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The value of information acquisition and sharing on an online intermediary platform

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Abstract

In practice, the online intermediary platform (e.g., JD.com and Amazon.com) not only acts as an e-tailer but also provides a marketplace for online retailers. This study investigates an online intermediary’s incentive for sharing private market demand information with the manufacturer when it is costly for the online retailer to acquire that information from the intermediary. We consider a hybrid-format supply chain consisting of a manufacturer, an online retailer, and an online intermediary. The online intermediary and retailer engage in Bertrand (price) competition. The retailer purchases products from the manufacturer and sells them through the online intermediary by paying an agency fee, while the online intermediary resells them as an e-tailer. Our findings indicate that the online intermediary may be willing to disclose market demand information to the manufacturer voluntarily under certain conditions. Interestingly, we find that when the information-acquisition cost is moderate, the retailer’s information-acquisition decision depends on the online intermediary’s information-sharing decision. Further, we demonstrate that sharing information is beneficial for the manufacturer, retailer, and online intermediary when the information-acquisition cost is moderate if and only if the competition intensity coefficient is low and the agency fee is high, or the competition intensity coefficient is high. However, when the information-acquisition cost is low or high, sharing information with the manufacturer would be detrimental to the retailer, i.e., a win–win–win outcome might not be achieved. These findings suggest that the three interacting forces of the information-acquisition cost, competition intensity coefficient, and agency fee steer equilibrium decisions for supply chain members.

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Notes

  1. Walmart’s e-commerce strategy is beginning to take effect. The retail giant recently surpassed Apple to become the world’s third largest online retailer, accelerated its quarterly e-commerce growth to a staggering 43%, and was expecting even stronger growth in 2019. Source: https://tinuiti.com/blog/walmart/walmart-ecommerce/ (accessed on June 23, 2021).

  2. Source: https://mall.jd.com/index-732842.html (accessed on June 8, 2022). Another example is that YYsports operates Nike, Adidas, and Converse, and has its own flagship store on the JD platform; source: https://mall.jd.com/index-57298.html (accessed on June 8, 2022).

  3. In practice, the online intermediary platforms may charge manufacturers and retailers a fixed fee. Without loss of generality, we normalize this fee to zero, which is in line with [30].

  4. We formulate the utility of a representative consumer as \(U=(a+\theta )-\frac{1}{2}(q_{R}^2+2bq_{R}q_{I}+q_{I}^2)-(p_{R}q_{R}+p_{I}q_{I})\).

  5. Source: https://retaillink.wal-mart.com (accessed on June 8, 2022).

  6. If the retailer determines to acquire the information, then she must pay a fixed fee F and the online intermediary obtains a revenue F. Note that due to the huge volume of the platform, this revenue is insignificant for the platform and can be ignored, but it is very important for retailers. Therefore, in what follows, this revenue F will not appear in the intermeidary’s payoff function when the retailer acquires the information.

  7. https://www.sohu.com/a/547480200_121394426 (accessed on June 13, 2021).

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (72072041, 71801053), the Chinese National Funding of Social Science (19BGL094), Guangdong Basic and Applied Research Foundation (2020A1515010407), and China Scholarship Council (202108440378).

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Correspondence to Rui Hou.

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Appendix

Appendix

Proof of Proposition 1

According to the equilibrium outcomes as presented in Tables 3 and 4. We first provide the optimal profits of supply chain members under cases NN and AN, respectively.

$$\begin{aligned}&E[\pi _{R}^{NN}]=\frac{-(-1 + b) (2 + b)^2 (-1 + \lambda ) (a + d)^2}{32 (1 + b) (-2 + b^2 (1 + \lambda ))}\\&E[\pi _{I}^{{NN}} ] = \frac{{\begin{array}{*{20}l} \begin{gathered} 3a^{2} ( - 4 + b( - 2 + b + 2\lambda + 3b\lambda ))( - 4 + b(2 - 10\lambda + b(1 + \lambda )(3 + b( - 1 + 4\lambda )))) \hfill \\ \; + 6a( - 4 + b( - 2 + b + 2\lambda + 3b\lambda ))( - 4 + b(2 - 10\lambda + b(1 + \lambda )(3 + b( - 1 + 4\lambda ))))d \hfill \\ \; + (112 + b( - 64 + 96\lambda + 8b^{2} (8 + (2 - 15\lambda )\lambda ) - 4b(31 + \lambda (16 + 15\lambda )) \hfill \\ \; + b^{3} (1 + \lambda )(31 + \lambda (13 + 24\lambda )) + b^{4} (1 + \lambda )( - 19 + \lambda ( - 13 + 36\lambda ))))d^{2} \hfill \\ \; + 6(2 + b)\lambda ( - 2 + b^{2} (1 + \lambda ))( - 6 + b + b^{2} (3 + 2\lambda ))(a + d)^{2} \hfill \\ \end{gathered} \hfill \\ \end{array} }}{{192(1 + b)( - 2 + b^{2} (1 + \lambda ))^{2} }}\\&E[\pi _{M}^{NN}]=-\frac{(8 - 4 \lambda + b (4 - 8 \lambda + b (-3 - b + (-5 + b) \lambda + 4 (1 + b) \lambda ^2))) (a + d)^2}{16 (1 + b) (-2 + b^2 (1 + \lambda ))} \\&E[\pi _{R}^{AN}]=\frac{(-1 + b) (2 + b)^2 (-1 + \lambda ) (3 a^2 + 6 a d + 7 d^2)}{96 (1 + b) (-2 + b^2 (1 + \lambda ))}\\& E[\pi _{I}^{{AN}} ] = \frac{{\begin{array}{*{20}l} \begin{gathered} 2(2 + b)\lambda ( - 2 + b^{2} (1 + \lambda ))(3a^{2} ( - 6 + b + b^{2} (3 + 2\lambda )) \hfill \\ \; + 6a( - 6 + b + b^{2} (3 + 2\lambda ))d + ( - 26 + b(7 + b(13 + 6\lambda )))d^{2} ) \hfill \\ \; + ( - 4 + b( - 2 + b + 2\lambda + 3b\lambda ))(3a^{2} ( - 4 + b(2 - 10\lambda + b(1 + \lambda )(3 + b( - 1 + 4\lambda )))) \hfill \\ \; + 6a( - 4 + b(2 - 10\lambda + b(1 + \lambda )(3 + b( - 1 + 4\lambda ))))d \hfill \\ \; + ( - 28 + b(14 - 38\lambda + b(1 + \lambda )(21 + b( - 7 + 12\lambda ))))d^{2} ) \hfill \\ \end{gathered} \hfill \\ \end{array} }}{{192(1 + b)( - 2 + b^{2} (1 + \lambda ))^{2} }} \\&E[\pi _{M}^{AN}]=-\frac{(8 - 4 \lambda + b (4 - 8 \lambda + b (-3 - b + (-5 + b) \lambda + 4 (1 + b) \lambda ^2))) (a + d)^2}{16 (1 + b) (-2 + b^2 (1 + \lambda ))} \end{aligned}$$

Next, we obtain \(F^{N}\) by calculating \(\pi _{R}^{AN}-\pi _{R}^{NN}=-\frac{(-1 + b) (2 + b)^2 (-1 + \lambda ) d^2}{24 (1 + b) [-2 + b^2 (1 + \lambda )]}\). \(\square\)

Proof of Corollary 1

Given that the online intermediary does not share the information, we give the following results:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial F^{N}}{\partial d}=-\frac{(-1 + b) (2 + b)^2 (-1 + \lambda ) d}{12 (1 + b) (-2 + b^2 (1 + \lambda ))}>0,\\ \frac{\partial F^{N}}{\partial d}=\frac{(2 + b) (-1 + \lambda ) (2 + b^3 + b (-2 + (-2 + b) b) \lambda ) d^2}{12 (1 + b)^2 (-2 + b^2 (1 + \lambda ))^2}<0,\\ \frac{\partial F^{N}}{\partial \lambda }=-\frac{(-2 + b + b^2)^2 d^2}{12 (-2 + b^2 (1 + \lambda ))^2}<0. \end{array}\right. } \end{aligned}$$

Clearly, according to the parameters’ setting, which reduces the result as shown in Corollary 1. \(\square\)

Proof of Proposition 2

According to the equilibrium outcomes as presented in Tables5 and 6. We first provide the optimal profits of supply chain members under cases NS and AS, respectively.

$$\begin{aligned}& E[\pi _{R}^{NS}]=\frac{(-1 + b) (2 + b)^2 (-1 + \lambda ) (-3 a^2 - 6 a d + d^2)}{96 (1 + b) (-2 + b^2 (1 + \lambda ))}\\& E[\pi _{I}^{{NS}} ] = \frac{{\begin{array}{*{20}l} \begin{gathered} 3a^{2} ( - 4 + b( - 2 + b + 2\lambda + 3b\lambda ))( - 4 + b(2 - 10\lambda + b(1 + \lambda )(3 + b( - 1 + 4\lambda )))) \hfill \\ \, + 6a( - 4 + b( - 2 + b + 2\lambda + 3b\lambda ))( - 4 + b(2 - 10\lambda + b(1 + \lambda )(3 + b( - 1 + 4\lambda ))))d \hfill \\ \, + (64 + 16b( - 1 + 8\lambda ) + b^{4} (1 + \lambda )(19 + \lambda + 24\lambda ^{2} ) - 4b^{2} (19 + \lambda (4 + 15\lambda )) \hfill \\ \, - 8b^{3} ( - 2 + \lambda (8 + 19\lambda )) + b^{5} ( - 7 + \lambda ^{2} (51 + 44\lambda )))d^{2} \hfill \\ \, + 2(2 + b)\lambda ( - 2 + b^{2} (1 + \lambda ))(3a^{2} ( - 6 + b + b^{2} (3 + 2\lambda )) \hfill \\ \, + 6a( - 6 + b + b^{2} (3 + 2\lambda ))d + ( - 26 + b(3 + b(13 + 10\lambda )))d^{2} ) \hfill \\ \end{gathered} \hfill \\ \end{array} }}{{192(1 + b)( - 2 + b^{2} (1 + \lambda ))^{2} }} \\& E[\pi _{M}^{{NS}} ] = \frac{{\begin{array}{*{20}l} \begin{gathered} - 3a^{2} (8 - 4\lambda + b(4 - 8\lambda + b( - 3 - b + ( - 5 + b)\lambda + 4(1 + b)\lambda ^{2} ))) \hfill \\ \, - 6a(8 - 4\lambda + b(4 - 8\lambda + b( - 3 - b + ( - 5 + b)\lambda + 4(1 + b)\lambda ^{2} )))d \hfill \\ \, + ( - 44 + 28\lambda + b( - 16 + 32\lambda + b(7(3 + b) + (11 - 7b)\lambda - 16(1 + b)\lambda ^{2} )))d^{2} \hfill \\ \end{gathered} \hfill \\ \end{array} }}{{48(1 + b)( - 2 + b^{2} (1 + \lambda ))}} \\& E[\pi _{R}^{AS}]=-\frac{(-1 + b) (2 + b)^2 (-1 + \lambda ) (3 a^2 + 6 a d + 4 d^2)}{96 (1 + b) (-2 + b^2 (1 + \lambda ))}\\& E[\pi _{I}^{{AS}} ] = \frac{{\begin{array}{*{20}l} \begin{gathered} (16 + 48\lambda + 48b\lambda - 4b^{3} \lambda (9 + 14\lambda ) \hfill \\ \, - 4b^{2} (5 + \lambda (13 + 15\lambda )) + b^{5} (1 + \lambda )( - 1 + \lambda (7 + 16\lambda )) \hfill \\ \, + b^{4} (1 + \lambda )(5 + \lambda (13 + 16\lambda )))(3a^{2} + 6ad + 4d^{2} ) \hfill \\ \end{gathered} \hfill \\ \end{array} }}{{192(1 + b)( - 2 + b^{2} (1 + \lambda ))^{2} }} \\& E[\pi _{M}^{{AS}} ] = - \frac{{\begin{array}{*{20}l} \begin{gathered} (2( - 1 + b\lambda )( - 4 + b( - 2 + b + 2\lambda + 3b\lambda ))d(3a^{2} + 6ad + 4d^{2} ))/( - 2 + b^{2} (1 + \lambda )) \hfill \\ \, + (2 + b)( - 1 + \lambda )( - a^{3} + (a + 2d)^{3} ) \hfill \\ \end{gathered} \hfill \\ \end{array} }}{{96(1 + b)d}} \end{aligned}$$

Next, we obtain \(F^{S}\) by calculating \(\pi _{R}^{AS}-\pi _{R}^{NS}=-\frac{(-1 + b) (2 + b)^2 (-1 + \lambda ) d^2}{24 (1 + b) [-2 + b^2 (1 + \lambda )]}\). \(\square\)

Proof of Corollary 2

When the online intermediary shares the information, we give the following results:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial F^{S}}{\partial d}=-\frac{5(-1 + b) (2 + b)^2 (-1 + \lambda ) d}{48 (1 + b) (-2 + b^2 (1 + \lambda ))}>0,\\ \frac{\partial F^{S}}{\partial b}=\frac{(2 + b) (-1 + \lambda ) (2 + b^3 + b (-2 + (-2 + b) b) \lambda ) d^2}{48 (1 + b)^2 (-2 + b^2 (1 + \lambda ))^2}<0,\\ \frac{\partial F^{S}}{\partial \lambda }=-\frac{5(-2 + b + b^2)^2 d^2}{48 (-2 + b^2 (1 + \lambda ))^2}<0. \end{array}\right. } \end{aligned}$$

Clearly, according to the parameters’ setting, which reduces the result as shown in Corollary 2. \(\square\)

Proof of Theorem 1

From here, according to the optimal profits of supply chain members, we first derive the retailer’s optimal information acquisition decision and then present the online intermediary’s information sharing decision.

Information acquisition decision For Bertrand competition, given that the online intermediary does not disclose the information to the manufacturer, in equilibrium, the retailer acquires the information if and only if \(E[\pi _{R}^{AN}]\ge E[\pi _{R}^{NN}]\), which leads to \(F\le F^{N}\). When the online intermediary shares the information with the manufacturer, in equilibrium, the retailer acquires the information if and only if \(E[\pi _{R}^{AS}]\ge E[\pi _{R}^{NS}]\), which leads to \(F\le F^{S}\). It is worth noting that where \(F^{S}>F^{N}\) holds.

Information sharing decision According to the retailer’s information-acquisition decision, for Bertrand competition,

  • if \(F\le F^{N}\), i.e., the retailer always acquires the information, then the online intermediary shares the information if and only if \(E[\pi _{I}^{AS}] \ge E[\pi _{I}^{AN}]\), which reduces \(b_{0}<b<1\) and \(\lambda _{0}(b)<\lambda <\lambda _{1}(b)\). Where the values of \(b_{0}\), \(\lambda _{0}(b)\), and \(\lambda _{1}(b)\) are complex, we omit them here. Note that we also show that the following relationship holds: \(E[\pi _{R}^{AS}]< E[\pi _{R}^{AN}]\) and \(E[\pi _{M}^{AS}]> E[\pi _{M}^{AN}]\).

  • If \(F> F^{S}\), i.e., the retailer does not acquire the information, then the online intermediary shares the information with the manufacturer if and only if \(E[\pi _{I}^{NS}] \ge E[\pi _{I}^{NN}]\), which leads to \(0<b<\frac{1}{3}(\sqrt{13}-1)\) and \(\lambda _{0}^{B}<\lambda <1\), or \(\frac{1}{3}(\sqrt{13}-1)\le b <1\). Note that we also show that the following relationships hold: \(E[\pi _{M}^{NS}]> E[\pi _{M}^{NN}]\) and \(E[\pi _{R}^{NS}]< E[\pi _{R}^{NN}]\).

  • If \(F^{N}<F\le F^{S}\), i.e., the retailer’s information-acquisition decision is dependent on the online intermediary’s information-sharing decision, then the online intermediary shares the information with the manufacturer if and only if \(E[\pi _{I}^{NN}]<E[\pi _{I}^{AS}]\), which reduces to \(0<b<\frac{1}{3}(\sqrt{13}-1)\) and \(\lambda _{2}(b)<\lambda <1\), or \(\frac{1}{3}(\sqrt{13}-1)\le b <1\). Note that we also show that the following relationships hold: \(E[\pi _{M}^{AS}]> E[\pi _{M}^{NN}]\) and \(E[\pi _{R}^{AS}]>E[\pi _{R}^{NN}]\).

\(\square\)

Proof of Proposition 3

It is easy obtain the results according to Theorem 1, we thus omit it here. \(\square\)

Proof of Sect. 6.2

By adopting the similar calculation process, we here provide the optimal profits of supply chain members under this scenario as follows. We employ symbol bar to distinguish the results presented in main model.

$$\begin{aligned}& E[\bar{\pi _{R}^{NN}}]=\frac{(-1 + b) (-1 + \lambda ) (2 + a b + (2 + b) d)^2}{32 (1 + b) (-2 + b^2 (1 + \lambda ))}\\& E[\overline{{\pi _{I}^{{NN}} }} ] = \frac{{\begin{array}{*{20}l} \begin{gathered} 144\lambda + 12b^{2} (1 - 5\lambda (2 + 3\lambda ) + b^{2} (1 + \lambda )^{2} ( - 1 + 4\lambda )) \hfill \\ \, + 3a^{2} (16 + 12b^{2} ( - 2 + 3\lambda ) + b^{6} (1 + \lambda )( - 1 + \lambda (7 + 16\lambda )) \hfill \\ \, + b^{4} (9 - \lambda (26 + 55\lambda ))) + 12(1 + b)(24\lambda + b(4 + b^{2} ( - 3 + \lambda )(1 + \lambda ) \hfill \\ \, - 2b(1 + 5\lambda (2 + 3\lambda )) + b^{3} (1 + \lambda )(1 + \lambda (5 + 8\lambda ))))d \hfill \\ \, + (1 + b)(16b( - 4 + 9\lambda ) + 16(7 + 9\lambda ) - 4b^{3} ( - 1 + 2\lambda )(16 + 21\lambda ) \hfill \\ \, - 4b^{2} (31 + 5\lambda (11 + 9\lambda )) + b^{5} (1 + \lambda )( - 19 + \lambda (5 + 48\lambda )) \hfill \\ \, + b^{4} (1 + \lambda )(31 + \lambda (55 + 48\lambda )))d^{2} + 6a(16d + 12b^{2} ( - 2 + 3\lambda )d \hfill \\ \, + b^{6} (1 + \lambda )( - 1 + \lambda (7 + 16\lambda ))d + b^{4} (9 - \lambda (26 + 55\lambda ))d \hfill \\ \, + 8b(1 + 6\lambda )(1 + d) + 2b^{5} (1 + \lambda )(1 + \lambda (5 + 8\lambda ))(1 + d) \hfill \\ \, - 2b^{3} (5 + \lambda (22 + 29\lambda ))(1 + d)) \hfill \\ \end{gathered} \hfill \\ \end{array} }}{{192(1 + b)( - 2 + b^{2} (1 + \lambda ))^{2} }} \\& E[\overline{{\pi _{M}^{{NN}} }} ] = - \frac{{\begin{array}{*{20}l} \begin{aligned} & ( - 1 + \lambda )(2 + ab + (2 + b)d)(1 + d + b(a + d)) \\ & \quad + \frac{{(b( - 1 + \lambda ) + a( - 1 + b^{2} \lambda ) + (1 + b)( - 1 + b\lambda )d)(a( - 4 + b^{2} (1 + 3\lambda )) - 4d + 2b( - 1 + \lambda )(1 + d) + b^{2} (d + 3\lambda d))}}{{ - 2 + b^{2} (1 + \lambda )}} \\ \end{aligned} \hfill \\ \end{array} }}{{16(1 + b)^{2} }}\\& E[\overline{{\pi _{I}^{{AN}} }} ] = \frac{{\begin{array}{*{20}l} \begin{gathered} 144\lambda + 12b^{2} (1 - 5\lambda (2 + 3\lambda ) + b^{2} (1 + \lambda )^{2} ( - 1 + 4\lambda )) \hfill \\ \, + 3a^{2} (16 + 12b^{2} ( - 2 + 3\lambda ) + b^{6} (1 + \lambda )( - 1 + \lambda (7 + 16\lambda )) \hfill \\ \, + b^{4} (9 - \lambda (26 + 55\lambda ))) + 12(1 + b)(24\lambda + b(4 + b^{2} ( - 3 + \lambda )(1 + \lambda ) \hfill \\ \, - 2b(1 + 5\lambda (2 + 3\lambda )) + b^{3} (1 + \lambda )(1 + \lambda (5 + 8\lambda ))))d + (1 + b)(144b\lambda + 16(7 + 13\lambda ) \hfill \\ \, - 4b^{3} \lambda (23 + 42\lambda ) - 4b^{2} (35 + 75\lambda + 57\lambda ^{2} ) + b^{5} (1 + \lambda )( - 7 + \lambda (17 + 48\lambda )) \hfill \\ \, + b^{4} (1 + \lambda )(35 + \lambda (91 + 48\lambda )))d^{2} + 6a(16d + 12b^{2} ( - 2 + 3\lambda )d \hfill \\ \, + b^{6} (1 + \lambda )( - 1 + \lambda (7 + 16\lambda ))d + b^{4} (9 - \lambda (26 + 55\lambda ))d + 8b(1 + 6\lambda )(1 + d) \hfill \\ \, + 2b^{5} (1 + \lambda )(1 + \lambda (5 + 8\lambda ))(1 + d) - 2b^{3} (5 + \lambda (22 + 29\lambda ))(1 + d)) \hfill \\ \end{gathered} \hfill \\ \end{array} }}{{192(1 + b)( - 2 + b^{2} (1 + \lambda ))^{2} }} \\& E[\overline{{\pi _{M}^{{AN}} }} ] = - \frac{{\begin{array}{*{20}l} \begin{aligned} & ( - 1 + \lambda )(2 + ab + (2 + b)d)(1 + d + b(a + d)) \\ & \quad + \frac{{(b( - 1 + \lambda ) + a( - 1 + b^{2} \lambda ) + (1 + b)( - 1 + b\lambda )d)(a( - 4 + b^{2} (1 + 3\lambda )) - 4d + 2b( - 1 + \lambda )(1 + d) + b^{2} (d + 3\lambda d))}}{{ - 2 + b^{2} (1 + \lambda )}} \\ \end{aligned} \hfill \\ \end{array} }}{{16(1 + b)^{2} }} \\& E[\bar{\pi _{R}^{NS}}]=\frac{(-1 + \lambda ) (-3 (-1 + b) (2 + a b)^2 - 6 (2 + a b) (-2 + b + b^2) d + (-1 + b) (2 + b)^2 d^2)}{96 (1 + b) (-2 + b^2 (1 + \lambda ))}\\& E[\overline{{\pi _{I}^{{NS}} }} ] = \frac{{\begin{array}{*{20}l} \begin{gathered} (144\lambda + 12b^{2} (1 - 5\lambda (2 + 3\lambda ) + b^{2} (1 + \lambda )^{2} ( - 1 + 4\lambda )) \hfill \\ \, + 3a^{2} (16 + 12b^{2} ( - 2 + 3\lambda ) + b^{6} (1 + \lambda )( - 1 + \lambda (7 + 16\lambda )) \hfill \\ \, + b^{4} (9 - \lambda (26 + 55\lambda ))) + 12(1 + b)(24\lambda + b(4 + b^{2} ( - 3 + \lambda )(1 + \lambda ) \hfill \\ \, - 2b(1 + 5\lambda (2 + 3\lambda )) + b^{3} (1 + \lambda )(1 + \lambda (5 + 8\lambda ))))d + (1 + b)(64 + 208\lambda \hfill \\ \, + b( - 16 + 208\lambda - 4b^{2} ( - 4 + \lambda (39 + 58\lambda )) - 4b(19 + \lambda (59 + 61\lambda )) \hfill \\ \, + b^{4} (1 + \lambda )( - 7 + \lambda (33 + 64\lambda )) + b^{3} (1 + \lambda )(19 + \lambda (59 + 64\lambda ))))d^{2} \hfill \\ \, + 6a(16d + 12b^{2} ( - 2 + 3\lambda )d + b^{6} (1 + \lambda )( - 1 + \lambda (7 + 16\lambda ))d \hfill \\ \, + b^{4} (9 - \lambda (26 + 55\lambda ))d + 8b(1 + 6\lambda )(1 + d) \hfill \\ \, + 2b^{5} (1 + \lambda )(1 + \lambda (5 + 8\lambda ))(1 + d) - 2b^{3} (5 + \lambda (22 + 29\lambda ))(1 + d)) \hfill \\ \end{gathered} \hfill \\ \end{array} }}{{192(1 + b)( - 2 + b^{2} (1 + \lambda ))^{2} }} \\& E[\overline{{\pi _{M}^{{NS}} }} ] = \frac{{\begin{array}{*{20}l} \begin{gathered} 3(4( - 1 + \lambda )( - 1 + b^{2} \lambda ) + 4ab( - 1 + \lambda )( - 3 + b^{2} (1 + 2\lambda )) \hfill \\ \, + a^{2} (4 + b^{2} (1 - 9\lambda ) + b^{4} ( - 1 + \lambda + 4\lambda ^{2} ))) + 6(1 + b)(a(4 - b( - 2 + b + b^{2} ) \hfill \\ \, + b( - 6 + ( - 3 + b)b)\lambda + 4b^{3} \lambda ^{2} ) + 2( - 1 + \lambda )( - 2 + b( - 1 + b + 2b\lambda )))d \hfill \\ \, + (1 + b)(44 - 28\lambda + b(16 - 32\lambda + b( - 7(3 + b) + ( - 11 + 7b)\lambda + 16(1 + b)\lambda ^{2} )))d^{2} \hfill \\ \end{gathered} \hfill \\ \end{array} }}{{48(1 + b)( - 2 + b^{2} (1 + \lambda ))}} \\& E[\bar{\pi _{R}^{AS}}]=-\frac{(-1 + b) (-1 + \lambda ) (3 (2 + a b)^2 + 6 (2 + b) (2 + a b) d + 4 (2 + b)^2 d^2)}{96 (1 + b) (-2 + b^2 (1 + \lambda ))}\\& E[\overline{{\pi _{I}^{{AS}} }} ] = \frac{{\begin{array}{*{20}l} \begin{gathered} 3a^{2} (16 + 12b^{2} ( - 2 + 3\lambda ) + b^{6} (1 + \lambda )( - 1 + \lambda (7 + 16\lambda )) \hfill \\ \, + b^{4} (9 - \lambda (26 + 55\lambda ))) + 4(36\lambda + 3b^{2} (1 - 5\lambda (2 + 3\lambda ) \hfill \\ \, + b^{2} (1 + \lambda )^{2} ( - 1 + 4\lambda )) + 3(1 + b)(24\lambda + b(4 + b^{2} ( - 3 + \lambda )(1 + \lambda ) \hfill \\ \, - 2b(1 + 5\lambda (2 + 3\lambda )) + b^{3} (1 + \lambda )(1 + \lambda (5 + 8\lambda ))))d \hfill \\ \, + (1 + b)(16 + 48\lambda + 48b\lambda - 4b^{3} \lambda (9 + 14\lambda ) - 4b^{2} (5 + \lambda (13 + 15\lambda )) \hfill \\ \, + b^{5} (1 + \lambda )( - 1 + \lambda (7 + 16\lambda )) + b^{4} (1 + \lambda )(5 + \lambda (13 + 16\lambda )))d^{2} ) \hfill \\ \, + 6a(16d + 12b^{2} ( - 2 + 3\lambda )d + b^{6} (1 + \lambda )( - 1 + \lambda (7 + 16\lambda ))d \hfill \\ \, + b^{4} (9 - \lambda (26 + 55\lambda ))d + 8b(1 + 6\lambda )(1 + d) \hfill \\ \, + 2b^{5} (1 + \lambda )(1 + \lambda (5 + 8\lambda ))(1 + d) - 2b^{3} (5 + \lambda (22 + 29\lambda ))(1 + d)) \hfill \\ \end{gathered} \hfill \\ \end{array} }}{{192(1 + b)( - 2 + b^{2} (1 + \lambda ))^{2} }} \\& E[\overline{{\pi _{M}^{{AS}} }} ] = - \frac{{\begin{array}{*{20}l} \begin{gathered} 3a^{2} (4 + b^{2} (1 - 9\lambda ) + b^{4} ( - 1 + \lambda + 4\lambda ^{2} )) \hfill \\ \, + 6a(4d + b^{4} ( - 1 + \lambda + 4\lambda ^{2} )d - 6b( - 1 + \lambda )(1 + d) \hfill \\ \, + 2b^{3} ( - 1 - \lambda + 2\lambda ^{2} )(1 + d) + b^{2} (d - 9\lambda d)) \hfill \\ \, + 4(3 + (6 + 9b - 3b^{3} )d + (8 + 12b + b^{2} - 4b^{3} - b^{4} )d^{2} \hfill \\ \, + b^{2} \lambda ^{2} (3 + 6(1 + b)d + 4(1 + b)^{2} d^{2} ) + \lambda ( - 3 - 6d - 4d^{2} + b^{4} d^{2} \hfill \\ \, - 3bd(3 + 4d) - b^{3} d(3 + 4d) - b^{2} (3 + 6d + 13d^{2} ))) \hfill \\ \end{gathered} \hfill \\ \end{array} }}{{48(1 + b)^{2} ( - 2 + b^{2} (1 + \lambda ))}} \end{aligned}$$

Finally, we confirm that the equilibirum results including thresholds remain vaild under asymmetric base demand between two channels. \(\square\)

Proof of Proposition 4

According to our previous results, we can easily to obtain that the online intermediary is willing to share information if and only if \(b_{0}<b<1\) and \(\lambda _{0}(b)<\lambda <\lambda _{1}(b)\). However, under this condition, as verified in Theorem 1, the retailer always prefers not to acquire information. Second, it is easy to obtain that the online intermediary is unwilling to share information if and only if \(0<\lambda <\frac{3}{4}\) and \(0<b<\frac{3-4\lambda }{3+4\lambda }\). Similarly, the retailer never acquires information under such conditions. Third, when the agency fee is moderate, by comparing \(E[\pi _{R}^{AS}]\) and \(E[\pi _{R}^{N}]\), we have that the retailer acquires information if and only if \(F<F^{***}\), and the online intermediary prefers to information sharing under acquisition, but prefers not to share information under non-acquisition. Note that \(F^{***}=-\frac{(-1 + b) (2 + b)^2 (-1 + \lambda ) d^2}{96 (1 + b) [-2 + b^2 (1 + \lambda )]}\). \(\square\)

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Zhao, Y., Chen, J., Shi, P. et al. The value of information acquisition and sharing on an online intermediary platform. Electron Commer Res 24, 2849–2875 (2024). https://doi.org/10.1007/s10660-022-09620-1

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