Downstream Information Leaking and Information Sharing Between Partially Informed Retailers

Abstract

Retailers benefit under certain conditions from horizontal information sharing, sharing information with competing retailers. However, these benefits could be hindered by the mediation of the manufacturer. Information leaking occurs when the manufacturer filters information from one retailer to the other. We focus on analyzing the impact of horizontal information sharing and information leaking on the profits of the manufacturer and retailers. We develop an analytical model with partial and asymmetric demand signals of customers’ valuation. Three scenarios are revised: no information sharing and no information leaking, information sharing, and information leaking. The originality of this study is the use of a demand process with distribution uncertainty, which imitates the information conditions of retailers who join a new market or start selling new products. These retailers own partial information but cannot determine if they are in a better information position than the other retailer. The results indicate that horizontal information sharing increases profits for the retailer with a higher demand signal, but it does not benefit the retailer with a lower demand signal. Additionally, retailers encounter their least preferred scenario if they do not agree to share information horizontally because the manufacturer will always respond by leaking information from the retailer with a higher demand signal to the other retailer. Managers of competing firms facing ambiguity about their demand information position should share information to benefit from a better demand estimation, or at least, prevent the manufacturer to use information leaking to his private benefit.

Appendices

Appendix

Proof of Lemma 1

In the no sharing no leaking scenario, we focus on the profit of retailer U in Eq. (4) to find the solutions of the inequality

$$ {\pi_{R1}}^{\ast }=\frac{1}{36b}\left(-4{a_{RU}}^2+8{a}_{RU}{a}_{RO}-4{a}_{RO}c+{c}^2\right)>0. $$

Since b > 0, we multiply both sides of the inequality times 36b and divide by 4,

$$ -{a_{RU}}^2+2{a}_{RU}{a}_{RO}-{a}_{RO}c+\frac{c^2}{4}>0. $$

We factorize the expression,

$$ \left[\frac{c}{2}-{a}_{RU}\right]\left[\frac{c}{2}+{a}_{RU}\right]-2{a}_{RO}\left[\frac{c}{2}-{a}_{RU}\right]>0, $$

$$ \left[\frac{c}{2}-{a}_{RU}\right]\left[\frac{c}{2}+{a}_{RU}-2{a}_{RO}\right]>0. $$

We deduce the solutions from the two factors,

$$ 0<\frac{c}{2}<{a}_{RU}<2{a}_{RO}-\frac{c}{2}. $$

For the profit of retailer O,

$$ 0<\frac{c}{2}<{a}_{RO}<2{a}_{RU}-\frac{c}{2}. $$

Assuming that retailer O is the overestimating retailer a_RU < a_RO,

$$ 0<\frac{c}{2}<{a}_{RU}<{a}_{RO}<2{a}_{RU}-\frac{c}{2}, $$

$$ 0<c<{a}_{RU}+\frac{c}{2}<{a}_{RO}+\frac{c}{2}<2{a}_{RU}. $$

In the information sharing scenario, this lemma also guarantees the positive profit of the manufacturer.

$$ {\pi_M^S}^{\ast }=\frac{1}{3b}\left({a}_{RU}+{a}_{RO}\right)\left({a}_{RU}+{a}_{RO}-c\right), $$

${\pi_M^S}^{\ast }>0$ if c < a_RU + a_RO.

The lemma establishes that the production cost c is less than two times the demand information of the underestimating retailer a_RU. Therefore, c < 2a_RU < a_RU + a_RO guarantees${\pi_M^S}^{\ast }>0$.

Proof of Theorem 1

We equate the two profits of retailers in Eqs. (4) and (5),

$$ {\pi_{RU}}^{\ast }={\pi_{RO}}^{\ast } $$

$$ \frac{1}{36b}\left(-4{a_{RU}}^2+8{a}_{RU}{a}_{RO}-4{a}_{RO}c+{c}^2\right)= $$

$$ \frac{1}{36b}\left(-4{a_{RO}}^2+8{a}_{RU}{a}_{RO}-4{a}_{RU}c+{c}^2\right), $$

We factorize to find solutions,

$$ {a_{RO}}^2-{a_{RU}}^2+{a}_{RU}c-{a}_{RO}c=0, $$

$$ \left({a}_{RO}-{a}_{RU}\right)\left({a}_{RU}+{a}_{RO}\right)-c\left({a}_{RO}-{a}_{RU}\right)=0, $$

$$ \left({a}_{RO}-{a}_{RU}\right)\left({a}_{RU}+{a}_{RO}-c\right)=0. $$

Following Lemma 1, (a_RU + a_RO − c) is always positive. Therefore,

π_RU^∗ > π_RO^∗, if a_RO > a_RU,

Proof of Theorem 2

Assuming that retailer O is the overestimating retailer; we compare the profit of retailer O in the no sharing no leaking and the information sharing scenario,

$$ {\pi_{RO}^S}^{\ast }-{\pi_{RO}^{NS}}^{\ast }=0, $$

$$ \frac{1}{36b}{\left({a}_{RU}+{a}_{RO}-c\right)}^2-\frac{1}{36b}\left(-4{a_{RO}}^2+8{a}_{RU}{a}_{RO}-4{a}_{RU}c+{c}^2\right)=0, $$

$$ \left({a_{RU}}^2+{a_{RO}}^2+{c}^2+2{a}_{RU}{a}_{RO}-2{a}_{RU}c-2{a}_{RO}c\right)- $$

$$ \left(-4{a_{RO}}^2+8{a}_{RU}{a}_{RO}-4{a}_{RU}c+{c}^2\right)=0, $$

$$ \left({a_{RO}}^2-2{a}_{RU}{a}_{RO}+{a_{RU}}^2\right)+\left(4{a_{RO}}^2-4{a}_{RU}{a}_{RO}\right)-2c\left({a}_{RO}-{a}_{RU}\right)=0, $$

$$ \left({a}_{RO}-{a}_{RU}\right)\left(5{a}_{RO}-{a}_{RU}-2c\right)=0. $$

The factor (a_RO − a_RU) is positive because retailer O is the overestimating retailer. The factor (5a_RO − a_RU − 2c) is always positive using Lemma 1, this is$0<\frac{a_{RU}+2c}{5}<{a}_{RU}+\frac{c}{2}<{a}_{RO}+\frac{c}{2}<2{a}_{RU}$. Consequently, the overestimating retailer earns higher profit in the information sharing scenario than in the no sharing no leaking scenario${\pi_{RO}^S}^{\ast }>{\pi_{RO}^{NS}}^{\ast }$.

Proof of Theorem 3

Assuming that the information is leaked from the overestimating retailer O to the underestimating retailer U, we compare the profits of both retailers in the information leaking scenario,

$$ {\pi_{RU}^L}^{\ast }-{\pi_{RO}^L}^{\ast }=0, $$

$$ \left(2{a_{RU}}^2+{c}^2-3c{a}_{RU}-c{a}_{\mathrm{R}O}+2{a}_{RU}{a}_{RO}\right)+ $$

$$ \left(6{a_{RO}}^2-{c}^2+5{a}_{RU}c-{a}_{RO}c-10{a}_{RU}{a}_{RO}\right)=0, $$

$$ 2\left({a}_{RU}-{a}_{RO}\right)\left({a}_{RU}-3{a}_{RO}+c\right)=0. $$

The factor (a_RU − 3a_RO + c) is always negative using Lemma 1. The factor (a_RU − a_RO) is negative because retailer O is the overestimating retailer a_RO < a_RU. Consequently, the underestimating retailer earns higher profit than the overestimating retailer in the information leaking scenario${\pi_{RU}^L}^{\ast }>{\pi_{RO}^L}^{\ast }$.

Proof of Theorem 4

Assuming that the information is leaked from the overestimating retailer O to the underestimating retailer U, we compare the profit of retailer O in the no sharing no leaking and the information leaking scenario,

$$ {\pi_{RO}^{NS}}^{\ast }-{\pi_{RO}^L}^{\ast }=0, $$

$$ \frac{1}{36b}\left(-4{a_{RO}}^2+8{a}_{RU}{a}_{RO}-4{a}_{RU}c+{c}^2\right)- $$

$$ \frac{1}{36b}\left(-6{a_{RO}}^2+{c}^2-5{a}_{RU}c+{a}_{RO}c+10{a}_{RU}{a}_{RO}\right)=0 $$

$$ \left(-4{a_{RO}}^2+8{a}_{RU}{a}_{RO}-4{a}_{RU}c+{c}^2\right)+ $$

$$ \left(6{a_{RO}}^2-{c}^2+5{a}_{RU}c-{a}_{RO}c+10{a}_{RU}{a}_{RO}\right)=0, $$

$$ 2{a}_{RO}\left({a}_{RO}-{a}_{RU}\right)-c\left({a}_{RO}-{a}_{RU}\right)=0, $$

$$ \left({a}_{RO}-{a}_{RU}\right)\left(2{a}_{RO}-c\right)=0. $$

The factor (2a_RO − c) is always positive using Lemma 1. The factor (a_RO − a_RU) is positive since retailer O has a higher level of demand information a_RO > a_RU. Consequently, retailer O is better off in the no sharing no leaking information scenario than in the scenario in which her information is leaked${\pi_{RO}^{NS}}^{\ast }>{\pi_{RO}^L}^{\ast }$.

Proof of Theorem 5

Assuming that the information is leaked from the overestimating retailer O to underestimating retailer U, we compare the profit of retailer O in the information sharing and the information leaking scenario,

$$ {\pi_{RO}^S}^{\ast }-{\pi_{RO}^L}^{\ast }=0, $$

$$ \frac{1}{36b}{\left({a}_{RU}+{a}_{RO}-c\right)}^2- $$

$$ \frac{1}{36b}\left(-6{a_{RO}}^2+{c}^2-5{a}_{RU}c+{a}_{RO}c+10{a}_{RU}{a}_{RO}\right)=0, $$

$$ {a_{RU}}^2-8{a}_{RU}{a}_{RO}+7{a_{RO}}^2+3{a}_{RU}c-3{a}_{RO}c=0, $$

$$ \left({a}_{RU}-{a}_{RO}\right)\left({a}_{RU}-7{a}_{RO}+3c\right)=0. $$

The factor (a_RU − 7a_RO + 3c) is always negative using Lemma 1. The factor (a_RU − a_RO) is negative since retailer O is the overestimating retailer a_RO > a_RU.Consequently, retailer O is better off in the information sharing than in the scenario in which her information is leaked${\pi_{RO}^S}^{\ast }>{\pi_{RO}^L}^{\ast }$.

Proof of Theorem 6

Assuming that the information is leaked from the overestimating retailer O to underestimating retailer U, we compare the profit of manufacturer in the information leaking and the no sharing no leaking scenario,

$$ {\pi_M^L}^{\ast }-{\pi_M^{NS}}^{\ast }=0, $$

$$ \frac{1}{12b}\left[{\left({a}_{RU}+{a}_{RO}\right)}^2+4{a_{RO}}^2-2c\left(3{a}_{RO}+{a}_{RU}-c\right)\right]- $$

$$ \frac{1}{3b}\left[{\left({a}_{RU}-\frac{c}{2}\right)}^2+{\left({a}_{RO}-\frac{c}{2}\right)}^2\right]=0, $$

$$ \frac{{\left({a}_{RU}+{a}_{RO}\right)}^2-4{a_{RU}}^2-2c\left({a}_{RO}-{a}_{RU}\right)}{12b}=0 $$

Evaluating this expression in the case where a_RO = a_RU, the value is exactly zero. Since a_RO > a_RU from Lemma 1, this expression will always be greater than zero. Consequently, ${\pi_M^L}^{\ast }>{\pi_M^{NS}}^{\ast }.$

Proof of Theorem 7

We compare the profit of manufacturer in the no sharing no leaking scenario and information sharing scenario,

$$ {\pi_M^{NS}}^{\ast }-{\pi_M^S}^{\ast }=0, $$

$$ \frac{1}{3b}\left[{\left({a}_{RU}-\frac{c}{2}\right)}^2+{\left({a}_{RO}-\frac{c}{2}\right)}^2\right]-\frac{1}{6b}{\left({a}_{RU}+{a}_{RO}-c\right)}^2= $$

$$ \frac{{\left({a}_{RO}-{a}_{RU}\right)}^2+2c\left({a}_{RO}+{a}_{RU}\right)}{6b}=0 $$

This expression is always greater than zero. Consequently, ${\pi_M^{NS}}^{\ast }>{\pi_M^S}^{\ast }.$

Sensibility Analysis Appendix

Table 5 Results of sensibility analysis changing uncertainty