Supplier’s strategy: align with the dominant entrant retailer or the vulnerable incumbent retailer?

Abstract

The emergence and rapid development of large-sized retail outlets creates opportunities for dominant retailers to align with the businesses in the supply chain, in which they have previously never been involved. When facing a vulnerable incumbent retailer and a dominant entrant retailer, which one should the supplier choose to align with? To address this problem, we model a supply chain composed of a dominant entrant retailer, a weak incumbent counterpart, and a common supplier from which both retailers source products. The retailers compete on quantity, and the dominant retailer is entitled to determine the wholesale price at which she purchases, while the weak retailer accepts the price offered by the supplier. Furthermore, the incumbent retailer is assumed to hold private information about market demand. We investigate the information strategy for the supplier of either aligning with the dominant entrant retailer, i.e., leaking the incumbent’s demand information to the entrant, or with the vulnerable incumbent retailer, i.e., concealing that information from the entrant. Our result reveals that all participants’ preferences depend on subtle considerations of multiple factors such as the state of terminal market demand, demand volatility, expected market demand and the dominant retailer’s wholesale price. We were surprised to find that the supplier would rather align with the vulnerable incumbent when fluctuations in terminal market demand are severe. In other words, the supplier does not always benefit from information leakage. Moreover, the incumbent and entrant do not always prefer a simplex information strategy. In addition, the supplier’s alignment strategies could realize the desirable opportunity of “win–win” outcomes with the incumbent or entrant in certain circumstances.

Appendix: Proofs

Proof of Proposition 1

From model (1), when the supplier leaks the incumbent’s actual market demand information to the entrant, the two retailers play the Cournot competition with complete information. As the research processes of the high and low market demand states are similar, we consider only one of them. When the market demand state is high, the high-type incumbent solves

$$\begin{aligned}&\max \limits _{q_{i\!H}^{Y}\ge 0}\left[ \left( a+\alpha -q_{iH}^{Y}-q_{eH}^{Y}\right) q_{iH}^{Y}-w_{iH}^{Y}q_{iH}^{Y}\right] \nonumber \\&\quad \Rightarrow q_{iH}^{Y} =\frac{a+\alpha -q_{eH}^{Y}-w_{iH}^{Y}}{2}, \end{aligned}$$

(9)

and the high-type entrant solves

$$\begin{aligned}&\max \limits _{q_{e\!H}^{Y}\ge 0}\left[ \left( a+\alpha -q_{iH}^{Y}-q_{eH}^{Y}\right) q_{eH}^{Y}-w_{e}q_{eH}^{Y}\right] \nonumber \\&\quad \Rightarrow \quad \quad q_{eH}^{Y} =\frac{a+\alpha -q_{iH}^{Y}-w_{e}}{2}. \end{aligned}$$

(10)

Above two Eqs. (9) and (10) are simultaneously solved, we have

$$\begin{aligned} q_{iH}^{Y}=\frac{a+\alpha -2w_{iH}^{Y}+w_{e}}{3}, \end{aligned}$$

and

$$\begin{aligned} q_{eH}^{Y}=\frac{a+\alpha +w_{iH}^{Y}-2w_{e}}{3}. \end{aligned}$$

And then, the high-type supplier solves for

$$\begin{aligned}&\max \limits _{w_{i\!H}^{Y}\ge 0}\left[ \frac{w_{iH}^{Y}\left( a+\alpha -2w_{iH}^{Y}+w_{e}\right) }{3}\right. \nonumber \\&\quad \quad \left. +\frac{w_{e}\left( a+\alpha +w_{iH}^{Y}-2w_{e}\right) }{3}\right] \nonumber \\&\quad \Rightarrow \quad \quad w_{iH}^{*Y} =\frac{a+\alpha +2w_{e}}{4}. \end{aligned}$$

(11)

Taking Eq. (11) of wholesale price that the supplier sets for the incumbent back to Eqs. (9) and (10), we have the incumbent’s order quantity

$$\begin{aligned} q_{iH}^{*Y}=\frac{a+\alpha }{6}, \end{aligned}$$

(12)

and the entrant’s order quantity

$$\begin{aligned} q_{eH}^{*Y}=\frac{5a+5\alpha -6w_{e}}{12}. \end{aligned}$$

(13)

Therefore, all participants’ profits are realized under the information leakage scenario as follows

$$\begin{aligned}\pi _{sH}^{Y}= & {} \frac{(a+\alpha )^{2}+12w_{e}(a+\alpha -w_{e})}{24},\\ \pi _{iH}^{Y}= & {} \left( \frac{a+\alpha }{6}\right) ^{2}, \\ \pi _{eH}^{Y}= & {} \left( \frac{5a+5\alpha -6w_{e}}{12}\right) ^{2}. \end{aligned}$$

Proof of Corollary 1

We analyze the difference of the two retailers’ order quantities under the scenario of information leakage. When the market demand state is high, the difference value of their order quantities

$$\begin{aligned} q_{iH}^{*Y}-q_{eH}^{*Y}=\frac{a+\alpha }{6}-\frac{5a+5\alpha -6w_{e}}{12} =\frac{-a-\alpha +2w_{e}}{4}. \end{aligned}$$

(14)

Obviously, when $w_{e}<\displaystyle \frac{a+\alpha }{2}$, we have $q_{iH}^{*Y}<q_{eH}^{*Y}$. The incumbent’s order quantity dominates the entrant’s. Otherwise, when $w_{e}\ge \displaystyle \frac{a+\alpha }{2}$, we have $q_{iH}^{*Y}\ge q_{eH}^{*Y}$. The entrant’s order quantity dominates the incumbent’s.

Proof of Proposition 2

When the terminal market demand state is low, we have the according solution procedure and conclusions.

Proof of Corollary 2

We analyze the difference of the two retailers’ order quantities under the scenario of information leakage. When the market demand state is low, the difference value of their order quantities

$$\begin{aligned} q_{iL}^{*Y}-q_{eL}^{*Y}=\frac{a-\alpha }{6}-\frac{5a-5\alpha -6w_{e}}{12} =\frac{-a+\alpha +2w_{e}}{4}. \end{aligned}$$

(15)

Obviously, when $w_{e}<\displaystyle \frac{a-\alpha }{2}$, we have $q_{iH}^{*Y}<q_{eH}^{*Y}$. The incumbent’s order quantity dominates the entrant’s. Otherwise, when $w_{e}\ge \displaystyle \frac{a-\alpha }{2}$, we have $q_{iH}^{*Y}\ge q_{eH}^{*Y}$. The entrant’s order quantity dominates the incumbent’s.

Proof of Propositions 3 and 4

From model (5), when the supplier conceals the incumbent’s actual market demand information from the entrant, the two retailers play the Cournot competition with incomplete information. When the market demand state is high, the high-type incumbent solves

$$\begin{aligned}&\max \limits _{q_{i\!H}^{N}\ge 0}\left[ \left( a+\alpha -q_{iH}^{N}-q_{e}^{N}\right) q_{iH}^{N}-w_{iH}^{N}q_{iH}^{N}\right] \nonumber \\&\quad \Rightarrow q_{iH}^{N} =\frac{a+\alpha -q_{e}^{N}-w_{iH}^{N}}{2}, \end{aligned}$$

(16)

and the entrant solves

$$\begin{aligned}&\max \limits _{q_{e}\ge 0} \left\{ \left[ p\left( a+\alpha -q_{iH}^{N}-q_{e}^{N}\right) \right. \right. \nonumber \\&\left. \quad \left. +(1-p)\left( a-\alpha -q_{iL}^{N}-q_{e}^{N}\right) \right] q_{e}^{N}-w_{e}q_{e}^{N}\right\} \nonumber \\&\quad \Rightarrow \quad \quad q_{e}^{N} =\frac{a-\alpha -q_{iL}^{N}-w_{e}+p\left( 2\alpha -q_{iH}^{N}+q_{iL}^{N}\right) }{2}. \end{aligned}$$

(17)

Above two Eqs. (16) and (17) are simultaneously solved, we have

$$\begin{aligned} q_{iH}^{N}=\frac{a+3\alpha +q_{iL}^{N}-2w_{iH}^{N}+w_{e}-p\left( 2\alpha +q_{iL}^{N}\right) }{4-p}, \end{aligned}$$

and

$$\begin{aligned} q_{e}^{N}=\frac{2a-2\alpha -2q_{iL}^{N}-2w_{e}-p\left( a-3\alpha -w_{iH}^{N}-2q_{iL}^{N}\right) }{4-p}. \end{aligned}$$

And then, the high-type supplier solves for

$$\begin{aligned}&\max \limits _{w_{i\!H}^{N}\ge 0}\left[ \frac{w_{iH}^{N}\left[ a+3\alpha +q_{iL}^{N}-2w_{iH}^{N}+w_{e}-p\left( 2\alpha +q_{iL}^{N}\right) \right] }{4-p}\right. \nonumber \\&\quad \quad \left. +\frac{w_{e}\left[ 2a-2\alpha -2q_{iL}^{N}-2w_{e}-p\left( a-3\alpha -w_{iH}^{N}-2q_{iL}^{N}\right) \right] }{4-p}\right] \nonumber \\&\quad \Rightarrow \quad \quad w_{iH}^{N} =\frac{a+3\alpha +q_{iL}^{N}+w_{e}-p\left( 2\alpha +q_{iL}^{N}-w_{e}\right) }{4}. \end{aligned}$$

(18)

Taking Eq. (18) of wholesale price that the supplier sets for the incumbent back to Eqs. (16) and (17), we have the expression of incumbent’s order quantity

$$\begin{aligned} q_{iH}^{N}=\frac{a+3\alpha +q_{iL}^{N}+w_{e}-p\left( 2\alpha +q_{iL}^{N}+w_{e}\right) }{2(4-p)}, \end{aligned}$$

(19)

and the expression of entrant’s order quantity

$$\begin{aligned} q_{e}^{N}=\frac{8\left( a-\alpha -q_{iL}^{N}-w_{e}\right) -p\left( 3a-15\alpha -9q_{iL}^{N}-w_{e}\right) -p^{2}\left( 2\alpha +q_{iL}^{N}-w_{e}\right) }{4(4-p)}. \end{aligned}$$

(20)

On the other hand, when the market demand state is low, the low-type incumbent solves

$$\begin{aligned}&\max \limits _{q_{i\!L}^{N}\ge 0}\left[ \left( a+\alpha -q_{iL}^{N}-q_{e}^{N}\right) q_{iL}^{N}-w_{iL}^{N}q_{iL}^{N}\right] \nonumber \\&\quad \Rightarrow q_{iL}^{N} =\frac{a+\alpha -q_{e}^{N}-w_{iL}^{N}}{2}. \end{aligned}$$

(21)

Under the situation of low-type market demand, the solving process of the entrant is same as Eq. (17). We solve two Eqs. (21) and (17) simultaneously, and have

$$\begin{aligned} q_{iL}^{N}=\frac{a-\alpha +w_{e}-2w_{iL}^{N}-p\left( 2\alpha -q_{iH}^{N}\right) }{p+3}, \end{aligned}$$

and

$$\begin{aligned} q_{e}^{N}=\frac{a-\alpha -2w_{e}+w_{iL}^{N}+p\left( a+3\alpha -w_{iL}^{N}-2q_{iH}^{N}\right) }{p+3}. \end{aligned}$$

And then, the low-type supplier solves for

$$\begin{aligned}&\max \limits _{w_{i\!L}^{N}\ge 0}\left[ \frac{w_{iL}^{N}\left[ a-\alpha +w_{e}-2w_{iL}^{N}-p\left( 2\alpha -q_{iH}^{N}\right) \right] }{p+3} \right. \nonumber \\&\quad \left. +\frac{w_{e}\left[ a-\alpha -2w_{e}+w_{iL}^{N}+p\left( a+3\alpha -w_{iL}^{N}-2q_{iH}^{N}\right) \right] }{p+3}\right] \nonumber \\&\quad \Rightarrow \quad \quad w_{iL}^{N} =\frac{a-\alpha +w_{e}-p\left( 2\alpha +q_{iH}^{N}-w_{e}\right) }{4}. \end{aligned}$$

(22)

Taking Eq. (22) of wholesale price that the supplier sets for the incumbent back to Eqs. (21) and (17), we have the expression of incumbent’s order quantity

$$\begin{aligned} q_{iL}^{N}=\frac{a-\alpha -p\left( 2\alpha -q_{iH}^{N}-w_{e}\right) }{2(p+3)}, \end{aligned}$$

(23)

and the expression of entrant’s order quantity

$$\begin{aligned} q_{e}^{N}=\frac{\left( 5a-5\alpha -6w_{e}\right) +p\left( 3a+11\alpha -7q_{iH}^{N}-3w_{e}\right) +p^{2}\left( 2\alpha -q_{iH}^{N}+w_{e}\right) }{4(p+3)}. \end{aligned}$$

(24)

Equations (18), (19), (22), and (23) are jointly solved, and then we have the high-type supplier’s wholesale price set for the incumbent

$$\begin{aligned} w_{iH}^{*N}\!=\!\frac{(28a\!+\!68\alpha \!+\!24w_{e})\!-\!(3a\!+\!45\alpha \!-\!30w_{e})p\!-\!(a\!+\!\alpha \!+\!6w_{e})p^2\!+\!2\alpha p^3}{6\left( 16+p-p^2\right) }. \end{aligned}$$

The high-type incumbent orders

$$\begin{aligned} q_{iH}^{*N}=\frac{(7a+17\alpha +6w_{e})+(a-7\alpha -3w_{e})p+(2\alpha +3w_{e})p^2}{3\left( 16+p-p^2\right) }. \end{aligned}$$

When the market demand state is low, the supplier offers the wholesale price

$$\begin{aligned} w_{iL}^{*N}\!=\!\frac{(24a\!-\!24\alpha \!+\!48w_{e})\!+\!(5a\!-\!41\alpha \!-\!18w_{e})p\!-\!(a\!+\!5\alpha \!+\!6w_{e})p^2\!+\!2\alpha p^3}{6\left( 16+p-p^2\right) }. \end{aligned}$$

The low-type incumbent orders

$$\begin{aligned} q_{iL}^{*N}=\frac{(8a-8\alpha )-(a+11\alpha -9w_{e})p+(2\alpha -3w_{e})p^2}{3\left( 16+p-p^2\right) }. \end{aligned}$$

Taking the incumbent’s low-type order quantity to Eq. (20), or taking the high-type order quantity to Eq. (24), we could both get the entrant’s order quantity no matter what state the actual market demand is.

$$\begin{aligned} q_{e}^{*N}\!=\!\frac{(40a\!-\!40\alpha \!-\!48w_{e})\!+\!(5a\!+\!79\alpha \!-\!18w_{e})p\!-\!(5a\!-\!3\alpha \!-\!18w_{e})p^2\!-\!2\alpha p^3}{6\left( 16+p-p^2\right) }. \end{aligned}$$

In consideration of all decision variables’ obtained results, the supplier’s profits under the information concealment scenario are realized and listed as follows

$$\begin{aligned} \pi _{sH}^{N}&=\bigg \{\left[ 4\left( 7a+17\alpha \right) ^2+48w_{e}\left( 47a-23\alpha -45w_{e}\right) \right] \\&\quad +\Big [7a^2-1241\alpha ^2-(494a -3540w_{e})\alpha +492aw_{e}\\&\quad -900w_{e}^2\Big ]p-\Big [10a^2-162\alpha ^2+\left( 104a-66w_{e}\right) \alpha \\&\quad +438aw_{e}-756w_{e}^2\Big ]p^2-\Big [a^2-131\alpha ^2\\&\quad -\left( 26a-192w_{e}\right) \alpha +24aw_{e} -36w_{e}^2\Big ]p^3\\&\quad -\Big [12\alpha ^2-\left( 4a-6w_{e}\right) \alpha -18w_{e}\left( a-2w_{e}\right) \Big ]\\&\quad \quad p^4 -4\alpha ^2p^5\bigg \}\bigg /18\left( 16+p-p^2\right) ^2 \end{aligned}$$

and

$$\begin{aligned} \pi _{sL}^{N}&=\bigg \{\left[ 192\left( a-\alpha \right) ^2+2304w_{e}\left( a-\alpha -w_{e}\right) \right] \\&\quad +\Big [16a^2+592\alpha ^2-(608a-3072w_{e})\alpha \\&\quad +384aw_{e}-576w_{e}^2\Big ]p -\Big [13a^2-443\alpha ^2\\&\quad -\left( 2a+546w_{e}\right) \alpha +402aw_{e}-648w_{e}^2\Big ]p^2\\&\quad +\Big [a^2-43\alpha ^2+(42a-216w_{e})\alpha -48aw_{e} +108w_{e}^2\Big ]p^3\\&\quad -\Big [32\alpha ^2+\left( 4a-6w_{e}\right) \alpha -18w_{e}\left( a-2w_{e}\right) \Big ]p^4\\&\quad +4\alpha ^2p^5\bigg \}\bigg / 18\left( 16+p-p^2\right) ^2, \end{aligned}$$

respectively, in high-type and low-type market demands. In addition, the high-type incumbent’s profit is

$$\begin{aligned} \pi _{iH}^{N}=\left[ \frac{(7a+17\alpha +6w_{e})+(a-7\alpha -3w_{e})p+(2\alpha +3w_{e})p^2}{3\left( 16+p-p^2\right) }\right] ^2. \end{aligned}$$

The low-type incumbent’s profit is

$$\begin{aligned} \pi _{iL}^{N}=\left[ \frac{(8a-8\alpha )-(a+11\alpha -9w_{e})p+(2\alpha -3w_{e})p^2}{3\left( 16+p-p^2\right) }\right] ^2. \end{aligned}$$

The entrant’s profit is

$$\begin{aligned} \pi _{e}^{N}\!=\!\left[ \frac{(40a\!-\!40\alpha \!-\!48w_{e})\!+\!(5a\!+\!79\alpha \!-\!18w_{e})p\!-\!(5a\!-\!3\alpha \!-\!18w_{e})p^2\!-\!2\alpha p^3}{6\left( 16+p-p^2\right) }\right] ^2 \end{aligned}$$

regardless of the market demand state.

Proof of Corollary 3

We analyze the differences of all participants’ optimal decisions between the scenario of information concealment and the scenario of information leakage. When the market demand state is high, the difference value of the supplier’s wholesale price sets for the incumbent

$$\begin{aligned} w_{iH}^{*N}-w_{iH}^{*Y}&=\frac{(28a+68\alpha +24w_{e})-(3a+45\alpha -30w_{e})p-(a+\alpha +6w_{e})p^2+2\alpha p^3}{6\left( 16+p-p^2\right) } -\frac{a+\alpha +2w_{e}}{4} \nonumber \\&=\frac{8a+88\alpha -48w_{e}-p(9a+93\alpha -54w_{e})+p^2(a+\alpha -6w_{e})+4\alpha p^3}{12\left( 16+p-p^2\right) }. \end{aligned}$$

(25)

After verification, the numerator and denominator of Eq. (25) are both positive. Therefore, we have $w_{iH}^{*N}>w_{iH}^{*Y}$, which implies that the supplier offers the incumbent the wholesale price in the scenario of information concealment is higher than that in the scenario of information leakage.

Next, we compare the incumbent’s order quantity under the information concealment scenario with that under the information leakage scenario.

$$\begin{aligned}&q_{iH}^{*N}-q_{iH}^{*Y}\nonumber \\&\quad =\frac{(7a+17\alpha +6w_{e})+(a-7\alpha -3w_{e})p-(2\alpha +3w_{e})p^2}{3\left( 16+p-p^2\right) }\nonumber \\&\quad \quad -\frac{a+\alpha }{6} \nonumber \\&\quad =\frac{p^2(a-3\alpha -6w_{e})+p(a-15\alpha -6w_{e})-2a+18\alpha +12w_{e}}{6\left( 16+p-p^2\right) }. \end{aligned}$$

(26)

After verification, when $\alpha \le \displaystyle \frac{(p+2)(a-6w_{e})}{3(p+6)}$, Eq. (26) is nonpositive. We have $q_{iH}^{*N}\le q_{iH}^{*Y}$, which means that the incumbent’s order quantity in the scenario of information concealment is no more than that in the scenario of information leakage. Inversely, when $\alpha >\displaystyle \frac{(p+2)(a-6w_{e})}{3(p+6)}$, Eq. (26) is positive. We have $q_{iH}^{*N}>q_{iH}^{*Y}$, which implies that the incumbent’s order quantity in the scenario of information concealment exceeds that in the scenario of information leakage.

Besides, we investigate the entrant’s order quantity between two scenarios—information concealment and information leakage. The difference of the entrant’s order quantity

$$\begin{aligned} q_{e}^{*N}-q_{eH}^{*Y}&=\frac{(40a-40\alpha -48w_{e})+(5a+79\alpha -18w_{e})p-(5a-3\alpha -18w_{e})p^2-2\alpha p^3}{6\left( 16+p-p^2\right) } -\frac{5a+5\alpha -6w_{e}}{12} \nonumber \\&=\frac{160\alpha -p(5a+153\alpha -30w_{e})+p^2(5a-11\alpha -30w_{e})+4\alpha p^3}{12\left( 16+p-p^2\right) }. \end{aligned}$$

(27)

After verification, when $\alpha \le \displaystyle \frac{5p(a-6w_{e})}{160+7p-4p^2}$, Eq. (27) is nonnegative. We have $q_{e}^{*N}\ge q_{eH}^{*Y}$, which means that the entrant’s order quantity in the scenario of information concealment is more than that in the scenario of information leakage. Inversely, when $\alpha >\displaystyle \frac{5p(a-6w_{e})}{160+7p-4p^2}$, Eq. (27) is negative. We have $q_{e}^{*N}<q_{eH}^{*Y}$, which implies that the entrant’s order quantity in the scenario of information concealment is less than that in the scenario of information leakage.

In the following, we illustrate the differences of three participants’ optimal decisions similarly when the market demand state is low. The difference value of the supplier’s wholesale price sets for the incumbent

$$\begin{aligned} w_{iL}^{*N}-w_{iL}^{*Y}&=\frac{(24a-24\alpha +48w_{e})+(5a-41\alpha -18w_{e})p-(a+5\alpha +6w_{e})p^2+2\alpha p^3}{6\left( 16+p-p^2\right) } -\frac{a-\alpha +2w_{e}}{4} \nonumber \\&=\frac{p\left[ 7a-79\alpha -42w_{e}+p(a-13\alpha -6w_{e})+4\alpha p^2\right] }{12\left( 16+p-p^2\right) }. \end{aligned}$$

(28)

After verification, when $\alpha \le \displaystyle \frac{(p+7)(a-6w_{e})}{79+13p-4p^2}$, Eq. (28) is nonnegative. We have $w_{iL}^{*N}\le w_{iL}^{*Y}$, which means that the supplier offers the incumbent the wholesale price in the scenario of information concealment is more than that in the scenario of information leakage. Inversely, when $\alpha >\displaystyle \frac{(p+7)(a-6w_{e})}{79+13p-4p^2}$, Eq. (28) is negative. We have $w_{iL}^{*N}>w_{iL}^{*Y}$, which implies that the supplier offers the incumbent the wholesale price in the scenario of information concealment is less than that in the scenario of information leakage.

Next, we compare the incumbent’s order quantity under the information concealment scenario with that under the information leakage scenario.

$$\begin{aligned}&q_{iL}^{*N}-q_{iL}^{*Y}\nonumber \\&\quad =\frac{(8a-8\alpha )-(a+11\alpha -9w_{e})p+(2\alpha -3w_{e})p^2}{3\left( 16+p-p^2\right) }\nonumber \\&\quad \quad -\frac{a-\alpha }{6} \nonumber \\&=\frac{p\left[ p(a+3\alpha -6w_{e})-3a-21\alpha +18w_{e}\right] }{6\left( 16+p-p^2\right) }. \end{aligned}$$

(29)

After verification, the numerator and denominator of Eq. (29) are both negative. Therefore, we have $q_{iL}^{*N}<q_{iL}^{*Y}$, which means that the incumbent’s order quantity in the scenario of information concealment is lower than that in the scenario of information leakage.

Besides, we investigate the entrant’s order quantity between two scenarios—information concealment and information leakage. The difference of the entrant’s order quantity

$$\begin{aligned} q_{e}^{*N}-q_{eL}^{*Y}&=\frac{(40a-40\alpha -48w_{e})+(5a+79\alpha -18w_{e})p-(5a-3\alpha -18w_{e})p^2-2\alpha p^3}{6\left( 16+p-p^2\right) } -\frac{5a-5\alpha -6w_{e}}{12} \nonumber \\&=\frac{p(5a+163\alpha -30w_{e})-p^2(5a-\alpha -30w_{e})-4\alpha p^3}{12\left( 16+p-p^2\right) }. \end{aligned}$$

(30)

After verification, the numerator and denominator of Eq. (30) are both positive. Therefore, we have $q_{e}^{*N}>q_{eL}^{*Y}$, which implies that the entrant’s order quantity in the scenario of information concealment is higher than that in the scenario of information leakage.

Proof of Proposition 5

We analyze the supplier’s strategies based on the differences of his profits between the scenario of information concealment and the scenario of information leakage. When the market demand state is high, the difference value of the supplier’s profit

$$\begin{aligned}&\pi _{sH}^{N}-\pi _{sH}^{Y} \nonumber \\&\quad =\bigg \{\left[ 4\left( 7a+17\alpha \right) ^2+48w_{e}\left( 47a-23\alpha -45w_{e}\right) \right] \nonumber \\&\quad +\Big [7a^2-1241\alpha ^2-(494a -3540w_{e})\alpha +492aw_{e}-900w_{e}^2\Big ]p\nonumber \\&\quad -\Big [10a^2-162\alpha ^2+\left( 104a-66w_{e}\right) \alpha +438aw_{e}-756w_{e}^2\Big ]p^2\nonumber \\&\quad -\Big [a^2-131\alpha ^2 -\left( 26a-192w_{e}\right) \alpha +24aw_{e}-36w_{e}^2\Big ]p^3 \nonumber \\&\quad -\Big [12\alpha ^2-\left( 4a-6w_{e}\right) \alpha -18w_{e}\left( a-2w_{e}\right) \Big ]p^4-4\alpha ^2p^5\bigg \} \bigg / \nonumber \\&\quad 18\left( 16+p-p^2\right) ^2-\frac{(a+\alpha )^{2}+12w_{e}(a+\alpha -w_{e})}{24} \nonumber \\&\quad =\left( 1-p\right) \bigg \{\left[ 3856\alpha ^2+\left( 2272a-13632w_{e}\right) \alpha +16\left( a-6w_{e}\right) ^2\right] \nonumber \\&\quad -\Big [1204\alpha ^2-\left( 104a-624w_{e}\right) \alpha +52\left( a-6w_{e}\right) ^2\Big ]p \nonumber \\&\quad -\Big [463\alpha ^2+\left( 126a-756w_{e}\right) \alpha -\left( a-6w_{e}\right) ^2\Big ]p^2 \nonumber \\&\quad +\Big [67\alpha ^2-\left( 10a-60w_{e}\right) \alpha +3\left( a-6w_{e}\right) ^2\Big ]p^3\nonumber \\&\quad +16\alpha ^2 p^4\bigg \}\bigg /\left[ 72\left( 16+p-p^2\right) ^2\right] . \end{aligned}$$

(31)

After verification, when

$$\begin{aligned}&\alpha <\left( 4 \ \sqrt{76{,}800+21{,}120p-12{,}468p^2-3339p^3 +672p^4+174p^5-12p^6-3p^7}\right. \\&\quad \left. -1136-52p+63p^2+5p^3\right) (a-6w_{e}) \Big /\left( 3856-1204p-463p^2+67p^3+16p^4\right) , \end{aligned}$$

Equation (31) is negative. We have $\pi _{sH}^{N}<\pi _{sH}^{Y}$, which means that the supplier’s profit in the scenario of information concealment is less than that in the scenario of information leakage, leading the supplier prefers to leak the incumbent’s actual market demand information to the entrant. Inversely, when

$$\begin{aligned} \alpha\ge & {} \left( 4 \ \sqrt{76{,}800+21{,}120p-12{,}468p^2-3339p^3 +672p^4+174p^5-12p^6-3p^7}\right. \\&\quad \left. -1136-52p+63p^2+5p^3\right) (a-6w_{e}) \Big /\left( 3856-1204p-463p^2+67p^3+16p^4\right) , \end{aligned}$$

Equation (31) is nonnegative. We have $\pi _{sH}^{N}\ge \pi _{sH}^{Y}$, which implies that the supplier’s profit in the scenario of information concealment surpasses that in the scenario of information leakage, making the supplier prefers to keep absolutely the incumbent’s actual market demand information secret from the entrant.

On the other hand, we compare the supplier’s profits under two information strategies when the market demand state is low. The difference value of the supplier’s profit

$$\begin{aligned}&\pi _{sL}^{N}-\pi _{sL}^{Y} \nonumber \\&\quad =\bigg \{\left[ 192\left( a-\alpha \right) ^2+2304w_{e}\left( a-\alpha -w_{e}\right) \right] \nonumber \\&\quad +\Big [16a^2+592\alpha ^2-(608a-3072w_{e})\alpha +384aw_{e} -576w_{e}^2\Big ]p\nonumber \\&\quad -\Big [13a^2-443\alpha ^2-\left( 2a+546w_{e}\right) \alpha +402aw_{e}-648w_{e}^2\Big ]p^2\nonumber \\&\quad +\Big [a^2-43\alpha ^2+(42a-216w_{e})\alpha -48aw_{e}+108w_{e}^2\Big ]p^3 \nonumber \\&\quad -\Big [32\alpha ^2+\left( 4a-6w_{e}\right) \alpha -18w_{e}\left( a-2w_{e}\right) \Big ]p^4+4\alpha ^2p^5\bigg \}\bigg / \nonumber \\&\quad 18\left( 16+p-p^2\right) ^2-\frac{(a-\alpha )^{2}+12w_{e}(a-\alpha -w_{e})}{24} \nonumber \\&\quad =p\bigg \{\Big [32\left( 6w_{e}+\alpha -a\right) \left( a+71\alpha -6w_{e}\right) \Big ] \nonumber \\&\quad +\Big [1865\alpha ^2-\left( 178a+1068w_{e}\right) \alpha +41\left( a-6w_{e}\right) ^2\Big ]p\nonumber \\&\quad +\Big [(5a+83\alpha -30w_{e}))*(2a-2\alpha -12w_{e}\Big ]p^2 \nonumber \\&\quad -\Big [131\alpha ^2+(10a-60w_{e})\alpha +3(a-6w_{e})^2\Big ]p^3 \nonumber \\&\quad +16\alpha ^2 p^4\bigg \}\bigg /\left[ 72\left( 16+p-p^2\right) ^2\right] . \end{aligned}$$

(32)

After verification, when

$$\begin{aligned}&\alpha <\left( 4 \ \sqrt{82{,}944+10{,}368p-16{,}956p^2 -744p^3+1257p^4-39p^5-33p^6+3p^7}\right. \\&\quad \left. +1120+89p-78p^2+5p^3\right) (a-6w_{e}) \Big /\left( 2272+1865p-166p^2-131p^3+16p^4\right) , \end{aligned}$$

Eq. (32) is negative. We have $\pi _{sL}^{N}<\pi _{sL}^{Y}$, which means that the supplier’s profit in the scenario of information concealment is less than that in the scenario of information leakage, leading the supplier prefers to leak the incumbent’s actual market demand information to the entrant. Inversely, when

$$\begin{aligned}&\alpha \ge \left( 4 \ \sqrt{82{,}944+10{,}368p-16{,}956p^2-744p^3+1257p^4-39p^5-33p^6+3p^7}\right. \\&\quad \left. +1120+89p-78p^2+5p^3\right) (a-6w_{e}) \Big /\left( 2272+1865p-166p^2-131p^3+16p^4\right) , \end{aligned}$$

Equation (32) is nonnegative. We have $\pi _{sL}^{N}\ge \pi _{sL}^{Y}$, which implies that the supplier’s profit in the scenario of information concealment surpasses that in the scenario of information leakage, making the supplier prefers to keep absolutely the incumbent’s actual market demand information secret from the entrant.

Proof of Proposition 6

We observe that the expressions of two retailers’ profits are the square of that of their corresponding order quantities no matter what state the actual market demand is. Therefore, the comparisons of the retailers’ profits are same as the conditions of (i) parts (b) and (c), and (ii) parts (b) and (c) in Corollary 3. The detailed proof is omitted here.