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An ascending price procurement auction for multipleitems with unit supplyDebasis Mishra a & Dharmaraj Veermani ba Center for Operations Research and Econometrics , Université Catholique de Louvain ,1348 Louvain-la-Neuve, Belgium E-mail:b Department of Industrial and Systems Engineering , University of Wisconsin-Madison ,Madison, WI, 53706, USA E-mail:Published online: 15 Aug 2006.

To cite this article: Debasis Mishra & Dharmaraj Veermani (2006) An ascending price procurement auction for multiple itemswith unit supply, IIE Transactions, 38:2, 127-140, DOI: 10.1080/074081791009040

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IIE Transactions (2006) 38, 127–140Copyright C© “IIE”ISSN: 0740-817X print / 1545-8830 onlineDOI: 10.1080/074081791009040

An ascending price procurement auction for multiple itemswith unit supply

DEBASIS MISHRA1,∗ and DHARMARAJ VEERMANI2

1Center for Operations Research and Econometrics, Universite Catholique de Louvain, 1348 Louvain-la-Neuve, BelgiumE-mail: mishra@core.ucl.ac.be2Department of Industrial and Systems Engineering, University of Wisconsin-Madison, Madison, WI 53706, USAE-mail: raj@engr.wisc.edu

Received September 7, 2004 and accepted March 21, 2005

In this paper, a supply chain scenario is considered in which an original equipment manufacturer wishes to procure a set of itemsfrom a set of suppliers with private costs. Each supplier can provide at most one item. Two ascending price auctions are proposedto implement an efficient allocation for this model. The first converges to a unique competitive equilibrium price of the economy ifsuppliers bid truthfully. However, because no equilibrium strategy exists for the suppliers, a second auction is designed based on thefirst, which converges to a second unique competitive equilibrium price. Truthful bidding is a Bayesian Nash equilibrium strategy forsuppliers in this auction. We show several practical advantages of our ascending auctions over traditional reverse auctions. Ascendingauctions perform better than reverse auctions in two main aspects: (i) information revelation; and (ii) bidder cost determination.Simulation results are reported to validate these claims.

1. Introduction

In this paper, we study a class of supply chain problemsin which an original equipment manufacturer (henceforthreferred simply as the manufacturer) wants to procure a setof indivisible items from a set of suppliers. In our model,suppliers have privately known costs to supply an item andeach supplier can supply a maximum of one item. An itemis merely an indivisible entity. It can be a collection of 50homogeneous items, such as 50 computer processors, ora heterogeneous item. The only restriction is that an itemcannot be divided, i.e., if 50 computer processors constitutean item, then the entire 50 processors must be procured asa single item from a single supplier. The manufacturer hasa “reservation cost” for procuring each item. This is themaximum amount he is willing to pay to procure the item.We can also view this as a cost of manufacturing the itemin-house. The payoff of a supplier for supplying an itemis the difference between the price he is paid and the costof supplying the item. The payoff of a manufacturer byprocuring an item is the difference between the reservationcost and the price he pays. The payoff of not supplying anyitem to a supplier is zero and the payoff of not procuringan item to the manufacturer is also zero.

∗Corresponding author

In this model, we seek to design iterative auction mech-anisms with the objective of maximizing the total surplusof suppliers and manufacturer, i.e., an (economically) ef-ficient allocation. One approach to find an efficient allo-cation is to ask every supplier his cost function and solvea centralized optimization problem. However, our modeldiffers from the traditional assignment models in the op-timization literature in the sense that suppliers are selfishand may lie about their costs unless sufficient incentives(in terms of payments) are provided. This is also knownas mechanism design in economics. A mechanism is simi-lar to an algorithm that takes information from participat-ing agents (suppliers in our case) and outputs two things:(i) allocation; and (ii) payment. We are interested in de-signing decentralized mechanisms (as are auctions) whichare more suitable than centralized mechanisms due to:(i) transparency (Cramton, 1998); (ii) their ability to beimplemented in a distributed manner; and (iii) their bet-ter elicitation properties (Parkes, 2004). We design auction-based mechanisms to implement an efficient assignment ofitems to suppliers. In particular, we design ascending priceauctions, in contrast to traditional reverse auctions used inpractice for procurement.

The reasons for designing ascending price auctions forour model are manifold. First, ascending price auctionsin a single-buyer model are fast and have less informationrevelation properties. (This issue is discussed in detail inSection 7.1.) Second, due to the descending price nature

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of the reverse auctions, suppliers feel that reverse auctions“eat” into their profits by exploiting the competition inthe market. An ascending price auction may eliminate thisproblem. Our ascending price auction has a simple strategywhich is a Bayesian Nash equilibrium strategy for suppliersto follow. Lastly, reverse auctions are criticized by suppliersfor their high participation costs (i.e., bidders are requiredto calculate bids many times which increases their trans-action costs). As we show, ascending auctions have lowertransaction costs than their reverse auction counterparts.

Although the unit supply restriction on cost functionsdoes not sound suitable in many practical situations, wefind two reasons for its consideration.

� Auctions designed for simple cost functions, such as theunit supply cost function, can provide building blocksfor the design of auctions for more complicated costfunctions. The ideas and insights gained from our auc-tion design for a unit supply cost function seem to havenatural generalization for complicated cost function set-tings. For example, the price adjustment directions inour auctions are derived from the basic principles of animbalance in supply and demand. For the unit supplycase, we have derived the appropriate price adjustmentdirection using the concept of an “undersupply”. Thisconcept seems to be extendible to more complicated costfunctions.

� Our experience with local industry shows that in manymanufacturing settings the unit cost function assump-tion holds. Often, the manufacturer wants to procuremultiple units of different parts. To maintain uniformityin quality and ease of procurement, manufacturers preferto procure all units of the same part from a single sup-plier. Also, in many settings, a supplier has the capabilityto supply only one (type of) part because of capacity (la-bor and machine hours) constraints. This automaticallycreates a unit supply cost function. Manufacturers alsohave restrictions on the volume of business (either in dol-lar terms or in quantity terms) from a single supplier (seethe discussion on these type of “business constraints” inBichler et al. (2005)). Unit supply restrictions enforcethese constraints in a natural way.

2. The single-seller and the single-buyer models

Most of the work in iterative auction design has been inthe single-seller setting. In a single-seller model, there is aseller who wants to sell a set of indivisible items to a set ofbuyers. The payoff to a buyer on an item is its value minusthe price he pays for the item. Contrast this with our set-ting, a single-buyer setting, where a manufacturer (buyer)wants to procure a set of items from a set of suppliers (sell-ers) who incur costs for providing them. The payoff to asupplier is the price he is paid for supplying the item mi-nus its cost. It should be apparent that both problems are

some sort of “dual” of one another. Moreover, an ascending(descending) auction designed in one setting can be trans-formed to a descending (ascending) auction for the othersetting due to this dual nature. As an example, consider thefamous English auction in a single-seller model setting. Inthis situation, the seller starts the auction at a low price andincreases the price until there is exactly one buyer interestedin the item. In the single-seller model, the demand for theitem is high at the low price. In the English auction, thedemand is lowered until it matches the supply.

Contrast this with the well known reverse auction in pro-curement (single-buyer model) where the price starts at ahigh level (where supply is high) and is lowered continu-ously until there is exactly one supplier willing to providethe item (supply matches demand). Thus, a careful trans-pose of ascending (descending) price auctions in one settingwill produce descending (ascending) price auctions in theother setting.

The literature reports ascending price auctions for thesingle-seller setting which can be transposed to descend-ing price (reverse) auctions for our setting. We note herethat the term “reverse auction” is overloaded with differ-ent meanings. In this paper, whenever we refer to a reverseauction we mean a descending price auction in the single-buyer (procurement) setting where the price is lowered inresponse to supplier bids.

3. Related work

The economics literature mainly focuses on the Vickrey-Clarke-Groves (VCG) mechanism (Vickrey, 1961; Clarke,1971; Groves, 1973) which implements an efficient alloca-tion and has a dominant strategy for participating agents.In its traditional form, the VCG mechanism asks suppliersto report their costs, calculates the efficient allocation, andpays suppliers an amount equal to their marginal product.The marginal product of a supplier is the increase in the to-tal cost of efficient allocation when that supplier is removedfrom the economy. A sealed-bid VCG mechanism of thisform has the property that submitting true costs is a dom-inant strategy for the suppliers. Krishna and Perry (1998)justify the use of VCG mechanisms in terms of revenuemaximization (minimization of price of procurement in oursetting).

However, suppliers can often be reluctant to reveal theirtrue costs directly, irrespective of the incentives. Also, costrevelation of suppliers may not be appropriate from a busi-ness or strategic standpoint. This directs our attention to-wards iterative or indirect mechanisms, such as dynamicauctions, in which supplier costs are not directly revealed.

Iterative auctions have been designed for the single-sellermodel. Leonard (1983) shows that the minimum competi-tive equilibrium price in a single-seller model correspondsto the VCG payments. In a single-buyer model, the max-imum competitive equilibrium price corresponds to the

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Industrial procurement auctions 129

VCG payment. Demange et al. (1986) designed an iter-ative ascending price auction for the single-seller model.This translates to a descending price/reverse auction forthe single-buyer model. The authors showed that such adescending price auction converges to the maximum com-petitive equilibrium price in the single-buyer model andthus each supplier gets his VCG payoff (Leonard, 1983).

Recently, the work of Demange et al. (1986) has been ex-tended to the multiple-item demand case in a single-sellermodel. Ausubel and Milgrom (2002), de Vries et al. (2003)and Parkes et al. (2004) have proposed ascending price auc-tions implementing the VCG outcome for a single-sellermodel when buyers demand multiple items. A proper trans-pose of these auctions will give us descending price auc-tions or reverse auctions for our model. The work of Bert-sekas (1992) on designing auction-based algorithms forthe assignment problem is also worth noting, althoughhe sidesteps the incentive issues which is our primaryconcern.

An excellent survey of the current state of procurementauctions in industry is provided by Elmaghraby (2004).Among other things, he discussed the reluctance of suppli-ers to participate in the reverse auctions of freemarkets.combecause of information revelation, and resulting collusionissues. To alleviate this problem, freemarkets.com has re-sorted to several measures to control the information re-vealed by the suppliers. This also appeals to an ascend-ing price auction design (as we show these auctions havebetter information revelation properties than the reverseauctions).

Two other research articles by Elmaghraby and Ke-skinocak (2003a; 2003b) describe the state of the art incombinatorial auctions for procurement. They point outthat very little research in combinatorial auctions has beendone keeping issues in procurement in mind. Other procure-ment auction design literature includes Chen et al. (2001),who discuss three sealed-bid auctions in procurement set-tings. Their auctions are incentive compatible and incor-porate transportation costs and other variables involved inproduction. Gallien and Wein (2005) discuss a “smart” auc-tion mechanism for procurement when suppliers are capac-ity constrained. Parkes and Kalagnanam (2004a) discussissues in designing procurement auctions. They underlinethe importance of “side constraints” in procurement auc-tions such as the maximum amount that a supplier can beawarded and the number of suppliers to be selected. Parkesand Kalagnanam (2004b) designed a multi-attribute reverseauction in a procurement setting. Their auction allows com-binatorial bids and has many desirable economic properties.

Ascending price auctions for our model have not beendesigned. For the single-item case, the well known Dutchauction can be transposed into our model as follows: startthe price from zero and increase it until some supplier iswilling to provide the item. This ascending price auction isnot strategy-proof. However, if the suppliers bid truthfully,then the auction converges to the minimum competitive

equilibrium price. We propose an ascending price auctionthat is a natural generalization of this auction to our unitsupply model and that is NOT strategy-proof, but convergesto the minimum competitive equilibrium price if suppliersbid truthfully.

3.1. Our contribution

In this research, we design two ascending price auctions forour model. In the first auction, the prices start from zerofor all items. At each iteration, the suppliers submit theirrespective supply sets (set of items which maximize theirpayoff). This constitutes the bids of the suppliers in eachiteration. Based on the bids, the unsupplied set of items anda minimally undersupplied set of items are identified. This iscalled an active set. If no active set exists, the auction stops;otherwise, the prices on the active set of items are raised byunity. The auction terminates at the minimum competitiveequilibrium price (a competitive equilibrium price which isthe most favorable to the manufacturer among all the com-petitive equilibrium prices) if suppliers submit supply setstruthfully. However, suppliers have no incentives to act inthis manner. Because there is no equilibrium strategy forsuppliers, we extend the auction by maintaining the alloca-tion of the first auction and starting the bidding from its fi-nal price. We identify a set of marginally undersupplied itemsand increase their prices. We stop when the marginally un-dersupplied set is empty. We show that this extended ascend-ing price auction converges to the maximum competitiveequilibrium price (a competitive equilibrium price which isthe most favorable to the suppliers among all the competi-tive equilibrium prices). The suppliers get their VCG payofffrom this price, thus demonstrating that behaving truth-fully is an ex post Bayesian Nash equilibrium strategy. Ourextended ascending price auction has the unique propertythat it discovers both the extreme competitive equilibriumprices (and hence the extreme core points) of our model.

The design of our ascending price auction is relatedto a primal-dual algorithm as described by Bikhchan-dani et al. (2001), Bikhchandani and Ostroy (2002a) andde Vries et al. (2003). Whereas we interpret our ascend-ing price auction as a primal-dual algorithm, the termi-nation of the primal-dual algorithm only gives us ourfirst auction (which does not have an equilibrium strat-egy for suppliers in which they behave truthfully). Toconverge to the right point (specifically, to the competi-tive equilibrium price in which the VCG payoff to sup-pliers can be obtained), we keep on adjusting the pricesappropriately. This stage of the auction does not have aprimal-dual algorithm interpretation. Whereas the generalidea in primal-dual-based auction design (as in de Vrieset al. (2003)) is to stop the auction as soon as we con-verge to a primal optimal solution, we show that this maynot be sufficient to achieve good incentive properties. Weshow that a careful price adjustment inside the competi-tive equilibrium space after we terminate the primal-dual

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algorithm takes care of incentive issues. This approach isunique in the literature.

Finally, we justify the design and use of ascending priceauctions over reverse auctions in two ways: (i) using aninformal argument to show how ascending price auctionsreveal less information than their reverse auction coun-terparts; and (ii) using simulation to show how the biddetermination by suppliers in each iteration of the auc-tions is less in ascending auctions than their reverse auctioncounterparts.

The rest of the paper is organized as follows. In Section4, we introduce our model formally. In Section 5, we intro-duce our first ascending price auction which converges tothe minimum competitive equilibrium price if the suppliersbid truthfully. Section 6 extends the ascending price auc-tion in Section 5 by adjusting prices carefully. In Section7, we give various practical advantages of using ascend-ing price auctions over reverse auctions. Section 8 summa-rizes the results and provides some directions for futureresearch.

4. The model

We consider an economy with a single manufacturer, msuppliers denoted by the set B = {1, . . . , m}, and n itemsdenoted by the set A = {1, . . . , n}. The manufacturer wantsto procure the items from the suppliers. For i ∈ B, the costof supplying item j ∈ A is cij. The cost of supplying the nullitem is ci∅ = 0 for all i ∈ B. The manufacturer has an upperlimit on the amount he wishes to pay for each item. Weimplement this idea through a dummy supplier, supplier 0,whose cost for an item j ∈ A is c0j. So, if an item cannot beprocured then it is provided by the dummy supplier. Thiscan also be interpreted as the cost of producing the itemin-house. Denote B0 = B ∪ {0} and B−i = B \ {i}.

An allocation is an assignment of items to suppliers. Weassume unit supply, i.e., each supplier in B can supply atmost one item. So, a feasible allocation assigns each itemto a supplier in B0 and no supplier in B is assigned morethan one item. The cost of a feasible allocation is the totalcost of suppliers in B0 to supply the items in the assign-ment. An efficient allocation is a feasible allocation thatminimizes the cost across all feasible allocations. The prob-lem of finding an efficient allocation can be formulated asa linear program. Let

xij ={

1 when supplier i ∈ B0 is supplying item j ∈ A,0 otherwise.

In the formulation, we minimize the total cost of procure-ment subject to the conditions that: (i) every item needs to beprocured; and (ii) no supplier in B can provide more thanone item with the exception of the dummy supplier. Thisis slightly different than the traditional one-to-one assign-ment problem in the literature but has similar polyhedral

properties:(P)

min∑i∈B0

∑j∈A

cijxij,

subject to ∑i∈B0

xij = 1 ∀ j ∈ A,

∑j∈A∪∅

xij = 1 ∀ i ∈ B,

xij ≥ 0 ∀ i ∈ B0, ∀ j ∈ A ∪ ∅.

It is easy to see that the constraint matrix is totally uni-modular and thus problem (P) has integral extreme points.Now, consider the dual of the above formulation:(DP)

max∑j∈A

pj −∑i∈B

πi,

subject topj ≤ c0j ∀ j ∈ A,

pj − πi ≤ cij ∀ i ∈ B, ∀ j ∈ A,

πi ≥ 0 ∀ i ∈ B.

If x∗ is an optimal solution of problem (P) and (p∗, π∗)is an optimal solution of problem (DP), we can write theComplementary Slackness (CS) conditions as follows:

CS1: x∗0j(c0j − p∗

j ) = 0 ∀ j ∈ A;CS2: x∗

ij(cij − p∗j + π∗

i ) = 0 ∀ i ∈ B, ∀ j ∈ A;CS3: x∗

i∅π∗i = 0 ∀ i ∈ B,

where p denotes the price on items and π denotes the max-imum payoff of suppliers over items. Denote Ap = {j : pj <

c0j}, i.e., the set of items whose prices are below those of thedummy supplier. From dual feasibility, if j /∈ Ap, pj = c0j.We can embed the CS condition and dual feasibility intothe primal problem. Define the supply set of supplier i ∈ Bat price p as follows: Si(p) = {j : j ∈ A ∪ {∅}, pj − cij ≥pk − cik for all k ∈ A ∪ {∅}}, i.e., the set of items that maxi-mize the utility or payoff of a supplier. We will also assumethat if

∑j∈A xij = 0 for a supplier i ∈ B, then i is assigned

a “∅” on which he always has a zero payoff. Now, condi-tion CS1 says that if x0j = 1, then j /∈ Ap or c0j = pj and ifj ∈ Ap then x0j = 0. Condition CS2 says that if xij = 1, thenj ∈ Si(p) and if j /∈ Si(p) then xij = 0. Condition CS3 saysthat if a supplier is assigned the “∅”, then his payoff shouldbe zero.

Now consider the following set of inequalities which com-bine the CS conditions with primal and dual feasibility intoone set of linear inequalities:(CE) ∑

i∈B

xij = 1 ∀ j ∈ Ap,

∑i∈B0

xij = 1 ∀ j /∈ Ap,

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Industrial procurement auctions 131∑

j∈Si(p)

xij = 1 ∀ i ∈ B,

xij ≥ 0 ∀ i ∈ B, ∀ j ∈ Si(p),xij = 0 ∀ i ∈ B, ∀ j /∈ Si(p).

Problem (CE) assigns every supplier an item from his sup-ply set. Also, an item from the set Ap is assigned to a non-dummy supplier and other items are assigned to some sup-plier (may be the dummy supplier). A feasible solution (p,x) to the above set of equations is also called a competitiveequilibrium. Formally, a competitive equilibrium is definedas follows.

Definition 1. (Competitive equilibrium.) A competitive equi-librium is a tuple (p, x), where x is a feasible allocation (fea-sible solution to problem (P)) and p ∈ R

A+ is a price vector

on items (feasible solution to problem (DP)), such that:� Every supplier i ∈ B is assigned an item from Si(p) in the

allocation x.� pj ≤ c0j for all j ∈ A and if item j ∈ A is not assigned in

x, then pj = c0j.

Simply, (p, x) is a competitive equilibrium if x is a fea-sible solution to problem (CE) at p. The price vector pis called a Competitive Equilibrium (CE) price. Anotherrelated concept to CE is the concept of core. For thiswe will follow Bikhchandani and Ostroy (2002b). DefineV (T) = ∑

j∈A c0j − C(T), where C(T) is the objective valueof the optimal solution of problem (P), when we only con-sider T ⊂ B set of suppliers and the dummy supplier inproblem (P). Note that if T = ∅, V (T) = 0.

Definition 2. (Core.) Let π ∈ RB+ be a vector on suppliers

and πm ∈ R+. (π, πm) is in the core of our model, denotedby (π, πm) ∈ core, if:∑

i∈T

πi + πm ≥ V (T) ∀ T ⊆ B,

∑i∈B

πi + πm = V (B).

We can interpret πi to be the payoff of supplier i and πm to bethe payoff of the manufacturer. A payoff vector in the coreindicates that no set of suppliers have any incentive to forma coalition of their own and gain more payoff. Shapley andShubik (1972) (and later Bikhchandani and Ostroy (2002b)for a generic model) showed that (π, πm) ∈ core if and onlyif there is a competitive equilibrium (p, x) which gives thesuppliers a payoff that is equal to π and the manufacturer apayoff that is equal to πm. They further show that the coreforms a lattice. This also implies that the set of CE pricesform a lattice; i.e., if p1 and p2 are two CE price vectors, thenso are min(p1, p2) and max(p1, p2). This means that thereis a unique minimum CE price (pmin) and a unique maxi-mum CE price (pmax). pmax corresponds to the core pointwhich is most favorable to the suppliers and pmin corre-sponds to the core point which is most favorable to the man-ufacturer. Furthermore, Leonard (1983) has shown that the

payoffs to the suppliers at pmax correspond to their VCGpayoff.

5. Ascending price auction

In this section, we will design an ascending price auctionfor our model. For this, we will adopt a primal-dual algo-rithm approach. We can also describe our auction withoutusing primal-dual algorithms but this approach is easier tounderstand. Using the primal-dual algorithm, we will de-fine the price adjustment directions. The algorithm startsthe auction at a low price where there is a low supply. So,the price adjustment direction identifies these items that nosupplier is willing to supply and a set of items which are“undersupplied” in some minimal sense. These price ad-justments guide the primal-dual algorithm to convergence.Throughout the section, we will use the following exampleto illustrate ideas.

Example 1. There are three items A = {1, 2, 3} and twosuppliers B = {1, 2}. The reservation cost of the manufac-turer for all the items is five. The costs of the suppliers are asfollows: c11 = 3, c12 = 2, c13 = 4, c21 = 4, c22 = 3, c23 =2. The efficient allocation is to procure item 2 from supplier1, item 3 from supplier 2 and produce item 1 in-house (orthe dummy supplier supplies item 1).

We seek a feasible solution of problem (CE). Consider thefollowing modification to problem (CE), where we now in-troduce artificial variables. For every A′ ⊆ Ap, we define thefollowing formulation:(RP):

Z(A′, p) = min∑j∈A′

zj,

subject to ∑i∈B

xij + zj = 1 ∀ j ∈ Ap,

∑i∈B0

xij = 1 ∀ j /∈ Ap,

∑j∈Si(p)

xij = 1 ∀ i ∈ B,

zj ≥ 0 ∀ j ∈ Ap,

xij ≥ 0 ∀ i ∈ B0, ∀ j ∈ Si(p),xij = 0 ∀ i ∈ B, ∀ j /∈ Si(p).

Observe that the feasible region of problem (RP) is inde-pendent of the choice of A′ in the objective function. Also,a feasible solution to problem (CE) exists at a price p ifZ(Ap, p) = 0. It is not difficult to see that if Z(S, p) = 0,then Z(S′, p) = 0 for all S′ ⊆ S. However, it is possible thatwe do not have a feasible solution to problem (CE) at pricep. Moreover, a feasible solution to problem (RP) may notexist. Consider the following definition.

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Definition 3. Undersupply holds at price p if problem (RP)has a feasible solution and Z(Ap, p) > 0.

Informally, undersupply requires that we have a feasiblesolution to problem (RP) but no feasible solution to prob-lem (CE). In example 1, at price p = (0, 0, 0) all the itemsare in Ap and no supplier is willing to supply any item. So,Z(Ap, p) = 3. So, undersupply holds at this price. When anitem is not in the supply set of any supplier, we call such anitem unsupplied.

Definition 4. An item j ∈ A is unsupplied if j ∈ Ap and theredoes not exist a supplier i ∈ B such that j ∈ Si(p). If j is notunsupplied, we say it is supplied.

It is not enough that items are merely supplied. Even if theitems are supplied, the aggregate supply may be less thanthe demand. In other words, a few suppliers may have manyitems in their supply set. In such a case, although the itemsare supplied, actual supply is less than the demand. Thisidea is captured in the following definition.

Definition 5. A set of items S ⊆ Ap is undersupplied at pricep if undersupply holds at price p and Z(S, p) > 0. A set ofitems S ⊆ Ap is minimally undersupplied at price p if S isundersupplied and there does not exist a S′ ⊂ S such thatS′ is undersupplied. A minimally undersupplied set of itemsare supplied if every item in that set is supplied.

This means that if S is minimally undersupplied thenZ(S′, p) = 0 for all S′ ⊂ S. Clearly, at price p, if an itemj is unsupplied, then it is a minimally undersupplied set.In example 1, consider the price p = (3, 2, 2). At this pricesupplier 1 can supply S1(p) = {∅, 1, 2} and supplier 2 cansupply S2(p) = {∅, 3}. If we want to assign items 1 and 2to suppliers, we will not be able to assign them simultane-ously to suppliers who are willing to supply them (since onlysupplier 1 has both of them in his supply set). So, {1, 2} is aset of undersupplied items. It is also minimally undersup-plied because only item 1 or only item 2 can be assigned tosupplier 1 and thus there is no smaller undersupplied set ofitems.

The idea behind any (trading) mechanism is to create abalance in “supply” (suppliers interest in items) and “de-mand” (items). At any price, the unsupplied items create alow supply and their price needs to be increased to increasesupply. Besides these items, the minimally undersuppliedset of items provides a group of minimal items which createan imbalance in the supply and demand and whose pricesneed to be increased to offset such an imbalance. This leadsto the idea of an active set of items.

Definition 6. (Active set.) Consider a price p in which under-supply holds. The active set of items at price p (denoted asT(p)) is a set of items which are unsupplied at p (denoted asU(p)) and a minimally undersupplied set of items (denotedas M(p)) which are supplied.

Since undersupply holds, either there is a set of itemswhich are unsupplied or there is a minimally undersuppliedset of items which are supplied. So, if undersupply holds,then the active set is non-empty. If there is no minimally un-dersupplied set of items which are supplied, then the activeset consists of items which are unsupplied. Next, we provetwo important lemmas. We will denote the set of suppliershaving items from S ⊆ A in their supply set at price p asO(S, p).

Lemma 1. If S is minimally undersupplied at price p, thenZ(S, p) = 1.

Proof. By Definition 5, for every S′ ⊂ S, Z(S′, p) = 0.Consider any j ∈ S and S′ = S \ {j}. Since Z(S′, p) = 0.Since the feasible solution of problem (RP) is the same inZ(S′, p) and Z(S, p), Z(S, p) ≤ 1. Since undersupply holds,Z(S, p) > 0. This gives, Z(S, p) = 1. �

Lemma 2. If T(p) = U(p) ∪ M(p) is an active set of items atprice p, where U(p) is the set of unsupplied items and M(p)is a minimally undersupplied set of items which are supplied,then the number of suppliers having items from T(p) in theirsupply set is equal to max(0, |M(p)| − 1).

Proof. Realize that no supplier has any item from U(p) intheir supply set (from Definition 4). This means, if M(p) isempty, we are done.

If M(p) is not empty, from Lemma 1, the number ofitems from M(p) that can be allocated in the optimal so-lution of Z(M(p), p) is |M(p)| − 1. Denoting the set of sup-pliers having items from S in their supply set at price pas O(S, p), we can write, |O(M(p), p)| ≥ |M(p)| − 1. Con-sider any subset S of M(p). By Definition 5, Z(S, p) = 0.This means |O(S, p)| ≥ |S|. Since Z(M(p), p) = 1 (Lemma1), by Hall’s theorem (Hall, 1935), |O(M(p), p)| < |M(p)|.(Hall’s marriage theorem says the following: Let A1, . . . , Atbe finite sets and let the union of any k (1 ≤ k ≤ t) of themcontain f (k) distinct elements. It is possible to select t dis-tinct elements, one from each of the sets, iff f (k) ≥ k forall 1 ≤ k ≤ t .) This means that |O(M(p), p)| = |M(p)| − 1.Since items in U(p) are not in the supply set of any supplier,we have |O(T(p), p)| = |M(p)| − 1. �

From the above two lemmas, we prove the followingproposition.

Proposition 1. Let T(p) = U(p) ∪ M(p) be an active set ofitems. If M(p) �= ∅, then the set of suppliers having itemsfrom T(p) in their supply set is the set of suppliers assignedto items in M(p) in the optimal solution of problem (RP),considering items from M(p) in the objective function.

Proof. Since M(p) is non-empty and minimally undersup-plied, Z(M(p), p) = 1 (from Lemma 1). This means, thereare |M(p)| − 1 suppliers who are assigned items from M(p)in the optimal solution of problem (RP), considering itemsfrom M(p) in the objective function. From Lemma 2, these

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are the only suppliers who have items from T(p) in theirsupply set. �

From Proposition 1, we will use O(T(p), p) to denote theset of suppliers who have items from T(p) in their supply setand also to denote the set of suppliers assigned to items inT(p) in the optimal solution of problem (RP), consideringitems from T(p). This leads us to define the price adjustmentin our auction.

Definition 7. If undersupply holds at an iteration, with pricep ∈ R

|A|:

Step 0. Collect supply sets of suppliers at price p.Step 1. Determine an active set of items T(p).Step 2. Set pj = pj + 1 for all j ∈ T(p).

Let T(p) be an active set of items at price p. Now, weshow that the price adjustment in the auction in Definition7 can be derived from the dual of the restricted primal prob-lem (RP), where T(p) is the set of items considered in theobjective function. For this consider the dual problem ofproblem (RP), where T(p) is the set of items considered inthe objective function.(DRP):

Z(T(p), p) = max∑i∈B

αi +∑j∈A

θj,

subject toθj ≤ 1 ∀ j ∈ T(p),

θj + αi ≤ 0 ∀ i ∈ B, ∀ j ∈ Si(p),θj ≤ 0 ∀ j /∈ Ap.

Lemma 3. Let T(p) be an active set of items at price p. Thereexists an optimal solution of problem (DRP) in which θj = 1for all j ∈ T(p), θj = 0 for all j /∈ T(p).

Proof. T(p) includes the set of all unsupplied items andthe set of minimally undersupplied items that are supplied.From Lemma 2 and Proposition 1 the optimal value of theobjective function of problem (RP) is equal to Z(T(p), p) =|U(p)| + 1, where U(p) is the set of unsupplied items at pricep.

Consider the solution θj = 1 for all j ∈ T(p), θj = 0 forall j /∈ T(p), αi = −1 for all i ∈ O(M(p), p), αi = 0 for alli /∈ O(M(p), p). Note that T(p) ⊆ Ap. So, θj = 1 for allj ∈ T(p) and zero otherwise, is feasible. If i ∈ O(M(p), p),then αi = −1 and the proposed solution is feasible. Else,j ∈ Si(p) ⇒ j /∈ T(p). This means that θj = 0. Hence, this isalso feasible. The objective function value from this feasiblesolution is |T(p)| − |O(M(p)| = |T(p)| − |M(p)| + 1 (fromLemma 2). From the definition of T(p) = U(p) ∪ M(p)(with M(p) ∩ U(p) = ∅), we get the objective function valuefor this feasible solution as |U(p)| + 1. This is optimal fromstrong duality theorem. �

Lemma 3 shows that the auction with a price adjustmentin Definition 7 is a primal-dual algorithm. The θ vectordenotes the direction and amount of the price adjustment.

As can be seen, this is exactly the price adjustment done inDefinition 7. Typically, in a primal-dual algorithm, a fea-sible dual solution is considered. A feasible dual solutionin our case is a price vector. From the feasible dual so-lution, a “restricted primal” is constructed by consideringCS conditions. This is done in problem (RP) by introducingartificial variables. If the artificial variables are zero in theoptimal solution of the restricted primal problem, then wehave a feasible solution of the main primal problem. Thisis optimal from the fact that for a feasible dual solution,feasible primal solutions that satisfy the CS conditions areoptimal. If we do not have an optimal solution, we look forthe dual of the restricted primal problem to change the dualvariables. This is done as shown in Lemma 3.

Now, we are ready to state one of our main results fordesigning the ascending price auction using the price ad-justment in Definition 7.

Theorem 1. If undersupply holds at a price p, after a price ad-justment using Definition 7, either we find a feasible solutionto problem (CE) or undersupply holds.

Proof. Let T(p) be an active set of items at price p whoseprices are raised using Definition 7. Consider the optimalsolution to problem (RP), where the objective function con-siders items from T(p). By Proposition 1, only suppliersassigned to items in T(p) have items from T(p) in their sup-ply set. Thus, by increasing the price of items in T(p) only(using Definition 7), every supplier in B still has the itemto which it was assigned in the optimal solution of problem(RP) in his supply set at the new price. This means that theoptimal solution of problem (RP) is a feasible solution toproblem (RP) at price p′, where p′ is the new price after ad-justing the price using Definition 7. This means, either wefind a feasible solution to problem (CE) at price p′ (this willhappen if Z(Ap′, p′) = 0) or undersupply holds. �

Theorem 1 allows us to define the ascending price auctionformally.

Definition 8. (Ascending price auction.) The ascending priceauction is defined as follows:

Step 0. Start the auction from an integral price p where un-dersupply holds. One such price is the zero vector.

Step 1. Perform price adjustment as defined in Definition7 by identifying an active set of items.

Step 2. If a feasible solution to problem (CE) exists, STOP,with the final allocation being any such feasibleallocation and final payment of the suppliers be-ing the final price in the auction. Else, go back toStep 1.

Theorem 2. The auction described in Definition 8 terminateswith a price p and an allocation x such that (p, x) is a CEif suppliers submit true supply sets in each iteration of theauction. Moreover, x is an efficient allocation.

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Proof. By Definition 8, the starting price is integral andwill remain integral by the price adjustment described inDefinition 7. Also, assuming (strictly) positive costs for allsuppliers (including the dummy supplier), starting the priceat zero indicates Ap = A and Z(A, p) = −|A|. This meansundersupply holds. By Theorem 1, either we find a feasiblesolution to problem (CE) or undersupply holds after eachprice adjustment. Since the price in the auction is upper-bounded by the cost of the dummy supplier, the auctiondescribed in Definition 8 will terminate in a feasible solution(p, x) of problem (CE) in finite number of iterations. Thismeans that (p, x) is a CE. Since the allocation in a CE is anoptimal solution of problem (P), x is an efficient allocationtoo. �

Theorem 2 only says that the auction converges to a CEprice. Next, we show that the final price is the minimum CEprice (pmin).

Theorem 3. Let p be the price at the end of the auction inDefinition 8. p = pmin if the suppliers submit true supply setsin each iteration of the auction.

Proof. From Theorem 2 and due to lattice nature of theCE price space, p ≥ pmin. �

Consider the first iteration t , at which the price of someitems reaches pmin. Let the price in iteration t be p. Let S ={j : pj = pmin

j }. This means that pj < pminj for all j ∈ A \ S.

We will show that there is no item in S that is part of anactive set of items at price p. Consider the following lemma.

Lemma 4. If any j ∈ S is in some active set of items at pricep, then j is NOT unsupplied.

Proof. If an item j ∈ S is in some active set of items atprice p, then j ∈ Ap. Since pj = pmin

j and pj < c0j (j ∈ Ap),in any feasible solution of problem (CE) at pmin, j shouldbe assigned to some supplier i ∈ B. Since p ≤ pmin, i shouldalso have j in his supply set at price p. This means, j is NOTunsupplied. �

Assume for contradiction, that there exists a set ofitems T ⊆ S which are part of some active set of itemsat price p. From Lemma 4, each item in T is NOT un-supplied. This means T ⊆ S′, where S′ is minimally un-dersupplied and each item in S′ is supplied. Let M be theset of suppliers assigned to items in T in a feasible allo-cation of problem (CE) at price pmin. Since pmin

j > pj forall j ∈ S′ \ T , suppliers in M will only have items from Tin their supply set and no items from S′ \ T at price p.From Lemma 2, |O(S′, p)| = |S′| − 1, where O(S′, p) de-notes the set of suppliers having items from S′ in their sup-ply set. Simplifying, |O(S′ \ T, p)| + |O(T, p)| = |S′| − 1.The suppliers in M have items from T only in their sup-ply set at price p. This gives |O(T, p)| ≥ |M| = |T |. Simpli-fying further, |O(S′ \ T, p)| = |S′| − 1 − |O(T, p)| ≤ |S′| −1 − |T | = |S′ \ T | − 1 < |S′ \ T |.

Since, S′ is minimally undersupplied and T ⊆ S′, S′ \T is not undersupplied, i.e., Z(S′ \ T, p) = 0. This means,|O(S′ \ T, p)| ≥ |S′ \ T |. This gives us a contradiction.

Unfortunately, this auction does not have an equilibriumstrategy for suppliers to submit a true supply set in each it-eration. Suppliers can manipulate prices in this auction andgain a larger payoff. So, the ascending price auction in Def-inition 8 is not strategy-proof. The intuition behind thisauction being not strategy-proof is that it does not imple-ment the VCG outcome. In our model, every strategy-proofand efficient mechanism should implement the VCG out-come (Green and Laffont, 1977). The maximum CE pricecorresponds to the VCG payoff in our model (Leonard,1983). So, we extend our auction further to discover themaximum CE price. This is the point where our auction de-parts from other iterative auctions in the literature. To ourknowledge, all iterative auctions (designed on the basis ofprimal-dual concepts) in the literature stop at the first CEprice to which the auction converges. As we showed here,this need not be sufficient for incentive compatibility andwe need to search through the CE space for the appropriatepoint.

6. Extended ascending price auction

In this section, we design an ascending price auctionwith good incentive properties that discovers both extremepoints of the core. As discussed earlier, good incentive prop-erties can be achieved by converging to the maximum CEprice. The key to converging to the maximum CE price isto maintain the feasibility of problem (CE), discovered inthe ascending price auction of Definition 8, while adjust-ing the prices. We seek a CE price where we can remove asupplier from the economy and still maintain CE. Such anidea stems from the fact that the maximum CE price cor-responds to the VCG payoff which is based on the idea ofmarginal products. Also, observe the following characteri-zation of the maximum CE price, pmax.

Lemma 5. For any S ⊆ Apmax , |O(S, pmax)| > |S|, whereApmax = {j : pmax

j < c0j} and O(S, pmax) denotes the set of sup-pliers who have items from S in their supply sets at pmax.

Proof. Since S ⊆ Apmax , pmaxj < c0j for all j ∈ S. This means

that all the items in S are assigned to suppliers in B in afeasible solution of problem (CE). So, |O(S, pmax)| ≥ |S|.If |O(S, pmax)| = |S|, then O(S, pmax) consists of supplierswho are assigned in a feasible solution of problem (CE).In that case, we can increase the price of items in S bya sufficiently small amount and the current feasible so-lution of problem (CE) will still be feasible. This meansthat pmax is not the maximum CE price and we get acontradiction. �

Based on Lemma 5, we define the concept of a marginallyundersupplied set of items.

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Definition 9. (Marginally undersupplied.) A set of items S ⊆Ap is marginally undersupplied at price p if |O(S, p)| ≤ |S|.In example 1, at price p = (3, 2, 2), only supplier 1 is will-ing to supply items 1 and 2. So, for S = {1, 2}, we have|O(S, p)| = 1 < 2 = |S|. This means that S is marginallyundersupplied. Now, consider a CE price p = (5, 4, 3). ForS = {2, 3}, O(S, p) = {1, 2}. So, |O(S, p)| = 2 = |S|. Thismeans that S is marginally undersupplied. If we can finda CE price at which there is no marginally undersupplieditems, we will be able to remove one supplier and still be ableto assign all the items at this price. This price is necessaryto implement a VCG outcome which is based on the ideaof marginal products. Consider the following lemma.

Lemma 6. If p is a price at which there is a feasible solution toproblem (CE) and S ⊆ Ap is marginally undersupplied, thenO(S, p) is the set of all suppliers assigned to S in a feasiblesolution of problem (CE).

Proof. Since S ⊆ Ap and there is a feasible solution toproblem (CE), each item in S is assigned to some sup-plier in B. This means that |O(S, p)| ≥ |S|. Since S ismarginally undersupplied, |O(S, p)| ≤ |S|. This means that|O(S, p)| = |S|. So, O(S, p) is the set of all suppliers as-signed to S in a feasible solution of problem (CE). �

Now, consider the following price adjustment.

Definition 10. If a feasible solution to problem (CE) existsat an iteration, with price p ∈ R

|A|

Step 0. Collect supply sets of suppliers at price p.Step 1. Determine a marginally undersupplied set of items

S ⊆ Ap.Step 2. Set pj = pj + 1 for all j ∈ S.

Now, we state our main result in extending our ascendingprice auction.

Theorem 4. If (p, x) is a feasible solution to problem (CE),then (p′, x) is a feasible solution to problem (CE), where p′ isthe price after adjusting p using Definition 10.

Proof. Let S be the marginally undersupplied set of itemson which prices are raised in the price adjustment. FromLemma 6, O(S, p) is the set of all suppliers assigned to itemsin S in the feasible solution of problem (CE). Due to unitprice increase, suppliers not in O(S, p) still have the itemsassigned to them in their supply set. This means (p′, x) isstill a feasible solution to problem (CE). �

Theorem 4 enables us to describe the extended ascendingprice auction.

Definition 11. (Extended ascending price auction.) The ex-tended ascending price auction is defined as follows:

Step 0. Run the ascending price auction as defined in Defi-nition 8. Start from the final price, p and allocation,x of the ascending price auction.

Step 1. Collect supply sets of suppliers at price p and de-termine a set of marginally undersupplied items.

Step 2. If no set of items are marginally undersupplied,STOP, with the final allocation being x and the fi-nal payment of suppliers being the final price in theauction. Else, adjust prices using Definition 10 (in-creasing price of a marginally undersupplied set ofitems) and go back to Step 1.

Now, we are ready to state our main result.

Theorem 5. The extended ascending price auction, describedin Definition 11, terminates at the maximum CE price ifsuppliers submit their true supply sets in each iteration.

Proof. Consider the following lemma first.

Lemma 7. pmax (the maximum CE price) is the only CE priceat which no set of items are marginally undersupplied.

Proof. From Lemma 5, at pmax there does not exist a set ofitems that are marginally undersupplied. Assume for con-tradiction that there exists a CE price p at which there existsno set of items that are marginally undersupplied. By the lat-tice nature of CE price space (Shapley and Shubik, 1972),p ≤ pmax. Let S = {j : pj < pmax

j }. Clearly, S is non-empty(since p �= pmax). Also, S is not marginally undersupplied.This means that |O(S, p)| > |S|. From p to pmax, prices ofitems only in S are higher. This means that suppliers inO(S, p) have items from only S in their supply set at pmax.This means that at price pmax, the number of suppliers hav-ing items only from S in their supply set is (strictly) greaterthan |S|. By Hall’s theorem, such a price cannot be a CEprice. This gives us a contradiction. �

Starting from the final price of the ascending price auc-tion in Definition 8, the extended ascending price auctionin Definition 11 maintains the feasibility of problem (CE)in every iteration from Theorem 4. The price increase can-not go on indefinitely since the prices are bounded aboveby the cost of the dummy supplier. So, the auction shouldterminate at a CE price at which there is no set of items thatare marginally undersupplied. From Lemma 7, such a CEprice is unique and is pmax. �

6.1. Incentive compatibility

Our extended ascending price auction implements a VCGoutcome if suppliers submit true supply sets in each itera-tion. A sealed-bid VCG outcome has a dominant strategyfor suppliers in which they submit true cost functions. Thus,the sealed-bid VCG mechanism is strategy-proof. However,in an iterative auction, dominant strategy implementationis not generally possible (Gul and Stacchetti, 2000; Parkesand Ungar, 2000; Ausubel and Milgrom, 2002; de Vrieset al., 2003). However, Bayesian Nash equilibrium (or, expost Nash equilibrium) implementation is possible by im-posing appropriate bidding rules. For a detailed discussionon the need for bidding rules, readers are referred to the

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articles by Parkes and Ungar (2000), Ausubel and Milgrom(2002) and de Vires et al. (2003), who provide an excellentexplanation at this topic. The main idea behind these ac-tivity rules is that every supplier should always play a strat-egy that is consistent with some unit supply cost function(it need not be his true cost function). Thus, any devia-tion of the supplier is restricted to the first cost functiondomains only and the auction still converges to a CE ofthese false cost functions. Parkes (2000) and Ausubel andMilgrom (2002) propose methods that use this consistencyin the bidding. We succinctly describe one such method.Combinatorial bids in which suppliers can submit bundlesof items are not allowed. For every supplier, his currentpayoff, as observed by the manufacturer, is monitored. Thebids by the supplier should be consistent with this observedpayoff. Initially (when the supplier only submits the nullitem as the supply set), this payoff is zero. As soon as thesupplier submits a supply set in which the null item is notpresent, this payoff is increased. The payoff of each item tothe supplier can be observed similarly. Thus, the maximumpayoff (payoff from the supply set) should be consistentwith the items in the supply set. With these kinds of re-strictions, every bidding strategy can be mapped to a unitsupply cost function (it may be a false one). Observe thatsuch restrictions on bidding should be enforced through-out the extended ascending price auction, which includesStep 0, where the ascending price auction is run.

Theorem 6. Submitting true supply sets in each iteration of theextended ascending price auction is an ex post Bayesian Nashequilibrium for the suppliers under appropriate consistencyrequirements.

Proof. Let supplier i ∈ B follow some strategy other thanthe submission of true supply sets. The bidding strategy ofevery supplier is consistent with a unit supply cost func-tion. Let this cost function vector be c. From Theorem 5,the auction will converge to the maximum CE correspond-ing to c, which also corresponds to the VCG payoffs thatcorrespond to c. From the revelation principle, this payoffis less than the VCG payoff from the true cost functions. So,submitting true supply sets is the best strategy for suppliers.This is not a dominant strategy because a supplier is play-ing this strategy after conditioning on the fact that othersuppliers are using some strategy which can be mapped toa unit cost function. This is an (ex post) Bayesian Nashequilibrium because irrespective of the beliefs about othersuppliers strategies, the best strategy for a supplier is tobehave truthfully. �

6.2. An example

Now, we will illustrate our auctions in Definition 8 and inDefinition 11 using example 1. The steps of the ascendingprice auction in Definition 8 are given in Table 1.

It is easy to verify that (5, 4, 3) is the minimum CE priceof the example. Now, we describe in Table 2 the steps of

Table 1. The steps of the ascending price auction

p Ap S1(p); S2(p) U(p) M(p) T(p)

(0, 0, 0) {1, 2, 3} {∅}; {∅} {1, 2, 3} ∅ {1, 2, 3}(1, 1, 1) {1, 2, 3} {∅}; {∅} {1, 2, 3} ∅ {1, 2, 3}(2, 2, 2) {1, 2, 3} {∅, 2}; {∅, 3} {1} ∅ {1}(3, 2, 2) {1, 2, 3} {∅, 1, 2}; {∅, 3} ∅ {1, 2} {1, 2}(4, 3, 2) {1, 2, 3} {1, 2}; {∅, 1, 2, 3} ∅ {1, 2, 3} {1, 2, 3}(5, 4, 3) {2, 3} {1, 2}; {1, 2, 3} ∅ ∅ ∅

the extended ascending price auction in Definition 11 forthis example. We will denote G(p) as a set of marginallyundersupplied items as defined in Definition 9.

It is easy to verify that (5, 5, 5) is the maximum CE priceof the example and our auction converges to that point.

7. Practical benefits of ascending price auctions

The current literature mainly focuses on designing ascend-ing price auction for single-seller models. These auctions,appropriately transposed to our model (a single-buyermodel), result in descending price/reverse auctions. How-ever, ascending price auctions offer several practical ad-vantages over descending price auctions in a single-buyermodel. We provide following observations to validate thisand discuss two of the observations in detail later:

� In current industrial practice, reverse auctions(e.g., Internet-based auctions such as Freemarkets(http://www.freemarkets.com)) are used in variousprocurement activities. However, suppliers often viewreverse auctions with distaste due to the perception thatit triggers competitive underbidding behavior amongthe suppliers in a manner that reduces the profitabilitythrough price erosion. In contrast, in an ascending priceauction, the suppliers can decide their own profit marginand enter the auction when they are guaranteed thisprofit. Since, profits in ascending price auctions increasefrom iteration to iteration, this provides a promisingframework to attract the suppliers.

� As we will discuss in Section 7.1, our ascending priceauction has desirable information revelation properties.Most of the losing suppliers never reveal their true costson any item whereas descending price auctions do havethe property that all the losing suppliers reveal their truecosts on most of the items. Depending on the size and

Table 2. The steps of the extended ascending price auction

p Ap S1(p); S2(p) G(p)

(5, 4, 3) {2, 3} {1, 2}; {1, 2, 3} {2, 3}(5, 5, 4) {3} {2}; {2, 3} {3}(5, 5, 5) ∅ {2}; {3} ∅

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competition of the economy, the number of (supplier,item) tuples whose costs are revealed may vary in theauctions. However, in large economies with high com-petition, an ascending price auction will have better in-formation revelation properties. A supplier may signal(through bids placed) to other suppliers during the auc-tion initiating collusion. By revealing less (bid) infor-mation, such collusive behavior can be minimized. In asense, an ascending price auction is an iterative auction(decentralized algorithm) similar to a reverse auction butpossessing some of the desirable properties of sealed-bidauction (RFQ-based processes) such as collusion avoid-ance.

� We will also show in Section 7.2 that ascending priceauctions have significantly smaller overheads for bidcomputation than reverse auctions. This is significantpractically because smaller overhead costs will encour-age supplier participation in the auctions.

7.1. Information revelation

If suppliers follow a straightforward strategy (i.e., submitsupply sets truthfully) in the extended ascending price auc-tion of Definition 11, they may reveal the costs of someitems in the auction. Each supplier will reveal the cost ofall the items it submits in his supply set throughout theauction. To understand why, consider the first iteration inwhich a supplier submits an item A in his supply set. Atthat iteration, he should also submit “∅” in his supply set(due to the unit price adjustment in the auction). Thus, themanufacturer knows the maximum payoff of the supplieron items in each iteration. Since the manufacturer can seethe prices, he can know the costs of items in the supplyset of each supplier in each iteration. However, if a sup-plier always submits “∅” as his supply set in the auction,he never reveals the cost of any item. Let M be the set ofsuppliers who are never assigned any item in an efficient al-location. Since prices are non-decreasing and we convergeto a CE, the suppliers in M will never have any item from Ain their supply set in our auction. This means suppliers inM will never reveal any cost information. As the economysize grows and with more competition (the number of sup-pliers being significantly higher than the number of items),the size of set M will increase and thus many suppliers willnot reveal any cost information. Moreover, suppliers in Mrarely participate in our auction.

The information revelation in the reverse auction (suchas Demange et al. (1986)) is as follows. If suppliers bidtruthfully (i.e., submit true supply sets in each iteration),the only iteration in which a manufacturer knows the costof a supplier is when he submits “∅” in his supply set. So,all the suppliers who submit “∅” as their supply set in someiteration of the auction, reveal the costs for all the itemswhich they have submitted as part of their supply set inany previous iteration of the auction. Let N be the set ofsuppliers who are allocated some item in A in all the effi-

cient allocations. Since we converge to a CE and the priceis non-increasing, suppliers in N will never submit “∅” inany iteration of the reverse auction. So, suppliers in N neverreveal any cost information.

Depending on the size and competition of the economy,reverse auctions and ascending price auctions have differ-ent information revelation properties. As the competitiongrows and for larger economies, ascending price auctionshave better information revelation properties.

7.2. Costs of bid determination

In this section, we measure and compare the cost of submit-ting bids per supplier in our auction and in the traditionalreverse auction. For this, we define two types of queries asupplier needs to ask so as to construct his bid (supply set):(i) a price query; (ii) and a supply query. In a price query, thesupplier finds out if the price of every item is greater thanthe corresponding cost. In the supply query, the supplierfinds out the set of items that maximizes the payoff of thesupplier (i.e., constructs the supply set). Clearly, the resultsof a price query can also be achieved using a supply query(by noting whether or not the null item is included in thesupply set). However, a price query is much simpler to exe-cute than a supply query and the information from a pricequery may not be sufficient to construct the supply set ofa supplier. The number of price queries and supply queriesperformed in an auction can measure the “cost” (measuredsimply by the number of queries) of the bid determina-tion of suppliers. For example, in our ascending auctionsuppliers perform price queries initially and supply queriestowards the end of the auction. On the other hand, in thereverse auction of Demange et al. (1986), suppliers initiallyperform supply queries. More supply queries per supplier inan auction means a higher cost of bid determination. Thesecosts of bid determination can be thought of as transaction(overhead) costs of the suppliers.

We remark that there is no price query in the reverse auc-tion. This is because the reverse auction starts with highpayoffs to the supplier. As the price decreases, the payoffdecreases. The suppliers perform supply queries in each it-eration and stop querying once their payoff reaches zero.Thus, there is no need for price queries in the reverse auc-tions.

We also comment that although the supply query is notvery computationally expensive for the unit supply case, itbecomes computationally expensive for the generic combi-natorial case. So, our results for the unit supply case maygive insights for the computationally challenging combina-torial cases. Also, irrespective of the computational burdenof supply queries for the unit supply case, the level of supplyqueries and price queries reflect the transaction costs in theauctions. This may be a valuable input in making manage-rial decisions about choosing between auction types.

The simulation setup is as follows. We draw integer costsof participating suppliers uniformly from [0, 100]. We can

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model our setting as a bipartite graph with suppliers as ver-tices on one side and items as vertices on the other side.An edge between a supplier and an item represents that thesupplier incurs a finite positive cost for supplying that item.These edges are created randomly. We measure the densityof the graph as the average number of items to which eachsupplier is connected. If every supplier is connected to everyitem, the density is one (the highest achievable density) andevery supplier can supply all the items. For each supplier, werandomly assign positive values to a fraction k (0 < k ≤ 1)of the items and zero values to the remaining (1 − k) frac-tion of the items. k measures the density of the economy(the density increases with k). Note that k is normalizedover items. The bid increment in the reverse auction is oneand the bid decrement in the extended ascending price auc-tion is also one. The starting price of the reverse auctionis 100 whereas the starting price of the extended ascendingprice auction is zero (for all items). We run 1000 iterationsfor every setup.

With this experimental setup, we measure the effect ofdensity, number of items, number of suppliers, and size ofeconomy on the average number of price queries and sup-ply queries by suppliers in both the auctions. As discussedearlier, there is no price query in the reverse auction. So, wemeasure the number of price queries in the Extended As-cending Price (EAP) auction and supply queries in the EAPauction and reverse auction. The number of queries mea-sured in the simulations are the average number of queriesmade per supplier.

7.2.1. DensityWe keep the number of items at 15 and number of sup-pliers at 20 and increase the density of the economy. Notethat a higher density means that each supplier can supplymore items. Thus, competition in the economy increaseswith density. This means more bidding in the reverse auc-tion which leads to more supply queries. This explains theincreasing number of supply queries in the reverse auctionwith an increase in density shown in Fig. 1.

To understand the EAP auction curves, we need to un-derstand the price trajectory in the EAP auction. The EAPauction price trajectory has two components:

Fig. 1. The effect of density on queries.

1. Out of equilibrium. In this region, the EAP auction pricelies below the equilibrium price region. The queries bysuppliers in this region are mostly price queries withvery few supply queries being performed. As the priceincreases in the EAP auction, it enters the equilibriumprice region.

2. Inside equilibrium. In this region, the EAP auction pricetravels through the equilibrium price region. The queriesin this region are mostly supply queries with very fewprice queries being performed. The price in the EAPauction travels inside the equilibrium region and finallyconverges to a unique maximum competitive equilib-rium price.

As the competition increases (with an increase in the den-sity), the overall equilibrium region is moved downwards(i.e., the values of the equilibrium price decrease) which re-duces the out-of-equilibrium region. This reduces the num-ber of price queries in Fig. 1. Also, the size of the equilibriumprice region decreases with higher competition levels. Thisleads to fewer supply queries in Fig. 1. When the density isreally low, even though the equilibrium region is large, thebidding by the suppliers is very small (since everyone canonly supply a small fraction of the items and is happy withtheir initial provisional allocation). This leads to fewer sup-ply queries when the density is very small (≤0.2) in Fig. 1.However, this effect disappears for density values > 0.2 (seeFig. 1).

7.2.2. Number of suppliersWe maintain 20 items in the economy and increase the num-ber of suppliers. We say that the demand and supply arebalanced when the number of suppliers in the economyequals the number of items. There are two distinct regionsin the graph in Fig. 2. Region 1 which is the region to theleft of the line where demand and supply balance (i.e., thenumber of suppliers ≤20) and region 2 which is the re-gion to the right of the line where demand and supply bal-ance (i.e., number of suppliers ≥20). In region 1, since thenumber of suppliers is less than the number of items, someitems are assigned to the dummy supplier at the end of theauction. By the definition of a CE price, the price of suchitems (unassigned items) is very high. This means that thereis no bidding on those items in the reverse auction. This

Fig. 2. The effect of the number of suppliers.

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explains the higher number of supply queries in the reverseauction in region 2 than in region 1. Inside region 1, as thenumber of suppliers increase, the number of items assignedto the dummy supplier decreases and thus the number ofsupply queries in the reverse auction increases. In region 2,as the number of suppliers increases, the equilibrium pricesincrease gradually and thus the number of demand queriesdecreases as the number of suppliers increases.

In region 1, the equilibrium region is large and this leadsto a higher number of supply queries in the EAP auction,whereas in region 2, the equilibrium region is smaller, lead-ing to fewer supply queries in the EAP auction. The numberof price queries in the EAP auction shows a slight increasewith the increase in the number of suppliers since moresuppliers inevitably results in more price queries.

7.2.3. Number of itemsWe keep the number of suppliers at 30 and increase thenumber of items. This creates a graph (Fig. 3) which is amirror image of Fig. 2. There are two regions (as in Fig. 2)in this graph: region 1 in which the number of items is lessthan the number of suppliers (=30) and region 2 in whichthe number of items is more than the number of suppliers. Interms of competition, region 1 of Fig. 3 is similar to region2 of Fig. 2 and region 2 of Fig. 3 is similar to region 1 ofFig. 2. Using the explanation given for Fig. 2, the plots inFig. 3 can also be understood.

7.2.4. Size of economyWe keep the ratio of the number of suppliers to the numberof items at 4/3 (i.e., 4× number of items = 3× number ofsuppliers) and gradually increase the number of items. Thiseffectively increases the size of the economy. Note that weare effectively in region 2 of Fig. 2 (region 1 of Fig. 3), wherethe number of items is less than the number of suppliers (webelieve that this is a more realistic scenario). In this setting,as the size of the economy grows, the equilibrium region ispushed downwards, leading to an increase in the number ofsupply queries in the reverse auction and fewer price queriesin the EAP auction (Fig. 4). Also, the equilibrium priceregion becomes smaller, leading to fewer supply queries inthe EAP auction (Fig. 4).

Fig. 3. The effect of the number of items.

Fig. 4. The effect of the size of the economy.

7.2.5. Summary of resultsThe results of our simulation can be summarized as follows.

� When the number of items is less than the number of sup-pliers, the EAP auction has fewer supply queries than thereverse auction. Since price queries are very inexpensive,the cost of bid determination is smaller in the EAP auc-tion in this setting.

� When the number of items is more than the number ofsuppliers, the reverse auction has fewer supply queriesthan the EAP auction. This means that the cost of biddetermination is smaller in the reverse auction in thissetting.

� If the number of suppliers is greater than the number ofitems and if the size of the economy increases, the numberof supply queries in the reverse auction increases and thenumber of price queries and supply queries in the EAPauction decreases. Thus, for large economies, the cost ofbid determination is higher for the reverse auction thanfor the EAP auction.

� In economies with a high density, the reverse auctionhas a higher number of supply queries. Thus, the EAPauction has a smaller cost of bid determination in thesesettings than the reverse auction.

8. Summary and future research

We have designed two ascending price auctions for procur-ing multiple items simultaneously when suppliers canprovide at most one item. Truthful behavior in one of theauctions is a Bayesian Nash equilibrium strategy for thesuppliers. This auction discovers both the extreme corepoints of the economy. We interpreted our auction as aprimal-dual algorithm until it achieves efficiency and thenas an appropriate price adjustment procedure that main-tains the optimality of a linear program. We discussed howascending auctions can be preferred over traditional reverseauctions in many settings due to better information revela-tion and cost of bid determination.

An immediate future research direction is to extend ourauction in Definition 11 to generic multi-item models inwhich suppliers can provide more than one item. Also worthpursuing is the case of homogeneous items where suppliers

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can provide multiple items but have non-additive unit costs.As discussed earlier, ascending price auctions can have bet-ter information revelation properties than reverse auctionsin certain settings. Detailed research on information revela-tion of ascending and descending price auctions is necessaryto evaluate the two types of auctions based on informationrevelation.

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Biographies

Dr. Debasis Mishra is a post-doctoral fellow at the Center for Opera-tions Research and Econometrics (CORE) at Universite Catholique deLouvain in Belgium. He received his Ph.D. in Industrial Engineeringfrom University of Wisconsin, Madison. He holds a Bachelors of Tech-nology degree in Industrial Engineering and Management from IndianInstitute of Technology, Kharagpur, and a Masters of Science degree inIndustrial Engineering from University of Wisconsin, Madison. His pri-mary research interests are in game theory, combinatorial auction design,discrete optimization and its application to game theory.

Dr. Raj Veeramani is a Professor at the University of Wisconsin-Madisonwith joint appointments in the College of Engineering and the Schoolof Business. He is the Director of the campus-wide UW E-BusinessInstitute and UW E-Business Consortium. Dr. Veeramani’s areas ofresearch interest and expertise include e-business models and decisiontechnologies, radio frequency identification (RFID), Internet-aided sup-ply chain management, quick response quoting and manufacturing, andcollaborative product development. His work has embodied active col-laboration with leading companies in a variety of industries, helpingthem develop e-business, supply chain and RFID strategies and im-plementation roadmaps. Dr. Veeramani received his PhD and MS de-grees in Industrial Engineering from Purdue University and his BS de-gree in Mechanical Engineering from the Indian Institute of Technology,Madras. He joined the faculty of the University of Wisconsin-Madison in1992.

Contributed by the Industrial Engineering Department

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