pricing in a supply chain for auction bidding under information asymmetry

European Journal of Operational Research xxx (2014) xxx–xxx

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European Journal of Operational Research

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Production, Manufacturing and Logistics

Pricing in a supply chain for auction bidding under informationasymmetry

http://dx.doi.org/10.1016/j.ejor.2014.02.0510377-2217/� 2014 Elsevier B.V. All rights reserved.

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Please cite this article in press as: Lorentziadis, P. L. Pricing in a supply chain for auction bidding under information asymmetry. European Journal oftional Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.051

Panos L. Lorentziadis ⇑Athens University of Economics and Business, Greece

a r t i c l e i n f o

Article history:Received 6 August 2013Accepted 24 February 2014Available online xxxx

Keywords:Auctions/biddingSupply chain managementEquilibrium mark-upInformation asymmetryDouble marginalizationInformation sharing

a b s t r a c t

We examine a supply chain in which a manufacturer participates in a sealed-bid lowest price procure-ment auction through a distributor. This form of supply chain is common when a manufacturer is activein an overseas market without establishing a local subsidiary. To gain a strategic advantage in the divisionof profit, the manufacturer and distributor may intentionally conceal information about the underlyingcost distribution of the competition. In this environment of information asymmetry, we determine theequilibrium mark-up, the ex-ante expected mark-up and expected profit of the manufacturer and theequilibrium bid of the distributor. In unilateral communication, we demonstrate the informed agent’sadvantage resulting to higher mark-up. Under information sharing, we show that profit is equally sharedamong the supply chain partners and we explicitly derive the mark-up when the underlying cost distri-bution is uniform in [0,1]. The model and findings are illustrated by a numerical example.

� 2014 Elsevier B.V. All rights reserved.

Introduction

Supply chain competition is increasingly replacing the firmversus firm framework of market competition (Ha & Tong, 2008).Ideally, partners in a supply chain should cooperate and proceedtogether with coordinated decisions against competing supplychains (Gumus & Guneri, 2007). In a manufacturer–retailerscheme, the two agents will have to determine the inventory level,the order quantity, the transfer sales price and the market price,ensuring profit maximization (Hu, Beil, & Duenyas, 2013; Lovejoy,2010). Typically, the market demand function involves uncertainparameters and the supply chain partners have asymmetric infor-mation about the possible states of the demand (Ha & Tong, 2008).

We extend this framework examining the purchase of a productor service for which there is no active market and, therefore, theend-user initiates a procurement auction. When the quantity of asupply is large, even if the product is sold in the retail market ona retail, the procurer may run an auction mechanism to purchasethe object at a bid price, which is lower than the retail price. Publictenders or bidding in a Request for Quotation are common exam-ples of such auctions. We consider a procurement mechanism ofa sealed-bid auction, where the awarding criterion is the lowestbid. This form of auction is commonly used in public procurementcontracts both in the EU (European Parliament and Council

Directive, 2004) and the U.S. (Federal Acquisition Regulation,2005).

In our setting, the upstream agent of the supply chain is amanufacturer, who collaborates with a downstream distributor.For example, the Austrian firm Rosenbauer International AG, aleading fire fighting vehicle and equipment manufacturer, hasestablished a worldwide network of distributors and provides itscustomer support ‘‘via a global sales and services organization withthe cooperation of selected partners’’ (Rosenbauer Group website).Clearly, the vast majority of state and municipal procurement forthese products is through a tender process.

We examine the environment in which the manufacturer isunwilling to participate directly to the procurement auction.Instead, the manufacturer prefers to supply the auctioned item tothe distributor, who eventually bids in the auction. There are manyreasons for which a manufacturer may decide to sell through a dis-tributor. In many cases, the manufacturer is located overseas andhe is unfamiliar with the local market, which may present variousbarriers for entry. The terms of the supply often require that thesupplier should provide maintenance and repair service, whichthe manufacturer is unable to offer to the end user without thecooperation of a local firm. Moreover, the manufacturer mayevaluate the establishment of a local subsidiary company asunprofitable, since the local customer pool is usually limited andthe outcome of the particular procurement auction remains uncer-tain, whereas the investment and the corresponding fixed cost arein most cases considerably high. This supply chain framework isconsistent with the estimate that 60% of US companies and 70%

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2 P.L. Lorentziadis / European Journal of Operational Research xxx (2014) xxx–xxx

of European companies use foreign-based agents or distributors toexport their products (Jobber, 2010). The appointment of localdistributors, who will promote a product and penetrate the localmarket, appears to be preferable when the profit is expected tobe low or the risk involved is high (Arnold, 2000). In general,manufacturers may decide to make their products available toend-users using distributor channels because of the expertise andcapacity of the distributors in the promotion, sale and technicalsupport of the corresponding products.

In public procurement, only authorized distributors or manu-facturers are usually allowed to participate in a supply or serviceauction. For example, the US government typically requires fromdistributors, who want to become suppliers to the General ServicesAdministration (GSA), to provide a letter from the correspondingmanufacturer which certifies their status as authorized distribu-tors (Letter of Supply in US GSA website). Moreover, manufacturershave to provide confirmation that, if their authorized distributor isawarded the procurement contract, they will properly supply tothe distributor the auctioned item. Authorized distributors arecarefully selected on the basis of geographic location or withregard to product or market segmentation. In general, the partnermanufacturer and the authorized distributor are bound by a formalcontractual agreement, which extends beyond the scope of aparticular auction project. Therefore, in a procurement auction,the agents of the supply chain have established a long lasting col-laboration and, for this reason, the respective manufacturers willtypically refrain from cooperating with other bidders.

Moreover, participants in public procurement auctions are usu-ally not allowed to propose variants to their tenders. For example,EU regulations specifically prohibit variant offers, when the crite-rion of awarding a supply or service contract is the lowest price(European Parliament and Council Directive, 2004; Guide to theCommunity Rules on Public Supply Contracts, Guide to the Com-munity Rules on Public Service Contracts). Consequently, in publicprocurement, distributors are legally bound to present only oneproposal coming from a single manufacturer. Further, most manu-facturers expect from their foreign authorized distributors not tosell competing products (Ministry of Economic Development andTrade, Canada). Therefore, it is reasonable to assume that in theprocurement auction environment that we examine, a distributorwill choose to cooperate with a single upstream agent.

We embed our supply chain model in a setting of informationasymmetry. In a manufacturer–retailer supply chain with down-stream market competition instead of an auction, the asymmetryof information is reflected in the representation of the market de-mand curve and the unit cost of the retailer. Typically, in the liter-ature, the market demand function includes random parametersand, as a result, the supply chain partners have asymmetric infor-mation about the possible states of the demand. For example, Haand Tong (2008) consider a manufacturer who does not observethe intercept of a linear demand curve, although this parameteris known to the retailer. Wang, Lau, and Ling (2008) considered amanufacturer–retailer supply chain, where the downstream agentsells in an active market, which is described by a linear demandcurve, instead of participating in a downstream auction. Themanufacturer has incomplete knowledge about the stochasticparameters of a linear demand function and the retailer’s cost,whereas the retailer is fully aware about the form of the demandcurve. Wang et al. (2008) showed that in this setting it is preferablefor the distributor to withhold or distort information and increasethe manufacturer’s uncertainty about the parameters of thedemand. We pose a similar question in the environment of a down-stream procurement auction and we investigate how the twoagents interact by exchanging information.

We model supply chain information asymmetry in an innova-tive way through the underlying distribution of the cost of the

Please cite this article in press as: Lorentziadis, P. L. Pricing in a supply chain fortional Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.051

competition. The probability distribution of cost plays a crucial rolein the determination of the bid price and, consequently, thetransfer price. The manufacturer, who is active in internationalmarkets, usually has a more comprehensive and global view ofthe competition. In contrast, the distributor possesses more accu-rate information about the local market competition. Therefore,each supply chain partner estimates differently the probability dis-tribution of the cost of competing bidding supply chains.

Collaborative partnership in the supply chain requires trust andcommitment for long term cooperation (Kiefer & Carter, 2009). Ingeneral, however, within a manufacturer–distributor supply chainand under conditions of unequal power, the party which possessesthe greater relative power tends to behave exploitatively towardsthe less powerful agent (Ghauri & Usunier, 2003). Our analysis con-tributes in the understanding of this behavior. In our setting,although both supply chain partners share the same goal, namelyto win in the auction, there is an adversarial relationship on howthe profit made will be divided among them. Consequently, theagents may decide to avoid an open and truthful exchange of infor-mation between them. Intuitively, the motive of each partner is togain a strategic advantage, which will allow a higher mark-up forthe particular agent while maintaining a competitive bid. Thesource of this conflict of interests may be traced to the dual roleof the distributor, who simultaneously acts as a purchasing agentfor the manufacturer and as a selling agent for the downstreamcustomer (Jobber, 2010).

In our framework of information asymmetry, we examine therole of communication in the determination of the mark-up of eachagent and the division of the total potential profit. We explore theinteraction among agents in an auction with intermediaries, withheterogeneous beliefs and limited communication across the sup-ply chain. To the best of our knowledge, this supply chain structurehas not been previously examined in the existing literature of auc-tion theory. Initially, we consider the case of limited exchange ofinformation within the two partners, who do not disclose theirknowledge and beliefs about the competition. When the upstreamagent believes that the competition is less intense compared tohow the downstream player expects his opponents to behave, weestablish analytically that the manufacturer is motivated to mis-lead the distributor about the level of competition, as describedby the cost probability distribution. The distributor acts in a similarway, when the manufacturer engages in one-way upstream com-munication. Unilateral upstream or downstream exchange of infor-mation, where one agent reveals to the other one the underlyingcost probability distribution, provides an information advantageto the informed party, which we formally derive and express inthe form of a higher mark-up. In information sharing, which occursunder an open, complete and truthful upstream and downstreamexchange of information, both partners share the same view onthe underlying probability distribution of the cost of the competi-tion and profit is equally divided.

Our results are useful to supply chain management, explainingthe fragility of partnership with respect to the division of profits,generated by information asymmetry in an auction setting. Ourwork raises the need to design contractual agreements withinthe supply chain, which induce truthful information sharing andequal allocation of profits.

The rest of the paper is organized as follows. A review of theexisting literature is examined in Section ‘Literature review’. Themodel description is formally presented and the assumptionsmade are explicitly stated in Section ‘The model’. In Section ‘Equi-librium mark-up and bidding strategies’, under information asym-metry within the supply chain, we derive the equilibrium mark-upfor the manufacturer and the equilibrium bid for the distributor,who participates in the downstream auction. Further, inSection ‘Upstream one-way communication in the supply chain’,

auction bidding under information asymmetry. European Journal of Opera-


P.L. Lorentziadis / European Journal of Operational Research xxx (2014) xxx–xxx 3

we examine the supply chain with upstream communication andwe analytically establish the manufacturer’s advantage over thedistributor in the division of profits. In Section ‘Downstream one-way communication in the supply chain’, we look at the symmetriccase of downstream one-way communication, and we find that thedistributor enjoys an advantage in terms of the mark-up. In Sec-tion ‘Information sharing across the supply chain’, we show that,with information-sharing, the profit is equally shared among bothsupply chain partners. Moreover, when the underlying cost distri-bution is the uniform in [0,1] distribution, and there is informationsharing, we derive explicitly the equilibrium mark-up and weinvestigate the effect of double marginalization on the bid price.In Section ‘Expected mark-up and profit of the manufacturer’, wepresent an expression of the expected ex ante mark-up and profitof the manufacturer, under the beliefs of the manufacturer. Sec-tion ‘Numerical illustration of the model’ illustrates the modeland discusses the findings by a numerical example. A discussionwith conclusions can be found in Section ‘Discussion – Conclusion’.All proofs are deferred to Appendix A.

Literature review

The seminal paper of Maskin and Riley (2000) introduced asym-metry in auctions of two participants, by assuming that the probabil-ity distribution of one bidder’s valuation stochastically dominatesthe respective probability distribution of the other player. Intui-tively, one bidder tends to consider ‘‘higher’’ values compared tohis opponent. This heterogenous environment has been extendedby Kirkegaard (2012) to account for other forms of stochastic domi-nance. In our setting, although auction bidders are assumed to besymmetric, asymmetry rests within the bidding supply chain itself.The asymmetric beliefs of the supply chain agents of our model arerepresented in a setting of stochastic dominance among the respec-tive probability distributions of the cost of the competition.

When the winning bidder of an auction is not the player withthe highest valuation, a resale process, after the auction, leads tonew profitable trade opportunities. In this two-stage framework,the auction winner acts as an intermediary agent. Therefore, theallocation of the auctioned object generates double marginaliza-tion. In the symmetric model of Haile (2003), the auction winnerruns a new second price auction to resell the object to his oppo-nents. Hafalir and Krishna (2008) constructed equilibrium biddingstrategies in a two bidder asymmetric first price auction with val-uations which are stochastically ranked, whereas the resale stage isin the form of take-or-leave-it offer. Conditions for the design ofseller optimal auctions with resale were developed by Zheng(2002). Bikhchandani and Huang (1989) developed a model ofcompetitive bidding in Treasury-Bill primary auctions, in whichbidders participate for the purpose of resale in a secondary market.In our model, conditionally on being the auction winner, thedistributor supplies to the auctioneer the object that has been pre-viously purchased from the manufacturer. Therefore, our settingcan be viewed as a resale auction in a procurement environment,where the trade following the auction is made among agentsalready committed to the resale.

Auctions often appear in different supply chain schemes. Theauction is usually placed upstream in the supply chain and anactive market exists downstream. The price information disclosedby the auction has an impact on the downstream demand. Chenand Vulcano (2009) examined a supply chain where an upstreamsupplier – for example, a manufacturer – auctions to two biddershis inventory or capacity, which is viewed as bundle. Additionalunits can be purchased at a fixed unit cost. Subsequently, thetwo players act as downstream resellers competing in an activemarket, in which the demand curve is described by a linear func-tion with random intercept. Mokherjee and Tsumagari (2004)


examined a supply chain network with double marginalization,where two suppliers deliver a quantity of input to a principal,while subcontracting and collusion is possible among the agents.The auction in our setting is positioned downstream and the com-petition is, in fact, among bidding supply chains.

In the environment of online auctions, ad agencies act as inter-mediaries in an auction for online display advertisements produc-ing a double marginalization. Each intermediary first runs acontingent auction selling the display to advertisers. Subsequently,based on the contingent price received, the intermediaries bidamong themselves in a downstream central auction for the pur-chase of an ad slot impression. Feldman, Mirronki, Muthukrishnan,and Pai (2010) considered the case when there is only one buyerfor each intermediary and the central auction has the form of asecond price auction. In this setting, they found that offering thedisplay to the buyer at a take-it-or-leave-it price is a dominantstrategy. This transfer price serves as the bid price of the interme-diary at the central auction. At a symmetric equilibrium, the inter-mediary should select the price level randomly within an intervalwhich depends on the reserve price of the central auction. Whenthere are multiple buyers for each intermediary, the authors con-jecture that the optimal mechanism for the contingent auctionhas the form of a second price auction. Although, the model ofFeldman et al. (2010) and our procurement auction both examinea downstream auction in the presence of intermediaries, thereare two profound differences that should be stressed. First, Feld-man et al. (2010) considered a second price downstream auctionin which it is a dominant strategy for the players to bid their value.Our setting is a sealed bid lowest price auction, in which biddersadd a mark-up on their cost. Therefore, in our environment, theasymmetry of information about the competition creates animportant interaction among agents in the determination of themark-up of each agent, which is the focus of our analysis. The sec-ond distinctive difference stems from the role of the intermediaryparty. In Feldman et al. (2010), it is the intermediary who deter-mines the transfer price in a take-it-or-leave-it offer to the buyer,while the transfer price is also the bid price. In our setting, how-ever, it is the upstream agent (the manufacturer) who sets thetransfer price, whereas the intermediary (the distributor) addsthe final mark-up to determine the bid price. Therefore, our modeldescribes a fundamentally different auction environment, whichposes completely new and challenging questions.

The model

Problem formulation and notation

We consider a supply chain where an auctioneer wants to pur-chase an indivisible item, which a manufacturer will supplythrough a distributor. The auction environment is an independentprivate value sealed bid auction. Each bidder knows precisely hisown cost and assumes that the cost values of his opponents areindependent and identically distributed random variables. In thesymmetric setting that we consider, all players believe that theunderlying cost distribution is the same for all auction participants.The auctioned item is awarded to the bidder who sells at the low-est price. This type of auction is widely used in public procurementand it is also common in industrial requests for quotations (RFQ).In terms of monetary payoffs, all bidders are risk-neutral sellers.The auctioned item can be either a product or a service.

The manufacturer incurs a cost c to provide the item to thedistributor. This cost is perfectly known to the manufacturer butremains unknown to the distributor. The manufacturer determinesthe transfer price, at which the item is made available to the dis-tributor, by adding an appropriate mark-up m to the cost c. Hence,the cost of the distributor is m + c.




Downstream the supply chain, the distributor participates inthe procurement auction, submitting a sealed bid b. It is assumedthat no cost is born by either the manufacturer or the distributorby participating in the auction. Therefore, any cost due to the prep-aration of the offers, or a possible submission of sample productsor a product presentation, or any financial expenses, as for exam-ple, for issuing a letter of bank-guarantee for participation to theauction, is regarded nonexistent or negligible. The distributor bidsagainst n opponents in the auction. It is assumed that n P 2. Themanufacturer and the distributor have both decided to co-operatein this procurement project.

The symmetric equilibrium bid b is assumed to be an increasingfunction of the cost of each bidder. The distributor sets the bidprice assuming that the cost of each one his opponents is anindependent and identically distributed random variable, whichfollows a cumulative probability distribution U with support inthe interval [0,x] and having a continuous density u. When thebid is b = b(c + m), the probability of winning the auction isFðb�1ðbÞÞ ¼ Unðb�1ðbÞÞ, where U ¼ 1�U The hazard rate associ-ated with the distribution U is given by rU = u/U. The reversehazard rate associated with F is denoted by kF = f/F, where f is thederivative of F.

Following a game-theoretic approach, the distributor selects thebid price b, which maximizes his expected profit: P(b,c + m|F) = (b � c �m)F(b�1(b)).

The reserve price, above which no sale takes place, is denotedby K. Following Krisna (2010), at the reserve price K, we haveb(K) = K. For simplicity, we will assume that K = x.

The distributor does not know the actual cost c of the manufac-turer and observes only the non-negotiable transfer price c + m,which constitutes his cost. Moreover, the distributor believes thatthe probability distribution of the underlying cost of the competi-tion is best described by U. The distributor will determine the bidprice by applying a mark-up l to the transfer price, so thatb = c + m + l.

Upstream the supply chain, the manufacturer assumes that allauction participants, including the distributor, select their bids ina game theoretic manner. Moreover, he believes that the cost ofeach opponent of the distributor follows a cumulative probabilitydistribution W with support in the interval [0,x] and density w.The random variables of the cost of each player are considered tobe independent. Although the manufacturer and the distributorboth share some information about the bid prices of the previoussupplies in the local and the international market, their beliefsabout the cost of the competition do not coincide. The manufac-turer, who is typically active in overseas markets, usually has moreextensive and more recent knowledge about the level of competi-tion for the auctioned item in the international environment. Themanufacturer is unwilling to disclose to the distributor the contentof this information and he may even deliberately refrain from pro-viding information to the distributor. The motive of the upstreamagent is to falsely create the impression that competition is tough-er in an effort to force a lower mark-up l in the final bid of thedistributor. Consequently, the distributor remains unaware of theprobability distribution W considered by the manufacturer. In gen-eral, under information asymmetry, the probability distribution Wwill be different from U.

At the same time, the distributor may also be unwilling to com-municate to the manufacturer accurate or up-to-date informationabout the competition, which shows up in the local market. Itmay also happen that the distributor intentionally overstates thelevel of competition in an attempt to lower the transfer price,which will either lead to a more competitive bid or allow a highermargin for the distributor. As a result, the manufacturer incorrectlybelieves that the distributor determines the bid price consideringan underlying cost probability distribution C rather than U.


In the special case of upstream one-way exchange of informa-tion, the distributor reveals to the manufacturer the underlyingdistribution U, and, therefore, U = C. In downstream unilateralcommunication, the manufacturer informs the distributor aboutW and believes that the downstream agent will employ C = W.When the manufacturer and the distributor engage in perfect, openand truthful information sharing, both partners agree about theunderlying cost probability distribution of the competition. There-fore, under information sharing, W = U = C and the agents of thesupply chain achieve a perfect alignment of incentives and goalconvergence.

Suppose that the bid price is b = b(c + m). According to the man-ufacturer, the probability that he will make a profit m isGðb�1ðbÞÞ ¼ Wnðb�1ðbÞÞ, where W ¼ 1�W. Similarly to the previousnotation, the hazard rate of the distribution W is rW ¼ w=W and thereverse hazard rate associated with G is kG = g/G, where g denotesthe derivative of G. In similar notation, we write C ¼ 1� C,H ¼ Cn;rC ¼ c=C and kH = h/H, where h is the derivative of H.

The expected profit of the manufacturer is: p(m, b, c|G,H) = mG(b�1(b)). The manufacturer chooses the mark-up m whichmaximizes p(m, b, c|G, H), under the condition that the distributorselects the optimal bid b. We summarize the notation used inTable 1.

Fig. 1 presents the sequence of decisions and events, depictingthe interaction of the supply chain agents.

Model assumptions

We proceed to present in detail the assumptions, which areneeded to derive the equilibrium pricing strategy of the supplychain agents. In general, the manufacturer has the opportunity tobe involved in a large number of procurement auctions of the sameobject, which take place in the international market. Therefore, themanufacturer has recent information about the level of competi-tion coming from a large pool of various auctions for the procure-ment of the same or similar items. Moreover, he is fully aware ofthe exact cost structure for producing the item and, consequently,he can accurately factor in the bid prices of the competition theexpected adjustment of different cost components accounting forfactors such as the change of the price of raw materials or labor.On the other hand, the distributor does not have access to thisinformation and possesses data only about the competition inthe limited number of auctions that have occurred in the local mar-ket. It is often the case, due to the nature of the item, which is pro-cured, that such local auctions are infrequent and past data are oldand less reliable. At the same time, the manufacturer is also awareof the data from these local procurement auctions, since he mostlikely participated in all of these auctions in the past, possiblythrough the same distributor. For this reason, the manufactureris confident, perhaps erroneously, that his estimate of the probabil-ity distribution of the underlying cost of the competition is moreaccurate than the corresponding estimate, which is available tothe local distributor. Therefore, according to the manufacturer’spoint of view, the distributor incorrectly believes that the probabil-ity distribution of the underlying cost of his opponents is C, whilethe actual probability distribution is W. Never-the-less, the distrib-utor actually estimates the corresponding probability distributionby U rather than C, and it is possible that, eventually, U may bea more precise description of the true probability distribution ofthe cost of the competition.

Suppose that the manufacturer does not engage in a perfect,open and truthful exchange of information with the distributorregarding the level of competition, and, therefore, C – W. Themotive of the manufacturer is to mislead the distributor, in an ef-fort to make him believe that competition is more aggressive. Inthis way, the manufacturer forces the distributor to apply a lower



Table 1Notation key.

Symbol

U Probability distribution of underlying cost of competition as considered by the distributorU 1 �U

u Density of UF Un

f Derivative of FrU Hazard rate of U: u=UkF Reverse hazard rate of F: f/FW Probability distribution of underlying cost of competition as considered by the manufacturerW 1 �W

w Density of WG Wn

g Derivative of GrW Hazard rate of W: w=WkG Reverse hazard rate of G: g/GC Probability distribution of underlying cost of competition that the manufacturer believes that the distributor considers in the determination of the

bidC 1 � C

c Density of CH Cn

h Derivative of HrC Hazard rate of C: c=CkH Reverse hazard rate of H: h/Hc Cost of the manufacturerm Mark-up of the manufacturern Number of opponent biddersb Bid priceK Reserve price of the auctionl Mark-up of the distributor: b � c �mp(m � b � c|G � H) Expected profit of the manufacturerP(b � c + m|F) Expected profit of the distributor


mark-up in the bid price, producing a more competitive offer. Themanufacturer will pursue this route, only if the incorrect probabil-ity distribution C generates lower cost values for the distributor’sopponents compared to distribution W. Consequently, it is reason-able to assume that W should dominate C in some form of stochas-tic ordering, since otherwise the manufacturer will choose tocommunicate to the distributor that W rather than C should beemployed. We make this assumption explicit and we further re-quire that the underlying densities are continuously differentiable.

Assumption 1.

(A) For all x < y wðxÞcðxÞ

1�CðxÞ1�wðxÞ 6

wðyÞcðyÞ

1�CðyÞ1�wðyÞ.

(B) The densities w and c are both continuously differentiable.

Stochastic dominance has been widely used in auction theory toreflect asymmetry among players (Maskin & Riley, 2000). The sto-chastic ranking of the underlying distributions captures the asym-metry of beliefs among the supply chain agents. In this respect,Assumption 1(A) states that distribution W dominates distributionC in terms of the ratio of hazard rates. We explore further the con-nection of this form of stochastic dominance to other more stan-dard types of stochastic ordering (Shaked & Shanthikumar, 2010).In particular, we show that our assumptions results to dominancein hazard rate and first-order stochastic dominance.

Proposition 1. Under Assumption 1,

(a) Distribution W dominates distribution C in terms of the hazardrate:

Pleasetional

wðxÞ1�WðxÞ 6

cðxÞ1� CðxÞ for all x in ð0;xÞ:

cite this article in press as: Lorentziadis, P. L. Pricing in a supply chain forResearch (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.051

(b) Distribution W first-order stochastically dominates distributionC: W(x) 6 C(x) for all x in (0,x).

The stochastic monotonicity of W over C of Assumption 1(A)produces a form of stochastic ordering for the functions G and H,in terms of the monotonicity of the ratio of hazard rates kG/kH.

Proposition 2. Under Assumption 1(A),

(a) kGkHðxÞ 6 kG

kHðyÞ; for all x < y.

(b) For all x in (0,x), kG(x) P kH(x) and G(x) 6 H(x).

Corollary 1. Under Assumption 1(A), for all x in (0,x), kGðxÞkHðxÞðxÞ 6 1.

We will also need one additional assumption, which capturesthe intuitive fact that when the cost is high, bidders should lowertheir bid price.

Assumption 2. The function JðxÞð1�CðxÞÞn is non-increasing in x, where

JðxÞ ¼Rx

x ð1� CðyÞÞndy:

Assumption 2 is equivalent to the statement that: I(x|H)/H(x) = J(x)/(1 � C(x))n is a non-increasing function, whereIðxjHÞ ¼

Rxx HðyÞdy We further note that, as it will be shown in The-

orem 1, below, the quantity I(x|H)/H(x) is the mark-up applied bythe distributor in the formulation of the bid, when the underlyingcost distribution is H. Therefore, the intuition behind Assumption 2is that when the cost of a bidder increases, the correspondingmark-up must decrease, in an effort to keep the bid competitive.

Assumptions 1 and 2 are sufficient to provide the findings of ouranalysis. Never-the-less, we may derive the same results with thefollowing weaker assumption.



Fig. 1. Time sequence of decisions and events.


Assumption 3. The densities w and c are both continuously differ-entiable and the distribution functions W and C satisfy the followingcondition

w0ðxÞwðxÞ �

c0ðxÞcðxÞ P ðnþ 1ÞrCðxÞ � rWðxÞ þ

JðxÞð1� CðxÞÞn

ð1Þ

for all x in (0,x), where JðxÞ ¼Rx

x ð1� CðyÞÞndy.Condition (1) can be expressed equivalently, in terms of the

functions G and H.

Proposition 3. Condition (1) is equivalent to the condition:

g0ðxÞgðxÞ �

h0ðxÞhðxÞ P ðkGðxÞ � kHðxÞÞ �

HðxÞðIðxjHÞÞ þ kHðxÞ� �

ð2Þ

for all x in (0,x), where IðxjHÞ ¼Rx

x HðyÞdy.


Assumption 3 can also be restated in an equivalent manner,which allows us another interpretation of condition (1). For anyfunction n(x), we denote by e(n(x)) the elasticity of the function n.

Proposition 4. Condition (1) is equivalent to each one of thefollowing conditions:

(i) e gh ðxÞ� �

P e GH ðxÞ� �

þ e eHH ðxÞ� �

;

for all x in (0,x), where eHðxÞ ¼ �IðxjHÞ.

(ii) For x < y; kGkHðxÞ H

I ðxÞPkGkHðyÞ H

I ðyÞ; where IðxÞ ¼ IðxjHÞ:




It is interesting to note that eHðxÞ is the anti-derivative of H(x).The inequality in the second part of Proposition 4 involves the ratioof the reverse hazard rates kG/kH and should be compared with themonotonicity of the ratio of reverse hazard rates established inProposition 2. Assumptions 1 and 2 are sufficient to ensure thevalidity of Assumption 3.

Proposition 5. Assumptions 1 and 2 imply Assumption 3.

Equilibrium mark-up and bidding strategies

In this section we establish the symmetric equilibrium bid ofthe distributor and the symmetric equilibrium mark-up of themanufacturer.

Theorem 1. Under Assumption 3, the symmetric equilibrium bid ofthe distributor b and the equilibrium optimal mark-up of themanufacturer m are determined by the following equations:

b ¼ bðc þmjFÞ ¼ c þmþ Iðc þmjFÞFðc þmÞ ð3Þ

and mkGðc þmÞ ¼ kHðc þmÞ Iðc þmjHÞHðc þmÞ ð4Þ

where Iðc þmjFÞ ¼Rx

cþm FðxÞdx and Iðc þmjHÞ ¼Rx

cþm HðxÞdx. More-over, for all c in [0,x],

�1 6 m0ðcÞ 6 0 ð5ÞIn view of Eq. (3), when the underlying probability distribution

of the cost of the competition is U and F = (1 �U)n, the mark-up ofthe distributor is l(c + m, F) = I(c + m|F)/F(c + m). Upstream, themanufacturer’s mark-up m is given by the distributor’s mark-upl(c + m, H) under H, inflated by the factor kH/kG. By Corollary 1, itis known that the factor kH/kG exceeds 1. We deduce that themark-up l(c + m, H), which the downstream agent applies whenacting consistently to the belief of the manufacturer and he em-ploys C, is lower compared to the mark-up m of the upstreamagent. Never-the-less, the actual mark-up of the distributor is infact l(c + m, F), rather than l(c + m, H). Hence, to accurately evalu-ate the division of profit between the two agents, the comparisonshould be made between m and l(c + m, F).

Further, the upstream agent should evaluate the loss of compet-itiveness of the two agent supply chain in comparison to the casewhere the manufacturer participates directly in the auction with-out the involvement of any intermediary, by examiningh = m + l(c + m, F) � l(c, G). In particular, h expresses the differenceof mark-up when bidding directly against the scheme of participat-ing indirectly through a distributor. Due to double marginalization,the sign of h will, in general, be positive. A low value of h impliesthat the loss in the overall competitiveness of the final bid, dueto the involvement of intermediary parties, is low.

Eq. (5) suggests that the manufacturer is sensitive to any in-crease of the cost and will try to decrease the mark-up m appliedto the transfer price. However, this decrease of the mark-up m will,in general, be less than the corresponding increase of the cost, and,as a result, the transfer price c + m will necessarily become higher.

Since Assumptions 1 and 2 imply Assumption 3, we have alsoestablished the following result:

Corollary 2. Theorem 1 is valid under Assumptions 1 and 2.We can also establish an upper bound for the equilibrium mark-

up m of the manufacturer.

Proposition 6. Under Assumptions 1 and 2,


m 6 � 1kGðc þmÞ ð6Þ

To interpret this result, we note that:

l0ðc;GÞ¼ ddcðIðcþGÞÞ

GðcÞ

� �¼�G2ðcÞ� IðcþGÞgðcÞ

G2ðcÞ¼�1�kGðcÞlðc;GÞ

and, therefore, kG(c + m) = �1 � l0(c + m, G). Then, we may writeinequality (6) in the form:

m 6lðc þm;GÞ

l0ðc þm;GÞ þ 1

Suppose that m remains fixed. We derive equivalently:ddc ðlðc þm;GÞ þmþ cÞ 6 lðcþm;GÞ

m .Next, we observe that l(c + m, G) + m + c is the bid price

b(c + m|G), when the probability distribution of the underlying costof the competition is in accordance to the belief of the manufac-turer and the transfer price is c + m. Consequently, we concludethat inequality (6), when keeping m fixed, takes the form:

ddc

bðc þmjGÞ 6 lðc þm;GÞm

ð7Þ

The interpretation of inequality (7) is that, when m remains fixedand the bidding strategy is consistent with the perception of themanufacturer about the competition, the rate of change of the bidprice in terms of the cost c is at most as high as the proportion ofthe distributor’s mark-up l(c + m, G) with respect to the manufac-turer’s mark-up m. Therefore, with m fixed, when the cost c in-creases by one monetary unit, the increase in the bid price doesnot exceed the distributor’s mark-up of the bid price on a per mon-etary unit of m basis.

Upstream one-way communication in the supply chain

We examine the case where the distributor accurately andtruthfully communicates to the manufacturer all the informationthat he has about the competition in the local market. In thisframework, the manufacturer becomes fully aware of the availableinformation and the method of analysis that the distributor appliesfor the determination of the probability distribution of the cost ofthe other auction participants. Consequently, the manufacturerknows precisely the probability distribution U, which the distribu-tor employs to determine the bid price. This unilateral upstreamcommunication corresponds, in our previous notation, to the caseC = U. Theorem 1 provides the following result about the equilib-rium margin of the manufacturer and the corresponding biddingstrategy.

Proposition 7. Under Assumption 3, in unilateral upstream com-munication, the equilibrium manufacturer’s mark-up m and distrib-utor’s bid b are given by

m ¼ kFðc þmÞkGðc þmÞ

Iðc þmjFÞFðc þmÞ ð8Þ

b ¼ c þ 1þ kFðc þmÞkGðc þmÞ

� �Iðc þmjFÞFðc þmÞ ð9Þ

The mark-up applied by the distributor in the determination ofthe bid b is l = b � c �m. Based on Proposition 7, we can directlycompare the mark-up m of the manufacturer with the mark-up lof the distributor.

Corollary 3. Under Assumption 3, when there is unilateral upstreamcommunication, m P l.




Corollary 3 states that the manufacturer applies a higher mark-up compared to the distributor. The reason for the advantageousposition of the upstream agent can be traced in Assumption 1.The manufacturer, although aware of the distribution U that thedistributor considers when determining the bid, knows that theactual probability distribution of the competition is W. Never-the-less, the manufacturer does not inform the distributor aboutthe probability distribution of the cost of the competition. FromProposition 1, W stochastically dominates C, which coincides withU. Therefore, the manufacturer leaves the distributor under thefalse impression that the cost for the other auction participantswill be lower in a stochastic ordering sense. Consequently, themanufacturer misleads the downstream agent to apply a lowermark-up l, which will ensure that the final bid price is competi-tive. At the same time, in light of this fact and knowing that thecost distribution of the competition is actually less aggressive,the manufacturer will set a higher mark-up m in the formulationof the transfer price. In the end, the upstream unilateral communi-cation generates a monetary benefit in the division of profit for theupstream supplier.

Downstream one-way communication in the supply chain

In unilateral downstream communication, the manufacturertruthfully communicates to the distributor his estimate W aboutthe probability distribution of the cost of the competition. Themanufacturer believes that the downstream agent will adopt Wand, as a result, he feels confident that the C coincides with W.Although the distributor is fully aware of W, he is not obliged to ac-cept W as the underlying probability distribution. In one-waydownstream communication, the distributor does not reveal hisown estimate U. We distinguish two cases. In the first case, themanufacturer’s estimate W is fully adopted by the distributor,and, consequently, W = U. In this setting the supply chain takesthe same form as under information sharing, which will be treatedin the next section. We now examine the setting of the second case,where the distributor decides to employ U rather than W, andW – U.

From Eq. (4) and by setting W = C, it is immediate that the man-ufacturer will apply a mark-up m given by:

m ¼ Iðc þmjGÞGðc þmÞ ð10Þ

Moreover, the mark-up of the distributor is given from Eq. (5)

l ¼ lðc þm; FÞ ¼ Iðc þmjFÞFðc þmÞ ð11Þ

We have already seen that Assumption 2 ensures that l(x, G) isdecreasing in x. We will assume further that l(x, F) is decreasing inx. Therefore, we require that when the cost is high, the distributorwill decrease his mark-up in an effort to make his bid more com-petitive against his opponents. In this setting, we can establishthe following result.

Proposition 8. Suppose that l(x, F) is decreasing in x and Assump-tions 3 holds. Then, in a supply chain with unilateral downstreamcommunication, we have m < l.

According to Proposition 8, the distributor achieves a highermark-up compared to the manufacturer. In view of the results inthe case of one-way upstream communication, we may concludethat the agent who acquires more comprehensive informationabout the cost distribution will ultimately enjoy a higher marginand an advantage in the division of the total profit of the supplychain. Therefore, each one of the two agents has no incentive toshare his private information with his counterpart. This is in some


extent similar to the ‘‘prisoner’s dilemma’’ in a two person gameand results to poor overall performance for the supply chain as awhole.

Information sharing across the supply chain

We, now, consider the case where the upstream and down-stream supply chain agents engage in an open and truthful two-way exchange of information. The motive of the agents is to com-bine their knowledge and to develop a common belief about thecompetition in the auction. Information sharing leads to one singleprobability distribution, which is mutually agreed by both supplychain partners as best describing the probability distribution ofthe cost of the auction participants. Consequently, under informa-tion sharing W = C = U.

Proposition 9. In information sharing suppose that the cumulativeprobability distribution W = C = U is twice continuously differentiable

and FðxÞIðxjFÞ P �kFðxÞ. Then, the equilibrium manufacturer’s mark-up m

and distributor’s bid b are given by:

m ¼ Iðc þmjFÞFðc þmÞ ð12Þ

b ¼ c þ 2Iðc þmjFÞFðc þmÞ ð13Þ

We observe that each agent in the supply chain applies thesame mark-up I(c + m|F)/F(c + m). We may formally state this resultas follows:

Corollary 4. In information sharing, suppose that F(x)/I(x|F) P �kF

(x) and the cumulative probability distribution W = C = U is twicecontinuously differentiable. Then, m ¼ 1

2 ðb� cÞ and l ¼ m

Information sharing across the supply chain ensures that themark-up of the manufacturer is equal to the margin, which is ap-plied by the distributor when determining the bid, and profit isequally shared by both partners. With two-way exchange of infor-mation, no agent is more informed within the supply chain and,therefore, there is no longer an information advantage which isachieved at the expense of the less informed player. The supplychain operates coherently and the goals of both the manufacturerand the distributor are perfectly aligned and focused on winningthe auction. In this way, the predictive ability of the supply chainas a whole is enhanced as the two agents can share informationand formulate together a common belief about the probability dis-tribution of the cost of their competition.

If the manufacturer had decided to bid directly in the auctionwithout any intermediary supply agent, the equilibrium bid Bwould be: B = c + I(c|F)/F(c). Therefore, under information sharing,bidding through the distributor raises the final bid price by anamount equal to:

b� B ¼ 2Iðc þmjFÞFðc þmÞ �

IðcjFÞFðcÞ :

It is clear that, due to double marginalization, the bid price,when auction participation involves an intermediary, is highercompared to bidding directly. In the environment of informationsharing and when the underlying distribution of the cost is the uni-form distribution in [0,1], the mark-up strategy of the manufac-turer and the bid price of the distributor can be easily calculatedand explicitly expressed.




Proposition 10. Suppose that W = C = U is the cumulative distribu-tion of the uniform distribution in [0,1]. Then, m ¼ 1�c

nþ2 and

b ¼ nnþ2 c þ 2

nþ2.

In the case of a uniform in [0,1] underlying distribution, whenno intermediary is involved, the optimal bid is: B = c+(1 � c)/(n + 1). Consequently, participation in the auction through a dis-tributor increases the bid price by:

b� B ¼ 21� cnþ 2

� 1� cnþ 1

¼ ð1� cÞ nðnþ 1Þðnþ 2Þ ¼

nnþ 2

ðB� cÞ

The additional mark-up, which is produced by double marginaliza-tion, proportionally to the mark-up, when the manufacturer bids di-rectly, is: b�B

B�c ¼ nnþ2.

We observe that, for n = 2, the presence of intermediaries gener-ates an additional mark-up, which is 50% of the mark-up underdirect participation. As n becomes large, the additional mark-upb – B caused by the two-agent supply chain becomes approxi-mately equal in magnitude to the mark-up without intermediariesB � c.

Further, the total mark-up of the manufacturer–distributorsupply chain is: b � c = (b � B) + (B � c).Therefore, for n large,b � c becomes approximately twice the mark-up of the supplychain without any intermediary B � c. In practice, the manufac-turer should weigh the derived benefit of participation in theauction though the distributor against the additional mark-up inthe final bid b, which comes with a corresponding drop of probabil-ity of success in the auction.

Expected mark-up and profit of the manufacturer

In this section, we derive the ex-ante expected manufacturermark-up E(m) and the ex ante manufacturer profit E(p), when theunderlying probability distribution of the cost of the auctionbidders is W. Therefore, the ex-ante expectation of the mark-upand profit of the manufacturer are calculated according to the be-liefs of the manufacturer regarding the cost distribution of thecompetition.

Suppose that X(n) = min Xi where Xi, with i = 1, . . . ,n, are indepen-dent and identically distributed random variables that follow theprobability distribution C. Similarly, we write Y(n+1) = min Yi, whereYi, for i = 1, . . . ,n + 1, are independent random variables, which areall distributed according to the probability distribution W. Let Y bea random variable, which follows distribution W.

Proposition 11. (A) Under Assumption 3, the expected mark-up ofthe manufacturer is:

EðmÞ ¼ 1n½EðYÞ þ EðXðnÞjXðnÞ P YÞ � EðXðnÞÞ� ð14Þ

(B) Under Assumption 3, the expected profit of the manufactureris:

EðpÞ ¼ 1n½EðY ðnþ1ÞÞ þ EðXðnÞjXðnÞ P Y ðnþ1ÞÞ � EðXðnÞÞ� ð15Þ

We note that the scale factor 1/n comes from the symmetry ofthe bidders in our auction environment. Proposition 10 allows usto distinguish two additive components for the expected mark-up of the manufacturer. For an event A, I[A] denotes the indicatorfunction of A, taking the value of 1 when A is realized and 0 other-wise. Eq. (14) can be expressed in the form:

EðmÞ ¼ 1=nfEðXðnÞjXðnÞ P YÞ � EðXðnÞI½XðnÞP Y�Þg þ 1=nfEðYÞ � EðXðnÞI½XðnÞ < Y �Þg ð16Þ

The first term is related to the event X(n) P Y, which occurs whenthe manufacturer and the distributor win the auction, as the cost


Y is below the minimum cost of the competition. Therefore, intui-tively, this component represents the mark-up which is achievedby the manufacturer from the bidding process. The second termin Eq. (16) expresses the margin which is generated by the lack ofexchange of information between the supply chain agents. Whenthe minimum cost of all competitors X(n) stands below the cost Y,the manufacturer can still obtain a potential profit due to the erro-neous perception of the distributor about the competition. Accord-ing to the manufacturer, the distributor employs C as theunderlying probability distribution rather than W, and, for thisreason, he bids more aggressively compared to the bid price underW. The manufacturer–distributor supply chain can be successful inthe auction even though Y exceeds the minimum cost of the compe-tition. Therefore, the second component in Eq. (16) captures thisadditional contribution to the manufacturer mark-up.

Similarly to Eq. (16), we may represent the expected profit ofthe manufacturer in Eq. (15) as the sum of two components:

EðpÞ ¼ 1=nfEðXðnÞjXðnÞ P Y ðnþ1ÞÞ � EðXðnÞI½XðnÞP Y ðnþ1Þ�Þg þ 1=nfEðY ðnþ1ÞÞ � EðXðnÞI½XðnÞ < Y ðnþ1Þ�Þg ð17Þ

As before, intuitively, the first component expresses the contribu-tion to the manufacturer’s profit by the auction bidding process,while the second term captures the profit which is achieved bythe lack of communication within the supply chain.

Numerical illustration of the model

In this section we illustrate the model investigating differentscenarios of communication among the agents and employing par-ticular forms for the underlying probability distributions.

Supply chain without exchange of information among agents

We consider first the case where the two agents are not willingto inform their supply chain partner about their belief for the prob-ability distribution of the cost of the competition. In the scenariothat we investigate, the manufacturer believes that the cost x ofthe other bidders follows distribution W, with cumulative distribu-tion function W(x) = (1 � ex)/(1 � e), for x in [0,1]. Moreover, themanufacturer is of the opinion that the uniform distribution in[0,1] best describes the belief of the distributor about the compe-tition. Consequently, C(x) = x, for x in [0,1].

We observe that distribution W first-order stochastically domi-nates distribution C, as W(x) = (1 � ex)/(1 � e) 6 C(x) = x. It is easyto verify that the conditions of Assumption 1 are satisfied for thesetwo distributions. In particular, for Assumption 1(A), we note that

the ratio of the hazard rates: wðxÞcðxÞ

1�CðxÞ1�wðxÞ ¼ �

exð1�xÞex�e

and � exð1� xÞex � e

� �0¼ � exþ1ðex�1 � xÞ

ð1� exÞ26 0 for x > 0:

Assumption 1(B) holds with c(x) = 1, c0(x) = 0, w(x) = �ex(1 � e) andw0(x) = �ex(1 � e). Moreover, we note that: J(x)/(1 � C(x))n=(1 � x)/(n + 1) is a non-increasing in x, and, as a result, Assumption 2 isvalid in our scenario. Finally, we can also confirm, in consistencywith Proposition 5, that Assumption 3 also holds, since

g0ðxÞgðxÞ �

h0ðxÞhðxÞ � ðkGðxÞ � kHðxÞÞ þ

HðxÞIðxjHÞ � kHðxÞ� �

¼ 11� ex�1 P 0

for x in (0,1). To find the equilibrium mark-up m, we applyTheorem 1. In particular, from Eq. (4), m is the solution of thefollowing equation:



Table 3Manufacturer’s mark-up m and distributor’s mark-up l for different values of c and nin the supply chain with upstream one-way communication.

c n = 2 n = 3 n = 4 n = 5

m l m l m l m l

0.1 0.284 0.205 0.236 0.166 0.201 0.140 0.177 0.1210.2 0.246 0.185 0.204 0.149 0.174 0.125 0.152 0.1080.3 0.210 0.163 0.173 0.132 0.147 0.111 0.128 0.0950.4 0.176 0.141 0.144 0.114 0.122 0.096 0.106 0.0820.5 0.143 0.119 0.117 0.096 0.099 0.080 0.086 0.0690.6 0.111 0.096 0.090 0.077 0.077 0.065 0.066 0.0560.7 0.081 0.073 0.066 0.059 0.056 0.049 0.048 0.0420.8 0.053 0.049 0.043 0.039 0.036 0.033 0.031 0.0280.9 0.026 0.025 0.021 0.020 0.018 0.016 0.015 0.014


m ¼ kHðc þmÞkGðc þmÞ

Iðc þmjHÞHðC þmÞ ¼ �

ðecþm�1 � 1Þð1� ðc þmÞÞecþm�1

1� ðc þmÞnþ 1

¼ 1nþ 1

ðe�c�mþ1 � 1Þ

Table 2 provides the mark-up m when n takes values from 2 to 5and for different values of c. We observe that as c increases themarkup m drops, which is, of course, expected from Eq. (5) of The-orem 1. Further, as the number n of competing bidders increases,competition becomes more aggressive and the mark-up m becomeslower.

In the scenario that we examine and for illustrative pur-poses, the downstream agent believes that the probability dis-tribution of the cost of the competition is best described byU(x) = (1 � ex/2)/(1 � e1/2), for x in [0,1]. To determine the bidprice, the distributor applies a mark-up l which, according toEq. (3), is

l ¼ 1ðeðcþm=2Þ � e1=2Þn

Xn

i¼1

nið�1Þn�i2eðn�iÞ=2 ei=2

i� eiðcþmÞ2

i

� �(þð�1Þnen=2ð1� c �mÞ

�It is clear that, in the absence of exchange of information, the

beliefs of the two supply chain agents remain completely different.The mark-up of the distributor l for the corresponding values of cand ni can also be found in Table 2.

One-way upstream communication

We proceed with a scenario of one-way upstream communica-tion. In particular, the distributor believes that the underlyingprobability distribution U of the cost of the competition is bestdescribed by the uniform in [0,1] distribution and he unilaterallyinforms the manufacturer about this belief. As a result, the up-stream agent is certain that C is the uniform in [0,1] distribution.Never-the-less, the manufacturer believes, without informing thedistributor, that the actual probability distribution of the competi-tion is W(x) = (1 � ex)/(1 � e) for x in [0,1]. The mark-up of themanufacturer m and the distributor’s margin l, when c rangesfrom 0.1 to 0.9 and for different values of n, appear in Table 3.

Table 3 demonstrates that the value of l is, in all cases, lowerthan the corresponding value of m in consistency with Proposition7. The manufacturer enjoys an information advantage, whichallows him to apply a higher margin and capture a larger shareof the overall profit of the supply chain. It is also clear from Table 3that as the cost c increases, the impact of the information advan-tage becomes less intense and the difference between m and l de-creases. In essence, the higher value of c forces both supply agentsto apply relatively low margins, shrinking the effect of the informa-tion advantage of the manufacturer. A similar remark can be madefor the difference in the two margins as the number n of opponents

Table 2Manufacturer mark-up m for different values of c and n in the supply chain withoutexchange of information.

c n = 2 n = 3 n = 4 n = 5

m l m l m l m l

0.1 0.284 0.221 0.236 0.183 0.201 0.157 0.177 0.1370.2 0.246 0.198 0.204 0.163 0.174 0.139 0.152 0.1210.3 0.210 0.173 0.173 0.142 0.147 0.121 0.128 0.1060.4 0.176 0.149 0.144 0.122 0.122 0.103 0.106 0.0900.5 0.143 0.124 0.117 0.101 0.099 0.086 0.086 0.0740.6 0.111 0.100 0.090 0.081 0.077 0.068 0.066 0.0590.7 0.081 0.075 0.066 0.061 0.056 0.051 0.048 0.0440.8 0.053 0.050 0.043 0.040 0.036 0.034 0.031 0.0290.9 0.026 0.025 0.021 0.020 0.018 0.017 0.015 0.014


increases. The presence of more aggressive competition lowers theeffect of the advantageous position of the informed agent.

One-way downstream communication

We now turn our attention to a scenario of downstream com-munication. In this setting, the manufacturer believes that theunderlying probability distribution of the cost of the competitionis W(x) = (1 � ex)/(1 � e) for x in [0,1] and communicates this infor-mation to the downstream agent. The manufacturer believes thatthe distributor determines the bid price using the probability dis-tribution C(x) = W(x). We consider the case in which the distribu-tor is confident about his own estimate U(x) = (1 � e2x)/(1 � e2) forx in [0,1] and, consequently, he will not adopt the probability dis-tribution, which the manufacturer proposes. At the same time,communication is one-way and the distributor does not communi-cate to the upstream agent his beliefs.

We observe that U(x) first order stochastically dominates W(x).Intuitively, the cost of the competition tends to take lower valuesunder W(x) and, therefore, under W(x) the manufacturer is forcedto apply a lower mark-up m in the transfer price, to ensure thatthe final bid price is competitive. Proposition 8 allows us to com-pute the corresponding mark-up, which the each one of the twosupply chain agents will employ. In particular, m is the solutionof the following equation:

m ¼ 1ðeðcþmÞ � enÞ

Xn

i¼1

nið�1Þn�ien�i ei

i� eiðcþmÞ

i

� �(

þð�1Þnenð1� c �mÞ)

ð16Þ

where ðniÞ ¼ n!i!ðn�iÞ!. The mark-up of the distributor is given by:

l ¼ 1ðe2ðcþmÞ�e2 Þn

Xn

i¼1

¼ nið�1Þn�ie2ðn�iÞ e2i

2i� e2iðcþmÞ

2i

� �(þð�1Þne2nð1� c �mÞ

�
Table 4Manufacturer’s mark-up m and distributor’s mark-up l for different values of c and nin the supply chain with downstream one-way communication.
c n = 2 n = 3 n = 4 n = 5

m l m l m l m l

0.1 0.252 0.290 0.211 0.252 0.181 0.225 0.160 0.2030.2 0.222 0.251 0.184 0.217 0.158 0.192 0.139 0.1730.3 0.192 0.215 0.159 0.184 0.136 0.162 0.119 0.1450.4 0.162 0.179 0.134 0.152 0.114 0.133 0.100 0.1180.5 0.133 0.145 0.109 0.122 0.093 0.106 0.081 0.0940.6 0.105 0.113 0.086 0.094 0.073 0.081 0.063 0.0710.7 0.078 0.083 0.063 0.068 0.053 0.058 0.046 0.0510.8 0.051 0.053 0.041 0.044 0.035 0.037 0.030 0.0320.9 0.025 0.026 0.020 0.021 0.017 0.018 0.015 0.015




The values for the mark-up m and l, when the number of compo-nents n is from 2 to 5 and c ranges from 0.1 to 0.9, can be foundin Table 4.

Table 4 clearly illustrates the information advantage of thedownstream agent by the unilateral downstream communication,as it is evident from the fact that the distributor is able to enjoya higher mark-up compared to the manufacturer. When the costc increases, the difference in mark-up for the informed agentdiminishes. Intuitively, the distributor will have to counter thehigh values of the cost c with a lower mark-up l.

Information sharing

Finally, we consider a scenario in which the manufacturer andthe distributor engage in truthful bilateral information sharing.The two supply chain partners both agree that the underlying dis-tribution describing the cost of the auction participants is given byU(x) = C(x) = W(x) = (1 � ex)/(1 � e) for x in [0,1]. From Proposition9, the manufacturer and the distributor apply the same mark-up m,which we obtain by solving Eq. (16). Table 5 provides the mark-upm when n ranges from 2 to 5 and for different values of c.

Comparing Table 5 with Tables 3 and 4, we observe that, whenthere is bilateral exchange of information, no agent enjoys an infor-mation advantage and the mark-up of the informed player is dras-tically reduced, while, at the same time, the margin of theuninformed supply chain partner increases. The drop of themark-up of the informed agent, once bilateral communication isestablished, appears to be higher than the raise of the margin ofthe uninformed player. The reason for this asymmetry is thatagents revise their mark-up differently. The uninformed agent,who had expected more aggressive competition and, now, learnsthat the cost of each one of his opponents is in fact ‘‘higher’’ in astochastic sense, is able to increase his margin and avoid unneces-sarily low pricing. The informed agent, on the other hand, will seehis margin suffer because he can no longer exploit the erroneousperception of his supply chain partner about the underlying com-petition. As c and n increases, both agents are forced to lower theirmark-ups in order to ensure that the final bid price remains com-petitive, and, therefore, information sharing tends to provide lessintense changes in the margins of the two players compared tothe respective values in one-way communication.

We may also compare the margins in information sharing, as itappears in Table 5, with the respective mark-ups of Table 2, wherethere is no exchange of information. It is evident that both agentsapply a lower mark-up once they communicate to each other theirbeliefs about the competition. In bilateral communication, themanufacturer can no longer expect that the distributor will errone-ously overrate the aggressiveness of the competition and, conse-quently, the transfer price will necessarily drop. On the otherhand, the distributor revises his initial belief about the competitionand, in our scenario he fully adopts the perspective of the manufac-turer. Due to information sharing, the distributor realizes that thecost of his competitors is ‘‘lower’’ in a stochastic sense than what

Table 5Mark-up m = l for different values of c and n in the supply chain with informationsharing.

c n = 2 n = 3 n = 4 n = 5

0.1 0.252 0.211 0.181 0.1600.2 0.222 0.184 0.158 0.1390.3 0.192 0.159 0.136 0.1190.4 0.162 0.134 0.114 0.1000.5 0.133 0.109 0.093 0.0810.6 0.105 0.086 0.073 0.0630.7 0.078 0.063 0.053 0.0460.8 0.051 0.041 0.035 0.0300.9 0.025 0.020 0.017 0.015


he had anticipated, and, therefore, he is forced to decrease his mar-gin. Never-the-less, although the mark-up of both agents isreduced, the supply chain partners, under information sharing,can more accurately predict the behavior of the other bidders.

In our numerical illustration, we have considered the casewhere, under information sharing, the distributor fully adopts theperspective of the manufacturer about the competition. In practice,the bilateral exchange of information will enable both agents to re-vise their initial probability distributions for the underlying cost,until they reach a common perspective about the competition.

Discussion – conclusion

We have examined a supply chain with double marginalizationleading to a sealed bid lowest price procurement auction withprivate values. In this environment, following a game-theoreticapproach, we derived the optimal mark-up of the upstream anddownstream supplier. The proposed bidding strategies maximizethe expected profit of each agent. In our analysis we have assumedthat all players are risk neutral with respect to monetary payoffsand know precisely their own cost. Moreover, bidders are assumedsymmetric in terms of the underlying cost distribution and costvalues are independently distributed. In future research, each oneof these assumptions could be relaxed by expanding the modelwithin the standard methodology of auction theory.

We have assumed that each agent of the supply chain cooper-ates with only one player and we have argued that this assumptionis plausible in procurement environments, such as public procure-ment. In general, however, the manufacturer or distributor maydecide to cooperate simultaneously with other competing agents.For example, the manufacturer may offer the same product tomore than one distributor with or without price differentiation.It will, then, be important to clearly define the information struc-ture of such a supply network. For example, the model should clar-ify which, if any, of the competing distributors are aware of the factthat the manufacturer is willing to supply the same product to dif-ferent agents, as well as to what extent the different transfer pricesare known. Moreover, conditionally on the information about thedifferent transfer prices, bidders will asymmetrically revise theirprior probability distribution of the cost of the competition, gener-ating an asymmetric auction environment.

A more complex network would allow some distributors tocooperate with multiple manufacturers and some manufacturersto supply more than one distributor (Elmaghraby, 2000). Clearly,this form of supply chain introduces another layer of competitionbetween manufacturers and distributors, and it could be arguedqualitatively that it will lead to more aggressive bidding. Informa-tion asymmetries become more involved and partnership commit-ments are no longer in place. In this scheme, it will be necessary tospecify how distributors will choose the manufacturer that theywill eventually cooperate with in case of winning. One alternativewould be to conduct a contingent auction to select the manufac-turer, who will supply the product. This contingent auction couldtake place before or after the central downstream auction. Simi-larly, the manufacturer may implement a contingent auctionmechanism to select the distributor who is willing to pay the high-est transfer price to purchase the product. The investigation of theoptimal mark-up and the equilibrium bidding strategies of thesecomplex supply chain arrangements constitutes a challengingdirection of future research. In conclusion, we believe that, in fu-ture research, the supply chain structure that we developed couldserve as the basis for the analysis of more complex supply chainenvironments, with various types of interactions among agentsand under different communication systems.

Our findings demonstrated that the form of supply chain thatwe examined is sensitive to information asymmetry between




agents. Concealing information about the level of competition pro-vides an advantage in the division of the total profit. On the con-trary, information sharing results to an equal distribution of thetotal profit among the supply chain partners. Information asymme-try was based exclusively on lack of information sharing. Agents inour model may choose to conceal or, alternatively, share informa-tion but, clearly, another and, certainly, unethical, if not illegal, op-tion would be to deliberately distort information, which iscommunicated to the other party. In such a general scheme, eachagent communicates the most preferable for the agent probabilitydistribution of the underlying cost of the competition. Of course, itremains unclear whether the particular probability distributionwill be eventually adopted by the other supply chain agent.Although we have assumed in our analysis that the manufacturerand distributor do not engage in such unethical behavior, inten-tional distortion of information may actually appear in practice.

In general, it is not easy to enforce and monitor the exchange ofinformation between supply chain partners. The type of informa-tion required will have to be clearly defined and agreed betweenthe agents involved. Ideally, there should be an agreement on thetype of data, which will be shared, and the statistical techniquesthat will be used to estimate the underlying cost distribution ofopponent bidders. In this respect, a number of obstacles may pro-hibit information sharing. Data gathered from past auctions is typ-ically filtered in many ways with regard to its relevance to futuresupplies. Moreover, the estimate about the underlying probabilitydistribution of competitors’ cost is, usually, not formally recordedin a company’s knowledge base. Consequently, the design andnegotiation of contracts leading to partnership relations with infor-mation sharing and cooperative decision making is a challengingtask. Information models of supply chain networks (Fiala, 2005),combining non-cooperative and cooperative behavior, may be use-ful in this direction. The optimal information sharing mechanismmay eventually involve bilateral negotiation. In order to induceeach agent to truthfully report to his partner the available privateinformation, each agent’s profit can be set to be his Shapley value,which provides the payoff that each player should reasonably ex-pect in a co-operative game (Osborne & Rubinstein, 1994). Intui-tively, the Shapley value reflects the notion that each agent’spayoff from bargaining depends on the agent’s marginal contribu-tion to the total supply chain performance. Exploring bargainingmechanisms along this framework remains an open question.

Finally, the auction setting could be extended to include multi-dimensional competition as, for example, auctions for which priceand quality are both evaluated by a scoring rule (Beil & Wein,2003; Che, 1993). In such an environment, the structure of infor-mation asymmetry in the manufacturer–distributor supply chainbecomes more complex and the exchange of information more dif-ficult to achieve. It would be of interest to investigate whether sim-ilar results to our findings may be derived in a multi-dimensionalauction setting.

Acknowledgements

The authors would like to thank two anonymous referees fortheir generous advice, insightful comments and helpfulsuggestions.

Appendix A

Proof of Proposition 1.

(a) Assumption 1(B) ensures that for all x in (0,x)

Pleasetional

limy!x

wðyÞcðyÞ

1� CðyÞ1� wðyÞ

� �P

wðxÞcðxÞ

1� CðxÞ1� wðxÞ :

cite this article in press as: Lorentziadis, P. L. Pricing in a supply chain forResearch (2014), http://dx.doi.org/10.1016/j.ejor.2014.02.051

Using De L’Hospital rule and, Assumption 1(A), we have that

auctio

limy!x

wðyÞcðyÞ

1� CðyÞ1� wðyÞ

� �¼ ½wð1� CÞ�0ð1Þ½cð1� wÞ�0ð1Þ

¼ w0ð1Þð1� Cð1ÞÞ � wð1Þcð1Þc0ð1Þð1� wð1ÞÞ � cð1Þwð1Þ

¼ 0� wð1Þcð1Þ0� cð1Þwð1Þ ¼ 1

It follows thatwðxÞ

1� wðxÞ 6cðxÞ

1� CðxÞ ðA:1Þ

(b) Stochastic dominance in terms of hazard rate, as given in(A.1), implies first-order stochastic dominance (Shaked &Shanthikumar, 2010). Therefore, W(x) 6 C(x) for all x in(0,x). h

Proof of Proposition 2.

(a) We note that: gðxÞGðxÞ ¼ �n wðxÞ

1�wðxÞ and hðxÞHðxÞ ¼ �n cðxÞ

1�CðxÞ

Therefore;kG

kHðxÞ ¼ gðxÞ

GðxÞHðxÞhðxÞ ¼

wðxÞ1�WðxÞ

1� CðxÞcðxÞ

It follows from Assumption 1(B) that kG/kH(x) is non-decreasingin x.

(a) We observe that: kGðxÞ ¼ gðxÞGðxÞ ¼ �n wðxÞ

1�wðxÞ P �n cðxÞ1�CðxÞ ¼

hðxÞHðxÞ ¼

kHðxÞ:

Hence, GðxÞ ¼ exp �R x

0 kGðxÞdy� �

6 exp �R x

0 kHðxÞdy� �

¼ HðxÞ. h

Proof of Corollary 1. As kG(x) 6 0 for all x in (0,x), the result fol-lows directly from Proposition 2. h

Proof of Proposition 3. We note that g(x) = �nw(x)(1 �W(x))n�1

and g0(x) = �nw0(x)(1 �W(x))n�1 + n(n � 1)w2(x)(1 �W(x))n�2.

Therefore;g0ðxÞgðxÞ ¼

w0ðxÞwðxÞ � ðn� 1Þ wðxÞ

1� wðxÞ ¼w0ðxÞwðxÞ � ðn� 1ÞrwðxÞ:

Further; kGðxÞ ¼gðxÞGðxÞ ¼ �n

wðxÞ1� wðxÞ ¼ �nrwðxÞ

and; similarly;h0ðxÞhðxÞ ¼

c0ðxÞcðxÞ � ðn� 1Þ cðxÞ

1� CðxÞ

¼ c0ðxÞcðxÞ � ðn� 1ÞrCðxÞ

while; kGðxÞ ¼hðxÞHðxÞ ¼ �n

cðxÞ1� CðxÞ ¼ �nrCðxÞ:

The result follows immediately since J(x) = I(x|H). h

Proof of Proposition 4. To prove (i) we proceed as follows:

eðghðxÞÞ ¼ x

hgðxÞ g

h

0ðxÞ ¼ x

hðxÞgðxÞ

h0ðxÞgðxÞ � hðxÞg0ðxÞh2ðxÞ

¼ xh0ðxÞgðxÞ � hðxÞg0ðxÞ

hðxÞgðxÞ ¼ xg0ðxÞgðxÞ �

h0ðxÞhðxÞ

� �

Similarly; eGHðxÞ

� �¼ x

G0ðxÞGðxÞ �

H0ðxÞHðxÞ

� �¼ x

gðxÞGðxÞ �

hðxÞHðxÞ

� �¼ xðkGðxÞ � kHðxÞÞ:

n bidding under information asymmetry. European Journal of Opera-



Finally; e~HHðxÞ

!¼ x

~H0ðxÞ~HðxÞ

� H0ðxÞHðxÞ

" #¼ �x

hðxÞHðxÞ þ

HðxÞIðxjHÞ

� �¼ �x

HðxÞIðxjHÞ þ kHðxÞ� �

:

To show (ii), we note that for any function n, with n(x) > 0 for all

x in (0,x), it holds true that eðfðxÞÞ ¼ ½inðfðxÞÞ�0

½lnðxÞ�0 . Therefore, (i) becomes:

½lnðgh ðxÞÞ�0 P ½lnðGH ðxÞÞ�

0 þ ½lnð IH ðxÞÞ�

0

where I(x) = I(x|H) and I/H(x) P 0 for all x in (0,x). The last

inequality is equivalently written as: ln g=Gh=H

HI

ðxÞ

h i0P 0.

It follows, equivalently, that ln g=Gh=H

HI

ðxÞ is non-decreasing. Due

to the monotonicity of the logarithmic function, we conclude that

the function wðxÞ ¼ ln g=Gh=H

HI ÞðxÞ ¼

kGkHðxÞ H

I ðxÞ must be non-decreasing. h

Proof of Proposition 5. Suppose that Assumptions 1 and 2 hold.Let x < y. Then, from Proposition 2, kG/kH(x) 6 kG/kH(y). Further,

from Assumption 2, JðxÞð1�CðxÞÞn

h i�16

JðyÞð1�CðyÞÞn

h i�1.

Given that all the quantities involved are non-negative, we mayhave:

kG

kHðxÞH

IðxÞ ¼ kG

kHðxÞ JðxÞð1� CðxÞÞn� ��1

6kG

kHðyÞ JðyÞð1� CðyÞÞn� ��1

¼ kG

kHðyÞH

IðyÞ:

Therefore, it follows immediately from Proposition 4(ii) thatAssumption 1 holds. h

Proof of Theorem 1. We use the following lemma to derive theresult. h

Lemma 1. For every c, we can locally solve Eq. (4) so that m = m(c).Moreover,

m0ðcÞ ¼ �1þ 11þ Lðc þmÞ where Lðc þmÞ

¼ kHðc þmÞkGðc þmÞ 1þ Iðc þmjHÞ

Hðc þmÞ Kðc þmÞ� �

and

Cðc þmÞ ¼ g0ðc þmÞgðc þmÞ �

h0ðc þmÞhðc þmÞ � kGðc þmÞ þ 2kHðc þmÞ:

Proof of Lemma 1. Let Aðc;mÞ ¼ m gðcþmÞGðcþmÞ �

hðcþmÞHðcþmÞ

IðcþmjCÞHðcþmÞ .

Eq. (4) is equivalent to the implicit equation A(c, m) = 0. Weemploy the Implicit Function Theorem, according to which in aneighborhood of (c, m), we can write m = m(c). Moreover, wecompute:

@Aðc;mÞ@m

¼ gðc þmÞGðc þmÞ þm

g0ðc þmÞGðc þmÞ � g2ðc þmÞG2ðc þmÞ

� h0ðc þmÞIðc þmjHÞH2ðc þmÞ � hðc þmÞH3ðc þmÞ � 2h2ðc þmÞIðc þmjHÞHðc þmÞH4ðc þmÞ

To simplify the notation, we drop the argument c + m, in theremaining of the proof, and we write, equivalently,


@Aðc;mÞ@m

¼ gGþm

g0

Gm

g2

G2 �h0I

H2 þhHþ 2

h2I

H3 ðA:2Þ

From (4), we have that m ¼ Gg

hH

IH. Hence, substitution in (A.2) yields:

@Aðc;mÞ@m

¼ gGþm

hI

H2

g0

g� ghI

GH2 �h0I

H2 þhHþ 2

h2I

H3

¼ kG þ kH þhi

H2

g0

g� g

G� h0

hþ 2

hH

� �¼ kG þ kH þ

hI

H2

g0

g� h0

h� kG þ kH þ

hI

H2 K� �

where K ¼ g0

gh0

h � kG þ 2kH . Similarly,

@Aðc;mÞ@c

¼ mg0

G�m

g2

G2 �h0IH2 � hH3 � 2h2IH

H4

¼ hI

H2

g0

g� hI

H2

gG� h0I

H2 þhHþ 2

h2I

H3

¼ hI

H2

g0

g� h0

h� kG þ 2kH

� �þ kH ¼ kH þ

hI

H2 K

From the Implicit Function Theorem we may deduce that:

dmdc¼

@Aðc;mÞ@c

@Aðc;mÞ@m

¼ �kH þ hI

H2 K

kG þ kH þ hIH2 K

¼ �1þ kG

kG þ kH þ hIH2 K

¼ �1þ 11þ kH

kGþ hI

H2 K¼ �1þ 1

1þ Lwith

L ¼ kH

kG1þ I

HK

� ��

We, now, show the validity of Eq. (5). From Lemma 1, we have:1 + m0(c) = (1 + L(c + m))�1. We need to show that L(c + m) P 0.Dropping the argument c + m, we have:

L ¼ kH

kG1þ 1

HK

� �¼ kH

kG1þ I

Hg0

g� h0

hkG þ 2kH

� �� ¼ kH

kG

IH

g0

g� h0

h� kG þ 2kH þ

HI

� �From Proposition 3, g0

g � h0

h � kG þ 2kH þ HI P 0

and since kHkG

P 0 and IH P 0 , it follows that: L ¼ kH

kG

IH

g0

g � h0

h

P 0:

The equilibrium strategy of the distributor is derived by settingthe derivative of P(b, c + m|F)=(b � c �m)F(b�1(b)) with respect tob equal to 0. In particular,

f ðc þmÞ½b0ðc þmÞ��1ðb� c �mÞ þ Fðc þmÞ ¼ 0

which implies, equivalently,

f ðc þmÞðb� c �mÞ þ b0ðc þmÞ ¼ 0

Consequently, [F(c + m))b]0 = f(c + m)(c + m), and integrating wehave:

b ¼ bðc þmÞ ¼ � 1Fðc þmÞ

Z x

cþmxf ðxÞdx

Integration by parts yields: b ¼ bðc þmÞ ¼ c þmþ 1Fðc;mÞ

Rxcþm FðxÞdx,

which is Eq. (3).




To prove that Eq. (3) provides, indeed, the symmetric equilib-rium strategy, we note that:

PðbðzÞ; c þmjFÞ ¼ ½bðzÞ � c �m�FðzÞ

¼ ðz� c �mÞFðzÞ þZ x

zFðxÞdx

Consequently, Pðbðc þmÞ; c þmjFÞ � pðbðzÞ; c þmjFÞ ¼�ðz� c �mÞFðzÞ þ

R zcþm FðxÞdx P 0

for all values of z. Therefore, b = b(c + m) is the symmetric equi-librium strategy of the distributor.

We proceed to the derivation of the symmetric equilibrium ofthe manufacturer. Taking the derivative of p(m, b, c|G,H) = mG(b�1(b)) with respect to m and setting it equal to 0 yields:

Gðb�1ðbÞÞ þmgðb�1ðbÞÞ @bðc;mÞ@m

� ��1

¼ 0

Since b�1(b) = c + m, it follows, equivalently, that:

Gðc þmÞ @bðc þmÞ@m

þmgðc þmÞ ¼ 0 ðA:3Þ

It is noted that:

@bðc þmÞ@m

¼ 1þ H2ðc þmÞ � Iðc þmjHÞhðc þmÞH2ðc þmÞ

¼ � hðc þmÞHðc þmÞ

Iðc þmjHÞHðc þmÞ ðA:4Þ

and substituting in (A.3) provides: �Gðc þmÞ hðcþmÞHðcþmÞ

IðcþmjHÞHðcþmÞ þ

mgðc þmÞÞ ¼ 0Rearranging the terms we have:m gðcþmÞ

@m ¼ hðcþmÞHðcþmÞ

IðcþmjHÞHðcþmÞ , which is Eq. (4).

The mark-up that satisfies Eq. (4) is denoted by m⁄. Supposethat m is another manufacturer mark-up value. The correspondingprofit of the manufacturer is p(m, b, c|G, H) = mG(b�1(b)) and, as be-fore, we have:

@p@m¼ Gðb�1ðbÞÞ þmgðb�1ðbÞÞ @bðc þmÞ

@m

� ��1

¼ Gðc þmÞ �mgðþmÞ hðc þmÞHðc þmÞ

Iðc þmjHÞHðc þmÞ ¼

¼ Gðc þmÞ 1�mkGðc þmÞkHðc þmÞ Iðc þmjHÞHðc þmÞ

� �Suppose that z is the cost of the manufacturer for whichc + m = z + m(z) and

mðzÞkGðzþmðzÞÞ ¼ kHðzþmðzÞÞ IðzþmðzÞjHÞHðzþmðzÞÞ

Notice that m⁄ = m(c). It follows that

@p@m¼ Gðc þmÞ 1� m

m�mðcÞmðzÞ

� �ðA:5Þ

Consider the case where m P m⁄. Suppose that z < c. Eq. (5) en-sures that the function z + m(z) is non-decreasing and the functionm(z) is non-increasing. Then, z + m(z) 6 c + m(c), and, equivalently,c + m 6 c + m⁄, which is a contradiction. Consequently, we havez P c. It follows from Eq. (5) that m(z) 6m(c). Finally, given thatG(c + m) P 0, it follows from equation (A.5) that @p

@m 6 0.Proceeding with a similar argument, it may be shown that for

m 6m⁄, it holds that @p@m P 0, which completes the proof.

Proof of Proposition 6. Since kG(c + m) 6 0, inequality (6)becomes: mkG(c + m) P �1.

Therefore, using Eq. (4), we write (6), equivalently, in the form:


1þ hðc þmÞIðc þmjHÞH2ðc þmÞ

P 0 ðA:6Þ

Consider the function : fðmÞ ¼ Iðc þmjHÞHðc þmÞ ¼

JðxÞð1� CðxÞÞn

where JðxÞ ¼Rx

x ð1� CðyÞÞndy. From Assumption 2, f(m) is non-increasing in m. Therefore,

f0ðmÞ ¼ �H2ðc þmÞ � 1ðc þmjHÞhðc þmÞH2ðc þmÞ

¼ �1� hðc þmÞIðc þmjHÞH2ðc þmÞ

6 0:

which is inequality (A.6). h

Proof of Proposition 7. The proof is immediate from Theorem 1.When C = U and, therefore, H = F, Eqs. (3) and (4) become equiva-lently Eqs. (8) and (9). h

Proof of Corollary 3. From Eqs. (8) and (9), mb�c�m ¼

kF ðcþmÞkGðcþmÞ.

Given that F = H, the result follows from Corollary 1. h

Proof of Proposition 8. Suppose that mo is the mark-up which themanufacturer would employ, if he had used U rather than W. Inparticular, we have: mo ¼ Iðcþmo jFÞ

FðcþmoÞ .

The distributor chooses not to disclose to the manufacturer thathis estimate for the probability distribution of the underlying costis U, because otherwise, the mark-up applied by the manufacturerwill increase. In other words, we have m < mo.

Since IðxjFÞFðxÞ is decreasing in x, it is immediate that:

IðcþmjFÞFðcþmÞ P Iðcþmo jFÞ

FðcþmoÞ and, therefore, l P mo. As mo > m, it follows that

l > m. h

Proof of Proposition 9. When W = C = U, Assumption 3 becomesequivalent to the condition that the underlying cumulative distribu-tion is twice continuously differentiable and H(x)/I(x|H) P �kH(x).

Consequently, the result follows from Eqs. (8) and (9). h

Proof of Proposition 10. It is clear that the uniform probabilitydistribution function is twice continuously differentiable and that

FðxÞIðxjFÞ ¼

nþ 11� x

P � � n1� x

¼ kFðxÞ

Moreover, from Proposition 9, m ¼ IðcþmjFÞFðcþmÞ ¼

1�ðcþmÞnþ1 . Therefore,

m ¼ 1�cnþ1. Further,

b ¼ c þ 2m ¼ c þ 21� cnþ 2

¼ nnþ 2

c þ 2nþ 2

: �

Proof of Proposition 11. First, we prove Eq. (14).

EðmÞ ¼Z x

0mwðxÞdx ¼

Z x

0

kHðxÞIðxÞ}kGðxÞHx

wðxÞdx

¼Z x

0

cðxÞð1�WðxÞÞ1ðxÞ}wðxÞð1� CðxÞÞð1� CðxÞÞn

wðxÞdx

¼Z x

0

cðxÞð1� CðxÞÞnþ1 ð1�WðxÞÞ

Z x

xð1� CðyÞÞndy dx

¼Z x

0ð1� CðyÞÞn

Z y

0ð1�WðxÞÞ cðxÞ

ð1� CðxÞÞnþ1 dx dy




We note that: cðxÞð1�CðxÞÞnþ1 ¼ n�1½ð1� CðxÞÞ�n�0 and, hence, integration

by parts providesZ y

0ð1�WðxÞÞ cðxÞ

ð1� CðxÞÞnþ1 dx ¼ 1n

1�WðyÞð1� CðyÞÞn

� 1n

þ 1n

Z y

0

WðxÞð1� CðxÞÞn

dx

Substituting in the expression of E(m) we have:

EðmÞ¼1n

Z x

0ð1�CðyÞÞn 1�WðyÞ

ð1�CðyÞÞndy�1

n

Z x

0ð1�CðyÞÞndy

þ1n

Z x

0ð1�CðyÞÞn

Z y

0

WðxÞð1�CðxÞÞn

dxdy¼1n

Z x

0ð1�WðyÞÞdy

�1n

Z x

0ð1�CðyÞÞndyþ1

n

Z x

0ð1�CðyÞÞn

Z y

0

WðxÞð1�CðxÞÞn

dxdy

ðA:7ÞIt is easy to see that: 1

n

Rx0 ð1�WðyÞÞdy ¼ 1

n

Rx0 ywðyÞdy ¼ 1

n EðYÞ and1n

Rx0 ð1� CðyÞÞndy ¼ 1

n EðXðnÞÞ.

Further;1n

Z x

0ð1� CðyÞÞn

Z y

0

wðxÞð1� CðxÞÞn

dxdy

¼ 1n

Z x

0

Z y

0

ð1� CðyÞÞn

ð1� CðxÞÞnwðxÞdxdy ¼ 1

n

Z x

0

Z y

0PrðXðnÞÞP yjXðnÞ

P xÞwðxÞdxdy ¼ 1n

Z x

0PrðXðnÞ P yjXðnÞ P YÞdy

¼ 1n

EðXðnÞjXðnÞ P YÞ:

Substituting in (A.7) yields Eq. (14). We proceed to the proof of part(B).

EðpÞ ¼Z x

0mGðxÞwðxÞdx ¼

Z x

0

kHðxÞIðxjHÞkGðxÞHðxÞ

GðxÞwðxÞdx

¼Z x

0

cðxÞð1�WðxÞÞnþ1

wðxÞð1� CðxÞÞnþ1 IðxjHÞwðxÞdx

¼Z x

0

cðxÞð1� CðxÞÞnþ1 ð1�WðxÞÞnþ1

Z x

xð1� CðyÞÞndydx

¼Z x

0ð1� CðyÞÞn

Z y

0ð1�WðxÞÞnþ1 cðxÞ

ð1� CðxÞÞnþ1 dxdy

Similarly to the proof of part (A),Z y

0ð1�WðxÞÞnþ1 cðxÞ

ð1� CðxÞÞnþ1 dx ¼ 1nð1�WðyÞÞnþ1

ð1� CðyÞÞn� 1

n

þ nþ 1n

Z y

0

wðxÞð1�WðxÞÞn

ð1� CðxÞÞndx

Consequently, substituting in the expression of E(p),

Ep ¼ 1n

Z x

0ð1� CðyÞÞn ð1�WðyÞÞnþ1

ð1� CðyÞÞndy� 1

n

�Z x

0ð1� CðyÞÞndyþ nþ 1

n

Z x

0ð1� CðyÞÞn

Z y

0

� wðxÞð1�WðxÞÞn

ð1� CðxÞÞndxdy ðA:8Þ

Further, we compute the three integrals in the expression of E(p).

1n

Z x

0ð1� CðyÞÞn ð1�WðyÞÞnþ1

ð1� CðyÞÞndy ¼ 1

n

Z x

0ð1�WðyÞÞnþ1dy

¼ 1n

EðY ðnþ1ÞÞ

1n

Z x

0ð1� CðyÞÞndy ¼ 1

nEðXðnÞÞ


and

nþ 1n

Z x

0ð1� CðyÞÞn

Z y

0

wðxÞð1�WðxÞÞn

ð1� CðxÞÞndxdy

¼ nþ 1n

Z x

0

Z y

0

ð1� CðyÞÞn

ð1� CðxÞÞnð1�WðxÞÞnwðxÞdxdy

1n

Z x

0

�Z y

0PrðXðnÞ P yjXðnÞ P xÞðnþ 1Þð1�WðxÞÞnwðxÞdxdy

¼ 1n

Z x

0PrðXðnÞ P yjXðnÞ P Y ðnþ1ÞÞdy ¼ 1

nEðXðnÞjXðnÞ�gesY ðnþ1ÞÞ

Substitution in (A.8) provides Eq. (15). h

References

Arnold, D. (2000). Seven rules of international distribution. Harvard Business Review(November, December), 131–137.

Beil, D. R., & Wein, L. M. (2003). An inverse-optimization-based auction mechanismto support a multiattribute RFQ process. Management Science, 49(11),1529–1545.

Bikhchandani, S., & Huang, C. (1989). Auctions with resale markets: An exploratorymodel of treasury bill markets. Review of Financial Studies, 2(3), 311–339.

Che, Y. K. (1993). Design competition through multidimensional auctions. RandJournal of Economics, 24(4), 668–680.

Chen, Y. J., & Vulcano, G. (2009). Effects of information disclose under first- andsecond-price auctions in a supply chain setting. Manufacturing and ServiceOperations Management, 11(2), 299–316.

Elmaghraby, W. J. (2000). Supply contract competition and sourcing policies.Manufacturing and Service Operations Management, 2(4), 350–371.

European Parliament and Council (2004). Directive 2004/18/EC of the EuropeanParliament and of the Council of 31 March 2004 on the coordination ofprocedures for the award of public works contracts, public supply contracts andpublic service contracts. Official Journal of European Union. <http://eur-lex.europa.eu/LexUriServ/LexUriServ.do?uri=OJ:L:2004:134:0114:0240:EN:PDF>.

Federal Acquisition Regulation (2005). General services administrations (Vol. 1, Part14, Issue March 2005). Department of Defense U.S.A. <https://www.acquisition.gov/far/05-53/pdf/FAR.book.pdf>.

Feldman, J., Mirronki, V. S., Muthukrishnan, S., & Pai, M. M. (2010). Auctions withintermediaries: Extended abstract. In Proceedings of the 11th ACM conference onelectronic commerce (pp. 23–32).

Fiala, P. (2005). Information sharing in supply chains. Omega, 22, 419–423.Ghauri, P. N., & Usunier, J.-C. (2003). International business negotiations (2nd ed.).

Oxford, UK: Pergamon Elsevier.Guide to the Community Rules for Service Contracts. European Commission. Public

Procurement in the European Union, Directive 92/50/EEC. <http://ec.europa.eu/internal_market/publicprocurement/docs/guidelines/services_en.pdf>.

Guide to the Community Rules for Supply Contracts. European Commission. PublicProcurement in the European Union, Directive 93/36/EEC. <http://ec.europa.eu/internal_market/publicprocurement/docs/guidelines/supply_en.pdf>.

Gumus, A. T., & Guneri, A. F. (2007). Multi-echelon inventory management in supplychains with uncertain demand and lead times: Literature review from anoperational research perspective. Proceeding of the Institution of MechanicalEngineers, Part B, Journal of Engineering and Manufacture, 221(B10), 1553–1570.

Ha, A. Y., & Tong, S. (2008). Contracting and information sharing under supply chaincompetition. Management Science, 54(4), 701–715.

Hafalir, I., & Krishna, V. (2008). Asymmetric auctions with resale. American EconomicReview, 98(1), 87–112.

Haile, P. A. (2003). Auctions with private uncertainty and resale opportunities.Journal of Economic Theory, 108, 72–110.

Hu, B., Beil, D. R., & Duenyas, I. (2013). Price quoting strategies of an upstreamsupplier. Management Science, 59, 2093–2110.

Jobber, D. (2010). Principles and practice of marketing (6th ed.). New York, NY:McGraw Hill.

Kiefer, L., & Carter, D. (2009). Global marketing management (2nd ed.). Oxford, UK:Oxford University Press.

Kirkegaard, R. (2012). A mechanism design approach to ranking asymmetricauctions. Econometrica, 80(5), 2349–2364.

Krisna, V. (2010). Auction theory. Burlington, MA: Academic Press.Lovejoy, W. S. (2010). Bargaining chains. Management Science, 56(12), 2282–2301.Maskin, E., & Riley, J. G. (2000). Asymmetric auctions. Review of Economic Studies, 67,

413–438.Ministry of Economic Development and Trade, International trade and Marketing

Division, Toronto, Ontario, Canada (2010). Getting ready to export guide.<http://www.sse.gov.on.ca/medt/ontarioexports/Documents/English/Getting_Ready_to_Export_Guide.pdf>.

Mokherjee, D., & Tsumagari, M. (2004). The organization of supplier networks:Effects of delegation and intermediation. Econometrica, 72(4), 1179–1219.

Osborne, M. J., & Rubinstein, A. (1994). A course in game theory. Cambridge, MA: MITPress.


http://refhub.elsevier.com/S0377-2217(14)00186-6/h0165














http://www.eur-lex.europa.eu/LexUriServ/LexUriServ.do?uri=OJ:L:2004:134:0114:0240:EN:PDF



https://www.acquisition.gov/far/05-53/pdf/FAR.book.pdf

https://www.acquisition.gov/far/05-53/pdf/FAR.book.pdf




http://www.ec.europa.eu/internal_market/publicprocurement/docs/guidelines/services_en.pdf

http://www.ec.europa.eu/internal_market/publicprocurement/docs/guidelines/services_en.pdf

http://www.ec.europa.eu/internal_market/publicprocurement/docs/guidelines/supply_en.pdf

http://www.ec.europa.eu/internal_market/publicprocurement/docs/guidelines/supply_en.pdf























http://www.sse.gov.on.ca/medt/ontarioexports/Documents/English/Getting_Ready_to_Export_Guide.pdf

http://www.sse.gov.on.ca/medt/ontarioexports/Documents/English/Getting_Ready_to_Export_Guide.pdf







Rosenbauer Group website. <http://www.rosenbauer.com/fr/group/unternehmen/standorte/vertriebs-und-servicestandorte.html>.

Shaked, M., & Shanthikumar, J. G. (2010). Stochastic orders. Springer series instatistics. New York, NY: Springer.

U.S. General Services Administration. Instructions for offerors – Submission of offers.Letter of supply (sample). <http://www.sws.gsa.gov/sws- search/view


SolDocument.do?method=view&solNum=M0ZOSi1DMS0wMDAwMDEtQg==&solRefresh=MTk=&solDoc=MDkwMDMwMzk4MDIwZDYxYw==>.

Wang, J. C., Lau, H. S., & Ling, A. H. L. (2008). How a retailer should manipulate adominant manufacturer’s perception on market and cost parameters.International Journal of Production Economics, 116, 43–60.

Zheng, C. H. (2002). Optimal auction with resale. Econometrica, 70(6), 2197–2224.


http://www.rosenbauer.com/fr/group/unternehmen/standorte/vertriebs-und-servicestandorte.html

http://www.rosenbauer.com/fr/group/unternehmen/standorte/vertriebs-und-servicestandorte.html



http://www.sws.gsa.gov/sws-%20search/viewSolDocument.do?method=view%26solNum=M0ZOSi1DMS0wMDAwMDEtQg==%26solRefresh=MTk=%26solDoc=MDkwMDMwMzk4MDIwZDYxYw==






