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A double-sided multiunit combinatorial auction for substitutes: Theory and algorithms

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Abstract

Combinatorial exchanges have existed for a long time in securities markets. In these auctions buyers and sellers can place orders on combinations, or bundles of different securities. These orders are conjunctive: they are matched only if the full bundle is available. On business-to-business (B2B) exchanges, buyers have the choice to receive the same product with different attributes; for instance the same product can be produced by different sellers. A buyer indicates his preference by submitting a disjunctive order, where he specifies the quantity he wants of each particular good and what limit price he is willing to pay for each good, thus providing a subjective valuation of each attribute. Only the goods with the best prices will be traded. This article considers a doubled-sided multiunit combinatorial auction for substitutes, that is, a uniform price auction where buyers and sellers place both types of orders, conjunctive (AND orders) and disjunctive (XOR orders). We show that linear competitive prices exist. We also propose an algorithm to clear the market, which is particularly efficient when the number of traders is large, and the goods are divisible.

Introduction

To quote de Vries and Vohra (2003) “because of complementarities or substitution effects between different assets, bidders have preferences not just for particular items, but for sets of items, sometimes called bundles”, or packages, or combinations. Combinatorial auctions are becoming more and more popular with the Internet revolution, because, as noted by Klein (1997), one effect of the Internet is the ability to conduct complex auctions.

This paper focuses on the description of a particular type of “double-sided, multiunit, combinatorial, for substitutes” auction, or DMCS auction.1 Roughly speaking,2 any order in our DMCS auction consists of “OR of XOR of AND bids”: a combination order is a conjunctive, or “AND” bid on complement items; the “OR” operator implements the (possible) multiunit feature of any order, which can be partially matched, whereas the disjunctive, or XOR part, allows to choose across substitutes. The choice between butter and cannons can be expressed as “butter XOR (2 wheels AND 1 gun)”, expressing the fact that a cannon is made of 3 parts: 2 wheels and 1 gun. Pekec and Rothkopf (2003) observe that such a bidding language is fully expressive. We believe that any bill of materials can be implemented in such a fashion, and that this language is a good compromise between flexibility and practicability (see Wohl (1997) for a discussion). In a manufacturing environment, it is also quite natural to express the “utility function” of the goods needed in terms of limit orders: the reservation or limit price(s), for instance can be equated to the shadow prices, which most production planning systems calculate anyway. The exchange then calculates a clearing (or equilibrium) price for all the goods, at which demand equals supply. Our equilibrium concept is a generalization of the concept of competitive equilibrium to combinatorial auctions, which we already developed for DMC auctions.3

There are four main originalities in our design. The first one is the “commoditization” of identical goods, which entails the law of one price: the clearing price offered by the exchange for the first wheel has to be identical to the price of the second wheel (we call such prices uniform or nondiscriminatory). This is the reason why we call our auction “multiunit”, as opposed to most other designs, which are mostly variants of the DC, or CS auctions. As the author observed while working for the options exchange Eurex, it is much easier to implement a continuous-time combinatorial exchange (which, in terms of pricing algorithm, resembles a DC auction), than the “opening-of-the day”, or batch clearing system (i.e., a DMC auction). The second originality is the fact that, due to commoditization, combination vectors contain all possible integer values, and not only {1, −1, 0}4: in our example, a cannon needs 2 wheels. Third, clearing prices are integers. Finally, clearing prices should not only be uniform for the same good, but also linear or consistent. In other terms, the clearing price of a combination should be the weighted sum of the clearing price of its parts. If it were not the case, an astute traded would for instance see that the combination good he bought is underpriced, “unbundle” it and try to sell (in another round of trading) the primitive goods at the current equilibrium prices; in other terms, dismantle the cannon and resell the wheels and the gun. This would not only result in an unfair profit for him, but also defeat one of the major functions of an auction, which is, to provide information on what should be the “fair” price of a good.

Our main contributions are to show that in a DMCS auction with perfectly divisible goods: (i) linear competitive prices exist, (ii) there is an efficient algorithm to calculate them. In other terms, we prove the existence of a linear competitive equilibrium on a market with XOR and combination orders. In order to show that, we had to overcome two main difficulties.

First, we needed to develop technical tools to handle the clearing of combinatorial goods. In our DCMS auction the market clearing equations are a system of linear equations. This is to be compared to the traditional market clearing equations in economics, which are a collection of monovariate equations, namely demand equals supply for each good.

Second, limit orders express discontinuous (and thus non-convex) preferences. This makes the mathematical treatment of DM auctions slightly more complicated, since demand and supply functions become discontinuous. The complication is compounded in the case of DCMS auctions. The disjunctive constraint (of a XOR order) makes demand even more discontinuous.

The limitations of our present design have been widely discussed with practitioners. They are: lack of built-in incentive compatibility, possible absence of competitive prices when indivisibilities are significant, and the necessity of a tie-breaking mechanism. We briefly address these issues below.

Incentive compatibility is in general difficult to implement. For one-sided auctions, Rothkopf and Harstad (1994) already pointed out the “limitations of traditional game-theoretic approaches to auction design”. Pekec and Rothkopf (2003) mention that, for combinatorial auctions, “any serious strategic analysis […] of optimal bidding strategies is impossible without mastery of the determination of auction winners”; in particular the Vickrey–Clarke–Groves, (Vickrey, 1961, Clarke, 1971, Groves, 1973) scheme is “impractical”, among other reasons because it is based on a simple model of beliefs, and that it does not prevent shill bidding. The difficulties of implementing incentive compatibility are compounded in the more general setting of double-sided auctions. Such a mechanism exists for DC auctions for public goods, see Ba et al. (2001). Gul and Stachetti (1999) establish that the Vickrey mechanism cannot be implemented in an English auction when buyers are strategic and values are interdependent. In a DC auction, a Vickrey-type mechanism may not be budget-balanced (Parkes et al., 2001), i.e., the Vickrey payments may absorb more than the surplus. Pekec and Rothkopf (2003) point out the greater “cooperative” (as opposed to collusive) behavior of bidding in combinatorial auctions, which mitigates the incentive compatibility effect. As for us, we do not claim that it is either possible or impossible in our DMCS auction to build a mechanism to ensure incentive compatibility, but think that the winner determination problem should be addressed first. An incentive compatible mechanism should anyway, like in McAfee (1992) for DM auctions, first calculate uniform prices (provided they exist) and then offer side-payments.

When goods are indivisible, competitive prices may not exist (see e.g., Gul and Stachetti, 1999, Bikhchandani and Ostroy, 1997). Nondiscriminatory linear prices may not be competitive, i.e., some orders remain unfulfilled although their limit price is strictly better than the clearing price. The function of nondiscriminatory linear prices is thus mostly informative. We argue that this function of price discovery is important, especially since auctions are usually a repetitive business. We also contend that a high auction volume has a “convexifying effect”, which reduces the number of unmatched orders at clearing and renders prices almost competitive. In the context of simple auctions, Dierker, 1971, Broome, 1972 investigated this convexifying effect of a large number of traders, which explains why “various approximation algorithms are likely to produce solutions which are not far from optimal” (Pekec and Rothkopf, 2003). Satterthwaite and Williams (1993) prove that the indivisible equilibrium converges to the perfectly divisible equilibrium when the number of traders increases in a DM auction. This robustness effect was also observed by Isaac and James (1998). Parkes et al. (2001) gives a systematic survey on the effect of indivisibilities in combinatorial auctions. Pekec and Rothkopf (2003) note that linear (or additive) prices are the most popular approximation method. Interestingly, another positive effect of a large number of traders, besides making prices competitive, is to make uniform price double auctions incentive compatible (Wilson, 1985).

Pekec and Rothkopf (2003) note that ties in an auction are undesirable. In a DMCS auction, ties across substitutes turn out to be as delicate an issue as ties across participants. We manage ties by restricting clearing prices to be discrete, as mentioned in Milgrom (2003, p. 317).

So far, academic research has concentrated on less general auctions than ours, mostly single-sided combinatorial auctions, that is, auctions with one seller, or auctions with only one buyer (the procurement auction). Pinker et al. (2003) in their survey are “not aware of any true double online auctions beyond the financial and commodity markets”, with the exception of www.chemconnect.com for the chemical industry. Pekec and Rothkopf (2003) point out the paucity of “documentation and public information on details of combinatorial auction design and implementations in the E-business arena”. It is well-known that the main criticisms of procurement auctions is that they are focussed on “squeezing the suppliers” (Rothkopf and Harstad, 1994). However, as Milgrom (2000) notes, “if the mechanism is designed to extract all the entire surplus from the sellers, it will be difficult to attract sellers to the auction site”. Designs for double-sided B2B auctions include among others Beil and Wein (2003) for a MS auction, and Ba et al. (2001) for a DC auctions for public goods.

The paper is structured as follows. In Section 2, we describe the auction model. For readability, we first present a complete specification of the model and then describe some features of the model. This latter part can be skipped at first reading. In Section 3, we present our main results. In the appendix, we prove two lemmas. We present an example on our web site (Schellhorn, 2006), as well as a discussion of the complexity of the algorithm. Our DMCS auction can be used to generalize Wohl’s (1997) price-contingent mechanism on securities markets in a particular direction. Since that example is special to securities markets, we discuss it in another paper (Schellhorn, 2006).

Section snippets

Auction model

Notation 1

We denote by vt the transpose of a column vector v. We write e for the vector of ones. We write ei for the unit vector in the direction i, i.e., eji=1 if i = j, and zero otherwise. The dimension of these vectors is clear from the context. We will define most variables and functions on the buyer side only. For the seller side, we refer the reader to the Matlab program available in the online supplement (Schellhorn, 2006). Buyer-specific vectors are identified by a superscript B and seller-specific

Main result

We prove the existence of a competitive equilibrium. The algorithm is based on the existence proof, which is constructive. Since the proof is quite detailed, we motivate its derivation by first describing it at an intuitive geometrical level.

Conclusion

We showed in this article that nondiscriminatory consistent (linear) prices exist, and can be used to clear a DMCS auction. When goods are perfectly divisible it is difficult to argue against this equilibrium concept, especially if an incentive-compatible mechanism is superposed to the clearing mechanism: this equilibrium is not only efficient, nondiscriminatory (and therefore not illegal) but, if participants bid their true reservation value, reveals the information about good value. We also

Acknowledgements

We would like to thank Christine Lang, John Ledyard, Joseph Ostroy, Avi Wohl and participants in seminars at UCLA, the University of Lausanne, and Claremont Graduate University. All errors are ours.

References (26)

  • J. Broome

    Approximate equilibrium in economies with indivisible commodities

    Journal of Economic Theory

    (1972)
  • S. Ba et al.

    Optimal investment in knowledge within a firm using a market mechanism

    Management Science

    (2001)
  • D. Beil et al.

    An inverse-optimization-based auction mechanism to support a multiattribute RFQ process

    Management Science

    (2003)
  • C. Berge

    Espaces topologiques et fonctions multivoques

    (1959)
  • Bikhchandani, S., Ostroy, J., 1997. The package assignment model. Working paper,...
  • E. Dierker

    Equilibrium analysis of exchange economies with indivisible commodities

    Econometrica

    (1971)
  • E.H. Clarke

    Multipart pricing of public goods

    Public Choice

    (1971)
  • S. de Vries et al.

    Combinatorial auctions: A survey

    INFORMS Journal on Computing

    (2003)
  • T. Groves

    Incentives in teams

    Econometrica

    (1973)
  • F. Gul et al.

    Walrasian equilibrium with gross substitutes

    Journal of Economic Theory

    (1999)
  • R.M. Isaac et al.

    Robustness of the incentive compatible combinatorial auction

    (1998)
  • S. Klein

    Introduction to electronic auctions

    Electronic Markets

    (1997)
  • P. McAfee

    A dominant strategy double auction

    Journal of Economic Theory

    (1992)
  • Cited by (0)

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