Influences of information sharing and online recommendations in a supply chain: reselling versus agency selling

Abstract

This paper investigates the effects of reselling and agency contracts on the platform’s incentive to share information under two scenarios, i.e., the recommendation scenario and non-recommendation scenario. Through analyzing the equilibrium solutions, we derive the following results. First, comparing the recommendation and non-recommendation scenarios, the platform prefers the recommendation scenario under the reselling contract. However, contrary to traditional wisdom, when the recommendation efficiency is relatively more efficient, the platform chooses the non-recommendation scenario in the agency contract; as the recommendation efficiency becomes less efficient, the recommendation scenario is preferred in the agency contract. Second, for the reselling contract, the platform prefers to withhold demand information in the recommendation and non-recommendation scenarios. However, the platform voluntarily shares this information under the agency contract for the two scenarios. Third, the platform’s optimal strategy depends on the commission rate, the platform’s recommendation efficiency, and the level of information accuracy. Intuitively, a lower commission rate induces the platform to prefer the reselling contract in the recommendation and non-recommendation scenarios. However, a higher commission rate does not always make the platform choose the agency contract. The recommendation efficiency and the level of information accuracy affect the platform’s optimal strategy. Furthermore, given the platform’s optimal strategy, there are Pareto optimal regions that make the manufacturer better off. That is, the platform and the manufacturer achieve a win–win situation. Finally, in an extension of the model, this paper shows the platform’s optimal strategy when a hybrid contract is used, which also demonstrates the robustness of the results.

Appendix

Proof of Lemma 1

Under mode RN, the expected profit of the platform is given as:

$$ \pi_{P}^{RN} = E\left[ {\left( {p^{RN} - w^{RN} } \right)q^{RN} - \frac{\eta }{2}\left( {s^{RN} } \right)^{2} \left| f \right.} \right] $$

Taking the first order derivative of $\pi_{P}^{RN}$ with respect to $p^{RN}$ and $s^{RN}$, we can derive

$$ \frac{{\partial \pi_{P}^{RN} }}{{\partial p^{RN} }} = E\left[ {\alpha \left| f \right.} \right] - 2p^{RN} + w^{RN} + \gamma s^{RN} ,\frac{{\partial \pi_{P}^{RN} }}{{\partial s^{RN} }} = \gamma \left( {p^{RN} - w^{RN} } \right) - \eta s^{RN} . $$

Let $\frac{{\partial \pi_{P}^{RN} }}{{\partial p^{RN} }} = 0$, and $\frac{{\partial \pi_{P}^{RN} }}{{\partial s^{RN} }} = 0$, we have

$$ p^{RN} \left( {w^{RN} } \right) = \frac{{\eta E\left[ {\alpha \left| f \right.} \right] + w^{RN} \left( {\eta - \gamma^{2} } \right)}}{{2\eta - \gamma^{2} }},\,{\text{and}}\,s^{RN} \left( {w^{RN} } \right) = \frac{{\gamma \left( {E\left[ {\alpha \left| f \right.} \right] - w^{RN} } \right)}}{{2\eta - \gamma^{2} }}. $$

The Hessian matrix is:

$$ H = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{P}^{RN} }}{{\partial \left( {p^{RN} } \right)^{2} }}} & {\frac{{\partial^{2} \pi_{P}^{RN} }}{{\partial p^{RN} \partial s^{RN} }}} \\ {\frac{{\partial^{2} \pi_{P}^{RN} }}{{\partial s^{RN} \partial p^{RN} }}} & {\frac{{\partial^{2} \pi_{P}^{RN} }}{{\partial \left( {s^{RN} } \right)^{2} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 2} & \gamma \\ \gamma & { - \eta } \\ \end{array} } \right],\,{\text{and}}\,\left| H \right| = 2\eta - \gamma^{2} . $$

To guarantee the profit function is concave and quadratic with $p^{RN} \left( {w^{RN} } \right)$ and $s^{RN} \left( {w^{RN} } \right)$, $\eta > \eta_{{0}}$ should be satisfied, where $\eta_{{0}} { = }\frac{{\gamma^{2} }}{2}$.

Under mode RN, the manufacturer does not know the demand information, and the manufacturer decides the wholesale price based on the expected demand $E\left[ {q^{RN} } \right]$. The expected profit of the manufacturer is $\pi_{M}^{RN} = E\left[ {w^{RN} q^{RN} } \right]$. Substituting $p^{RN} \left( {w^{RN} } \right)$ and $s^{RN} \left( {w^{RN} } \right)$ into the expected profit of the manufacturer, we obtain that $\frac{{\partial \pi_{M}^{RN} }}{{\partial w^{RN} }} = \frac{{\eta \left( {a_{0} - 2w^{RN} } \right)}}{{2\eta - \gamma^{2} }}$, and $\frac{{\partial^{2} \pi_{M}^{RN} }}{{\partial \left( {w^{RN} } \right)^{2} }} = - \frac{2\eta }{{2\eta - \gamma^{2} }}$. Since $\eta > \eta_{{0}}$, it is obvious $\frac{{\partial^{2} \pi_{M}^{RN} }}{{\partial \left( {w^{RN} } \right)^{2} }} < 0$. The expected profit of the manufacturer is concave in $w^{RN}$, and we have the optimal wholesale price is $w^{RN * } = \frac{{a_{0} }}{2}$.

Substituting $w^{RN * } = \frac{{a_{0} }}{2}$ into $p^{RN} \left( {w^{RN} } \right)$ and $s^{RN} \left( {w^{RN} } \right)$, the equilibrium solutions of mode RN are $p^{RN * } = \frac{{a_{0} \left( {\eta \left( {3 - 2z} \right) - \gamma^{2} } \right) + 2\eta zf}}{{2\left( {2\eta - \gamma^{2} } \right)}}$ and $s^{RN * } = \frac{{\gamma \left( {a_{0} \left( {1 - 2z} \right) + 2zf} \right)}}{{2\left( {2\eta - \gamma^{2} } \right)}}$. Based on the equilibrium solutions of mode RN, the ex ante profits are $\pi_{M}^{RN * } = \frac{{\eta a_{0}^{2} }}{{4\left( {2\eta - \gamma^{2} } \right)}}$ and $\pi_{P}^{RN * } = \frac{{\eta \left( {a_{0}^{2} + 4z\sigma_{0}^{2} } \right)}}{{8\left( {2\eta - \gamma^{2} } \right)}}$.

The proof of equilibrium solutions under mode RS is similar to that under mode RN. Under mode RS, the expected profit of the platform is $\pi_{P}^{RS} = E\left[ {\left( {p^{RS} - w^{RS} } \right)q^{RS} - \frac{\eta }{2}\left( {s^{RS} } \right)^{2} \left| f \right.} \right]$. And we have $p^{RS} \left( {w^{RS} } \right) = \frac{{\eta E\left[ {\alpha \left| f \right.} \right] + w^{RS} \left( {\eta - \gamma^{2} } \right)}}{{2\eta - \gamma^{2} }}$, and $s^{RS} \left( {w^{RS} } \right) = \frac{{\gamma \left( {E\left[ {\alpha \left| f \right.} \right] - w^{RS} } \right)}}{{2\eta - \gamma^{2} }}$.

The Hessian matrix is:

$$ H = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} \pi_{P}^{RS} }}{{\partial \left( {p^{RS} } \right)^{2} }}} & {\frac{{\partial^{2} \pi_{P}^{RS} }}{{\partial p^{RS} \partial s^{RS} }}} \\ {\frac{{\partial^{2} \pi_{P}^{RS} }}{{\partial s^{RS} \partial p^{RS} }}} & {\frac{{\partial^{2} \pi_{P}^{RS} }}{{\partial \left( {s^{RS} } \right)^{2} }}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} { - 2} & \gamma \\ \gamma & { - \eta } \\ \end{array} } \right],\,{\text{and}}\,\left| H \right| = 2\eta - \gamma^{2} . $$

When $\eta > \eta_{{0}}$, $\left| H \right| = 2\eta - \gamma^{2} > 0$. And the objective function is concave. Under mode RS, the platform discloses the demand information. Therefore, the manufacturer makes decisions conditional on the demand signal $E\left[ {q^{RS} \left| f \right.} \right]$. The expected profit of the manufacturer is:

$$ \pi_{M}^{RS} = E\left[ {w^{RS} q^{RS} \left| f \right.} \right] $$

Substituting $p^{RS} \left( {w^{RS} } \right)$ and $s^{RS} \left( {w^{RS} } \right)$ into the expected profit of the manufacturer, we obtain that $\frac{{\partial \pi_{M}^{RS} }}{{\partial w^{RS} }} = \frac{{\eta \left( {E\left[ {\alpha \left| f \right.} \right] - 2w^{RS} } \right)}}{{2\eta - \gamma^{2} }}$ and $\frac{{\partial^{2} \pi_{M}^{RS} }}{{\partial \left( {w^{RS} } \right)^{2} }} = - \frac{2\eta }{{2\eta - \gamma^{2} }} < 0$.

Let the first order condition $\frac{{\partial \pi_{M}^{RS} }}{{\partial w^{RS} }} = 0$, the optimal wholesale price is derived, that is $w^{RS * } = \frac{{a_{0} \left( {1 - z} \right) + zf}}{2}$. Substituting $w^{RS * } = \frac{{a_{0} \left( {1 - z} \right) + zf}}{2}$ into $p^{RS} \left( {w^{RS} } \right)$ and $s^{RS} \left( {w^{RS} } \right)$, the equilibrium solutions and the ex ante profits of mode RS are derived as follows.

$$ \begin{aligned} p^{RS * } & = \frac{{\left( {a_{0} \left( {1 - z} \right) + zf} \right)\left( {3\eta - \gamma^{2} } \right)}}{{2\left( {2\eta - \gamma^{2} } \right)}},\,s^{RS * } = \frac{{\gamma \left( {a_{0} \left( {1 - z} \right) + zf} \right)}}{{2\left( {2\eta - \gamma^{2} } \right)}}; \\ \pi_{M}^{RS * } & = \frac{{\eta \left( {a_{0}^{2} + z\sigma_{0}^{2} } \right)}}{{4\left( {2\eta - \gamma^{2} } \right)}},\,\pi_{P}^{RS * } = \frac{{\eta \left( {a_{0}^{2} + z\sigma_{0}^{2} } \right)}}{{8\left( {2\eta - \gamma^{2} } \right)}}. \\ \end{aligned} $$

Proof of Proposition 1

In the reselling contract, $w^{RN * } - w_{B}^{RN * } = 0$, $p^{RN * } - p_{B}^{RN * } = \frac{{\gamma^{2} \left( {a_{0} \left( {1 - 2z} \right) + 2fz} \right)}}{{4\left( {2\eta - \gamma^{2} } \right)}}$, $q^{RN * } - q_{B}^{RN * } = \frac{{\gamma^{2} \left( {a_{0} \left( {1 - 2z} \right) + 2fz} \right)}}{{4\left( {2\eta - \gamma^{2} } \right)}}$; $w^{RS * } - w_{B}^{RS * } = 0$, $p^{RS * } - p_{B}^{RS * } = \frac{{\gamma^{2} \left( {a_{0} \left( {1 - z} \right) + fz} \right)}}{{4\left( {2\eta - \gamma^{2} } \right)}}$, $q^{RS * } - q_{B}^{RS * } = \frac{{\gamma^{2} \left( {a_{0} \left( {1 - z} \right) + fz} \right)}}{{4\left( {2\eta - \gamma^{2} } \right)}}$. Based on the equilibrium solutions in Lemmas 1 and 3, we can derive that $w^{RN * } = w_{B}^{RN * }$, $p^{RN * } > p_{B}^{RN * }$, $q^{RN * } > q_{B}^{RN * }$; $w^{RS * } = w_{B}^{RS * }$, $p^{RS * } > p_{B}^{RS * }$, $q^{RS * } > q_{B}^{RS * }$.

In the agency contract, $p^{AN * } - p_{B}^{AN * } = \frac{{a_{0} \theta \gamma^{2} }}{{2\left( {\eta - \theta \gamma^{2} } \right)}}$, $q^{AN * } - q_{B}^{AN * } = 0$; $p^{AS * } - p_{B}^{AS * } = \frac{{\theta \gamma^{2} \left( {a_{0} \left( {1 - z} \right) + fz} \right)}}{{2\left( {\eta - \theta \gamma^{2} } \right)}}$, $q^{AS * } - q_{B}^{AS * } = 0$. It is easy to derive that $p^{AN * } > p_{B}^{AN * }$, $q^{AN * } = q_{B}^{AN * }$; $p^{AS * } > p_{B}^{AS * }$, $q^{AS * } = q_{B}^{AS * }$.

Proof of Proposition 2

Based on the equilibrium solutions in Lemmas 1–3, when $0 < \eta \le \eta_{0}$, the platform can only choose the non-recommendation scenario under the reselling contract. When $\eta > \eta_{0}$, we can obtain that $\pi_{P}^{RN * } - \pi_{BP}^{RN * } = \frac{{\gamma^{2} \left( {a_{0}^{2} + 4z\sigma_{0}^{2} } \right)}}{{16\left( {2\eta - \gamma^{2} } \right)}} > 0$, $\pi_{P}^{RS * } - \pi_{BP}^{RS * } = \frac{{\gamma^{2} \left( {a_{0}^{2} + z\sigma_{0}^{2} } \right)}}{{16\left( {2\eta - \gamma^{2} } \right)}} > 0$. Under the agency contract, the platform can only choose the non-recommendation scenario when $0 < \eta \le \eta_{1}$. When $\eta > \eta_{1}$, we can derive that $\pi_{P}^{AN * } - \pi_{BP}^{AN * } = \frac{{a_{0}^{2} \theta^{2} \gamma^{2} \left( {\eta - 2\theta \gamma^{2} } \right)}}{{8\left( {\eta - \theta \gamma^{2} } \right)^{2} }}$, $\pi_{P}^{AS * } - \pi_{BP}^{AS * } = \frac{{\theta^{2} \gamma^{2} \left( {\eta - 2\theta \gamma^{2} } \right)\left( {a_{0}^{2} + z\sigma_{0}^{2} } \right)}}{{8\left( {\eta - \theta \gamma^{2} } \right)^{2} }}$. As a relust, if $\eta_{1} < \eta \le \eta_{2}$, $\pi_{P}^{AN * } < \pi_{BP}^{AN * }$, $\pi_{P}^{AS * } < \pi_{BP}^{AS * }$; otherwise, $\pi_{P}^{AN * } > \pi_{BP}^{AN * }$, $\pi_{P}^{AS * } > \pi_{BP}^{AS * }$, where $\eta_{0} = \frac{{\gamma^{2} }}{2}$, $\eta_{1} = \frac{{3\theta \gamma^{2} }}{2}$, $\eta_{2} = 2\theta \gamma^{2}$.

Proof of Proposition 3

Based on the equilibrium solutions of the reselling contract, the value of information sharing under the recommendation scenario and non-recommendation scenario are obtained. Under the recommendation cenario, we have $v_{P}^{R} = \pi_{P}^{RS * } - \pi_{P}^{RN * } = - \frac{{3z\eta \sigma_{0}^{2} }}{{8\left( {2\eta - \gamma^{2} } \right)}}$, $v_{M}^{R} = \pi_{M}^{RS * } - \pi_{M}^{RN * } = \frac{{z\eta \sigma_{0}^{2} }}{{4\left( {2\eta - \gamma^{2} } \right)}}$, and $v^{R} = \left( {\pi_{M}^{RS * } + \pi_{P}^{RS * } } \right) - \left( {\pi_{M}^{RN * } + \pi_{P}^{RN * } } \right) = - \frac{{z\eta \sigma_{0}^{2} }}{{8\left( {2\eta - \gamma^{2} } \right)}}$; and under the non-receommendation scenario, we have $v_{BP}^{R} = \pi_{BP}^{RS * } - \pi_{BP}^{RN * } = - \frac{{3z\sigma_{0}^{2} }}{16}$, $v_{BM}^{R} = \pi_{BM}^{RS * } - \pi_{BM}^{RN * } = \frac{{z\sigma_{0}^{2} }}{8}$, and $v_{B}^{R} = \left( {\pi_{BM}^{RS * } + \pi_{BP}^{RS * } } \right) - \left( {\pi_{BM}^{RN * } + \pi_{BP}^{RN * } } \right) = - \frac{{z\sigma_{0}^{2} }}{16}$. Since $\eta > \eta_{0}$ in Lemma 1, we have $v_{P}^{R} < 0$, $v_{M}^{R} > 0$, $v^{R} < 0$, $v_{BP}^{R} < 0$, $v_{BM}^{R} > 0$, and $v_{B}^{R} < 0$.

Proof of Proposition 4

Based on the results given in the proof of Proposition 3, and $z = {{\sigma_{{0}}^{{2}} } \mathord{\left/ {\vphantom {{\sigma_{{0}}^{{2}} } {\left( {\sigma_{{0}}^{{2}} + \sigma_{r}^{{2}} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\sigma_{{0}}^{{2}} + \sigma_{r}^{{2}} } \right)}}$, it is easy to verify the results in Table

Table 3 The effects of $\sigma_{0}$ and $\sigma_{r}$ on the value of information sharing

3.

As shown in Table 3, we can prove the Proposition 4. For the manufacturer, $\frac{{\partial v_{BM}^{R} }}{{\partial \sigma_{0} }} > 0$, $\frac{{\partial v_{BM}^{R} }}{{\partial \sigma_{r} }} < 0$, $\frac{{\partial v_{M}^{R} }}{{\partial \sigma_{0} }} > 0$, $\frac{{\partial v_{M}^{R} }}{{\partial \sigma_{r} }} < 0$. For the platform, $\frac{{\partial v_{BP}^{R} }}{{\partial \sigma_{0} }} < 0$, $\frac{{\partial v_{BP}^{R} }}{{\partial \sigma_{r} }} > 0$, $\frac{{\partial v_{P}^{R} }}{{\partial \sigma_{0} }} < 0$, $\frac{{\partial v_{P}^{R} }}{{\partial \sigma_{r} }} > 0$. For the supply chain, $\frac{{\partial v_{B}^{R} }}{{\partial \sigma_{0} }} < 0$, $\frac{{\partial v_{B}^{R} }}{{\partial \sigma_{r} }} > 0$, $\frac{{\partial v^{R} }}{{\partial \sigma_{0} }} < 0$, $\frac{{\partial v^{R} }}{{\partial \sigma_{r} }} > 0$.

Proof of Proposition 5

Based on the equilibrium solutions of the agency contract, the value of information sharing under the recommendation scenario and non-recommendation scenario are obtained. Under the recommendation cenario, we have $v_{P}^{A} = \pi_{P}^{AS * } - \pi_{P}^{AN * } = \frac{{z\eta \theta \sigma_{0}^{2} \left( {2\eta - 3\theta \gamma^{2} } \right)}}{{8\left( {\eta - \theta \gamma^{2} } \right)^{2} }}$, $v_{M}^{A} = \pi_{M}^{AS * } - \pi_{M}^{AN * } = \frac{{z\eta \sigma_{0}^{2} \left( {1 - \theta } \right)}}{{4\left( {\eta - \theta \gamma^{2} } \right)}}$, and $v^{A} = \left( {\pi_{M}^{AS * } + \pi_{P}^{AS * } } \right) - \left( {\pi_{M}^{AN * } + \pi_{P}^{AN * } } \right) = \frac{{z\eta \sigma_{0}^{2} \left( {2\eta - \theta \gamma^{2} \left( {2 + \theta } \right)} \right)}}{{8\left( {\eta - \theta \gamma^{2} } \right)^{2} }}$; and under the non-receommendation scenario, we have $v_{BP}^{A} = \pi_{BP}^{AS * } - \pi_{BP}^{AN * } = \frac{{z\theta \sigma_{0}^{2} }}{4}$, $v_{BM}^{A} = \pi_{BM}^{AS * } - \pi_{BM}^{AN * } = \frac{{z\sigma_{0}^{2} \left( {1 - \theta } \right)}}{4}$, and $v_{B}^{A} = \left( {\pi_{BM}^{AS * } + \pi_{BP}^{AS * } } \right) - \left( {\pi_{BM}^{AN * } + \pi_{BP}^{AN * } } \right) = \frac{{z\sigma_{0}^{2} }}{4}$.

To ensure that the manufacturer and the retailer participating in the game, the ex ante profits in Lemma 2 should satisfy the condition $\eta > \eta_{1}$, and it is easy to derive $v_{P}^{A} > 0$,$v_{M}^{A} > 0$, $v^{A} > 0$, $v_{BP}^{A} > 0$, $v_{BM}^{A} > 0$, and $v_{B}^{A} > 0$.

Proof of Proposition 6

Based on the results given in the proof of Proposition 5, and $z = {{\sigma_{{0}}^{{2}} } \mathord{\left/ {\vphantom {{\sigma_{{0}}^{{2}} } {\left( {\sigma_{{0}}^{{2}} + \sigma_{r}^{{2}} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\sigma_{{0}}^{{2}} + \sigma_{r}^{{2}} } \right)}}$, it is easy to verify the results in Table

Table 4 The effects of $\sigma_{0}$ and $\sigma_{r}$ on the value of information sharing

4.

As shown in Table 4, we can prove the Proposition 6. For the platform, $\frac{{\partial v_{BP}^{A} }}{{\partial \sigma_{0} }} > 0$, $\frac{{\partial v_{BP}^{A} }}{{\partial \sigma_{r} }} < 0$, $\frac{{\partial v_{P}^{A} }}{{\partial \sigma_{0} }} > 0$, $\frac{{\partial v_{P}^{A} }}{{\partial \sigma_{r} }} < 0$. For the manufacturer, $\frac{{\partial v_{BM}^{A} }}{{\partial \sigma_{0} }} > 0$, $\frac{{\partial v_{BM}^{A} }}{{\partial \sigma_{r} }} < 0$, $\frac{{\partial v_{M}^{A} }}{{\partial \sigma_{0} }} > 0$, $\frac{{\partial v_{M}^{A} }}{{\partial \sigma_{r} }} < 0$. For the supply chain, $\frac{{\partial v_{B}^{A} }}{{\partial \sigma_{0} }} > 0$, $\frac{{\partial v_{B}^{A} }}{{\partial \sigma_{r} }} < 0$, $\frac{{\partial v^{A} }}{{\partial \sigma_{0} }} > 0$, $\frac{{\partial v^{A} }}{{\partial \sigma_{r} }} < 0$.

Proof of Proposition 7

From Propositions 3 and 5, we can derive that the potential optimal strategies for the platform are mode BRN, mode BAS, mode RN, and mode AS. Since $0 < \theta < \frac{1}{2}$, we derive that $\left\{ {\begin{array}{*{20}c} {\eta_{0} > \eta_{2} > \eta_{1} ,} & {0 < \theta < \frac{1}{4}} \\ {\eta_{2} > \eta_{0} > \eta_{1} ,} & {\frac{1}{4} < \theta < \frac{1}{3}} \\ {\eta_{2} > \eta_{1} > \eta_{0} ,} & {\frac{1}{3} < \theta < \frac{1}{2}} \\ \end{array} } \right.$.

• (i) When $0 < \theta < \frac{1}{4}$, $\eta_{0} > \eta_{2} > \eta_{1}$, the platfrom’s optiaml strategy is given as follows:

  • (a) When $0 < \theta < \frac{1}{4}$, $0 < \eta \le \eta_{1}$, the platform can only choose the non-recommendation scenario in two contracts. Therefore, the platfrom’s optiaml strategy can be obtained by comparing mode BRN and mode BAS. From Lemma 3, we have the following result: $\pi_{BP}^{RN * } - \pi_{BP}^{AS * } = \frac{{a_{0}^{2} \left( {1 - 4\theta } \right) + 4z\sigma_{0}^{2} \left( {1 - \theta } \right)}}{16}$. Since $0 < \theta < \frac{1}{4}$, $0 < \eta \le \eta_{1}$, it is easy to obtain $\pi_{BP}^{RN * } > \pi_{BP}^{AS * }$.

  • (b) When $0 < \theta < \frac{1}{4}$, $\eta_{1} < \eta \le \eta_{2}$, the platform can only choose non-recommendation scenario in the reselling contract, but non-recommendation scenario and recommendation scenario are meaningful in the agency contract. The results in Proposition 2 show that the platform still prefers the non-recommendation scenario when $\eta_{1} < \eta \le \eta_{2}$ in the agency contract. Hence, $\pi_{BP}^{RN * } > \pi_{BP}^{AS * }$ still holds.

  • (c) When $0 < \theta < \frac{1}{4}$, $\eta_{2} < \eta \le \eta_{0}$, the platform still prefers mode BRN under the reselling contract. However, under the agecny contract, we have $\pi_{P}^{AS * } > \pi_{BP}^{AS * }$. Therefore, by comparing mode BRN and mode AS, we have $\pi_{BP}^{RN * } - \pi_{P}^{AS * } = \frac{1}{16}\left( {a_{0}^{2} + 4z\sigma_{0}^{2} - \frac{{2\eta \theta \left( {2\eta - 3\theta \gamma^{2} } \right)\left( {a_{0}^{2} + z\sigma_{0}^{2} } \right)}}{{\left( {\eta - \theta \gamma^{2} } \right)^{2} }}} \right) > 0$. As a result, when $0 < \theta < \frac{1}{4}$, $0 < \eta \le \eta_{0}$, the platform prefers mode BRN.

  • (d) When $0 < \theta < \frac{1}{4}$, $\eta > \eta_{0}$, we have $\pi_{P}^{RN * } > \pi_{BP}^{RN * }$ and $\pi_{P}^{AS * } > \pi_{BP}^{AS * }$. Therefore, by comparing mode RN and mode AS, we obtain that $\pi_{P}^{RN * } - \pi_{P}^{AS * } = \frac{1}{8}\eta \left( {\frac{{\theta \left( {3\theta \gamma^{2} - 2\eta } \right)\left( {a_{0}^{2} + z\sigma_{0}^{2} } \right)}}{{\left( {\eta - \theta \gamma^{2} } \right)^{2} }} + \frac{{a_{0}^{2} + 4z\sigma_{0}^{2} }}{{2\eta - \gamma^{2} }}} \right) > 0$. As a result, when $0 < \theta < \frac{1}{4}$, $0 < \eta \le \eta_{0}$, the platform will choose mode BRN; when $\eta > \eta_{0}$, mode RN is preferred.

• (ii) When $\frac{1}{4} < \theta < \frac{1}{3}$, $\eta_{1} < \eta_{0} < \eta_{2}$, the conditions of the platform’s optimal strategy are complex.

  • (a) When $\frac{1}{4} < \theta < \frac{1}{3}$, $0 < \eta \le \eta_{1}$, the platform can only choose non-recommendation scenario in two contracts. When $\frac{1}{4} < \theta < \frac{1}{3}$, $\eta_{1} < \eta \le \eta_{0}$, the platform prefers non-recommendation scenario in two contracts though both non-recommendation scenario and recommendation scenario are meaningful in the agency contract. Hence, we have $\pi_{BP}^{RN * } - \pi_{BP}^{AS * } = \frac{{a_{0}^{2} \left( {1 - 4\theta } \right) + 4z\sigma_{0}^{2} \left( {1 - \theta } \right)}}{16}$. Let $z_{BP} = \frac{{a_{0}^{2} \left( {4\theta - 1} \right)}}{{4\sigma_{0}^{2} \left( {1 - \theta } \right)}}$, then $\pi_{BP}^{RN * } < \pi_{BP}^{AS * }$ if $0 < z \le \min \{ z_{BP} ,{\kern 1pt} {\kern 1pt} 1\}$; $\pi_{BP}^{RN * } > \pi_{BP}^{AS * }$ if $\min \{ z_{BP} ,{\kern 1pt} {\kern 1pt} 1\} < z < 1$.

  • (b) When $\frac{1}{4} < \theta < \frac{1}{3}$, $\eta_{0} < \eta \le \eta_{2}$, the platform can choose the non-recommendation scenario and recommendation scenario in both contracts. Since $\pi_{P}^{RN * } > \pi_{BP}^{RN * }$ and $\pi_{P}^{AS * } < \pi_{BP}^{AS * }$, we have $\pi_{BP}^{AS * } - \pi_{P}^{RN * } = \frac{1}{8}\left( {2\theta \left( {a_{0}^{2} + z\sigma_{0}^{2} } \right) - \frac{{\eta \left( {a_{0}^{2} + 4z\sigma_{0}^{2} } \right)}}{{2\eta - \gamma^{2} }}} \right) < 0$. Hence, mode RN is chosen.

  • (c) When $\frac{1}{4} < \theta < \frac{1}{3}$, $\eta > \eta_{2}$, we have $\pi_{P}^{RN * } - \pi_{P}^{AS * } = \frac{\eta }{8}\left( {\frac{{\theta \left( {3\theta \gamma^{2} - 2\eta } \right)\left( {a_{0}^{2} + z\sigma_{0}^{2} } \right)}}{{\left( {\eta - \theta \gamma^{2} } \right)^{2} }} + \frac{{a_{0}^{2} + 4z\sigma_{0}^{2} }}{{2\eta - \gamma^{2} }}} \right)$.

    Let $z_{P} = \frac{{a_{0}^{2} \left( {2\theta^{2} \gamma^{4} - 6\eta \theta^{2} \gamma^{2} - \eta^{2} \left( {1 - 4\theta } \right)} \right)}}{{\sigma_{0}^{2} \left( {\theta^{2} \gamma^{4} + 4\eta^{2} \left( {1 - \theta } \right) - 6\eta \theta \gamma^{2} \left( {1 - \theta } \right)} \right)}}$, when $\eta > \eta_{2}$ and $0 < z < \min \{ \max \{ 0,{\kern 1pt} {\kern 1pt} z_{P} \} ,{\kern 1pt} {\kern 1pt} 1\}$, $\pi_{P}^{RN * } < \pi_{P}^{AS * }$; when $\eta > \eta_{2}$ and $\min \{ \max \{ 0,{\kern 1pt} {\kern 1pt} z_{p} \} ,{\kern 1pt} {\kern 1pt} 1\} < z < 1$, $\pi_{P}^{RN * } > \pi_{P}^{AS * }$.

• (iii) When $\frac{1}{3} < \theta < \frac{1}{2}$, $\eta_{0} < \eta_{1} < \eta_{2}$, the conditions of the platform’s optimal strategy are given as follows.

  • (a) When $\frac{1}{3} < \theta < \frac{1}{2}$, $0 < \eta \le \eta_{0}$; the platform prefers non-recommendatin scenario under two contracts. Hence, $\pi_{BP}^{RN * } < \pi_{BP}^{AS * }$ if $0 < z < \min \{ z_{BP} ,{\kern 1pt} {\kern 1pt} 1\}$; $\pi_{BP}^{RN * } > \pi_{BP}^{AS * }$ if $\min \{ z_{BP} ,{\kern 1pt} {\kern 1pt} 1\} < z < 1$.

  • (b) When $\frac{1}{3} < \theta < \frac{1}{2}$, $\eta_{0} < \eta \le \eta_{1}$; or $\frac{1}{3} < \theta < \frac{1}{2}$, $\eta_{1} < \eta \le \eta_{2}$, the platform always prefers mode RN under the reselling contract and mode BAS under the agency contact. Hence, $\pi_{BP}^{AS * } - \pi_{P}^{RN * } = \frac{1}{8}\left( {2\theta \left( {a_{0}^{2} + z\sigma_{0}^{2} } \right) - \frac{{\eta \left( {a_{0}^{2} + 4z\sigma_{0}^{2} } \right)}}{{2\eta - \gamma^{2} }}} \right) < 0$.

  • (c) When $\frac{1}{3} < \theta < \frac{1}{2}$, $\eta > \eta_{2}$, the results are same as proved above. Hence, we obtain the results in Proposition 7.

Proof of Corollary 1

Given the platform’s optimal strategy, if the manufacturer can better off, the manufacturer will accept this contract.

• (i) When $0 < \theta < \frac{1}{4}$, the platform’s optiaml strategy is mode BRN or mode RN.

  • (a) When $0 < \theta < \frac{1}{4}$, $0 < \eta \le \eta_{1}$, the platfrom’s optiaml strategy is mode BRN. However, given the platform’s strategy, $\pi_{BM}^{RN * } - \pi_{BM}^{AS * } = - \frac{{a_{0}^{2} \left( {1 - 2\theta } \right) + 2z\sigma_{0}^{2} \left( {1 - \theta } \right)}}{8} < 0$. Hence, the manufacturer is worse off in mode BRN. Similarly, when $0 < \theta < \frac{1}{4}$, $\eta_{1} < \eta \le \eta_{2}$, we have $\pi_{BM}^{RN * } - \pi_{BM}^{AS * } < 0$. When $0 < \theta < \frac{1}{4}$, $\eta_{2} < \eta \le \eta_{0}$, we have $\pi_{BM}^{RN * } - \pi_{M}^{AS * } = \frac{{a_{0}^{2} \left( {\theta \left( {2\eta - \gamma^{2} } \right) - \eta } \right) - 2z\eta \sigma_{0}^{2} \left( {1 - \theta } \right)}}{{8\left( {\eta - \theta \gamma^{2} } \right)}} < 0$. Hence, when $0 < \eta \le \eta_{0}$, the platform and the manufacturer can not reach a consensus.

  • (b) When $0 < \theta < \frac{1}{4}$, $\eta > \eta_{0}$, we have $\pi_{M}^{RN * } - \pi_{M}^{AS * } = \frac{{\eta a_{0}^{2} \left( {\gamma^{2} - \eta } \right)\left( {1 - 2\theta } \right)}}{{4\left( {2\eta - \gamma^{2} } \right)\left( {\eta - \theta \gamma^{2} } \right)}} - \frac{{z\sigma_{0}^{2} \eta \left( {1 - \theta } \right)}}{{4\left( {\eta - \theta \gamma^{2} } \right)}}$. Let $z_{M} = \frac{{a_{0}^{2} \left( {\gamma^{2} - \eta } \right)\left( {1 - 2\theta } \right)}}{{\sigma_{0}^{2} \left( {2\eta - \gamma^{2} } \right)\left( {1 - \theta } \right)}}$, when $0 < z < \min \{ \max \{ 0,{\kern 1pt} {\kern 1pt} z_{M} \} ,{\kern 1pt} {\kern 1pt} 1\}$, $\pi_{M}^{RN * } > \pi_{M}^{AS * }$; when $\min \{ \max \{ 0,{\kern 1pt} {\kern 1pt} z_{M} \} ,{\kern 1pt} {\kern 1pt} 1\} < z < 1$, $\pi_{M}^{RN * } < \pi_{M}^{AS * }$. As a result, when $0 < \theta < \frac{1}{4}$, $\eta > \eta_{0}$, $0 < z < \min \{ \max \{ 0,{\kern 1pt} {\kern 1pt} z_{M} \} ,{\kern 1pt} {\kern 1pt} 1\}$, the platform’s and manufacturer’s optimal strategy is mode RN, and they can achieve a win–win situation.

• (ii) When $\frac{1}{4} < \theta < \frac{1}{2}$, the proof process is similar, and we obtain the results as follows.

  • (a) When $\frac{1}{4} < \theta < \frac{1}{2}$, $0 < \eta \le \eta_{0}$, and $0 < z < \min \{ z_{BP} ,{\kern 1pt} {\kern 1pt} 1\}$, mode BAS makes the manufacturer better off. In addition, when $\frac{1}{4} < \theta < \frac{1}{2}$, $\eta_{0} < \eta \le \eta_{2}$, mode RN makes the manufacturer better off.

  • (b) When $\frac{1}{4} < \theta < \frac{1}{2}$, $\eta > \eta_{2}$, the optimal strategy for the platform and the manufacturer is affected by the information accuracy. For the platform, if $0 < z < \min \{ \max \{ 0,{\kern 1pt} {\kern 1pt} z_{P} \} ,{\kern 1pt} {\kern 1pt} 1\}$, $\pi_{P}^{RN * } < \pi_{P}^{AS * }$; if $\eta > \eta_{2}$ and $\min \{ \max \{ 0,{\kern 1pt} {\kern 1pt} z_{p} \} ,{\kern 1pt} {\kern 1pt} 1\} < z < 1$, $\pi_{P}^{RN * } > \pi_{P}^{AS * }$. For the manufacturer, if $0 < z < \min \{ \max \{ 0,{\kern 1pt} {\kern 1pt} z_{M} \} ,{\kern 1pt} {\kern 1pt} 1\}$, $\pi_{M}^{RN * } > \pi_{M}^{AS * }$; if $\min \{ \max \{ 0,{\kern 1pt} {\kern 1pt} z_{M} \} ,{\kern 1pt} {\kern 1pt} 1\} < z < 1$, $\pi_{M}^{RN * } < \pi_{M}^{AS * }$. Through analyzing, we obtain that $z_{P} < z_{M}$ if $\eta_{2} < \eta < \eta_{3}$; and $z_{P} > z_{M}$ if $\eta > \eta_{3}$. Therefore, when $\frac{1}{4} < \theta < \frac{1}{2}$, $\eta_{2} < \eta < \eta_{3}$, $\min \{ \max \{ 0,{\kern 1pt} {\kern 1pt} z_{p} \} ,{\kern 1pt} {\kern 1pt} 1\} < z < \min \{ \max \{ 0,{\kern 1pt} {\kern 1pt} z_{M} \} ,{\kern 1pt} {\kern 1pt} 1\}$, mode RN makes the manufacturer better off; When $\frac{1}{4} < \theta < \frac{1}{2}$, $\eta > \eta_{3}$, $\min \{ \max \{ 0,{\kern 1pt} {\kern 1pt} z_{M} \} ,{\kern 1pt} {\kern 1pt} 1\} < z < \min \{ \max \{ 0,{\kern 1pt} {\kern 1pt} z_{P} \} ,{\kern 1pt} {\kern 1pt} 1\}$, mode AS makes the manufacturer better off, where $\eta_{3} = \frac{{\gamma^{2} }}{4}\left( {3 + \sqrt {\frac{{9 - 33\theta + 32\theta^{2} }}{1 - \theta }} } \right)$.

Proof of Corollary 2

It is easy to derive that $\frac{{\partial z_{BP} }}{\partial \eta } = 0$, $\frac{{\partial z_{M} }}{\partial \eta } = - \frac{{a_{0}^{2} \gamma^{2} \left( {1 - 2\theta } \right)}}{{\sigma_{0}^{2} \left( {2\eta - \gamma^{2} } \right)^{2} \left( {1 - \theta } \right)}} < 0$, and $\frac{{\partial z_{P} }}{\partial \eta } = \frac{{6a_{0}^{2} \gamma^{2} \theta \left( {\eta - \theta \gamma^{2} } \right)\left( {\eta \left( {1 - \theta } \right) - \theta \gamma^{2} \left( {2 - 3\theta } \right)} \right)}}{{\sigma_{0}^{2} \left( {4\eta^{2} \left( {1 - \theta } \right) - 6\eta \theta \gamma^{2} \left( {1 - \theta } \right) + \gamma^{4} \theta^{2} } \right)^{2} }} > 0$.

1.1 Proof of the equilibrium solutions for mode BHN and mode BHS

Given the hybrid contract of the non-recommendation scenario, there exist equilibrium solutions for mode BHN and mode BHS, respectively. The optimal decisions are as follows:

• (i) $w_{B}^{HN * } = \frac{{a_{0} \left( {1 - \theta } \right)}}{{2\left( {1 + \beta } \right)}}\left( {\frac{\beta }{1 - \beta } + \frac{{\phi \left( {8 + \beta \left( {8\theta - \beta^{2} \left( {1 + \theta } \right)^{2} } \right)} \right)}}{{8\left( {1 - \theta } \right) + \beta^{2} \left( {1 + \theta } \right)^{2} }}} \right)$, $p_{BM}^{HN * } { = }\frac{{a_{0} }}{{2\left( {1 - \beta^{2} } \right)}}\left( {1 - \phi + \frac{{\beta \phi \left( {2\left( {5 - 3\theta } \right) - \beta^{2} \left( {1 - \theta^{2} } \right)} \right)}}{{8\left( {1 - \theta } \right) + \beta^{2} \left( {1 + \theta } \right)^{2} }}} \right)$, $p_{BP}^{HN * } = \frac{1}{2}\left( {\phi \left( {a_{0} \left( {1 - z} \right) + zf} \right) + \frac{{a_{0} }}{1 + \beta }\left( {\frac{\beta }{1 - \beta } + \frac{{\phi \left( {4\left( {1 - \theta } \right)\left( {1 - \beta } \right) + \beta^{2} \left( {1 + \theta } \right)^{2} } \right)}}{{8\left( {1 - \theta } \right) + \beta^{2} \left( {1 + \theta } \right)^{2} }}} \right)} \right)$;

• (ii) $w_{B}^{HS * } { = }\frac{{\left( {1 - \theta } \right)\left( {a_{0} \left( {1 - z} \right) + zf} \right)}}{{2\left( {1 + \beta } \right)}}\left( {\frac{\beta }{1 - \beta } + \frac{{\phi \left( {8 + \beta \left( {8\theta - \beta^{2} \left( {1 + \theta } \right)^{2} } \right)} \right)}}{{8\left( {1 - \theta } \right) + \beta^{2} \left( {1 + \theta } \right)^{2} }}} \right)$, $p_{BM}^{HS * } = \frac{{\left( {a_{0} \left( {1 - z} \right) + zf} \right)}}{{2\left( {1 - \beta^{2} } \right)}}\left( {1 - \phi + \frac{{\beta \phi \left( {2\left( {5 - 3\theta } \right) - \beta^{2} \left( {1 - \theta^{2} } \right)} \right)}}{{8\left( {1 - \theta } \right) + \beta^{2} \left( {1 + \theta } \right)^{2} }}} \right)$, $p_{BP}^{HS * } { = }\frac{{\left( {a_{0} \left( {1 - z} \right) + zf} \right)}}{2}\left( {\phi + \frac{1}{1 + \beta }\left( {\frac{\beta }{1 - \beta } + \frac{{\phi \left( {4\left( {1 - \theta } \right)\left( {1 - \beta } \right) + \beta^{2} \left( {1 + \theta } \right)^{2} } \right)}}{{8\left( {1 - \theta } \right) + \beta^{2} \left( {1 + \theta } \right)^{2} }}} \right)} \right)$.

1.2 Proof of the equilibrium solutions for mode HN and mode HS

Given the hybrid contract of recommendation scenario, there exist equilibrium solutions for mode HN and mode HS, respectively. The optimal decisions are as follows:

• (i) $w^{HN * } = \frac{{a_{0} \eta \left( {1 - \theta } \right)\left( {H_{4} + H_{5} + H_{6} + H_{7} } \right)}}{{2\left( {H_{1} + H_{2} + H_{3} } \right)}}$, $p_{M}^{HN * } = \frac{{a_{0} \eta \left( {2 - \phi \left( {2 - \beta } \right)} \right)}}{{H_{8} }} + \frac{{w^{HN * } \left( {\beta \eta \left( {3 - \theta } \right) - \phi \gamma^{2} \left( {\beta \phi \left( {2 - \theta } \right) + \left( {1 - \phi } \right)\left( {1 - \theta } \right)} \right)} \right)}}{{H_{8} \left( {1 - \theta } \right)}}$, $s^{HN * } = \frac{{\gamma \left( {\phi \left( {\phi \left( {a_{0} \left( {1 - z} \right) + zf} \right) - w^{HN * } } \right) + p_{M}^{HN * } \left( {\beta \phi + \theta \left( {2 - \phi \left( {2 - \beta } \right)} \right)} \right)} \right)}}{{2\eta - \gamma^{2} \phi^{2} }}$, $p_{P}^{HN * } = \frac{{\eta \phi \left( {a_{0} \left( {1 - z} \right) + zf} \right) + w^{HN * } \left( {\eta - \gamma^{2} \phi^{2} } \right) + p_{M}^{HN * } \left( {\beta \eta \left( {1 + \theta } \right) + \phi \theta \gamma^{2} \left( {1 - \phi } \right)} \right)}}{{2\eta - \gamma^{2} \phi^{2} }}$;

• (ii) $w^{HS * } = \frac{{\eta \left( {1 - \theta } \right)\left( {a_{0} \left( {1 - z} \right) + zf} \right)\left( {H_{4} + H_{5} + H_{6} + H_{7} } \right)}}{{2\left( {H_{1} + H_{2} + H_{3} } \right)}}$, $p_{M}^{HS * } = \frac{{\eta \left( {a_{0} \left( {1 - z} \right) + zf} \right)\left( {2 - \phi \left( {2 - \beta } \right)} \right)}}{{H_{8} }} + \frac{{w^{HS * } \left( {\beta \eta \left( {3 - \theta } \right) - \phi \gamma^{2} \left( {\beta \phi \left( {2 - \theta } \right) + \left( {1 - \phi } \right)\left( {1 - \theta } \right)} \right)} \right)}}{{H_{8} \left( {1 - \theta } \right)}}$, $\begin{gathered} s^{HS * } = \frac{{w^{HS * } \gamma }}{{H_{8} }}\left( {\frac{{\beta \theta \left( {3 - \theta } \right)\left( {1 - \phi } \right) + \phi 2\beta^{2} }}{{\left( {1 - \theta } \right)}} - \phi \left( {2 - \theta \beta^{2} } \right)} \right) \\ + \frac{{\gamma \left( {a_{0} \left( {1 - z} \right) + zf} \right)\left( {\phi \left( {\beta + \left( {2 - \beta } \right)\phi } \right) + 2\theta \left( {1 - \phi } \right)\left( {1 - \phi \left( {1 - \beta } \right)} \right)} \right)}}{{H_{8} }} \\ \end{gathered}$, $\begin{gathered} p_{P}^{HS * } = \frac{{\eta \left( {a_{0} \left( {1 - z} \right) + zf} \right)\left( {\beta \left( {1 + \theta } \right)\left( {1 - \phi } \right) + 2\phi } \right)}}{{H_{8} }} \\ + \frac{{w^{HS * } }}{{H_{8} }}\left( {2\eta - \theta \gamma^{2} + \frac{{\eta \beta^{2} \left( {1 + \theta } \right) + \phi \gamma^{2} \left( {\left( {\theta \left( {2 + \beta - \theta \left( {2 - \beta } \right)} \right) - \beta } \right) - \phi \left( {1 + \left( {1 - \beta } \right)\left( {1 - \theta - \theta^{2} } \right)} \right)} \right)}}{1 - \theta }} \right) \\ \end{gathered}$.

Where $H_{1} = - \eta^{2} \left( {1 - \beta^{2} } \right)\left( {8\left( {1 - \theta } \right) + \beta^{2} \left( {1 + \theta } \right)^{2} } \right)$,

$$ \begin{aligned} H_{2} & = \gamma^{4} \left( {1 - \theta + \phi \left( {2\beta - 1 + \theta \left( {1 - \beta } \right)} \right)} \right)\left( \begin{gathered} \phi^{2} \left( {1 - \beta^{2} } \right)\left( {1 - \phi } \right) - 3\theta \phi \left( {1 - \phi \left( {1 - \beta } \right)} \right)\left( {\beta + \phi \left( {1 - \beta } \right)} \right) \hfill \\ - 2\left( {1 - \phi } \right)\left( {\theta \left( {1 - \phi \left( {1 - \beta } \right)} \right)} \right)^{2} \hfill \\ \end{gathered} \right), \\ H_{3} & = \eta \gamma^{2} \left( \begin{gathered} 2\phi^{2} \left( {2 - \beta^{2} - \beta^{4} } \right) + \theta^{3} \left( {\beta \left( {1 - \phi \left( {1 - \beta } \right)} \right)} \right)^{2} - 2\theta^{2} \left( {1 - \phi \left( {1 - \beta } \right)} \right)\left( {4 - \beta^{2} - \phi \left( {1 - \beta } \right)\left( {4 + \beta } \right)} \right) \hfill \\ + \theta \left( {8 + \beta^{2} - 2\phi \left( {1 - \beta } \right)\left( {8 - \beta } \right) + \phi^{2} \left( {4 - \beta \left( {18 - 17\beta + 3\beta^{3} } \right)} \right)} \right) \hfill \\ \end{gathered} \right), \\ H_{4} & = - \phi \eta \beta^{4} \left( {1 + \theta } \right)^{2} - \beta^{3} \left( {1 + \theta } \right)\left( {1 - \phi } \right)\left( {\eta \left( {1 + \theta } \right) + \phi^{2} \gamma^{2} \left( {1 + 2\theta } \right)} \right), \\ H_{5} & = \phi \beta^{2} \left( {8\eta \theta - \gamma^{2} \left( {1 + 4\theta^{2} \left( {1 - \phi } \right)^{2} - \phi \left( {2 + \phi } \right) + \theta \left( {5 - \phi \left( {10 - 9\phi } \right)} \right)} \right)} \right), \\ H_{6} & = - 2\beta \left( {1 - \phi } \right)\left( {4\eta \left( {1 - \theta } \right) + \gamma^{2} \left( {\theta \left( {1 + \theta } \right)\left( {1 - 2\phi } \right) + \phi^{2} \left( {\theta \left( {3 + \theta } \right) - 5} \right)} \right)} \right), \\ H_{7} & = - 4\phi \left( {2\eta - \gamma^{2} \left( {1 - 2\phi \left( {1 - \phi } \right)} \right)} \right), \\ H_{8} & = \eta \left( {4 - \beta^{2} \left( {1 + \theta } \right)} \right) - \gamma^{2} \left( {2\theta \left( {1 - \phi } \right)\left( {1 - \phi \left( {1 - \beta } \right)} \right) + \phi \left( {\beta + \phi \left( {2 - \beta } \right)} \right)} \right). \\ \end{aligned} $$