Auctions with bidder-determined allowable combinations

Abstract

Combinatorial auctions are desirable as they enable bidders to express the synergistic values of a group of assets and thus may lead to better allocations. Compared to other types of auctions, they keep bidders from being exposed to risks (of receiving only parts of combinations that would be valuable to them) or from being overly cautious (in order to minimize such risks). However, computation time needed to determine the set of optimal winning combinations in a general combinatorial auction may grow exponentially as the auction size increases, and this is sometimes given as a reason for not using combinatorial auctions. To determine the winning allocation in a reasonable time, a bid taker might try to limit the kinds of allowable combinations, but bidders may disagree on what combinations should be allowed, and this may make limiting the allowable combinations politically infeasible.

This paper proposes and tests successfully a new approach to managing the computational complexity of determining the set of winning combinations. The main idea is to let bidders themselves determine and prioritize the allowable combinations. Using bidder-determined combinations has two nice properties. First, by delegating the decision on what is biddable to the bidders who know what combinations are important to them, the bid taker is able to be (and appear) fair. Second, since bidders know their economics and have the incentive to get important combinations included, bidder prioritization of combinations will tend to assure that the most economically-important combinations are included in determining the winning set of bids if the bid taker is not able to consider all of the combinations submitted by bidders. The proposed auction process is useful in situations, such as government auctions, in which the bid taker is reluctant to limit the allowable combinations.

Introduction

Game-theoretic auction theory developed first with models of single, isolated auctions (McAfee and McMillan, 1987). In the past few years, the academic literature has begun to pay attention to the design of auctions for selling large numbers of items with interrelated values. The FCC spectrum auctions, which involved the sale of thousands of licenses worth billions of dollars, helped motivate this attention (see Cramton, 1995, Cramton, 1997; Cramton and Schwartz, 2000; McAfee and McMillan, 1996; McMillan, 1994), and the rise of auctions in electronic commerce is now also a motivator (Huhns and Vidal, 1999).

One of the issues facing those designing auctions for multiple items with interrelated values is deciding whether to allow bids on combinations of items, and if so, how to decide which combinations should be biddable. This presents a dilemma. Bidders may strongly prefer to bid on combinations when their values are for the combinations rather than for the individual items, and bid takers may prefer an auction in which bidders can make such bids. However, the mathematical problem of finding the revenue-maximizing set of bids is NP-complete when bids on all possible combinations are allowed. Rothkopf et al. (1998) suggested two approaches to managing the computational complexity. When there is sufficient time, the auction's fairness can be maintained by letting all bidders as well as the bid taker have a chance to find solutions to the integer-programming problem and then selecting the solution that gives the highest total revenue. When there is not sufficient time or when the economically significant combinations are generally known, then combinations may be able to be limited so that the problem is guaranteed to be computationally manageable. They discussed and gave a variety of examples of the kinds of combinations that are guaranteed to be computationally manageable and of the kinds that are not.¹ They suggested that the bid taker might get the assistance of the bidders in identifying economically important combinations as the US Department of the Interior used to get nominations from oil companies of which offshore oil tracts to offer for sale.

While bidders may sometimes agree on the economically important combinations, there may be at times essential disagreements. Some bidders may have synergies between some kinds of items and others may have completely different kinds of synergies. In addition, bidders who have no synergies may fear competition from those who do have them; they oppose the use of bids on combinations or push for allowing bids on combinations that would be difficult to handle computationally in conjunction with their competitors' sincerely desired ones. In addition, for strategic reasons bidders may be reluctant to reveal truthfully the combinations of greatest interest to them.² Bid takers who need to maintain the appearance of fairness may, thus, face difficulty in determining the most important combinations and a dilemma even if they do know which are economically significant. This paper discusses a possible solution to this dilemma––an auction process in which the bidders themselves determine in the auction the allowable combinations.

The auction process we propose is a one-time standard sealed bid auction in which bidders pay the amount of their accepted bids. Thus, it is an alternative to the simultaneous progressive multi-round auction used by the FCC. This has the advantage of limiting the opportunities for signaling and tacit collusion among bidders, but does not have the information sharing advantage (in the face of affiliated values) of the progressive format. As discussed in the final section below, however, it is possible to adapt our approach to simultaneous progressive auctions.

While auction theory has rigorous models of the incentives of bidders in single, isolated auctions, relatively little is known about the incentives of bidders in practical auctions involving multiple assets with interrelated values. We do not propose to deal with that issue in this paper. Rather, we deal with the issue of computational limitations on auctions that allow bids on combinations. Allowing such bids is desirable when there are significant synergies, but not when there are no synergies. (The econometric work on British auctions of bus route contracts (Cantillon and Pesendorfer, 2001) suggests that even small synergies are economically significant.) Some think that combating tacit collusion and its associated inefficiencies requires a one-time auction such as we envision. (See Robinson, 1985; Graham and Marshall, 1987; Rothkopf and Harstad, 1994; Cramton and Schwartz, 2000.) Others believe that simultaneous progressive auctions are preferable and allow better coordination by bidders unaware of each other's preferences, especially if bids on combinations are not allowed (McAfee and McMillan, 1996; Ausubel and Milgrom, 2002). We present our approach in the context of a one-time auction, but, as noted, it is adaptable to the progressive format.

The basic idea of our approach is to let the bidders decide on and prioritize the combinations. Bidders will submit with their bids on individual items a prioritized list of combinations upon which they wish to bid along with bids on those combinations. The bid taker will first solve the revenue-maximization problem using the individual bids but no combination bids. This calculation involves no computational difficulties. The bid taker will then attempt to solve the revenue maximization problem using, in addition to the individual bids, each bidder's first priority combination bid. The bid taker will have a time limit for this and subsequent calculations. If the time limit is reached before this calculation is completed, the bid taker will return to the last of the previously completed calculations and announce it as the results of the auction. If this calculation is completed before the time limit is reached, the bid taker will add each bidder's next priority bid to the bids to be considered and repeat the calculation. This process will continue until either the time limit is reached or all of the combination bids of all of the bidders have been considered.³ Fig. 1 shows a flow diagram of this process.

This paper describes our approach in detail and reports and discusses a set of tests of it. After initial tests with smaller problems, we put together a test problem roughly patterned on the FCC spectrum auctions involving the sale of 153 assets to three bidders with differing interrelated values for them. First, we prepared our own set of bids for the bidders. The bidders had 15, 22, and 25 combinations on their priority lists. It turned out that computation was not a serious issue. The entire process, which involved the solution of 42 integer-programming problems, was completed using 3.9 seconds of CPU time on a 550MHz Pentium machine using CPLEX 7.1. In our view, a potential drawback of this test was that we made up the bids, and we knew every bidder's values when we did so. To remedy this deficiency, we enlisted the help of several economists who served as advisors to bidders in the FCC auctions. We asked each to bid for one of the bidders, knowing that bidder's values and only qualitative information about the situation of the other bidders.

In the following sections, we provide a detailed description of our algorithm and experimental results. Section 2 surveys the related work on combinatorial auctions, and discusses the current algorithms for winner determination in combinatorial auctions. Section 3 describes our algorithm in detail. Section 4 describes the test problem and experimental results. Section 5 concludes the paper. Appendix A shows the letter sent to the test bidders.