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An Iterative Generalized Vickrey Auction: Strategy-Proofness Without Complete Revelation

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An Iterative Generalized Vickrey Auction: Strategy-Proofness Without Complete Revelation From: AAAI Technical Report SS-01-03. Compilation copyright © 2001, AAAI (www.aaai.org). All rights reserved. An Iterative Generalized Vickrey Auction: Strategy-Proofness without Complete Revelation David C. Parkes Department of Computerand Information Science, University of Pennsylvania, 200 South 33rd Street, Philadelphia, PA19104 dparkes @unagi.cis.upenn.edu Abstract sible combinationsof items. This can be very difficult for bounded-rational agents, with limited or costly computation Thegeneralized Vickreyauction (GVA)is a strategy-proof combinatorialauction, in whichtruthful biddingis the opti- and hard valuation problems. The complete information re- mal strategy for an agent. In this paperwe address a fun- quirementarises becauseof the single-shot nature of the auc- damentalproblem with the GVA,which is that it requires tion: every agent submits a sealed-bid to the auctioneer from agents to computeand reveal their values for all combina- which the allocation is computed.Without an option to ask tions of items. Thiscan be verydifficult for bounded-rational an agent for more information a mechanismcan only com- agents with limited or costly computation.We propose an ex- pute the efficient allocation in every probleminstance with perimentaldesign for an iterative combinatorialauction. We complete information up-front about agents’ valuation func- havea theoretical proof that the the auction implementsthe tions. outcomeof the Vickreyauction in special cases, and initial In comparison, an iterative GVAcan terminate with the experimentalresults supportour conjecturethat the auction implementsthe outcomeof the Vickreyauction in all cases. same outcome (allocation and payments) but with less in- Theauction has better informationproperties than the sealed- formation revelation. Aniterative auction can elicit infor- bid GVA:in each roundagents mustonly bid for the set of mation from agents dynamically, as required to determine bundlesthat maximizetheir utility givencurrent ask prices, the efficient allocation. Terminatingwith the Vickrey out- whichdoes not require agents to computetheir exact values comeprovides an iterative procedure with muchof the same for everybundle. strategy-proofness as the sealed-bid GVA.The design of an iterative GVAis stated as an important open problem in the auction design literature (Bikchandani & Ostroy 1998; Introduction Milgrom2000). However, iterative Vickrey auctions are A central problem in distributed open systems with self- only known for special cases (Kelso & Crawford 1982; interested agents, each with private information, is to com- Demange,Gale, & Sotomayor 1986; Gul & Stacchetti 2000; pute optimal system-widesolutions to a global problemthat Ausubel2000), with restrictions on agent valuation func- involves all the agents. Market-based methods have been tions. proposed to solve problemsof this type; e.g., distributed Wepropose an experimental design for an ascending- scheduling (Weilmanet al. 2000), supply-chain (Walsh et price combinatorial auction. Wehave a theoretical proof al. 2000), course registration (Graves et al. 1993), airport that the auction terminates with the Vickreyoutcome in spe- scheduling (Rassenti et aL 1982), and air-conditioning for cial cases, and initial experimentalresults support our con- an office building (Huberman& Clearwater 1995). jecture that the auction implements the Vickrey outcome The Generalized Vickrey Auction (GVA) (Varian in all cases. The auction extends/Bundle (Parkes 1999a; MacKie-Mason1995) has received wide attention in the Parkes & Ungar 2000b) and builds on the ideas introduced literature, e.g. Wellmanet al. (2000) and Hunsberger in "proxy and adjust" (Parkes & Ungar 2000c). The goal Grosz (2000), because of its incentive properties; truthful to implement the Vickrey outcomewith best-response agent bidding is the optimal strategy for an agent in the GVA.The strategies, i.e. if agents bid in each roundfor the bundle(s) GVAsolves the combinatorial allocation problem (CAP) that maximizetheir utility. (Rothkopf et al. 1998; de Vries & Vohra 2000), in which The auction has two distinct phases. The first phase is there is a discrete set of items to allocate to agents that have used to determine the efficient (value-maximizing)alloca- values for combinations of items, e.g. "I only want A if I tion, while the second-phase is used to determine Vickrey also get B". The goal in the CAPis to computethe alloca- payments. However,this transition from Phase I to Phase tion that maximizestotal value. Expressingcontingencies of II is hidden from participants. The auction design is quite this kind is important in manyapplications; for example,to novel: bid for a compatiblepair of take-off and landing slots in an airport scheduling problem. 1. First, we adjust agents’ paymentsafter the auction termi- In this paper we address a fundamentalshortcoming of the nates, so that agents do not pay their final bid prices. This GVA,which is that it requires complete information from allows the implementationof non-equilibrium solutions, agents, i.e. every agent must reveal its value for all pos- which is important because the GVAoutcome can not al- 78 ways be supported in equilibrium (Bikchandani & Ostroy principle states that in the design of mechanismswe can re- 1998). strict attention to "direct revelation" mechanismswhich re- 2. Second, we introduce "dummyagents" during Phase II quest complete information about an agent’s valuation func- to force continued bidding from winning agents and to tion. However,the revelation principle assumes unlimited elicit enough information to adjust agent paymentsto- computationalresources, both for agents in submitting val- wards Vickrey payments. uation functions, and for the auctioneer in computing the outcome of a mechanism(Ledyard 1989). Our methodologyis to construct an auction to implement Beforewe continue, it is interesting (and perhaps surpris- a primal-dual algorithm for the GVAwith best-response ing) to note that there is one sense in whichagent bounded- agent strategies. Best-responsebids from agents provide in- rationality can help in mechanismdesign. Perhaps we can formation about the complementary-slacknessconditions in design mechanismsthat cannot be manipulated unless an a primal-dual formulation, and can be used to adjust towards agent can solve an NP-hard computational problem. Nisan an optimal solution. Bertsekas (1990) makesa similar con- & Ronen(2000) follow this line of reasoning, and describe nection in his AUCTIONalgorithm for the simple assignment the concept of "feasible truthfulness" in mechanismdesign. problem. The CombinatorialAllocation Problem Outline In this paper we focus on mechanismsfor the Combinatorial In the next section we provide a brief discussion of the con- Allocation Problem(CAP), whichis a quite general resource sequences of agent bounded-rationality on mechanismde- allocation problem in which agents have values for combi- sign. Wealso present an application of a combinatorial nations of items. Manyresource allocation problems can auction with hard valuation problems, the packagedelivery be modeled as a CAP, including the job-shop scheduling, problem. We define the GVA,and show by example how it course registration and airport scheduling problems men- is possible to computethe outcomeof the auction without tioned in the introduction. complete value information. Wediscuss two methods to re- Formally,let 6 represent a set of discrete items to allocate duce information revelation in the GVA:bidding programs, to Z agents. Each agent has a value vi(S) >_ for bundles and dynamic methods. We then describe our new auction ,_q C G of items. The CAPis to allocate items to maximize procedure, and illustrate the mechanismon a couple of ex- total value over all agents: amples. Wepresent initial experimentalresults that confirm (CAP) our conjecture that the auction implementsthe outcomeof E vl (Si) the GVA.Finally, wepresent a brief theoretical justification for the design of the auction. Weclose with conclusions and s.t. s~nsj=O,vi,jel, i#j suggestions for future work. It is well knownthat the CAPis NP-hard,it is equivalent to the maximumweighted set packing problem (Rothkopf et MechanismDesign with Bounded-Rational al. 1998). Agents However,this straightforward statement of complexity hides another very important computational problemin the Wewill begin with a brief discussion of the implications CAP, which arises in the case that a hard computational of agent bounded-rationality on mechanismdesign. Nisan problem must be solved by each agent to computeits value & Ronen (2000) provide a good introduction to related for each bundle. This is quite likely in manyapplications, concerns that follow from the computational requirements for examplewhenever an agent must solve a local optimiza- placed on the mechanisminfrastructure. tion problemto computeits value for different outcomes.As Onevery important consideration is the amountof value an example, consider the packagedelivery problem. information that is required by a mechanism.Single-shot mechanisms,such as the GVA,can fail whenit is too expen- Example: The package delivery problem. The package sive for agents to computeand reveal their value for every delivery problemis an exampleof a distributed optimization combination of items. Wedescribe an example below in problem in which agents have hard valuation problems. which agents have hard valuation problems. In comparison, The problem can be modeled as a combinatorial alloca- an iterative mechanismcan
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