Pricing combinatorial auctions - Semantic Scholar

European Journal of Operational Research 154 (2004) 251–270 www.elsevier.com/locate/dsw

O.R. Applications

Pricing combinatorial auctions Mu Xia a

a,*

, Gary J. Koehler

b,1

, Andrew B. Whinston

c,2

Department of Business Administration, College of Commerce and Business Administration, University of Illinois at Urbana-Champaign, Champaign, IL 61820, USA b Department of Decision and Information Sciences, Warrington College of Business Administration, 351 STZ, P.O. Box 117169, University of Florida, Gainesville, FL 32611, USA c Center for Research in Electronic Commerce, Department of Management Science and Information Systems, University of Texas at Austin, Austin, TX 78712, USA Received 9 October 2001; accepted 5 September 2002

Abstract Single-item auctions have many desirable properties. Mechanisms exist to ensure optimality, incentive compatibility and market-clearing prices. When multiple items are offered through individual auctions, a bidder wanting a bundle of items faces an exposure problem if the bidder places a high value on a combination of goods but a low value on strict subsets of the desired collection. To remedy this, combinatorial auctions permit bids on bundles of goods. However, combinatorial auctions are hard to optimize and may not have incentive compatible mechanisms or market-clearing individual item prices. Several papers give approaches to provide incentive compatibility and imputed, individual prices. We find the relationships between these approaches and analyze their advantages and disadvantages.
2002 Elsevier B.V. All rights reserved. Keywords: Combinatorial auction; Bidding; Pricing; Incentive compatibility

1. Introduction to combinatorial auctions Most auctions, both online or offline, are used to trade individual items, one at a time. Either there are multiple buyers competing for one unit of a good from a seller, or there are multiple sellers competing for the right to sell a unit of good to a

single buyer, in the form of a reverse auction. 3 Either way, there is only one good involved in the auction and on one side of the market there is only one trader. The buyer with the highest bid or the seller with the lowest bid gets to buy or sell the good. However, there are circumstances where it is not efficient to hold only single-item auctions. One such scenario is when there exist complementarities between different goods. When positive

*

Corresponding author. Tel.: +217-333-2878; fax: +217244-7969. E-mail addresses: [email protected] (M. Xia), koehler@ufl.edu (G.J. Koehler), [email protected] (A.B. Whinston). 1 Tel.: +352-846-2090; fax: +352-392-5438. 2 Tel.: +512-471-8879; fax: +512-471-0587.

3

Reverse auctions are often used in government procurement, in which a government agency specifies exactly what it wants, and invites companies to bid for the right to do business. Usually, whoever can meet the specification with the lowest price gets the contract.

0377-2217/$ - see front matter
2002 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(02)00678-1

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M. Xia et al. / European Journal of Operational Research 154 (2004) 251–270

complementarities exist, two goods are worth more to a bidder if she acquires both than the sum of the individual value of each to her alone. Such a bidder desires the complete bundle of goods. If she bids on and acquires components individually, rather than as a bundle or combination of goods, she faces a possible exposure (Rothkopf et al., 1998). The exposure problem can result when a bidder places a high value on a combination of goods but a low value on strict subsets of the desired collection. She may pay more for a subset of goods in individual auctions than they are worth to her in an unsuccessful attempt to obtain her desired bundle. One way researchers have proposed to solve the exposure problem is to allow combinatorial bidding (Rothkopf et al., 1998). In a market where heterogeneous goods are to be traded, each bidder can specify a combination of goods she wants to acquire and a price she would pay for the combination. The market tries to allocate the set of goods so as to maximize the total revenue of the auction, which is in line with the overall social welfare. However, there are many challenges associated with combinatorial auctions. The first challenge is solving the winner determination problem (WDP). It is well known that the problem can be formulated as a multi-dimensional knapsack problem (MDKP). Although this can be a difficult problem, it can be solved up to moderate sizes using a variety of different approaches such as optimization, intelligent search and heuristics. Although it is computationally challenging when the size becomes large, it is not hard to conceptually understand. When the objective of the WDP, the profit for the seller, is maximized, the allocation is also efficient, as the goods are allocated to bidders that value them most. 4 However, this is true only when the biddersÕ bids reveal their true valuation. Thus, the overall efficiency is dependent on the biddersÕ reported valuation. If any bid does not reflect a

4

However, in general it is not always true that a revenue maximizing allocation is efficient. They are often competing objectives.

bidderÕs true valuation, the resulting revenue may not be the best the seller could expect. A bidder can potentially better her utility by misrepresenting her true valuation on a bundle. When this happens, the optimization result of the WDP is no longer overall efficient. Therefore, one has to consider how to ensure the truthfulness of all bidders, in addition to solving the WDP to optimality. Mechanisms that ensure this are termed incentive compatible. Thus, a second objective of auction mechanisms is to ensure incentive compatibility. An effective way to achieve incentive compatibility is to adjust winning bidderÕs final payoffs so they would not gain any utility by misrepresenting their true valuation. In other words, bidding truthfully becomes their dominant strategy. Finally, after the completion of an auction, determining prices for individual goods is also valuable because: (1) they help explain the auction result––why a certain bid lost and another won; and (2) they can serve as a price guide for future auctions. 5 Ideally, these individual prices should satisfy a market-clearing condition. First, the sum of all prices for the goods in a bundle of a winning bid should be greater than or equal to the winning bundle price (i.e., the initial bid price before any incentive compatible adjustments). Likewise, the sum of all prices for goods in a bundle of a losing bid should be less than or equal to the bid price. Hence, the ideal auction mechanism for combinatorial auctions provides the following three features: • An efficient winner determination mechanism. • Incentive compatible bid pricing mechanism. • A way to determine imputed prices for goods. When all the goods are divisible, the WDP is a linear programming problem. It is known that bundle prices computed from the dual model of the LP are asymptotically incentive compatible

5 However, as an anonymous reviewer points out, since allocations are based on bundle bids, prices for individual items cannot accurately reflect the conditions that make one bid win and another lose.

M. Xia et al. / European Journal of Operational Research 154 (2004) 251–270

(Fan et al., 2000). The value of LP dual variables also gives individual prices. So all three criteria are easily (asymptotically) satisfied by the WDP. However, the indivisible case is not so accommodating. When goods are indivisible, the wellknown duality gap of integer programming (IP) assures us that a solution and corresponding dual prices satisfying all three desired characteristics exist only in special cases, one of which is when the integer solution is also a solution to the linear programming relaxation achieved by removing the integrality constraint. Moore et al. (1972) proposes a pricing algorithm for resource allocation in a non-convex economy, which can also be applied to combinatorial auctions. This algorithm, however, either finds a set of prices with the desired characteristics or results in an infinite loop. In general, it is still desirable to specify procedures that attempt to achieve some of the three desired features while perhaps compromising on others. This paper will examine the existing approaches of pricing combinatorial auctions, the relationships among these approaches, as well as their ties with traditional auction pricing theory such as Generalized Vickrey auctions (GVAs). Our goal is to provide a comparison of methods to help researchers understand the role of various pricing mechanisms in combinatorial auctions. We start by reviewing each of the three aforementioned criteria in more detail.

WDP1

1.1. The winner determination problem

Z ¼ max

Z ¼ max

m 1 n 2X X

i¼1

s:t:

X

253

pij xij

j¼1

ð1Þ

wj xij 6 1

i;j

X

xij 6 1

j ¼ 1; . . . ; 2m  1

ð2Þ

i

xij ¼ 0; 1 i ¼ 1; . . . ; n

and

m

j ¼ 1; . . . ; 2  1 xij is a binary variable, indicating whether bundle j is awarded to bidder i. pij is the bid price for bundle j from bidder i. wj is a vector of size m where ðwj Þk is one if good k is part of bundle wj and zero otherwise. These bundles range over all 2m  1 possibilities (with the zero-valued bundle being ignored). Constraint (1) is the resource availability constraint for each good––only one unit of each item is available for sale. Constraint (2) reflects the condition that each bidder gets at most one bundle. The second model places no restrictions on how many bundles each bidder can obtain as long as the availability constraint is satisfied. Thus, we can regard each bundle as coming from a unique bidder. Therefore, the number of bidders is the same as the number of bids, n. The model is as follows: WDP2 n X

p j xj

j¼1

Arguably, the most important challenge with combinatorial auction is solving the WDP. Hence, this aspect has received most attention from researchers. For simplicity, we assume only one unit of each good is available in the auction. From the literature, there are two general models for the WDP. The first type (e.g., see Wurman and Wellman, 1999) limits the number of bundles a winner may win to, at most, one bundle. The second approach (e.g., see DeMartini et al., 1999) allows multiple winning bundles per winner. Let m be the number of unique goods being traded and n the total number of bidders. The first model is as follows:

s:t:

n X

wj xj 6 1

j¼1

xj ¼ 0; 1 j ¼ 1; . . . ; n xj is a binary variable that indicates whether bundle j gets traded. pj is the bid price for bundle j. wj is a bundle vector of size m formed as described above. All bid/bundle submissions are in the formulation so a particular bidder may have several xj Õs associated with her. Or equivalently, each bundle can be regarded as coming from a unique bidder. While the two models have different assumptions, their goals are the same, i.e. to determine which bundles should be selected for trade

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in order to maximize the social surplus. Furthermore, WDP1 is just a special case of WDP2 because constraints (1) and (2) can be combined and the resulting columns re-interpreted as bundles with m additional dummy goods corresponding to the original (2) constraints. WDP1 has an advantage over WDP2 in describing XOR bids. XOR bids are bids sharing an ‘‘XOR’’ relationship, i.e. a bidder is interested in getting only one bundle out of a given set. When such bids are allowed WDP2 has to add a new constraint (which can be considered as a dummy good) to represent the XOR relationship among such bids in the same form of the other item availability constraints. WDP1 can accommodate such relationship because each bidder gets only one bundle in the first place. Moreover, when complete valuations are present, i.e. every bidder has a valuation for every possible bundle, WDP1 has an added advantage of ruling out the possibility of any bidder getting her preferred allocation in more than one bundle (for a lower total price) and then re-assembles them. WDP2, however, cannot prevent any bidder from doing this from the model design perspective. WDP2 represents the most widely studied single-unit (each item is unique and there is only one unit for sale each), single-sided (one seller and multiple buyers) case. It is a set packing problem (SPP), a well-known NP-complete problem (Garey and Johnson, 1979), which is difficult to solve when the problem size is large. While researchers such as Hoffman and Padberg (1993) have developed various algorithms and techniques to solve some moderately-sized or special cases of SPPs, in general there is no guarantee the problem can be solved to optimality in an acceptable time. If we relax the single-unit constraint, the WDP problem becomes the MDKP. Lin (1998) offered an excellent survey article on solving the MDKP. Much of recent research on solving the WDP has been carried out by applying artificial intelligence (AI) techniques such as intelligent search (Sandholm, 1999; Sandholm et al., 2001; Fujishima et al., 1999; Leyton-Brown et al., 2000). These papers test their algorithms on generated data sets and claim to have good results when solving single-unit, singlesided combinatorial auctions. Andersson et al.

(2000) show using a commercial IP solver (CPLEX) one can also solve the WDP reasonably fast, while Gonen and Lehmann (2000) apply the well-known branch-and-bound procedure to solve the WDP as an IP problem. A natural question is: how do the AI approaches compare to the optimization approaches? A recent paper (Xia et al., 2001) discusses, both theoretically and experimentally, the differences of the two approaches. In any case, examining algorithmic approaches to WDP is not our goal. We are primarily interested in determining incentive compatible pricing and market-clearing prices. We focus on WDP only insofar as it impacts these issues. 1.2. Incentive compatibility After the WDP is solved, an important issue is how to price the winning bundles to achieve incentive compatibility. Although it is straight forward to sell the winning bidder her desired bundle at the bid price, it may leave room for bidders to mask their true valuations and possibly get the bundle at a lower price. As a result, the optimization allocation may not be efficient overall. Thus an important issue is how to devise a rule for pricing the bundles in order to induce each bidder to state their true valuation. Termed incentive compatibility, it is the most desirable feature in mechanism design (Mas-Collel et al., 1995). In general, as is the case in most auction literature, if we assume each bidderÕs utility function to be quasilinear, we can denote the utility for any agent as: U ðB; qÞ ¼ lðBÞ  q where B denotes the bundle, lðBÞ the value of the bundle and q the price paid by the agent to get the bundle. To ensure incentive compatibility, the monetary transfer to each bidder has to be set so that the expected utility of bidding truthfully is always greater than or equal to the utility when the valuation is misrepresented. By adjusting selling prices for each bundle traded (and even those not winning), effectively we are creating monetary transfers between bidders, which can also be considered as redistribution of the trade surplus. Hence, we will approach the incentive compatibility issue by focusing on the final pricing of winning bundles.

M. Xia et al. / European Journal of Operational Research 154 (2004) 251–270

One of the earliest works on auction pricing (and auctions in general), is the seminal paper by Vickrey (1961). Vickrey proved that in a sealedbid, single-item auction, where each bidder has her own private valuation, if the highest bidder wins the good and pays the highest losing price, it is incentive compatible in that bidding oneÕs true valuation is a weakly dominant strategy for each bidder. We denote the highest bid as p1 and the second highest bid as p2 . Taking the highest losing bid as the price in effect redistributes part of the trade surplus, p1  p2 , from the seller to the winner, thus making underbidding (in hope of paying less for the good if one is a winner) unnecessary. From the buyerÕs point of view, the winner is awarded the surplus she brings to the auction, exactly the difference between her bid and the highest losing bid. The Vickrey auction offers great insight into single-item auctions. It is powerful in that it makes few assumptions other than private valuation and a sealed bid––there is no assumption on the distribution of the biddersÕ valuations. Using this mechanism, each bidderÕs truthful bidding of their valuation is a weakly dominant strategy. A mechanism with these characteristics is very desirable from an economic perspective. Naturally economists have extended the concept to encompass more general models. The Groves–Clarke mechanism, proposed separately by Clarke (1971) and Groves (1973), is a generalization of the Vickrey auction to social choice problems. By considering the auction outcome, i.e. the allocation of the indivisible resources, as a social choice, a combinatorial auction is a special case of the social choice problem. When applied in an auction setting, the Groves–Clarke mechanism is also called the GVA. The basic idea of the Vickrey auction carries over––the price should be set such that oneÕs bid can only impact oneÕs payoff by affecting the social choice outcome, but have no effect on the price it pays. The winning bidder is rewarded with the surplus she contributes to the trade. In particular, in the combinatorial setting, if we adopt WDP2, an incentive compatible price for winning bundles, using the Groves–Clarke mechanism is straightforward. First WDP2 is solved, then

255

for any winning bundle j, the price is set at: qj ¼ pj  ðZ  Zj Þ, where Z is the optimal auction revenue and Zj is the optimal revenue for the auction without bundle j included. The main challenge of solving the pricing problem using this approach is the computational burden––one has to solve (n þ 1) WDPs optimally to get the price vector, one for the original WDP, and one for each bidder. Taking into account the complexity of solving a WDP, it is thus usually impractical to implement. 1.3. Imputing individual item prices Another issue of interest is imputing prices for individual items from winning bundle prices. First we define prices that clear the market. Definition 1. For combinatorial auctions, a set of item prices is called market clearing or equilibrium if all the winning bids are greater than or equal to the total price of the bundle items and all the losing bids are less than or equal to the total price of the bundle items. Individual prices of combinatorial auction items can provide insights into a combinatorial auction, which may be useful for understanding the value of bundles. The prices for bundles or individual goods, however, are not readily computed when the WDP is solved. This is due to the duality gap of IP caused by the indivisibility of the goods. Equilibrium prices may not exist (Nemhauser and Wolsey, 1988). In addition, the two different WDP models may yield different prices. For WDP1, if there are m unique goods, there is a price for each of the 2m  1 bundles. For WDP2, it is unlikely that all 2m  1 bundles receive bids, thus in order to compute prices for each bundle, one has to have prices for individual goods. Therefore, how to best approximate such individual prices becomes the focus of researchers. In this paper, we will review some notable explorations of imputing item prices in combinatorial auctions and compare advantages and disadvantages of using various approaches. To gain more insight into the matter, we examine the relationships between these models.

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The paper is organized as follows. In Section 2 we focus on the incentive compatibility issue by examining bundle pricing approaches found in the literature. In Section 3 we focus on the imputed prices for individual goods. In both cases we start by reviewing their approach, and then compare, contrast and look for commonalities in these approaches. Finally, we end with a conclusion.

2. Bundle pricing approaches Bundle pricing, as its name suggests, is used to compute a final price for each bundle. Wurman and Wellman (1999), Bikhchandani and Ostroy (2001) and Parkes (2001) take this approach. Ba et al. (2001) compute bundle prices in a public good combinatorial double auction. By having a price for each bundle but not each individual component, one does not have to assume anything about the complementarity or substitutability among the components in a bundle. However, as bundles overlap, one does have to have some additional assumptions beyond the individual pricing approach. These assumptions range from auction rules to distribution of valuations. They include: • Every bidder must bid on every bundle: Without this constraint, all bundles may not receive bids. It would be hard to price a bundle that receives one bid or no bids at all. However, having to bid on every one of the 2m  1 bundles could be very burdensome for a bidder who is interested in only one or a few bundles, unless some rule is used to automatically generate consistent bids on other bundles. Typically, to avoid having to specify all the valuations one by one, a bidder can report only valuations for interesting bundles and have a computerized agent (or the auctioneer) fill in valuations for the remaining bundles according to some rule. 6 For example, in a combinatorial auction to sell three goods X ,

Y and Z, if bidder 1 is interested in getting only bundle ðX ; Y Þ for $10 and it is free to dispose of unnecessary goods, then the valuation for bundle ðX ; Y ; ZÞ might also be set at $10 and the valuations of ðX Þ, ðY Þ and ðZÞ might be set at zero. • Each bidder gets, at most, one bundle in the resulting allocation: This is an auction rule imposed on all bidders. It is aimed to prevent bidders from underbidding when they value a set of complementary goods. Without this constraint, bidders may submit only single-item bids even when they value a combination much more than the sum of all individual component items, attempting to acquire all the individual items separately at a lower price and pocket the value of the complementarity without having to pay for it. While the constraint automatically accommodates exclusive-or (XOR) bids, 7 the same feature can be achieved in combinatorial auctions by adding a dummy good corresponding to the XOR constraint. The goals for determining bundle prices for combinatorial auctions are: • Market clearing: Under these prices, the total surplus is maximized (so the allocation is efficient), and it is a competitive equilibrium in that not every bidder can be better off selecting another bundle to trade other than what she is assigned. Note the definition of market clearing when imputing individual prices, shown in Definition 1, is different. • Incentive compatibility: Given the prices, there is no incentive for any individual bidder to misrepresent her valuation in order to better her outcome. One incentive compatibility research effort on a model similar to WDP1 was done by Leonard (1983). The paper investigated incentive compatible prices of the well-studied assignment problem in operations research. His model is:

6

A similar scheme of automatically adjusting bid prices for other bundles is also proposed by Wurman and Wellman (2000) in the AkBA mechanism.

7 When bids from the same bidder are XOR bids, the bidder is only interested in getting at most one bundle out of the set.

M. Xia et al. / European Journal of Operational Research 154 (2004) 251–270

VIP ¼ max

m X m X i¼1 m X

s:t:

pij xij

j¼1

xij 6 1

i ¼ 1; . . . ; m

xij 6 1

j ¼ 1; . . . ; m

j¼1 m X

257

Section 2.5, we provide comparisons based on the review and our results. In the following, denote the price for bundle j as qj ; the winning bundle set Bþ ; the losing bundle set B ; and the bidder set A with winning bidder set Aþ and losing bidder set A . Let ZðA n iÞ be the optimal revenue without bidder i.

i¼1

xij ¼ 0; 1 i; j ¼ 1; . . . ; m The assignment problem is an IP problem that is totally unimodular, and, hence, can be solved as a linear programming problem. Its dual is: ! m m X X min ui þ vj i¼1

s:t:

j¼1

ui þ vj P pij i; j ¼ 1; . . . ; m ui ; vj P 0 i; j ¼ 1; . . . ; m

Due to degeneracy of the primal problem, however, the shadow prices associated with positions, i.e. the dual variables vj , are not unique. LeonardÕs paper identifies a set of shadow prices, Pm which maximizes j¼1 vj , that not only clears the market but also provides incentive compatibility. Regard i as the index for individuals and j as the index for positions. The shadow price for position j is the difference between the optimal value to all individuals of all positions plus another position of type j, VIPþj , and the value of the current positions, VIP : qj ¼ VIPþj  VIP . An individual to which position j is assigned in the optimal solution, as denoted by i, will not be better off if she misrepresents her values. If the represented value does not change the assignments, then the result will remain the same. Otherwise, if the assignment result is different, the individual can only be worse off. LeonardÕs result is regarded as either a special case (in Bikhchandani et al., 2001) or a basis (in Wurman and Wellman, 1999) in the bundle pricing approach. Although the above goals and assumptions are the most common in the bundle pricing approach, each of the following models differs slightly from the others. Next we study each in more detail. In Sections 2.1–2.4, we first review these approaches in a unified framework, followed by discussions and, in some cases, new results that help to elucidate the relationship between approaches. In

2.1. Wurman and Wellman (1999) Wurman and Wellman (1999) (henceforth WW) discuss the bundle pricing problem based on the winner determination model WDP1 as listed earlier. We first review the proposed algorithm. 2.1.1. Model and algorithm review The WW bundle pricing algorithm has four stages. 1. Solve the WDP. Since there is at most one bundle won by each bidder, there are two sets of bidders in the optimal allocation: those who win a bundle and those who do not. There are also two sets of bundles, those assigned to some bidder and those unassigned. 2. Create a dummy good /i for each losing bidder i 2 A . Set each bidderÕs valuation for every dummy bundle to zero. Let U ¼ f/i ji 2 A g. Consider the assignment problem, illustrated in Fig. 1, which matches the bundle set G ¼ U [ Bþ with A ¼ Aþ [ A . Each bundle in Bþ is assigned to the corresponding winning bidder in Aþ ; each dummy bundle in U is matched with its corresponding bidder in A . LeonardÕs (1983) price determination method discussed earlier can be applied to this assignment problem.

Fig. 1. The pseudo-assignment problem in WW.

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M. Xia et al. / European Journal of Operational Research 154 (2004) 251–270

3. The following two problems are solved: LPmin Q ¼ argmin

X

qg

fqg jg2Bþ g

g2G

s:t:

si þ qg P pi;g

8i 2 A and g 2 G

si ; qg P 0 8i 2 A and g 2 G X X si þ qg ¼ Z i2A

g2G

and LPmax Q ¼ argmin

X

qg

fqg jg2Bþ g

g2G

s:t:

si þ qg P pi;g

8i 2 A and g 2 G

si ; qg P 0 8i 2 A and g 2 G X X si þ qg ¼ Z i2A

g2G

Z is the optimal objective value of WDP1; pi;g is iÕs bid on bundle g 2 G; qg is the price for bundle g and si is bidder iÕs maximum surplus. Both Q and Q are vectors of jBþ j. They are marketclearing prices of winning bundles for the pseudo-assignment problem. (LPmin ) is a direct application of LeonardÕs (1983) model, while (LPmax ) gives an upper bound of the marketclearing prices. WW proves that any linear combination of the two prices, kQ þ ð1  kÞQ ð0 6 k 6 1Þ, is also market-clearing. 4. For each remaining unassigned bundle g, the price pg is calculated by qg ¼ maxi2A ðpi;g  si Þ. With such a price, no bidder would be better off if she chose to buy this bundle, because 8i 2 A; si P pi;g  qg . In other words, given such prices, no bidder, winner or loser, will have any incentive to deviate from her allocation determined by the optimal solution of the combinatorial auction. 2.1.2. Discussion WW prices are market clearing, because the prices support the optimal allocation, and nobody will choose a different bundle to buy given these prices. From step 2, there are multiple market clearing prices for the auction problem. However,

they are not incentive compatible. In step 2, Q is LeonardÕs incentive compatible price for the pseudo-assignment problem. But because this pseudo-assignment problem comprises dummy bundles for the losing bidders, even if we take Q as the price for winning bundles, once the losing bundles are added back to the problem, the prices, although still market clearing because of step 3, are no longer incentive compatible. In other words, the Leonard approachÕs incentive compatibility result does not carry over to this case, due to the transformation of step 2. After the WW procedure, every bundle, winning or losing, is assigned a price. But are these prices equally informative? Probably not, as the prices for winning bundles are the result of the dual of an assignment problem, but those for the losing bundles only need to satisfy one constraint (in step 4). In fact, the prices for losing bundles can be infinitely large so long as they prevent all bidders from buying them. As a result, prices for the losing bundles are less indicative of the true value of the bundles than those for the winning ones. Thus, without the information as to which bundles are won and which ones are not, these prices have little guidance value for helping the potential bidders valuate different bundles. Wurman and Wellman claim the prices for different bundles are anonymous, as opposed to GVA, which is discriminative pricing because it is based on the agent that wins the bid. They define anonymity as ‘‘every agent has the opportunity to purchase the same object at the same price’’. They give an example in which GVA would lead to different prices for the same bundle for different buyers. This may be an important issue if bidders perceive fairness, defined as ‘‘every agent has the opportunity to purchase the same object at the same price’’, as a requirement for auctions, such as in government procurement auctions. 8 However, the notion of anonymous price for a bundle may have limited meaning in the bundle auction setting. Suppose bidder i is the winner of Bundle g, and the price is pg , then if another bidder j bids the same price pg , would he/she be able to get the bundle? It

8

We thank an anonymous referee for pointing this out.

M. Xia et al. / European Journal of Operational Research 154 (2004) 251–270

is not clear, because the whole auction is changed and needs to be solved again. Very likely the auction result will also change, which leads to a new price for the bundle. 2.2. Bikhchandani and Ostroy (2001) Bikhchandani and Ostroy (2001) (henceforth BO) discuss the package assignment problem. They call the bundles ‘‘packages’’, which can be reserved for a specific bidder instead of everyone. When the assignment is a ‘‘first-order assignment’’, which is a special case of the package assignment problem, it is indeed exactly the WDP1 formulation.

In this model, l denotes both a partition of the set of goods and an assignment of the components of the partition to bidders. Let C denote all such partition-assignment pairs, dl ¼ 1 if the partitionassignment pair l is selected. (4) dictates that for bundle j to be assigned to bidder i in the auction, the pair (i; j) has to belong to a partition-assignment pair l that is selected. (5) reflects the fact that no more than one partition-assignment pair can be selected. The formulation is stronger than WDP1 in that (4) and (5) implies (2). This can be shown by adding (4) for all bundle–bidder pairs: X X X X xij 6 dl 6 dl 6 1 i;j

2.2.1. Model and algorithm review To get prices using LP duality, they add auxiliary variables to the original IP problem. In addition to all the assumptions of WW, they provide a sufficient and necessary condition for the packages to have market-clearing and incentive compatible prices. This requires valuations to satisfy a ‘‘buyers are substitutes’’ condition. The term, ‘‘buyers are substitutes’’, first used by Shapley (1962), means that the marginal product of any buyer subset is greater than the sum of its individual buyerÕs. X 8S A : ZðAÞ  ZðA n SÞ P ðZðAÞ  ZðA n iÞÞ i2S

(ZðAÞ denotes the total auction revenue with all buyers, while ZðA n iÞ is the auction revenue without bidder i.) Based on this assumption, BO construct the following model: BOWDP ZðAÞ ¼ max

XX

s:t:

X

wj xij 6 1

j2B

xij 6 X

pij xij

j2B

i2A

X

dl

8i 2 A

ð3Þ

8j 2 B; 8i 2 A ð4Þ

l:ði;jÞ2l

ð5Þ

dl 6 1

l2C

xij ; dl ¼ 0; 1

8j 2 B and l 2 C

259

ði;jÞ l:ði;jÞ2l

l2C

But because each allocation will have partitionassignment pairs, every solution for WDP1 will also be a solution for BOWDP. Hence, the two models are equivalent. BO prove that under the condition that buyers are substitutes, the LP relaxation of BOWDP gives integer solutions. Moreover, the optimal value of dual variables associated with condition (3), denoted by vi , is the Vickrey discount for each bidder, i.e., vi ¼ ZðAÞ  ZðA n iÞ. 2.2.2. Discussion For each winning bidder i, assuming she is assigned bundle j in the auction, the price she pays the auctioneer to get j is exactly qj ¼ pij  vi where pij is iÕs bid for bundle j. For each losing bidder i, the payment is zero, as ZðAÞ ¼ ZðA n iÞ. We can use qj as the price for the winning bundle j. It is not only market clearing, but also incentive compatible, since it is the Vickrey price. While the paper does not compute prices for losing bundles, it can be easily done using the last step of WWÕs procedure. Although BO imposes an additional constraint, the paper does get a very powerful result. It is a direct application of the GVA (MacKie-Mason and Varian, 1994). Another paper by Bikhchandani et al. (2001) observes that BO is not the only way to add auxiliary variables to WDP1 to get the Vickrey discount. They propose another model as follows:

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M. Xia et al. / European Journal of Operational Research 154 (2004) 251–270

BOWDP2 ZðAÞ ¼ max

XXX r2P j2r

s:t:

X

pij xij

i2A

xrij 6 zr

8i 2 A; 8r 2 P

ð6Þ

xrij 6 zr

8j 2 r; 8r 2 P

ð7Þ

j2r

X i2A

X

ð8Þ

zr 6 1

r2P

XX

wj xrij 6 zr

8i 2 A

ð9Þ

r2P j2r

xrij P 0

8j 2 r; 8i 2 A

zr P 0

8r 2 P

Here, P is the set of partitions of the goods. For any partition r 2 P, j 2 r means bundle j is part of the partition r. Let zr ¼ 1 if partition r is selected, zero otherwise. Set xrij ¼ 1 to mean that r is selected and bundle j is assigned to bidder i in partition r. Let vi be the dual variable associated with (9). Bikhchandani et al. (2001) prove BOWDP2 has an integral optimal solution. Under the condition that ‘‘buyers are substitutes’’, the dual variable for (9), vi ¼ ZðAÞ  ZðA n iÞ, is exactly the Vickrey discount of the bidder i. One can easily compute prices for winning bundles as before, by subtracting the Vickrey discount from the winnerÕs bid. Since both BO and WW base their pricing procedure on WDP1, it is worthwhile to compare the two approaches. First, WW prices are market clearing but not incentive compatible, while BO prices are both market clearing and incentive compatible, given that buyers are substitutes. Second, BO prices support the optimal allocation of the pseudo-assignment problem in step (2) of WW. This is shown as follows. Theorem 1. Under the assumption of buyer substitutes, the Vickrey price for the winning bundles also supports the pseudo-assignment problem’s allocation. Proof. Let Q denote the sets of prices for winning bundles, i.e., Q ¼ fqj ; j 2 Bj9i 2 A, s.t. xij ¼ 1g in the original combinatorial auction problem. Since Q is the Vickrey price for the original auction problem, it is also competitive. Thus, 8i s:t: xij ¼ 1,

pij  qj P pij0  qj0 8j0 6¼ j, j0 2 B. In WWÕs pseudoassignment problem, the new bundle set G ¼ Bþ [ U. As shown by WW, the prices for the dummy bundles are zero, i.e., qð/i0 Þ ¼ 0, 8i0 2 A . Since pi/ 0 ¼ 0 8i 2 A, i0 2 A , pig  qg ¼ 0, 8g 2 U, i i 2 A. Therefore, 8j 2 Bþ and xij ¼ 1, g0 2 U pij  qj ¼ vi P 0 ¼ pig0  qg0 .  Although the Vickrey price for the winning bundles from BO supports WWÕs pseudo-assignment problem allocation, it may not be either of the two prices given in WW, or a linear combination of the two prices. We can show this through an example. Consider the following combinatorial auction where {X ; Y ; W } is the set of individual goods for sale and {1,2,3} is the set of bidders. Each bidderÕs valuation for every bundle is listed in the table. A best allocation, apparently, is assigning items X and Y to bidder 1 and item W to bidder 2 with bidder 3 getting nothing, as highlighted in Table 1. The auctionÕs optimal revenue is 16. The Vickrey prices for the two winning bundles are qðXY Þ ¼ p1 ðXY Þ  ðZðAÞ  ZðA n 1ÞÞ ¼ 10  ð16  16Þ ¼ 10 qðW Þ ¼ p2 ðW Þ  ðZðAÞ  ZðA n 2ÞÞ ¼ 6  ð16  15Þ ¼ 5 The pseudo-assignment problem of WW, is assigning G ¼ fðXY Þ; ðW Þ; /3 g to the three bidders. Solving ðLPmin Þ and ðLPmax Þ, one can get Q ¼ fqfXY g ; qfW g g ¼ f6; 4g and Q ¼ fqfXY g ; qfW g g ¼ f10; 6g. WW proves that any price that is a linear combination of Q and Q is also an equilibrium price that supports the allocation and is competitive. If we denote the Vickrey price for the original auction problem as QV , it is apparent that QV 6¼ kQ þ ð1  kÞQ 80 6 k 6 1. That is, the Vickrey Table 1 A combinatorial auction example Bidders

1 2 3

Items X

Y

W

XY

XW

YW

XYW

6 7 4

6 3 2

5 6 4

10 8 5

7 9 7

8 6 9

12 11 12

M. Xia et al. / European Journal of Operational Research 154 (2004) 251–270

261

price, although supporting the allocation, is not a linear combination of the two WW prices.

ted out in the paper, the value of an approximate incentive compatible outcome is also uncertain.

2.3. Parkes (2001)

2.4. Ba et al. (2001)

Parkes (2001) (PARK) proposes another bundle pricing scheme as part of an iterative ascending bundle auction model.

The three approaches we outlined above all deal with private-good combinatorial auctions. It is interesting to consider a public good auction. In another related paper, Ba et al. (2001) proposes a market mechanism for knowledge production and allocation within an organization. The knowledge goods are traded in bundles; thus, possible complementarities among components can be taken into account. Since the knowledge components within a firm must be considered as public goods, once a product is purchased, it can be freely shared by all departments within the firm. The combinatorial auction model is different from WDP1. The combinatorial auction model is:

2.3.1. Model and algorithm review In his approach, he imposes the same set of assumptions as WW does. However, he argues that while the GVA gives incentive compatible prices, it needs complete information of bidder evaluation on every bundle desired. Yet, it may be too costly for bidders to evaluate all the bundles. Moreover, bidders may not be willing to reveal their full valuation. In such cases, GVA would not work because of incomplete information. Parkes proposes a two-stage procedure with iterative ascending combinatorial auctions to approximate the GVA. In the first stage, an iterative combinatorial auction is held for all bundles. At the end of this stage, the auction terminates with an outcome close to the optimal allocation. A part of the Vickrey discount is calculated. In the second stage, the rest of the Vickrey discount is computed, for those winning bidders whose absence from the auction would cause some other winning bidders to lose also. 2.3.2. Discussion Using this approach, bidders do not need to reveal complete information of their valuation at the beginning of the auction. This is an attractive feature as bidders may be either unwilling to do so, or they may not absolutely certain about their valuation at the beginning. In the latter case, as the auction continues, bidders have more information to help determine the valuation. While the process does give incentive compatible prices in some special cases, it is still an approximation to the original GVA mechanism. The approximation is perfect only when the bid increment goes to zero, yet, that in itself can make the iterative bidding process too long to hold and participate in. Just as a bidderÕs submitting approximate valuation makes the GVA outcome inefficient, as poin-

MP max

n X

pj xj

j¼1

s:t:

max fwi;j xj g þ min fwi;j xj g 6 0

j¼1;...;n

j¼1;...;n

8i ¼ 1; . . . ; m xj ¼ f0; 1g

8j ¼ 1; . . . ; n

This is a combinatorial double auction, in that there can be multiple buyers and sellers, and possibly, hybrid traders that both buy and sell in one bundle. The model is similar to WDP2, with the only difference being the goodÕs public nature. The vector, wj ; represents the content of a bundle: if wk;j ¼ 1, the kth good is to be bought in bundle j; if wk;j ¼ 1, the kth good is to be sold in bundle j; otherwise the value is 0 meaning item k is not part of the bundle. The constraint ensures that for a component to be bought, it has to be supplied by a bundle. Here, the issue is how to price the knowledge bundles so that all bids are incentive compatible. The difficulty lies in both the combinatorial nature and the free-riding problem brought about by the public-good nature, since bidders may try to under-represent their true value on a knowledge component in the hope of getting it for free.

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M. Xia et al. / European Journal of Operational Research 154 (2004) 251–270

The original formulation MP is a non-linear programming problem. To get prices, an additional assumption has to be made. The paper assumes that each knowledge component is provided by only one seller (a unique provider). Under this assumption, the pricing problem becomes the dual of a network flow problem. The dual variables correspond to prices for knowledge bundles. Moreover, they are incentive compatible. It is shown that it is an ap-

plication of the Groves–Clarke (Groves, 1973 and Clarke, 1971).

mechanism

2.5. Comparison of the four bundle pricing approaches To compare the above four approaches of bundle pricing, we create Table 2 listing their characteristics.

Table 2 Comparison of different bundle pricing approaches Wurman and Wellman (1999) (WW)

Bikhchandani et al. (2001) (BO)

Parkes (2001) (PARK)

Ba et al. (2001) (BA)

Bundle auction model

Private goods; direct revelation combinatorial auctions

Private goods; direct revelation combinatorial auctions

Public goods; combinatorial double auctions

Optimization formulation

ðWDP1ÞZ ¼ max

Private goods; iterative ascending combinatorial auctions Buyers bid on all bundles iteratively; auctioneer needs to solve WDP1 for provisional winners Same as WW

Unique provider (UP)

Yes

Yes

No transformation. Leave burden of optimization with bidders

With the UP assumption, the problem is the dual of a network flow problem. However, prices do not correspond to individual knowledge components due to the transformation

In general, no; in special cases, yes No

Yes

Approximation of GC; implements GC only in special cases

Application of GC

XX i2A

s:t:

X

pij xij

j2B

Same as WDP1

wj xij 6 1 8i 2 A

j2B

X

xij 6 1 8j 2 B

i2A

xij ¼ 0; 1 8i 2 A; 8j 2 B Assumptions

Bidder bids on every bundle; at most one bid is awarded to each bidder

Need to solve Yes the WDP first? Transformation Constructing pseudo-assignment problem to get prices by keeping only the winning bundles and adding dummy bundles for the losing bidders; then getting prices for the assignment problem

Prices incentive compatible?

No

Prices for losing bundles? Relationship with Groves– Clarke Mechanism (GC)

Yes

None

Buyer substitute in addition to all assumptions in WW Yes Adding auxiliary variables to make the WDP unimodular so the LP relaxation can be solved to get prices Yes

No, but such prices can be easily derived An implementation of the GC under special assumptions (buyer substitute)

max

X pj xj

s:t:

max wj xj þ min wj xj 6 08i

j2B j2B

j2B

xj ¼ 0; 1 8j 2 B

No

M. Xia et al. / European Journal of Operational Research 154 (2004) 251–270

2.6. Advantages and disadvantages of the bundle pricing approach The advantages of bundle pricing are: 1. There is no assumption on the bidderÕs bundle evaluations; 2. With a price for every bundle, by comparing prices, one can gain some insight into the value of complementarity in different scenarios. For example, if for bundles X and Y , qðX Þ ¼ 5, but qðX [ Y Þ ¼ 25, then there is a very strong complementarity between A and B. However, if qðX [ Y Þ ¼ 5 instead, then very little complementarity exists. The disadvantages of bundle pricing are: (1) The number of combinations any bidder has to evaluate is exponential to the number of goods. For example, when m ¼ 4, one has to bid on 15 bundles. It is extremely burdensome and impractical to ask anybody to evaluate so many bundles, especially those she is not interested in at all. A way to get around the tedious bid evaluation and expression process is to preset all bids to zero, so an agent needs to bid only on bundles she is interested in. One way of relieving bidders from the burden of having to bid on every bundle is to have a computer program that automatically updates the bid for all the bundles according to a rule given by the bidder every time a bid is submitted. For instance, the rule may be that the bidderÕs valuation is monotonous on bundles, i.e. bidder iÕs valuation on a bundle j is not less than that of any of its subsets j0 j. Therefore, if the current bid for a superset is less than the bid, then it is set to the latter. More specifically, when a bidder i bids pi on bundle j, then 8j0 , s.t. wj0 < wj iÕs utility on j0 must be updated to be at least pij , i.e. pij0 maxðpij0 ; pij Þ. (2) Not every bundleÕs price is equally informative: prices for unassigned bundles carry less information than those for assigned bundles, as they need only to ensure that bidders do not get distracted from getting the bundle they are assigned. If one is given only price information without knowing which price is for winning bundles and which is for losing bundles, the value of

263

this information is very limited. Take WW for example. The losing bundleÕs price can be anything greater than the pg proposed and all the results still hold, while for the winning bids, prices are within a finite closed range. (3) Since the prices are for bundles, there is little information on the value of individual goods that make up a bundle. This is especially true if there is no single-item bundle in the optimal allocation. (4) For bundle pricing to be effective, it has to be strictly enforced that: first, the bidders will pay the bundle prices dictated by the seller; second, there is no opportunity for collusion, in which a group of buyers submit one bid for a set of goods then divide them up later to improve their payoffs. 3. Individual pricing The bundle pricing approach gives prices only for bundles. So, if no winning bundle contains only a single component of interest, bundle pricing does not provide a way to determine an individual component price. Because individual component prices can serve as benchmarks for combinatorial bids, they are desirable. Next, we review one proposed approach and another general IP approach and compare them to find their relationship. In Section 3.1 and the first part of Section 3.2, we review the models and algorithms of the DeMartini and OÕNeill approaches. In the latter part of Section 3.2, we compare their relationship. Section 3.3 discusses the advantages and disadvantages of using the individual pricing approaches. In the following, let M ¼ f1; . . . ; mg represent the set of items being auctioned. Here we restrict bidders to bidding on one bundle and notationally represent the bidder set as equivalent to the bundle set. 9 As earlier, we denote this set by A. The vector corresponding to individual imputed item prices is denoted as p. 9

Actually, a bidder can place two different bids, and can possibly win both if the bids do not overlap––we can treat the bidder as two different bidders, one for each bundle. In that regard, there is no restriction on how many bids a bidder can place.

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M. Xia et al. / European Journal of Operational Research 154 (2004) 251–270

3.1. DeMartini et al. (1999)

solve (K ¼ U)

DeMartini et al. (1999) proposes a pricing scheme for all individual goods in a combinatorial auction, by approximating the prices in a divisible case. Their scheme first solves the WDP2. The paper proposes the following three-step process to determine prices. 10 Step 1: Given pj and the values Aþ , A , and xj resulting from the solution to WDP2, find the lowest z that bounds the discrepancy: min

z

s:t:

p0 wj ¼ pj

z;p;g

p wj þ gj P pj gj 6 z

j 2 A

z; p; gj P 0 If z ¼ 0, go to step 3. Let K be the empty set and J ¼ fjjj 2 A and gj ¼ z g. If J ¼ A , go to step 3. Otherwise go to step 2. Step 2 : Given pj and the values Aþ , A , and xj resulting from the solution to WDP2, and J , solve z;p;g

p0 wj ¼ pj

j 2 Aþ

p0 wj þ gj P pj 0

p wj þ gj ¼ pj

s:t:

p0 wj ¼ pj j 2 Aþ p0 wj þ gj P pj j 2 A p 0 1i P y i 2 M p0 1i ¼ pi i 2 K y; p P 0

M n K. If M is Let K fi : pi ¼ y  g and M empty, stop. Otherwise go to step 3 again.

02 3 1 1 B6 7 C ðw1 ; p1 Þ ¼ @4 1 5; 30A; 0 02 3 1 0 B6 7 C ðw2 ; p2 Þ ¼ @4 1 5; 30A; 1 02 3 1 1 B6 7 C ðw3 ; p3 Þ ¼ @4 0 5; 30A; 1 02 3 1 1 B6 7 C ðw4 ; p4 Þ ¼ @4 1 5; 39A

min z s:t:

y

y;p

We present a simple example to illustrate the process. The bids are:

j 2 Aþ

0

max

j2J

1 

j2A nJ

gj 6 z z; p; gj P 0 J [ fj 2 A n If z ¼ 0, go to step 3. Let J    J : z ¼ gj g. If J ¼ A , go to step 3. Otherwise go to step 2 again. Step 3 : Given bj and the values Aþ , A , and xj resulting from the solution to WDP2, and J from step 1,

10 We rewrite some of the original notation using a vector format.

Winner determination gives 2 3 0 607  7 x ¼6 405 1 DeMartini Prices are then determined as follows.

Aþ ¼ f4g; A ¼ f1; 2; 3g and M ¼ f1; 2; 3g

M. Xia et al. / European Journal of Operational Research 154 (2004) 251–270

3.2. O’Neill et al. (2001)

Step 1: Solve: min

z

z;p;g P 0

p1 þ p2 þ g1 P 30

s:t:

p2 þ p3 þ g2 P 30 p1 þ p3 þ g3 P 30 p1 þ p2 þ p3 ¼ 39 g1 6 z; pP0

g2 6 z;

g3 6 z

The solution is: 2

3 13 p ¼ 4 13 5 13





z ¼ 4;

and

2 3 4 g ¼ 445 4 

so J ¼ A , K ¼ U. Step 3: Solve: y

s:t:

p1 þ p2 P 26 p2 þ p3 P 26

s:t:

n X

wj xj 6 1

xj 2 f0; 1g

p2 P y;

p3 P y

2

y ¼ 13 and

p j xj

j¼1

The solution is: 

n X j¼1

p1 þ p3 P 26 p1 þ p2 þ p3 ¼ 39 p1 P y;

OÕNeill et al. (2001) discusses pricing for the more general resource allocation problem with non-convex objective functions. The main idea is to associate a cost with each positively valued integer variable. In particular, after the resource allocation problem is solved, add a new equality constraint for each positively valued, optimal integer variable, setting the variable to its optimal value. The new LP problem has the same optimal solution as the original IP. Next solve the dual for the new LP. The price is the sum of all the composing items plus the additional dual variable corresponding to the integer variable. We apply the approach to combinatorial auctions. It produces prices by the following process. First solve WDP2 max

max

y;p P 0

265

3 13 p ¼ 4 13 5 13

Stop. The DeMartini approach attempts to compute a set of prices such that (1) for any winning bundle, the sum of the prices of its comprising individual items is equal to the bid; (2) for losing bundles, the discrepancy between the sum of individual item prices and the bid, if the bid is greater than the sum, is minimized. The model in step 1 results in a set of prices with the above features. But because the model may have multiple solutions, steps 2 and 3 iteratively eliminate multiple solutions and ensure the uniqueness.

j ¼ 1; . . . ; n

(pj is the bid price for bundle j; wj is a vector representing the items in bundle j.) Then solve min

p 0 1 þ g 0 x

s:t:

p0 wj þ gj P pj

p;g

j ¼ 1; . . . ; n

pP0 p is the price vector for individual goods. The same example presented in Section 3.1 gives min

p 0 1 þ g4

s:t:

p1 þ p2 þ g1 P 30

p;g

p2 þ p3 þ g2 P 30 p1 þ p3 þ g3 P 30 p1 þ p2 þ p3 þ g4 P 39 pP0

266

M. Xia et al. / European Journal of Operational Research 154 (2004) 251–270

and has a solution 2

2

3

9 p ¼4 9 5 21 

Proposition 3. When some goods are unallocated in the optimal solution, if the DeMartini prices for the unallocated items are zero, then the DeMartini price is also an O’Neill price.

3

12 6 0 7 7 g ¼6 4 0 5 0 

and

Some observations on the above two individual pricing approaches are presented in a sequence of theorems as follows. Proposition 1. O’Neill price is not unique.

Proof. Let us rearrange the indices of items so that the first set of indices is for items sold in the optimal solution (denote as A), and the second set for unsold items (denote as U). From the DeMartini model, the optimal auction revenue X X X pj ¼ p0L wj ¼ p0L wj Z ¼ j2Aþ

In the above example, any price that satisfies p1 þ p2 þ p3 ¼ 39 and g4 ¼ 0 is an OÕNeill price. Proposition 2. When all the goods are sold in the optimal allocation, it is always true that the DeMartini price is an O’Neill price. Proof. When all the goods are sold in the optimal allocation, let the DeMartini price be pL and the optimal revenue be Z  . We have, in the DeMartini procedure pj ¼ p0L wj

8j 2 Aþ

Thus X X X Z ¼ pj ¼ p0L wj ¼ p0L wj ¼ p0L 1 j2Aþ

j2Aþ

 ¼

0 pj  p0L wj

ð10Þ

j2Aþ

j 2 Aþ j 2 K n Aþ

where K is the set of all goods. Then, when j 2 Aþ , p0L wj þ gj ¼ p0L wj ¼ pj and when j 2 K n Aþ , p0L wj þ gj ¼ p0L wj þ pj  p0L wj ¼ pj . Therefore, ðpL ; g Þ satisfies p0L wj þ gj P pj

j2Aþ

In the OÕNeill model, we can still set  0 j 2 Aþ  gj ¼ 0 p j  p L w j j 2 K n Aþ Then, when j 2 Aþ , p0L wj þ gj ¼ p0L wj ¼ pj and when j 2 K n Aþ , p0L wj þ gj ¼ p0L wj þ pj  p0L wj ¼ pj . Thus ðpL ; g Þ is a solution to the OÕNeill model. And the corresponding objective value is X X X 0 p0L 1 þ g x ¼ p0L 1 ¼ pLi þ pLi ¼ pLi ¼ Z  i2A

(The last ‘‘ ¼ ’’ is due to the fact that all goods are sold.) Conversely, in the OÕNeill procedure, if we set gj

j2Aþ

8j 2 K

and

i2U

i2A

Since Z  is the optimal objective value of the OÕNeill model, ðpL ; g Þ is an optimal solution to the OÕNeill model. Therefore, pL is also an OÕNeill price.  Proposition 4. When some of the goods are not allocated in the optimal solution, if the DeMartini price for the unallocated items is positive, rearrange the indices of items as before and write the DeMartini price as   pA pL ¼ pU where pU is the price vector corresponding to the unallocated items. We claim   pA pR ¼ 0

p0L 1 þ g0 x ¼ p0L 1 þ 0 ¼ Z 

is an O’Neill price.

so the DeMartini price, pL , is also an OÕNeill price. 

Proof. From the DeMartini model, the optimal revenue

M. Xia et al. / European Journal of Operational Research 154 (2004) 251–270

Z ¼

X

pj ¼

j2Aþ

X

p0L wj ¼ p0L

j2Aþ

X

wj

j2Aþ

For the OÕNeill model, if we let  0 j 2 Aþ  gj ¼ 0 p j  pR w j j 2 K n A þ þ

then when j 2 A ,  0  0 pA pA p0R wj þ gj ¼ wj ¼ wj 0 pU (The last equality is due to the fact that unallocated items do not appear in any winning bundle.) But  0 pA wj ¼ p0L wj ¼ pj pU Therefore p0R wj þ gj ¼ pj , 8j 2 Aþ . When j 2 Kn Aþ , p0R wj þ gj ¼ p0R wj þ pj  p0R wj ¼ pj So pR ¼



pA 0



is a feasible solution to the OÕNeill model. Moreover, the corresponding objective value is  0 X X pA 0 0 0 pR 1 þ g x ¼ pR 1 ¼ 1¼ pLi ¼ p0L wj 0 i2A j2Aþ ¼ Z

for this unallocated item is 1. Denote this itemÕs price as p1 ðp1 > 0Þ. Denote the OÕNeill price for the rearranged indices as   p1 pR ¼ p2 (Note p2 is a vector of (m  1).) Because it is an OÕNeill price, there exists g ; such that ðpR ; g Þ is an optimal solution for the OÕNeill problem. Next we show that there is another feasible solution that yields a smaller objective. Denote N ¼ fj : w1j ¼ 1g as the set of bids that include the unallocated item. jN j ¼ k. Rearrange the indices of bids so the bids of N are the first k bids. Partition g as 

gN  gN



where gN is the vector consisting of gj Õs corre sponding to N and gN the rest of gj Õs. N  A , because the unallocated item cannot be part of a winning bid. Let 0 1 p1 gN ¼ gN þ @ . . . A p1 g ¼



gN  gN



and

Thus,   pA pR ¼ 0 is an OÕNeill price.

267

 pR0 ¼ 

Proposition 5. In the O’Neill procedure, the prices corresponding to the unallocated items are always zero. Proof. Suppose there exists an optimal solution to the OÕNeill problem in which not all prices for unallocated items are zero. Then there must exist at least one unallocated item, with a positive price. Rearrange the indices of the items, so the index

0 p2



then ðpR0 ; g Þ is also a feasible solution to the OÕNeill problem, because when:  0 0 þ 0  wj þ gj j 2 A ; pR0 wj þ gj ¼ p2  0 p1 ¼ wj þ gj P pj p2 (The last ‘‘ ¼ ’’ is due to the fact item 1 is unallocated thus not in any winning bundle; the last ‘‘ P ’’ is due to the feasibility of ðpR ; g Þ.) When

268

j2

M. Xia et al. / European Journal of Operational Research 154 (2004) 251–270

N ; p0R0 wj

þ

gj

 ¼  ¼

0 p2 p1 p2

0 0

wj þ gj þ p1 wj þ gj P pj

Since a price is associated with each unique good, one can use the price to compute a value for any bundle even if it had not received any bids in the last round. The disadvantages of individual prices are:

and when j 2 K n ðN [ Aþ Þ; p0R0 wj þ gj ¼

 

¼

0 p2 p1 p2

0 0

wj þ gj wj þ gj P pj

(The last ‘‘ ¼ ’’ is because item 1 does not appear in any bundle in j 2 K n ðN [ Aþ Þ.) At the same time, the objective value corresponding to ðpR0 ; g Þ is  0  0 0 0 1 þ g0 x ¼ 1 þ g0 x ¼ Z  p2 p2 (The first ‘‘ ¼ ’’ is because g differs from g g only in losing bids; the last ‘‘ ¼ ’’ is due to the feasibility of ðpR0 ; g Þ.) Thus, ðpR0 ; g Þ yields a smaller objective value than ðpR ; g Þ. This conflicts our assumption that ðpR ; g Þ is an optimal solution to the OÕNeill problem.  From the above propositions, we can see that for single-unit single-sided auctions such as WDP2, the DeMartini price is, in some sense, a special case of the OÕNeill price. 3.3. Advantages and disadvantages of individual prices for goods The advantages of having individual prices for goods are as follows: 1. The model does not place any restriction on the number of bundles a bidder can bid on or acquire. Thus a bidder is free to place bids on two disjoint bundles, and possibly win both, if there is no complementary existing between the goods in the two bundles. 2. Individual prices give guidance to bid formation and evaluation for new entrants and losing bidders. If the assumption of WDP2 is adopted, some possible bundles do not receive a bid.

1. Since a bundleÕs value is the sum of all components, complementarities between components are not accounted for in the individual price approach; 2. In general, incentive compatibility cannot be achieved due to the existing duality gap for IP; 3. If there are additional constraints in the bids, such as an XOR constraint, they can be modeled as dummy components so that the model is still in the form of WDP2. In addition, individual pricing approaches will assign a price for each constraint. Thus the prices corresponding to the additional fictional components will become part of the bundle price for those bundles involved in the constraint, e.g. an XOR constraint. Therefore, a bundleÕs price may not be the sum of all prices for the real components.

4. Summary and future research In this paper, we reviewed several pricing approaches for combinatorial auctions with indivisibilities. Prices using a bundle approach all clear the market, while aiming to achieve incentive compatibility. Such prices, when historical, help potential bidders evaluate the bundles before bidding when complementarities between products are unclear. Yet, because prices for losing bundles do not give as much information as those for winning bundles, the value of the prices for bundles without winning bids is greatly reduced. All of the bundle pricing schemes are either an application or an extension of the Groves–Clarke Mechanism. However, in order to achieve incentive compatibility, either more assumptions have to be made, or only approximations can be attained. The rest of the approaches cannot claim incentive compatibility at all. Moreover, the assumptions that one has to bid on all possible

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bundles and that win at most one are too restricted in a combinatorial auction. The above disadvantages, along with the fact that the bundle approach does not reflect the value of individual components within a bundle, make the individual component pricing approaches appealing. Under individual pricing, two approaches are examined. One (DeMartini et al., 1999) specifically aimed to provide imputed prices for combinatorial auctions while the other (OÕNeill et al., 2001) is a general IP method which one can apply to the combinatorial pricing problem. Papers using these approaches usually do not have any restrictions on how many bundles a bidder can bid on, and how many she can win. In some special cases, e.g. the single-item, single-sided case, we have found that prices from one approach (DeMartini et al., 1999) are a subset of the other approach (OÕNeill et al., 2001). Since the OÕNeill approach almost always gives non-unique prices, we feel the DeMartini approach is better. Nonetheless, the individual pricing approaches have disadvantages also. Among them is the inability to gain insight into item complementarities. However, due to the potential duality gap of IP, incentive compatibility cannot always be achieved in this approach. Moreover, if there are additional constraints in the model, for example, one that reflects the exclusive-OR relationship between two bids, the total price of a bundle will not be the sum of all individual item prices. Future research is needed to meet the shortcomings of these pricing approaches. Bundle pricing approaches could be modified to not only generate incentive, compatible prices but also provide estimates or bounds for individual prices. Furthermore, losing bundle bids could provide information, perhaps helping to provide tighter bounds on individual, item prices. Individual pricing approaches suffer several drawbacks that should be further investigated. Conceptually, of course, estimated item prices from a prior auction may have little predictive value for future auctions. Factors that diminish or enhance their predictive value could be investigated. The inability of individual pricing models to capture complementarities is also a severe restriction. Non-convex programming methods for

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working with duality gaps might prove useful for providing estimates and/or bounds on complementarities.

References Andersson, A., Tenhunen, M., Ygge, F., 2000. Integer programming for combinatorial auction winner determination. In: Proceedings of the Fourth International Conference on Multi-agent Systems (ICMAS), Boston, MA. Ba, S., Stallaert, J., Whinston, A.B., 2001. Optimal investment in knowledge within a firm using a market mechanism. Management Science 47 (9), 1203–1219. Bikhchandani, S., Ostroy, J., 2001. The package assignment model. Working Paper, UCLA. Bikhchandani, S., de Vries, S., Schummer, J., Vohra, R., 2001. Linear programming and Vickrey auctions. Working Paper. Clarke, E., 1971. Multipart pricing of public goods. Public Choice 11, 17–33. DeMartini, C., Kwasnica, A., Ledyard, J., Porter, D., 1999. A new and improved design for multi-object iterative auctions. Working Paper, California Institute of Technology. Fan, M., Stallaert, J., Whinston, A.B., 2000. Decentralized mechanism design for supply chain organizations using an auction market. Working Paper, CREC, UT Austin, forthcoming in Information Systems Research. Fujishima, Y., Leyton-Brown, K., Shoham, Y., 1999. Taming the computational complexity of combinatorial auctions: optimal and approximate approaches. In: Proceedings of the IJCAIÕ99, Stockholm, pp. 548–553. Garey, M.R., Johnson, D.S., 1979. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, CA. Gonen, R., Lehmann, D., 2000. Optimal solutions for multiunit combinatorial auctions: branch and bound heuristics. In: Proceedings of the ACM Conference on Electronic Commerce (ECÕ00), Minneapolis, MN. Groves, T., 1973. Incentives in teams. Econometrica 41, 617– 631. Hoffman, K., Padberg, M.W., 1993. Solving airline crew scheduling problems by branch and cut. Management Science 39, 657–672. Leonard, H.B., 1983. Elicitation of honest preferences for the assignment of individuals to positions. Journal of Political Economy 91 (3), 461–479. Leyton-Brown, K., Tennenholtz, M., Shoham, Y., 2000. An algorithm for multi-unit combinatorial auctions. In: Proceedings of the National Conference on Artificial Intelligence (AAAI), Austin, TX. Lin, E.Y.H, 1998. A bibliographical survey on some wellknown non-standard knapsack problems. INFOR 36 (4), 274–317. MacKie-Mason, J., Varian, H., 1994. Generalized Vickrey auctions. Technical Report. Department of Economics, University of Michigan.

270

M. Xia et al. / European Journal of Operational Research 154 (2004) 251–270

Mas-Collel, A., Whinston, M., Green, J., 1995. Microeconomic Theory. Oxford University Press, Oxford. Moore, J., Whinston, A.B., Wu, J., 1972. Resource allocation in a non-convex economy. Review of Economic Studies 39 (3), 303–323. Nemhauser, G.L., Wolsey, L.A., 1988. Integer and Combinatorial Optimization. John Wiley & Sons. OÕNeill, R., Sotkiewicz, P., Hobbs, B., Rothkopf, M., Stewart Jr., W., 2001. Equilibrium prices in markets with nonconvexities. Working Paper. Parkes, D., 2001. An iterative generalized Vickrey auction: strategy-proofness without complete revelation. AAAI Spring Symposium on Game Theoretic and Decision Theoretic Agents. Rothkopf, M.H., Pekec, A., Harstad, R.M., 1998. Computationally manageable combinational auctions. Management Science 44 (8), 1131–1147. Sandholm, T., 1999. An algorithm for optimal winner determination in combinatorial auctions. In: Proceedings of the IJCAIÕ99, Stockholm, pp. 542–547.

Sandholm, T., Suri, S., Gilpin A., Levine, D., 2001. CABOB: A fast optimal algorithm for combinatorial auctions. In: Proceedings of the Seventeenth International Joint Conference on Artificial Intelligence (IJCAI), Seattle, Washington, USA. Shapley, L.S., 1962. Complements and substitutes in the optimal assignment problem. Naval Research Logistics Quarterly 9, 45–48. Vickrey, W., 1961. Counterspeculation, auctions and competitive sealed tenders. Journal of Finance, 8–37. Wurman, P., Wellman, M., 1999. Equilibrium prices in bundle auctions. Working Paper. Department of Computer Science, The University of Michigan. Wurman, P., Wellman, M., 2000. AkBA: a progressive, anonymous-price combinatorial auction. Second ACM Conference on Electronic Commerce (ECÕ00), Minneapolis, pp. 21–29. Xia, M., Stallaert, J., Whinston, A.B., 2001. Solving combinatorial double auctions. CREC Working Paper, University of Texas at Austin.