m2}^{NN}={{\bar{\pi }}_{m2}}$ $\left( N, I \right)$ $q_{1}^{NI}=K\alpha$ $\pi _{e}^{NI}={{\bar{\pi }}_{e}}+\frac{\beta }{4}\delta$ $q_{2}^{NI}=K\alpha +\frac{E\left[ \theta |Y \right]}{2}$ $\pi _{m1}^{NI}={{\bar{\pi }}_{m1}}$ $\pi _{m2}^{NI}={{\bar{\pi }}_{m2}}+\frac{1-\beta }{4}\delta$ $\left( I, I \right)$ $q_{1}^{II}=K\alpha +KE\left[ \theta |Y \right]$ $\pi _{e}^{II}={{\bar{\pi }}_{e}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $q_{2}^{II}=K\alpha +KE\left[ \theta |Y \right]$ $\pi _{m1}^{II}={{\bar{\pi }}_{m1}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ $\pi _{m2}^{II}={{\bar{\pi }}_{m2}}\left( 1+\frac{\delta }{{{\alpha }^{2}}} \right)$ Note: $K=\frac{1}{2+\gamma }$, ${{\bar{\pi }}_{e}}=\frac{2\beta }{{{\left( 2+\gamma \right)}^{2}}}{{\alpha }^{2}}$, and ${{\bar{\pi }}_{m1}}={{\bar{\pi }}_{m2}}=\frac{\left( 1-\beta \right)}{{{\left( 2+\gamma \right)}^{2}}}{{\alpha }^{2}}$. We let ${{\bar{\pi }}_{e}}$, ${{\bar{\pi }}_{m1}}$ and ${{\bar{\pi }}_{m2}}$ be the deterministic payoffs (in the absence of uncertainty). Let $\delta = E\left[ {{\left( E\left[ \theta |Y \right] \right)}^{2}} \right]$ for simplicity.
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