Int. J. Production Economics 136 (2012) 352–365

Contents lists available at SciVerse ScienceDirect

Int. J. Production Economics

0925-52

doi:10.1

E-m

journal homepage: www.elsevier.com/locate/ijpe

Strategic information sharing between competing retailers in a supply chainwith endogenous wholesale price

Noam Shamir

Kellogg School of Management, Northwestern University, Evanston, IL 60208, United States

a r t i c l e i n f o

Article history:

Received 31 December 2010

Accepted 19 December 2011Available online 24 December 2011

Keywords:

Supply chain

Information sharing

Signaling game

73/$ - see front matter & 2011 Elsevier B.V. A

016/j.ijpe.2011.12.023

ail address: n-shamir@kellogg.northwestern.e

a b s t r a c t

This paper introduces a new motivation for information sharing in decentralized supply chains—as a

mechanism to achieve truthful information sharing and to reduce signaling costs. We study a two-

echelon supply chain with one manufacturer selling a homogenous product to n price-setting

competing retailers. Each retailer has access to private information about the potential market demand,

and the retailers have an ex-ante incentive to share this information with each other and to conceal

the information from the manufacturer. However, without a mechanism that induces the retailers to

truthful information exchange as their strategic choice, no information can be exchanged via pure

communication (cheap talk). To overcome this obstacle, two signaling games are analyzed: in the first

game, information is shared truthfully among the retailers; in the second game, information is also

shared truthfully with the manufacturer. We show that under some conditions sharing information

with the manufacturer results in a higher profit for the retailers.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

The estimated cost of stockouts and inventory are astronom-ical, ranging from $14 billion for the food service industry (Troyer,1996) to $30 billion for the grocery industry (Kurt SalmonAssociates, 1993). A key initiative commonly mentioned as apossible remedy to lower these costs is information sharingamong partners in supply chains. Recently, there has been moreresearch that emphasizes the value of information sharing foroperational effectiveness, such as improved inventory control, theelimination of the bullwhip effect and better matching of supplywith demand (Chen, 1998; Aviv and Federgruen, 1998; Lee et al.,2000; Cachon and Fisher, 2000 and Lee and Whang, 2000. Anexcellent survey of information sharing in supply chains can befound in Chen, 2003).

Although better information usually improves the perfor-mance of a supply chain, when the supply chain is comprised ofindependent profit-maximizing firms, some key obstacles exist increating an information-sharing agreement. First, in equilibrium,each firm must be better off sharing information than concealingit. Even when information sharing achieves the efficient outcomefor the firms in the supply chain, in many cases there is a tensionbetween efficiency and self-interest. This tension, which is a typeof the famous prisoners’ dilemma, can lead to an inefficient

ll rights reserved.

du

equilibrium, in which no-information is shared among competingfirms (e.g. Gal-Or, 1985 and Li, 2002).

Furthermore, when the accuracy of the shared informationcannot be verified, firms in the supply chain must select truthfulrevelation of their private information as a strategic choice. Forexample, Solectron, a major electronics supplier, had $4.7 billionin excess component inventory because of the inflated forecastsprovided by its customers (Engardio, 2001).

When the supply chain is comprised of competing retailers, anadditional challenge exists when a retailer shares informationwith his supplier, since once a retailer shares its private informa-tion with his supplier, its ability to control leakage of thisinformation to his competitors is compromised. Wal-Mart, forexample, announced that it would no longer share its sales datawith outside companies like Information Resources, Inc. andACNielsen, which paid Wal-Mart for the information and thensold it to other retailers (Hays, 2004). Two recent papers,Li and Zhang (2008) and Anand and Goyal (2009), studythe effects of information leakage on the incentives to shareinformation in a decentralized supply chain with downstreamcompetition.

The description above suggests that researchers view thecomplex structure of supply chains and the conflicting incentivesof firms within the supply chain as obstacles to achievinginformation-sharing agreements. However, in this paper, we offeran alternative view: we assert that the same complex nature of asupply chain can provide firms with a tool to facilitate informa-tion-sharing agreements in settings of asymmetric information.

N. Shamir / Int. J. Production Economics 136 (2012) 352–365 353

To accomplish this, we model a two-echelon supply chain withdownstream competition and present a new incentive for infor-mation sharing between the retailers and their mutual manufac-turer—as a substitute for the signaling cost. When the retailersshare hard information (information that cannot be manipulated)they have an ex-ante incentive to share private information aboutthe future market demand and to conceal this informationfrom their manufacturer. However, when the retailers share soft

information (information for which the credibility of the sharedinformation cannot be verified) each retailer has an incentive tomanipulate the information; each retailer is better-off pretendingthat the market condition is better than his private signalsuggests. Such information manipulation encourages the compet-ing retailers to set a higher price, which benefits the manipulatingretailer. As a result of this tendency to manipulate the sharedinformation, the information-sharing agreement unravels; noinformation can be shared via means of cheap-talk.

To overcome the problem of accountability and to allow theretailers to share information in a credible manner, we studytwo signaling games: in the first mechanism (referred to as thehorizontal signaling game), each retailer incurs some cost relatedto his shared information. The retailers design the mechanism insuch a way that the cost associated with the sent message signalsthat the retailers are accountable for the shared information. Inthis game, the retailers exchange information horizontally andconceal their private information from their manufacturer.

As an alternative mechanism to the horizontal signaling game,we investigate the effect of including the manufacturer in theinformation club (we denote this option as the public signaling

game) on the ability of the retailers to reach an information-sharing agreement. We demonstrate how the presence of themanufacturer in the information coalition affects the retailers’incentives to manipulate the shared information and conse-quently affects the signaling cost required to achieve truthtellingas the retailers’ strategic choice. The presence of the manufacturerallows the retailers to expand the set of available mechanismsthat result in truthtelling information sharing; that is, whenthe retailers share soft information, they benefit from the infor-mation-sharing agreement, but must incur some cost in order tobe accountable for the shared information. The retailers candecide to share information with the manufacturer when thecost of reaching a truthtelling equilibrium under this setting islower than the cost incurred in order to share informationcredibly when the manufacturer is not exposed to the sharedinformation.

The remainder of the paper is organized as follows. Section 2discusses some of the relevant literature. Section 3 introduces themodel, and Section 4 analyzes the incentives of the retailers toshare information (horizontally and vertically) when the accuracyof the information can be verified. Section 5 relaxes the assumptionthat information is exchanged truthfully and examines the incen-tives of the retailers to truthfully exchange their private informa-tion in the presence of the manufacturer and in the absence of themanufacturer from the information club. Section 6 analyzes thetwo signaling games that result in truthtelling as the retailers’strategic choice. We conclude and offer future research path inSection 7.

2. Literature review

Our study relies on three different bodies of research: the firstline of research focuses on the value of information sharing; thesecond group of studies investigates the incentives for truthfulinformation sharing and the mechanisms that induce truthfulinformation sharing, and the third explores the value of pure

communication (cheap talk) and its effect on the results of thestrategic interaction between supply chain participants.

As discussed above, the merits of information sharing insupply chains have been well documented. However, some ofthe research that studies the value of information ignores the roleof incentives to share information. The first stream of papers toconsider the incentives for information sharing under horizontalcompetition includes Novshek and Sonnenschein (1982), Vives(1984), Gal-Or (1985, 1986), Li (1985), Shapiro (1986) and Raith(1996). These papers demonstrate that the benefit from informa-tion sharing is increased precision about the market conditiondue to the pooled information. However, by sharing information,rivals also obtain more precise information. Thus, the positiveeffect of the increased precision on a firm’s profit might bebalanced against the increased precision of rivals’ information.The above papers consider the effects of the competition mode(Bertrand or Cournot), the type of products (substitutes orcomplements) and the type of information (common demandintercept or private production cost). In general, the researchersconclude that under a Bertrand (or Cournot, respectively) compe-tition with substitute products, firms will share (or not share)their private demand information, and under Bertrand (or Cour-not) with complementary products, firms will not share (or share)their private demand information. The key assumptions of thesemodels are that the decision to share information is madeex-ante, prior to observing the actual private information, andthat the information is exchanged truthfully. Our paper differsfrom this line of research in two ways: we study the retailers’incentives to share information—not only with each other, butalso with a mutual manufacturer. More importantly, we relax theassumption that information is exchanged truthfully and searchfor a mechanism that induces truthful information exchange asthe retailers’ strategic choice.

Ziv (1993) was among the first researchers to consider theproblem of ex-post truthful information exchange. He endo-genizes the incentives for the truthful exchange of private costinformation in a Cournot competition model with substituteproducts and considers the ex-post incentives of firms to truth-fully reveal their private cost information with horizontalcompeting firms. Ziv demonstrates that without commitmentto truthful information sharing, firms will manipulate theirannouncement and report lower costs than their actual costs. Asa result, each firm will discard its competitors’ report and act asthough no information had been exchanged.

Considering the incentives for information exchange in supplychains, Li (2002) studies the incentives for vertical informationexchange in an environment with horizontal competition. Lidevelops a model with multiple retailers and one mutual supplier,in which each retailer is endowed with a private demand signal.The retailers are engaged in a Cournot competition and can decidewhether to share their signals truthfully with the supplier. Thesupplier sets the wholesale price based on the shared informa-tion. Li demonstrates how, based on the wholesale price, theretailers who do not share their demand signals can infer thevalue of the average signal of the retailers who choose to sharetheir information with the supplier. As a result, sharing demandinformation with the supplier is equivalent to sharing it withcompeting retailers; consequently, the retailers would choose notto share their private information with the supplier. Using asimilar model, Zhang (2002) explores the incentives for verticalinformation sharing in a supply chain with duopoly retailers andreaches similar conclusions.

In a recent paper, Li and Zhang (2008) explore the incentivesfor vertical information sharing in a supply chain consisting ofone mutual manufacturer and multiple retailers engaged in aBertrand competition. The model is similar to that of Li (2002).

N. Shamir / Int. J. Production Economics 136 (2012) 352–365354

They consider three different levels of confidentiality between theretailers and the manufacturer (disclosing the information to allfirms in the supply chain; exchanging information only with themanufacturer and other retailers participating in the informationexchange and revealing the information only to the manufac-turer). Li and Zhang demonstrate that when information is sharedconfidentially between the retailers and the manufacturer, theretailers will choose to truthfully report their private informationunder some conditions. Our model is closely related to that of Liand Zhang. However, Li and Zhang focus on vertical informationsharing between the retailers and the manufacturer, whereas wefocus on horizontal information sharing between the retailers; theinformation shared with the manufacturer is used to facilitatecredible information exchange between the retailers. Anotherdifference between the models is that Li and Zhang assume thatwhen information is shared publicly, it is done truthfully. Con-trary to this assumption, we seek an equilibrium in which truth-telling is the retailers’ strategic choice.

Research in credible information exchange led to another lineof research that studies the mechanisms that induce truthfulinformation sharing. This stream of research includes the modelsin which the uninformed firm offers a menu of contracts to theinformed firm (a screening model) as well as the models in whichthe informed firm undertakes a costly action in order to reveal itsprivate information (a signaling game). An influential paper in theOM literature that considers the incentives for truthful verticalinformation sharing is that of Cachon and Lariviere (2001).Cachon and Lariviere analyze a signaling game in which theyconsider a manufacturer’s incentives to share private demandinformation with an upstream supplier. They establish a separat-ing equilibrium in which each type of manufacturer offers adistinct contract to the upstream supplier.

In our paper, firms are endowed with private demand informa-tion, and we seek mechanisms that induce them to truthfullyreveal this information. The issue of demand signaling has beenaddressed in various channel relationships, including the pairs ofmanufacturer–retailer, manufacturer–supplier, producer–consumer,and franchiser–franchisee. The signaling devices that have beenstudied in this context include wholesale prices, advertisements,slotting allowances, and various types of fees (Chu, 1992; Desaiand Srinivasan, 1995; Lariviere and Padmanabhan, 1997; Desai,2000). In most of the signaling models described above, thesetting includes an informed firm seeking to establish the cred-ibility of its announcement and an uninformed firm that estab-lishes a belief system based on the actions and announcements ofthe informed firm. Our setting is different from that of thetraditional signaling model in that we examine a setting withone uninformed firm (the manufacturer) and multiple partially-informed firms (the retailers).

On a spectrum between assuming truthful informationexchange and establishing a separating equilibrium in signalingor screening games lies another stream of research whichexplores the extent to which agents can communicate informa-tion credibly through costless messages—known as cheap talk (orpure communication). In a seminal work, Crawford and Sobel(1982) examine a setting in which a better-informed sendertransmits a signal to a receiver, who then takes an action basedon that signal; thus, the receiver’s action determines the payoffsfor both the sender and receiver. They show that a Bayesian Nashequilibrium takes a form in which the sender partitions the spaceof his private information into a finite number of sub-intervalsand reports the segment of the partition in which his privatevalues lie. Crawford and Sobel further demonstrate that ex-ante,the equilibrium with the greatest number of elements in itspartition is Pareto-superior to all other equilibria. Another impor-tant study in the area of pure communication and its effect on the

chosen equilibrium of a game was conducted by Farrell andGibbons (1989), who pose the question of why some claims aremade publicly whereas others are made privately. They consider amodel with a simple structure: an informed sender transmits amessage to two interested, but uninformed, receivers, who sub-sequently take actions based on their individual beliefs. Theystudy how costless, non-verifiable claims (cheap talk) can affectthese beliefs and how the incentives for truthful revelation to onereceiver are affected by the presence of the other.

Recently, the concepts of pure communication provided thebasis for some research studies in marketing and OM. Forinstance, Li (2005) examines the role of cheap talk in channelcoordination. In his model, an informed vendor contracts with anexclusive retailer to sell a new technology. Li studies the incen-tives provided by several contracts for the technology vendor totruthfully reveal the private information. As an alternative to acostly signaling game, he introduces contracts in which cheap talkcoordinates the channel. Two other recent papers in the opera-tions management literature also explore the extent to whichcheap talk can be used in supply chains are Allon and Bassamboo(2008) and Allon et al. (2007). Allon and Bassamboo (2008)consider a retail operations model in which customers arestrategic in both their actions and in the way they interpretinformation, whereas the retailer is strategic in the way itprovides information. They demonstrate that when there is onlya single retailer in the market, no information can be exchangedbetween the retailer and the customers, whereas when there areseveral retailers, complete information can be exchanged. In arelated paper, Allon et al. (2007) analyze a queuing model inwhich customers receive a reward from being served, but alsoincur a delay cost. In this case, the firm and the customers actstrategically: the firm may choose its delay announcement, whilethe customers interpret these announcements and decide whento join the system and when to balk.

Our paper lies at the intersection between the signalingliterature and the pure-communication literature. We first estab-lish the retailers’ incentives to share demand information anddemonstrate that without a mechanism to verify the credibility ofthe exchanged information, firms will discard the shared infor-mation and act as if no information had been shared. One possiblemechanism that we consider as a remedy to this situation is usinga costly action in addition to the exchanged information. Thisapproach is aligned with the signaling literature. We also consideran alternative mechanism, in which the demand information isshared with the manufacturer as well and examine the effect ofthe presence of the manufacturer on the retailers’ incentives tomanipulate their announcements.

3. The basic model

We consider a supply chain in which a manufacturer (she)provides an identical product to n retailers (he) at a wholesaleprice of w per unit. The retailers further process the product tomake partially substitutable products using different customiza-tion processes. We assume that all firms are risk neutral andmaximize their expected profits. The retailers are engaged in aBertrand price competition in the consumer market, where theirdemand function is given by

qi ¼ aþy�ð1þgÞpiþg

n�1

Xn

j ¼ 1,ja i

pj for every i¼ 1, . . . ,n: ð1Þ

In the above equation, the demand function qi is the realizeddemand for product i, pi is the price set by retailer i, a is theknown part of the potential market size, and y is the uncertainpart of the market size, with E½y� ¼ 0 and VarðyÞ ¼ s2, drawn from

N. Shamir / Int. J. Production Economics 136 (2012) 352–365 355

a continuous distribution over a finite support ½y,y�.1 For a givenprice pi, the demand for product i is influenced by the averageprice set by the other n�1 retailers. Products are assumed to beimperfect substitutes, and g captures the degree of substitution,where gZ0. If g¼ 0, the demand for product i is independent ofthe retail prices set by the other retailers, and each retailer has amonopoly power in his own local market. A higher level of gindicates a high degree of substitution and greater intensity ofcompetition among the retailers. In addition to the wholesaleprice w, the retailers incur an identical constant marginal cost forcustomization, which we normalize to zero.

Given this structure, retailer i’s profit is

piðpi,p�i,wÞ9 aþy�ð1þgÞpiþg

n�1

Xn

j ¼ 1,ja i

pj

0@

1Aðpi�wÞ ð2Þ

the supplier’s profit is

pMðp,wÞ9Xn

i ¼ 1

aþy�ð1þgÞpiþg

n�1

Xn

j ¼ 1,ja i

pj

0@

1A

24

35w ð3Þ

and the total supply chain profit is

pSCðp,wÞ9Xn

i ¼ 1

ðaþy�ð1þgÞpiþg

n�1

Xn

j ¼ 1,ja i

pjÞpi

24

35, ð4Þ

where p¼ ðp1, . . . ,pnÞ and p�i ¼ ðp1, . . . ,pi�1,piþ1, . . . ,pnÞ.We next characterize the information with which each retailer

is endowed.

3.1. Information structure

At time 0, the manufacturer and the retailers are endowed withthe common prior of the demand intercept y. Because of theproximity of the retailers to the consumer market, each retailer canobtain a private signal Yi about y, and the signals are assumed to bei.i.d. conditional on y. We further model the signals to be unbiasedand to satisfy the affine conditional expectations assumption; i.e.E½Yi9y� ¼ y, where i¼ 1, . . . ,n, and E½y9Y1, . . . ,Yn� ¼

Pni ¼ 1 aiYi,

where ai is a constant for every i. There are many interestingconjugate pair distributions for the demand intercept that satisfythese assumptions, such as Normal–Normal, Gamma–Poisson, andBeta–Binomial (see Ericson, 1969 for a detailed discussion). We defines9E½VarðYi9yÞ�=s2, as the expected precision of the common prior,relative to the signal accuracy.2 We denote by KDN a subset ofretailers, where 9K9¼ k, and the set of their signals by Yk. Thisinformation structure implies that3:

E½y9Yk� ¼1

sþk

XiAK

Yi,

E½Yj9Yk� ¼ E½y9Yk� where j =2 K ,

E½Yl9Yj,Yk� ¼ E½y9Yj,Yk� ¼ðkþsÞE½y9Yk�þYj

kþsþ1where l,j =2 K and la j,

E½E½y9Yk,Yj�9Yk� ¼ E½y9Yk� where j =2 K ,

E½YiYj� ¼ s2 where ia j,

E½Y2i � ¼ s

2ðsþ1Þ: ð5Þ

1 To avoid algebraic difficulties, we assume that the sold quantities are

positive for each set of realized signals fYigni ¼ 1 and realized value of y.

2 A similar information structure was used by Li (1985, 2002), Zhang (2002),

Shin and Tunca (2008), Li and Zhang (2008), Ha et al. (2008), and Jain et al. (2008).3 See Li (1985, Lemma 1) and Zhang (2002).

We denote the profit of a retailer by pR and the ex-ante profitof a retailer prior to signal realization by PR. Similarly, we use pM

to represent the manufacturer’s profit and PM for the manufac-turer’s ex-ante profit prior to signal realization.

3.2. Sequence of events

We study a three-stage game in which the sequence of eventsis as follows. In the first stage, each retailer observes Yi—the valueof his signal; then, information is shared based on the differentscenarios examined in this paper. We focus on the followingmethods of sharing information:

Horizontal information sharing—In the horizontal information-sharing setting, if information is shared, it is shared only among theretailers. The manufacturer is not privy to the shared information.

Public information sharing—In the public information-sharingsetting, when information is revealed, it is revealed to both thecompeting retailers and the manufacturer.4

When the retailers are committed to truthful information sharing,each retailer reveals his true signal (if the information is shared).However, when the retailers are not restricted to truth-telling, retaileri can choose to use any message miA ½y,y�, which may differ from hisobserved signal Yi. We further assume that messages are sentsimultaneously. During the second stage of the game, we allow eachretailer to undertake a costly action to be used as a signaling device,denoted by ti. After information is exchanged, each receiver of theinformation forms a belief (the structure of the belief system will bedefined shortly) about the true signal, given the received message mi

and the action ti (if such action was taken).At the second stage of the game, the manufacturer sets the

wholesale price, based on the information available to her fromthe information exchange stage of the game as well as her beliefin the authenticity of this information.

At the final stage of the game, upon learning the wholesaleprice set by the manufacturer and based on the availableinformation and belief system, the retailers establish their pricessimultaneously. Finally, the demand is realized, and the produc-tion is completed to satisfy the demand.

3.3. Belief system

The game between the retailers and the manufacturer belongs tothe class of games with incomplete information; this model istypically solved using the concept of a Perfect Bayesian equilibrium.A perfect Bayesian equilibrium includes a strategy profile and a belief

system, such that the strategies are sequentially rational, given thebelief system. A belief system is defined as an assignment ofprobabilities to each decision node in an extensive form game, suchthat the sum of probabilities in any information set is 1. A system ofbeliefs can be thought of as specifying a probabilistic assessment ofthe relative likelihood of being at each of the information set’s variousdecision nodes for each information set, conditional upon the playthus far. In PBE, whenever a player makes a move, the player’s actionmust be optimal, given the player’s belief at that point.

In our setting, signals are realized first; then information isshared. When information is shared truthfully, each recipient ofthe information uses the shared information to determine theretail or wholesale price. However, when information is sharedwithout a mechanism to verify the authenticity of the sharedinformation, the recipient of the information might be subjectto information manipulation to increase the sender’s profit by

4 Li and Zhang (2008) studied several additional information sharing possibi-

lities, including the option for a retailer to share information with the manufac-

turer alone and the option to share information only with the other retailers who

share information.

N. Shamir / Int. J. Production Economics 136 (2012) 352–365356

inducing the recipient to set a non-optimal price. Consequently,the recipient of the information assesses the accuracy of theshared information based on his available information.

We now develop the notation for the firm’s belief system in thegame. A firm’s belief system about the opponents’ signals is givenby l, which means that for each retailer ia j, retailer j has a beliefmj,i about Yi. Note that mj,i differs from E½Yi9y�. The latter is the beliefthat retailer j has about Yi at the beginning of the first stage, prior toobserving the signal realization Yj and receiving the set of messagesm�j, whereas mj,i is the belief of retailer j about Yi after observing Yj

and receiving m�j. We formally denote the belief system as

mj,iðm�j,YjÞ ¼ E½Yi9m,Yj�: ð6Þ

If retailer i uses a truth-telling strategy, a sequential equili-brium requires that mj,iðm�j,YjÞ ¼mi; if all retailers use a constantannouncement for every observed signal, the belief systemevolves according to the rule mj,iðm�j,YjÞ ¼ E½Yi9Yj�.

Finally, we use mj ¼ ðmj1, . . . ,mjj�1,mjjþ1, . . . ,mjnÞ as the vector ofassessments retailer j has about his opponents’ signals, and wesummarize the updating function for all retailers simultaneously as amatrix l¼ ðmT

1, . . . ,mTnÞ, where T denotes the transpose operator. We

use a similar notation mM for the belief system of the manufacturer.

4. Motivation for information sharing

In this section, we study the incentives of the retailers toexchange information, assuming that the accuracy of the infor-mation can be verified; i.e. mi ¼ Yi for every i. We first establishthe incentives of the retailers to exchange information horizon-tally, and in Section 4.2 we demonstrate that when informationcan be verified, the retailers choose to conceal their informationfrom the manufacturer. This section serves as a benchmark inorder to contrast between the incentives of the retailers to shareinformation with the manufacturer when the accuracy of theshared information can be verified, and their incentives to shareinformation with the manufacturer when the accuracy of theinformation cannot be verified. Therefore, this section builds uponprevious results in the information sharing literature.

4.1. Horizontal information sharing

At the third stage of the game, each retailer establishes hisprice. Suppose a subset K of N retailers choose to share theirsignals truthfully, where 9K9¼ k. Let YK be the set of sharedsignals available to all retailers, and let w be the wholesale priceset during the second stage of the game. We use the notation piðkÞ

to denote the profit of a retailer who chooses to share his privateinformation when other k�1 retailers choose to share theirinformation as well, and we denote by p iðkÞ the profit of a retailernot revealing his private information when k retailers share theirinformation. A retailer i chooses pi to maximize

piðkÞ ¼ aþE½y9YK ,Yi��ð1þgÞpiþg

n�1

XN

j ¼ 1,ja i

E½pj9YK ,Yi�

0@

1Aðpi�wÞ: ð7Þ

The following result, which is based on Li and Zhang (2008),represents the ex-ante profit of the retailers in this game:

Lemma 1.

PHTi ðkÞ ¼

1þg4ð2þgÞ2

a2þ4k

kþss2

� �for 1rkrn,

PHT

i ðkÞ ¼1þg

4ð2þgÞ2a2þ

1þg4ð2þgÞ2

4k

kþss2

þð1þgÞðakÞ

2ðkþ1þsÞs

ð2þgÞ2ðkþsÞs2 for 0rkrn�1,

where

ak ¼ð2þgÞðn�1Þ

2ð1þgÞðn�1Þðkþsþ1Þ�gðn�k�1Þ:

PHTM ¼

nð1þgÞ4ð2þgÞ a

2:

All proofs are given in the Appendix.

The superscript HT (horizontal truthful) is used to denote the

scenario in which retailers exchange information truthfully, andonly among the retailers. When the manufacturer is not exposedto the shared information, his pricing decision and expected profitare independent of the number of retailers that choose to shareinformation.

When a retailer i does not share his information, he establishesthe value of E½y9Yi,YK � based on two sources—the pool of sharedinformation YK , and his own private signal Yi. The coefficient1�ak is the weight a retailer who does not share his privateinformation places on the pooled information, and the ak is theweight such a retailer puts on his private observed signal. It canbe verified that ak is decreasing in the number of retailerschoosing to share their private information; as the number ofretailers that share their private information increases, the preci-sion of the pooled information relative to the private informationincreases as well, and the non-participating retailer puts greaterweight on the shared information.

It is worth emphasizing that when a retailer decides not toshare information, he is still exposed to the information of the setof retailers who are willing to share their private information.A similar modeling approach was used by researchers whoexamined incentives for information sharing, such as Li (1985,2002) and Gal-Or (1985). An alternative approach can be toassume that a retailer who decides not to share information isnot exposed to the information provided by other retailers. Theanalysis of this approach is left for future work.

Equipped with these results, we are ready to examine theretailers’ incentives to exchange information. The following pro-position establishes the result that there exists an equilibrium inwhich all retailers choose to exchange their private information.

Proposition 1. (a) PHTR ðkþ1ÞZP

HT

R ðkÞ for any k¼ 0, . . . ,n�1.A retailer who chooses to reveal his private information becomes

better off.

(b) PHTR ðkþ1ÞZPHT

R ðkÞ for any k¼ 1, . . . ,N�1. A retailer becomes

better off as more retailers share their private information.

The first part of the proposition suggests that each retailer isbetter off sharing his private information, regardless of thenumber of retailers who choose to share their information aswell. In fact, when information is shared truthfully, choosing toreveal the private information is a dominant strategy for eachretailer. The second part of the proposition asserts that as thenumber of retailers who choose to share their private informationincreases, it benefits all other retailers who also choose to sharetheir private information. Since it is a dominant strategy to shareinformation, in equilibrium, all retailers choose to share theirprivate information and the outcome is Pareto-optimal.

4.2. Public information sharing

We now assume that when information is revealed, it isrevealed publicly to both the competing retailers and themanufacturer.

The retailers’ pricing decision during the third stage of thegame is not affected by the decision to share the privateinformation with the manufacturer since the wholesale price w

N. Shamir / Int. J. Production Economics 136 (2012) 352–365 357

is taken as given. During the second stage of the game, themanufacturer is endowed with better information to determineher wholesale price:

w¼aþE½y9Yk�

2: ð8Þ

Plugging the value of w into the pricing decision of theretailers, the following ex-ante profit of the retailers and themanufacturer can be derived:

Lemma 2. The retailers’ ex-ante profit in the public truthful (PT)setting is:

PPTi ðkÞ ¼

1þg4ð2þgÞ2

a2þk

kþss2

� �or 1rkrn,

PPT

i ðkÞ ¼1þg

4ð2þgÞ2a2þ

1þg4ð2þgÞ2

k

kþss2

þð1þgÞðakÞ

2ðkþ1þsÞs

ð2þgÞ2ðkþsÞs2 for 0rkrn�1,

PPTM ðkÞ ¼

nð1þgÞ4ð2þgÞ

a2þk

kþss2

� �for k¼ 0;1, . . . ,n:

The following proposition summarizes the retailers’ incentivesto share information publicly:

Proposition 2. (a) PPTi ðkþ1ÞrP

PT

i ðkÞ. A retailer becomes worse off

as he shares his demand information publicly.

(b) PPTM ðkÞrPPT

M ðkþ1Þ. The manufacturer becomes better off as

more information is revealed publicly.

As a result of the above proposition, a retailer has no incentiveto share information publicly. Although the retailer gains fromsharing his information with the other retailers, the manufactureruses the shared information to adjust the wholesale price to themarket condition and hurts the retailer in such a way that hisexpected net payoff from information sharing is negative.

5. Cheap talk and truthfulness

In Section 4, we fixed the retailers’ strategy with regard to theability to manipulate the shared information and established theretailers’ incentives to share their information. In this section, weexplore the game played between the retailers and the manufac-turer when each retailer is allowed to use any message in theinformation-sharing stage of the game. We examine whetherthe goal of information exchange can indeed be achieved whenthe retailers are not committed to truthful information exchangeand only use pure communication (cheap talk) to exchange theirprivate information. Section 5.2 examines the retailers’ incentivesto truthfully reveal their private information when information isexchanged only among the retailers, and Section 5.3 studies theretailers’ motivation to share their information truthfully when themanufacturer is also exposed to the shared information.

Before analyzing the game between the retailers and the man-ufacturer, in Section 5.1 we provide a definition of the equilibrium inthis setting and introduce two types of information-sharing levels:the first is an informative equilibrium; the second is a quasi-

informative equilibrium. In an informative equilibrium, the retailersshare the value of their observed signal, whereas in a quasi-

informative equilibrium, the retailers report only an interval withintheir signal observation lies.

5.1. Equilibrium concept and definitions

We start by formally describing the equilibrium concept whenthe retailers are not restricted to announcing their signals truth-fully. Let M represent the set of all feasible messages a retailer canuse, such that m : Y�!M represents the strategy of a retailerduring the first stage of the game. A retailer observes a signalrealization drawn from the space Y and provides an announce-ment miðYiÞ. Let w : Mn

�!W represent the strategy of themanufacturer. When information is shared with her, the manu-facturer receives a set of m messages; wðm,mMÞ represents thepricing decision for the manufacturer based on her availableinformation and belief system mM . Finally, let p : Y�MN

�

W�!P represent the pricing decision of each retailer during thethird stage of the game, such that piðYi,m,w,miÞ determines aretailer’s pricing decision after observing signal realization Yi,receiving the set of messages m and forming a belief system mi.

In the following analysis, the interim profit of a retailer (i.e. theprofit a retailer expects to earn given his observed signal and priorto observing the information received from the other retailers)plays an important role, hence, we introduce new notation for theinterim profit:

VjðYi,miÞ ¼ E½pj

iðmiÞ9Yi� where jAfH,Pg: ð9Þ

The retailer’s interim profit is defined as the expected profit ofa retailer observing a signal Yi and reporting a message mi. Here,the function V is calculated prior to observing the informationshared by the other retailers; hence, the E operator is taken on theuncertain part of the market demand y and the signal realizationof the other retailers Y�i.

Definition 1. ðmn,wn,pn,lnÞ forms a Bayesian Nash equilibrium inthe pure communication game if the following conditions hold:

(1) For a given wholesale price w, an observed signal Yi shared

messages m�i and a belief system mi,

pn

i ðYi,m�i,w,miÞA arg maxpi

E

� aþy�ð1þgÞpiþg

n�1

Xn

j ¼ 1,ja i

pn

j

0@

1Aðpi�wÞ9Yi,m�i,mi

24

35,

where the expectation is taken over y.

(2) For a given set of shared messages m and belief system mM ,

wnðmÞA arg maxw

E

�Xn

i ¼ 1

aþy�ð1þgÞpn

i þg

n�1

Xn

j ¼ 1,ja i

pn

j

0@

1Aw9m,mM

24

35,

where the expectation is taken over y.

(3) For every Yi,

mn

i ðYiÞA arg maxmi

VjðYi,miÞ where jAfH,Pg:

The above definition requires that neither the manufacturernor each retailer has any unilateral profitable deviation duringeach stage of the game. Specifically, the first condition in thedefinition requires that fixing the strategy of the remainder of theretailers, based on his available information, retailer i sets hisprice based on the function pn. Similarly, the second conditionrequires that, given the shared messages provided to her, themanufacturer maximizes her profit by using strategy wn. Thethird condition requires that each retailer determines hisannouncement to maximize his profits.

The primary objective of this section is to examine whether anequilibrium that includes truth-telling during the informationexchange stage of the game exists in this pure communication

N. Shamir / Int. J. Production Economics 136 (2012) 352–365358

setting. In a truth-telling equilibrium, the retailer’s interimprofit is maximized by reporting the true value of his privateinformation.

We next set the framework for a few definitions about theamount of information the retailers can share in the pure com-munication game.

Definition 2. A function f is responsive with respect to retailer i’smessage mi if there exists a pair of messages m1

i and m2i such that

f ðm1i Þa f ðm2

i Þ:

A function is called responsive if there is at least one pair ofmessages sent by retailer i that affects the value of the function f

when all other problem parameters are held constant. An illus-trative example of this concept is each retailer’s pricing decisionwhen information is shared truthfully. In this case, the pricingstrategy is responsive since each retailer takes the messages sentby his opponents and updates his belief about the potentialmarket demand and the pricing of his competitors into account.

The following two definitions are related to the level ofinformation exchange the retailers can achieve in equilibrium.

Definition 3. ðmn, wn, pn,lnÞ forms an informative equilibrium if:

(a) ðmn, wn, pn,lnÞ is an equilibrium as defined above.

(b) mðYiÞ ¼ Yi for every Yi and every i.

An informative equilibrium is an equilibrium in which allretailers report their signal observations truthfully for all possiblesignal realizations. Naturally, an informative equilibrium providesan upper bound on the amount of information the retailers canshare. However, as we will demonstrate, an informative equili-brium might be difficult to obtain, hence, we define below a lessrestrictive concept of information sharing—the quasi-informative

equilibrium.We partition ½y,y� by choosing z points, such that x0 ¼ y,

x1, . . . ,xz�1,xz ¼ y so that x0ox1o � � �oxz�1oxz. Denote such apartition by I and use the notation I0 to represent the nullpartition that includes only one interval. We use the functionIðYiÞ to denote the interval containing the observation Yi. In aquasi-informative equilibrium, a retailer is required to report onlya sub-interval from the partition I, rather than the exact realiza-tion of his signal. In the quasi-informative game, we need to makeanother adjustment to our prior definitions. Let the function mðYiÞ

denote the sub-interval that a retailer announces after observingYi. We are now ready to define the quasi-informative equilibrium.

Definition 4. Given a partition I, ðmn, wn, pn,lnÞ forms a quasi-informative equilibrium if:

(a) ðmn, wn, pn,lnÞ is an equilibrium as defined above.

(b) mðYiÞ ¼ IðYiÞ for every Yi and every i.

Therefore, in a quasi-informative equilibrium, all retailerstruthfully reveal the interval within which their observed signallies.

5.2. Horizontal information sharing

We start by assuming, as the benchmark suggests, that theretailers share information horizontally. In order to examine theexistence of an informative equilibrium, we first fix the strategiesof all retailers ja i to truthtelling; we further assume that allretailers ja i use a responsive function with respect to mi. Aninformative equilibrium exists if under these conditions retailer i

chooses to report his signal truthfully. The next lemma charac-terizes the effects on the sold quantity and pricing decisions ofretailer ja i and retailer i in this case.

Lemma 3. Assume that all retailers ja i report their signals truth-

fully and use a responsive pricing function with respect to the

message mi then the following hold:

pj ¼ pHTj þ

gðmjiðmiÞ�YiÞ

ð2þgÞðsþnÞð1þgÞfor ja i,

qj ¼ qHTj �

gðmjiðmiÞ�YiÞ

ð2þgÞðsþnÞð1þgÞ for ja i,

pi ¼ pHTi þg

Pj a i

mjiðmiÞ

n�1 �Yi

2ð2þgÞðsþnÞð1þgÞ,

qi ¼ qHTi þg

Pj a i

mjiðmiÞ

n�1 �Yi

2ð2þgÞðsþnÞð1þgÞ:

The pricing decision of retailer j is increasing in mjiðmiÞ, and ifmjiðmiÞ4Yi, retailer j sets his price above the retail price deter-mined in the horizontal truthful (HT) setting. As a result, retailer i isalso able to increase his price; when retailer i is able to create theimpression that his observed signal is better than the actual signalby manipulating mi, retailer i is able to set his price higher than theone used in the horizontal truthful setting. Since the retailers areengaged in pricing competition, when one retailer increases hisprice, his opponent’s best response is to follow and increase hisprice as well. It is also worth mentioning that as mjiðmiÞ�Yi (whichis the difference between the belief retailer j has about Yi and theactual observed signal YiÞ increases, retailer j becomes worse off,not only in the expected value of his profit, but for every realizationof y compared with the profit retailer j earns in the HT setting; asmjiðmiÞ�Yi increases, retailer i becomes better off for every realiza-tion of y compared with the setting in which information isrevealed truthfully in a horizontal manner truthfully.

After observing the pricing decision of retailer i, all other retailersare able to infer that retailer i provided them with a false report,since the pricing decision of retailer i takes into account both thefact that all other retailers are not pricing their product in an optimalmanner; furthermore, retailer i uses his own signal Yi when pricinghis product. If the retailers only meet in the marketplace once, suchopportunistic behavior might be expected. However, if, as iscommon in many market interactions, the retailers expect to meetagain in the future, retailer i might be reluctant to expose the factthat he has provided his rivals with false information. In this case,retailer i might provide a false report ~mi and price his product as ifYi ¼ ~mi. Given such behavior, the competing retailers cannot inferthat retailer i has manipulated his message, based solely on the pricepi. A detailed analysis of this case is not presented in this paper.

The next result, which is a direct consequence of the abovelemma, shows that without a mechanism that verifies the sharedinformation, the retailers cannot reach an informative equilibrium.

Proposition 3. (a) For any responsive pricing function pj, with respect

to the message vector m�j, each retailer ia j announces mi such that

miðYiÞA arg maxmAM

pjð�,miÞ for every Yi and every i:

(b) An informative equilibrium does not exist in the pure commu-

nication game with horizontal information sharing.

If a retailer uses a responsive pricing function with respect to theannounced messages m, each retailer has an incentive to manip-ulate his announcement to induce his competitors to increase theirretail prices. Consequently, the retailers ignore the messages andmake their pricing decisions regardless of any information provided.In this cheap talk game, full credibility cannot be achieved.

After establishing the fact that an informative equilibrium cannotbe achieved, we investigate whether or not a quasi-informative

N. Shamir / Int. J. Production Economics 136 (2012) 352–365 359

equilibrium can provide the retailers with some level of informationsharing.

Proposition 4. There exists a quasi-informative equilibrium only for

I¼ I0.

The above proposition provides another negative result withrespect to the retailers’ ability to exchange information. Anypartition of the signal space into two or more sub-intervalsresults in an equilibrium that is not quasi-informative. Similarto the results derived for the informative equilibrium, if theretailers’ pricing decision is responsive to the message sent by acompeting retailer, the sender retailer has an incentive to reportthe maximum sub-interval in the partition (i.e. a retailer reportsthe right most sub-interval in the partition).

At this point, it is worth contrasting this result with theexistence of a quasi-informative equilibrium in Crawford andSobel (1982). Sobel and Crawford assume that the sender has aconcave utility function with a unique maximizer that lies withinthe interior of the message space. As a result, it is possible toconstruct an equilibrium in which the sender truthfully reportsthe interval containing his signal observation. In our model, foreach partition and for each observed signal, a retailer does nothave an interior maximizer; a retailer maximizes his profit byannouncing the right-most sub-interval within the partition.Consequently, there is no equilibrium in which the retailersprovide different announcements when receiving different signals.

5.3. Incentives for truthful public information sharing

How does the presence of the manufacturer affect the incen-tives of the retailers to manipulate their announcements? With-out the presence of the manufacturer, we observed that eachretailer is tempted to inflate his demand signal thereby creatingan overly optimistic belief for his competitors. However, when theshared information is also available to the manufacturer, inflatingthe demand signal has a dual effect: reporting a high demandsignal induces the competing retailers to increase their retailprices, which has a positive effect on the retailer’s profit; onthe other hand, since the manufacturer is exposed to thisinformation as well, the same over-optimistic announcementencourages the manufacturer to set a higher wholesale price,which has a negative effect on the retailer’s profit. When a retailerdecides on an announcement mi he must consider the trade-offbetween the desire to induce the competing retailers to increasetheir prices, and the undesired result of higher wholesale price. Inthis section, we examine whether these two effects can counter-balance each other, such that reporting truthfully results in anequilibrium.

The following lemma provides the effect on the sold quantitiesand retail prices when the manufacturer and retailers useresponsive pricing functions.

Lemma 4. Assume that all retailers ja i and the manufacturer use a

responsive pricing function and report their signals truthfully, then

the following hold:

pi�w¼ pPTi �wPTþ

ðYi�mjiðmiÞÞ

2ðsþnÞð2þgÞð1þgÞ ,

qi ¼ qPTi þ

ðYi�mjiðmiÞÞ

2ðsþnÞð2þgÞð1þgÞ :

In contrast with the incentives to manipulate the sharedinformation in the horizontal information exchange setting,when the manufacturer has access to the exchanged information,a retailer has incentives to under-report his signal value. Althoughsuch a report results in the other retailers setting their price

lower than in the public-truthful (PT) setting, the manufactureralso sets a lower wholesale price. The desire to reduce thewholesale price outweighs the incentive to increase the compe-titors’ retail prices. Jain et al. (2008) investigated the incentives totruthfully report private demand information in an environmentwith Cournot-competing retailers and one mutual manufacturerand found that without a mechanism that verifies the authenti-city of the shared information, a uniform wholesale price resultsin the retailers under-reporting their signal observation. When aquantity-setting retailer creates the impression for the manufac-turer and his competing retailers that the demand is low, itresults in a low wholesale price and a low quantity produced bythe competing retailers—both of these results have a positiveeffect on the profit of the reporting retailer. When the retailers areprice-setters, the result of reduced wholesale and retail priceshave opposite effects on the reporting retailer’s profit. This lemmasuggests that the effect of the manufacturer’s wholesale price onthe retailer’s profit is stronger than the effect of the competingretail pricing; hence, a retailer has incentives to under-report hissignal observation.

The next proposition confirms that an informative equilibriumdoes not exist in the public-information sharing game.

Proposition 5. (a) For any responsive pricing function w, retailer i

announces mi such that

miðYiÞA arg min wð�,miÞ for every Yi:

(b) An informative equilibrium does not exist in the pure commu-

nication game with public information sharing.

(c) There exists a quasi-informative equilibrium only for I¼ I0.

Regardless of the intensity of the competition between theretailers, the effect of information sharing on the wholesale price(the negative effect from the retailers’ perspective) is always higherthan the positive effect of the ability to increase the competitors’retail prices. The fact that the manufacturer is also part of theinformation exchange coalition changes the retailers’ incentives toinflate their signal observation; the presence of the manufacturer inthis coalition reduces the retailers’ incentives to inflate their signal tosuch an extent that the retailers prefer to provide low estimates oftheir observed signal. Since the retailers choose to manipulate theirsignal, the manufacturer and the retailers will discard the informationthat is shared in the supply chain, and all parties in the supply chainset their prices as though no information had been exchanged.

6. Signaling games to induce truthful information sharing

The results obtained above imply that although informationsharing is desired, without a mechanism that holds the retailersaccountable for their reports, the retailers wish to provide falsereports; consequently, no information can be exchanged. Tomitigate this problem, we next consider two signaling gamesthat result in truthful information sharing. In the signaling game,when a retailer provides his report mi, he also incurs a cost that isassociated with the value of his report. We first characterize theequilibrium that induces truthful information sharing in thehorizontal information-sharing game; then, we study the signal-ing game that induces truthful information sharing in the publicinformation game, followed by a discussion of the properties ofthe equilibrium in each game.

6.1. Inducing truthful horizontal information sharing

We consider a mechanism that induces truthful informationsharing among the retailers. In order to achieve credibility, thesender has to incur a cost related to his shared information.

5 A function j : X �F�!R has the single crossing property if jX exists, and is

strictly increasing in yAF for all xAX.

N. Shamir / Int. J. Production Economics 136 (2012) 352–365360

The signaling costs is denoted by THðYiÞ where the superscript H

is used for the horizontal information exchange. An alternativeinterpretation is to assume that the signaling cost is observable bythe manufacturer as well, but the manufacturer sets thewholesale price prior to observing the signaling cost, thus, theinformation inferred by the manufacturer does not affect herpricing decision. Our model assumes general signaling costs.This cost can represent charity, advertisement investment orany other observable cost that does not affect the demandthe retailers face. There are other papers that consider thesignaling role of advertisement, and also assume that advertisingmay not be demand-enhancing (e.g. Nelson, 1974; Milgrom andRoberts, 1986).

In the signaling game, the retailers search for a signaling costTH such that the following optimization problem is solved:

maxfTHðmiÞg

E½VHðYi,miÞ�TH

ðmiÞ�

s:t: YiA arg maxmi

VHðYi,miÞ�TH

ðmiÞ for every YiA ½y,y�: ð10Þ

The above optimization ensures that every retailer reports hisprivate information truthfully when using the signaling costfunction TH and that the retailers cannot find any other signalingcost that induces truthful information exchange at a lower cost.

The structure of this problem resembles a mechanism-designprogram. In a mechanism-design problem a principal, endowedwith inferior information, maximizes his profit subject to truth-telling constraints of an agent possessing private information.In our setting, the constraint also ensures that the agents (the setof retailers in our model) report their private information truth-fully. However, the objective function is also taken from theretailers’ perspective—the retailers search for a signaling cost thatresults in maximum profit, subject to truth-telling constraints. Toput it differently, the retailers, prior to observing any signal,decide on a signaling cost TH. After each retailer observes hissignal Yi, he sends a message mi and incurs a cost TðmiÞ. Thesignaling cost is designed so that each retailer reports his signaltruthfully.

In our modeling approach, we assume that in order to beaccountable for the shared information, a retailer must incur acost when sharing his information with the other retailers. Weassume that the retailers cannot commit to pricing their productdifferently from way the Nash equilibrium of the third stage ofthe game. An alternative mechanism considers a differentiatedpricing scheme in which the price set by each retailer is differentfrom his Nash equilibrium pricing decision. In such a mechanism,a retailer reporting a high signal might be asked to set his pricehigher than his Nash equilibrium price in order to convey hiscredibility. This type of mechanism requires the retailers to have acertain level of commitment, which we assume they lack. As aresult, we assume that following the information exchange stage,the outcome of stages 2 and 3 is identical to the one studied inSection 4 when information can be verified.

We next characterize an informative equilibrium using asignaling cost TH

Proposition 6. ðmH , wH , pH ,mH ,THÞ is an informative equilibrium in

the horizontal-signaling game, if the following set of equations hold:

mn

i ðYiÞ ¼ Yi for every i and yi,

wH ¼wHT ,

pH ¼ pHT ,

mHi,jðm�i,YiÞ ¼mj for every i and mj,

THðmiÞ ¼

a

2ð2þgÞ2ðsþnÞð1þgÞgðmi�yÞ

þg

2ð2þgÞ2ðsþnÞðsþ1Þð1þgÞðmi�yÞ2:

In Section 5, it was demonstrated that each retailer has anincentive to inflate his report in order to encourage his compe-titors to increase their retail prices. As a result, the signaling costaligns this incentive by imposing a high cost on a retailer whoreport a high signal observation; this cost ensures that a retailerobserving a lower signal is not tempted to report a higher signal.It will be shown in Section 6.3 that the function TH

ðmiÞ isincreasing in the term mi. A retailer who reports a high observa-tion must be accountable for his message by incurring a highsignaling cost. A retailer who provides the lowest possiblemessage y does not incur any signaling cost.

Following the information exchange stage, all retailers areidentical with respect to their information; hence, their profit atthe third stage of the game (i.e. excluding signaling costs) isuniform. In this situation, the net profit of a retailer observing ahigh signal is lower than that of a retailer observing a low signal;the payoff to both retailers in the third stage is identical, but thesignaling cost of a retailer observing a high signal is higher. Thisraises the question ‘‘Why is a retailer observing a high signalwilling to incur a higher cost than a retailer observing a lowersignal?’’ The answer lies in the nature of the retailer’s profitfunction, which exhibits the single crossing property.5 The singlecrossing property suggests that a retailer who observes a highersignal enjoys a marginal increase in retail prices greater than aretailer observing a lower signal; consequently, a retailer obser-ving a higher signal is willing to incur a higher signaling cost thana retailer observing a lower signal.

6.2. Inducing truthful public information sharing

We consider an alternative mechanism that will induce truth-ful information exchange as well. Under the setting considered inthis section, the information is revealed to both the competingretailers and the manufacturer. We observed in Section 5, thatsuch a setting alone cannot induce truthful information exchange;hence, we added a signaling cost. We seek to ascertain howsharing information with the manufacturer affects the cost aretailer must incur in order to convey credibility. When retailer i

announces mi, he also incurs a cost TPðmiÞ. We are looking for the

minimum cost that induces a retailer to announce truthfully thatmi ¼ Yi for every possible realization of the signal Yi. In Section 5,we observed that when the retailers share the informationpublicly, the presence of the manufacturer affects the retailers’incentives to manipulate the shared information; each retailerhas an incentive to induce the manufacturer to set her wholesaleprice as though the observed information were lower than theactual observation.

Similar to the analysis of the horizontal-signaling game, thepublic signaling cost is found by solving the following optimiza-tion problem:

max E½VPðYi,miÞ�TP

ðmiÞ�

s:t YiA arg maxmi

VPðYi,miÞ�TP

ðmiÞ,

for every YiA ½y,y� and every iAN: ð11Þ

N. Shamir / Int. J. Production Economics 136 (2012) 352–365 361

The following proposition characterizes an informative equili-brium in the public information-sharing game:

Proposition 7. ðmP , wP , pP ,mP ,TPÞ is an informative equilibrium in

the public signaling game, if the following set of equations hold:

mn

i ðYiÞ ¼ Yi for every i and Yi,

wP ¼wPT ,

pP ¼ pPT ,

mPi,jðm�i,YiÞ ¼mj for every i and mj,

TPi ðmiÞ ¼

a

2ð2þgÞ2ð1þgÞðsþnÞðy�miÞþ

ðy�miÞ2

4ðsþnÞðsþ1Þð2þgÞ2ð1þgÞ:

It is demonstrated in Section 6.3 that the function TPi ðmiÞ is

increasing in the term ðy�miÞ. As was demonstrated above wheninformation is shared publicly, a retailer has an incentive to under-state his observed signal. The signaling function aligns the incentiveof the retailer to report his signal truthfully by imposing a high coston a retailer who reports a low signal observation. When a retailerreports mi ¼ y, he does not incur any signaling costs.

6.3. Discussion

This section discusses the properties of the signaling costs inboth games.

Proposition 8. (a) TPðmÞ is a convex decreasing function.

(b) THðmÞ is a convex increasing function.

In the public information-sharing game, when there are nosignaling costs, the retailer’s incentive is to provide an under-estimate of his observed signal. The signaling cost aligns theretailer’s incentive to report his signal truthfully by imposing acost for ‘‘bad news’’—the public signaling cost fines the retailerfor reporting low signal realization. In contrast, in the horizontalgame the retailer’s incentive was to inflate his report. As a result,the signaling cost induces truthtelling by imposing a cost for‘‘good news’’—a retailer incurs a high signaling cost when heprovides a high signal.

The number of retailers effect. As the number of retailersincreases, the effect of a single message on the pricing strategyof the firms in the supply chain decreases. As a result, as thenumber of retailers increases, a retailer has reduced incentives toprovide a false; the signaling cost for both the horizontal andpublic information-sharing setting decreases in the number ofretailers. The following proposition formalizes this result.

Proposition 9. Let N1 and N0 be the number of retailers in the

supply chain, such that N0oN1o1. Then,

TPðmi,N

1ÞoTP

ðmi,N0Þ for every miAM,

THðmi,N

1ÞoTH

ðmi,N0Þ for every miAM:

The support effect. It is interesting to note that the signalingfunction for both the public and horizontal games is not affected bythe density of the posterior or prior distribution of y. The horizontalsignaling function is affected by the lower bound of the support of y,since without signaling costs, a retailer has an incentive to providethe message y. In order to align a retailer’s incentives, announcing ahigh mi must be accompanied by high signaling costs.

In the public information-sharing game, absent signaling costs,a retailer has an incentive to provide the lowest message y. Inorder to align a retailer’s incentive and to reach a truth-telling

equilibrium announcing a high mi must be accompanied by highsignaling costs.

Proposition 10. Let Y1 and Y2 be two possible signal spaces.

(a)

if y1oy2. Then,

THðmi,y

1Þ4TH

ðmi,y2Þ for every miAY1

\Y2:

(b)

if y14y

2. Then,

TPðmi,y

1Þ4TP

ðmi,y2Þ for every miAY1

\Y2:

The competition effect. The competition factor (measured by gÞhas two effects on the incentives of the retailers to mis-reporttheir observed signals. As the competition factor increases, theproducts become closer substitute; as a result, the retailers’ profitdecreases. On the other hand, an increase in the competitionfactor also increases the incentive to mis-report the observedsignal in the horizontal game, since inducing the rival retailers toincrease their retail prices has a larger impact on the profit whenthe competition factor is high. Similarly, when the competitionfactor increases, the retailers have less incentive to mis-reporttheir signals in the public game. The following propositionsummarizes the effect of the competition factor on the signalingcost.

Proposition 11. Let g1 and g0 be the competition levels between the

retailers, such that g14g0. Then, the following hold:

(a)

TPðmi,g1ÞrTP

ðmi,g0Þ for every miAM.

(b) There exists a threshold value gn such that @TH=@g40 for grgn,

and @TH=@go0 for g4gn.

In the public game, the two effects of increased competition(reduced retail profit and reduced incentives to mis-report theobserved signal) have the same effect on the public signaling cost:as the competition between the retailers increases, the signalingcost decreases. However, in the horizontal game, the two effectsof increased competition have the opposite effects on the incen-tive of the retailers to mis-report their signals. An intensecompetition reduces the retailers’ profit; consequently, there isless incentive to mis-report. However, at the same time, anintense competition implies a higher correlation between theretailers’ pricing strategies which creates an incentive for theretailers to mis-report their signals.

The next natural question is what type of information-sharingmechanism do the retailers choose. The next proposition providesan answer under which conditions the retailers prefer to use thepublic information-sharing mechanism over the horizontal one.

Proposition 12. There exists a threshold value X, such that if

9y9�y4X:

PPi 4PT

i :

The above proposition suggests that if the support of thedemand distribution is asymmetric so that it is skewed to theleft, the retailers would find it beneficial to share their informa-tion publicly rather than to share it horizontally. The intuition forthis conclusion is that in the horizontal setting, the retailers’signaling cost is a function of y, whereas in the public setting it isa function of y.

The traditional signaling model involves an efficiency loss: thesender of the information needs to take a costly action to conveythe credibility of his shared information. However, in our settingwhen the retailers choose the public information setting, a portionof their signaling costs involve sharing their private information

N. Shamir / Int. J. Production Economics 136 (2012) 352–365362

with the manufacturer. In this case, the signaling cost is actuallynot wasted, but rather it is used by a different firm in the supplychain—the manufacturer.

7. Concluding remarks

Although information sharing can increase efficiency, indepen-dent firms endowed with private information often face a majorchallenge in adopting a scheme that induces truthtelling informa-tion sharing. In this paper, we introduce a new motivation forinformation sharing between a set of retailers and their mutualmanufacturer—as a substitute for signaling cost. We present a novelmodel in which a set of competing retailers search for a mechanismto share their private demand information. The first option weconsider is the retailers engaging in a signaling game. As analternative mechanism, we examine the effect of including themanufacturer in the information club on the ability of the retailersto share information. We demonstrate that sharing informationwith the manufacturer affects the incentives of the retailers tomanipulate the shared information; as a result, under some condi-tions, the retailers can reach an information sharing equilibrium at alower cost. The model we present can be viewed as an example of amore general idea: by carefully including or excluding firms in thesupply chain from the information club, the firms in the supplychain can change their signaling costs and reach an informationexchange agreement for lower signaling costs.

Further research in this area can evolve in several directions:the first involves generalizing the model to different types ofcompetition and private information. For example, one can con-sider a set of retailers who compete by setting quantities withprivate cost information. As discussed in Section 4, it is agreedamong researchers that the retailers in this setting wish to sharethis private cost information (e.g. Gal-Or, 1985). However, with-out a commitment mechanism, each retailer will be tempted toreport low production costs (Ziv, 1993) in order to induce thecompeting retailer to produce less quantity of the product. Apossible mechanism to induce truthful information sharing in thiscase is to share the private cost information with the manufac-turer as well; reporting low production costs will result in a highwholesale price; hence, the incentives of a retailer to mis-reporthis true production costs can be balanced when the manufactureris exposed to this information.

Another possible way to extend our basic model is to assumethat a retailer’s decision to share his information publicly orhorizontally is made after observing his private information. In thecurrent model, we compare two settings: in the first, all retailersshare their demand information horizontally; and in the secondsetting, all retailers share their demand information publicly. Thus,an important assumption in our modeling approach is that thedecision to include the manufacturer (or not) in the informationclub is made ex-ante prior to observing the signal realization. As analternative, we can adopt a different approach in which each retailerfirst observes his demand realization and only then decides if hewishes to share this information horizontally or publicly.

The view of information sharing as a cost can also haveimplications for anti-trust regulation. Several economists (Kuhnand Vives, 1995; Kuhn, 2001; Motta, 2004) have recently arguedthat inferring collusion from market data is virtually impossible.Consequently, most competition authorities around the worldhave adopted the so-called parallelism plus rule. This policy allowsthe prosecution of collusive behavior only in cases where suspi-cion can be supported by hard evidence of facilitating practices,such as information exchange. Based on this view, Kuhn (2001)has suggested that the information exchange of costs or demandshould be considered to be an illegal restriction of the

competition if the competition authorities demonstrate that nosignificant efficiency gains can be expected. However, based onthe possible view of information sharing as a cost, we suggest thatthe gains from information sharing should be examined based onwhat we define as the total effect of information sharing on thesupply chain. Although the direct effect of information sharingcan result in losses for the sender of the information (in ourmodel a retailer sharing information with his manufacturer) thetotal effect of this information sharing is positive since it enablesthe retailers to exchange information as well.

Appendix A

Proof of Lemma 1. The lemma is based on Li and Zhang (2008).For a more detailed derivation of this result the reader can consulttheir paper.

Assume a set of retailers KDN choose to share their private

information horizontally. The FOC are given by

aþE½y9YK ,Yi�þg

n�1

XE½pj9YK ,Yi�þð1þgÞw�2ð1þgÞpi ¼ 0: ð12Þ

It can be verified that the following pricing strategy satisfies the

FOC conditions:

pi ¼

aþE½y9YK �þwð1þgÞ2þg

if iAK ,

aþð1�akÞE½y9YK �þakYiþwð1þgÞ2þg if i =2 K ,

8>>><>>>:

ð13Þ

where ak is given by

ak ¼ð2þgÞðn�1Þ

2ð1þgÞðkþsþ1Þðn�1Þ�gðn�k�1Þ: ð14Þ

The conditional expected quantity sold by each retailer is

E½qi� ¼ ð1þgÞðpi�wÞ: ð15Þ

The manufacturer maximizesPn

i ¼ 1 E½qi�w. Using (13) we derive

that when the manufacturer is not endowed with any private

information, she will set the wholesale price w¼ a=2.

Plugging the wholesale price to Eqs. (13) and (15) we can find

the ex-ante profit of the retailers E½E½pi9Yk,Yi�� ¼ ð1þgÞE½ðpi�wÞ2�,

which is given in the result of Lemma 1. &

Proof of Proposition 1. (a)

PHTR ðkþ1Þ�PHT

R ðkÞ

¼ð1þgÞð2þgÞ2

s2 kþ1

kþ1þs�

k

kþs�ðakÞ

2ðkþ1þsÞs

kþs

" #

¼ð1þgÞs2s½1�ðakÞ

2ðkþ1þsÞ2�

ð2þgÞ2ðkþsÞðkþ1þsÞ:

Li and Zhang (2008) showed that ðakÞðkþ1þsÞo1 which com-pletes the proof of this part.

(b)

ðPHTR ðkþ1Þ�PHT

R ðkÞ ¼ð1þgÞ

4ð2þgÞ24s2½ðkþ1ÞðkþsÞ�kðkþ1þsÞ�

¼ð1þgÞ

4ð2þgÞ24s2s40: &

Proof of Lemma 2. Assume as before that a set K with 9K9¼ k

share information with all the other retailers and the manufac-turer. The pricing decision of the retailers is the same as the one

N. Shamir / Int. J. Production Economics 136 (2012) 352–365 363

derived in Lemma 1 and it is given by (13). When the manufac-turer sets the wholesale price she estimates that

E½Q9YK� ¼ð1þgÞ2þg

½aþE½y9YK��w�: ð16Þ

As a result the manufacturer sets the wholesale price to

w¼aþE½y9YK�

2: ð17Þ

Plugging (17) to (13)

pi�w¼

1

2þgaþE½y9YK�

2

� �, iAK ,

1

2þgaþE½y9YK�

2þakðYi�E½y9YK�

� �, i =2 K:

8>>><>>>:

The retailers’ conditional expected profit is given by

E½pi9YK� ¼1þgð2þgÞ2

aþE½y9YK�

2

� �2

for iAK ,

E½pi9YK,Yi� ¼1þgð2þgÞ2

aþE½y9YK�

2þakðYi�E½y9YK�

� �2

for i =2 K:

ð18Þ

Taking expectations over the signal realizations we get the

retailers’ and manufacturer ex-ante profit. &

Proof of Proposition 2. (a)

PPT

i ðkÞ�PPTi ðkþ1Þ

¼ s2 k

kþs�

kþ1

kþ1þs

� �1þg

4ð2þgÞ2þð1þgÞðakÞ

2ðkþ1þsÞs

ð2þgÞ2ðkþsÞ

" #:

Li and Zhang (2008) have demonstrated that akðkþ1þsÞ41=2.Therefore,

PPT

i ðkÞ�PPTi ðkþ1Þ ¼ s2 ð1þgÞs

4ð2þgÞ2ðkþsÞðkþ1þsÞ

�½4ðakðkþ1þsÞÞ2�1�40:

(b)

PPTM ðkþ1Þ�PPT

M ðkÞ ¼nð1þgÞ4ð2þgÞs

2 kþ1

kþ1þs�

k

kþs

� �40: &

Proof of Lemma 3. Assume all retailers ja i report their signalstruthfully, and retailer i chooses to report mi. In this case

pj ¼að3þgÞ2ð2þgÞ

þE½y9Y1, . . . ,Yi�1,mjiðmiÞ,Yiþ1, . . . ,Yn�

ð2þgÞ

for every ja i. At the third stage of the game retailer i maximizeshis price, taking the other retailers’ price as given, and m also asgiven. Retailer �ı’s problem is

maxpi

E

�

aþy�ð1þgÞpi

þg

n�1

Pja i

að3þgÞ2ð2þgÞ þ

E½y9Y1, . . . ,Yi�1,mjiðmiÞ,Yiþ1, . . . ,Yn�

ð2þgÞ

!0BB@

1CCA

2664 ðpi�wÞ9Yn

3775:ð19Þ

The solution to the FOC is given by

pi ¼að3þgÞ2ð2þgÞ þ

E½y9Yn�

2ð1þgÞ þgE½y9Y1, . . . ,Yi�1,mi,Yiþ1, . . . ,Yn�

2ð1þgÞð2þgÞ , ð20Þ

which can also be written as

pHTi þg

1n�1

Pja imjiðmiÞ�Yi

� �2ð1þgÞð2þgÞðnþsÞ

:

Substituting (20) to the pricing decision of the other retailers we

get that the sold quantities are

qj ¼ qHTj �g

ðmjiðmiÞ�YiÞ

ð1þgÞð2þgÞðnþsÞ

and

qi ¼ qHTi þg

1n�1

Pja imjiðmiÞ�Yi

� �ð1þgÞð2þgÞðnþsÞ

: &

Proof of Proposition 3. (a) During the first stage of the game,retailer i solves the following optimization problem:

maxmi

E qHTi þg

1n�1

Pja imjiðmiÞ�Yi

� �2ð1þgÞð2þgÞðnþsÞ

0@

1AðpHT

i �wHT Þ

24

þg1

n�1

Pja imjiðmiÞ�Yi

� �2ð1þgÞð2þgÞðnþsÞ

������Yi

35: ð21Þ

Note that qHTi , pi

HT and wHT are not affected by the choice of mi,

and that the problem in (21) is actually convex in 1=ðn�1ÞPja imjiðmiÞ, and hence the solution is a boundary solution. By

Lemma 3, maximizing 1=ðn�1ÞP

ja imðmiÞ results in increased

quantity and price, and thus the proposition summarizes that

retailer i chooses mi to maximize the average belief in his report.

Part (b) is a direct result of the part (a). &

Proof of Proposition 4. Assume that all retailers but retailer i

report truthfully the interval containing their signal realization.

For notational purposes assume that each retailer’s announce-

ment denotes the expected value y given the sub-interval contain-

ing his signal observation. Denote the vector of announcements

ðY1, . . . ,Yi�1,Yiþ1, . . . ,YnÞ by Y�i and the announcement of retailer i

by mi.

We will show that the retailer i’s profit is increasing in his

announcement. Define the function

HðpjÞ ¼maxpi

ðaþE½y9Y��ð1þgÞpiþgpjÞðpi�wÞ:

By the Envelope Theorem the function HðpjÞ is increasing in pj,

and hence retailer i wishes to send an announcement which

maximizes the price all other retailers set in the market. Since pj is

increasing in mi, retailer i will announce the highest possible sub-

interval. &

Proof of Lemma 4. (a) Assume all retailers ja i report truthfullyand use the message provided by their competitors to determinetheir pricing decision. In this case, during the third stage of thegame the pricing decision of the retailer ja i is given by

pj ¼ðaþE½y9Y1, . . . ,Yi�1,mi,Yiþ1, . . . ,Yn�Þð3þgÞ

2ð2þgÞ

and the pricing decision of retailer i is given by

pi ¼ ðaþE½y9YN�Þð3þgÞ

2ð2þgÞ þðmi�YiÞ

4ð2þgÞðnþsÞð1þgÞ�½ð3gþg2þð1þgÞð2þgÞ� ð22Þ

and

w¼aþE½y9YN�

2þðmi�YiÞ

2ðnþsÞ

N. Shamir / Int. J. Production Economics 136 (2012) 352–365364

and therefore the marginal profit can be expressed as

pi�w¼ pPTi �wPTþ

ðYi�miÞ

2ð2þgÞðnþsÞð1þgÞ:

(b) In a similar manner to the proof of part (a). &

Proof of Proposition 5. (a) From Lemma 4 we know that qi isdecreasing in mi and the marginal profit also decreases in mi.Therefore, during the first stage of the game retailer i maximizesthe following function by choosing mi:

maxmi

E½qiðpi�wÞ9Yi�,

where the expectation is taken over the set Y�i (i.e. the signal reali-zations of the other retailers and the value of yÞ. The FOC is given by

qi

@ðpi�wÞ

@miþðpi�wÞ

@qi

@mi:

Since qi40, pi�w40 and @ðpi�wÞ=@mio0 and @qi=@mio0 weconclude that the FOC is negative for any realization of Yi. As a resultretailer i chooses to report mi ¼ y.

(b) This is a direct result of part (a). Since retailer i chooses a

constant report regardless of his signal realization no information

exchange can be reached.

(c) The proof is similar to that of part (a). Assume all retailer

share their information truthfully and use the information pro-

vided by all other retailers to determine their retail price. In this

case, a retailer is better-off deviating and reporting the lowest

sub-interval as part (a) suggests. As a result there does not exist a

truthful equilibrium with a non-degenerated partition. &

Proof of Proposition 6. In order to reach the conclusion that theTH induces truthtelling we need to show that the interim profit ismaximized by announcing mi ¼ Yi for any Yi.

Using the results from Lemma 3 the interim expected profit of

retailer i is given by

VHðYi,miÞ ¼ E½pHT

i 9Yi�þa

2ð2þgÞ þE½E½y9YN�9Yi�

� �

� g ðmi�YiÞ

2ð2þgÞðnþsÞð1þgÞ

� �þ

gðmi�YiÞ

2ð2þgÞðnþsÞð1þgÞ

� �2

:

The first element is the interim profit of retailer i when

information is revealed truthfully, and it is independent of the

chosen message mi.

The FOC taken over VHðYi,miÞ�TH

ðmiÞ is

gð2þgÞðnþsÞð1þgÞ

a

2ð2þgÞ þE½y9Yi�

� �

þg

2ð2þgÞðnþsÞð1þgÞ

� �2

2ðmi�YiÞ

�ga

2ð2þgÞ2ðnþsÞð1þgÞ�

gmi

ð2þgÞ2ðnþsÞð1þgÞðsþ1Þ:

Note that indeed the FOC equals zero at the point mi ¼ Yi. &

Proof of Proposition 7. In order to reach the conclusion that theTP induces truthtelling we need to show that the interim profit ismaximized by announcing mi ¼ Yi for any Yi.

Using the results from Lemma 4 the interim expected profit of

retailer i is given by

VPðYi,miÞ ¼ E½pPT

i 9Yi�þa

ð2þgÞþ

Yi

ð2þgÞðsþ1Þ

� �

�Yi�mi

2ð2þgÞðnþsÞð1þgÞ

� �þ

Yi�mi

2ð2þgÞðnþsÞð1þgÞ

� �2

:

The first element is the interim profit of retailer i when

information is revealed truthfully, and it is independent of the

chosen message mi.

The FOC taken over VPðYi,miÞ�TP

ðmiÞ is

@ VPðYi,miÞ�TP

ðmiÞ

� �@mi

¼�1

2ð2þgÞðnþsÞð1þgÞ

�a

ð2þgÞþ

Yi

ð2þgÞðsþ1Þ

� ��

1

2ð2þgÞðnþsÞð1þgÞ

� �2

2ðYi�miÞ

þa

2ð2þgÞ2ðnþsÞð1þgÞþ

mi

2ð2þgÞ2ðnþsÞð1þgÞðsþ1Þ:

Note that indeed the FOC equals zero at the point mi ¼ Yi. &

Proof of Proposition 8. (a)

@TP

@m¼�

a

2ð2þgÞ2ðnþsÞð1þgÞþ

mi

2ð2þgÞ2ðnþsÞð1þgÞðsþ1Þ

" #

¼sign�

a

2ð2þgÞ þmi

2ð2þgÞðsþ1Þ

� �:

The above expression can be written as �E½qPTi 9Yi ¼mi�, i.e. the

expected sold quantity by retailer i in setting PT when he observes

the signal value of mi. By assumption (A1) the sold quantity is always

positive, and hence �E½qi9Yi ¼mi�o0, which completes the proof.

(b)

@TH

@m¼

ga

2ð2þgÞ2ðnþsÞð1þgÞþ

gmi

ð2þgÞ2ðnþsÞð1þgÞðsþ1Þ

¼sign a

2ð2þgÞ þmi

ð2þgÞðsþ1Þ:

The above expression can be written as E½qHTi 9Yi ¼mi�, i.e. the

expected quantity sold by retailer i in setting HT. By assumption

(A1) this is positive. &

Proof of Proposition 9. (a)

@TP

@n¼�

1

ðsþnÞTP o0:

(b)

@TH

@n¼�

1

ðsþnÞTH o0: &

Proof of Proposition 10. (a)

@TH

@y¼

sign�

a

2ð2þgÞþ

yð2þgÞðsþ1Þ

� �:

Note that the term in the square brackets is the expected soldquantity when observing the signal realization y in the setting HT.Using assumption (A1) this term is positive, and hence thederivative is negative.

(b)

@TP

@y¼

sign a

2ð2þgÞþ

y2ð2þgÞðsþ1Þ

:

This term is the expected sold quantity of a retailer observing

signal realization y. By assumption (A1) this term is positive. &

Proof of Proposition 11. (a) After some simple algebraic manip-ulation @TP=@g can be written as

�gð2þgÞð1þgÞ

TP o0:

N. Shamir / Int. J. Production Economics 136 (2012) 352–365 365

(b) TH can be written as

gð2þgÞ2ð1þgÞ

aðmi�yÞ2ðsþnÞ

þm2

i �y2

2ðsþnÞðsþ1Þ

" #:

The term in the square brackets is positive and independent of g.

Therefore, the sign of the derivative depends only on the term

g=ð2þgÞ2ð1þgÞ. Taking derivative results in 4�5g2�2g3. Define

gn40 as the root of the equation 4�5g2�2g3 ¼ 0. When the

competition between the retailers is moderate ðgognÞ, an

increase in the competition rate results in increased signaling

costs, and when the competition is intense ðg4gnÞ the signaling

cost is decreasing in the competition rate. &

Proof of Proposition 12.

E½TH� ¼

a

2ð2þgÞ2ðsþnÞð1þgÞg9y9

þg

2ð2þgÞ2ðsþnÞðsþ1Þð1þgÞðs2ðsþ1Þþy2

Þ,

E½TPi � ¼

a

2ð2þgÞ2ð1þgÞðsþnÞy

þ1

4ðsþnÞðsþ1Þð2þgÞ2ð1þgÞðs2ðsþ1Þþy

2Þ:

By carefully choosing y and y we can create a scenario in which

the increased signaling cost of the horizontal scenario is higher

than the signaling cost in the public setting and the cost of

sharing this information with the manufacturer. &

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