Analyzing online B2B exchange markets: Asymmetric cost and incomplete information

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Decision Support

Analyzing online B2B exchange markets: Asymmetric cost

and incomplete information

Yung-Ming Li

a,⇑

, Jhih-Hua Jhang-Li

b

a_{Institute of Information Management, National Chiao Tung University, Hsinchu 30010, Taiwan} b

Department of Information Management, Hsing Wu Institute of Technology, New Taipei City 24452, Taiwan

a r t i c l e

i n f o

Article history:

Received 31 December 2009 Accepted 16 May 2011 Available online 24 May 2011 Keywords: Auctions Economics Game theory Online exchanges Supplier competition

a b s t r a c t

This research applies the discriminating auction to analyze the online B2B exchange market in which a single buyer requests multiple items and several suppliers having equal capacity and asymmetric cost submit bids to compete for buyer demand. In the present model, we examine the impact of asymmetric cost and incomplete information on the participants in the market. Given the complete cost information, each supplier randomizes its price and the lower bound of the price range is determined by the highest marginal cost. In addition, the supplier with a lower marginal cost has a larger considered pricing space but ultimately has a smaller equilibrium one than others with higher marginal costs. When each supplier’s marginal cost is private information, the lowest possible price is determined by the number of suppliers and the buyer’s reservation price. Comparing these two market settings, we find whether IT is beneficial to buyers or suppliers depends on the scale of the bid process and the highest marginal cost. When the number of suppliers and the difference between the highest marginal cost and the buyer’s reservation price are sufficiently large, each supplier can gain a higher profit if the marginal costs are pri-vate information. On the contrary, when the highest marginal cost approaches the buyer’s reservation price, complete cost information benefits the suppliers.

1. Introduction

In most real-world markets, information technology is used to analyze data and manipulate information. As the world’s economy becomes increasingly competitive, more and more transactions are being processed by computers. With the growth of the Internet and commercial web-based applications, a B2B (Business to Business) exchange has been universally recognized as an online platform that creates a trading marketplace linked by the Internet and offers

substantial cost savings to buyers (Zhu, 2004). Based on a recently

released IDC report, it is estimated that business transactions on

the internet will swell to 450 billion a day by 2020 (Johnston,

2010). Currently, there are many successful B2B exchange portals,

such as BusyTrade.com, Global Sources, and IndiaMarkets. In these B2B exchange portals, a buyer can post his/her offer, and suppliers bid their selling prices to compete for demand.

The bidding process, also known as online procurement auction, usually composed of one buying ﬁrm and many suppliers, is one of the popular exchange mechanisms helping business buyers pro-cure better contract terms and prices for the goods and services they purchase. In online procurement auctions, buyers host the

on-line auction and invite potential suppliers to bid on announced

request-for-quotations (RFQs) (Parente et al., 2001). Empirical

evi-dences have revealed the importance of online procurement auc-tions on B2B exchange markets. For example, software maker Ariba helps a major fast food company set up an online auction

for suppliers to bid, reducing the total cost to $30 million (Yang,

2009). Before adopting the auction, the company was spending

$40 million on 13 different suppliers with about 30 different con-tracts. Recently, many public departments have also adopted the same approach to achieve competitive pricing for public projects and outsourcing contracts, such as street maintenance projects

and power purchase agreements (Louisville, 2009; Isensee, 2009).

Bandyopadhyay et al. (2005)study a buy-centric market, which is common in online B2B exchanges, in which suppliers are bidding for the business of large buyers. By examining IndiaMarkets, a B2B

exchange portal, Bandyopadhyay et al. (2005) conﬁrm that the

supplier submitting the lowest bid price will be ﬁrst invited to car-ter to the demand, followed by the supplier with the second-low-est bid price, and so on, until the demand is satisﬁed. Through the mechanism of reverse auctions implemented on the B2B exchange portal, a buyer can select the best offer in a time bound manner, and suppliers can lower the cost for selling and marketing. A series of studies have been conducted by them to provide a theoretical basis for understanding the effect of costs and capacities on B2B

exchanges (Bandyopadhyay et al., 2006, 2008). Procurement

auc-tions usually involve a sealed bid due to difﬁcult bid evaluation

⇑Corresponding author. Address: Institute of Information Management, National Chiao Tung University, Management Building 2, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, ROC. Tel.: +886 3 5712121x57414; fax: +886 3 5723792.

E-mail addresses:yml@mail.nctu.edu.tw,ymli@iim.nctu.edu.tw(Y.-M. Li).

Contents lists available atScienceDirect

European Journal of Operational Research

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(Cramton, 1998). Within a buy-centric B2B exchange framework, they analyze the case of competition among sellers, each of which cannot fulﬁll the entire market’s demand, and focus on the mech-anism of a one-sided, sealed-bid reverse auction.

Since no supplier can fulﬁll the entire market’s demand, in the two-supplier case where overall capacity is greater than demand, the low priced supplier can sell its entire capacity, whereas the high priced supplier has residual demand which is less than its capacity. Much recent economic research in residual demand is associated with the discriminating auction format, in which each winning supplier sells its capacity at the actual price bid, rather

than a single price common to all suppliers (Fabra et al., 2006).

For example, in the England and Wales electricity market, the dis-criminating auction is adopted as the solution for determining electricity supply prices. Before each period that the market is open, the generating companies, such as National Power and Nu-clear Electric, submit their bids to the National Grid Company. When demand is sufﬁciently large, the generators have incentive to raise bids above their marginal costs, and thus prices cannot

get driven down toward marginal costs (Fehr and Harbord, 1993).

1.1. Motivation and problems

Motivated by the research initiated by Bandyopadhyay et al.

(2005, 2006, 2008), we explore the impact of cost information on participants in online B2B exchange markets, in which suppliers submit their bids under the consideration of residual demand. When each supplier’s cost information is common knowledge, the discriminating auction is essentially a Bertrand-Edgeworth

game (Deneckere and Kovenock, 1996). However, the problem is,

as pointed out by Edgeworth, an equilibrium (in pure strategies)

in such a game may not exist (Vives, 1999). Bandyopadhyay

et al. (2008)model the competition between identical suppliers vying for the same business in the discriminating auction and ﬁnd the suppliers may randomize their prices to form a mixed strategy Nash equilibrium. In their setting, each supplier has identical mar-ginal cost, which is common knowledge among all participants in the auction.

In the present study, we consider each supplier can have asym-metric marginal cost and the information can be common knowl-edge or privately owned. In reality, each supplier in B2B marketplaces cannot know other potential competitors’ cost struc-tures in advance unless information transparency can be improved. Prior studies indicate that information systems can improve

infor-mation transparency (Zhu, 2004; Singh et al., 2005; Granados et al.,

2006). For example, each supplier in B2B marketplaces may infer

other potential competitors’ cost structures in advance by exploit-ing the technique of data minexploit-ing and artiﬁcial reasonexploit-ing, both of which rely on information systems such as data warehousing and expert systems. In addition, a buyer may have incentive to learn cost information and implicitly distribute it by inviting buyers with cost advantage to participate in the online procurement auction. The investment in IT could be expensive; as a result, it is critical for buyers and suppliers to better understand the value of IT in the auction. Therefore, the purpose of this study is to analyze the effect of cost information in a tractable model designed to capture some of the key features of an online bid process. In addition, we seek to provide valuable analytical insights into how the auction is executed in B2B marketplaces.

1.2. Findings and contributions

Comparing these two market settings, we have the following ﬁndings. First, given the complete cost information, each supplier would randomize its price and the lower bound of the price range is determined by the highest marginal cost. Thus, when the highest

marginal cost approaches the buyer’s reservation price, each sup-plier would submit a sufficiently high bid which is close to the buyer’s reservation price. In addition, the supplier with a lower marginal cost has a larger considered pricing space but ultimately a smaller equilibrium one than others with higher marginal costs. Second, given the incomplete cost information, each supplier bids a price only according to its individual marginal cost and the number of suppliers in the bid process would significantly affect their prof-its. When the number of suppliers in the bid process increases, it is possible each supplier submits a higher bid. The bids would con-verge to the buyer’s reservation price when the number of suppli-ers is sufficiently large. Further, we find a supplier would have a higher profit when its capacity increases or its competitors’ capac-ities decrease.

Comparing the suppliers’ profits in the two different market set-tings, we find whether IT is beneficial to buyers or suppliers de-pends on the scale of the bid process and the highest marginal cost. When the number of suppliers and the difference between the highest marginal cost and the buyer’s reservation price are suf-ficiently large, each supplier can gain a higher profit if the marginal costs are private information. On the contrary, when the highest marginal cost approaches the buyer’s reservation price, complete cost information is beneficial for the suppliers. The rest of this pa-per is organized as follows. In the next section, we review prior

re-search and highlight their contribution. In Section3, we introduce

our model. In Sections 4 and 5, we derive mixed strategy Nash

Equilibrium and Bayesian Nash equilibrium, respectively. In

Sec-tion6, we analyze the impact of the cost information on

partici-pants in the online B2B exchange market. Finally, we conclude

our research and address future study in Section7.

2. Related work 2.1. B2B market

In the past few years, many neutral B2B exchanges have strug-gled to survive and many successful B2B markets are buyer-owned (Yoo et al., 2007). A successful business to business (B2B) elec-tronic market has to attract enough participants. In a single buyer electronic market, lower price plays an important role as an order

winning criterion (Lee et al., 2009). Assuming electronic

market-places provide positive network effects,Yoo et al. (2007)examine

the role of the ownership structure of electronic marketplaces on various participants in the marketplace. Their study shows a buyer-owned marketplace where some buyers jointly own the marketplace can provide greater beneﬁts to participants than a

neutral marketplace owned by an independent third party. Aron

et al. (2008)point out the creation of a private electronic exchange by the large producer will result in signiﬁcant welfare loss when upstream suppliers are highly efﬁcient.

2.2. Online auctions and residual demand

For games in which the payoff functions are discontinuous,

Dasgupta and Maskin (1986a)proved the existence of a mixed-strategy equilibrium in such games. The empirical evidence of the mixed-strategy equilibrium can be found in retail markets in

which each store changes its price over time (Varian, 1980).Fabra

et al. (2006)studied bidding behavior and market outcomes in the uniform-price and discriminating electricity auctions. In their basic duopoly model, two independent suppliers with asymmetric mar-ginal costs and limited capacities must submit a single price offer for its entire capacity. If demand is low, there is a pure equilibrium in the two auction formats in which both suppliers submit offer prices equaling the cost of the inefﬁcient supplier (i.e., the high

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cost). However, only mixed-strategy equilibria exist in high-de-mand realizations, in which the higher-bidding supplier’s capacity is then dispatched to serve the residual demand. As for online auc-tion environments, policy makers should focus on the design and frequency of the auction or the amount of real-time information

made available to market participations (Puller, 2007).

If a buyer decides to procure multiple items at one go in a single auction, there are many possibilities, such as competitive auctions (also known as uniform-price auctions), discriminating auctions

(also known as pay-your-bid auctions), and Vickrey auctions.

Har-ris and Raviv (1981)show the expected revenues generated from the discriminating and the competitive auctions are the same if the bidders are risk neutral. Moreover, some studies have pointed out collusion is less likely under discriminatory auctions than

un-der uniform-price auctions (Klemperer, 2002; Fabra, 2003). In

recent years there has been considerable interest in the multi-item auction in which different products or services are auctioned

simultaneously (Jin et al., 2006; Teich et al., 2006; Mishra and

Veeramani, 2007). Dasgupta and Maskin (1986b) illustrate an application of their research in which the ﬁrm quoting the lower price serves the entire market up to its capacity, and the residual demand is met by the other ﬁrm. The existence of equilibria for

all combination of capacities is founded on a mixed-strategy.

Ban-dyopadhyay et al. (2008)analyze the scenario in which several identical suppliers are vying for business from a single larger buyer in a B2B exchange framework. By simulating suppliers in a syn-thetic environment, they ﬁnd the agents altering their pricing strategy over time do indeed converge toward the theoretical Nash

equilibrium (Bandyopadhyay et al., 2006).

2.3. Unique feature comparing with existing literatures

In the present study, we analyze a buyer-owned B2B market in which a buyer endeavors to procure multiple identical items sup-plied by several ﬁrms with heterogeneous marginal costs, and each can submit a sealed bid in the buyer’s procurement process. A un-ique feature of our approach is each supplier’s cost information can be common knowledge or privately owned. Compared with the

model studied by Bandyopadhyay et al. (2008), our model has

the following different features. First, to measure the value of IT in online B2B exchange markets, we analyze the impact of sharing cost information on buyer’s expected cost and supplier’s expected profit. Second, because each supplier’s marginal cost is not identi-cal in the present study, the impact of asymmetric cost on each supplier’s pricing strategy can be investigated. In addition, we also discuss the impact of heterogeneous capacity on suppliers’ pricing strategies when each supplier has private cost information. Most economics literatures utilize the cost to the buyer and profit to the supplier to measure the performance of auction mechanisms (Teich et al., 2004). We also adopt the same criteria to measure the impact of cost information on buyer’s expected cost and sup-plier’s expected profit in the auction. Based on our derived results, we can further understand the value of information technology and the function of residual demand in online B2B exchange markets.

3. The model

We consider the online B2B exchange market in which there are one buyer and n suppliers vying for the cumulative demand Q. For the level of demand, some of prior studies associated with

Ber-trand-Edgeworth equilibrium consider it a function of price (Allen

and Hellwig, 1986; Osborne and Pitchik, 1986; Deneckere and Kovenock, 1996), while some of them consider that it is

indepen-dent of the market price (Bandyopadhyay et al., 2005; Fabra

et al., 2006; Sakellaris, 2010). One of the reasons to model price-inelastic demand can be attributed to the fact that observation in electricity markets shows that the price elasticity of demand is very low, which forms a common assumption in electricity market

literature (Fehr and Harbord, 1993; Crampes and Creti, 2005; Fabra

et al., 2006; Sakellaris, 2010). Another reason is that the assump-tion of inelastic demand has been veriﬁed by IndiaMarkets.com,

India’s largest exchange market.Bandyopadhyay et al. (2005)

con-ﬁrm that the demand for materials posted on the B2B portal will tend to be inelastic. In essence, the present study concentrates on online B2B exchanges, so we enhance their model and inherit the veriﬁed assumption on the level of demand.

For simplicity, the suppliers are denoted as Siwhere 1 6 i 6 n. A

summary of all the variables used in the present paper appears in

Table 1. The supplier Si’s individual marginal cost is denoted as ci.

The buyer’s reservation price is denoted as r and the suppliers have equal capacities k. The assumption of equal capacities in the

pres-ent model is the same as that studied byBandyopadhyay et al.

(2008). The buyer will not accept any bid higher than its reserva-tion price.

Here, we consider the capacity constraint (n 1)k < Q < nk. That is, each supplier can sell its entire capacity other than the one sub-mitting the highest bid supplies to the residual demand. Given the complete cost information, if nk 6 Q, each supplier can sell its en-tire capacity as long as its bid is not larger than the buyer’s reser-vation price. As a result, each supplier bids the buyer’s reserreser-vation price. In the case where (m 1) < Q < mk with n > m, (n m) sup-pliers do not supply anything in this bidding game. It is definitely a profitable move for a new supplier to join the bidding. Accord-ingly, the positive equilibrium profits for some suppliers are de-stroyed. However, in any equilibrium system marginal price cannot exceed the marginal cost of the m + 1st most efficient

sup-plier if n > m (Fehr and Harbord, 1993, p. 535). Since the case where

only a fraction of the suppliers can sell is enough to result in a

Bertrand competition (Bandyopadhyay et al., 2008), we are

inter-ested in the case (the supplier with the highest price is at least able to sell the residual) where pure equilibrium does not exist. For con-sistency and comparability, we adopt the capacity constraint throughout the present study to examine the impact of cost infor-mation on the buyer and suppliers.

Therefore, the supplier Sibidding a price pican gain the proﬁt

(pi ci)(Q (n 1)k) or (pi ci)k, depending on whether its price

is the highest. If the suppliers submit different bids, the lower-bid-ding supplier’s capacity is dispatched ﬁrst. If there are multiple suppliers submitting equal bids, we adopt the same approach as

proposed byFabra et al. (2006). That is, when the case arises, the

service rank is determined by the supplier’s marginal cost and the supplier with a lower marginal cost is ranked ﬁrst. In addition, in most auction literature, when the suppliers bidding the same

Table 1 Notations.

Notation Description

Q Total quantity demanded

k Capacity of each supplier

ci Constant marginal cost of each supplier i r The reservation price of the buyer

F Supplier’s cumulative density function of prices f Supplier’s probability density function of prices fYj The jth order statistic

n The number of suppliers

UI ,UC

Buyer’s expected cost

p_ðÞ;_p

i Supplier’s expected proﬁt

pi A speciﬁc threshold in which Gi(pi) = 1, where 1 6 i 6 n 1 p⁄

() The symmetric equilibrium bidding function

Fij Supplier i’s cumulative density function of prices in[pj1, pj) Gi() Supplier i’s cumulative density function

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price have the same marginal costs, a random draw will determine their service rank. In fact, because there are no point masses in the equilibrium bidding functions when the suppliers have the same marginal cost, we can omit the consideration.

4. An online B2B exchange market with public cost information In the section, we consider the setting in which each supplier’s marginal cost is common knowledge. To begin with, we

demon-strate the simple case in which the two suppliers, SHand SL,

com-pete for the cumulative demand Q. The idea of the example is helpful to build the mixed strategy Nash equilibrium in the B2B ex-change market with n suppliers.

4.1. A two-supplier auction game

In the two-supplier auction game, both suppliers have equal capacity k and the inequality k < Q < 2k holds. The winner can sup-ply its total capacity k, whereas the loser can only sell the residual

demand Q k. The supplier SH’s (SL’s) bid and marginal cost are

denoted as pH(pL) and cH(cL), respectively. The supplier SHhas the

highest marginal cost; that is, 0 6 cL< cH6r. There can be no

pure-strategy equilibrium for the two-supplier auction game if both suppliers have the same marginal costs, which has been

proved by Bandyopadhyay et al. (2008). If both suppliers have

asymmetric marginal costs in the two-supplier auction game, we

can derive the same result by considering ^p cHþ ðr cHÞðQ

kÞ=k.

Therefore, we seek the mixed strategy Nash equilibrium in the two-supplier auction game and then examine suppliers’ pricing strategies. In the mixed-strategy equilibrium, the goal is to ﬁnd

cumulative distribution functions FH() and FL() so the suppliers,

SHand SL, are able to gain indifferent expected proﬁts by

submit-ting their bids according to FH() and FL(), respectively. We adopt

the approach similar to that inFabra et al. (2006) to construct

the cumulative distribution functions as follows. For conciseness,

given the supplier Si, we use Fi() to denote the cumulative

distri-bution function of the other supplier. If the two suppliers bid

according to FH() and FL(), their expected proﬁts are given by

p

iðpÞ ¼ ðp ciÞðkð1 FiðpÞÞ þ ðQ kÞFiðpÞÞ; where

i 2 fL; Hg: ð4:1Þ

Lemma 1. In the two-supplier auction game, FLð^pÞ ¼ FHð^pÞ ¼ 0 and

FL(r) = FH(r) = 1 under the mixed-strategy equilibrium. All the proofs

can be found in the Appendix.

A necessary condition for the supplier SHto be indifferent

be-tween any price in ½^p; r is, for all p 2 ½^p; r;

p

HðpÞ ¼

p

HðrÞ. Note, the

supplier SHwould certainly supply the residual demand by setting

the price at r. That is,

p

H(r) = (Q k)(r cH). Using(4.1)and the

nec-essary condition, we can obtain the expression for FL(p) as follows:

FLðpÞ ¼ðp cHÞk ðr cHÞðQ kÞ

ðp cHÞð2k Q Þ

: ð4:2Þ

Note, FL(p) is a continuous function on ½^p; r. Indeed, the

prop-erty of FL(p) ﬁts our requirement given in Lemma 1. However,

p cLþ ðr cLÞðQ kÞ=k is the lower limit of pL. Thus, a

straight-forward evaluation of FH(p) with FHð^pÞ ¼ 0 would imply FH(r) < 1.

To compensate for this, Pr (pH= r) has a positive probability h, i.e.,

FH(r) = 1 h. Now, using this value and the equation

p

LðpÞ ¼

limp!r

p

LðpÞ, we have FHðpÞ ¼ ðpcLÞkðrcLÞðQkÞð2kQ ÞðrcLÞh ðpcLÞð2kQÞ ; ^p 6 p < r; 1; p ¼ r; ( ð4:3Þ where h ¼cHcL

rcL is derived by solving FHð^pÞ ¼ 0. According to the

rules described in Section3, the supplier SLis dispatched ﬁrst when

both suppliers bid p = r. Therefore, the equilibrium proﬁts become

p

H¼ ðr cHÞðQ kÞ ¼ ð^p cHÞk;

p

L¼ ðr cLÞfPrðpH<rÞðQ kÞ þ PrðpH¼ rÞkg ¼ ð^p cLÞk:

ð4:4Þ

Proposition 1. In the two-supplier auction game, the probability the

supplier with the higher marginal cost bids pH= r increases with the

difference between the two suppliers’ marginal costs. Formally, @h/ oD> 0, whereD cH cL.

As inBandyopadhyay et al. (2008), both suppliers would bid

according to the same cumulative distribution function when they

have the same marginal costs. Note, supplier SLcan bid a price

low-er than ^p if cH> cLholds. However, in this case, supplier SHwould

always obtain the residual demand. That is, because of the

disad-vantage of the higher marginal cost, the supplier SH has to bid

pH= r with a positive probability to ensure the supplier SLhas no

incentive to bid a price lower than ^p. Thus, supplier SHalso has a

chance of supplying its total capacity by randomizing its price on

½^p; rÞ. As for the supplier SL, it just simply randomizes its price on

½^p; r to ensure the supplier SH can receive indifferent expected

proﬁt for all pH2 ½^p; r.

Because the supplier SH’s proﬁt under the mixed strategy Nash

equilibrium is positively proportional to the residual demand, we

have o

p

H/ok < 0. However, examining o

p

L/ok, we have the following

interesting ﬁnding.

Proposition 2. In the two-supplier auction game, the supplier SL’s

equilibrium proﬁt may increase (decrease) with the given capacity

when the supplier SH’s marginal cost is higher (smaller) than a speciﬁc

threshold. Formally, o

p

L/ok P 0 if and only if cHP(cL+ r)/2.

In fact, the equilibrium proﬁt of the supplier SLcan be rewritten

as follows:

p

L¼

p

Hþ ðcH cLÞk: ð4:5Þ

That is, the proﬁt of the supplier SLis the proﬁt of the supplier SH

plus (cH cL)k. As the given capacity increases, supplier SLhas to

randomize its price within a pricing space containing smaller prices

than before. In other words, @^p=@k < 0. Since the proﬁt of supplier

SHis inversely proportional to the given capacity, supplier SL’s proﬁt

may increase or decrease with the given capacity, depending on whether its cost advantage can dominate the effect of competition. 4.2. Extending the model to a n-supplier auction game (n P 2)

Suppose there are n suppliers competing for the cumulative de-mand Q. All suppliers have equal capacities k and the inequality

(n 1)k < Q < nk holds. Supplier Si’s marginal cost is given by ci

where 0 6 c1< c2< < cn6r. Based on the approach of the

two-supplier auction game, we can determine the bottom of the range of the active bids for all suppliers as follows:

p0 cnþ ðr cnÞðQ ðn 1ÞkÞ=k: ð4:6Þ

Lemma 2. Suppose each supplier bids a random price according to

the cumulative density function Gi(), forming a mixed strategy Nash

equilibrium. Denoting Pias the support of Gi(), we have Pi#[p0, r]

for 1 6 i 6 n.

Because ofLemma 2, we can only put attention on the speciﬁc

closed set [p0, r] to ﬁgure out the formulation of Gi(). In the

follow-ing, we ﬁrst give the complete formulation of Gi(), and then show

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if each supplier Sibids a random price on [p0, pi] according to the

cumulative density function Gi(p). The formal formulation of Gi()

is given by

8

i ¼ 1; 2; . . . ; n GiðpÞ ¼ Fi;jðpÞ; pj16p < pj; 1; pi6p; _ð4:7Þ where 1 6 j 6 minfi; n 1g; ð4:8Þ Fi;jðpÞ ¼ kðp p0Þðp ciÞnj1 ðnk Q ÞQd–i;dPjðp cdÞ !1 nj ; ð4:9Þ

Fi;iðpiÞ ¼ 1 for 1 6 i 6 n 1 and pn¼ r: ð4:10Þ

Clearly, Gi() is a piecewise function composed of Fi,j(p), in which

1 6 j 6 min{i, n 1}. To examine the validity of Gi(), we have to

check whether the relation p0< p1< < pn1holds.

Lemma 3. p0< p1< < pn1= r if Fi,i(pi) = 1 for 1 6 i 6 n 1.

Moreover, Fi,j(p) is an increasing function.

The function Fi,j(p) is the key component of the mixed strategy

Nash equilibrium in the n-supplier auction game. The basic idea

of the piecewise functions is as follows. To ﬁnd Gi() so the

equilib-rium proﬁt of each supplier is indifferent among all prices higher

than p0, we can follow the approach of the two-supplier auction

game to derive each Gi(). However, if Gi() is not a piecewise

func-tion, we can ﬁnd p1 which satisﬁes G1(p1) = 1 is less than the

buyer’s reservation price r when n P 3, implying Gi() must be a

piecewise function. Thus, we have to adopt the piecewise approach

to construct Gi(). The difference between the n-supplier auction

game (n P 3) and 2-supplier auction game is we only consider

par-tial suppliers in each segment other than the ﬁrst segment [p0, p1].

For example, we solve all suppliers in the ﬁrst segment [p0, p1], but

solve the only two suppliers, Sn1 and Sn, in the ﬁnal segment

[pn2, pn1]. In the case of the ﬁnal segment, because we have

limp!rFn;n1ðpÞ < 1, supplier Snhas to set its price at p = r with a

po-sitive probability to compensate this. Because supplier Snhas the

same pricing space as supplier Sn1, we use the constraint(4.8)

to treat with the concern.

Next, because p0is the bottom of the range of active bids for all

suppliers and each supplier’s proﬁt has to be indifferent among the support of prices, each supplier’s expected proﬁt is given by

p

i¼ kðp0 ciÞ: ð4:11Þ

Lemma 4. For the suppliers Sj, Sj+1, . . . , and Sn, if they are the only

suppliers bidding on [pj1, pj) and each supplier Si, where j 6 i 6 n,

bids p

[pj1, pj) according to the cumulative distribution function Fi,j

(p), then each supplier’s expected proﬁt on [pj1, pj) is

p

i.

Relying onLemma 4, the ﬁnal task is to check whether each

supplier has no incentive to deviate the mixed strategy Nash

equi-librium derived from Gi(p).

Proposition 3. In the n-supplier auction game, there is a mixed

strategy Nash equilibrium if each supplier Sibids a random price in

[p0, pi] according to the cumulative density function Gi(p).

Compared with the case discussed by Bandyopadhyay et al.

(2008), our results display an interesting ﬁnding. It is considered the supplier with cost advantage would bid as high as possible.

However, for the case of n P 3, supplier Si, where i 6 n 2, with

a cost advantage wouldn’t bid a price higher than the speciﬁc

threshold pi. Actually, a price higher than the speciﬁc threshold pi

would increase the proﬁt of supplier Siwhen it is not the highest

bidder; however, the price also results in a higher probability of

getting residual demand for supplier Si. When supplier Si’s bid is

higher than pi, the increase in the proﬁt resulting from a higher

price is dominated by the loss of getting residual demand.

There-fore, the highest possible bid submitted by supplier Siis less than

that submitted by other suppliers whose marginal costs are higher than supplier Si’s.

Note, if there are multiple suppliers having the same marginal costs, they would bid prices according to the same cumulative dis-tribution functions. Recently, with two suppliers, the case of heter-ogeneity in both costs and capacities has been studied by

Bandyopadhyay et al. (2005), in which the lowest support price de-pends on the relative rankings of indifference prices, where each supplier has its own indifference price, being indifferent between bidding the highest price to sell residual demand and the

indiffer-ence price to sell all capacity. Moreover,Fabra et al. (2006)

demon-strate the interesting case in which three suppliers differ in both costs and capacities. In this case, there exist multiple pure equilib-ria when uniform auction is adopted. Generally speaking, if all sup-pliers have unequal capacities and asymmetric costs, this complicates the analysis although it may introduce some sort of new phenomena. The analysis for the case becomes more com-plex and less tractable for more than the scenario of two suppliers (Bandyopadhyay et al., 2005); therefore, in this research we only treat the issue of heterogeneity in costs when extending to the n-supplier case. The extension to the n-n-supplier case of heterogeneity in both costs and capacities remains an open question in various

domains. A numerical example of Gi(p) is shown inFig. 1in which

ci= i/10, n = 5, k = 5, r = 1, and Q = 21. Subsequently, the buyer’s

ex-pected cost is given by

U

C¼X n i¼1

p

iþ Xn1 i¼1 Z pi p0 ci p0 ci p ci kdGiðpÞ þ lim ~ p!r Z ~p p0 cn p0 cn p cn kdGnðpÞ þ cnðcn cn1ÞðQ ðn 1ÞkÞ r cn1 ð4:12Þ

The ﬁrst term ofUC_{is the sum of each supplier’s expected}

prof-it, whereas the second term ofUCis the sum of each supplier’s

ex-pected cost. Note, the supplier with the highest marginal cost

would randomize its price in [p0, r) and bid r with a positive

prob-ability; therefore, we need the third and fourth terms to calculate

Sn’s expected cost. On the other hand, because Gi(p) is a piecewise

function, there is no general closed form for the buyer’s expected cost in the n-supplier auction game. Thus, by numerical integra-tion, we can derive the suppliers’ expected costs and then utilize the results to ﬁgure out the suppliers’ expected revenues (i.e., the

buyer’s expected cost). For the suppliers (Si, 1 6 i 6 n), their

ex-pected costs are given by 0.4964, 0.9777, 1.3851, 1.5579, and

Price

G1 G2 G3 G4 G5

p

₀

p

₁

p

₂

p

₃

p

₄ Fig. 1. Graph of Gi() p0= 0.6, p1= 0.62, p2= 0.65, p3= 0.77, p4= 1.

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1.2637, respectively. Further, the buyer’s average expected cost

(i.e., UC_{/Q) is about 0.628 in the case. We can observe supplier}

S5’s expected cost is less than S4’s and S3’s, although supplier S5’s

marginal cost is higher than S4’s and S3’s. The result stems from

supplier S5 having more chance than others to get residual

de-mand; thus, we have this interesting ﬁnding.

5. An online B2B exchange market with private cost information

In reality, supplier’s cost information should be private and each supplier only knows the distribution of the supplier’s marginal cost. In the online auction, each supplier submits a sealed-bid dur-ing a speciﬁc period of time. Here, we emphasize the construction of a strictly increasing bidding function that is optimal for each supplier to employ, given all other suppliers also employ this bid-ding function. The symmetric equilibrium bidbid-ding function is

de-noted as p(). Clearly, for every i, ci6p(ci) 6 r.

We assume the marginal costs of the suppliers are independent and identically distributed, drawn from the common distribution

F(ci) with F(0) = 0, F(r) = 1 and F(ci) strictly increasing and

differen-tiable over the interval [0, r]. The assumption was ﬁrst presented by Vickrey, and has been frequently employed in the bidding liter-ature. Based on the approach presented by Vickrey, the probability

bisubmitted by the supplier Siexceeds the bids of all other

suppli-ers is given by Fn1_(b

i). Thus, if the supplier Sichooses bias its bid,

the expected proﬁt of the supplier Siis given by

p

i¼ ðpðbiÞ ciÞfð1 Fn1ðbiÞÞk þ Fn1ðbiÞðQ ðn 1ÞkÞg

¼ ðpðbiÞ ciÞfk þ Fn1ðbiÞðQ nkÞg: ð5:1Þ

The Bayesian Nash equilibrium is given by solving o

p

i/obi= 0 and

setting bi= ci. Letting f denote the probability distribution function

associated with the cumulative distribution function F. Then, we have the following equation:

@ @ci

pðciÞfk þ Fn1ðciÞðQ nkÞg ¼ ciðn 1ÞFn2ðciÞf ðciÞðQ nkÞ:

ð5:2Þ Integrating both sides, we have

pðciÞfk þ Fn1ðciÞðQ nkÞg ¼ ðn 1ÞðQ nkÞ

Z ci

0

Fn2_{ðxÞxf ðxÞdx þ C:} _ð5:3Þ

Applying the condition p⁄_{(r) = r, we can derive C as follows:}

C ¼ rk þ ðQ nkÞ r Z r 0 xdFn1ðxÞ : ð5:4Þ

Thus, the symmetric equilibrium bidding function is given by p_ðc iÞ ¼ rk þ ðQ nkÞðr þRci 0 xdF n1 ðxÞ R0rxdF n1 ðxÞÞ k þ Fn1ðciÞðQ nkÞ : ð5:5Þ

To derive the analytical results, we assume each supplier’s marginal cost is uniformly distributed on [0, r]. Consequently, the strictly increasing symmetric bidding function is

p_ðc iÞ ¼ rn_{k þ ðQ nkÞ} ðn1Þcni n þ rn n rn1_{k þ c}n1 i ðQ nkÞ : ð5:6Þ

Relying on the symmetric bidding function, the supplier Si’s

ex-pected proﬁt is

p

_ðc iÞ ¼ ðr ciÞk þ ðQ nkÞ rn_cn i nrn1 : ð5:7Þ

Proposition 4. In the auction in which each supplier has private cost information drawn from common uniform distribution, each supplier’s price and proﬁt decrease with capacity k but increase with cumulative

demand Q. Formally, op⁄_(c

i)/ok 6 0, o

p

⁄(ci)/ok 6 0, op⁄(ci)/oQ P 0,

and o

p

⁄_(c

i)/oQ P 0.

Under the inequality (n 1)k < Q < nk, if capacity k increases, the amount of residual demand would decrease. Therefore, each supplier in the case would lower its price to prevent from becom-ing the highest bidder. Because each supplier adopts the same

strategy, the supplier Si’s expected proﬁt would decrease when

capacity k increases. On the contrary, when the cumulative de-mand Q increases, we have the opposite results because the amount of residual demand increases. Given r = 1, Q = 21, n = 5,

and k = 5, the numerical example of p⁄_(c

i) and

p

⁄(ci) are plotted

inFigs. 2 and 3.Fig. 2shows the bidding function p⁄_(c

i) is a strictly

increasing one, whereasFig. 3shows each supplier’s expected

prof-it decreases wprof-ith prof-its marginal cost.

Although we only consider equal capacities in the present study, we can, in fact, relax the assumption of equal capacities when each supplier has private cost information, discussing its implications in the following. If each supplier has asymmetric

capacity denoted as ki, the symmetric equilibrium bidding function

and the supplier Si’s expected proﬁt are given by

p_ðc iÞ ¼ rnkiþ Q Xn j¼1 kj ! ðn 1Þcn i n þ rn n ( ) , rn1_k iþ cn1i Q Xn j¼1 kj ! ( ) ; ð5:8Þ

p

_ðc iÞ ¼ ðr ciÞkiþ Q Xn j¼1 kj ! rn_cn i nrn1 : ð5:9Þ

Proposition 5. In the auction in which each supplier has asymmetric capacity and private cost information drawn from common uniform distribution, the impact of exogenous parameters on the symmetric equilibrium bidding function and each supplier’s expected proﬁt is

summarized inTable 2and can be summarized as follows:

1. Each supplier Si’s bid p⁄(ci) decreases with its capacity, ki, and

the capacities of others, kj, but increases with cumulative

demand Q and its marginal cost, ci.

2. Each supplier Si’s expected proﬁt

p

⁄(ci) decreases with its

mar-ginal cost, ci, and the capacities of others, kj, but increases with

its capacity, ki, and cumulative demand Q.

Margian cost

Fig. 2. Graph of p⁄ ().

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When each supplier can have asymmetric capacity, most results

in Proposition 5 remain the same as inProposition 4; however, for

supplier Si, it’s proﬁt increases with individual capacity but

de-creases with its competitor’s. Formally, o

p

⁄_(c

i)/okiP0 and

o

p

⁄_(c

i)/okj60. The reason for the results is as follows. No matter

which supplier’s capacity increases, each supplier lowers its prices

to respond. If one of the suppliers’ capacities increases (e.g., kj),

supplier Si’s proﬁt, of course, would decrease. However, if only

sup-plier Si’s capacity increases, the positive effect of the increasing

capacity dominates the negative effect of the decreasing price.

Therefore, supplier Sican gain a higher proﬁt when only its

capac-ity increases.

Subsequently, we want to know how much the buyer would pay in the online auction. From the order statistics, the buyer’s ex-pected total cost is given by

U

I¼ Z r 0 Xn1 j¼1 p_{ðxÞ k f} YjðxÞ þ p _{ðxÞðQ ðn 1ÞkÞf} YnðxÞ ( ) dx ¼ Z r 0 kX n1 j¼1 fYjðxÞ þ ðQ ðn 1ÞkÞfYnðxÞ ( ) p_ðxÞdx; ð5:10Þ where fYjðxÞ ¼ n! ðnjÞ!ðj1Þ!F j1_{ðxÞð1 FðxÞÞ}nj f ðxÞ.

The following formulation based on the binomial theorem is

helpful to derive the explicit form of(5.10).

Xn j¼1 fYjðxÞ ¼ Xn j¼1 n n 1 j 1 Fj1ðxÞð1 FðxÞÞnjf ðxÞ ¼n r: ð5:11Þ Because of fYnðxÞ ¼ n r x r n1

, the buyer’s expected total cost can be rewritten as follows:

U

I¼ Z r 0 kn r 1 x r n1 þ ðQ ðn 1ÞkÞnx n1 rn p_ðxÞdx ¼ r n n þ 1ð2Q kðn 1ÞÞ: ð5:12Þ We ﬁnd the buyer’s expected total cost increases with cumulative demand Q.

When cumulative demand Q increases, each supplier would bid a higher price since each can sell a higher amount of residual units even if losing the auction. This leads to an increase in the buyer’s expected total cost. Ideally, a small residual demand can serve as a strategy to make each supplier bid a higher price. The numerical

example ofUIis shown inFig. 4, in which the parameters are given

by r = 1, Q = 21, n = 5, and 4.2 < k < 5.25. Clearly, when the amount of each supplier’s capacity sold increases (i.e., the residual demand decreases), the buyer’s expected total cost decreases.

In the following, we consider the impact of the number of sup-pliers on the buyer’s expected total cost. Intuitively, we can

differ-entiateUI_{with respect to the number of suppliers, n, to examine}

the impacts. However, because the capacity constraint

(n 1)k < Q < nk must hold for all n P 2, the value of capacity would vary with the number of suppliers to satisfy the capacity constraint. Thus, we have to adopt other approaches to examine the effect of the number of suppliers in the auction.

Proposition 6. In the auction in which each supplier has private cost information, the buyer’s expected cost decreases with capacity k. Moreover, if the buyer can choose capacity k and the number of suppliers so the capacity constraint (n 1)k < Q < nk holds for all n P 2, the expected average cost per unit approaches the buyer’s reservation price when the number of suppliers is sufﬁciently large.

When the amount of capacity increases, each supplier hazards the decrease in the residual demand; therefore, the suppliers par-ticipating in the auction would lower their prices. From the buyer’s point of view, the number of suppliers should be as small as possi-ble to derive a competitive price, which is contrary to the ﬁndings of general competition study. For example, in Cournot competition, when the number of suppliers approaches inﬁnity, the price would fall to the level of the suppliers’ marginal costs. The reason for the difference is as follows. In the assumption of our model, only the highest bidder would obtain the residual demand. However, when the number of suppliers increases, the probability of becoming the highest bidder decreases; therefore, each supplier raises its bid to increase the buyer’s expected total cost. Thus, the buyer should limit the number of suppliers and raise the amount of capacity supplied by each bidder.

6. Managerial implications: the design of online B2B exchange market

The goal of this study is to provide a theoretical basis for under-standing the effect of asymmetric cost and incomplete information on B2B exchanges. In this research, in order to derive an explicit closed form of bidding function for further analytical analysis, we adopt the classic assumption of uniform distribution, which is Margian Cost

Fig. 3. Graph ofp⁄ ().

Table 2

Impact of exogenous parameters on bidding function and expected proﬁt.

ki kj Q ci p⁄ (ci) + + p⁄ (ci) + + + increase; decrease.

Capacity

Fig. 4. Graph ofUI_.

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frequently used in auction literature, although other distribution of the unit cost may result in different comparative results. Examin-ing the two settExamin-ings which differ in cost information, we ﬁnd that the buyer has an incentive to introduce as much competition as possible; however, our results also show the buyer’s expected cost may increase with the number of suppliers when the assumption of capacity constraints must hold (i.e., (n 1)k < Q < nk). In addi-tion, regardless of whether the supplier’s cost information is known, the impact of residual demand on supplier’s pricing strat-egy is the same. That is, each supplier would bid as high as possible when the highest bidder can supply nearly its entire capacity and lower the price when the residual demand is a small fraction of the suppliers’ capacity. Currently, the private exchange, a privately built electronic market by a single buyer attracting many suppliers into the buyer’s market, such as Wal-Mart, forms the largest part of

e-commerce (Laudon and Traver, 2008; Lee et al., 2009). To

strengthen the relationship between the buyer and suppliers, the function of residual demand can be considered the participation motivation of the suppliers. This research also points out residual demand can serve as a strategy to make each supplier bid a lower price in the online B2B exchange market.

For each supplier Si, when the cost information is private, its

proﬁt is only associated with its individual marginal cost. However, if the cost information is known, its proﬁt is not only associated with its individual marginal cost but also associated with the

high-est marginal cost. Therefore, comparing

p

⁄

(ci) with

p

i, we ﬁnd

p

i>

p

ðciÞ when n(r cn) < r ci(ci/r)n1 holds and

p

i<

p

ðciÞ

when the opposite holds true. The result shows complete cost

information beneﬁts supplier Siwhen the highest marginal cost

ap-proaches the buyer’s reservation price. Because the lower bound of the bid range of each supplier depends on the highest marginal cost when the cost information is known, the buyer would pay more in that case.

On the contrary, when the number of suppliers is large enough and the highest marginal cost is less than the buyer’s reservation

price, sharing cost information cannot improve the supplier Si’s

ex-pected proﬁt. The reason for this result is as follows. When mar-ginal cost is not common knowledge, each supplier bids according to its individual marginal cost; however, when the num-ber of suppliers grows, the chance of getting the residual demand is reduced. As a result, as long as the number of suppliers is

sufﬁ-ciently large, each supplier can gain a high expected proﬁt

p

⁄

(ci)

approximating (r ci)k. Compared with

p

i¼ ðp0 ciÞk, clearly,

when the required condition is satisﬁed,

p

i<

p

ðciÞ holds.

Consequently, from the supplier’s point of view, investing in IT to improve information transparency beneﬁts them is uncertain because it depends on the scale of the auction and the highest mar-ginal cost. On the other hand, from the buyer’s point of view, it can collect a large amount of suppliers’ data from the Internet in ad-vance and then invite qualiﬁed suppliers with cost advantages to submit their bids. Our research also suggests restraining the num-ber of suppliers is also helpful if the cost of IT investment is expen-sive, which is consistent with the traditional market where most raw materials or components buyers demand are supplied by a limited set of suppliers.

7. Conclusion

In the present study, we examine the function of residual de-mand and the value of IT in the online B2B exchange market in which the buyer requests multiple items and the suppliers have differing marginal costs. In the online B2B exchange market, each supplier can sell its entire capacity other than the one bidding the highest price can only supply the residual demand. To strengthen the relationship between the buyer and suppliers in

the supply chain, residual demand plays an important role in the motivation for participation of the suppliers because each supplier can make a proﬁt. On the other hand, residual demand can serve as a strategy for deriving competitive prices from the suppliers be-cause no supplier wants to be the loser supplying the residual de-mand. Our research points out the buyer should only keep a limited set of suppliers if it wants to both maintain a long-term partner relationship and procure materials at lower prices.

To understand the value of IT in online B2B exchange markets, we consider the suppliers’ cost information may be common knowledge or private information. Whether IT in the online B2B exchange market benefits buyers or suppliers depends on the scale of the auction and the highest marginal cost. Public cost informa-tion can prevent suppliers from submitting high bids even if the number of participants in the online auction is sufficiently large; however, when the highest marginal cost is sufficiently high, the buyer would have a higher expected total cost because each sup-plier would bid high after they know there is a competitor having such a high marginal cost as to barely constitute a threat at all.

Some future directions of research are as follows. First, we only consider each supplier has equal capacity in the current research. Although we can relax the assumption when each supplier has pri-vate cost information, the general case in which the suppliers have varied capacity can be further studied. Second, although a buyer, indeed, has incentive to aggregate more suppliers to lead a Ber-trand competition in prices, time cost can be a plausible reason so the buyer prefer receiving materials as soon as possible rather than waiting. This can form a future research that studies whether a buyer should wait (that is, the buyer set a longer deadline to add more bidders to the supplier pool). Third, some common knowl-edge given in the study, such as the buyer’s reservation price and the number of suppliers, could be incomplete information in the future research. In addition, based on the interest of mathematics, it would be worthy to examine whether the mixed strategy Nash equilibrium given in the present study is unique. Fourth, a more realistic model should include the number of suppliers, supplier’s capacity, and time variation to gather a different number of suppli-ers because these factors will certainly affect buyer’s decision. In addition, better product quality and reputation, which improve product reliability and buyer’s trust, are beneﬁcial to suppliers; thus, future directions can include the analysis of product quality and supplier’s reputation mechanism after auctions. Finally, although the mathematical result shows the less residual demand, the lower the procurement cost, whether suppliers have an incen-tive to participate in the bidding process remains to be solved. That is, the buyer has to ensure the motivation for participation of the suppliers by offering a sufﬁciently high residual demand. On the other hand, the lower residual demand can reduce the procure-ment costs. Thus, how to design an optimal residual demand is worthy of further study.

Appendix A. Proof of Lemma 1

Clearly, the suppliers cannot submit any bid higher than r,

implying FL(r) = FH(r) = 1. For the supplier SH, if the supplier SL

ran-domizes its price in ½^p; r, the pure strategy pH= r dominates the

choices pH< ^p, implying FHð^pÞ ¼ 0. For the supplier SL, if the

sup-plier SHonly bids in ½^p; r, the pure strategy pL< ^p is a strictly

dom-inated strategy since its proﬁt is strictly increasing in pLup to ^p,

implying FLð^pÞ ¼ 0. h

Appendix B. Proof of Propositions 1 and 2

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Appendix C. Proof of Lemma 2

Clearly, each supplier’s bid cannot be larger than r. Moreover, the supplier with the highest marginal cost has no incentive to

bid p < p0because the pure strategy p = r dominates the choices

p < p0, implying Pn#[p0, r]. Again, similar toLemma 1, all

suppli-ers whose marginal costs are less than cnhave no incentive to bid

p < p0; consequently, we have Pi#[p0, r]. h

Appendix D. Proof of Lemma 3

Because Fn1,n1(r) = 1 holds, we have pn1= r. Suppose we have

found pnj so Fnj,nj(pnj) = 1. Then, we can easily verify

Fnj1,nj1(pnj) > 1. Because of Fnj-1,nj1(p0) = 0, we can ﬁnd

pnj1so Fnj1,nj1(pnj1) = 1 holds, where p0< pnj1< pnj. If

there are multiple values satisfying the equation, we choose the

smallest one. h

D.1. The increasing property of Fi,j(p)

Subsequently, it is easy to verify that oFn,n1(p)/op > 0 and

oFn1,n1(p)/op > 0; therefore, we consider the other cases as

fol-lows. Because of Fi,j(p) > 0 for p

e

(p0, r] and Fi,j(p0) = 0, we can

en-sure that Fi,j(p) increases with p when p0< p < p0+ e; otherwise,

Fi,j(p) < 0, which violates the fact that Fi,j(p) > 0. Next, we examine

the properties of Fi,j(p) where p > p0by the following approach:

sign@Fi;jðpÞ @p ¼ sign @ @p ðp p0Þðp ciÞnj1 Q d–i;dPjðp cdÞ ¼ sign @ @pln ðp p0Þðp ciÞnj1 Q d–i;dPjðp cdÞ :

That is, we can examine the following function instead:

C

i;jðpÞ 1 ðp p0Þ þ ðn j 1Þ 1 ðp ciÞ X d–i;dPj 1 ðp cdÞ :

Notice that limp!p0Ci;jðpÞ > 0 holds. Moreover, Ck,i(p) PCi,i(p)

where i 6 k 6 n and i

e

{1, 2, . . . , n 2}. Because

1

ðpp0Þþ ðn j 1Þ 1 ðpciÞand

P

d–i;dPjðpc1dÞare decreasing convex

func-tions in p and limp!1_px1 ¼ 0, both functions can at most cross once.

Accordingly, it is possible that Ci,j(p) is less than zero when p

reaches a certain value; however,Ci,j(p) < 0 holds forever as long

as p is higher than the value. Subsequently, we discuss the case that Ci,i(p) < 0 when p < pi.

IfCi,i(p) < 0 when p < pi, we have Fi,i(pi) < 1; however, we have

shown the result that there exists pisuch that Fi,i(pi) = 1 in the

pre-vious part, which leads contradiction. That is,Ci,i(p) < 0 only holds

when p > pi. Therefore, Fi,i(p) increases with p in the given interval.

Moreover, Fi,j(p) increases with p in the given interval due to

Ck,i(p) PCi,i(p) > 0, which completes the proof. h

Appendix E. Proof of Lemma 4

Given the range pj16p < p_j, there are n j + 1 suppliers

bid-ding in the range. Therefore, if the supplier Si, where j 6 i 6 n, bids

p

e

[pj1, pj) according to the cumulative distribution function Gi(p),

the supplier’s proﬁt is

p

iðpÞ ¼ ðp ciÞ 1 Y m–i;mPj GmðpÞ ! k þ Y m–i;mPj GmðpÞðQ ðn 1ÞkÞ ( ) :

To ensure the supplier Si’s proﬁt equals

p

i, we have the equation

(p ci){k +Qm–i,mPjGm(p)(Q nk)} = (p0 ci)k, which can be

rewritten as follows: Q_m–i;mPjGmðpÞ ¼_nkQk pp_pc0

i n o . Because of Q j6i6n Q m–i;mPjGmðpÞ ¼ QmPjGmðpÞ nj and Gi(p) =QmPjGm(p)/ Q

m–i,mPjGm(p), we can ﬁnd Gi(p) = Fi,j(p).

Appendix F. Proof of Proposition 3

ByLemma 2–4, we can conﬁrm each supplier Sican gain the

ex-pected proﬁt

p

iby bidding p

e

[p0, pi]. In addition, the supplier with

the highest marginal cost can also gain the expected proﬁt

p

nby

bidding p

e

[p0, r]. However, because of Fn,n1(p) < 1, to compensate

for this, the supplier n bids the price r with a positive probability

(cn cn1)/(r cn1). Examining Gi(p), we ﬁnd they are continuous

functions other than Gn(p) having a jump at p = r. For the case of

n = 2, we have completed the proof. For the case of n P 3, due to

Di#[p0, r], we only need to check whether each supplier Si, other

than supplier Snand Sn1, can gain more proﬁt by bidding any price

in [pi, r]. For supplier Si bidding in [pj1, pj) in which

pi6pj1< pj6r, its proﬁt is given by

p

iðpÞ ¼ ðp ciÞ k þ Y mPj GmðpÞðQ nkÞ ( ) :

Suppose the proﬁt is larger than

p

i, because term (Q nk) is

nega-tive, the following inequality must be satisﬁed. Y mPj GmðpÞ 6 k nk Q p p0 p ci :

However, by the formulation of Gi(p) given in(4.7)–(4.10), we can

easily ﬁnd Y mPj GmðpÞ ¼ k nk Q njþ1 _Y j6m6n p p0 p cm !1 nj > k nk Q p p0 p ci ; which is a contradiction. h

Appendix G. Proof of Proposition 4 and 5

By ﬁrst-order conditions, we have the results. h Appendix H. Proof of Proposition 6

By ﬁrst-order conditions, we have oUI_{/ok < 0. Moreover, as n}

in-creases, the capacity k decreases. Otherwise, the capacity

con-straint cannot hold. Consider L < Q < R. Consider L = Q

a

k and

R = Q + bk where

a

+ b = 1,

a

P0, and b P 0. Clearly, for any

n P 2, we can ﬁnd

a

, b, and k to satisfy the capacity constraint,

(n 1)k < Q < nk. Since the value of b is bounded and the capacity k decreases with the number of suppliers, R approaches Q when n is sufﬁciently large. This implies each supplier would bid a price close to the buyer’s reservation price as long as the number of sup-pliers is sufﬁciently large.

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