comparison of the group-buying auction and the fixed pricing mechanism

43 (2007) 445–459www.elsevier.com/locate/dss

Decision Support Systems

Comparison of the group-buying auction and thefixed pricing mechanism

Jian Chen a,⁎, Xilong Chen a, Xiping Song b

a Department of Management Science, School of Economics and Management, Tsinghua University, Beijing, 100084, Chinab Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, China

Received 21 December 2004; received in revised form 10 August 2006; accepted 5 November 2006Available online 29 December 2006

Abstract

With the development of electronic commerce, online auction plays an important role in the electronic market. This paperanalyzes the seller's pricing strategy with the group-buying auction (GBA), a popular form of online auction, which is designed toaggregate the power of buyers to gain volume discounts. Based on the bidders' stochastic arrival process and optimal strategy withindependent private value model, this paper analyzes the sellers' optimal price curve of the GBA in the uniform unit cost case andin some supply chain coordination contracts. We find that the best discount rate is zero, which implies the optimal GBA isequivalent to the optimal fixed pricing mechanism (FPM). Then we compare the GBA with the FPM—in two special cases, theeconomies of scale and risk-seeking seller, and find that (1) when economies of scale are considered, the GBA outperforms theFPM; (2) when the seller is risk-seeking, the GBA also outperforms the FPM.© 2006 Elsevier B.V. All rights reserved.

Keywords: Group-buying auction; On-line auction; Fixed pricing mechanism

1. Introduction

With the development of E-business, online auctionplays an important role. Johnson et. al. [13] point outthat the online consumer auction sales in the US willreach $65 billion by 2010, accounting for nearly one-fifth of all online retail sales. The popularity of theonline auctions creates many new kinds of price mech-anisms, where the consumers participate more and morein the price-setting process. The group-buying auction(GBA) is one of them. As a homogeneous multi-unitauction, the GBA has many users on the sites such as

⁎ Corresponding author. Tel.: +86 10 62789896; fax: +86 1062785876.

E-mail address: [email protected] (J. Chen).

0167-9236/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.dss.2006.11.002

LetsBuyIt.com and Ewinwin.com. In contrast to thetraditional auction, where bidders compete against oneanother to be the “winner” with the highest price, theGBA enables bidders to aggregate to offer a lower priceat which they all “win” [11].

The GBA was once regarded as a promising novelauction mechanism, e.g., the GBAwebsites Mercata.comand Mobshop.com were once widely recognized asinternational market leaders. Partly because they misusedthe mechanism in the B2C market, Mercata closed downin January 2001 [6], and Mobshop changed its strategicdirection to the B2B market [5]. Since then, many group-buying websites have failed or reoriented themselves andhave given up the mechanism. However, some (e.g.,LetsBuyIt.com and Ewinwin.com) still use the GBA astheir pricing mechanism and have passed the winter of the

mailto:[email protected]

http://dx.doi.org/10.1016/j.dss.2006.11.002

446 J. Chen et al. / Decision Support Systems 43 (2007) 445–459

E-commerce. The GBA is based on the idea that “globallylocate, encourage and enable all buyers wishing topurchase a particular product or service within a giventime frame to join forces in a buying group formedspecifically to accomplish the desired purchase” [11]. Ifthis idea does work, why do so many websites fail?Comparing to the traditional FPM, in which scenario canthe GBA bring us more profit? These questions motivateus to study the GBAwith a theoretical analysis.

The study on the GBA is rare. Most existing paperson the GBA is experimental. They use the data from theGBA websites to study the customer behaviors. Kauff-man and Wang [14,15] analyze the changes in thenumber of orders for Mobshop-listed products overvarious periods, and find the “positive participationexternality effect,” the “price drop effect,” and the“ending effect”. These experimental studies on consum-er behavior are very useful for building the analyticalmodels to study the GBA. Anand and Aron [1] devise ananalytical model to study the GBA. Based on the as-sumptions that the revenue function is derived from adeterministic demand curve, they compare the fixedpricing mechanism (i.e., FPM) with the GBA in dif-ferent scenarios, e.g., uncertain demand regime andeconomies of scale. Using the simple analytic model,they finds that the GBA outperforms the FPM when 1)the seller faces uncertain market with two possibleintersecting demand regimes and 2) the seller sets theprice vector before production in combination with scaleeconomies. With Anand and Aron's results, we knowthat the value of the GBA depends on the nature of theuncertainty about the demand regimes. If the sellers doknow about in which demand regime they are operating,they are almost always better off by running a posted-price market. So, it is critical for the seller to know aboutthe advantage of the GBA over the FPM before using it.

Although similar to Anand and Aron's paper we alsofocus on the conditions that favor the GBA, we dealdifferent scenarios.

First, the demand assumption is different in thefollowing ways.

i) Different from the deterministic demand curve inAnand and Aron's paper, the demand in ourmodel is based on the rational strategy of thebidders in the GBA. According to Gupta andBapna [9], the customers in the online auctionsbecome involved in the price-setting process andtheir bidding strategy becomes very important todetermine the trading price. Hence, studying theGBA based on the customers' behavior becomesmore practical. Our paper based on Chen et. al.'s

paper [4], which builds a dynamic game model forthe GBA and studies the bidders' optimal strategy.It proves that the mechanism is incentive com-patible for bidders under the IPV (independentprivate values) assumption.

ii) We consider a different demand uncertainty case.In Anand and Aron's paper, they consider theseller operates in two different demand regimes. Ineach demand regime, the demand is deterministic.In reality, even if we are clear of the customersegmentation (demand regime), the visibility involatile B2C markets is sharply limited because somany different variables are in play. According toSull [21], in volatile markets, many variables areindividually uncertain and they interact with oneanother to create unexpected outcomes, the ran-domness in the demand is inevitable. Hence, weconsider the randomness in the regime. Pinker etal. [19] point out that for the online auction, thebidding process is an important issue. Our modelintroduces the bidding process, which simulatesthe randomness in the demand regime and thebidders' strategy in the GBA.

Second, we study the GBA in the case when the riskseeking seller is considered. Astebro [2] points out thatrisk-seeking is one of several plausible reasons why somany inventors proceed to develop their inventions whileonly a small fraction can reasonably expect to earn positivereturns on their efforts. Risk seeking also applies to theE-business world. Many websites care more about thepossible explosive expanding market as Ebay encountersthan the expected profit. Hence the resulting insights onthe risk seeking sellers can be of value to the sellers whoseobjective is not only on the expected revenue.

This paper considers a situation that one auctioneeruses the GBA to sell products to the individual onlineconsumers. As what happens in the online auction, inour paper, the consumers with different values to theobjects may arrive to the auction with a stochasticprocess. They make their decision according to theirrational strategy. The auctioneers may face uncertaindemand, supply chain contracts, economies of scale andrisk seeking scenario. The objective of our paper is toanswer the following questions: Can the GBA bringmore profit than the fixed pricing mechanism (FPM)? Inwhat scenarios does the GBA perform better?

The rest of this paper is organized as follows. InSection 2, we explain the GBA and the FPM. Section 3describes the problem that we study. In Section 4, theequivalence of the optimal GBA and FPM is proved.Section 5 studies the GBA in various scenarios. In

Fig. 1. The trading price in the GBA.

447J. Chen et al. / Decision Support Systems 43 (2007) 445–459

Section 6, we summarize the paper and highlightdirections for future study.

2. The group-buying auction and the fixed pricingmechanism

Although there exist plenty of studies on traditionalauctions (see Klemperer (1999) [16] for surveys of theauction literature), these results cannot be used in theGBA directly because the GBA has the followingproperties. 1) It is more often used in the B2C and B2Bmarkets. Hence, we should pay more attention to itsmulti-unit property. 2) Unlike traditional auctions, inwhich the number of bidders is fixed, in the GBA thenumber is random and the seller does not know howmany bidders will take part. 3) It is widely used bydifferent companies whose risk attitudes are quitedifferent. Hence, we should not only consider the risk-neutral profit-maximizing seller, as do most studies ofthe traditional auction. Paying more attention to thedifferent objectives will be more practical.

We list the main notation in Appendix F forreference. A typical GBA includes two rounds, theoffer round and the bidding round. The auction period Tand the number of objects N are exogenously given. Theseller acts first in the offer round, and determines theprice curve P=( p1, p2,…, pN), p1≥p2≥ ···≥pN, wherepi denotes the price if the sold quantity is i. Then thebidding round begins at time 0. In this round, the buyersarrive with a stochastic process. The arriving buyerdetermines whether and how much to bid. Once thebuyer bids, he is not permitted to change the bid. Theauction will end when either of the following two casesoccurs: 1) there are N bids; or 2) the time reaches T. Todistinguish the buyers who bid and the buyers who donot bid, we call all of the buyers who come to the GBAbefore T as the potential buyers and the buyers who bidin the auction as the bidders. It should be noted that onlythe potential buyers who come before the auction endshave the opportunity to bid because the auction may endbefore T. Although the number of potential buyers isindependent of the price curve P, because the buyers'bidding strategy is related to the price setting, thenumber of the bidders is related to the price level. N andP are common knowledge. Only discrete bids areallowed, i.e., bidders can only bid pi, i∈ [1, N].

According to the mechanism of the GBA, the numberof the winners and the trading price in the GBA can bedetermine by Rule I, which is equivalent to Chen et. al.(2002)'s paper [4]:

Rule I: If the bidding vector is B=(b1, b2,…, bn), n≤N,then the trading price is p(B) = pq, where q ¼

maxðijPnj¼1 Hðbj−piÞziÞ, and where H is the heavyside

function, i.e.,

HðxÞ ¼ 1 if xz00 if xb0

�With the trading price pq, the bidders bidding higher

than or equal to pqwill get the object and pay unit price pq.Those who don't bid or bid lower than pq will get nothingand pay nothing.

Rule I is illustrated in Fig. 1. Let the demand curvedenote the number of bids bidding higher than or equal tothe price. The demand curve and the price curve's rightpoint of intersection, K( p, q), denotes the trading priceand the number of winners. In Fig. 1, K's vertical coor-dinate, p, is the trading price and horizontal coordinate, q, isthe number of winners. The bidders bidding higher than orequal to p will get one object with unit price p each. Thebidders bidding lower than p will get nothing and paynothing.

The FPM can be described as a special case of the GBAwith the horizontal price curve, i.e., P=( p, …, p). Thebidders get the products when and only when they pay theunit price p.

3. Problem description

The GBA can be considered as a two-stage gamemodel: the offer stage for the seller and the bidding stagefor the bidders. The bidders' strategies in the bidding stagehave been studied by Chen et al. [4], based on which ourstudy emphasizes the seller's optimal strategy in the offerstage in various scenarios. The argument is based on thefollowing assumptions.

Assumption 1. The independent private values modelapplies, i.e. each potential buyer gives the object avaluation, and the valuations are statistically independent.


Assumption 2. The potential buyers' valuations are inmonetary units, and the potential buyers are risk neutral.

Assumption 3. The potential buyers are symmetric, i.e.all potential buyers draw their valuations from the sameprobability distribution. Let F(·) denote the cdf of thedistribution and f (·) the pdf respectively.

Assumption 4. There is no collusion between potentialbuyers.

Assumption 5. The potential buyers' arrival times arestatistically independent and have no correlation withtheir values to the object and bids.

Assumption 6. Each potential buyer's demand is 1.Segev et al. [20] point out that in online auction, approx-imately 76% of the purchases made were for one item.

Assumption 7. When a potential buyer decides not tobid, he silently “drops out” forever, i.e., he leaves theauction without record and never returns to it.

Assumption 8. The interval between the bidder'sarrival and bidding is omitted, i.e., the arrival timeequals the bidding time.

Assumption 9. The buyer chooses to buy the objectwhenthere is no difference between buying and not buying.

Assumption 10. The buyers are individual rational.

With these assumptions, Chen et al. [4] prove thatstrategy S is weakly dominant.

Strategy S: If a potential buyer's valuation of theobject is, ν, then he bids b=θ(ν), where

hðmÞup1; mzp1pj; pj−1Nmzpj; ja½2;N �0; mbpN

8<: ð1Þ

Here bidding 0 means that the bidder doesn't bid anddrops out forever. Strategy S implies that the biddershould drop out when his value is lower than the lowestprice level pN. Otherwise, he should bid the price levelwhich is just below his valuation to the object, ν.

Because strategy S is the buyers' weakly dominantstrategy, it is rational for the bidders to take it. It is trivialthat in the FPM, if a buyer's valuation of the object is noless than p, he will bid p and get one object withpayment p. Otherwise he will not buy the object.

According to rule I and strategy S, the sold quantityq(Vn,P) is

qðVn;PÞ ¼ maxðijXnr¼1

HðhðmrÞ−piÞziÞ ð2Þ

where Vn=(ν1, ν2,…, νn) denotes n potential buyers'valuations of the object, where vi denotes potential buyeri's valuation, and the profit of the seller is q(Vn,P) ·( pq(Vn,P)− c), where pq(Vn,P) denotes the trading price.

In both the GBA and the FPM, the seller knows thebuyers' arrival process distribution and their valuationdistribution. The seller's objective is to set the optimalprice curve P to maximize his utility. In differentscenarios, the seller may have different utilityfunctions, e.g., the expected profit, the expected profitwith economies of scale, and the risk-seeking utility,which will be studied in the following sections.

4. The seller's expected profit

In this section, we suppose that the seller is risk-neutral and his objective is to maximize the expectedprofit. If a seller sells k identical products at price p, andthe unit cost is c, then his profit is k⁎ ( p−c). The sellerfaces a trade-off between losing a sale because of a highprice and losing the customer's surplus because of a lowprice.

If there are n potential buyers, the seller's expectedprofit πn(P) is:

pnðPÞ ¼XNr¼0

Prqðr; n;PÞd rd ðpr−cÞ ð3Þ

where Prq(r, n, P) denotes the probability that there arer sold units with n potential buyers in the auction,0≤ r≤N.

Lemma 1. Any given P, n, ∃p, πn (L(p,N))≥πn(P),where Lðp; kÞ ¼ ð p; p; N ; pÞ|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

k

denotes a k-dimensional

vector with k identical elements, p.

Lemma 1 shows that if the seller knows exactly howmany potential buyers are in the auction, then theoptimal price curve is horizontal. In traditional auctions,the number of bidders is open to the seller. Hence, thislemma implies that using the GBA in a traditionalmanner will not bring the seller more profit. However, inan online GBA, potential buyers come to the auction in astochastic process. Hence, the seller does not knowexactly how many buyers will come. In the GBA,according to the total probability formula, the sellers'expected profit is pT ðPÞ ¼

Pln¼0 PrAðT ; nÞd pnðPÞ,

where PrA(T, n)denotes the probability that there are npotential buyers in GBA with period T.

Table 1Comparison of expected revenues under the different price curves

kth Element in price curve P (1≤k≤N ) Expected revenue

pk=p⁎=4.67 116.4pk=4.67−0.05 · (k−1) 95.8pk=5.6−0.1 · (k−1) 76.8pk=5.6−0.02 · (k−1) 113.7pk=4.67−0.05 · (k−1)1.1 77.1pk=5.8−0.1 · (k−1)1.1 42.28pk=5.8−0.02 · (k−1)1.1 96.2pk=4.94−0.05 · (k−1)0.9 92.4pk=5.5−0.1 · (k−1)0.9 87.2pk=5.5−0.02 · (k−1)0.9 96.3


Theorem 1. ∀ P, π T (P)≤ πT (L(p⁎ ,N)), wherep* ¼ argmax

ppT ðLðp;NÞÞ:

Theorem 1 implies even when the potential buyers'arrival process is stochastic and not known to the seller,the GBA cannot outperform the FPM either. Because theGBA can mimic an FPM by setting a horizontal pricecurve, the GBA is at least as good as the FPM. Hence,the optimal GBA and the optimal FPM are equivalent.Why can't the GBA utilize the discount to bring theseller more profit? According to Chen et al. [4], theGBA cannot motivate the bidders bidding higher. Andalso, according to Assumption 6, because the demandfor each bidder is only one, the GBA cannot motivatethe bidders bidding more. Can the GBA play as a pricediscrimination mechanism based on the number of bids?Because the GBA asks for a higher unit price when themarket is worse (less sold quantity) and asks for a lowerunit price when the market is better (more sold quantity),this mechanism cannot be an excellent price discrimi-nation mechanism intuitively. It follows that if thewebsite only considers maximizing its expected profit, itshould not use a GBA, because it cannot bring moreprofit. It should be noted that if c=0, then this theoremimplies that should the seller's objective be to maximizetheir revenue, the GBA is still not a good mechanism.

We now use a numerical example to illustrate theresult that the discount of the GBA cannot bring theseller more profit than the FPM. Suppose that thebidders' arrival process is a Poisson one with arrival rateλ=1/3. The auction period is T=100. The quantity ofavailable objects is N=40. The distribution of thebidders' valuations is a normal distribution with meanμ=6 and standard variance σ=2. The products' unitcost c is 0.

In the Poisson arrival process case, we have

pT ðLðp;NÞÞ

¼ p N−e−kTð1−FðpÞÞXN−1

k¼0

ðN−kÞðkTð1−FðpÞÞÞkk!

!

ð4ÞBy solving

ApT ðLð p;NÞÞAp

¼ N d 1−CðN þ 1; ð1−FðpÞÞkTÞ

CðN þ 1Þ� �

þ kTðCðN ; ð1−FðpÞÞkTÞCðNÞ ð1−FðpÞ−pd f ðpÞÞ ¼ 0;

ð5Þ

where Cðn; zÞ ¼ Rþlz tn−1e−tdt;CðnÞ ¼ Rþl

0 tn−1e−tdt,we can get that p⁎=4.67437.

Table 1 shows that although different families ofprice curves lead to different revenues for the seller,none of them outmatch the optimal price level q⁎. Thisresult confirms Theorem 1, i.e. under the assumptionsthe optimal price curve is horizontal in the GBA.

This conclusion may partly explain why somewebsites with the GBA failed in the B2C e-market.Products commonly sold in this market are CDs,cameras, and other electronic products, the coststructures of which are similar to this model. Hence,those websites that use the GBAwith a positive discountrate can hardly outperform their competitors.

Coordination plays an important role in the supplychain management, where the sellers, called the retailerin the Cachon et al.'s paper [3], may face different profitfunctions [3].

We suppose that if there are r sold units, where0≤ r≤N the seller's profit is

prðr; prÞ ¼ rðapr−bcmÞ−gcf ð6Þ

where cf is the fixed cost, cν is the variable cost and α, β,and η∈ [0,1] are the divison ratio of the unit revenue,variable cost, and fixed cost respectively for the seller. Forexample, α implies the division ratio of the unit revenuebetween the seller and his upstream supplier is α:(1−α).

The seller's expected profit with this kind of linearprofit function is

pcðPÞ ¼Xln¼0

PrAðT ; nÞXNr¼0

Prqðr; n;PÞd rd ðapr−bcÞ−gcf

ð7ÞThis profit function can describe the profit of the

sellers in many typic supply chain contracts.


For example, in revenue-sharing contract, the seller'sprofit can be expressed as

pðPÞ ¼ /ðSðN ;PÞd ðpSðN ;PÞÞÞ−ðcR þ wÞd N

¼Xln¼0

PrAðT ; nÞXNr¼0

Prqðr; n;PÞd rd /pr−ðcR þ wÞd N ð8Þ

where 0≤u≤1 is the seller's share of revenue, S(N, P)is the sold quantity with price vector P and total orderquantity N, cR is the unit cost for the seller and w is thewholesale price. Suppose the order quantity N isdetermined before selling, the order cost is fixed whendetermine the optimal price vector.

If we set a ¼ /; b ¼ 0; g ¼ ðcR þ wÞNcf

, which impliesthat the seller will get u share of the revenue and pay thefixed cost caused in the selling side, i.e., (cR+w)N,Eq. (7) is the same as Eq. (8), which implies that therevenue-sharing contract is a special case of the contractconsidered in the form of Eq. (6).

Theorem 2. ∀P, πC (P)≤πC (L(p⁎, N)), where p⁎ is theoptimal fixed price when contract (in the form of Eq. (6))is considered.

Theorem 2 extends the result of Theorem 1. WithTheorem 2, it is sufficient to say that the GBA cannotoutperform the FPM with linear profit function. Hence,with the supply chain coordination, if the sellers facethis kind of linear profit function, (e.g., revenue sharingcontracts, buy back contracts,) they should not use theGBA as a pricing mechanism.

5. The GBA in other scenarios

5.1. Scenario 1: economies of scale

Demand aggregation is at the core of the GBA. Bybringing together as many bidders as possible, web-sites can negotiate lower prices with merchant partnersor manufacturers (e.g., Letsbuyit.com, Ewinwin.com).For GBA websites, more sold units imply less unitcost, i.e., economies of scale are important. In thissection we introduce the economies of scale into ourmodel when we compare the GBA with the FPM. Herewe use the all units quantity discounts schedule mo-del. Suppose that a seller's utility function is π(r,pr)=r (pr−cr), where r is the number of the sold units, pris the final trading price, and cr is the unit cost whenthe volume is r.

For simplification, we only consider the price curvewith two price levels and assume that the supply canalways fulfill the demand. Suppose that the bidders'

arrival process is a Poisson process with arrival rate λ.The price curve is P ¼ ðp1; N ; p1|fflfflfflfflffl{zfflfflfflfflffl}

l

; p2; p2; N :Þ. The cost

function is C=(c1,c2,l), where c1Nc2 are the cost levelsand l is threshold to get the economies of scale. That is ifthe sold units are less than or equal to l, the unit cost isc1; if the sold units are more than l, the economies ofscale are achieved and the unit cost for all products is c2.

The seller's expected profit with economies ofscale is

pG;EðPÞ ¼Xlk¼0

ðkTÞke−kTk!

d ðXky¼0

Xyx¼0

zpðk; x; yÞ

�ðxd ðp1−c1ÞÞÞ þXþl

k¼lþ1

ðkTÞke−kTk!

�ðXly¼0

Xyx¼0

zpðk; x; yÞðxd ð p1−c1ÞÞÞ

þXþl

k¼lþ1

ðkTÞke−kTk!

�ðXky¼lþ1

Xyx¼0

zpðk; x; yÞðyd ðp2−c2ÞÞÞ ð9Þ

where

zpðk; x; yÞ ¼ k!ð1−Fðp1ÞÞxðFðp1Þ−Fðp2ÞÞy−xFðp2Þk−yx!ð y−xÞ!ðk−yÞ!

With some algebraic transactions, the expected profitcan be formulated as follows,

pG;EðPÞ ¼ kTð1−Fð p1ÞÞð p1−c1Þdwðp2Þþ kTð1−Fðp2ÞÞð p2−c2Þð1−wð p2ÞÞ ð10Þ

where w( p) = Γ(l, λT(1− F(p))) / Γ(l)∈ (0,1)) and

Cðn; zÞ ¼ Rþlz tn−1e−tdt;CðnÞ ¼ Rþl

0 tn−1e−tdt.If the seller posts a fixed price, p instead, it can be

deduced from Eq. (4) that the seller's expected profit is

pF;EðpÞ ¼ kTð1−FðpÞÞð p−c1Þd wðpÞþ kTð1−Fð pÞÞð p−c2Þð1−wðpÞÞ ð11Þ

By optimizing Eqs. (5) and (6), we get the optimalGBA price curve and the optimal FPM, with which wecan compare the expected profit of the GBA and that ofthe FPM.


Theorem 3. maxP

pG;EðPÞNmaxp

pF;EðpÞ

Theorem 3's implication is interesting. Differentfrom the former scenario without economies of scale,the GBA now strictly outperforms the FPM. This meansthat economies of scale are important for GBAwebsites.Because the GBA automatically asks for a higher pricewhen the unit cost is higher and a lower one whenthe unit cost is lower, it can outperform the fixedmechanism intuitively.

To illustrate this result, we provide a numerical ex-ample. Suppose that the potential buyers' arrival processis a Poisson process with λ=10; their valuations aredrawn from the same uniform distribution in interval[0,1], T=7, l=10, 20, 30, c1=0.8, and c2=0.7, 0.6,respectively. With Mathematica 4.0, by searching themaximal point in Eqs. (10) and (11) with the aboveparameters, we get the optimal price and maximal profit,which are shown in Table 2.

As these numbers suggest, the GBA outperforms theFPM when we consider economies of scale, whichconfirms Theorem 3. And also we can clearly observethat the improvement of the seller's profitability in theGBA depends on the expected demand (λT) and thethreshold (l). Comparing to the expected demand, a toolong or too short threshold provides a less improvementthan do the medium one. Because the GBA outperformsthe FPM through its adaptability, i.e., the GBAautomatically asks for a higher price when the unitcost is higher and a lower one when the unit cost islower. A too long or too short threshold implies theuncertainty of the unit cost is low, i.e., we are clear if wecan use the economies of scale or not. A mediumthreshold implies the uncertainty of the unit cost is high,i.e., we don't know if we can use the economies of scaleor not before the sales realized. Hence, in this scenario,the adaptability of the GBA values higher.

Table 2Comparison of GBA and FPM with economies of scale

c2 p1⁎, p2⁎ πG,E p⁎ πF,E

l=100.7 0.900, 0.817 1.356 0.815 1.2700.6 0.900, 0.775 2.656 0.771 2.604

l=200.7 0.900, 0.770 0.783 0.900 0.7000.6 0.900, 0.702 1.562 0.900 0.700

l=300.7 0.900, 0.754 0.701 0.900 0.7000.6 0.900, 0.662 0.790 0.900 0.700

If the seller can obtain volume discount from thesupplier, then the seller should choose the GBA as thepricing mechanism because this choice will bring moreprofit. However, as Kauffman and Wang [15] shows,many GBAwebsites in B2C markets cannot achieve thecritical mass, i.e. which implies that the threshold is toolong for the demand, which ultimately induces theirfailure. To solve this problem, trying to attract more con-sumers in different regions through the Internet is aneffective way for sellers to expand their consumer groups.And it is also a good idea to sign with the supplier somecontract through which the GBA websites can get thediscount with a small threshold.

5.2. Scenario 2: risk seeking seller

Another focus of the networked economy is risk-seeking. The players' different risk attitudes play animportant role on the application of the auction. In theexpanding E-business environment, there exists a fairlylarge group of risk-seeking persons, who believe that noventure no gain. AsGourville [8] points out, especially forthe sellers of the new products, gains have a greater impacton them than similarly sized losses. Hence, though theymay not have sufficient sale scale in the past, they still takea risk and hope to get it at some point in the future.Quadratic utility function is popular used in the literaturesto study different risk attitudes [12]. In this paper, we alsouse a simple quadratic utility function to model the risk-seeking sellers, where we assume that the seller's utilityfunction is r2pr, instead of the revenue being rpr, where ris the number of sold units and pr is the trading price. Weonly consider the two-level price curve and suppose thatthe supply always fulfills the demand. We also supposethat the bidders' arrival process is a Poisson process witharrival rate λ. Hence, the utility of the seller in GBAwithprice curve P ¼ ðp1; N ; p1|fflfflfflfflffl{zfflfflfflfflffl}

l

; p2; p2; N :Þ is

pG;RðPÞ ¼Xþl

k¼1

Xkz¼1

Xzy¼1

ðkTÞke−kTk!

� k!ð1−Fð p1ÞÞyðFð p1Þ−Fðp2ÞÞz−yFð p2Þk−zy!ðz−yÞ!ðk−zÞ! y 2p1

−Xþl

k¼1

Xkz¼lþ1

Xzy¼1

ðkTÞke−kTk!

� k!ð1−Fðp1ÞÞyðFð p1Þ−Fðp2ÞÞz−yFðp2Þk−zy!ðz−yÞ!ðk−zÞ! y2p1

þXþl

z¼lþ1

ðkTÞzð1−Fðp2ÞÞzz!

e−kTð1−Fðp2ÞÞz2p2 ð12Þ


With some algebraic transactions, the expected utilityis formulated as,

pG;RðPÞ ¼ ðh22 þ h2Þp2þ ðh21p1−h22 p2Þ

Cðl−1; h2ÞCðl−1Þ

þ ðh1p1−h2p2ÞCðl; h2ÞCðlÞ ; ð13Þ

where hi≡λt(1−P(qi)), i=1, 2When the seller uses the FPM, the expected utility is

pF;RðpÞ ¼ ktð1−FðpÞÞd ðktð1−FðpÞÞ þ 1Þd p ð14Þ

where p is the fixed price.With these formulae, we can now compare the

expected utility of the GBA and that of the FPM. LetP⁎aargmaxP pG;RðPÞ.

Theorem 4. For the optimal price curve P⁎ ¼ð p1⁎; N ; p1⁎|fflfflfflfflffl{zfflfflfflfflffl}

l

; p2⁎; p2⁎; N :Þ, where lN0, p1⁎Np2⁎.

Theorem 4 implies that for the risk-seeking sellers,the optimal price curve in the GBA is no longer ho-rizontal. Because the GBA can mimic the FPM bysetting a horizontal price curve, this theorem alsoinfers that the optimal GBA will outperform the FPMfor the risk-seeking sellers. Because the optimal pricemechanism is to find out the best tradeoff betweenthe sold quantity and the unit price and comparingwith the risk neutral sellers, in the marginal utilitypoint of view, the risk-seeking one weighs more andmore on selling one more product than on getting ahigher unit price (the power 2 on r in the objectivefunction r2 pr shows a heavier weight on r when rbecomes larger.). The GBA automatically adjusts thetradeoff between the sold quantity and unit price thatcan definitely benefit the seller. Different from theeconomies of scale case where exists an exogenousthreshold l to get the discount, here, l is endogenetic,yet how to set a suitable l in the optimal GBA is ourfuture work.

6. Conclusions and future studies

In this paper, we study the GBA mechanism inthree scenarios. First we consider the seller's expectedprofit when he is risk-neutral. By comparing the GBAwith the FPM, we find that the optimal GBA isequivalent to the optimal FPM. This result also appliesto the seller with linear profit division contracts withhis suppliers. Second, we introduce economies ofscale into our model, and find that the GBA willoutperform the FPM in this scenario. This resultimplies that GBA is more suitable for selling the goodsthat produce a learning effect; it also infers that there isa critical mass for the GBA. If the GBA websites cannever achieve a critical mass of units sold no matterthe market is good or bad, then they can hardly out-perform their competitors. Finally, we consider theGBA in the e-economy context, where do exist a largeamount of risk seeking sellers. We find that the GBAoutperforms the FPM in this scenario, which impliesthat the GBA is a suitable mechanism in the expandingE-business environment, where the seller is risk-seeking.

There is still much work to do. It has been pointedout that the GBA, in contrast to the ordinary salemechanism, can attract buyers with different objec-tives [7]. Because these buyers may take differentactions, asymmetric buyers will have to be considered.This paper models the GBA in the B2C markets,where the individual consumers buy the products fortheir own use. When we want to extend the GBA to theB2B world, we should pay more attention to thecommon value model or an affiliate value model [18].And also, as we point out, achieving more consumersis a key factor for the GBAwebsites to get the discountfrom the supplier. Collusion, cooperation of the bu-yers, may make the existing consumers to invite morefriends to join the GBA. Hence considering the coo-peration in the GBA is useful. Almost all auctionforms are susceptible to collusion, but degrees ofincentives vary [10] and collusion may heavily affectthe efficiency of the auction [17]. Because the GBAmay use collusion to attract more consumers and moreparticipants benefit the seller and the buyers both, in-troducing collusion to the GBA may lead to someinteresting results.

Acknowledgments

This work was supported in part by the NationalScience Foundation of China under Grant No. 70321001,NSFC/RCG Joint Research Grant 79910161987.


Appendix A. Proof of lemma 1

Before proving this lemma, we need first prove some lemmas and a corollary for technical reasons.
Notation. Π(Vn, P)≡ q(Vn,P) · ( pq(Vn,P)− c), which denotes the seller's profit when the price curve P and n potential buyers' valuations Vn are given.
.
Lemma A1. For any given n and any price curve P ¼ ðpk−1; pk−1; N ; pk−1|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
k−1

; pk ; N ; pN Þ, let P V¼ ð pk ; pk ; N ; pk|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}k

; pkþ1; N ;

pN Þ , if πu( pk,c)≥πu( pk−1,c), where πu( p,c)=(1−F(p))(p−c) denotes the >unit expected profit, then πn(P′)≥πn(P).

Let us discuss the cases according to r, r=q(Vn,P′).C1) In the case that r∈ [0, k): because pk− 1≥pk, it follows that q(Vn, P)= j≤ r.

PrðqðVn;PÞ¼ jjqðVn;P VÞ¼ rÞ ¼ Cjrd ð1−Fðpk−1ÞÞ jd ðFðpk−1Þ−FðpkÞÞr−j

ð1−FðpkÞÞr

EðjðVn;PÞjqðVn;P VÞ ¼ rÞ ¼Xri¼1

id ð pk−1−cÞd PrðqðVn;PÞ ¼ ijqðVn;P VÞ ¼ rÞ

¼ðpk−1−cÞd ð

Xri¼0

idCird ð1−Fðpk−1ÞÞid ðFðpk−1Þ−Fð pkÞÞÞr−j

ð1−FðpkÞÞr ¼ ðpk−1−cÞd rd ð1−Fð pk−1ÞÞ1−FðpkÞ

SoEðjðVn;PÞjqðVn;P VÞ ¼ rÞEðjðVn;P VÞjqðVn;P VÞ ¼ rÞ ¼

puðpk−1; cÞpuðpk ; cÞ V1

Hence, E(Π(Vn, P)|q(Vn, P′)= r)≤E(Π(Vn,P′)|q(Vn, P′)= r), r∈[0, k).C2) In the case that r≥k: With the definition of q(·), q(Vn, P)= r.

EðjðVn;PÞjqðVn;P VÞ ¼ rÞVEðjðVn;P VÞjqðVn;P VÞ ¼ rÞ; rzk

Integrating C1) and C2),

pnðP VÞ ¼XNr¼0

PrðqðVn;P VÞ ¼ rÞd EðjðVn;P VÞjqðVn;P VÞ ¼ rÞzXNr¼0

PrðqðVn;P VÞ ¼ rÞd EðjðVn;PÞjqðVn;P VÞ¼ rÞ ¼ pnðPÞ: □

.Lemma A2. For any given n and price curve P ¼ ðpk−1; pk−1; N ; pk−1 ; pk ; N ; pN Þ, kbN, let P V¼ ðpk−1; pk−1; N ; pk−1 ;
|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
k−1|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

k
pkþ1; N ; pN Þ, if πu( pk, c)bπu ( pk−1, c), then πn (P′)Nπn (P).
Proof. Let us discuss the cases according to r, r=q(Vn,P).
C1) In the case that r∈ [0, k):Because the price elements in P′ are not lower than those in P, r≥q(Vn,P′).With Rule I, there are r bidders whose values are not lower than qk−1, which implies that q(Vn,P′)≥ r.Thus, q(Vn, P)= r=q(Vn,P′).Hence, E(Π(Vn, P)|q(Vn, P)= r)≤E(Π(Vn,P′)|q(Vn,P)= r).C2) In the case that r=k:
PrðqðVn;P VÞ ¼ jjqðVn;PÞ ¼ rÞ ¼ Cjrd ð1−Fðpk−1ÞÞ jd ðFðpk−1Þ−FðpkÞÞr−j

ð1−FðpkÞÞr :


EðjðVn;P VÞjqðVn;PÞ ¼ rÞ ¼Xrj¼0

jd pk−1d PrðqðVn;P VÞ ¼ jjqðVn;PÞ ¼ rÞ

¼ðpk−1−cÞd ð

Xrj¼0

jd Cjrd ð1−Fð pk−1ÞÞ jd ðFðpk−1Þ−FðpkÞÞr−j

ð1−FðpkÞÞr ¼ ð pk−1−cÞd rd ð1−Fð pk−1ÞÞð1−FðpkÞÞ :

It follows thatEðjðVn;PÞjqðVn;PÞ ¼ rÞEðjðVn;P VÞjqðVn;PÞ ¼ rÞ ¼

puðpk ; cÞpuð pk−1; cÞ b1.

Hence, E(Π(Vn,P′)|q(Vn,P)=r)NE(Π(Vn,P)|q(Vn,P)=r).C3) In the case that rNk: with the definition of q(·), q(Vn, P′)=q(Vn, P)= r.Hence E(Π(Vn,P′)|q(Vn,P)=r)=E(Π(Vn,P)|q(Vn,P)=r).Integrating C1), C2) and C3), with total probability formula,

pnðP VÞ ¼XNr¼0

PrðqðVn;PÞ ¼ rÞd EðjðVn;P VÞjqðVn;PÞ ¼ rÞ

zXNr¼0

PrðqðVn;PÞ ¼ rÞd EðjðVn;PÞjqðVn;PÞ ¼ rÞ ¼ pnðPÞ:

Corollary A1. For any given n and price curve P, let B ¼ ðpxN−1 ; N ; pxN−1|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}N−1

; pN Þ, where pxN−1aargmaxpjðpuðpj; cÞÞ; 1VjVN−1; pnðPÞVpnðBÞ:

.Proof. Lemmas A1 and A2 imply ∀kbN, πn(P)≤En(P′), where P ¼ ðpk−1; pk−1; N ; pk−1|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}k−1

; pk ; N ; pN Þ and P V¼

ðpx; px; N ; px|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}k

; pkþ1; N ; pN Þ with xaargmaxjðpu ðpj; cÞÞ; j ¼ k or k−1. It follows that

pnðPÞVpnððpx2 ; px2|fflfflffl{zfflfflffl}2

; N ; pN ÞÞVpnððpx3 ; px3 ; px3|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}3

; N ; pN ÞÞV N VpnððpxN−1 ; N ; pxN−1|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}N−1

; pN ÞÞ ¼ pnðBÞ;

Where pxN−1aargmaxpjðpuðpj; cÞÞ; 1VjVN−1. □

.Lemma A3. For any given n and price curve B ¼ ðpN−1; N ; pN−1|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}; pN Þ, if πu(pN,c)bπu(pN−1,c), let B′=L(pN−1,N), πn(B′)≥πn (B).
N−1
Proof. Let us discuss the cases according to r, r=q(Vn,B).
C1) In the case that r∈ [0, N), similar with Lemma A2 C1).
EðjðVn;B VÞjqðVn;BÞ ¼ rÞ ¼ EðjðVn;BÞjqðVn;BÞ¼ rÞC2) In the case that r=N:This implies that there are at least N potential buyers who value the objects at no less than qN. Because the price

elements in B′ are not lower than those in B, q(Vn,B′)≤q(Vn,B).Suppose that there are i(i≥N) potential buyerswho value the objects at no less than qN, which is denoted byAR= i, then

EðjðVn;B VÞjAR ¼ iÞ ¼XN−1

j¼0

PrfqðVn;B VÞÞ ¼ jjAR ¼ igd jd ð pN−1−cÞ þ PrfqðVn;B VÞ ¼ N jAR ¼ ig

� N d ðpN−1−cÞ ¼

XN−1

j¼0

jd ðpN−1−cÞd Cjið1−FðpN−1ÞÞ jðFðpN−1Þ−FðpN ÞÞi−j

ð1−Fð pN ÞÞi

þ

Xij¼N

N d ðpN−1−cÞd Cjið1−FðpN−1ÞÞ jðFð pN−1Þ−FðpN ÞÞi−j

i
ð1−FðpN ÞÞ


i) If aka½N ; i�;

Xij¼k

ðpN−1−cÞdCjið1−FðpN−1ÞÞ jðFðpN−1Þ−FðpN ÞÞi−j

ð1−FðpN ÞÞiNðpN−cÞ, then

Xij¼k

N d ðpN−1−cÞd Cjið1−FðpN−1ÞÞ jðFðpN−1Þ−FðpN ÞÞi−j

ð1−Fð pN ÞÞiNN d ð pN−cÞ:

Hence, E(Π(Vn,B′)|AR= i)

¼

XN−1

j¼0

jd ð pN−1−cÞdCjið1−Fð pN−1ÞÞ jðFðpN−1Þ−Fð pN ÞÞi−j

ð1−Fð pN ÞÞiþ

X k−1

j¼N

N d ð pN−1−cÞdCjið1−FðpN−1ÞÞ jðFðpN−1Þ−Fð pN ÞÞi−j

ð1−Fð pN ÞÞi

þ

Xij¼k

N d ð pN−1−cÞdC ji ð1−Fð pN−1ÞÞ jðFð pN−1Þ−Fð pN ÞÞi−j

ð1−FðpN ÞÞiN

Xij¼k

N d ðpN−1−cÞdCjið1−Fð pN−1ÞÞ jðFð pN−1Þ−FðpN ÞÞi−j

ð1−FðpN ÞÞiN N d ð pN−cÞ ¼ EðjðVn;BÞjAR ¼ iÞ:

ii) If 8ka½N ; i�;

Xij¼k

ð pN−1−cÞdCjið1−FðpN−1ÞÞ jðFðpN−1Þ−FðpN ÞÞi−j

ð1−FðpN ÞÞiV pN−c, then

Xik¼Nþ1

Xij¼k

ðpN−1−cÞdCjid ð1−FðpN−1ÞÞ jd ðFð pN−1Þ−Fð pN ÞÞi−j

ð1−FðpN ÞÞiVði−NÞd ðpN−cÞ:

BecauseXik¼Nþ1

Xij¼k

ðpN−1−cÞdC jid ð1−FðpN−1ÞÞ j

d ðFðpN−1Þ−Fð pN ÞÞi−jð1−FðpN ÞÞi

¼Xi

j¼Nþ1

ðpN−1−cÞdð j−NÞdC jid ð1−FðpN−1ÞÞjd ðFðpN−1Þ−FðpN ÞÞi−j

ð1−FðpN ÞÞi;

Hence,

Xij¼Nþ1

ð pN−1−cÞd ð j−NÞdCjid ð1−FðpN−1ÞÞ jd ðFðpN−1Þ−Fð pN ÞÞi−j

ð1−Fð pN ÞÞiVði−NÞd ð pN−cÞ ðA1Þ

As

Xij¼0

jd ðpN−1−cÞdCjid ð1−Fð pN−1ÞÞ jd ðFðpN−1Þ−FðpN ÞÞi−j

ð1−Fð pN ÞÞi¼ id ðpN−1−cÞd ð1−FðpN−1ÞÞ

ð1−FðpN−1ÞÞ and

puðpN−1; cÞ ¼ ðpN−1−cÞð1−Fð pN−1ÞÞzðpN−cÞð1−FðpN ÞÞ ¼ puðpN ; cÞ;

Xij¼0

jd ð pN−1−cÞdCjid ð1−Fð pN−1ÞÞ jd ðFðpN−1Þ−Fð pN ÞÞi−j

ð1−Fð pN ÞÞiNid ðpN−cÞ ðA2Þ


With in Eqs. (A1) and (A2), it can be deducted that

XNj¼1

jd ðpN−1−cÞdC jidð1−FðpN−1ÞÞ jdðFðpN−1Þ−FðpN ÞÞi−j þ

Xij¼Nþ1

N dð pN−1−cÞdC jidð1−Fð pN−1ÞÞ jdðFð pN−1Þ−FðpN ÞÞi−j

ð1−Fð pN ÞÞiNN d ð pN−cÞ

i.e., E(Π(Vn, B′)|AR= i)NE(Π(Vn,B)|AR= i).

Thus, in both cases

EðjðVn;B VÞjAR ¼ iÞNEðjðVn;BÞjAR ¼ iÞ;

Because q(Vn,B)=N if and only if ∃i≥N, s.t. AR= i,

EðjðVn;B VÞjqðVn;BÞ ¼ NÞ ¼Xþl

i¼N

PrfAR ¼ igEðqðVn; LðpN−1;NÞÞjAR ¼ iÞ

PrfqðVn;BÞ ¼ Ng

z

Xþl

i¼N

PrfAR ¼ igEðqðVn;BÞjAR ¼ iÞPrfqðVn;BÞ ¼ Ng ¼ EðjðVn;BÞjqðVn;BÞ ¼ NÞ;

Hence,

EðjðVn;B VÞjqðVn;BÞ ¼ NÞzEðjðVn;BÞjqðVn;BÞ ¼ NÞ

Thus, with total probability formula

pnðBÞ ¼XNr¼0

PrfqðVn;BÞ ¼ rgdEðjðVn;BÞjqðVn;BÞ ¼ rÞ

VXNr¼0

PrfqðVn;BÞ ¼ rgd EðjðVn;B VÞjqðVn;BÞ ¼ rÞ ¼ pnðB VÞ: □

Proof of Lemma 1. According to Corollary A1, let B ¼ ðpxN−1 ; N ; pxN−1 ; pN|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}N−1

Þ, where pxN−1aargmaxpjðpuðpj; cÞÞ; 1VjVN−1; pnðPÞVpnðBÞ. Hence,

1) if πu( pN,c)≥πu ( pN−1,c), then πn(L( pN,N))≥πn(B)≥πn(P) with Lemma A1,2) if πu ( pN, c)bπu ( pN−1, c), then πn (L( pN−1, N))≥πn (B)≥πn (P) with Lemma A3. □

Appendix B. Proof of Theorem 1

With Lemma 1, for any given P there exists p, whichis irrelated to n, s.t., for any n, πn(L( p, N))≥πn(P).

With total probability formula,

pT ðPÞ ¼Xln¼0

PrAðT ; nÞd pnðPÞ

VXln¼0

PrAðT ; nÞd pnðLðp;NÞÞ ¼ pT ðLðp;NÞÞ:

Suppose that p⁎aargmaxp pT ðLðp;NÞÞ i.e., for anyp, πT (L( p,N ))≤πT(L( p⁎,N )).

It follows that πT (P)≤πT(L( p⁎,N )). □

Appendix C. Proof of Theorem 2

The proof is very similar to the proofs ofLemma 1 and Theorem 1. The only difference is todefine πu( p, c)=(1−F( p)) (αp−βc) instead of πu(p,c)=(1−F(p)) ( p−c). □

ðl−1Þ!


Appendix D. Proof of Theorem 3

Suppose that p⁎aargmaxp pF;EðpÞ. It follows that

dpF;EðpÞ=dpp¼p⁎ ¼ 0 ¼ kTð−pf ðpÞ þ ðc1wðpÞþc2ð1−wð pÞÞÞf ð pÞ−ð1−FðpÞÞ�ðc1−c2Þw VðpÞ þ 1−FðpÞÞ

Because w′( p)=λ Te−λT (1−F( p)) (λT (1−F ( p )))l−1

f ( p) /Γ (l )≥0,

ð1−Fðp⁎ÞÞðc1−c2Þw Vðp⁎Þ ¼ 1−Fðp⁎Þ−p⁎f ðp⁎Þþ ðc1wðp⁎Þþ c2ð1−wðp⁎ÞÞ f ðp⁎Þz0

Suppose that p1⁎ ¼ argmaxp puðp; c1Þ. It follows that∂πu( p,c1) /∂p|p=p1⁎=0

Hence,

Apuðp; c1Þ=Apjp¼p⁎ ¼ 1−Fðp⁎Þ−p⁎f ðp⁎Þ þ c1 f ðp⁎Þ¼ 1−Fðp⁎Þ−p⁎f ð p⁎Þ þ c1ðwðp⁎Þ

þ 1−wðp⁎ÞÞ f ð p⁎ÞN1−Fðp⁎Þ−p⁎f ðp⁎Þþ ðc1wðp⁎Þ þ c2ð1−wðp⁎ÞÞ f ð p⁎Þ

¼ ð1−Fð p⁎ÞÞðc1−c2Þw Vðp⁎Þz0¼ Apuð p; c1Þ=Apjp¼p1

⁎

So p⁎bp1⁎.It follows that

maxP

pG;EðPÞzð p1⁎; N ; p1⁎|fflfflfflfflffl{zfflfflfflfflffl}l

; p⁎; N Þ

¼kTðð1−Fðp1⁎ÞÞð p1⁎−c1Þwðp⁎Þþ ð1−Fðp⁎ÞÞð p⁎−c2Þð1−wðp⁎ÞÞÞ

NkTðð1−Fð p⁎ÞÞð p⁎−c1Þwðp⁎Þþ ð1−Fðp⁎ÞÞð p⁎−c2Þð1−wðp⁎ÞÞÞ

¼ pF;Eðp⁎Þ ¼ maxp

pF;EðpÞ

It follows that maxP pG;EðPÞNmaxp pF; Eð pÞ. □

Appendix E. Proof of Theorem 4

Suppose that p⁎ ¼ argmaxp

pF;RðpÞDefine I( p)=πF,R( p) / (λT(1−F( p))+1)

With some algebraic transactions, the expected utilityof the seller in the GBA can be formulated as follows,

pG;RðPÞ ¼ pF;Rðp1ÞCðl−1; kTð1−Fðp2ÞÞÞCðl−1Þ þ pF;Rðp2Þ

� ð1−Cðl−1; kTð1−Fðp2ÞÞÞCðl−1Þ

þðIð p1Þ−Ið p2ÞÞ ðkTð1−Fð p2ÞÞl−1

ðl−1Þ! e−kTð1−Fð p2Þ:

Because

AIð pÞAp j

p¼p⁎

¼ pF;Rð pÞkTf ðpÞ þ pG;R Vð pÞðkTð1−FðpÞ þ 1ÞðkTð1−FðpÞ þ 1Þ2 j

p⁎

¼ pF;Rðp⁎ÞkTf ðp⁎ÞðkTð1−Fðp⁎Þ þ 1Þ2 N0

It follows that

ApG;RðPÞAp1 j

ðp1¼ p⁎;p2¼ p⁎Þ

¼ ðApF;Rðp1ÞA p1

Cðl−1; kTð1−Fðp2ÞÞCðl−1Þ

þAIð p1ÞA p1

ðkTð1−Fð p2ÞÞl−1ðl−1Þ! e−kTð1−Fð p2ÞÞj

ð p1¼p⁎; p2¼p⁎Þ

¼ 0þ ðkTð1−Fð p2ÞÞl−1e−kTð1−Fð p2Þðl−1Þ!

AIð p1ÞA p1 j

ð p1¼p⁎; p2¼p⁎ÞN0

ApG;RðPÞAp2 j

ð p1¼p⁎; p2¼ p⁎Þ

¼ fðpF;Rð p1Þ−pF;Rð p2ÞÞA Cðl−1; kTð1−Fð p2ÞÞ

Cðl−1Þ� �

=Ap2

þpF;R Vð p2Þ 1−Cðl−1; kTð1−Fð p2ÞÞ

Cðl−1Þ� �

þðIð p1Þ−Ið p2ÞÞA ðkTð1−Fð p2ÞÞl−1e−kTð1−Fð p2Þðl−1Þ!

!=Ap2

−ðkTð1−Fð p2ÞÞl−1e−kTð1−Fð p2Þ

ðl−1Þ! I Vð p2Þgjð p1¼p⁎; p2¼p⁎Þ

¼ −ðkTð1−Fðp2ÞÞl−1e−kTð1−Fð p2Þ

ðl−1Þ! I Vðp2Þjð p1¼p⁎; p2¼p⁎Þ

¼ −ðkTð1−Fðp⁎ÞÞl−1e−kTð1−Fð p⁎Þ

I Vð p⁎Þb0


Hence, there exists a feasible ascent direction at thepoint ( p⁎, p⁎,…) for πG,R(P)

It follows that

p⁎1 Np⁎2 □

Appendix F. The notation

N The number of objects to be sold in the GBAT The auction timeP=( p1, p2, ..., pN), p1≥p2≥…≥pN The price curve

in the GBA
Lðp; kÞ ¼ ðp; p; N ; p|fflfflfflfflffl{zfflfflfflfflffl}
k

Þ Denotes a k-dimensional vector

with k Identical elements, p.Vn=(v1, v2, …, vn) Denotes n potential buyers' valuations

iof the object, where vi denotes potential buyer i'svaluation.

PrA (T, n) Denotes the probability that there are nipotential buyers in GBA with period T

c Is the unit cost of the auctioned products in thescenario without economies of scale.

C=(c1,c2;l ) Where c1Nc2 are the cost levels and l is
ithreshold to get the economies of scale. That is ifthe sold units are less than or equal to l, the unitcost is c1; if the sold units are more than l, theeconomies of scale are achieved and the unit costfor all products is c2.
Given the price curve P and posted price p

q(Vn, P) Denotes the sold quantity in the auction with npotential buyers, Vn,

Prq (r, n, P) Denotes the probability that there are rsold units with n potential buyers in theauction, 0≤ r≤N

πu ( p, c)= (1−F( p))( p − c) Denotes the unit expected
iprofit, where for any given c, πu (p, c)isunimodal in p.
πn (P) Denotes the seller's expected profit with npotential buyers in the auction.

πT (P) Denotes the seller's expected profit in the GBAwith auction time T

πC (P) Denotes the seller's expected profit in the GBAwith some coordination contracts.

πG,E (P) Denotes the seller's expected profit in the GBAwith economies of scalewhere the cost function isC.

πF,E (P) Denotes the seller's expected profit in the FPMwith price p when the cost function is C

πG,R (P) Denotes the seller's utility in the GBA whenthe seller is risk seeking.

πF,R (P) Denotes the seller's utility in the FPM with theprice p when the seller is risk seeking.

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Jian Chen received the B.Sc. degree in Electrical Engineering fromTsinghua University, Beijing, China, in 1983, and the M.Sc. and thePh.D. degree both in Systems Engineering from the same University in1986 and 1989, respectively.He is a Professor and Chairman of Management Science Department,Director of Research Center for Contemporary Management, TsinghuaUniversity. Dr. Chen has over 100 journal papers and has been aprincipal investigator for over 30 grants or research contracts withNational Science Foundation of China, governmental organizationsand companies. His main research interests include modeling andcontrol of complex systems, decision support systems and informationsystems, forecast and optimization techniques, supply chain manage-ment, E-commerce.He is a senior member of IEEE and a member of INFORMS. He servesas Chairman of the Service Systems and Organizations TechnicalCommittee of IEEE Systems, Man and Cybernetics Society, vicepresident of China Society for Optimization and Overall Planning, amember of the Standing Committee of Systems Engineering Society ofChina, China Information Industry Association and Decision ScienceSociety of China. He is the recipient of Science and TechnologyProgress Awards of Beijing Municipal Government, 2000 and 2001;the Outstanding Contribution Award of IEEE Systems, Man andCybernetics Society, 1996; Science and Technology Progress Award ofthe State Educational Commission, 1994.He is the editor of “the Journal of Systems Science and SystemsEngineering”, an associate editor of “IEEE Transactions on Systems,Man and Cybernetics: Part A” and “IEEE Transactions on Systems,Man and Cybernetics: Part C”, and serves on the Editorial Board of“Systems Research and Behavioral Science”, “International Journal ofInformation Technology and Decision Making ” and “InternationalJournal of Electronic Business”. He is the Secretary General of the1996 IEEE International Conference on Systems, Man and Cyber-netics in Beijing, co-chair of the IPC of the 3rd (1998) and 4th (2003)International Conferences on Systems Science and Systems Engineer-ing, and Chair of the 1st Asian eBiz Workshop in 2001, co-chair of theInternational Conference on Service Systems and Service Manage-ment in 2004, Chair of the 4th International Conference on ElectronicBusiness in 2004.

Xilong Chen received the PhD. Degree from Tsinghua University,Beijing, China, in 2004. He is currently a doctoral student at NorthCarolina State University, USA.

Xiping Song received the M.Sc. Degree from Tsinghua University,Beijing, China, in 2003. He is currently a doctoral student at theChinese University of Hongkong, Hongkong, China.

comparison of the group-buying auction and the fixed pricing mechanism

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