Optimal rebate strategy for an online retailer with a cashback platform: commission-driven or marketing-based?

Abstract

This paper analyzes an emerging rebate mode where a cashback website not only serves as a promotional device in generating more sales for an online retailer, but also makes decisions on its marketing effort autonomously under certain product availability conditions set by the online retailer. Using a game-theoretic approach, this paper analyzes the optimal decision of both parties in commission-driven rebate mode and marketing-based rebate mode in a market dominated by the online retailer and cashback website respectively. Our main findings are as follows: (1) In the Retailer Stackelberg (RS) game, which rebate mode an online retailer will adopt is related to the marketing cost of the rebate channel and the consumer utility discount (UD) factor. (2) Both rebate modes will help the retailer to expand the market and increase profit because of price differentiation. (3) There is a possible price trap in marketing-based rebate mode. (4) In RS game, the effect of consumer UD factor on the optimal marketing effort is related to the product availability in the rebate channel. (5) In both RS game and Cashback-website Stackelberg (CS) game, the online retailer should strive to become a leader. However, the cashback website prefers the online retailer to be a leader when marketing cost is relatively low in the marketing-based mode. Our results, consistent with several real-world observations, have useful implications for marketers.

Appendices

Appendix

1.1 Proof of the equilibrium of the benchmark case (Retailer Without Cashback Channel)

The retailer’s profit function is given by: $\pi_{RN} (p_{N} ) = (p_{N} - c)\left( {1 - p} \right)$. Obviously, the second order condition holds. From the first order condition, we can yield $p_{N}^{*} = \frac{1 + c}{2}$. By substituting the maximum point into the demand function and profit function, we then have $D_{N}^{*} = \frac{1 - c}{2}$,$\pi_{N}^{*} { = }\frac{{\left( {1 - c} \right)^{2} }}{4}$.

1.2 Proof of the equilibrium of the Retailer Stackelberg (RS) Game

1.2.1 Commission-driven Mode in RS Game

For any given retail price $p$ and commission rate $\alpha$,the objective the cashback website is to determine the rebate rate given to customers to maximize its profit, which is given by $\pi_{CF} (\beta ) = p(\alpha - \beta )(\frac{\beta p}{{1 - \theta }} - \frac{{\left( {1 - \beta } \right)p}}{\theta })$. The second order derivative of $\pi_{CF}$ with respect to (w.r.t)$\beta$ is $\frac{{\partial^{2} \pi_{CF} }}{{\partial \beta^{2} }}{ = } - \frac{{2p^{2} }}{{\theta \left( {1 - \theta } \right)}}$. It is easy to see that $\frac{{\partial^{2} \pi_{CF} }}{{\partial \beta^{2} }} < 0$ and $\pi_{CF}$ is a concave function of $\beta$ given $p$ and $\alpha$. Hence, solving the first order condition of $\pi_{CF}$ w.r.t $\beta$, we can yield $\beta^{*} { = }\frac{1 + \alpha - \theta }{2}$.

Then, substituting this $\beta$ into the profit function of the retailer, we can express the function as $\pi_{RL} \left( {p,\alpha } \right) = \left( {p - c} \right)\left( {1 - \frac{{\left( {1 + \alpha - \theta } \right)p}}{{2\left( {1 - \theta } \right)}}} \right) + \frac{{\left( {1 - \alpha } \right)\left( {\alpha + \theta - 1} \right)p^{2} }}{{2\theta \left( {1 - \theta } \right)}}$. By solving the first order condition of $\pi_{RL} \left( {p,\alpha } \right)$ w.r.t $p$ and $\alpha$, the following results can be yield:$p_{dL}^{*} = \frac{1 + c}{2}$,$\alpha_{RS}^{*} = 1 - \frac{\theta }{1 + c}$. Then, we use the bordered Hessian matrix to prove whether $\pi_{RL} \left( {p,\alpha } \right)$ is jointly quasi-concave in $\left( {p_{dL}^{*} ,\alpha_{RS}^{*} } \right)$. The bordered Hessian matrix of $\pi_{RL} \left( {p,\alpha } \right)$ in $\left( {p_{dL}^{*} ,\alpha_{RS}^{*} } \right)$ is as follows: $H_{1} = \left( {\begin{array}{*{20}c} {\frac{{2\left( {1 + c} \right)^{2} - \left( {2 + c\left( {4 + c} \right)} \right)\theta }}{{\left( {1 + c} \right)^{2} \left( { - 1 + \theta } \right)}}} & {\frac{c}{{2\left( { - 1 + \theta } \right)}}} \\ {\frac{c}{{2\left( { - 1 + \theta } \right)}}} & {\frac{{\left( {1 + c} \right)^{2} }}{{4\left( { - 1 + \theta } \right)\theta }}} \\ \end{array} } \right)$. If the determinant of the matrix $\left| {H_{1} } \right|$ is greater than zero and the first order principal minor of the Hessian matrix $H_{1} \left[ {1,1} \right]$ is less than zero,$\pi_{RL} \left( {p,\alpha } \right)$ is quasi-concave. From $H_{1}$, it is easy to get $\left| {H_{1} } \right| = \frac{{\left( {1 + c} \right)^{2} }}{{2\left( {1 - \theta } \right)\theta }}$, $H_{1} \left[ {1,1} \right] = - \frac{{2\left( {1 + 2c} \right)\left( {1 - \theta } \right) + c^{2} \left( {2 - \theta } \right)}}{{\left( {1 + c} \right)^{2} \left( {1 - \theta } \right)}}$. It is obvious that the two conditions ($\left| {H_{1} } \right| > 0$, $H_{1} \left[ {1,1} \right] < 0$) for the negative determination of $H_{1}$ are satisfied. Thus,$\pi_{RL} \left( {p,\alpha } \right)$ is jointly concave in $\left( {p,\alpha } \right)$.

Finally, substituting $p_{dL}^{*}$ and $\alpha_{RS}^{*}$ into $\beta^{*}$, demand functions and profit functions, the equilibrium of commission-driven mode in the RS Game obtained (See Table 2).

1.2.2 Marketing-based Mode in RS Game

In marketing-based mode, the profit functions of the retailer and the cashback platform are as follows respectively:

$$\begin{gathered} \pi_{{RL^{\prime}}} (p,l) = (p - c)\left( {1 - \frac{ - \delta \lambda l + \lambda p - p - \gamma \lambda s}{{\theta \lambda - 1}}} \right){ + }(p - l)\left( {\frac{ - \delta \lambda l + \lambda p - p - \gamma \lambda s}{{\theta \lambda - 1}} - \frac{ - \delta l + p - \gamma s}{\theta }} \right), \hfill \\ \pi_{{CF^{\prime}}} (\delta ,s) = l\left( {1 - \delta } \right)\left( {\frac{ - \delta \lambda l + \lambda p - p - \gamma \lambda s}{{\theta \lambda - 1}} - \frac{ - \delta l + p - \gamma s}{\theta }} \right) - \frac{1}{2}\eta s^{2} . \hfill \\ \end{gathered}$$

According to the backward induction method, we calculate the optimal rebate strategy $\delta^{*}$ and marketing effort strategy $s^{*}$ of the cashback platform first for any given price p and marketing expense l. By solving the first order condition of $\pi_{CF^{\prime}} \left( {\delta ,l} \right)$ w.r.t $\delta$ and $s$, the following results can be yield:$\delta^{*} = \frac{{l\left( {\gamma^{2} - \eta \theta \left( {1 - \lambda \theta } \right)} \right) - p\eta \left( {1 - \theta } \right)\theta \left( {1 - \lambda \theta } \right)}}{{l\left( {\gamma^{2} - 2\eta \theta \left( {1 - \lambda \theta } \right)} \right)}}$, $s^{*} = \frac{{\gamma \left( {\left( {1 - \theta } \right)p - l} \right)}}{{\gamma^{2} - 2\eta \theta \left( {1 - \lambda \theta } \right)}}$. Similarly, we use the bordered Hessian matrix to prove whether $\pi_{CF^{\prime}} \left( {\delta ,s} \right)$ is jointly quasi-concave in $\left( {\delta^{*} ,s^{*} } \right)$. The bordered Hessian matrix of $\pi_{CF^{\prime}} \left( {\delta ,s} \right)$ in $\left( {\delta^{*} ,s^{*} } \right)$ is as follows: $H_{2} = \left( {\begin{array}{*{20}c} { - \frac{{2l^{2} }}{{\left( {1 - \lambda \theta } \right)}}} & { - \frac{\gamma l}{{\theta \left( {1 - \lambda \theta } \right)}}} \\ { - \frac{\gamma l}{{\theta \left( {1 - \lambda \theta } \right)}}} & { - \eta } \\ \end{array} } \right)$. Obviously, the first order principal minor of the Hessian matrix $H_{2} \left[ {1,1} \right]$ is less than zero. If the determinant of the matrix $\left| {H_{2} } \right|{ = - }\frac{{l^{2} \left( {\gamma^{2} - 2\eta \theta \left( {1 - \lambda \theta } \right)} \right)}}{{\theta^{2} \left( {1 - \lambda \theta } \right)^{2} }}$ is greater than zero,$\pi_{CF^{\prime}} \left( {\delta ,l} \right)$ is jointly quasi-concave in $\left( {\delta^{*} ,s^{*} } \right)$. Thus, we derive the condition $\gamma^{2} - 2\eta \theta \left( {1 - \lambda \theta } \right) < 0$.

Then, after substituting $\delta^{*}$ and $s^{*}$ into demand functions and the retailer’s profit function respectively, the retailer’s profit function is

$$\pi_{RL^{\prime}} \left( {p,l} \right) = \frac{{\left( {l^{2} \eta - c\left( {1 - p} \right)\left( {\gamma^{2} - 2\eta \theta } \right) + c\eta \theta \left( {p - l - \left( {2 - p} \right)\theta } \right)\lambda + p\left( {\left( {1 - p} \right)\gamma^{2} + \eta \left( {\left( {p - \left( {2 - p} \right)\theta )\left( {1 - \lambda \theta } \right) + l\left( {\theta + \lambda \theta - 2} \right)} \right)} \right)} \right)} \right)}}{{\gamma^{2} - 2\eta \theta \left( {1 - \lambda \theta } \right)}}.$$

From the first order condition of $\pi_{RL^{\prime}} \left( {p,l} \right)$ w.r.t $p$ and $l$, we have $\begin{gathered} p_{dL^{\prime}}^{*} = \frac{{2\left( {1 + c} \right)\gamma^{2} + \eta \theta \left( { - 4 + 4\lambda \theta + c\left( { - 4 + \lambda \theta \left( {3 + \lambda } \right)} \right)} \right)}}{{4\gamma^{2} + \eta \theta \left( { - 8 + \theta \left( {1 + \lambda \left( {6 + \lambda } \right)} \right)} \right)}}, \hfill \\ l_{RS}^{*} = \frac{{\left( {1 + c} \right)\left( { - 2 + \theta } \right)\left( { - \gamma^{2} + 2\eta \theta } \right) + \theta \left( {\left( { - 1 + c} \right)\gamma^{2} + \eta \theta \left( {6 + c - \left( {2 + c} \right)\theta } \right)} \right)\lambda + \eta \theta^{2} \left( {c + \left( { - 2 + c} \right)\theta } \right)\lambda^{2} }}{{4\gamma^{2} + \eta \theta \left( { - 8 + \theta \left( {1 + \lambda \left( {6 + \lambda } \right)} \right)} \right)}}. \hfill \\ \end{gathered}$.

Similarly, we use the bordered Hessian matrix to prove whether $\pi_{RL^{\prime}} \left( {p,l} \right)$ is jointly quasi-concave in $\left( {p_{dL^{\prime}}^{*} ,l_{RS}^{*} } \right)$. The bordered Hessian matrix of $\pi_{RL^{\prime}} \left( {p,l} \right)$ in $\left( {p_{dL^{\prime}}^{*} ,l_{RS}^{*} } \right)$ is as follows: $H_{3} = \left( {\begin{array}{*{20}c} { - \frac{{2\left( {\gamma^{2} - \eta \left( {1 + \theta } \right)\left( {1 - \lambda \theta } \right)} \right)}}{{\gamma^{2} - 2\eta \theta \left( {1 - \lambda \theta } \right)}}} & {\frac{{\eta \left( {\theta + \lambda \theta - 2} \right)}}{{\gamma^{2} - 2\eta \theta \left( {1 - \lambda \theta } \right)}}} \\ {\frac{{\eta \left( {\theta + \lambda \theta - 2} \right)}}{{\gamma^{2} - 2\eta \theta \left( {1 - \lambda \theta } \right)}}} & {\frac{2\eta }{{\gamma^{2} - 2\eta \theta \left( {1 - \lambda \theta } \right)}}} \\ \end{array} } \right)$. If the two conditions: $H_{3} \left[ {1,1} \right]{ = } - \frac{{2\left( {\gamma^{2} - \eta \left( {1 + \theta } \right)\left( {1 - \lambda \theta } \right)} \right)}}{{\gamma^{2} - 2\eta \theta \left( {1 - \lambda \theta } \right)}} < 0$ and $\left| {H_{3} } \right|{ = - }\frac{{\eta \left( {4\gamma^{2} + \eta \theta \left( { - 8 + \theta \left( {1 + \lambda \left( {6 + \lambda } \right)} \right)} \right)} \right)}}{{\left( {\gamma^{2} - 2\eta \theta \left( {1 - \lambda \theta } \right)} \right)^{2} }} > 0$ satisfy, then $\pi_{RL^{\prime}} \left( {p,l} \right)$ is jointly quasi-concave in $\left( {p_{dL^{\prime}}^{*} ,l_{RS}^{*} } \right)$. Thus, the premise condition that the equilibrium exists is $4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right) < 0$. By reducing the two conditions ($\gamma^{2} - 2\eta \theta \left( {1 - \lambda \theta } \right) < 0$ and $4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right) < 0$), the final condition that the optimal equilibrium of marketing-based rebate mode in RS Game exists is $4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right) < 0$.

Finally, bringing $p_{dL}^{*}$ and $\alpha_{RS}^{*}$ back to $\delta^{*}$,$s^{*}$, demand functions and profit functions, the equilibrium of marketing-based mode in the RS Game obtained (See Table 2).

1.3 Proof of the equilibrium of the CW Stackelberg (CS) game

1.3.1 Commission-driven Mode in CS Game

For any given commission $\alpha p$, the objective of the online retailer is to determine the product price p and rebate rate $\beta$ to maximize its profit. Like, Zhou et al. [53], we denote the commission to the cashback website ($\alpha p$) as l for convenience. Thus, the profit function of the online retailer is given by $\pi_{RF} \left( {p,\beta } \right) = \left( {p - c} \right)\left( {1 - \frac{\beta p}{{1 - \theta }}} \right) + \frac{{p\left( {l - p\left( {1 - \beta } \right)} \right)\left( {1 - \beta - \theta } \right)}}{{\theta \left( {1 - \theta } \right)}}$. By solving the first order condition of $\pi_{RF} \left( {p,\beta } \right)$ w.r.t $p$ and $\beta$, the following results can be yield:$p_{dF}^{*} = \frac{1 + c}{2}$,$\beta_{CS}^{*} = \frac{1 + c - l - \theta }{{1 + c}}$. Then, we use the bordered Hessian matrix to prove whether $\pi_{RF} \left( {p,\beta } \right)$ is jointly quasi-concave in $\left( {p_{dF}^{*} ,\beta_{CS}^{*} } \right)$. The bordered Hessian matrix of $\pi_{RF} \left( {p,\beta } \right)$ in $\left( {p_{dF}^{*} ,\beta_{CS}^{*} } \right)$ is as follows: $H_{4} = \left( {\begin{array}{*{20}c} {\frac{{2\left( {\left( {1 + c} \right)^{2} \theta + \left( {l + \theta } \right)\left( {l - \left( {1 + 2c} \right)\theta } \right)} \right)}}{{\left( {1 + c} \right)^{2} \left( { - 1 + \theta } \right)\theta }}} & {\frac{l - c\theta }{{\theta \left( {1 - \theta } \right)}}} \\ {\frac{l - c\theta }{{\theta \left( {1 - \theta } \right)}}} & {\frac{{\left( {1 + c} \right)^{2} }}{{2\left( { - 1 + \theta } \right)\theta }}} \\ \end{array} } \right)$. If the determinant of the matrix $\left| {H_{4} } \right|$ is greater than zero and the first order principal minor of the Hessian matrix $H_{4} \left[ {1,1} \right]$ is less than zero, $\pi_{RF} \left( {p,\beta } \right)$ is quasi-concave. From $H_{4}$, it is easy to get $\left| {H_{4} } \right| = \frac{{\left( {1 + c} \right)^{2} }}{{\left( {1 - \theta } \right)\theta }}$, $H_{4} \left[ {1,1} \right] = \frac{{2\left( {\left( {1 + c} \right)^{2} \theta { + }\left( {1 + \theta } \right)\left( {l - \left( {1 + 2c} \right)\theta } \right)} \right)}}{{\left( {1 + c} \right)^{2} \left( {1 - \theta } \right)\theta }}$. It is obvious that the two conditions ($\left| {H_{4} } \right| > 0$, $H_{4} \left[ {1,1} \right] < 0$) for the negative determination of $H_{4}$ are satisfied. Thus, $\pi_{RF} \left( {p,\beta } \right)$ is jointly concave in $\left( {p_{dF}^{*} ,\beta_{CS}^{*} } \right)$.

Then, after substituting $p_{dF}^{*}$ and $\beta_{CS}^{*}$ into demand functions and the cashback website’s profit function respectively, the profit function of the cashback website is $\pi_{CL} \left( l \right) = \frac{{l\left( {l - c\theta } \right)}}{{2\left( { - 1 + \theta } \right)\theta }}$. The second order derivative of $\pi_{CL}$ w.r.t $l$ is $\frac{{\partial^{2} \pi_{CL} }}{{\partial l^{2} }}{ = } - \frac{1}{{\theta \left( {1 - \theta } \right)}}$. It is easy to see that $\frac{{\partial^{2} \pi_{CL} }}{{\partial l^{2} }} < 0$ and $\pi_{CL}$ is a concave function of $l$ given $p$ and $\beta$. Hence, solving the first order condition of $\pi_{CL}$ w.r.t $l$, we can yield $l^{*} { = }\frac{c\theta }{2}$.

Finally, bringing this $l^{*}$ into $\alpha_{RS}^{*}$ back to $p_{dF}^{*}$,$\beta_{CS}^{*}$, demand functions and profit functions, the equilibrium in commission driven mode in the CS Game obtained (See Table 3).

1.3.2 Marketing-based Mode in CS Game

First, the cashback site chooses the marketing effort $s$ conditional on the price p and rebate rate $\beta$ to maximize its profit, that is $\pi_{CL^{\prime}} (s) = \frac{{l\left( {s\gamma + p\left( {\beta + \theta - 1} \right)} \right)}}{{\theta \left( {1 - \lambda \theta } \right)}} - \frac{{s^{2} \eta }}{2}$.The second order derivative of $\pi_{CL^{\prime}}$ w.r.t $s$ is $\frac{{\partial^{2} \pi_{CL^{\prime}} }}{{\partial s^{2} }}{ = } - \eta$. It is obvious that $\frac{{\partial^{2} \pi_{CL^{\prime}} }}{{\partial s^{2} }} < 0$ and $\pi_{CL^{\prime}}$ is a concave function of $s$ given $p$ and $\beta$. Hence, solving the first order condition of $\pi_{CL^{\prime}}$ w.r.t $s$, we can yield $s^{*} { = }\frac{l\gamma }{{\eta \theta - \eta \theta^{2} \lambda }}$.

Second, bring this $s^{*}$ into the retailer’s profit function, we can get $\pi_{RF^{\prime}} \left( {p,\beta } \right) = \frac{1}{{\eta \theta^{2} \left( {1 - \theta \lambda } \right)^{2} }}\left( \begin{gathered} \left( {p\left( {1 - \beta } \right) - l} \right)\left( {l\gamma^{2} - p\eta \theta \left( {1 - \beta - \theta } \right)\left( {1 - \theta \lambda } \right)} \right) \hfill \\ + \left( {c - p} \right)\theta \left( {l\gamma^{2} \lambda - \eta \theta \left( {1 - \theta \lambda } \right)\left( {1 - p + p\left( {1 - \beta } \right)\lambda - \theta \lambda } \right)} \right) \hfill \\ \end{gathered} \right)$. By solving the first order condition of $\pi_{RF^{\prime}} \left( {p,\beta } \right)$ w.r.t $p$ and $\beta$, one pair of $p^{*}$ and $\beta^{*}$ can be yield. As above, we use the bordered Hessian matrix to prove whether $\pi_{RF^{\prime}} \left( {p^{*} ,\beta^{*} } \right)$ is jointly quasi-concave in $\left( {p_{{}}^{*} ,\beta_{{}}^{*} } \right)$.

Then, bring the above $s^{*}$,$p^{*}$ and $\beta^{*}$ into the cashback website’s profit function to yield the optimal unit marketing expense l, that is $\pi_{CL^{\prime}} (l) = - \frac{{l\left( {2\eta \theta \left( {1 + c - \left( {1 - c} \right)\lambda } \right)\left( {1 - \lambda \theta } \right)^{2} + l\left( {\gamma^{2} \left( {1 - \lambda } \right)^{2} - 4\eta \left( {1 - \lambda \theta } \right)^{2} } \right)} \right)}}{{2\eta \theta \left( {1 - \theta \lambda } \right)^{2} \left( { - 4 + \theta \left( {1 + \lambda } \right)^{2} } \right)}}$. From the first order derivation of $\pi_{CL^{\prime}}$ w.r.t $l$, the final $l^{*} = \frac{{\eta \theta \left( {1 + c - \left( {1 - c} \right)\lambda } \right)\left( {1 - \theta \lambda } \right)^{2} }}{{ - \gamma^{2} \left( {1 - \lambda } \right)^{2} + 4\eta \left( {1 - \theta \lambda } \right)^{2} }}$ yields. Meanwhile, the second order derivation of $\pi_{CL^{\prime}}$ w.r.t $l$ must less than zero, that is the condition $\gamma^{2} \left( {1 - \lambda } \right)^{2} < 4\eta \left( {1 - \theta \lambda } \right)^{2}$ must be hold.

Finally, substituting the above $l^{*}$ into $s^{*}$,$p^{*}$ and $\beta^{*}$, demand functions and the retailer’s profit function respectively, the equilibrium in markeing-based mode in the CS Game obtained (See Table 4).

Proof of Proposition 1

Under the equilibrium of commission driven mode in RS Game (See Table 2), it is obvious that $p_{dL}^{*} = p_{N}^{*} = {{(c + 1)} \mathord{\left/ {\vphantom {{(c + 1)} 2}} \right. \kern-\nulldelimiterspace} 2}$. From the optimal sales price of the rebate channel $p_{rF}^{*} = \frac{{\left( {2 + c} \right)\theta }}{4}$, we can easily derive that $\frac{{\partial p_{rF}^{*} }}{\partial \theta } = \frac{2 + c}{4} > 0$. Therefore, Corollary 1 holds. From the equilibrium of commission driven mode in RS Game, it is straightforward to show that $D_{dL}^{*} = \frac{{2\left( {1 - \theta } \right) - c\left( {2 - \theta } \right)}}{{4\left( {1 - \theta } \right)}},D_{rF}^{*} = \frac{c}{{4\left( {1 - \theta } \right)}},\pi_{CF}^{*} = \frac{{c^{2} \theta }}{{16\left( {1 - \theta } \right)}}$, $\pi_{RL}^{*} = \frac{1}{8}\left( {c\left( {\frac{c}{1 - \theta } + c - 4} \right) + 2} \right)$. Thus,$D_{dL}^{*} + D_{rF}^{*} = \frac{2 - c}{4}$,$\pi_{RL}^{*} - \pi_{N}^{*} = \frac{{c^{2} \theta }}{{8\left( {1 - \theta } \right)}}$,$\frac{{\partial \pi_{RL}^{*} }}{\partial \theta } = \frac{{c^{2} }}{{8\left( {1 - \theta } \right)^{2} }}$,$\frac{{\partial \pi_{CF}^{*} }}{\partial \theta } = \frac{{c^{2} }}{{16\left( {1 - \theta } \right)^{2} }}$. Clearly,$D_{dL}^{*} + D_{rF}^{*} > D_{N}^{*}$,$\pi_{RL}^{*} > \pi_{N}^{*}$,$\frac{{\partial \pi_{RL}^{*} }}{\partial \theta } > 0$ and $\frac{{\partial \pi_{CF}^{*} }}{\partial \theta } > 0$. Both Corollaries 2 and 3 hold.

Proof of Proposition 2

By comparing $\pi_{RL}^{*}$ and $\pi_{RL^{\prime}}^{*}$ under the condition that the optimal equilibrium of marketing-based rebate mode in RS Game exists ($4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right) < 0$), we can draw the following conclusions:

When $\pi_{RL}^{*} > \pi_{RL^{\prime}}^{*}$ and $4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right) < 0$, the two conditions that.

$\eta > \frac{{4c^{2} \gamma^{2} }}{{\theta \left( {\lambda - 1} \right)\left( {c^{2} \left( {\theta \left( {\lambda - 1} \right) - 2\left( {\lambda + 3} \right)} \right) - 4c\left( {\theta - 1} \right)\left( {\lambda + 1} \right) + 2\left( {\theta - 1} \right)\left( {\lambda - 1} \right)} \right)}}$ and $\frac{{2\left( {c - 1} \right)\left( {c\left( {\lambda + 3} \right) - \lambda + 1} \right)}}{{\left( {c - 4} \right)c\lambda - c\left( {c + 4} \right) + 2\left( {\lambda - 1} \right)}} < \theta < 1$ should be satisfied;

When $\pi_{RL}^{*} > \pi_{RL^{\prime}}^{*}$ and $4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right) < 0$, we can derive the following two conditions: (1) $- \frac{{4\gamma^{2} }}{{\theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)}} < \eta < \frac{{4c^{2} \gamma^{2} }}{{\theta \left( {\lambda - 1} \right)\left( {c^{2} \left( {\theta \left( {\lambda - 1} \right) - 2\left( {\lambda + 3} \right)} \right) - 4c\left( {\theta - 1} \right)\left( {\lambda + 1} \right) + 2\left( {\theta - 1} \right)\left( {\lambda - 1} \right)} \right)}}$ and $\frac{{2\left( {c - 1} \right)\left( {c\left( {\lambda + 3} \right) - \lambda + 1} \right)}}{{\left( {c - 4} \right)c\lambda - c\left( {c + 4} \right) + 2\left( {\lambda - 1} \right)}} < \theta < 1$; (2) $\eta > - \frac{{4\gamma^{2} }}{{\theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)}}$ and $0 < \theta \le \frac{{2\left( {c - 1} \right)\left( {c\left( {\lambda + 3} \right) - \lambda + 1} \right)}}{{\left( {c - 4} \right)c\lambda - c\left( {c + 4} \right) + 2\left( {\lambda - 1} \right)}}.$

Through the above comparative analysis, we can get Proposition 2.

Proof of Proposition 3

From the equilibrium of marketing-based mode in RS Game, it is straightforward to show that $p_{{dL^{\prime}}}^{*} = \frac{{2\left( {c + 1} \right)\gamma^{2} + \eta \theta \left( {c\left( {\theta \lambda \left( {\lambda + 3} \right) - 4} \right) + 4\theta \lambda - 4} \right)}}{{4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)}}$ and $p_{rF^{\prime}}^{*} { = }\frac{{\eta \theta \left( {c\left( {4 - 2\theta \lambda } \right) + 3\theta \lambda + \theta - 4} \right) - 2\left( {c - 1} \right)\gamma^{2} }}{{4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)}}$. Since the relationship between them satisfies $p_{rF^{\prime}}^{*} = p_{dL^{\prime}}^{*} (1 - \beta )$, it is obvious that $p_{dL^{\prime}}^{*} > p_{rF^{\prime}}^{*}$. By comparing $p_{rF^{\prime}}^{*}$ and $p_{N}^{*}$ under the condition that the optimal equilibrium of marketing-based rebate mode in RS Game exists ($4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right) < 0$), we can conclude that the conditions for the existence of price traps are:$- \frac{{4\gamma^{2} }}{{\theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)}} < \eta < \frac{{2\gamma^{2} \left( {\left( {1 + c} \right)\left( {2 - \theta } \right) - \left( {1 - c} \right)\lambda \theta } \right)}}{{\theta \left( {8 + 2\lambda \theta^{2} \left( {3 + \lambda } \right) - \theta \left( {1 + \lambda } \right)\left( {7 + \lambda } \right) + c\left( {8 + 4\lambda \theta^{2} - \theta \left( {7 + \lambda \left( {4 + \lambda } \right)} \right)} \right)} \right)}}$. Thus, Proposition 3 holds.

From the equilibrium of marketing-based mode in RS Game, it is straightforward to show that $s_{RS}^{*} = - \frac{{\gamma \theta \left( {\left( {c - 1} \right)\lambda + c + 1} \right)}}{{4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)}}$, $\pi_{CF^{\prime}}^{*} = - \frac{{\eta \theta^{2} \left( {\left( {c - 1} \right)\lambda + c + 1} \right)^{2} \left( {\gamma^{2} + 2\eta \theta \left( {\theta \lambda - 1} \right)} \right)}}{{2\left( {4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)} \right)^{2} }}$ and $\pi_{RL^{\prime}}^{*} = \frac{{\left( {c - 1} \right)^{2} \gamma^{2} + \eta \theta \left( {\left( {c - 2} \right)\left( {c - 1} \right)\theta \lambda - c\left( {2c + \theta - 4} \right) - 2} \right)}}{{4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)}}$. Therefore, $\frac{{\partial s_{RS}^{*} }}{\partial \gamma } = - \frac{{\theta \left( {\left( {1 + \lambda } \right)c + 1 - \lambda } \right)\left( {\left( {4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)} \right) - 8\gamma^{2} } \right)}}{{\left( {4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)} \right)^{2} }}$, $\frac{{\partial \prod_{CF^{\prime}}^{*} }}{\partial \gamma } = - \frac{{\gamma \eta \theta^{2} \left( {1 - \lambda + c\left( {1 + \lambda } \right)} \right)^{2} \left( { - 4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda - 10} \right) + 1} \right) + 8} \right)} \right)}}{{\left( {4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)} \right)^{3} }}$ and $\frac{{\partial \prod_{RL^{\prime}}^{*} }}{\partial \gamma } = \frac{{2\gamma \eta \theta^{2} \left( {1 - \lambda + c\left( {1 + \lambda } \right)} \right)^{2} }}{{\left( {4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)} \right)^{2} }}$.

Since the equilibrium of marketing-based mode in RS Game exists only if $4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right) < 0$. Given the necessary equilibrium conditions, it is clear that $\frac{{\partial s_{RS}^{*} }}{\partial \gamma } = - \frac{{\theta \left( {\left( {1 + \lambda } \right)c + 1 - \lambda } \right)\left( {\left( {4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)} \right)} \right)}}{{\left( {4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)} \right)^{2} }} + \frac{{8\gamma^{2} \theta \left( {\left( {1 + \lambda } \right)c + 1 - \lambda } \right)}}{{\left( {4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)} \right)^{2} }} > 0$,$\frac{{\partial \pi_{CF^{\prime}}^{*} }}{\partial \gamma } = - \frac{{\gamma \eta \theta^{2} \left( {1 + c - \left( {1 - c} \right)\lambda } \right)^{2} \left( {2\eta \theta^{2} \left( {1 - \lambda } \right)^{2} - \left( {4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)} \right)} \right)}}{{\left( {4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)} \right)^{3} }} > 0$ and $\frac{{\partial \pi_{RL^{\prime}}^{*} }}{\partial \gamma } = \frac{{2\gamma \eta \theta^{2} \left( {1 + c - \left( {1 - c} \right)\lambda } \right)^{2} }}{{\left( {4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)} \right)^{2} }} > 0$. Therefore, Corollary 4 holds.

From the equilibrium of marketing-based mode in RS Game, it is straightforward to show that $\frac{{\partial s_{RS}^{*} }}{\partial \theta } = \frac{{\gamma \left( {\left( {1 + \lambda } \right)c + 1 - \lambda } \right)\left( { - 4\gamma^{2} + \eta \theta^{2} \left( {\lambda \left( {\lambda + 6} \right) + 1} \right)} \right)}}{{\left( {4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)} \right)^{2} }}$. Therefore, to determine the relationship between $s_{RS}^{*}$ and $\theta$ is to determine the positive or negative of $- 4\gamma^{2} + \eta \theta^{2} \left( {\lambda \left( {\lambda + 6} \right) + 1} \right)$ under the necessary equilibrium condition holds ($4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right) < 0$). Consequently, when $- 4\gamma^{2} + \eta \theta^{2} \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) > 0$ and $4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right) < 0$, $s_{RS}^{*}$ increases with $\theta$; when $- 4\gamma^{2} + \eta \theta^{2} \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) < 0$ and $4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right) < 0$,$s_{RS}^{*}$ decreases with $\theta$. By analyzing these conditions, we get Corollary 5.

Proof of Proposition 4

From the equilibriums of commission driven mode in RS Game and in CS game, it is obvious that $p_{dF}^{*} = p_{dL}^{*} = p_{N}^{*} = \frac{1 + c}{2}$, $p_{rL}^{*} = p_{rF}^{*} = \frac{{\left( {c + 2} \right)\theta }}{4}$.

Proof of Proposition 5

From the equilibriums of commission driven mode in RS Game and in CS game, we can get $\pi_{CF}^{*} = \frac{{c^{2} \theta }}{{16\left( {1 - \theta } \right)}}$,$\pi_{RL}^{*} = \frac{1}{8}\left( {c\left( {\frac{c}{1 - \theta } + c - 4} \right) + 2} \right)$,$\pi_{RF}^{*} = \frac{1}{16}\left( {c\left( {c\left( {\frac{1}{1 - \theta } + 3} \right) - 8} \right) + 4} \right)$ and $\pi_{CL}^{*} = \frac{{c^{2} \theta }}{8 - 8\theta }$. Thus,$\pi_{RL}^{*} - \pi_{RF}^{*} = \frac{{c^{2} \theta }}{{16\left( {1 - \theta } \right)}} > 0$, $\frac{{\pi_{CL}^{*} }}{{\pi_{CF}^{*} }} = 2 > 1$. Proposition 5 holds.

Proof of Proposition 6

From the equilibriums of marketing-based mode in RS Game and in CS game, we can get.

$\pi_{CF^{\prime}}^{*} = - \frac{{\eta \theta^{2} \left( {\left( {c - 1} \right)\lambda + c + 1} \right)^{2} \left( {\gamma^{2} + 2\eta \theta \left( {\theta \lambda - 1} \right)} \right)}}{{2\left( {4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)} \right)^{2} }}$, $\pi_{RL^{\prime}}^{*} = \frac{{\left( {c - 1} \right)^{2} \gamma^{2} + \eta \theta \left( {\left( {c - 2} \right)\left( {c - 1} \right)\theta \lambda - c\left( {2c + \theta - 4} \right) - 2} \right)}}{{4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)}}$, $\pi_{CL^{\prime}}^{*} = \frac{{\eta \theta \left( {\left( {c - 1} \right)\lambda + c + 1} \right)^{2} \left( {\theta \lambda - 1} \right)^{2} }}{{2\left( {\theta \left( {\lambda + 1} \right)^{2} - 4} \right)\left( {\gamma^{2} \left( {\lambda - 1} \right)^{2} - 4\eta \left( {\theta \lambda - 1} \right)^{2} } \right)}}$ and $\begin{gathered} \pi_{RF^{\prime}}^{*} = \frac{{\theta \left( {\left( {c - 1} \right)\lambda + c + 1} \right)^{2} \left( {\theta \lambda - 1} \right)\left( {\gamma^{2} + \eta \theta \left( {\theta \lambda - 1} \right)} \right)}}{{(4\eta \left( {\theta \lambda - 1} \right)^{2} - \gamma^{2} \left( {\lambda - 1} \right)^{2} )\theta \left( {\theta \left( {\lambda + 1} \right)^{2} - 4} \right)}} - \frac{{\left( {\left( {c - 1} \right)\lambda + c + 1} \right)^{2} \left( {\theta \lambda - 1} \right)^{2} \left( {\gamma^{2} + \eta \theta \left( {\theta \lambda - 1} \right)} \right)^{2} }}{{\left( {\gamma^{2} \left( {\lambda - 1} \right)^{2} - 4\eta \left( {\theta \lambda - 1} \right)^{2} } \right)^{2} \theta \left( {\theta \left( {\lambda + 1} \right)^{2} - 4} \right)}} \hfill \\ - \frac{{\theta \left( {\left( {c - 1} \right)\theta \lambda + c\left( {c + \theta - 2} \right) + 1} \right)}}{{\theta \left( {\theta \left( {\lambda + 1} \right)^{2} - 4} \right)}} \hfill \\ \end{gathered}$. If we make $\pi_{RL^{\prime}}^{*} - \pi_{RF^{\prime}}^{*} > 0$, we get the following condition:$\left( {\gamma^{2} - \eta \theta \left( {1 - \lambda \theta } \right)} \right)^{2} \left( {\gamma^{2} \left( {1 - \lambda } \right)^{2} - 4\eta \left( {1 - \lambda \theta } \right)^{2} } \right)^{2} \left( {4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)} \right) < 0$. Obviously, when the preconditions of the equilibrium solution are satisfied ($4\gamma^{2} + \eta \theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right) < 0$), the condition is also true. Thus,$\pi_{RL^{\prime}}^{*} > \pi_{RF^{\prime}}^{*}$. Similarly, we use the same method to obtain the following conditions when $\pi_{CL^{\prime}}^{*} > \pi_{CF^{\prime}}^{*}$: $- \frac{{4\gamma^{2} }}{{\theta \left( {\theta \left( {\lambda \left( {\lambda + 6} \right) + 1} \right) - 8} \right)}} < \eta < - \frac{{\gamma^{2} }}{{\theta^{2} \lambda - \theta }}$.