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Firms’ pricing strategies under different decision sequences in dual-format online retailing

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Abstract

E-commerce platforms are increasingly adopting a dual-format retailing mode, not only acting as a reseller, but also providing manufacturers a platform (i.e., marketplace) to access consumers. Based on platforms’ offers, manufacturers can choose to operate either dual-format or single-format retailing. To figure out this problem, we first investigate channel selecting strategy for a manufacturer based on consumer value and further consider the impacts of power structure and pricing timing on firms’ optimal pricing policies under dual-format retailing. Our findings suggest the manufacturer prefers to operate marketplace channel when utility discount factor is sufficiently high, while operate dual-format retailing channels when utility discount factor is moderate. An interesting observation is that more market power does not always create higher profit. Under some certain conditions, the e-commerce platform could obtain maximum profit when his rival is the leader, so does the manufacturer. In addition, we also explore leader’s optimal pricing timing under different power structures. If the manufacturer is the leader, she should either let platform pricing early or set her own retail price early, depending on estimated utility discount and given platform fee. Being the leader, the e-commerce platform shouldn’t set his retail price early in any case, which is consistent with “last mover” advantage.

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Notes

  1. The form of \(\Delta _s\) is very complicated, and thus, we describe it in “Appendix.” See “Appendix” for details.

  2. The forms of \(\Delta _{\mathrm{e}1}\) and \(\Delta _{\mathrm{e}2}\) are also very complicated, and thus, we describe them in “Appendix.” See “Appendix” for details.

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Acknowledgements

This study was funded by National Natural Science Foundation of China (No. 71371141).

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Correspondence to Hua Ke.

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Appendix

Appendix

1.1 Proof of Proposition 1

The backward induction is employed to solve the equilibrium solutions.

MSS model

Given w, the manufacturer and the EP decide their retail prices simultaneously in the second stage.

$$\begin{aligned} \pi _\mathrm{m}= & {} \left( (1-r) p_\mathrm{m}-c\right) \left( \frac{p_\mathrm{r}-p_\mathrm{m}}{1-\theta }-\frac{p_\mathrm{m}}{\theta }\right) \\&+(w-c) \left( 1-\frac{p_\mathrm{r}-p_\mathrm{m}}{1-\theta }\right) , \\ \pi _\mathrm{e}= & {} r p_\mathrm{m} \left( \frac{p_\mathrm{r}-p_\mathrm{m}}{1-\theta }-\frac{p_\mathrm{m}}{\theta }\right) \\&+\left( p_\mathrm{r}-w\right) \left( 1-\frac{p_\mathrm{r}-p_\mathrm{m}}{1-\theta }\right) ; \end{aligned}$$

we get

$$\begin{aligned} \frac{\partial \pi _\mathrm{m}}{\partial p_\mathrm{m}}= & {} \left( -\frac{1}{\theta }-\frac{1}{1-\theta }\right) \left( (1-r) p_\mathrm{m}-c\right) \\&+\frac{w-c}{1-\theta }+(1-r) \left( \frac{p_\mathrm{r}-p_\mathrm{m}}{1-\theta }-\frac{p_\mathrm{m}}{\theta }\right) ,\\ \frac{\partial \pi _\mathrm{m}^2}{\partial p_\mathrm{m}^2}= & {} 2 \left( -\frac{1}{\theta }-\frac{1}{1-\theta }\right) (1-r)<0,\\ \frac{\partial \pi _\mathrm{e}}{\partial p_\mathrm{r}}= & {} 1-\frac{p_\mathrm{r}-p_\mathrm{m}}{1-\theta }+\frac{\text {rp}_\mathrm{m}}{1-\theta }-\frac{p_\mathrm{r}-w}{1-\theta },\\ \frac{\partial \pi _\mathrm{e}^2}{\partial p_\mathrm{r}^2}= & {} -\frac{2}{1-\theta }<0. \end{aligned}$$

Therefore, \(\pi _\mathrm{m}\) is concave in \(p_\mathrm{m}\), and \(\pi _\mathrm{e}\) is concave in \(p_\mathrm{r}\). Let

$$\begin{aligned} \frac{\partial \pi _\mathrm{m}}{\partial p_\mathrm{m}}=\frac{\partial \pi _\mathrm{e}}{\partial p_\mathrm{r}}=0. \end{aligned}$$

We obtain

$$\begin{aligned} p_\mathrm{m}= & {} \frac{\theta (-\theta +r (\theta -w-1)+3 w+1)-2 c (\theta -1)}{(r-1) (\theta +\theta r-4)} \hbox { and }\\ p_\mathrm{r}= & {} \frac{2+\theta w+2 w-2 \theta -c (\theta -1) (r+1) +r (2 \theta +\theta w-2 w-2)}{(r-1) (\theta +\theta r-4)}. \end{aligned}$$

Substituting \(p_\mathrm{m}\) and \(p_\mathrm{r}\) into Eq. (5), we can get

Therefore, \(\pi _\mathrm{m}\) is concave in w. Let \(\frac{\partial \pi _\mathrm{m}}{\partial w}=0\). We obtain \(w^{MSS}=\frac{c+\theta (1-r)}{2}+\frac{4 (1-\theta ) (1-r)}{8-8 r\theta +\theta r^2+2 \theta r}\). Replacing w with \(w^{MSS}\) into \(p_\mathrm{m}\) and \(p_\mathrm{r}\), then we have

$$\begin{aligned} p_\mathrm{m}^{\text {MSS}}= & {} \frac{c}{2 (1-r)}+\frac{\theta }{2}+\frac{(1-\theta ) \theta (r+1)}{8-8 r\theta +\theta r^2+2 \theta r},\\ p_\mathrm{r}^{\text {MSS}}= & {} 1+\frac{c}{2 (1-r)}-\frac{\theta }{2}-\frac{2 (1-\theta ) (1-r)}{8-8 r\theta +\theta r^2+2 \theta r}. \end{aligned}$$

\(D^{MSS}_\mathrm{m}\) and \(D^{MSS}_\mathrm{e}\) are obtained by substituting \(p_\mathrm{m}^{\text {MSS}}\) and \(p_\mathrm{r}^{\text {MSS}}\) into demand functions (1) and (2).

MSE model

Given w and \(p_\mathrm{r}\), the manufacturer decides the retail price in marketplace channel to maximize profit

$$\begin{aligned} \pi _\mathrm{m}= & {} \left( (1-r) p_\mathrm{m}-c\right) \left( \frac{p_\mathrm{r}-p_\mathrm{m}}{1-\theta }-\frac{p_\mathrm{m}}{\theta }\right) \\&+(w-c) \left( 1-\frac{p_\mathrm{r}-p_\mathrm{m}}{1-\theta }\right) ; \end{aligned}$$

we get \(\frac{\partial \pi _\mathrm{m}}{\partial p_\mathrm{m}}=\left( -\frac{1}{\theta }-\frac{1}{1-\theta }\right) \left( (1-r) p_\mathrm{m}-c\right) +\frac{w-c}{1-\theta } +(1-r) \left( \frac{p_\mathrm{r}-p_\mathrm{m}}{1-\theta }-\frac{p_\mathrm{m}}{\theta }\right) \), \(\frac{\partial \pi _\mathrm{m}^2}{\partial p_\mathrm{m}^2}=2 \left( -\frac{1}{\theta }-\frac{1}{1-\theta }\right) (1-r)<0\).

Therefore, \(\pi _\mathrm{m}\) is concave in \(p_\mathrm{m}\). Let \(\frac{\partial \pi _\mathrm{m}}{\partial p_\mathrm{m}}=0\). We have

$$\begin{aligned} p_\mathrm{m}=\frac{c (\theta -1)+\theta (r-1) p_\mathrm{r}-\theta w}{2 (r-1)}. \end{aligned}$$

Substituting \(p_\mathrm{m}\) into Eq. (6), we get

$$\begin{aligned} \frac{\partial \pi _\mathrm{e}}{\partial p_\mathrm{r}}= & {} \frac{(r-1) (c (1-\theta )+r (2 \theta +\theta w-2 w-2)+2 (1-\theta +w))-(r-1)^2 p_\mathrm{r} (\theta (r+2)-4)}{2 (\theta -1) (r-1)^2}, \\ \frac{\partial \pi _\mathrm{e}^2}{\partial p_\mathrm{r}^2}= & {} -\frac{\theta (r+2)-4}{2 (\theta -1)}<0. \end{aligned}$$

Therefore, \(\pi _\mathrm{e}\) is concave in \(p_\mathrm{r}\).

Let \(\frac{\partial \pi _\mathrm{e}}{\partial p_\mathrm{r}}=0\). We get \(p_\mathrm{r}=\frac{c (1-\theta )-2 \theta +2 \theta r+\theta r w-2 r w-2 r+2 w+2}{(r-1) (2 \theta +\theta r-4)}\). Substituting \(p_\mathrm{m}\) and \(p_\mathrm{r}\) into Eq. (5), we have

$$\begin{aligned} \frac{\partial \pi _\mathrm{m}}{\partial w}= & {} \frac{4 \theta \left( (r-1) \left( \theta ^2 \left( r^2+4 r+2\right) -2 \theta (4 r+3)+8\right) +2 w \left( \theta ^2-5 \theta +\theta r^2+4 (\theta -2) r+8\right) \right) }{4 \theta (r-1) (\theta (r+2)-4)^2}\\&-\frac{4 c \theta \left( \theta ^2-5 \theta +\theta r^2+\left( \theta ^2-\theta -4\right) r+8\right) }{4 \theta (r-1) (\theta (r+2)-4)^2}, \\ \frac{\partial \pi _\mathrm{m}^2}{\partial w^2}= & {} \frac{2 \left( \theta ^2-5 \theta +\theta r^2+4 (\theta -2) r+8\right) }{(r-1) (\theta (r+2)-4)^2}<0; \end{aligned}$$

therefore, \(\pi _\mathrm{m}\) is concave in w. Let \(\frac{\partial \pi _\mathrm{m}}{\partial w}=0\). We get \(w^{MSE}=\frac{\left( \theta ^2-5 \theta +4\right) (r (c+\theta -2)-\theta +2)}{2 \left( \theta ^2-5 \theta +\theta r^2+4 (\theta -2) r+8\right) }+\frac{c+\theta (1-r)}{2}\). Replacing w with \(w^{MSE}\) into \(p_\mathrm{m}\) and \(p_\mathrm{r}\), we obtain \(p_\mathrm{m}^{\text {MSE}}=\frac{(1-\theta ) \theta (r+1) (c r-\theta +\theta r-2 r+2)}{2 (1-r) \left( \theta ^2-5 \theta +\theta r^2+4 \theta r-8 r+8\right) }+\frac{c}{2 (1-r)}+\frac{\theta }{2}\) and \(p_\mathrm{r}^{\text {MSE}}=-\frac{(1-\theta ) (-\theta -2 r+2) (c r-\theta +\theta r-2 r+2)}{2 (1-r) \left( \theta ^2-5 \theta +\theta r^2+4 (\theta -2) r+8\right) }+\frac{c}{2 (1-r)}-\frac{\theta }{2}+1\).

\(D^{MSE}_\mathrm{m}\) and \(D^{MSE}_\mathrm{e}\) can be obtained by substituting \(p_\mathrm{m}^{\text {MSE}}\) and \(p_\mathrm{r}^{\text {MSE}}\) into demand functions (1) and (2).

MSM model

Given w and \(p_\mathrm{m}\), the EP decides the retail price in e-tailer channel to maximize profit. \(\pi _\mathrm{e}=r p_\mathrm{m} \left( \frac{p_\mathrm{r}-p_\mathrm{m}}{1-\theta }-\frac{p_\mathrm{m}}{\theta }\right) +\left( p_\mathrm{r}-w\right) \left( 1-\frac{p_\mathrm{r}-p_\mathrm{m}}{1-\theta }\right) \), we have \(\frac{\partial \pi _\mathrm{e}}{\partial p_\mathrm{r}}=-\frac{p_\mathrm{r}-p_\mathrm{m}}{1-\theta }+\frac{\text {rp}_\mathrm{m}}{1-\theta }-\frac{p_\mathrm{r}-w}{1-\theta }+1\), \(\frac{\partial \pi _\mathrm{e}^2}{\partial p_\mathrm{r}^2}=-\frac{2}{1-\theta }<0\). Therefore, \(\pi _\mathrm{e}\) is concave in \(p_\mathrm{r}\). Let \(\frac{\partial \pi _\mathrm{e}}{\partial p_\mathrm{r}}=0\). We get \(p_\mathrm{r}=\frac{1}{2} \left( -\theta +r p_\mathrm{m}+p_\mathrm{m}+w+1\right) \). Substituting \(p_\mathrm{r}\) into Eq. (5), we can get

$$\begin{aligned} \frac{\partial \pi _\mathrm{m}}{\partial p_\mathrm{m}}= & {} \frac{2 c (\theta -1)+2 (r-1) p_\mathrm{m} (\theta +\theta r-2)+\theta (r-1) (-\theta +2 w+1)}{2 (\theta -1) \theta }, \\ \frac{\partial \pi _\mathrm{m}^2}{\partial p_\mathrm{m}^2}= & {} \frac{(1-r) (-\theta +\theta (-r)+2)}{(\theta -1) \theta }<0. \end{aligned}$$

Therefore, \(\pi _\mathrm{m}\) is concave in \(p_\mathrm{m}\). Let \(\frac{\partial \pi _\mathrm{m}}{\partial p_\mathrm{m}}=0\). We get \(p_\mathrm{m}=\frac{-2 c \theta +2 c-\theta ^2+\theta +\theta ^2 r-\theta r-2 \theta r w+2 \theta w}{2 (r-1) (\theta +\theta r-2)}\). Substituting \(p_\mathrm{r}\) and \(p_\mathrm{m}\) into Equation (5), we have \(\frac{\partial \pi _\mathrm{m}}{\partial w}=\frac{\theta (r-1) (8 (\theta r-1)+16 w)-8 c \theta (r-1)}{8 \theta (r-1) (\theta +\theta r-2)}\), \(\frac{\partial \pi _\mathrm{m}^2}{\partial w^2}=\frac{2}{\theta +\theta r-2}<0\). Therefore, \(\pi _\mathrm{m}\) is also concave in w. Let \(\frac{\partial \pi _\mathrm{m}}{\partial w}=0\). We obtain \(w^{\text {MSM}}=\frac{c+\theta (1-r)}{2}+\frac{1-\theta }{2}\). Replacing w with \(w^{MSM}\) into \(p_\mathrm{m}\) and \(p_\mathrm{r}\), we obtain \(p_\mathrm{m}^{\text {MSM}}=\frac{c}{2 (1-r)}+\frac{\theta }{2}\) and \(p_\mathrm{r}^{\text {MSM}}=\frac{c}{2 (1-r)}-\frac{\theta }{2}-\frac{1-\theta }{4}+1\). \(D^{MSM}_\mathrm{m}\) and \(D^{MSM}_\mathrm{e}\) are also obtained by substituting \(p_\mathrm{m}^{\text {MSM}}\) and \(p_\mathrm{r}^{\text {MSM}}\) into demand functions (1) and (2).

The proofs of other models are similar to those of the above; thus, we omit them.

1.2 Proof of Proposition 2

According to Table 2, we can get \(p^{i}_\mathrm{r}-p^{i}_\mathrm{m}>0\). We omit the straightforward algebraic steps here.

1.3 Proof of Proposition 3

Part (a).

Here \(w^{\text {ESS}}-w^{\text {VNS}}=\frac{(2-r)\theta -2}{24}<0\); \(p_\mathrm{m}^{\text {ESS}}-p_\mathrm{m}^{\text {VNS}}=-\frac{\theta r}{24 (1-r)}<0\); \(D_\mathrm{m}^{\text {ESS}}-D_\mathrm{m}^{\text {VNS}}=\frac{2-r}{24 (1-r)}>0\); thus, \(w^{\text {ESS}}<w^{\text {VNS}}\), \(p_\mathrm{m}^{\text {ESS}}<p_\mathrm{m}^{\text {VNS}}\), \(D_\mathrm{m}^{\text {ESS}}>D_\mathrm{m}^{\text {VNS}}\). From Table 2, we easily find \(D_\mathrm{e}^{\text {VNS}}>D_\mathrm{e}^{\text {ESS}}\), then \(D_\mathrm{e}^{\text {MSS}}-D_\mathrm{e}^{\text {VNS}}=\frac{2 \theta -\theta r^2+\theta r+2 r-2}{3 \left( 8-8 r+\theta +\theta r^2+2 \theta r\right) }>0\), and thus, \(D_\mathrm{e}^{\text {MSS}}>D_\mathrm{e}^{\text {VNS}}>D_\mathrm{e}^{\text {ESS}}\). Because of

$$\begin{aligned} \pi _\mathrm{m}^{\text {MSS}}= & {} \frac{c^2}{4 \theta (1-r)}-\frac{c}{2}+\frac{(1-\theta ) (1-r)}{\theta +\theta r^2+2 (\theta -4) r+8}\\&+\frac{\theta (1-r)}{4},\\ \pi _\mathrm{m}^{\text {VNS}}= & {} \frac{c^2}{4 \theta (1-r)}-\frac{c}{2}+\frac{-5 \theta -8 \theta r^2+2 (7 \theta +2) r-4}{36 (r-1)}, \end{aligned}$$

we can get

$$\begin{aligned} \pi _\mathrm{m}^{\text {MSS}}-\pi _\mathrm{m}^{\text {VNS}}=\frac{\left( -2 \theta +\theta r^2-(\theta +2) r+2\right) ^2}{36 (1-r) \left( \theta +\theta r^2+2 (\theta -4) r+8\right) }>0 \end{aligned}$$

and

$$\begin{aligned}&\pi _\mathrm{m}^{\text {ESS}}=\frac{c^2}{4 \theta (1-r)}-\frac{c}{2}\\&+\frac{-12 \theta -15 \theta r^2+28 \theta r+4 r-4}{64 (r-1)},\\&\pi _\mathrm{m}^{\text {VNS}}-\pi _\mathrm{m}^{\text {ESS}}=\frac{7 \left( 4 \theta +\theta r^2-4 \theta r+4 r-4\right) }{576 (r-1)}. \end{aligned}$$

In order to ensure that profit function of the manufacturer in the ESS model is concave on the decision variable \(p_\mathrm{m}\) and has maximum value, we have assumed \(4-4 \theta -\theta r^2+4 \theta r-4 r>0\); thus, \(\pi _\mathrm{m}^{\text {MSS}}>\pi _\mathrm{m}^{\text {VNS}}>\pi _\mathrm{m}^{\text {ESS}}\). Based on the equilibrium solutions, we have

$$\begin{aligned} \pi _\mathrm{e}^{\text {ESS}}= & {} -\frac{c^2 r}{4 \theta (r-1)^2}-\frac{c r^2}{8 (r-1)^2}\\&+\frac{-8 \theta {+}15 \theta r^3+(8-40 \theta ) r^2{+}16 (2 \theta -1) r+8}{64 (r-1)^2}, \\ \pi _\mathrm{e}^{\text {VNS}}= & {} -\frac{c^2 r}{4 \theta (r-1)^2}-\frac{c r^2}{6 (r-1)^2}\\&+\frac{-4 \theta +8 \theta r^3+(4-22 \theta ) r^2+(17 \theta -8) r+4}{36 (r-1)^2}; \end{aligned}$$

thus,

$$\begin{aligned}&\pi _\mathrm{e}^{\text {ESS}}-\pi _\mathrm{e}^{\text {VNS}}=\frac{24 c r^2-8 \theta +7 \theta r^3-8 \theta r^2+8 r^2+16 \theta r-16 r+8}{576 (r-1)^2}=\frac{8(1-\theta )(1-r)^2+24 c r^2+7 \theta r^3}{576 (1-r)^2}>0;\\&\pi _\mathrm{e}^{\text {MSS}}=\frac{\left( -11 \theta ^3+32 \theta ^2+92 \theta -32\right) r}{4 \left( 8+\theta +\theta r^2+2 (\theta -4) r\right) ^2}-\frac{c^2 r}{4 \theta (r-1)^2}-\frac{c (\theta -1) r (r+1)}{(r-1) \left( 8+\theta +\theta r^2+2 (\theta -4) r\right) }\\&\qquad \ \ \qquad +\frac{-4 (\theta -1) (\theta +2)^2+\theta ^3 r^5+4 (\theta -4) \theta ^2 r^4+2 \theta \left( \theta ^2-4 \theta +30\right) r^3-4 \left( 2 \theta ^3-13 \theta ^2+42 \theta -4\right) r^2}{4 \left( 8+\theta +\theta r^2+2 (\theta -4) r\right) ^2}, \end{aligned}$$

where

$$\begin{aligned} \Delta _s= & {} r^3 \left( 32 c \left( \theta ^2-12 \theta -16\right) \right. \\&\left. +97 \theta ^3-464 \theta ^2-576 \theta +1024\right) \\&-4 r^4 \left( 2 (10 c-59) \theta ^2-64 (2 c-1) \theta +17 \theta ^3+64\right) \\&+ 4 \theta ^2 r^6 (2 c+3 \theta -6)-8 \left( 7 \theta ^3+9 \theta ^2-48 \theta +32\right) \\&+\theta ^3 r^7-2 \theta r^5 \left( 16 c (\theta +2)+21 \theta ^2+16 \theta -64\right) \\&+16 (\theta -1) r \left( 4 c (\theta +8)-3 \theta ^2+28 \theta -64\right) \\&+8cr^2 \left( 17 \theta ^2-64 \theta +128\right) \\&+8 r^2(15 \theta ^3-47 \theta ^2+224 \theta -192). \end{aligned}$$

When \(\Delta _s>0\), \(\pi _\mathrm{e}^{\text {MSS}}-\pi _\mathrm{e}^{\text {ESS}}>0\), i.e., \(\pi _\mathrm{e}^{\text {MSS}}>\pi _\mathrm{e}^{\text {ESS}}\), otherwise, \(\pi _\mathrm{e}^{\text {MSS}}<\pi _\mathrm{e}^{\text {ESS}}\).

Then, the proofs of Part (b) and Part (c) are similar to this, and we here omit them and only give the forms of \(\Delta _{\text {e1}}\) and \(\Delta _{\text {e2}}\).

$$\begin{aligned}&\Delta _{\text {e1}}=\frac{c^2 \left( -\theta ^2+5 \theta +(\theta -2) \theta r^2-4 (\theta -2) r-8\right) }{(r-1) \left( 8+\theta ^2-5 \theta +\theta r^2+4 (\theta -2) r\right) }-\frac{\theta (r-1) \left( \theta ^2 r^2+4 (\theta -2) \theta r+4\right) }{8+\theta ^2-5 \theta +\theta r^2+4 (\theta -2) r}\\&\qquad \quad -\frac{2 c \theta \left( \theta ^2-5 \theta +\theta r^2-\left( \theta ^2-7 \theta +10\right) r+8\right) }{8+\theta ^2-5 \theta +\theta r^2+4 (\theta -2) r}+\frac{2 c \theta (\theta -1)^2 \left( 392 r^2-256 r+64\right) }{\left( 8-8 \theta +\theta r^3-8 (\theta -1) r^2+16 (\theta -1) r\right) ^2} \\&\qquad \quad -\frac{4 c^2 \left( -4 \left( 5 \theta ^2-9 \theta +4\right) r^3+(\theta -1)^2 \left( 48 r^2-48 r+16\right) +(\theta -1) \theta \left( 3 r^4-18 r^5\right) \right) }{\left( 8-8 \theta +\theta r^3-8 (\theta -1) r^2+16 (\theta -1) r\right) ^2}\\&\qquad \quad +\frac{2 c \theta \left( \theta ^2 r^6+2 \left( 53 \theta ^2-89 \theta +36\right) r^4-16 \left( 18 \theta ^2-35 \theta +17\right) r^3\right) }{\left( 8-8 \theta +\theta r^3-8 (\theta -1) r^2+16 (\theta -1) r\right) ^2}\\&\qquad \quad +\frac{\theta (r-1) \left( \theta ^3 r^6-16 (\theta -1) \theta ^2 r^5-8 \left( 31 \theta ^3-58 \theta ^2+25 \theta +2\right) r^3\right) }{\left( 8-8 \theta +\theta r^3-8 (\theta -1) r^2+16 (\theta -1) r\right) ^2}\\&\qquad \quad +\frac{4 \theta \left( 23 \theta ^2-38 \theta +15\right) r^4+(\theta -1)^2 \left( 16 (3 \theta +1)+4 (83 \theta +12) r^2-16 (13 \theta +3) r\right) }{\left( 8-8 \theta +\theta r^3-8 (\theta -1) r^2+16 (\theta -1) r\right) ^2}.\\&\Delta _{\text {e2}}=\frac{4 (\theta -1)^2+\theta ^2 r^4-8 (\theta -1) \theta r^3+4 \left( 5 \theta ^2-6 \theta +1\right) r^2-8 \left( 2 \theta ^2-3 \theta +1\right) r}{8-8 \theta +\theta r^3-8 (\theta -1) r^2+16 (\theta -1) r}\\&\qquad \quad +\frac{4 c (\theta -1) r (2 c+\theta r)}{\theta \left( -8 \theta +\theta r^3-8 (\theta -1) r^2+16 (\theta -1) r+8\right) }-\frac{2 c^2 \left( \theta ^4-12 \theta ^3+49 \theta ^2-86 \theta +64\right) r^2}{\theta (r-1)^2 \left( 8+\theta ^2-5 \theta +\theta r^2+4 (\theta -2) r\right) ^2}\\&\qquad \quad -\frac{2 c (\theta -1) r \left( \theta ^3-9 \theta ^2+28 \theta +(\theta -2) \theta r^3+\left( 3 \theta ^2-8 \theta +8\right) r^2-\left( \theta ^3-11 \theta ^2+26 \theta -16\right) r-24\right) }{(r-1) \left( 8+\theta ^2-5 \theta +\theta r^2+4 (\theta -2) r\right) ^2}\\&\qquad \quad -\frac{c^2 r \left( -\left( \theta ^2-5 \theta +8\right) ^2+(\theta -2) \theta ^2 r^4+4 \theta \left( \theta ^2-4 \theta +5\right) r^3-\left( \theta ^4-12 \theta ^3+43 \theta ^2-72 \theta +64\right) r^2\right) }{\theta (r-1)^2 \left( 8+\theta ^2-5 \theta +\theta r^2+4 (\theta -2) r\right) ^2}\\&\qquad \quad -\frac{\theta ^3 r^5+8 (\theta -2) \theta ^2 r^4+\theta \left( 3 \theta ^3+\theta ^2-40 \theta +60\right) r^3+4 \left( 4 \theta ^3-19 \theta ^2+27 \theta -8\right) r}{\left( 8+\theta ^2-5 \theta +\theta r^2+4 (\theta -2) r\right) ^2}\\&\qquad \quad -\frac{8 \left( \theta ^2-3 \theta +2\right) +2 \left( 5 \theta ^4-33 \theta ^3+84 \theta ^2-80 \theta +8\right) r^2}{\left( 8+\theta ^2-5 \theta +\theta r^2+4 (\theta -2) r\right) ^2}. \end{aligned}$$

1.4 Proof of Proposition 4

This proof is similar to that of Proposition 3; thus, we omit them.

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Liu, J., Ke, H. Firms’ pricing strategies under different decision sequences in dual-format online retailing. Soft Comput 24, 7811–7826 (2020). https://doi.org/10.1007/s00500-019-04399-0

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