Vickrey–Dutch procurement auction for multiple items

European Journal of Operational Research 180 (2007) 617–629

www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

Vickrey–Dutch procurement auction for multiple items

Debasis Mishra a, Dharmaraj Veeramani b,*

a Department of Computer Science and Automation, Indian Institute of Science, Bangalore 560012, Indiab Department of Industrial and Systems Engineering, University of Wisconsin-Madison,

266 B, Mechanical Engineering Building, 1513 University Avenue, Madison, WI 53706, USA

Received 10 November 2004; accepted 3 April 2006Available online 5 June 2006

Abstract

We consider a setting where there is a manufacturer who wants to procure multiple items from a set of suppliers each ofwhom can supply one or more of these items (bundles). We design an ascending price auction for such a setting whichimplements the Vickrey–Clarke–Groves outcome and truthful bidding is an ex post Nash equilibrium. Our auction main-tains non-linear and non-anonymous prices throughout the auction. This auction has a simple price adjustment step and iseasy to implement in practice. As offshoots of this auction, we also suggest other simple auctions (in which truthful biddingis not an equilibrium by suppliers) which may be suitable where incentives to suppliers are not a big concern. Computersimulations of our auction show that it is scalable for the multi-unit case, and has better information revelation propertiesthan its descending auction counterpart.� 2006 Elsevier B.V. All rights reserved.

Keywords: Iterative auctions; Procurement auctions; Vickrey–Dutch auctions; Winner determination problem

1. Introduction

Procurement is an integral part of supply chainoperation for many companies. The traditional pro-curement process involves sending request for quota-

tion (RFQ) to the suppliers and receiving quotes forthe RFQ from the suppliers. This is also termed assealed-bid auction procedure in the auction litera-ture. The other method, popularized by the Inter-net, is the reverse auction. In a reverse auction,suppliers iteratively lower the price on an item till

0377-2217/$ - see front matter � 2006 Elsevier B.V. All rights reserved

doi:10.1016/j.ejor.2006.04.020

* Corresponding author. Tel.: +1 608 262 0861; fax: +1 608 2628454.

E-mail addresses: [email protected] (D. Mishra), [email protected] (D. Veeramani).

the price reaches a level where there is only one sup-plier interested in supplying the item at that price.The popularity of reverse auctions is substantiatedby research that shows that iterative auctions canbe preferred over sealed-bid auctions due to bettertransparency, preference elicitation etc. (Cramton,1998; Parkes, 2005).

The multi-item setting is very different from a sin-gle item setting because items can be complementsor substitutes to the suppliers. So, the auctiondesign for procuring multiple items is no more atrivial generalization of the single item reverse auc-tion. The descending price reverse auctions for pro-curing multiple items are already known (Demangeet al., 1986; Ausubel, 2004; Ausubel and Milgrom,2002; de Vries et al., forthcoming; Mishra and

.

mailto:[email protected]

mailto:raj@ cae.wisc.edu

mailto:raj@ cae.wisc.edu

618 D. Mishra, D. Veeramani / European Journal of Operational Research 180 (2007) 617–629

Parkes, forthcoming).1 In this work, we propose anascending price auction for procurement that imple-ments the Vickrey–Clarke–Groves (VCG) outcome.We call our auction the Vickrey–Dutch auction(VDA) for procurement. This generalizes our priorwork on Vickrey–Dutch auctions for special settings(Mishra and Veeramani, 2006; Mishra and Parkes,2004a,b).2

Our Vickrey–Dutch auction is designed on thefoundations of universal competitive equilibrium

(UCE) prices introduced in Mishra and Parkes(forthcoming). By giving bonus to every supplierfrom the final price of the auction, we implementthe VCG outcome. These bonuses, which are essen-tially marginal contributions of the suppliers to theprice of procurement, give suppliers incentive toparticipate truthfully in the auction. In particular,bidding truthfully in an ex post Nash equilibriumfor suppliers in our auction. As offshoots of thisauction, we also suggest other simple auctions (inwhich truthful bidding is not an equilibrium by sup-pliers), which may be suitable where incentives tosuppliers are not a big concern.

The Vickrey–Dutch auction design is differentfrom the descending reverse auction design. This isbecause the prices in the descending reverse auctionstarts at a high price where many suppliers are will-ing to supply the items, i.e., supply is more thandemand. The price dynamics are designed todecrease supply to balance the supply and demand.Contrast this with the Vickrey–Dutch auction whereprices start low with suppliers not willing to supplyitems (i.e., supply is less than demand). Thus, theprice dynamics should be designed to increase sup-ply to balance supply and demand. Thus, the designof a Vickrey–Dutch auction is not a trivial general-ization of the design of a descending reverse auc-tion. In fact, a careful look at our auction andsome of the standard descending reverse auctions(Mishra and Parkes, forthcoming) reveal that theirunderlying price adjustment rules are very different.

The single-item Vickrey–Dutch auction appearedin Vickrey (1961) in the single-seller setting, where aseller is selling an item to multiple buyers. The

1 This literature is for the single-seller model where a singleseller is selling multiple items to buyers. But this can be easilyadapted to our procurement setting, a single-buyer model.

2 Research literature in iterative auction dealing exclusively inprocurement setting is scarce. An exception is Parkes andKalagnanam (2005) who design a descending price reverseauction that allows bidding for multiple attributes.

appropriate method to generalize Vickrey’s idea ofVickrey–Dutch auction to multiple items caseremains a puzzle in the literature. For instance, intheir work on the design of an ascending price auc-tion for the homogeneous items case in single-sellersetting, Bikhchandani and Ostroy (2006) observethe following while interpreting their auction as aprimal–dual algorithm: ‘‘The primal–dual algorithm

we describe starts at a low price where there is excess

demand. One could start the primal–dual algorithm at

a high price at which there would be excess supply,

but it is unlikely that this would converge to a mar-ginal pricing equilibrium’’.3 In that sense, this work,along with Mishra and Veeramani (2006), Mishraand Parkes (2004b,a), fill a void in the literature ofiterative auctions.

We highlight several benefits of a (reverse) Vick-rey–Dutch auction, e.g., speed, better preferenceelicitation, privacy etc. Elmaghraby (2004) discusshow such issues are even more important in pro-curement setting. In particular, we discuss the infor-mation revelation properties of our Vickrey–Dutchauction and compare it with a (descending price)reverse auction. Using simulation, we show thatour Vickrey–Dutch auction has better preferenceelicitation properties than its descending price auc-tion counterpart. An inherent drawback of iterativeauctions for multiple items is that they need tomaintain an exponential-sized price vector in everyiteration to implement the VCG outcome (this isnecessary in general—Mishra, 2004). But, for aspecial case, when all the items are of the same type,the size of the price vector in these auctions aremanageable. For this special case, we show, usingsimulation, that our Vickrey–Dutch auction is com-putationally scalable—our auction has an averagerunning time of under 2 minutes for 30 suppliersand 150 units.

The rest of the paper is organized as follows. Sec-tion 2 defines our model and introduces some pre-liminaries. In Section 3, we introduce our Vickrey–Dutch auction and prove its theoretical properties.We discuss some practical issues of our auction inSection 4 and its information revelation propertiesin Section 5. We conclude with a summary and dis-cussion in Section 6.

3 A marginal pricing equilibrium is a price where all bidders gettheir respective marginal products as payoffs.

4 We note that we do not consider the dummy supplier as astrategic agent. The dummy supplier (the manufacturer) acts as a‘‘social planner’’ whose sole objective is to implement an efficientallocation. There are inherent difficulties, in terms of impossibilityresults, in considering the manufacturer (along with all thesuppliers) as a strategic agent (see any standard textbook inmicroeconomics, e.g., Mas-Colell et al., 1995).

5 Such complex prices are necessary to implement the VCGmechanism using iterative auctions (Mishra, 2004).

D. Mishra, D. Veeramani / European Journal of Operational Research 180 (2007) 617–629 619

2. The model and preliminaries

2.1. The model

Let A = {1, . . . ,n} be the set of items that themanufacturer needs to procure. Let X = {S : S � A}be the set of bundles of items. There are m supplierswho can supply the items to the manufacturer. LetB = {0,1, . . . ,m} be the set of suppliers. Supplier 0is a dummy supplier who supplies the items thatcannot be supplied by other suppliers. In a sense,the dummy supplier represents the manufacturerwho can be thought to have ‘‘in-house’’ manufac-turing capability.

The cost of supplying a set of item S 2 X by asupplier i 2 B is denoted by ci(S). So, the cost ofin-house production of a bundle of items S isc0(S). We assume all costs to be non-negative inte-gers. The payoff of a supplier i 2 B�0 from supply-ing a bundle S at price p is p � ci(S). If themanufacturer procures items in S at price p fromsuppliers in B�0 and manufacturers the remainingitems in-house, then his payoff is �p � c0(AnS).An allocation X = (X0,X1, . . . ,Xm) is a vector onsuppliers with Xi denoting the bundle to be suppliedby supplier i and [i2BXi = A. The total cost of pro-curement from an allocation X is given byCðX Þ ¼

Pi2BciðX iÞ. Observe that the total cost of

procurement equals the total payoff of suppliers inB. An efficient allocation is an allocation X such thatCðX Þ ¼ minY CðY Þ, i.e., an allocation that minimizesthe total cost of procurement.

We will denote Bn{i} as B�i. Let B ¼ fB;B�1; . . . ;B�mg. We will often be interested in ‘‘mar-ginal economies’’ where a single supplier is absentfrom B�0. We will denote an economy with suppli-ers from M 2 B as E(M). Notice that every econ-omy E(M) contains the dummy supplier. We willdenote the total cost of procurement in an efficientallocation in economy E(M) as CðMÞ. We will calleconomy E(B) the main economy. We will call econ-omy E(M) for M 2 ðB n fBgÞ a marginal economy.

2.2. The VCG mechanism

In this research, we are interested in implement-ing an efficient allocation. Mechanism design litera-ture points to the VCG mechanism forimplementing an efficient allocation. In its naturalform, the VCG mechanism is a sealed-bid auctionin which bidders (suppliers in this case) are askedto submit their entire cost function. Based on this,

the efficient allocation is determined, and paymentsare given to suppliers such that their payoff equalstheir respective ‘‘marginal product’’. Specifically,the payoff of supplier i 2 B in the VCG mechanismis pvcg

i ¼ CðBÞ � CðB�iÞ. This makes the VCG mech-anism strategy-proof.4 The payment of a supplierassociated with the VCG mechanism will be referredto as the Vickrey payment of that supplier.

The VCG mechanism is not easy to implement inpractice. Its computational and preference elicita-tion problems has motivated researchers to designiterative auctions that overcome these problems(Parkes, 2005; Cramton, 1998). The fundamentalidea behind iterative auctions is that of prices. Inevery iteration, a price vector is announced and bid-ders are asked to report their ‘‘bids’’ against thisprice. The prices are adjusted given the current bids.The auction ends when there is no need to adjustprices. The natural question is how to find a pricevector that gives enough information to implementthe VCG mechanism. Using a modified notion of‘‘competitive equilibrium’’, Mishra and Parkes(forthcoming) characterized such prices.

A price vector p belongs to RjXj�ðmþ1Þþ , i.e., every

supplier sees a personalized price on every bundle.5

Given a price vector p and an economy E(M) forM 2 B, components of p corresponding to suppliersin M only are considered. We will always assumethat p0(S) = c0(S) for all S 2 X and for all p. Also,pi(;) = 0 for all i 2 B and for all p. Given a pricevector p, define the payoff of supplier i 2 B aspi(p) :¼ maxS2X[pi (S) � vi(S)], and his supply set

as Li(p) :¼ {S 2 X : pi(S) � ci(S)}. Notice thatp0(p) = 0 and L0(p) = X. Let XðMÞ denote all thefeasible allocations of economy E(M). Given a pricevector p, for every economy EðMÞðM 2 BÞ definethe price of procurement of the manufacturer aspmðM ; pÞ :¼ minX2XðMÞ

Pi2M piðX iÞ, and the demand

set as DðM ; pÞ :¼ fX 2 XðMÞ :P

i2M piðX iÞ ¼pmðM ; pÞg. Using these notions, we define a compet-itive equilibrium.


Definition 1. A price vector p 2 RjXj�ðmþ1Þþ and an

allocation X 2 XðMÞ are a competitive equilibrium

(CE) of economy E(M) if X 2 D(M,p) andXi 2 Li(p) for every i 2M. If (p,X) is a CE ofeconomy E(M), then p is called a CE price vector ofeconomy E(M). p is a universal CE (UCE) pricevector if it is a CE price vector of economy E(M) forevery M 2 B.

The UCE price concept is central to the design ofiterative auctions. This was shown in Mishra andParkes (forthcoming) who proved the followingresult.6

Theorem 1 Mishra and Parkes, forthcoming. Let

(p,X) be a CE of main economy. The Vickrey

payments of every supplier in B�0 can be calculated

from (p,X) if and only if p is a UCE price vector.

Moreover, if p is a UCE price vector then for every

supplier i 2 B�0, the Vickrey payment of i is

pvcgi ¼ piðX iÞ þ ½pmðB; pÞ � pmðB�i; pÞ�.

We will refer to the term [pm(B,p) � pm(B�i,p)] asthe bonus of supplier i at the UCE price vector p.

3. Vickrey–Dutch procurement auction

In this section, we describe our Vickrey–Dutchprocurement auction. To do so, we define some con-cepts first. Define the compatible demand set of themanufacturer in economy EðMÞðM 2 BÞ at a pricevector p as D*(M,p) :¼ {X 2 D(M,p) : Xi 2 Li [ A}.Every allocation in the compatible demand setbelongs to the demand set, and thus minimizes theprice of procurement. Also, every allocation in thecompatible demand set is such that every supplieris either allocated a bundle from his supply set orthe entire set of items. The motive behind definingsuch a notion will become clear as we analyze theproperties of the auction.

Definition 2. A price vector p is a restricted CE price

vector of economy EðMÞ ðM 2 BÞ if the compatibledemand set D*(M,p) is not empty.

Observe that a CE price vector is also a restrictedCE price vector.

Definition 3. Undersupply holds in economyEðMÞ ðM 2 BÞ at a price vector p if p is a restricted

6 The idea of giving bonuses to bidders can also be found inParkes and Kalagnanam (2005).

CE price vector but not a CE price vector ofeconomy E(M).

Using the notions of restricted CE price vectorand undersupply, we define our Vickrey–Dutch auc-tion for procurement.

Definition 4. The Vickrey–Dutch auction (VDA) for

procurement is an iterative auction with the follow-ing steps:

S0 Initialize the price vector to the zero pricevector.7

S1 Collect supply sets of suppliers at the currentprice vector p.

S2 If undersupply does not hold in economyE(M) for every M 2 B at the current price vec-tor p, go to Step (S3). Else, select an economyE(M) in which undersupply holds at the cur-rent price vector p and do the following priceadjustment.

7 Teclow prrestric

8 Weour rese. The

S21 For every i 2 B�0 and S 2 X, if S 62 Li(p)then increase price pi(S) by 1,8 else donot change the price pi(S). Go to Step(S1).

S3 The auction ends. If p* is the final price vectorin the auction, then a final allocationX 2 D(B,p*) is chosen such that the numberof suppliers who get a bundle from their sup-ply set is maximized. The final payment ofsupplier i is p�i ðX iÞ þ ½pmðB�i; p�Þ � pmðB; p�Þ�.

Since prices of bundles in the supply set of a sup-plier is never increased and prices of bundles not inthe supply set are increased by 1, the payoff of a sup-plier is unchanged by price adjustment. Since initialpayoff is zero, it remains zero throughout the auc-tion. Also, since the price of bundles in the supplysets never increase, once a bundle enters the supplyset it never leaves. This gives us the following lemmadirectly.

Lemma 1. The payoff of a supplier remains zero

throughout the auction and the supply sets of the

supplier (weakly) grows from iteration to iteration.

hnically, we require that the price vector be initialized to aice vector where every supplier has zero payoff and it is ated CE price vector of economy E(M) for every M 2 B.

can choose any arbitrary e > 0 as price increment and allults will hold with some error terms, which will depend onchoice of 1 as e makes the analysis less cumbersome.


An analogous lemma corresponding to the man-ufacturer is the following.

Lemma 2. Suppose undersupply holds in economy

E(M). Then, after a price adjustment the price of

procurement increases by 1 in economy E(M).

Proof. Since undersupply holds in economy E(M),in every allocation X 2 D(M,p), where p is the pricevector before price adjustment, there is a supplieri 2M such that Xi 62 Li(p). By the price adjustmentrule, price of Xi 62 Li(p) will increase by 1. Thisimplies that the price of procurement is increasedby at least 1 on every X 2 D (M,p). ConsiderX 2 D*(M,p) (D*(M,p) is non-empty since p is arestricted CE price vector). Since undersupplyholds, there is only one supplier k 2M such thatXk = A 62 Li(p) and for every other supplier i 5 k,Xi = ; 2 Li(p) (this follows from Lemma 1 and thefact that ; is in the supply set of every supplier ini-tially). Again, by the price adjustment rule, the priceof procurement from allocation X is increased byexactly 1. The payoff from any allocationX 62 D(M,p) is at least 1 above the payoff fromany allocation X 2 D(M,p). This implies that theprice of procurement increases by 1 in economyE(M) by price adjustment if undersupply holds ineconomy E(M). h

The following theorem says that we maintain arestricted CE price vector in every iteration of ourauction.

Theorem 2. Consider any iteration t in the auctionwhere undersupply holds in some economy. Let the

price in iteration t be pt in the auction. If pt is a

restricted CE price vector in economy E(M) for any

M 2 B, then pt+1 is a restricted CE price vector of

economy E(M).

Proof. Since pt is a restricted CE price vector ofeconomy E(M), there are two cases:

Case 1: Undersupply holds in economy E(M).Consider the allocation X 2 D*(M,pt). FromLemma 2, the price of procurement is increased by1 in economy E(M). By price adjustment, price ofprocurement from X is increased by 1. This impliesthat X 2 D(M,pt+1) after the price adjustment.

From Lemma 1, the payoff of all suppliersremain zero and the supply sets weakly increase ineach iteration. This implies that if Xi 2 Li(p

t) thenXi 2 Li(p

t+1) for every i 2M. So, X 2 D*(M,pt+1)after the price adjustment. So, pt+1 is a restricted CEprice vector of economy E(M).

Case 2: Undersupply does not hold in economyE(M). This implies that pt is a CE price of economyE(M). Let X 2 D*(M,pt) be an allocation such that(pt,X) is a CE of economy E(M). For every i 2M,Xi 2 Li(p

t). So, prices of bundles in X do notincrease, and X remains in D*(M,pt+1). FromLemma 1, a supplier i will continue to have Xi inhis supply set throughout the auction. This impliesthat (pt+1,X) is a CE of economy E(M) in iterationt + 1, and pt+1 is a CE price vector of economyE(M). h

Theorem 2 shows how we maintain a restrictedCE price vector for economy E(M) for everyM 2 B in every iteration of our auction. The follow-ing theorem shows that the auction achieves theVCG outcome.

Theorem 3. The Vickrey–Dutch auction for procure-

ment achieves the VCG outcome.

Proof. Starting price is a restricted CE price vectorof economy E(M) for every M 2 B. From Theorem2, every iteration in the auction is a restricted CEprice vector of economy E(M) for every M 2 B.By Lemma 1, the payoff of every supplier is zerothroughout the auction. This implies that once theprice of a bundle reaches its cost, it does notincrease anymore. This implies that the price cannotincrease forever and the auction will terminate. Theterminating condition ensures that the finalrestricted CE price vector is a CE price vector ofeconomy E(M) for every M 2 B. Thus, the finalprice vector in the auction is a UCE price vector,and the final allocation is an efficient allocation(because every allocation associated with a CE allo-cation of the main economy is an efficient alloca-tion). From Theorem 1, the payment of eachsupplier is his Vickrey payment. Thus, the Vick-rey–Dutch auction for procurement achieves theVCG outcome. h

3.1. Incentives

The sealed-bid format of the VCG mechanism isstrategy-proof. But in an iterative implementation(extensive form game), the VCG outcome may notsupport truthful bidding in a dominant strategyequilibrium (Ausubel and Milgrom, 2002; Bikh-chandani and Ostroy, 2006; de Vries et al., forth-coming; Parkes and Ungar, 2000). We can stillshow that truthful bidding is an ex post Nash equi-librium. For ex post Nash equilibrium to exist, we


require some simple bidding rules in the auction.For example, initially all suppliers should be willingto supply the ; bundle, the supply sets of the suppli-ers should weakly grow etc. These are ‘‘consistency’’requirements in bidding and are easy to implement.For example, Parkes and Ungar (2000) propose thenotion of ‘‘proxy agents’’ to implement such consis-tency requirements. With such consistency require-ments, one can map the bidding strategy of asupplier to some cost function (may be non-truth-ful). Given these consistency requirements are met,the outcome of the auction will be the VCGoutcome with respect to some profile of cost func-tions. Since the VCG mechanism is strategy-proof,truthful bidding is the best strategy for suppliersin our Vickrey–Dutch auction for procurementunder consistency requirements. Truthful biddingis not a dominant strategy for suppliers becauseevery supplier has to condition his strategy againstthe fact that every other supplier is playing a strat-egy that is consistent with some cost function. Thisgives us the following theorem immediately.

Theorem 4. Truthful bidding is an ex post Nash

equilibrium in the Vickrey–Dutch auction for pro-curement if consistency requirements are met.

Table 1An example

; {1} {2} {1,2}

0 0 1 1 1

1 0 3 3 32 0 2 3 63 0 2 4 4

Table 2Progress of auction in Definition 4 for the Example in Table 1

# Supplier 1 Supplier 2

{1} {2} {1,2} {1} {2} {1,2

1 0 0 0 0 0 02 1 1 1 1 1 13 2 2 2 (2) 2 24 (3) (3) (3) (2) (3) 3

CE of economy E(B), E(B�2), E(B�3) is achieved

5 (3) (3) (3) (2) (3) 4CE of economy E(B�1) is achieved. A UCE price is foundFinal allocation: Supplier 1 supplies both itemsFinal payment: Supplier 1 is paid 3 + (4 � 3) = 4. Other supp

3.2. An example

Consider an example with three (non-dummy)suppliers. The manufacturer needs to procure twoitems. The ‘‘in-house’’ costs for the manufacturerare represented by the dummy supplier. For simplic-ity, we have put 1 costs for in-house production.

The progress of Vickrey–Dutch auction for pro-curement, for the example in Table 1, is providedin Table 2. The columns corresponding to suppliersshow the personalized prices on bundles. The bun-dles which have prices in (Æ) are in the supply setof the suppliers. The manufacturer’s price of pro-curement in economy E(M) for every M 2 B isshown in every iteration in the last column. The auc-tion terminates at a UCE price vector in iteration 5.Note that bonuses are given to the winning suppliersaccording to the payment rule of the auction.

3.3. Alternate auctions

In this subsection, we will sidestep incentiveissues (assume a CE to be a ‘‘fair’’ scheme for sup-pliers to bid truthfully) and explore other possibleauctions. These auctions can be derived from slightmodification of the Vickrey–Dutch auction forprocurement.

• A1—Vickrey–Dutch auction for procurement:This is the exact auction in Definition 4 andachieves the VCG outcome.

• A2—Dutch auction for procurement with bonus:In this auction, the auction in Definition 4 isstopped as soon as a CE of economy E(B) isreached, but the final price vector in the auctionneed not be a UCE price vector. The final alloca-tion and payment schemes remain the same as

Supplier 3 Price of procurement

} {1} {2} {1,2} pm(Æ)

0 0 0 0, 0, 0, 01 1 1 1, 1, 1, 1(2) 2 2 2, 2, 2, 2(2) 3 3 3, 3, 3, 3

(2) (4) (4) 3, 4, 3, 3

liers are paid zero


Step (S3) in Definition 4. So, suppliers may getbonus at the end of the auction.

• A3—Dutch auction for procurement without

bonus: In this auction, like the auction in A2,the auction in Definition 4 is stopped as soonas a CE of economy E(B) is reached. The finalallocation remains the same as in Definition 4,but the final payment is the final price in theauction.

These auctions can be used in different scenarios.If incentive is a big concern, only auction A1 hasnice incentive properties (Theorem 4). But if incen-tive is not a concern and simplicity and speed is aconcern, auctions A2 and A3 should be considered.Auction A2 provides bonus to suppliers and thuscan be viewed as more fair to the suppliers than auc-tion A3.

4. Issues in practical implementation

A practical limitation of iterative auctions is thecomputational complexity of the winner determina-

tion problem (WDP) (Rothkopf et al., 1998). TheWDP is not the same for every iterative auction.To define the WDP in our case, we state some cor-ollaries of our previous results.

We state some corollaries to Lemma 2 and The-orem 2 that will help us simplify the WDP for ourcase. The first corollary is due to Theorem 2. Weassume that the first iteration of our auction is iter-ation 1.

Corollary 1. If undersupply holds in economy E(M)

in iteration t > 1 of the Vickrey–Dutch auction for

procurement, then it holds in every iteration t 0 < t.

Theorem 2 says more than Corollary 1.

Corollary 2. If the price vector in iteration t of the

Vickrey–Dutch auction is a CE price vector of

economy E(M), then so is the price vector in iterationt + 1 (if iteration t + 1 exists).

As an immediate corollary to Lemma 2 usingCorollaries 1 and 2 is the following result.

Corollary 3. Starting from iteration 1, consider an

iteration t P 1 of the Vickrey–Dutch procurement

auction. If undersupply holds in economy E(M) in

iteration t, then the price of procurement in iterationt + 1 in economy E(M) is t. Further, if t* is the

iteration in the auction where undersupply holds for

the last time in economy E(M), then the price of

procurement in any iteration t > t* in economy E(M)

is t*.

In any WDP of an iterative auction that imple-ments the VCG outcome, computation is done inevery iteration to check two things:

W1 The demand and supply needs to be balancedin the main economy and in every marginaleconomy (Mishra and Parkes, forthcoming).When bidders have special class of cost func-tions, balance of demand and supply need tobe checked only in the main economy (deVries et al., forthcoming; Ausubel and Mil-grom, 2002). But this is not true when suppli-ers have general cost functions.

W2 Once an imbalance in demand and supply isidentified in an economy, a group of biddersare selected whose prices need to be changed.In the auction in de Vries et al. (forthcoming),these are a minimal set of ‘‘undersupplied’’bidders. In the auction in Ausubel andMilgrom (2002), these are a minimal set of‘‘losing’’ bidders.

In some auctions, e.g., in one of the auctions inDemange et al. (1986) and the auction in Ausubeland Milgrom (2002), the computation of (W1) alsogives us price adjustment direction in (W2). This isnot true for the auction in de Vries et al. (forthcom-ing) and in an auction in Demange et al. (1986),where explicit calculations are needed for (W1)and (W2).

We note that our Vickrey–Dutch procurementauction has a very simple price adjustment direction(See Step S21 in Definition 4) and involves no com-putation in (W2). We will now discuss how to do thecomputation in (W1) for a special case.

4.1. The multi-unit case

In this section we consider the special case whereall the items are of the same type (homogeneousunits). We call this the multi-unit case. The multi-unit case is very typical in procurement settingwhere multiple units of the same item is often pro-cured simultaneously. This case is also tractablefrom a computation perspective because the numberof bundles to consider is simply the number of units.Contrast this with the general heterogeneous itemscase, where the number of bundles to consideris exponential in the number of items. We will


consider the winner determination problem in ourauction for the multi-unit case.

For this case, we will give an integer programmingformulation to check if undersupply holds in aneconomy. Let A = {1, . . . ,n} be the set of units thatneed to be procured. Let X = {0,1, . . . ,n} be the pos-sible bundles of units. At a price vector p, the supplyset of a supplier i 2 B, denoted by Li(p), will includeall the bundle of units that maximize his payoff atprice vector p. We note that Li(p) can include 0 whenthe maximum payoff of supplier i is zero over all thebundles in X at price vector p. Let xi(j) 2 {0,1} be abinary variable that is set to 1 if supplier i is assignedto supply j (j 2 Li(p) [ {n}) units at price vector p andset to zero otherwise. We remind that pm(M,p)denotes the price of procurement of economy E(M)at price vector p. Now, consider the following formu-lation for economy E(M) where M 2 B in iteration t

of our auction with price vector pt.

aðM ;ptÞ ¼minX

i2M :n 62LiðptÞxiðnÞ ðWDÞ

s:t:X

j2LiðptÞ[fngxiðjÞ ¼ 1 8i2M ; ð1Þ

X

i2M

X

j2LiðptÞ[fngjxiðjÞ ¼ n; ð2Þ

X

i2M

X

j2LiðptÞ[fngpt

iðjÞxiðjÞ ¼ pmðM ;ptÞ;

ð3ÞxiðjÞ 2 f0;1g 8i2M ;j2 LiðptÞ[fng:

ð4Þ

Call a supplier unsatisfied if he is allocated a bun-dle of units that is not in his supply set. Formulation(WD) minimizes the number of unsatisfied suppliersin M. The first set of constraints (1) ensure thatevery supplier is either allocated a bundle from hissupply set or all the n units. The second set of con-straints (2) ensure that the total number of unitsallocated is exactly n. The third set of constraints(3) ensure that the price of procurement from theallocation equals the actual price of procurement.We note that the price of procurement in iterationt in economy E(M), pm(M,pt), can be tracked bythe auctioneer throughout our auction using Corol-lary 3. We will elaborate more on this after provingthe following Theorem.

Theorem 5. Undersupply holds in economy

EðMÞðM 2 BÞ in iteration t of the VDA for procure-

ment if and only if a(M,pt) = 1.

Proof. If undersupply holds in economy E(M) initeration t then by definition, pt is a restricted CEprice vector but not a CE price vector of economyE(M). So, by definition D*(M,pt) is not empty. So,there exists an allocation that gives a price of pro-curement pm(M,pt) and allocates every supplier i abundle of units from Li(p

t) [ {n}. The x(Æ) variablescorresponding to this allocation is a feasible solu-tion to (WD). Since this is not a CE price vectorof economy E(M), there is some supplier i that isassigned all units n 62 Li(p

t) in this allocation. Byconstraint (2), suppliers other than i are assignedzero unit in this allocation, which is in their respec-tive supply sets (Lemma 1). Thus, the value of theobjective function corresponding to this feasiblesolution is 1. Clearly, this is the minimum value ofthe objective function when undersupply holds,implying a(M,pt) = 1.

Now, consider the case when a(M,pt) = 1. FromTheorem 2, pt is a restricted CE price vector ofeconomy E(M). If pt is a CE price vector then wecan find an allocation in D*(M,pt) such that everysupplier is allocated bundles of units from hissupply set, which implies a(M,pt) = 0. So, pt is not aCE price vector, which further implies that under-supply holds in economy E(M) in iteration t. h

To use Theorem 5 effectively, we need some mod-ifications in the VDA for procurement. We providethese modifications below:

• In the VDA for procurement, we can considereconomy E(M) for every M 2 B using somepre-determined order (without loss of generality,that order can be B�1, . . . ,B�m ,B). Once we startconsidering economy E(M), we continue toupdate prices using Step S21 in Definition 4 aslong as undersupply holds in economy E(M).Once undersupply does not hold in economyE(M), we do not need to consider that economyagain (Theorem 2).

• Using Corollary 3, we can only track the revenueof an economy whose undersupply held in theprevious iteration. But, when an economy E(M)is first considered for price update in an iterationt, it is not possible to say whether undersupplyheld in the previous iteration or not. For this iter-ation, i.e., the iteration in which an economy isfirst considered for price update using Step S21in Definition 4, we use a formulation similar toformulation (WD) to find the price of procure-ment. The only differences are: (i) constraints

0

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100

150

200

0 50 100 150 200

Auc

tion

Tim

e in

Sec

onds

Number of Units

Plot of average auction running time with number of units

m=10m=20m=30

Fig. 1. Average running time with number of units.


(3) are omitted, (ii) the objective function mini-mizes the price of procurement—min

Pi2M�P

j2LiðptÞ[fngptiðjÞxiðjÞ. Using arguments in the

proof of Theorem 5, it is easy to see that thismodification finds the price of procurement ofeconomy E(M) in iteration t. Once this price ofprocurement is calculated, we can use formula-tion (WD) to determine whether undersupplyholds or not. From that iteration onwards, theprice of procurement of economy E(M) can betracked using Corollary 3 (or, Lemma 2) as longas undersupply holds, and we need not solve aninteger program in every iteration to determinethe price of procurement.

Since the maximum number of binary variablesand constraints in the integer programming formu-lation (WD) are m · (n + 1) and m + 2 respectively,we expect the computational burden to of our auc-tion to be reasonable for the multi-unit case (since asupplier is likely to have only a subset of bundles ofunits from X in his supply set, the actual number ofbinary variables will be less than this maximumamount).

The descending reverse auctions (for example,the ones derived from Mishra and Parkes (forth-coming)) do not allow for such simpler formulationof winner determination problem in the multi-unitcase. The main difficulty is that it is not possibleto track the price of procurement in these descend-ing reverse auctions.9 So, we need an explicit formu-lation that finds the price of procurement, and thisinvolves more binary variables than formulation(WD).

To check the scalability of the VDA for procure-ment, we performed simulations where we used for-mulation (WD) in every iteration to determine priceadjustment.

The simulations were set up as follows. Theinputs to the simulation were number of suppliersand number of items. Cost values were randomlygenerated (using a uniform distribution) for eachsupplier. For every supplier and every unit, a perunit cost is drawn from a uniform distribution with

9 But it is possible to track the payoff of the suppliers in thesedescending reverse auctions. Thus, it makes the problem offinding the supply sets of suppliers easier. Since the number ofbundles involved in this setting is only n + 1, the problemof finding the supply sets of suppliers is not that difficult even inour auction.

range [10, 30]. The cost of k units is k times the perunit cost.

We measured the average run-time of our auc-tion (average taken over 1000 runs of our auction)when the number of suppliers (m) is 10, 20, and30. We increased the number of units in each ofthese cases, keeping at least twice the number ofsuppliers. The average run-times are shown inFig. 1. Fig. 1 shows that the average run-times areunder two minutes when the number of units areless than or equal to 150. In practice, we will expecta firm to select not more than 30 suppliers for bid-ding phase of procurement. Our VDA for procure-ment has good average running-time when thenumber of units are less than or equal to 150.

5. Information revelation

In this section, we will discuss (informally andexperimentally) information revelation properties,defined as the cost information (of suppliers)revealed due to bidding in the auction, of theVDA for procurement and its descending reverseauction counterpart. The descending reverse auc-tion for multi-item combinatorial procurement isderived from iBEA auction for single-seller settingin Mishra and Parkes (forthcoming). In this section,we will call this auction the descending reverse auc-

tion. Also, we will assume that the suppliers bidtruthfully in both the auctions as bidding truthfullyis an ex post Nash equilibrium in both the auctions(under mild consistency requirements).

We emphasize that the information revelationproperty of an auction is important because compa-nies are always reluctant to reveal their internal


costs to rival suppliers due to strategic reasons.Also, Elmaghraby (2004) discusses how too muchinformation revelation in the descending reverseauctions have discouraged participation of suppliersin E-marketplaces. Besides the strategic reason,Elmaghraby (2004) gives possibility of collusion

due to information revelation as another reasonfor non-participation of suppliers in E-market-places. Less participation of suppliers lead to higherprice of procurement. So, information revelationhas effect on price of procurement of amanufacturer.

5.1. Characterizing information revelation

The VDA has the property that no supplier haspayoff more than zero during the auction (Lemma1). So, a supplier reveals his cost information on abundle as soon as he reports that the bundle is inhis supply set. Since the VDA converges to a UCEprice vector, a supplier who is allocated ; in the effi-cient allocation of economy E(M) for every M 2 B,will have only ; in his supply set throughout theauction. Call such suppliers losing suppliers. So, los-ing suppliers reveal no cost information in theVDA. Observe that if a supplier is not a losing sup-plier, he may not reveal his entire cost information.But he may reveal some of his cost information.

Contrast this with the information revealed in thedescending reverse auction. In the descendingreverse auction, once a bundle is in the supply setof a supplier, it continues to be in his supply setthroughout the auction. So, once a supplier has ;in his supply set, he has zero payoff on all the bun-dles he has ever reported to be in his supply set dur-ing the auction. Since the prices of all the bundlesare known, the cost information of the bundles inthe supply set are revealed. A supplier who winssome bundle at positive payoff in the efficient alloca-tion of the main economy never has ; in his supplyset during the descending reverse auction. Call suchsuppliers winning suppliers. All winning suppliersreveal no cost information during the descendingreverse auction. Again, observe that if a supplier isnot a winning supplier, he may not reveal his entirecost information. But he may reveal some of his costinformation.

These discussions tell us that if the number of los-ing suppliers are more than the number of winningsuppliers, there is a good chance that the VDA willperform better in terms of information revelation.This can happen in large ‘‘competitive’’ economies,

where the number of suppliers are more comparedto the number of items.

In many situations, revealing the costs of the win-ning suppliers may not be considered a good idea.For example, revealing the cost information of thewinning supplier may embarrass a manufacturer ifhe comes to know that the winning supplier is mak-ing a lot of profit and he could have got the items ata lower price. The VDA does a bad job in such sit-uations but the descending reverse auction does notsuffer from this problem. But this problem can befixed in the VDA by imposing restrictions on infor-mation revealed during the auction. Most of themanufacturers use third party softwares for auction-ing, and such restrictions on information revelationis common in those softwares (Elmaghraby, 2004).We propose some such restrictions:

Auction as a ‘‘blackbox’’: In such a setting, auc-tion can be treated as a blackbox. The only informa-tion available to the suppliers and the manufacturerare the prices. Bids from suppliers are known onlyto the blackbox. At the end of the auction, the finalallocation and payments are disclosed. Such ablackbox can be implemented for both the VDAand the descending reverse auction and neither ofthem reveal any information about costs of thesuppliers.

Limited information revelation: The other optionis to broadcast only limited information from theauction. For example, in a non-anonymous pricesetting, suppliers see their own personalized priceonly, but bid information of every supplier is notmade public. This prevents a supplier from knowingthe cost information of other suppliers.

We conclude this discussion by observing thatthere are several (technological) methods to over-come the information revelation disadvantages inthe VDA and the descending reverse auction. Butif the identity of the supplier (i.e., a winner or loser)is not an issue, the VDA will likely outperform thedescending reverse auction in many settings in termsof information revelation. Combine this with fasterconvergence rate and possibility of minimizing col-lusion, the VDA can be preferred over the descend-ing reverse auction in many settings.

5.2. Experiment results

To validate our claim about the VDA having bet-ter (overall) information revelation properties, weconducted computer based experiments. We simu-lated the descending reverse auction and the VDA

0.4

0.5

0.6

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0.8

0.9

1

1.1

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

Rev

elat

ion

perc

enta

ge ti

mes

0.0

1

Density

Plot of revelation percentage versus density, m = 10, n = 6

Descending reverse auctionVDA for procurement

Fig. 2. Revelation percentage with density.

0.3

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0.5

0.6

0.7

0.8

0.9

1

1.1

4 5 6 7 8 9

Rev

elat

ion

perc

enta

ge ti

me

0.01

Number of suppliers

Plot of revelation percentage versus number of suppliers, n = 6, density = 0.5


Fig. 3. Revelation percentage with number of suppliers.


using computer programs and observed the infor-mation revelation properties of both the auctions.

The experiments were set up as follows. Theinputs to the experiment were number of suppliers,number of items, and density. Cost values were ran-domly generated (using a uniform distribution) foreach supplier. Density of the system determinesthe fraction of the total possible bundles (2n for asystem with n items) on which the suppliers areassigned costs randomly. For every supplier, thesebundles are chosen randomly and assigned costsfrom a uniform distribution, and the rest of the bun-dles are assigned high costs (indicating that the sup-plier cannot supply these bundles). To assign cost toa bundle, a random number is drawn from a uni-form distribution and multiplied by the number ofitems in that bundle. We note that although thenumber of items are less in our simulations, thenumber of bundles are reasonably high.

We observed the revelation percentage in both theauctions. The revelation percentage of an auction isthe percentage of (supplier, bundle) tuples whosecosts can be exactly known (to anyone who seesthe bid information), given that suppliers bid truth-fully. Realize that the cost on the ; bundle is zeroand universally known.

Density: We fixed the number of suppliers (m) at10 and number of items (n) at 6. So, each suppliercan have costs on 64 bundles. A density of 0.4means, costs for 25 (=0.4 · 64) bundles are drawnrandomly from a uniform distribution for each sup-plier and the rest of the bundles are assigned highcosts. We varied the density from 0.2 to 0.7 andplotted revelation percentages (times 0.01) of boththe auctions in Fig. 2.

The VDA has lower revelation percentage thanthe descending reverse auction. Also, revelation per-centage in the VDA shows a marginal increase withdensity whereas it remains almost constant in thedescending reverse auction. The intuition behindthis is as follows. In the VDA, winning suppliers

(defined earlier), reveal cost information of bundlesin their supply set. As the density increases, the sup-ply sets also increase and hence the revelation per-centage increases. For the descending reverseauction, the revelation percentages are already veryhigh (almost one) for lower densities and remainthat way.

Number of suppliers: We fixed the number ofitems at 6 (number of possible bundles 64), and var-ied the number of suppliers between 4 and 9. Theeffect of number of suppliers on revelation percent-

ages in both the auctions is shown in Fig. 3 for den-sity 0.5. We observed similar plots for densities 0.3and 0.7.

The VDA has lower revelation percentage thanthe descending reverse auction. The revelation per-centage of both the auctions increase with theincrease in number of suppliers. This can beexplained as follows. For the VDA, as the numberof suppliers increase, the number of winning suppli-ers increase and hence the revelation percentageincreases. But, after the number of suppliers reacha certain number (almost equal to number of items),the number of winning suppliers do not increasethat much and hence the revelation percentageremains almost constant after that. For the descend-ing reverse auction, when the number of suppliersare less, the competition in the economy is lessand competitive equilibrium in main economy andmarginal economies can be achieved without muchbidding. This leads to less information revelation


when number of suppliers are less and explains theincrease in revelation percentage with increase innumber of suppliers.

Number of items: We fixed the number of suppli-ers at 6 and varied the number of items from 2 to 6.The effect of number of items on revelation percent-age in both the auctions is shown in Fig. 4 for den-sity 0.5. We observed similar plots for densities 0.3and 0.7.

Again, the VDA does better than the descendingreverse auction in terms of revelation percentage.The graph shows that the revelation percentage inthe VDA decreases with the number of items. Withthe increase in the number of items, the number ofwinning suppliers increase and hence the revelationpercentage should increase. But this effect is domi-nated by the fact that the number of bundlesincrease (exponentially) with the increase in numberof items. Since winning suppliers do not reveal coston all bundles, this effect reduces the revelation per-centage with increase in number of items. We callthis the exponential bundle effect. The exponentialbundle effect has negative impact on the descendingreverse auction as losing suppliers reveal cost infor-mation on almost all bundles in this auction. As, wewill see next, to achieve a UCE price vector for lar-ger economies, the descending reverse auction revealcost information on almost all bundles of the losingsuppliers. This continues as the number of bundlesgrow exponentially and hence the revelation per-centage increases.

Size of economy: We fixed the ratio of number ofsuppliers to number of items at 4

3and varied the

number of suppliers (and the number of items).The effect of size of economy (number of suppliersand items) on revelation percentage in both the auc-

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2 3 4 5 6

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Number of items

Plot of revelation percentage versus number of items, m = 6, density = 0.5


Fig. 4. Revelation percentage with number of items.

tions is shown in Fig. 5 for density 0.5. We observedsimilar plots for densities 0.3 and 0.7.

The revelation percentages of the VDA got betterthan the descending reverse auction as the size of theeconomy (number of suppliers and items) increased.The decrease in the revelation percentage of theVDA and the increase in the revelation percentageof the descending reverse auction is again explainedby the exponential bundle effect. In larger econo-mies, the descending reverse auction requires almostall cost information to achieve a UCE price vectorand hence the revelation percentage increaseswith the increase in the size of economy. On theother hand, the exponential growth in number ofbundles reduces the number of bundles whose costinformation is revealed by winning suppliers in theVDA.

We summarize our findings from the experimentsbelow:

• In larger economies, the VDA has better infor-mation revelation property than the descendingreverse auction. In fact, the descending reverseauction requires (almost) complete revelation ofcost information to implement the VCG outcomein larger economies.

• Lower density (typical in many practical settings)favors the VDA, as revelation percentagedecreases with lower density.

• With higher densities, the revelation percentageof the descending reverse auction improves (spe-cially when the number of items or suppliers aresmall). But this setting is rare in practice.

These findings validate our claim that the VDA isbetter than the descending reverse auction in terms

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Number of items (=3/4 times number of suppliers)

Plot of revelation percentage versus size of economy, density = 0.5


Fig. 5. Revelation percentage with size of economy.


of information revelation, and does really well inthis regard in larger economies.

6. Conclusions

We have designed a Vickrey–Dutch auction,which is ascending in nature, for procuring multipleitems simultaneously. The auction implements theVCG outcome and truthful bidding is an ex postNash equilibrium. We showed that our auction iscomputationally scalable for the multi-unit caseand has better information revelation propertiesthan its descending price reverse auction counter-part. This, coupled with the inherent benefit ofspeed in the Vickrey–Dutch auctions, makes ourauction suitable for practical use in procurementsettings.

Acknowledgements

We thank two anonymous referees, David C.Parkes, and seminar participants at the Universityof Wisconsin-Madison for their comments on anearlier version of the paper. This work is based ona Chapter of the doctoral thesis of the first author,submitted at the University of Wisconsin-Madisonin 2004.

References

Ausubel, L.M., 2004. An efficient ascending-bid auction formultiple objects. American Economic Review 94 (5), 1452–1475.

Ausubel, L.M., Milgrom, P.R., 2002. Ascending auctions withpackage bidding. Frontiers of Theoretical Economics 1 (1), 1–42.

Bikhchandani, S., Ostroy, J., 2006. Ascending price Vickreyauctions. Games and Economic Behavior 55 (2), 215–241.

Cramton, P., 1998. Ascending auctions. European EconomicReview 42, 745–756.

de Vries, S., Schummer, J., Vohra, R.V., forthcoming. Onascending Vickrey auctions for heterogeneous objects. Jour-nal of Economic Theory. DOI: http://dx.doi.org/10.1016/j.jet.2005.07.010.

Demange, G., Gale, D., Sotomayor, M., 1986. Multi-itemauctions. Journal of Political Economy 94 (4), 863–872.

Elmaghraby, W., 2004. Pricing and auctions in E-marketplaces.In: Simchi-Levi, D., Wu, S. David, Max Shen, Z. (Eds.),Handbook of Quantitative Supply Chain Analysis: Modelingin the E-Business Era, International Series in OperationsResearch and Management Science. Kluwer Academic Pub-lishers, Norwell, MA.

Mas-Colell, A., Whinston, M.D., Green, J.R., 1995. Microeco-nomic Theory. Oxford University Press, New York, USA.

Mishra, D. 2004. Simple primal–dual auctions are not possible.In: Proceedings of 5th ACM Conference on ElectronicCommerce (EC’04), New York City, NY.

Mishra, D., Parkes, D.C., 2004a. Multi-item Vickrey–Dutchauction for unit demand preferences. Working Paper, Har-vard University.

Mishra, D., Parkes, D.C. 2004b. A Vickrey–Dutch clinchingauction. Working Paper, Harvard University.

Mishra, D., Parkes, D.C., forthcoming. Ascending price Vickreyauctions for general valuations. Journal of Economic Theory.DOI: http://dx.doi.org/10.1016/j.jet.2005.09.004.

Mishra, D., Veeramani, D., 2006. An ascending price procure-ment auction for multiple items with unit supply. IIETransactions 38 (2), 127–140.

Parkes, D.C., 2005. Auction design with costly preferenceelicitation. Annals of Mathematics and AI 44, 269–302.

Parkes, D.C., Kalagnanam, J., 2005. Models for iterativemultiattribute Vickrey auctions. Management Science 51,435–451.

Parkes, D.C., Ungar, L.H., 2000. Preventing strategic manipu-lation in iterative auctions: Proxy agents and price-adjust-ment. In: Proceedings of the 17th National Conference onArtificial Intelligence (AAAI-00), pp. 82–89.

Rothkopf, M.H., Pekec, A., Harstad, R.M., 1998. Computation-ally manageable combinatorial auctions. Management Sci-ence 44 (8), 1131–1147.

Vickrey, William, 1961. Counterspeculation, auctions, and com-petitive sealed tenders. Journal of Finance 16, 8–37.

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Vickrey–Dutch procurement auction for multiple items

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