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

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. TheCombinatorial Allocation 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 solve realistic problemswith less tion problem: items G represent packages for delivery, with information, eliciting only the minimalinformation required pick-up and delivery requirements(e.g. locations, times, pri- to computean efficient allocation. ority, etc.); agents 27 represent individual delivery compa- In addition to solving problemswithout complete infor- nies, each with its owntransportation resources, e.g. vans, airplanes, trucks, etc. mationrevelation it is also importantthat agents can participate without computingexact values for all bundles. It is An agent’s value, vi(S), for bundle of packages S C_ not enoughfor an auction to require less information from is computedas the payment it receives for delivering the agents if the agents must still computevalues for all bundles package minus its cost. Assumethat the paymentis a lin- to provide that information. ear sum of payments p(x) for packages x E S, and denote This focus on informationrevelation conflicts with a cen- cost C~(S). tral result in mechanismdesign, the "revelation principle" = p(x) - c (s) (Green & Laffont 1977; Myerson 1981). The revelation xES

79 The global objective in the package delivery problem, This interpretation is more consistent with the method to captured by this CAPformulation, is to determine an alloca- computeVickrey paymentsin the iterative GVAthat we in- tion that maximizesthe total value across all agents. This is troduce later in the paper. equivalent to allocating packages at minimalcost and drop- It is quite straightforwardto showthat the optimalstrategy ping packagesthat cannot be delivered profitably. of an agent in the GVAis to bid its true valuation function, It is reasonableto expectthe cost to an agent for the pick- 9i = vi, whateverthe bids of other agents. up and delivery of a particular package to depend on fac- tors such as: prior commitments(e.g. prescheduled pick- Theorem1. (Groves 1973) Truthful bidding is a dominant ups and deliveries), van capacities, delivery locations and strategy in the GVA. times. For example,the additional cost to deliver a package is muchless if a van is already scheduled to makea pick-up Furthermore, because V* > (V_i)* the discount is al- at the destination location. Formally,the cost for a partic- waysnon-negative and agents pay less than their bids, while ular set of packages (i.e. a commitmentto perform a set the discount is not so large that the adjusted price is ever of deliveries) might perhaps be computedas a muhi-vehicle negative, because (V-i)* > V* - ~i(S~) implies that _,/3 * capacity-constrained traveling salesperson problem, which Avick(/) i( i). is NP-hard. Computational Problems This example should give a sense in which value information can be hard to provide in distributed optimization The GVAhas unreasonable computational properties in problems. manyinteresting problems. Problem1. The GVAis intractable for the auctioneer. The GVA: Incentive-Compatibility with Once the auctioneer has received bids from each agent Complete Revelation it must solve multiple CAPinstances, once to compute In this section we define the generalized Vickrey auction V*, and once to compute(V-i)* for every agent i in the (GVA),which is an incentive-compatible and efficient mech- optimal allocation. anism for the combinatorial allocation problem. Weobserve Problem2. The GVAis intractable for agents with hard that the GVArequires complete information revelation from valuation problems. The GVArequires complete infor- agents, and demonstratethat it is in fact often possible to mation revelation from each agent. As soon as an agent computethe outcomeof the GVA(payments and allocation) submits approximateor missing information: (i) there with less information from each agent. someprobability that the agent will do worse- in terms Wedescribe two approaches to reduce information reve- of its value for the bundleit receives and/or the price it lation but still computethe outcomeof the GVA:(1) allow pays -than if it had revealed completeand accurate infor- agents to submit bidding programs,which the auctioneer can mation; and (ii) the allocation implementedin the GVA query to determine values for particular bundles; (2) involve mightnot be efficient. agents dynamically during the execution of an algorithm to The first problemhas received someattention, but only computethe Vickrey outcome, and request additional infor- in the context of sealed-bid auctions, and without address- mation on-the-fly as required. Westate a numberof draw- ing the second problem. In general, introducing approxi- backs with bidding programs,and suggest that iterative auc- mation algorithms for winner-determination can break the tions are a useful class of dynamicmethods. incentive-compatibility of the GVA(Kfir-Dahav, Monderer, & Tennenholtz 1998; Nisan & Ronen 2000). In compari- The Generalized Vickrey Auction son, the problem of information revelation in the GVAhas Each agent i E 17 submits a (possibly untruthful) valua- received little attention. Oneexception is the discussion of tion function, ~3i, to the auctioneer. The auctioneer solves bidding programsin Nisan (2000). the CAPwith these reported values, computing allocation S* = (S~’,... ,S~) with value V*. This is the allocation Solving the GVAwithout Complete Information implementedby the auction. The auctioneer also solves CAPwithout each agent in In this section we demonstratehow it is possible to compute turn, computingthe best allocation (S_~)* without agent and verify the optimality of a solution to the GVAwithout for value (V-i)*. The Vickrey paymentto agent i is com- complete information about agents’ values. Weignore for the momentthe question of howto elicit the required infor- puted as: mation, and simply demonstrate that we can computesolu- pviok(i)= (v_,)* tions with incomplete information. js~i Example1. Single-itemauction with 3 agents, and values vl = In words, an agent pays the marginal negative effect that 16, v2 = 10, va = 4. The Vickreyoutcome is to sell the item its participation has on the (reported) value of the other to agent 1 for agent 2’s value, i.e. for 10. Instead of information agents. Equivalently, the Vickrey payment can be com- {vl, v2, va} it is sufficient to know{vl > 10, v2 = 10, va < 10} puted as a discount Avick(i) from its bid price, ti(S*), i.e. to computethis outcome. Pvick (i) ---- Vi (S~) - Avick (i), for Vickreydiscount: Example2. Considerthe simple combinatorialallocation prob- Avick(i ) = V* -- (V-i)* lem instance in Table1, with items A, B and agents 1, 2, 3. The

8O A B AB agent for every bundle must be at least considered to com- Agent 1 0 a b pute the solution to a CAPinstance. Moreusefully, the agent Agent 2 10 0 10 might provide a program that can give approximate value Agent 3 0 0 15 information to the auctioneer, or that allows richer query- response modes. Table 1: Agentvalues in Example2. DynamicMethods. An alternative approach is to "open up" the algorithm for computing the outcome of the GVA, values of agent 1 for item B and bundleAB are stated as a _< b and involve agents dynamically in the computational pro- andb _< 15, but otherwiseleft undefined.Consider the following cess. cases: The algorithm might ask agents for the following types of [a < 5] In this case the GYAassigns bundleAB to agent3, with informationduring its execution: V* = 15, (V-3)* = max[10+ a, hi, so that the paymentfor ¯ Ordinal information, i.e. "whichbundle has highest value agent 3 is prick(3) 15- ( 15- max[10 + a,b]) =max[10 a, b]. It is sufficient to know{a < 5, b < 15, marx[10+ a, hi} out of S1, $2 and ,5’3 ?" to computethe outcome. ¯ Approximateinformation, i.e. "is your value for bundle [a > 5] In this case the GVAassigns item B to agent 1 anditem $1 greater than 100?" A to agent 2, with V*= 10 + a, (V-l)* = 15, and (V-2)* ¯ Best-response information, i.e. "which bundle do you 15. Thepayment for agent 1 is p,,ick(1) = a--(10+a--15) andthe paymentfor agent2 is p,,ick(2) = 10 -- (10 + a -- 15) want at prices p(S) 15 -- a. It is sufficient to know{a,b < 15} to computethe ¯ Equivalence-setinformation, i.e. "is there an item that is outcome. substitutable for A?" Notice that in all of these cases an agent can respondwithout Note that it is not necessary to computethe value of the computingits exact value for all bundles. optimal allocation S*; we only need to computethe allocation to each agent. Consider Example1. Wecan compute Iterative price-directed auctions provide a useful class of the optimal allocation (give the item to agent 1) with infor- dynamic methods to implement mechanisms. mation vl > {v2, v3}, and without knowingthe exact value First, iterative auctions can solve realistic problemswithout complete information from agents, and without agents ofvx. Also, it is not even necessary to computeV* and (V-i)* computingtheir exact values for all bundles. Consider the to compute Vickrey payments because commonterms can- classic English auction, whichis an ascending-price auction cel. In Example1, it is enoughto knowthe value of v2 to for a single item. It is sufficient to determinein each round computeagent 1 ’s Vickrey paymentbecause the value of vl whether two or more agents have value greater than the ask price; it is not necessary to knowthe exact value of every cancels: Pvick(1)---- Vx -- Avick(1)= Vl -- (Vl V2)= V2 agent for the item.1 Methods to Reduce Information Revelation Second, iterative auctions are quite transparent methods to implement the outcome of a mechanism. For example, Wewill consider two different methodologiesto reduce in- the information requested from agents in each round of the formation revelation but compute the GVAoutcome: (1) English auction is captured by a best-response to the current bidding programs, and (2) dynamicmethods. ask prices, i.e. bidding for the item while the price is less Bidding Programs. Instead of requiring agents to com- than the agent’s value. pute and reveal their valuation functions we might ask agents Third, agents can follow myopicbest-response strategies to provide a program (or oracle) that can be used by the in iterative price-directed auctions without computingexact auctioneer to computevalues for bundles on demand(Nisan values for all bundles. For example, an agent can follow a 2000). This will reduce agent computationif it is easier for best-response bidding strategy in a price-directed iterative an agent to construct the programthan it is to computeits auction with lower and upper bounds on its values for bun- explicit value for all bundles. dles. Myopicbest-response only requires that an agent bids However,this approach maynot be very useful in prac- for the bundle(s) with maximumutility (value - price) tice for a numberof reasons: (1) trust, agents might be re- each round. This utility-maximizing set of bundles can be luctant to reveal all the informationthat goes into assigning computedby refining the values on individual bundles until the utility of one or morebundles dominatesall other bun- a value to an outcome;(2) cost of constructing the program, 2 it might be costly to collect all necessaryinformation to de- dles. fine such an autonomousprogram; (3) computational bur- i In earlier work,Parkes (1999b) demonstrates that iterative auc- den, this merelyshifts computationto the auctioneer. tions can computebetter solutionsthan single-shot auctionsfor a Note, in particular, that if the only functionality provided simplemodel of agent bounded-rationality. by the bidding programis to computethe exact value, vi (S), 2Standardalgorithmic approachescan provide lower and up- for bundle ,5’ and agent i, the auctioneer must nowcompute per boundson values; e.g. anytimealgorithms such as heuristic the completethe value of every agent for all possible bundles search computelower-bounds on optimal solutions to maximiza- to compute the GVAoutcome. One can make a straightfor- tion problems,while introducing problem relaxations and solving ward information theoretic argumentthat the value of every easy special-cases can computeupper-bounds on value.

81 An Iterative Generalized Vickrey Auction also implementsVickrey payments, and is therefore an iter- In this section we describe our experimentalascending-price ative GVA: combinatorial auction. Wehave a theoretical proof that Proposition 1. The auction terminates with the outcome of the auction implements the outcomeof the GVAin special the GVAfor agents that follow a myopic best-response bid- cases, and experimental results support our conjecture that ding strategy, as the minimal bid increment ~ -40. the auction implements the Vickrey outcomein all cases. This is quite significant, because an iterative Vickreyauc- Initial experimentalresults support this conjecture. Our tion inherits muchof the strategy-proofness of the GVAbut belief in this statementalso follows fromthe theoretical un- with less information revelation from agents. derpinnings of the auction design, whichwe discuss later in Although hidden from participants, the auction has two the paper. distinct phases. PhaseI is identical to/Bundle, an ascending- The auction provably implements the outcome of the price combinatorial auction introduced in Parkes (1999a). GVAin the same cases that/Bundle with Adj us t* com- Phase I ends when/Bundleterminates, at which point the putes the Vickrey outcome. These special cases include: auctioneer stores the provisional allocation and computes initial price discounts, using methodAdj us t* introduced Theorem3. (Parkes & Ungar 2000c) The auction termi- in Parkes & Ungar (2000c). The allocation at the end nates with the outcomeof the GVAin the following special Phase I is implementedat the end of the auction. Then, the cases: assignment problem with unit-demands; with multi- auctioneer uses a simple test to determine whetherto execute ple identical items and subadditive valuation functions (i.e. Phase II, or terminate the auction immediately. decreasingreturns); with linear-additive valuation functions The purpose of Phase II is to collect enoughadditional in items. information to be able to compute Vickrey payments. At the end of Phase II, or after PhaseI if PhaseII is skipped, FromProposition 1, one might think that myopic best- the agents’ paymentsare computedas their bid prices at the response should be a dominantstrategy in the iterative auc- end of Phase I minusthe total discounts computedover both tion, because truthful bidding is a dominantstrategy in the phases. GVA.In fact manipulationremains possible (if perhaps difficult), with somestrategy other than best-response. Wewill first describe the theoretical properties of the auc- Wecan make the following statement about an iterative tion, both proven properties and conjectures. Then, we de- auction that myopically implementsthe outcomeof the GVA; scribe the auction in somedetail, and illustrate it with a couple of workedexamples. i.e. an auction that implementsthe outcomeof the GVAwith agents that follow truthful myopicbest-response strategies. Auction Properties Theorem4. (Gul & Stacchetti 2000) Truthful myopic bid- Let us first assumethat agents follow a truthful myopicbest- ding is a sequentially rational best-response to truthful my- response bidding strategy: opic bidding by other agents in an iterative auction ~4 that myopically-implements the Vickrey outcome. Definition 1. A truthful myopic best-response bidding strategy is to bid to maximizeutility in the current round, The proof follows quite directly from the incentive- taking prices as fixed. compatibility of the GVA.Basically, for any other strategy the agent selects a GVAoutcome for somenon-truthful val- Wewill later claim that the auction implementsthe Vick- uation function, which is less preferable than the GVAout- rey outcomewith truthful myopicbest-response, which jus- comefor its true valuation function. tifies this assumptionby Theorems4 and 5 (below). To make a stronger claim about strategy-proofness we The first result is that the allocation implementedin the must somehowrestrict the strategies that agents can follow. auction is efficient, at least for small enoughbid incre- In particular, it wouldbe useful to restrict agents to fol- ments. This result follows immediatelyfrom the efficiency low a (possibly untruthful) best-responsestrategy (i.e. best- of/Bundle with best-response strategies (Parkes & Ungar response for somev I ~ v): 2000b), because Phase I is identical to/Bundle and the final allocation is the allocation as computedat the end of Phase Theorem5. Truthful myopic bidding is a dominant strat- I. egy in an iterative auction ¢4 that myopically-implementsthe Vickrey outcome,if all agents are constrained to following a Theorem2. (Parkes & Ungar 2000b) The auction termi- (possibly untruthful) best-response bidding strategy. nates with an allocation within 3 mirt{l~I, Izl}~of the optimal solution, for myopicbest-response bidding strategies. Wepursued this idea in Parkes & Ungar (2000c), with the idea of "proxy bidding agents". A proxy bidding agent can sit betweena participant and the auction and makeonly where 191is the numberof items, IZl is the numberof best-response bids, based on value information provided by agents, and e is the minimalbid incrementin the auction, the participants. Crucially, participants can provide incremen- rate at which prices are increased (see below). Clearly tal information; proxy agents do not require complete in- e -4 0 the auction terminates with the optimal allocation. formation from agents. All that is required is that there is The following unprovedconjecture states that the auction enough information to compute a best-response to current

82 prices. The proxy agent checks that the information is con- unsuccessfully, i.e. in any round that the agent does not sistent over the courseof the auction. receive a bundlein the provisional allocation. Wecan make the following claim about the strategy- For every bundle S that received an unsuccessful proofness of a proxied iterative Vickreyauction: bid, the ask price in the next round is increased to max[ Pbid,~(S) q- e, Pask,i(S)], for minimalbid increment Theorem6. Truthful dynamic information revelation is a e. Theinitial ask prices are zero. sequentially rational best-response to truthful dynamic information revelation by other agents in an iterative auction Only bundles that have received unsuccessful bids are ex- .4 with best-response proxy bidding agents that myopically- plicitly priced. However,an additional consistency rule implements the GVA. states that the ask price on all bundlesis at least as high as the greatest ask price of any bundle they contain, i.e. Unfortunately,it is still not quite true that truthful incre- Pask(,S ’t) >_ Pask(S)for all S C S’. mental information revelation to proxy agents is a domi- Phase I. The termination condition at the end of Phase I is nant strategy. This is because agents do not committo a single valuation function at the start of the auction, but in- as follows: stead report incremental value information during the auc- ¯ Phase I termination. Phase I terminates when: [T1] all tion. As long as the informationis self-consistent, an agent agents submit the same bids in two consecutive rounds, can condition the values it chooses to report based on bids or [T2] all agents that bid receive a bundle. from other agents during the auction. There remains a slight This is the end of PhaseI. Let S* = (S~’,... , $7) denote "gap" in strategy-proofness that, theoretically at least, an the allocation at the end of Phase I, P* denote the auction- agent might be able to squeeze through. eer’s revenue, W*C Z denote the set of agents that receive a bundlein S*, and P~ald ---- (P~id,l’" " " ’P~id,I) denotethe Auction Description final bid price of each agent for the bundleit receives in the The auction is an ascending-price combinatorial auction in final allocation, i.e. * which agents can bid for bundles of items. The auctioneer Pbld,1----- Pbid,1(S~). increases prices on bundles as bids are received and com- Intermediate Computation. At the end of Phase I the putes a provisional allocation in each roundof the auction to auctioneer performs someintermediate computation to de- maximize revenue. termine: (i) initial discounts to each agent, and (ii) whether There are two distinct phases: Phase I, in which the fi- to enter Phase II or terminate immediately. nal allocation is determined, followed by Phase II, in which Let Ainit(i ) ~ 0 denote the initial discount, computedat final payments are determined. Intermediate computation this stage, and Aextra(i) _> 0 denote any additional discount performed at the end of Phase I determines whether or not computedduring Phase II. An agent’s final paymentat the Phase II can be safely skipped. end of the auction is its discounted bid price at the end of Both phases follow the price update rules, bidding Phase I: max[P;id,i- Airtit(i)- Aextra(i), /¯ J rules, and winner-determinationrules of iBundle(3) (Parkes Weneed the following definitions: 1999a), which is the version of/Bundle with separate ask prices for each agent throughout the auction. The termina- Definition 2. Let MAXREV(P_i)denote the second-best tion condition in PhaseII, and the additional steps performed revenue-maximizingallocation to the auctioneer at the cur- at the end of each roundin PhaseII, are new. rent ask prices; the revenue-maximizing allocation over CommonElements. First, let us describe the elements all allocations without agent i, i.e. MAXREV(P_i) that are commonto both phases: maxs Ej~i Pask,j (Sj) for feasible allocations S with Si = ¯ Bids. Agentscan place exclusive-or bids for bundles, e.g. ~). Also, let (W-i)* C (Z \ i) denote the set of agents that receive a bundlein this allocation. $1 XOR$2, to indicate than an agent wantseither all items in $1 or all items in $2 but not both S1 and $2. Each Definition 3. The dependents(~( i) of agent i are: bid is associated with a bid price, Pbld,i(S), from agent = w*\ Ui ), ifi¯ w* for bundle S. The bid price must either be within e of, ix(i) = O, otherwise. or greater than, the ask price, Pask,i(S), announced the auctioneer. Parameter e > 0 is the minimal bid in- In other words, the dependentsof agent i are agents that crement, the minimal price increase in the auction (see receive a bundle in allocation S* but do not receive a bundle below). Agents must repeat bids for bundles in the pro- in the revenue-maximizingallocation without agent i at the visional allocation, but otherwise are only committedto current ask prices. bids for a single round. ¯ Definition 4. The set ACTIVEC_ Z of agents is the set Winner-determination. The auctioneer computes a proof active agents, all agents that are the dependentof some visional allocation in each round, to maximizerevenue agent at the current ask prices; i.e. i E ACTIVE given agents’ bids. 3j i ¯ Price-update. The auctioneer maintains separate ask prices, Pask,i(S), for each agent. The prices on bundles At the end of Phase I the auctioneer computesinitial dis- to agent i are increased in any round that the agent bids counts, the sets of dependents, and determines the active

83 agents: 5. Introduce new dummyagents. First, for any agent that 5 I. Computeinitial discount has just droppedout of the auction. Second, if we detect a state of quiescencefor the active agents.6 In this case we Ainit(i ) = P* - MAXREV(P_I) use a simple rule to choose the agent to receive a dummy agent; select (1) an agent with no dummythat is not ac- for every agent i in allocation S*, or Ainit(i ) ---- 0 other- tive; or failing that (2) an active agent with no dummy; wise. failing that (3) an active agent that already has at least one 2. Computedependents c~(i) for every agent i E W*. dummyagent. Ainit(i) P~id,i then set c~(i) = O (i .e. re move all depen- The termination condition is: dents for the agent). Initialize the set of ACTIVEagents. ¯ Phase H termination. Terminate Phase II whenall active The following simple test is performed to determine 3 agents have droppedout of the auction, or whenthe set of whetheror not to enter Phase1I. ACTIVEagents is empty. ¯ Early TerminationTest. Phase II is skippedif there are no active agents. Final Price Adjustment. Finally, when the auction terminates: allocation S* as computedat the end of Phase I is If there is at least one active agent then moreinformation implemented; and agent payments is required to computeVickrey payments, and the auction continues into Phase II. pi = max[0,P~id,i -- Ainlt(i) -- Aextra(i)] Phase IL The purpose of Phase II is to computean addi- are computed. tional discount Aextra(i) > 0 to each agent with dependents. Weintroduce dummyagents to competewith agents in the Worked Examples allocation at the end of PhaseI. The auctioneer simulates the Wewill illustrate the auction on Example2, for different dummyagents, generating bids in each round. values of a and b. Wesay that an agent participates in the auction if it is either in the provisional allocation or bidding at least the Case (a = b = 3). PhaseI: S* = (0, 0, AB), P* = 13, W*= ask price for one or more bundles. Similarly, we say that {3}, P~id = (0, 0, 13). Intermediate Computation.(S-a)* an agent has droppedout of the auction if it neither in the (B, A, 0), MAXREV(P-3)= 13, (W-3)* = {1, 2}, c~(3) provisional allocation or submitting newbids. Air, it(3) = 13 -- 13 = 0. ACTIVE= 0. Skip PhaseII. Outcome: At the start of Phase II the auctioneer initializes Allocate bundleAB to agent 3 forp3 = 13 - (0 + 0) = 13. This Aextra(i) = 0 for all agents, and introduces a dummyagent is the Vickreypayment: priCk(3) = 15 -- (15 -- 13) = for any agent that droppedout of the auction in the last round Case (a = b = 10). PhaseI: S* = (A,B,O), P* = 15, W*= of PhaseI. {1, 2}, P~id = (S, 7, 0). Intermediate Computation.(S_~)* A simple method is used to construct dummyagents: (~,O, AB), MAXREV(P_~)= 15, (W-i)* = {3}, c~(1) {1,2} \ {3,1} = {2}, Ai.it(1) = 15- = O, (S- 2)* = Definition 5. The valuation function of a dummyagent for ($,O, AB), MAXREV(P-2)= 15, (W-2)* = {3}, c~(2) agent j is based on the ask prices of agent j: set v(S) {1,2} \ {3,2} = {1}, Ainit(2) = 15 -- 15 = ACTIVE = Pask,j ( S) + L for bundlesS with Pask,/(S) ~>0, and v( S) {1,2}. O for all other bundles, for somelarge constant L4 > 0. PhaseII. Introduce a dummyagent for agent 3, with values v4 = (0, 0, 15 + L) for a large L > 0. Asprices increase agent The auctioneer performsthe following additional steps at 1 drops out first, whenpl(B)> 10. At this time Aextra(2) 2 the end of each round of Phase II: becauseagent l’s bid has increased by 2 since the end of Phase I. A dummyagent is introduced for agent 1, with values v5 -- 1. Update the dependentsof agents. For example,if c~(i) (0, 10 + L, 10 + L). Finally, agent 2 dropsout whenp2(A) > 0 then compute MAXREV(P_i)and (W-s)*, restricting at whichtime Aextra(1)= 3 becauseagent 2’s bid has increased the allocation to all real agents except agent i. (i.e. with- by 3 since the endof PhaseI. out any dummyagents). Update a(i) based on W*from Outcome:Allocate item B to agent 1 for pt = 8 - (0 + 3) = the end of Phase I and (W-s)*. and item A to agent 2 for p2 = 7 - (0 + 2) = 5. Theseare the Vickreypayments: pg,,~(1) = pg,,~(2) = 10 - (20 - 15) 2. For each agent i E W*and with c~(i) ¢ 0, increment Aextra(i) by Y’]je~(i) Aincr(j) whereAincr(j ) ~" 0 is the Case (a = b = 20). Phase I and Intermediate Computation increase in bid price by agent j for bundle S~ since the is the sameas in case a = b = 10. PhaseIL Introduce a dummy previous round. agentfor agent3 with valuesv4 = (0, 0, 15 + L) for a large L > 3. If Atntt(i) + Aextra(i) >__p*, then set a(i) ~If a dummyagent already exists for this agent, then replace 4. Update the set of ACTIVEagents. with this newone. 6The precise definition of quiescenceis not too important.We 3Thisis a reinterpretationof the VickreyTest in Parkes& Ungar considerthat the auctionis in quiescenceif: (1) the sameactive (2000c). agentshave participatedin the auctionfor the past three rounds; 4Thevaluation function will satisfy "free disposal" (v(S) > and(2) all participatingactive agentshave been allocated the same v(S’) for all St C S) becauseof the price consistencymaintained (non-empty)bundle in the provisionalallocation in the past three acrossask prices. rounds,and for the sameprice.

84 Asprices increase agent 2 drops out first, whenp~(A) > 10 and 12 Aextra(J.) -- 3. Introducea dummyagent for agent 2 with value ,,--x PhaseI v5 = (10 + .L, 0, 10 + L). Finally, agent 1 enters (S-2)*, ~10 ~,...,.,,Initial Adjust pl (B) = 15 and Aextra(2) = 7. At this stage a(2) = 0 and . ------Phasell 1 is no longeractive. ~,.. x-.-.xMin CE Outcome:Allocate item B to agent 1 forpl = 8 - (0 + 3) = ";.~.,, and item A to agent 2 for p2 = 7 - (0 + 7) = 0. These are the Vickreypayments: pgva(1) = 20 - (30 - 15) = 5 and pgva(2) 10 - (30 - 20) =

Experimental Analysis In this section we describe initial experimental results, ,2 0.4 0.6 0.8 which support our conjecture that the ascending-price com- Fractionof AllocationsCorrect binatorial auction computesthe outcomeof the GVA. The auction is tested in a suite of probleminstances: prob- Figure 1: Distanceto VickreyPayments in PS1-12. lems PS 1-12 from (Parkes 1999a), and also problems De- cay, Weighted-random, Randomand Uniform in Sandholm w, 7 ¯ ,, ~--* PhaseI (1999). Each problemset defines a distribution over agents’ ¯,, .,-.,. LqitialA..djust = ", ~ PnaSl~ values for bundlesof items...... ~,,..... *’-’. Min CE We measure the distance between agent payments in the auction and GVApayments with an L2 norm, as o2[kv:...... ’" [~i(Pi --Pgva(i)) 2] 1/2. For a particular bid increment we computethe average distance to Vickrey paymentsover the %.! ---., 0 04 0 08 instances in which the auction terminates with the optimal ’~ractlon’ofAIIoc~ons Co~rect 0.2 0.4 0.6 0.8 Fractionof AllocationsCorrect allocation. Asthe bid incrementgets small this fraction ap- (a) Uniform. proaches 100%,and therefore the proportion of trials over (b) Decay. which we measure the distance to Vickrey payments ap- proaches8 100%. Figure 1 plots the distance betweenVickrey paymentsand auction paymentsagainst the "correctness" of the auction, I...... i.e. the fraction of instances in whichthe allocation at the end of Phase I is the efficient allocation. As we reduce the minimal bid increment correctness increases and we move to the right in the plots. The correspondingallocative ef- #.75 0’.s o.h5o19 o.~5 o.2 0.4 o.6 o.8 ficiency increases from around 90%at 23%correctness, to Fr.q,*.flnn of AIInnAfinn~~nrrAP.t Fractionof AllocationsCorrect almost 100%at correctness of 65%and above. (c) Random. (d) WeightedRandom Weplot the distance to GVApayments at the end of Phase I, after the initial price discounts at the end of PhaseI, and Figure 2: Distanceto VickreyPayments. at the end of Phase II. For comparison,we also computethe average distance between minimal competitive equilibrium (CE) prices and Vickrey payments(see the next section Figure 2 illustrates the performanceof the auction in prob- a9 discussion of the relevance of minimalCE prices). lems Uniform, Decay, Random, and Weighted Random. In all problems allocative efficiency approaches 100%for 7problemsizes (numagents, numitems, numbundles) are set small bid increments, and the distance to GVApayments ap- to (8, 50, 120)in Decay,(20, 50, 400)in Uniform,(8, 30, 80) proaches zero. Random,and (15, 50, 300) in WeightedRandom. Decay parameter Paymentsin the auction converge to GVApayments in all a = 0.85. Theresults are averagedover 40 trials. problems that we examined, as the minimal bid increment XThisprovides a moreuseful measureof distance than com- gets small and the correctness of the allocation in the auction puting the averageL2 distance over all trials, includingthose in approaches100%. The effect of Phase II is quite significant, whichthe wrongagents are in the final allocation. Asan alterna- while the initial price adjustmentat the end of PhaseI has a tive wemight compute distance with an L1norm over all trials; this smaller effect. is equivalentto comparingthe total revenuein the auctionwith the revenuein the GVA. It is also worth noting that the auction implementsthe 9Additionalstatistics for PS1-12: at 95%correctness the aver- Vickrey outcome even in problems in which the outcome age numberof roundsin PhaseI is 149, comparedto 18 in Phase is not supported in any competitive equilibrium; notice that I1; eachround in PhaseI takes an averageof 0.5s, comparedto 2.8s the distance between the minimal CE prices and the GVA in PhaseII; the agent valuationinformation provided at the endof paymentsis non-zero in all experiments. PhaseI is 77%,and increasesto 83%at the endof PhaseII, using a metric introducedin Parkes (1999a); and approximately1-in-3 agents receive a dummyagent duringPhase II.

85 Auction Design: Theoretical Motivation dependentsof that agent for the bundlethey receive in the In this section webriefly provide sometheoretical justifica- final allocation mustequal their value for the bundle. tion for the design of PhaseI and PhaseII of the auction. It is not hard to showthat these conditionsare all satisfied (to within e) at the end of Phase II. Wemake agents in Phase I. In Phase I we determine the efficient allocation, but not in (W-i)*continue to bid higher prices, and increase based on myopicbest-response bids from agents. Phase I is their ask prices, until they enter (W-i)* or drop out of the equivalent to/Bundle, whichimplements a primal-dual algorithm for the CAPwith best-response bids from agents, with auction. respect to linear program formulations LP(g3) and DLP(ga) The tricky part of the proof that we terminate with GVA introduced by Bikchandani & Ostroy (1998). payments is to showthat the competition from the dummy In each roundthe provisional allocation is a feasible pri- agents is sufficient to makePhase II terminate, i.e. to push mal solution and the ask prices are a feasible dual solution. up the bid prices high enoughof all active agents. Webe- The complementary-slacknessconditions, which are neces- lieve that the current protocol for detecting quiescencedur- sary and sufficient for optimality of the provisional alloca- ing Phase II and introducing new dummyagents is sufficient tion, can be given interpretations in the auction. In Parkes & for this purpose. Ungar (2000b) we prove that/Bundle terminates with prices Example. and an allocation that satisfy complementary-slacknessconditions, and therefore with an efficient allocation. For example,consider case (a = b = 10) in Example2. At the endof PhaseI wehave the followingCE prices: Pask,1= (0~ 8, 8), Intermediate Computation. The intermediate computa- Pask,2 = (7, 0, 7), and Pask,3 : (0, 0, 15). DuringPhase tion determines the initial price discounts to each agent. agents 1 and 2 bid continue to bid and face higher ask prices. In Parkes & Ungar (2000c) we demonstrate that given At the end of PhaseII, we havethe followingadditional sets of competitive-equilibrium prices (pl,... ,PI), prices (Pl CEprices: (i) Pask,1 = (0, 10, 10), Pask,2 (7,0,7), an Ainit,l,P2~. ¯ ¯ ,Pl) are also in competitive-equilibrium P~sk,a = (0, 0, 15), from which we can computeminimal (CE). CEprices are any set of prices that satisfy comple- prices Pask,1 : (0, 10, 10), Pask,2 :-- (5, 0, 5), and Pask,a = mentaryslackness with the optimal primal solution, i.e. any (0, 0, 15); and (ii) Pask,1= (0, 8, 8), Pask,2= (10, 0, 10), optimal dual solution. Note that they are not unique. Pask,3 = (0,0, 15), from which we can computeminimal CE Bikchandani & Ostroy (1998) prove that CE prices are prices Pask,1 : (0,5,5), Pask,2 (10,0, 10 ), an Pask ,3 = upper-boundson Vickrey payments, therefore adjusted price (0, 0, 15). Thefirst set of minimalCE prices supportsthe Vickrey P~id,i -- Ainit,i is an upper-boundon an agent’s Vickreypay- paymentto agent 2. Thesecond set of minimalCE prices supports ment. This is the adjusted paymentcomputed in Intermedi- the Vickreypayment to agent 1. ate Computationat the end of Phase I of the auction. In addition, we prove in Parkes & Ungar (2000c) (see Discussion Theorem9) that a sufficient condition for the adjusted pay- It is importantthat agents cannotidentify the transition from ments to equal Vickreypayments is that there are no active Phase I to Phase II of the auction because nothing that an agents; i.e. that all agents in the allocation at the end of agent bids in PhaseII will changeeither the final allocation Phase I are also in the revenue-maximizingallocation with- or its ownfinal payment.The only effect of an agent’s bids out any single agent. in Phase II is to reduce the final payment madeby other Phase II. The motivation for Phase II is to force ac- agents. If it is costly to participate in the auction an agent tive agents to bid higher prices for bundles received in wouldchoose to drop out after PhaseI. In addition, there are the optimal allocation. The ask prices to all agents re- opportunities for collusion betweenagents in Phase II (just main valid competitive equilibrium prices during Phase II. as the GVAitself is vulnerableto collusion). Therefore, we can still compute discount A(i) = P* Certainly, we must hide bids from dummyagents in MAXREV(P_i)for agent i as the auction continues. This Phase I (or give the dummyagents false identities). Each is computed dynamically in the auction, as the sum of agent only needs information about its ownask prices, and Ainit(i) and Aextr~(i). whetheror not it is receiving a bundlein the provisional al- In Parkes & Ungar (2000c) we prove necessary and suf- location. Agentsdo not need any information about the bids, ficient conditions for this discount to computeVickrey pay- prices, or allocations of other participants. Wemust also be ments: sure that agents cannot distinguish the competitive effects ¯ The ask prices must leave every agent not in the optimal of bids from dummyagents from the competitive effects of allocation indifferent between receiving no bundle and bids fromreal agents. This is our reasoning for constructing dummyagents to "clone" the real agents that compete for any bundleit receives in a second-bestallocation. items in PhaseI of the auction. ¯ The ask prices must leave every agent in the optimal allocation indifferent betweenthe bundleit receives in the Future Work optimal allocation and any bundle it receives in a second- There are a numberof interesting areas for future work. best allocation. First, it should be possible to reduce the computationalde- ¯ Every agent in the final allocation must either: (a) have mandson the auctioneer. For example, it would be useful an adjusted paymentof zero; or (b) the ask price to all to allow the auctioneer to recomputethe dependent agents

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