Expressing Preferences with Price-Vector Agents in Combinatorial Auctions: A Brief Summary

Transcription

1 Expressing Preferences with Price-Vector Agents in Combinatorial Auctions: A Brief Summary Robert W. Day PhD: Applied Mathematics, University of Maryland, College Park 1 Problem Discussion and Motivation A combinatorial auction is one in which preferences are expressed for combinations or bundles of items, rather than just for individual items. The importance of combinatorial auctions has grown tremendously in the past 15 years, which have seen an explosion in the use of auction mechanisms in both government allocation problems and businessto-business commerce. Chapter 2 of the dissertation outlines the many applications of combinatorial auction research (as do Anandalingam et al., 2005), including shipping-lane and procurement auctions in the private sector, as well as auctions for spectrum licenses by the FCC and airport landing slots by the FAA in the public sector. In each of these environments, the expression of aggregate information allows the bidders to realize synergies (e.g., economies-of-scale or owning complementary items) while the auction mechanism stimulates competition, aiding the seller of items with more competitive prices. There are, however, two major computational difficulties limiting the implementation of the combinatorial auction paradigm: (1) the underlying problem of finding an efficient allocation (called the winner-determination problem) is N P-hard, and (2) the amount of information necessary to describe a bidder s preferences for all bundles grows exponentially in the number of items. For each of the real-world environments for which combinatorial auctions have been used (or are being heavily considered for use, as in the airport landing-slot case) the number of items being auctioned is large enough for problem (2) to have a limiting effect. The exponential growth of bundles makes it difficult for bidders to express preferences for every bundle that may be of interest. Clearly, a bidding language is necessary to ease the burden of preference expression, helping bidders by placing bids on many bundles simultaneously. The development of an easier to use language of preference expression can increase the efficiency of the auction, both by achieving better outcomes, and by reducing the time spent by the bidders communicating their preferences. The development of a bidding language to aid the bidder with problem (2) should be coupled with the development of a practically implementable solution technique, simultaneously addressing the auctioneer s difficulty, (1) above. Viable solutions to this problem involve both restricted versions of the N P-hard winner-determination problem that remain N P-hard but are smaller in scale, and restricted versions that are polynomially solvable, but then are more 1

2 restrictive in the types of preferences that can be expressed. (New examples of each of these techniques are provided in this dissertation.) In addition to the bidding language problem, there is not yet a consensus as to the best method for setting prices given the bids that are submitted in a combinatorial auction. Using some form of sensible pricing, with payments less than the actual amount bid but large enough to beat out competitors, promises to encourage bidding with potential benefits to both the buyer (who can bid more aggressively with less risk) and the seller (who does best when the items go to those who value them the most, see Ausubel and Cramton, 1999). Currently, the concept of bidder-pareto-optimal pricing within the core offers the best candidate for sensible prices, but these are difficult to compute. This dissertation presents a major improvement on the computation of such prices. 2 Contribution of the Dissertation In this dissertation we investigate the problem of compact preference expression in a combinatorial auction. Focusing on problem (2) above, we introduce two new formats for writing down preferences, both of which are compact (growing quadratically in the number of items) and provide the opportunity for a positive incremental offer (i.e., the format gives the bidder the opportunity to offer more money for a bundle whenever a new item is added, regardless of which bundle.) The primary question addressed in this dissertation is: how can simple compact representations of preference (price-vectors) be combined to form more elaborate statements of preference? This question follows somewhat naturally from the economic literature on unit-demand bidders, which is well studied. Indeed, our own Theorems 3.1 and 3.5 generalize the results of Demange et al. (1986), establishing the gross substitutes property and demonstrating how to compute minimal Walrasian equilibria, from the case of unit-demand bidders to the case in which each bidder is represented by multiple unit-demand agents. The first noteworthy methodological contribution of this dissertation is therefore an extension of existing economic theory to include a more rich environment of preference expression for use in a combinatorial auction. This development leads naturally to the first of our new bidding formats, called bid tables, which has a polynomially solvable winner-determination problem, thus overcoming problem (1) above. With this polynomial solvability the format is necessarily restrictive, however, not allowing for a wide range of preference expression. For some applications, though, this restricted form of preference expression is appropriate if used as an initial revelation mechanism for individual item prices. We therefore incorporate bid tables into a three-stage hybrid auction, improving upon existing hybrid techniques with better individual item prices, revelation of bundle synergy information, and minimization of the information available for certain forms of strategic manipulation. Our second methodological contribution is a recognition of the limitations of bid tables and the introduction of a 2

3 Figure 1: New Formats for Preference Expression 10am 11am 12pm 1pm A Bid Table F1 F2 F am 11am 12pm 1pm A Matrix Bid technique to overcome this limitation. Price-vector agents are a natural idea for expressing multi-unit preferences; a bidding airline might express its decreasing marginal utility for landing-slots in the bid table of Figure 1. They are assured that at most one landing-slot will be awarded for their most profitable flight F1, only one will be awarded for their second most profitable flight F2, etc., with the rules of a bid table auction allowing the auctioneer to accept at most one entry per row (item) and at most one entry per column (agent). Unfortunately, although bid table preferences are natural and compact, we show that they cannot express even simple complementary preferences. If the airline would like to express that they would not pay for the most profitable single flight F1 unless both F2 and F3 are also accommodated, they cannot express this preference within a bid table but may instead use the accompanying matrix bid in Figure 1, where the matrix bid does not allow the acceptance of a bid from one agent (column) unless the (higher-ranked) agent immediately to its left receives an item from a higher (ranked) row. Thus our second format, called matrix bidding, allows for a much richer expression of preferences. The introduction of agents that are ranked and a bidder defined ranking of the items in the auction allows matrix bids to maintain the compactness and some of the ease of use properties from bid tables, while simultaneously allowing for a relative explosion in the range of possible preference expression (and hence computational complexity). Indeed, for matrix bidding the winner-determination problem remains N P-hard, but we show that the format is an assignment problem with side constraints, which is easier to solve in practice than the general winner-determination problem discussed in the literature. This innovation of agents using simple cooperation to express preferences and the accompanying computational techniques (specialized cover inequalities and perturbations) offer a new paradigm for rapidly achieving solutions to combinatorial auction problems with a compact representation of preferences. Relative to existing techniques, we show that if joined with logical connectives matrix bidding is equally as expressive as the best available compact logical bidding language (see Boutilier, 2002). Each individual matrix bid can express preferences that are difficult to read and cumbersome in that language, allowing us to interpret matrix bidding as an innovative reorganization of bid information into an intuitive system. In addition to the work on bid submission formats, our investigation uncovered other developments in the study 3

4 of combinatorial auctions in general. Most importantly, we introduce a new constraint generation technique for achieving bidder-pareto-optimal core outcomes in any combinatorial auction setting. This new technique provides a way to rapidly achieve final prices in a sealed-bid auction that satisfy a state-of-the-art notion of economic stability in the combinatorial auction environment. Stated differently, we provide evidence that we have found a better way to find the best prices for a combinatorial auction. This improved technique for computing prices in any combinatorial auction environment represents the third major methodological contribution of the dissertation. The technique is an application of the general theory of constraint generation from the OR literature, but as always, this paradigm may not be applied until the application is appropriately modeled. In this case, after clearly defining the competitive market situation and the desired core outcome, the key insight was the formulation of the separation problem at any intermediate point in price space. This formulation rests heavily on the economics of coalition formation and our introduction of a special version of the winner-determination problem in which each winning bidder is compensated her opportunity cost. Putting these economically inspired pieces together into an alternating sequence of LPs and IPs that achieve bidder-pareto-optimal core outcomes, we achieve an algorithm that is an order of magnitude faster than the best existing technique. 3 Impact of the Dissertation on Industry Over the course of the dissertation we demonstrate the ease of expression using the bid-table and matrix bid formats with concrete examples of preference expression for shipping-lane, procurement, spectrum license, and airport landing-slot auctions, among others, and in several cases demonstrate the relative difficulty of an equivalent expression in other languages from the literature. Chapter 4 in particular shows how to tailor our overall auction design using bid tables to the airport landing-slot environment, demonstrating how the techniques presented may be applied to a significant real world problem currently under investigation. Additionally, the technique presented in Chapter 5 for computing bidder-pareto-optimal core outcomes offers a significant improvement on the existing computational techniques available in the literature (Hoffman et al., 2005; Wurman et al., 2004). Ausubel et al. (2005) provide a framework through which these outcomes may be implemented, based on their wide range of experience running and designing actual electricity and spectrum auction implementations internationally via Market Design Inc. Given that these authors are among the foremost authorities on governmental allocation auctions, our technique for improving the speed and selection of these outcomes promises to have lasting significance. Given the significance of a compact format of preference expression, our matrix bidding format offers a new bidding language that with some experience can be read much more intuitively than the existing best compact language, and with no loss of expressivity. Since the majority of proposed and existing combinatorial auction applications involve 4

5 repeated play, high-stakes, and ongoing participation, sometimes occurring over weeks or months, a new format which is easier to use after some initial learning promises to more smoothly expedite auction participation, leading to more successful applications of the combinatorial auction paradigm. The research in the dissertation regarding bid tables also lays good theoretical groundwork for auction applications currently taking place or under development. As an indicator of this relevance, we note that a recent mock auction held by NEXTOR and the FAA to investigate the feasibility of a landing-slot auction, made use of a bid table auction with side constraints as its model for airline preferences. Our results on bid table auctions clearly define the strengths and limitations of such approaches, and the computational techniques we introduce may in fact provide a better set of linear feedback prices (minimal Walrasian equilibrium prices in our case) than the ones used in that experiment (and in the Ausubel et al., 2005, clock-proxy auction in general). Within the dissertation, we also demonstrate the ease of computation of these linear price signals, an interesting application of primal-dual theory in a new economic context. As further evidence of the dissertation s impact on industry, we have been informed that researchers at Decisive Analytics Inc. have implemented a version of our Core Constraint Generation algorithm for the computation of bidder-pareto-optimal core outcomes. This organization provides combinatorial auction design and analysis services to the U.S. government and plays an integral role in the FCC spectrum license auctions, as well as contributing research to the proposed FAA auctions. Thus, with further study and independent verification of its robustness, there is a strong possibility that our algorithm for price determination will have significant impact in one of the highest profile combinatorial auctions in the world. References Anandalingam, G., R. Day, and S. Raghavan The landscape of electronic market design. Management Science 51(3). Ausubel, L. and P. Cramton The optimality of being efficient. Working paper, University of Maryland. Ausubel, L., P. Cramton, and P. Milgrom The clock-proxy auction: A practical combinatorial auction design. In P. Cramton, Y. Shoham, and R. Steinberg, editors, Combinatorial Auctions, chapter 5. MIT Press, forthcoming. Boutilier, C Solving concisely expressed combinatorial auction problems. In AAAI/IAAI, pages Demange, G., D. Gale, and M. Sotomayor Multi-item auctions. J. of Political Economy 94(4), Hoffman, K., D. Menon, S. van den Heever, and T. Wilson Observations and near-direct implementations of the ascending proxy auction. In P. Cramton, Y. Shoham, and R. Steinberg, editors, Combinatorial Auctions, chapter 17. MIT Press, forthcoming. Wurman, P., J. Zhong, and G. Cai Computing price trajectories in combinatorial auctions with proxy bidding. Working paper, Computer Science Department, North Carolina State University. 5