Pricing combinatorial auctions

Transcription

1 European Journal of Operational Research 154 (24) O.R. Applications Pricing combinatorial auctions Mu ia a, *, Gary J. Koehler b,1, AndrewB. Whinston c,2 a Department of Business Administration, College of Commerce and Business Administration, University of Illinois at Urbana-Champaign, Champaign, IL 6182, USA b Department of Decision and Information Sciences, Warrington College of Business Administration, 351 STZ, P.O. Box , 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, T 78712, USA Received 9 October 21; accepted 5 September 22 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 lowvalue 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. Ó 22 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 * Corresponding author. Tel.: ; fax: addresses: mxia@uiuc.edu (M. ia), koehler@ufl.edu (G.J. Koehler), abw@uts.cc.utexas.edu (A.B. Whinston). 1 Tel.: ; fax: Tel.: ; fax: 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 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 /$ - see front matter Ó 22 Elsevier B.V. All rights reserved. doi:1.116/s (2)678-1

2 252 M. ia et al. / European Journal of Operational Research 154 (24) 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 lowvalue 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 allowcombinatorial 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 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 howto 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 4 However, in general it is not always true that a revenue maximizing allocation is efficient. They are often competing objectives. 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.

3 M. ia et al. / European Journal of Operational Research 154 (24) (Fan et al., 2). 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 The winner determination problem 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: WDP1 Z ¼ max s:t: n i¼1 2 m 1 j¼1 p ij x ij w j x ij 6 1 ð1þ i;j x ij 6 1 j ¼ 1;...; 2 m 1 ð2þ i x ij ¼ ; 1 i ¼ 1;...; n and j ¼ 1;...; 2 m 1 x ij is a binary variable, indicating whether bundle j is awarded to bidder i. p ij is the bid price for bundle j from bidder i. w j is a vector of size m where ðw j Þ k is one if good k is part of bundle w j and zero otherwise. These bundles range over all 2 m 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 Z ¼ max x j s:t: n j¼1 n j¼1 p j x j w j x j 6 1 x j ¼ ; 1 j ¼ 1;...; n is a binary variable that indicates whether bundle j gets traded. p j is the bid price for bundle j. w j 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 x j 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

4 254 M. ia et al. / European Journal of Operational Research 154 (24) 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 OR bids. OR bids are bids sharing an OR 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 OR 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., 21; Fujishima et al., 1999; Leyton-Brown et al., 2). 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. (2) showusing a commercial IP solver (CPLE) one can also solve the WDP reasonably fast, while Gonen and Lehmann (2) apply the well-known branch-and-bound procedure to solve the WDP as an IP problem. A natural question is: howdo the AI approaches compare to the optimization approaches? A recent paper (ia et al., 21) 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 Incentive compatibility After the WDP is solved, an important issue is howto 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 howto 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: UðB; qþ ¼lðBÞ 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.

5 M. ia et al. / European Journal of Operational Research 154 (24) 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 p 1 and the second highest bid as p 2. Taking the highest losing bid as the price in effect redistributes part of the trade surplus, p 1 p 2, 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 fewassumptions 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 for any winning bundle j, the price is set at: q j ¼ p j ðz Z j Þ, where Z is the optimal auction revenue and Z j 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 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 2 m 1 bundles. For WDP2, it is unlikely that all 2 m 1 bundles receive bids, thus in order to compute prices for each bundle, one has to have prices for individual goods. Therefore, howto 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.

6 256 M. ia et al. / European Journal of Operational Research 154 (24) 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 (21) and Parkes (21) take this approach. Ba et al. (21) 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 2 m 1 bundles could be very burdensome for a bidder who is interested in only one or a fewbundles, 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, Y and Z, if bidder 1 is interested in getting only bundle ð ; Y Þ for $1 and it is free to dispose of unnecessary goods, then the valuation for bundle ð ; Y ; ZÞ might also be set at $1 and the valuations of ð 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 (OR) bids, 7 the same feature can be achieved in combinatorial auctions by adding a dummy good corresponding to the OR 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 (2) in the AkBA mechanism. 7 When bids from the same bidder are OR bids, the bidder is only interested in getting at most one bundle out of the set.

7 M. ia et al. / European Journal of Operational Research 154 (24) V P I ¼ max s:t: m m i¼1 j¼1 m j¼1 m i¼1 x ij 6 1 x ij 6 1 x ij ¼ ; 1 p ij x ij i ¼ 1;...; m j ¼ 1;...; m i; j ¼ 1;...; m Section 2.5, we provide comparisons based on the reviewand our results. In the following, denote the price for bundle j as q j ; the winning bundle set B þ ; the losing bundle set B ; and the bidder set A with winning bidder set A þ and losing bidder set A. Let ZðA n iþ be the optimal revenue without bidder i Wurman and Wellman (1999) 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 min u i þ m v j i¼1 j¼1 s:t: u i þ v j P p ij i; j ¼ 1;...; m u i ; v j P i; j ¼ 1;...; m Due to degeneracy of the primal problem, however, the shadow prices associated with positions, i.e. the dual variables v j, are not unique. LeonardÕs paper identifies a set of shadowprices, which maximizes P m j¼1 v j, 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 shadowprice for position j is the difference between the optimal value to all individuals of all positions plus another position of type j, V Pþj I, and the value of the current positions, VI P: q j ¼ V Pþj I VI P. 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., 21) 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 , we first review these approaches in a unified framework, followed by discussions and, in some cases, newresults that help to elucidate the relationship between approaches. In Wurman and Wellman (1999) (henceforth WW) discuss the bundle pricing problem based on the winner determination model WDP1 as listed earlier. We first reviewthe proposed algorithm 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 ¼f/ i ji 2 A g. Consider the assignment problem, illustrated in Fig. 1, which matches the bundle set G ¼ U [ B þ with A ¼ A þ [ A. Each bundle in B þ is assigned to the corresponding winning bidder in A þ ; 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.

8 258 M. ia et al. / European Journal of Operational Research 154 (24) The following two problems are solved: LP min Q ¼ argmin q g fq gjg2b þ g g2g s:t: s i þ q g P p i;g 8i 2 A and g 2 G s i ; q g P 8i 2 A and g 2 G s i þ q g ¼ Z i2a g2g and LP max Q ¼ argmin fq gjg2b þ g g2g q g s:t: s i þ q g P p i;g 8i 2 A and g 2 G s i ; q g P 8i 2 A and g 2 G s i þ q g ¼ Z i2a g2g Z is the optimal objective value of WDP1; p i;g is iõs bid on bundle g 2 G; q g is the price for bundle g and s i is bidder iõs maximum surplus. Both Q and Q are vectors of jb þ j. They are marketclearing prices of winning bundles for the pseudo-assignment problem. (LP min ) is a direct application of LeonardÕs (1983) model, while (LP max ) gives an upper bound of the marketclearing prices. WW proves that any linear combination of the two prices, kq þð1 kþq ð 6 k 6 1Þ, is also market-clearing. 4. For each remaining unassigned bundle g, the price p g is calculated by q g ¼ max i2a ðp i;g s i Þ. With such a price, no bidder would be better off if she chose to buy this bundle, because 8i 2 A; s i P p i;g q g. 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 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 p g, then if another bidder j bids the same price p g, would he/she be able to get the bundle? It 8 We thank an anonymous referee for pointing this out.

9 M. ia et al. / European Journal of Operational Research 154 (24) 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 Bikhchandani and Ostroy (21) Bikhchandani and Ostroy (21) (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 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. 8S A : ZðAÞ ZðAnSÞP ðzðaþ ZðAniÞÞ 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 s:t: p ij x ij i2a j2b w j x ij 6 1 8i 2 A ð3þ j2b x ij 6 d l 8j 2 B; 8i 2 A ð4þ l:ði;jþ2l d l 6 1 ð5þ l2c x ij ; d l ¼ ; 1 8j 2 B and l 2 C 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, d l 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 ij 6 d l 6 d l 6 1 i;j 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 v i, is the Vickrey discount for each bidder, i.e., v i ¼ ZðAÞ ZðAniÞ 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 q j ¼ p ij v i where p ij is iõs bid for bundle j. For each losing bidder i, the payment is zero, as ZðAÞ ¼ZðA n iþ. We can use q j 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. (21) 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:

10 26 M. ia et al. / European Journal of Operational Research 154 (24) BOWDP2 ZðAÞ ¼ max s:t: p ij x ij r2p j2r i2a x r ij 6 z r 8i 2 A; 8r 2 P ð6þ j2r x r ij 6 z r 8j 2 r; 8r 2 P ð7þ i2a z r 6 1 ð8þ r2p w j x r ij 6 z r 8i 2 A ð9þ r2p j2r x r ij P z r P 8j 2 r; 8i 2 A 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 z r ¼ 1 if partition r is selected, zero otherwise. Set x r ij ¼ 1 to mean that r is selected and bundle j is assigned to bidder i in partition r. Let v i be the dual variable associated with (9). Bikhchandani et al. (21) prove BOWDP2 has an integral optimal solution. Under the condition that buyers are substitutes, the dual variable for (9), v i ¼ ZðAÞ ZðAniÞ, 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 ¼fq j ; j 2 Bj9i 2 A, s.t. x ij 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: x ij ¼ 1, p ij q j P p ij q j 8j 6¼ j, j 2 B.InWWÕspseudo- assignment problem, the newbundle set G ¼ B þ [ U. As shown by WW, the prices for the dummy bundles are zero, i.e., qð/ i Þ¼, 8i 2 A. Since p i/i ¼ 8i 2 A, i 2 A, p ig q g ¼, 8g 2 U, i 2 A. Therefore, 8j 2 B þ and x ij ¼ 1, g 2 U p ij q j ¼ v i P ¼ p ig q g. 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 { ; 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 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 qðy Þ¼p 1 ðy Þ ðzðaþ ZðAn1ÞÞ ¼ 1 ð16 16Þ ¼1 qðw Þ¼p 2 ðw Þ ðzðaþ ZðAn2ÞÞ ¼ 6 ð16 15Þ ¼5 The pseudo-assignment problem of WW, is assigning G ¼fðY Þ; ðw Þ; / 3 g to the three bidders. Solving ðlp min Þ and ðlp max Þ, one can get Q ¼ fq fy g ; q fw g g¼f6; 4g and Q ¼fq fy g ; q fw g g¼f1; 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 Q V, it is apparent that Q V 6¼ kq þð1 kþq 8 6 k 6 1. That is, the Vickrey Table 1 A combinatorial auction example Bidders Items Y W Y W YW YW

11 M. ia et al. / European Journal of Operational Research 154 (24) price, although supporting the allocation, is not a linear combination of the two WW prices Parkes (21) Parkes (21) (PARK) proposes another bundle pricing scheme as part of an iterative ascending bundle auction model 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 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 pointed out in the paper, the value of an approximate incentive compatible outcome is also uncertain Ba et al. (21) 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. (21) 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: MP max s:t: n j¼1 p j x j max fw i;jx j gþ min fw i;jx j g 6 j¼1;...;n j¼1;...;n 8i ¼ 1;...; m x j ¼f; 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, w j ; represents the content of a bundle: if w k;j ¼ 1, the kth good is to be bought in bundle j;if w k;j ¼ 1, the kth good is to be sold in bundle j; otherwise the value is 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 howto 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.

12 262 M. ia et al. / European Journal of Operational Research 154 (24) 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 flowproblem. The dual variables correspond to prices for knowledge bundles. Moreover, they are incentive compatible. It is shown that it is an application of the Groves Clarke mechanism (Groves, 1973 and Clarke, 1971) 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. (21) (BO) Parkes (21) (PARK) Ba et al. (21) (BA) Bundle auction model Optimization formulation Private goods; direct revelation combinatorial auctions ðwdp1þz ¼ max s:t: i2a p ij x ij j2b w j x ij 6 1 8i 2 A j2b x ij 6 1 8j 2 B i2a x ij ¼ ; 1 8i 2 A; 8j 2 B Private goods; direct revelation combinatorial auctions Same as WDP1 Private goods; iterative ascending combinatorial auctions Buyers bid on all bundles iteratively; auctioneer needs to solve WDP1 for provisional winners Public goods; combinatorial double auctions max j2b p j x j s:t: max w jx j þ min w jx j 68i j2b j2b x j ¼ ;1 8j 2 B Assumptions Need to solve the WDP first? Transformation to get prices Prices incentive compatible? Prices for losing bundles? Relationship with Groves Clarke Mechanism (GC) Bidder bids on every bundle; at most one bid is awarded to each bidder Buyer substitute in addition to all assumptions in WW Same as WW Yes Yes Yes Yes Constructing pseudo-assignment problem by keeping only the winning bundles and adding dummy bundles for the losing bidders; then getting prices for the assignment problem Adding auxiliary variables to make the WDP unimodular so the LP relaxation can be solved to get prices No transformation. Leave burden of optimization with bidders No Yes In general, no; in special cases, yes Yes No, but such No prices can be easily derived None An implementation of the GC under special assumptions (buyer substitute) Approximation of GC; implements GC only in special cases Unique provider (UP) 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 Yes No Application of GC

13 M. ia et al. / European Journal of Operational Research 154 (24) 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 and Y, qð Þ¼5, but qð [ Y Þ¼25, then there is a very strong complementarity between A and B. However, if qð [ Y Þ¼5 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 j 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 p i on bundle j, then 8j, s.t. w j < w j iõs utility on j must be updated to be at least p ij, i.e. p ij maxðp ij ; p ij 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 this information is very limited. Take WW for example. The losing bundleõs price can be anything greater than the p g 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 ¼f1;...; 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 howmany bids a bidder can place.

14 264 M. ia et al. / European Journal of Operational Research 154 (24) DeMartini et al. (1999) 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. 1 Step 1: Given p j and the values A þ, A, and x j resulting from the solution to WDP2, find the lowest z that bounds the discrepancy: min z;p;g z s:t: p w j ¼ p j j 2 A þ p w j þ g j P p j g j 6 z z; p; g j P If z ¼, go to step 3. j 2 A Let K be the empty set and J ¼fjjj 2 A and g j ¼ z g.ifj ¼ A, go to step 3. Otherwise go to step 2. Step 2: Given p j and the values A þ, A, and x j resulting from the solution to WDP2, and J, solve min z;p;g z s:t: p w j ¼ p j j 2 A þ p w j þ g j P p j p w j þ g j ¼ p j g j 6 z z; p; g j P j 2 J j 2 A n J If z ¼, go to step 3. Let J J [fj 2 A n J : z ¼ gj g.ifj ¼ A, go to step 3. Otherwise go to step 2 again. Step 3: Given b j and the values A þ, A,andx j resulting from the solution to WDP2, and J from step 1, solve (K ¼ U) max y;p y s:t: p w j ¼ p j j 2 A þ p w j þ g j P p j p 1 i P y p 1 i ¼ p i y; p P i 2 M i 2 K j 2 A Let K fi : p i ¼ y g and M M n K. IfM is empty, stop. Otherwise go to step 3 again. We present a simple example to illustrate the process. The bids are: B6 7 C ðw 1 ; p 1 Þ¼@ 4 1 5; 3A; B6 7 C ðw 2 ; p 2 Þ¼@ 4 1 5; 3A; B6 7 C ðw 3 ; p 3 Þ¼@ 4 5; 3A; B6 7 C ðw 4 ; p 4 Þ¼@ 4 1 5; 39A 1 Winner determination gives 2 3 x ¼ DeMartini Prices are then determined as follows. 1 We rewrite some of the original notation using a vector format. A þ ¼f4g; A ¼f1; 2; 3g and M ¼f1; 2; 3g