The Communication Complexity of EfficientAllocation Problems∗

Noam Nisan† Ilya Segal‡

November 21, 2001

Preliminary and Incomplete

Abstract

We analyze the communication complexity of surplus-maximizingallocations. We study both the continuous and discrete models ofcommunication, measuring its complexity with the dimensionality ofthe message space and the number of transmitted bits, respectively. Inboth cases, we offer a lower bound on the amount of communication.This bound is applied to the problem of allocating L heterogenousobjects among N agents, when agents’ valuations are (i) unrestricted,(ii) submodular, or (iii) homogeneous in objects. Efficiency requiresexponential communication in L in cases (i) and (ii), and in logL incase (iii). Furthermore, in case (i), polynomial communication in Lcannot guarantee a higher surplus than selling all objects as a bundle.On the other hand, in case (iii), an arbitrarily large share of the surpluscan be realized with only O(logL) bits. When agents’ valuationssatisfy the gross substitute property, efficiency can be achieved withdeterministic communication that is polynomial in L.

∗Acknowledgements to be added.†Institute of Computer Science, Hebrew University, Jerusalem, Israel, email:

noam@cs.huji.ac.il‡Department of Economics, Stanford University, Stanford, CA 94305; email:

ilya.segal@stanford.edu

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1 IntroductionWe have recently seen great interest in so-called combinatorial auctions, inwhich heterogeneous indivisible items are to be allocated among bidderswhose preferences for combinations of items can exhibit complementarityor substitutability. The objective an auction is to elicit enough informa-tion about bidders’ preferences so as to realize an efficient or approximatelyefficient allocation. (Recent important applications include auctions of spec-trum licenses and online procurement - see, e.g., Vohra and de Vries (2000)for an overview.) The mechanism design literature has examined the bid-ders’ incentives to reveal their valuations truthfully, appealing to the Reve-lation Principle to focus on direct revelation mechanisms. For example, theVickrey-Groves-Clarke direct revelation mechanism implements the efficientallocation and is incentive-compatible in the private-value environment. Atthe same time, it has been recognized that the full revelation of bidders’ pref-erences may require an enormous amount of communication. Indeed, everybidder has a valuation for each subset of the items, and the number of suchsubsets is exponential in the number of objects. With 30 items, full revela-tion of such preferences would require the communication of more than onebillion numbers, which is beyond the capabilities of any human or machine.1

Recognition of the communication problem has prompted researchers toexamine the properties of simpler mechanisms, which do not fully revealvaluations. For example, auctions that quote only per-item prices were shownto be efficient in some restrictive settings (Gul and Stacchetti (2000), Milgrom(2000)). For other settings, ascending-bid auctions combinatorial auctionshave been examined, in the hope of economizing on communication by askingbidders to announce their valuations only for the subsets revealed to berelevant in previous bidding (Ausubel and Milgrom (2001), Bikhchandani etal. (2001)). However, until now there has been no analysis of the minimumamount of communication required to achieve or approximate efficiency. Thepresent paper closes this gap, identifying the communication burden of theproblem of implementing efficient or approximately efficient allocations.We analyze the communication problem using techniques developed in

parallel in economics and computer science.2 In economics, these techniques

1Even if it is known a priori that the bidder is not interested in more than, say, sixitems, full revelation would require the communication of more than half a million numbers(¡306

¢).

2For an earlier discussion relating the economic and computer science approaches to

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were developed in the literature on the dimensionality of message spaces,originated by Hurwicz (1960) and Mount and Reiter (1974). This litera-ture measures the communication burden of an allocation problem with thenumber of real variables that need to be announced to verify that the rightallocation is implemented. The key achievement of that literature is a for-malization of Hayek’s (1945) idea that in standard “convex” environments,the Walrasian mechanism is “informationally efficient,” i.e., it realizes socialefficiency with the least amount of communication. The Walrasian mech-anism involves only the announcement of prices, along with the allocation,which is much more economical than full revelation of agents’ preferences. Innonconvex environments, however, a Walrasian equilibrium with linear pricesmay not exist, and much more extensive communication may be needed (seeCalsamiglia (1977) for an early example). The problem of combinatorialallocation of indivisible items is nonconvex, and a Walrasian equilibriumwith linear (per-item) prices in general does not exist (with some notableexceptions discussed below). The question examined in this paper can beformulated as how many real-valued “prices” need to be announced to verifyefficiency in such discrete allocation problems.A parallel approach to measuring communication has developed in the

computer science field of communication complexity, pioneered by Yao (1979)and described in Kushilevitz and Nisan (1997). The main difference fromthe economic approach is that it considers a discrete (finite) setting. All“inputs” (in our case, agents’ valuations) are assumed to be given with afinite precision, and the complexity of communication is measured with thenumber of bits transmitted.Our main result is a lower bound on the amount of communication re-

quired to implement efficiency in both the continuous and discrete cases.We obtain it by showing that all distinct states of the world in which thetotal surplus is constant across outcomes must give rise to different commu-nications. In the terminology of computer science, such states constitute a“fooling set.” A lower bound on communication is given in the continuouscase by the dimensionality of the fooling set, and in the discrete case by thebinary log of its cardinality.For the problem of allocating L heterogoneous objects to N agents whose

valuations over subsets of objects are unrestricted, our lower bound impliesthat the required communication is exponential in the number L of items

communication complexity, see Marschak (1996).

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to be allocated. Even if the agents’ valuations are known to be submodular(i.e., exhibit diminishing marginal utility of objects), exponential communi-cation in L is still required. On the other hand, when objects are identical(hence agents only care about the number of objects they consume), thecommunication of L numbers is required.These results are extended to the problem of finding approximately effi-

cient allocations, i.e., those whose total surplus is within some predeterminedfactor from optimal. Since arbitrarily close approximation can be obtainedwith a zero-dimensional finite communication, here we focus on the discretemeasure counting the communicated bits. Most of the approximation resultsfollow from the simple observation that in the discrete case where the agents’valuations are described with a given precision, sufficiently close approxima-tion is equivalent to exact efficiency. Hence, the lower bounds obtained forexact efficiency can be brought to bear on this case.In the case of unrestricted valuations, we show that any mechanism guar-

anteed to achieve more than 1/N of the maximum available surplus involvesthe communication of an exponential number of bits in L. On the otherhand, share 1/N of the surplus is attained by an auction selling all items asa bundle. Therefore, no “practical” mechanism can improve upon the simplebundled auction. For the case of submodular valuations, we were only able toestablish a far weaker result, ruling out so-called “Fully Polynomial Approx-imation,” but leaving open the possibility of arbitrarily close approximationwith polynomial communication. (A polynomial communication achieving50% of the surplus has been suggested by Lehmann et al. (2001).) Finally,for the case of homogeneous valuations, Fully Polynomial Approximation inlogL is possible. In particular, efficiency can be approximated arbitrarilyclosely using only O(logL) bits, as contrasted with the L real numbers re-quired for exact efficiency. This demonstrates that in some settings, includingCalsamiglia (1977), an enormous savings on communication can be achievedwith only a slight sacrifice in economic efficiency.The approximation results described above have required uniform ap-

proximation across states. However, we also have a result for the weakerrequirement of approximating expected surplus, given a probability distribu-tion over states. The result establishes that when the agents’ valuations aredrawn independently from a probability distribution, for some such distrib-utions, any mechanism using sub-exponential communication entails a sub-

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stantial (constant fraction) loss in expected surplus.3 The result is proven byreducing the problem to the so-called “disjointness problem” and adaptinga result of Babai, Frankl, and Simon (1986). While the constructed distri-butions may not be realistic, the point is that efficiency can be approachedwith sub-exponential communication only for a restricted class of probabilitydistributions over valuations.Our lower bounds on communication are obtained under what is called the

“verification” scenario in economics, and the “nondeterministic” scenario incomputer science. This scenario presupposes the existence of an omniscientoracle, who must prove to the agents that a given allocation should be imple-mented. (In economics, the oracle is known as the “Walrasian auctioneer”,who can “guess” the equilibrium prices.) The problem of communication inthe absence of such an oracle, called “deterministic communication,” is moredifficult. Deterministic communication can be facilitated by having multiplerounds, where at each round an agent only transmits information that isrevealed in the previous rounds to be relevant for the outcome. An exampleof this in economics is given by “tatonnement” processes used to attain thecompetitive equilibrium. Such multi-round combinatorial auctions have re-cently been considered by Ausubel and Milgrom (2001), Bikhchandani et al.(2001), Parkes (1999), Parkes and Ungar (2001). One claimed advantage ofthese auctions is that they economize on communication, by asking the bid-ders in each round to send only their valuations for a small number of subsetsthat have been previously revealed to be “relevant.” While such multi-roundmechanisms may indeed reduce the amount of deterministic communication,they are not of any help when nondeterministic communication is allowed.Indeed, an omniscient oracle can announce the whole equilibrium commu-nication sequence (in particular, the “relevant” subsets) upfront and verifythat no agent objects to it. Therefore, our lower bounds apply to communi-cation in any mechanism, be it nondeterministic or deterministic, single- ormulti-round.The communication problem examined here is different from the often

considered problem of computing the optimal allocation once valuations areknown. The computational problem remains intractable even if each bidderis known to be interested in only a single subset of items, while the com-munication burden in this case is light. However, it seems that the commu-

3An added strengthening is that the concept of “distributional complexity” used in thisresult counts the expected rather than worst-case number of bits transmitted.

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nication bottleneck may be more severe in practice than the computationalone. First, it “kicks in” already when there are a few dozen items, while thecomputational complexity can be handled for up to hundreds of items (andthousands of bids) optimally (Vohra and de Vries 2000, Sandholm et al. 2001)and thousands of items (with tens of thousand of bids) near-optimally (Zureland Nisan 2001). Second, the computational burden may be sidestepped bytransferring it to the bidders themselves, e.g., by asking the bidders to sug-gest allocations or matches to their package (Banks et al. 1989, Nisan andRonen 2000).Our results imply that any practical combinatorial auction can only ap-

proximate efficiency for some restricted class of valuations (or probabilitydistributions over valuations). For example, restrictions can be imposed toensure the existence of a Walrasian equilibrium with per-item prices, whichis a polynomial mechanism achieving efficiency. The relevant restrictionfor this is that the linear programming relaxation of the integer program-ming problem of efficient combinatorial allocation finds an optimal alloca-tion (Bikhchandani and Mamer 1997). (The “gross substitutes” property ofKelso and Crawford (1982) and Gul and Stacchetti (1999) is one conditionto ensure this, but other conditions are known (Nisan 2000, Vohra and deVries 2000)). Under this restriction, we actually suggest a polynomial de-terministic mechanism that achieves efficiency. (It works for a more generalclass of valuations than Ausubel’s (2000) auction, and it achieves exact effi-ciency in polynomial time, as opposed to only approximate efficiency.4) Thismechanism mimics a separation-based linear programming algorithm, and itcan be made incentive-compatible with little added communication. Whilethis mechanism is very unnatural, it does show that whenever a Walrasianequilibrium with per-item prices is known to exist, efficiency can also be ob-tained with a deterministic incentive-compatible mechanism with polynomialcommunication.The remainder of the paper is organized as follows. In Section 2, we

describe the general setup. In Section 3 we discuss the different notions ofcommunication complexity. In Section 4 we obtain a general lower boundon communication using the “fooling set” technique. In Section 5, this re-

4Formally, the mechanims of Kelso and Crawford (1982) and Ausubel (2000) take timethat is proportional to ²−1 to obtain an outcome that is ²-close to optimal (this is usuallycalled a FPAS), while we obtain what is considered “true polynomial time,” i.e., polynomialin the representation length of ², i.e. in log ²−1. The difference between these two notionsis demonstrated in our context in section 6.3.

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sult is applied to the problem of efficient combinatorial allocation for threeclasses of agents’ valuations. In Section 6, the same technique is appliedto the problem of approximate efficiency. Section 7 suggests a polynomialdeterministic mechanism that implements efficiency whenever a Walrasianequilibrium with per-item prices exists.

2 SetupLet N be the finite set of agents, and K be the set of outcomes. (With aslight abuse of notation, we will use the same letter to denote a set and itscardinality when this causes no confusion. At this point, we need not requirethe set K to be finite, though it will be in most applications.)An agent’s valuation is a vector in RK , which assigns real values to dif-

ferent outcomes. The class of possible valuations of agent i ∈ N will bedenoted by V i ⊂ RK . Agent i’s valuation vi ∈ V i is assumed to be his pri-vately observed “type.” A state is a profile of valuations:

¡v1, ..., vN

¢ ∈ V ≡V 1 × . . .× V N ⊂ RNK .A leading application considered in this paper is the problem of allocating

L items among the agents. The outcome set in this problem is K = NL.Thus, for each item l ∈ L, k(l) ∈ N denotes the agent holding it in outcomek. Correspondingly, k−1(i) ⊂ L is the set of objects held by agent i.The efficient choice correspondence K∗ : V ³ K is defined by

K∗(v) =argmaxk∈K

Xi∈N

vik ∀v ∈ V.

(This “Kaldor-Hicks” notion of efficiency is based on the implicit assump-tion that the agents’ utilities are quasilinear in monetary transfers, and thatsuch transfers can be used to compensate agents. For the most part we willabstract from incentive considerations, and therefore will not model the sidetransfers explicitly.)Observe that valuations (vi1, . . . , v

iK) and (v

i1 + a, . . . , v

iK + a) describe the

same preferences for any a ∈ R. It will be convenient to rule out such situ-ations, by assuming that each agent i’s class of valuations V i is normalized,meaning that it does not contain the two above valuations for any a 6= 0. Asimple way to normalize valuations is by assigning the value of zero to oneof the outcomes.

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3 The Communication Problem

3.1 Nondeterministic Communication

We begin with a formal definition:

Definition 1 A (nondeterministic communication) protocol is a triple hM,µ, hi,where M is the message set, µ : V ³M is the message correspondence, andh : M → K is the outcome function, and the message correspondence µsatisfies the following two properties:

(a) µ(v) 6= ∅ ∀v ∈ V ,(b) µ(v) = ∩i∈Nµi(vi) ∀v ∈ V , where µi : V i ³M for each i.

The protocol hM,µ, hi realizes correspondence F : V ³ K if h(µ(v)) ⊂F (v) ∀v ∈ V .

This definition can be interpreted using a scenario called “verification”in economics and “nondeterministic” in computer science. In the scenario,an omniscient oracle knows the state of the world v and correspondingly theset F (v) of “desirable” outcomes, but needs to prove to an outsider thatan outcome k ∈ F (v) is desirable. He does this by publicly announcing amessage m ∈M . Each agent i either accepts or rejects the message, makingthe decision on the basis of his own type vi. The message correspondenceµ(v) describes the set of messages acceptable to all agents in state v. Ifmessage m is accepted by all agents, then the oracle implements outcomeh(m).In this interpretation, condition (a) requires that there exist an acceptable

message in each state. Condition (b) follows from the fact that each agentdoes not observe other agents’ types when making his acceptance decision,thus the set of messages he accepts is a function µi(vi) that depends only onhis own type vi. This condition is known as “privacy preservation” in theeconomic literature.Conditions (a) and (b) can also be given a geometric interpretation. Con-

dition (a) can be interpreted as saying that the collection of sets {µ−1(m)}m∈Mis a covering of the state space V . In this interpretation, condition (b) saysthat each element of the covering is a product (“rectangular”) set. This iswhy condition (b) is known as the “rectangle property” in computer science.

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Finally, the definition of realization requires that acceptable messagesgive rise only to desirable outcomes, which is equivalent to saying that theoutcome h(m) is “proven” to be desirable whenever m is accepted by allagents.A famous nondeterministic communication protocol in economics is the

Walrasian equilibrium. The role of the oracle is played by the “Walrasian auc-tioneer,” who can guess the equilibrium prices and quantities.5 Each agentverifies that the quantities indeed constitute his optimal demand given theprices. This protocol realizes the Pareto efficient allocation correspondence.

3.2 Measures of Complexity

The economic literature considers problems with continuous type spaces Vi,which usually require protocols with continuous message spaces M . Theamount of communication is measured with the dimensionality by the mes-sage space M . A potential problem with this approach is that the messagecorrespondence could be chosen to be a one-to-one function µ : RNK → R(e.g., the inverse Peano function can be used for this purpose). Then anyamount of information can be “smuggled” in a one-dimensional messagespace. To eliminate such smuggling, it is customary to impose a regular-ity restriction on the message correspondence. We follow Walker (1977) andSato (1981) in requiring that the message space M be a Hausdorff topo-logical space, and that the message correspondence µ be locally threaded -which means that for every v ∈ V , it has a continuous selection on an openneighborhood of v.As suggested byWalker (1977), the “dimensionality” of the message space

can be measured by comparing it with Euclidean spaces using the Frechetordering, according to which M ≥F RS if RS can be homeomorphicallyembedded in M (in this case, we will also write dimM ≥ S).The computer science literature, in contrast, considers finite (discrete)

state spaces, in which case a finite message space M is obviously sufficient.The relevant complexity measure is then the number of bits needed to encodea message, which is log2 |M |.When the agents’ valuation classes V i are continuous, efficiency will typ-

ically require an infinite (continuous) message space, and the continuous

5The choice of quantities can often be delegated to the agents, but not always - forexample, when production exhibits constant returns to scale, the oracle needs to assignmarket shares.

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complexity measure will be used. On the other hand, when valuations arerounded off with a finite precision, we will use the discrete complexity mea-sure. A more interesting case, however, is one where only approximate effi-ciency is required. Observe efficiency can be approximated arbitrarily closelywith a finite message space, by announcing valuations with a finite precision.Thus, we will use the discrete measure in the analysis of approximation,but the precision with which agents’ valuations are revealed will be chosenendogenously in order to achieve a given approximation of efficiency.

3.3 Relation to Deterministic Communication

The notion of communication defined above does not describe a realisticprocess by which a state v results in a message m. For example, in thecase of the Walrasian protocol, it does not explain how the Walrasian pricesare obtained. A more realistic concept, called deterministic communication,defines communication as a sequence of messages sent by different agents(see Kushilevitz and Nisan 1997). An economic example of deterministiccommunication is a “tatonnement” process that converges to the equilibriumprices.In the language of game theory, deterministic communication is an extensive-

form message game form along with the agents’ strategies (complete planscontingent on their types and on observed history) specified in the game.6

Since incentives are not considered, without loss of generality the game cantaken to have perfectly observed histories.The computer science literature considers finite deterministic communi-

cation, and restricts attention to games with two actions at each node (in-terpreted as sending a bit of information). The (“worst-case”) deterministiccommunication complexity is then defined as the maximum depth of thegame tree, i.e., the maximum number of bits transmitted over all states.7

To show that deterministic communication is “harder” than nondeter-ministic, we argue that any deterministic protocol can be converted intonondeterministic without an increase in communication. The converted pro-

6We call it a game form rather than a game because the agents’ payoffs are not specified,and are indeed irrelevant here since we do not insist on incentive-compatibility, insteadassuming that the agents blindly follow the strategies assigned to them.

7One may instead be interested in the expected number of bits transmitted given acertain probability distribution over states, rather than the worst-case number. A lowerbound for such distributional complexity is offered in subsection 6.5.

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tocol works as follows: the oracle announces the equilibrium path of the gamein the given state. Each agent accepts a path if and only if at each node onit, his strategy corresponding to his type is to move along the path. Thisprotocol yields the same outcome as the original deterministic protocol. Itscomplexity equals the binary log of the number of terminal nodes in the gametree, which cannot exceed the maximum depth of the tree, which is the com-plexity of the deterministic protocol. Therefore, nondeterministic complexityoffers a lower bound on deterministic complexity. (A similar argument canbe made for continuous communication, where real-valued messages can besent at each node.)

3.4 Relation to “Black-Box” Computational Complex-ity

One can consider the problem of computing a selection from the efficientchoice correspondence K∗. However, in the cases where the computationalproblem might be difficult, such as that of combinatorial allocation with alarge number L of objects, the agents’ valuations vi that serve as inputs tocomputation hold a huge amount of information. In such cases, one mustconsider how an algorithm will access this information. The most generalmodel will allow the algorithm to ask a “black box” an arbitrary questionabout a valuation vi and receive an answer for any such question. Morerestricted models will define which queries the black box for vi will answer -e.g., the “valuation oracle” model will only answer queries of the form “whatis vi(k)”.A trivial, yet important, observation is that the general black box model

is equivalent to the deterministic communication complexity model. A stagein a deterministic communication protocol where player i has observed themessage history hm1...mti and responds with his own message mt+1 is equiv-alent to a black box that is asked the query hm1...mti and responds with theanswermt+1. In both casesmt+1 depends only on vi and hm1...mti. Thus, ourlower bounds on communication also imply lower bounds on computationalcomplexity in the general black box model.

3.5 Relation toWalrasian Equilibria: Messages as Prices

A Walrasian Equilibrium is the classical economic example of continuousnondeterministic communication. The efficiency of this protocol is ensured

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by the First Welfare Theorem. However, the existence condition (a) is tra-ditionally established under convexity assumptions that are unlikely to besatisfied in the combinatorial allocation setting. One way to ensure existenceis by allowing so-called Lindahl prices (pik)i∈N,k∈K — personalized prices thatdepend on the whole outcome. The Lindahl-Walras protocol works as fol-lows: the designer announces a price vector and an outcome k, and eachagent accepts if and only if k is his preferred outcome given the price vector(i.e., k ∈argmax

k0∈K[vik0 − pik0 ]). We should require that

Pi pik = 0 for all k ∈ K,

reflecting the fact that each outcome k is costless to the designer. Also, wecan normalize

Pk p

ik = 0 for all i ∈ N , since adding a constant to prices

for all outcomes does not changes agents’ choices. Thus, the dimensional-ity of the message space in the Lindahl-Walras protocol is (N − 1) (K − 1).It is easy to verify that the protocol is efficient and satisfies the existencecondition (a). The problem with the Lindahl-Walras protocol, however, isthat it requires a large amount of communication - equivalent to full reve-lation of valuations by all but one agent. The dimensionality of the pricespace in the combinatorial allocation problem can be reduced by restrictingthe prices to be, e.g., linear in objects and anonymous, but this will usuallyviolate the existence property. In this terminology, our search for the mes-sage space of minimum dimensionality can be interpreted as looking for theminimum-dimensional subset of the Lindahl-Walras price space that ensuresthe existence of a Walrasian equilibrium.

3.6 Relation to Mechanism Design

The mechanism design literature (specifically, its “partial implementation”branch) examines the incentive compatibility of choice rules and correspon-dences, appealing to the Revelation Principle to focus attention on direct rev-elation mechanisms. This ignores the communication requirements of choicecorrespondences. In this paper, we examine the communication requirementsof choice correspondences, for the most part ignoring their incentive compat-ibility (which would also depend on monetary transfers, ignored here). Wefind that the communication requirement by itself in some cases constitutesa “bottleneck” that prevents efficiency.If agents behave in their own interest rather than blindly following the

strategies suggested by the designer, this can only further constrain the de-signer. These constraints are apparent in the deterministic communication

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model, where they require that the agents’ strategies constitute an equilib-rium of the extensive-form message game. (It would be harder to imposeincentive-compatibility constraints in the nondeterministic communicationmodel, since it does not specify what an agent would obtain by “rejecting”the oracle’s message.) A normal-form equilibrium concept, such as that of theBayesian-Nash equilibrium, will impose the standard incentive-compatibilityconstraints on the social choice rule. Examples of a joint analysis of incentive-compatibility and communication complexity constraints can be found inReichelstein (1984) and Segal (1999, Section 4).

4 The Basic Lower BoundOur lower bound on communication is obtained by constructing a set that iscalled the “fooling set” in computer science and the “set with the uniquenessproperty” in economics. The simplest illustration of this construction obtainsin the case where a single indivisible object is to be allocated between twoagents. Letting vi denote agent i’s valuation for this object, a state of theworld is described by a pair (v1, v2). Efficiency requires giving the object tothe agent with the higher valuation.The basic idea is that an efficient protocol cannot use the same message

in two different “diagonal” states of the world, (v, v) and (w,w), where,say, w > v. Suppose in negation that there exists a single message thatoccurs in both states. Then each agent 1 and 2 accepts this message whenhis valuation is v or w. But then the “privacy preservation”/“rectangle”property of communication implies that both agents accept the same messagein states (w, v) and (v, w), and therefore the same outcome will obtain in thetwo states. But this contradicts efficiency, which requires giving the objectto agent 1 in state (w, v) and to agent 2 in state (v,w). Therefore, alldistinct diagonal states in the example must give rise to distinct messages.A set of states with this property is called a “fooling set.” Communicationof the fooling set bounds from below the number of messages required torealize efficiency. In particular, in the case where each agent’s valuationlies in a continuum [v, v], there is a continuum of feasible diagonal states,and therefore at least a continuum of possible messages is needed to realizeefficiency.Extending this argument to arbitrary many agents and outcomes yields

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Lemma 1 Suppose each V i is normalized, and v,w ∈ V 1×. . .×V N such thatv 6= w and K∗(v) = K∗(w) = K. Then in any efficient protocol hM,µ, hi,µ(v) ∩ µ(w) = ∅.

Proof. Suppose in negation that m ∈ µ(v) ∩ µ(w). Then by the rectangleproperty, we must also have for all i ∈ N ,m ∈ µ (vi, w−i) andm ∈ µ (wi, v−i).Hence, for outcome k∗ = h(m) ∈ K we must have

k∗ ∈argmaxk∈K

"vik +

Xj 6=iwjk

#and k∗ ∈argmax

k∈K

"wik +

Xj 6=ivjk

#.

But by assumption,P

j vjk and

Pj w

jk do not depend on k, which allows us

to rewrite the above display as

k∗ ∈argmaxk∈K

£vik − wik

¤and k∗ ∈argmax

k∈K

£wik − vik

¤=argmax

k∈K

£vik − wik

¤.

This is only possible when vik −wik does not depend on k ∈ K, and sinceV i is normalized, we must have vi = wi. Since this argument applies for eachi ∈ N , we must have v = w, contradicting our assumption.In words, the Lemma says that any two distinct states in which all out-

comes are indifferent from the social viewpoint must give rise to distinctmessages. Formally, the message correspondence in any efficient protocol isinjective on the set

F =©v ∈ V 1 × . . .× V N : K∗(v) = K

ª,

i.e., the set can serve as a “fooling set.” Efficiency requires at least as muchcommunication as that needed to communicate a point in the fooling set.For the discrete and continuous communication measures, this implies

Proposition 1 Suppose each V i is normalized, and hM,µ, hi is an efficientprotocol. Then log2 |M | ≥ log2 |F| and dimM ≥ dimF.

Proof. The first inequality follows immediately from Lemma 1. The secondinequality follows from Lemma 1 and our topological assumptions on theprotocol - see Lemma 1 in Sato (1981)

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5 Application to Combinatorial Allocation Prob-lems

In this section we apply Proposition 1 to the problem of allocating L itemsamong N agents. The outcome set is in this problem is K = NL, where k(l)denotes the agent holding object l ∈ L in outcome k ∈ K. We will maintainthe following standard assumptions on agents’ valuations:8

• No Externalities: For each agent i, vik = ui(k−1(i)), where ui : 2L →R. In words, each agent i cares only about the subset of items k−1(i)allocated to him. Let U i ⊂ R2L denote the class of agent i’s valuationsfor his own consumption.

• Monotonicity: For each ui ∈ U i, ui(S) is nondecreasing in S ⊂ L.This assumption can be justified by the agents’ ability to freely disposeof items.

• Normalization: ui(∅) = 0 for each agent i. This assumption is with-out loss of generality, as discussed in Section 2.

For the discrete case, we assume in addition that the agents’ valuationsare integers in the range 0..R, formally that ui(S) ∈ {0...R} for all S ⊂ L.This means that the valuation for each set can be encoded with log2R bits.(The results for the discrete case will also prove useful for the analysis ofapproximation in the next section, where the precision with which agents’valuations are announced can be chosen endogenously to achieve the desiredapproximation.) In the continuous case, the range of valuations is the realline R.We will examine in turn the communication requirements for three classes

of valuations:

(i) Unrestricted valuations: no restrictions are imposed other than those justdescribed.

(ii) Submodular valuations: in addition to the above restrictions, each agenti’s valuation ui is submodular, i.e., satisfies ui(S ∪ T ) + ui(S ∩ T ) ≤

8Note that relaxing these assumptions could only raise the communication burden, thuspreserving our lower bounds.

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ui(S)+ui(T ) for all S, T ⊂ L. It is well known that submodularity canbe alternatively defined by requiring that the agent’s marginal benefitui(S ∪ l)− ui(S) be nondecreasing in S ⊂ L (see, e.g., Lehmann et al.(2001)).

(iii) Homogeneous valuations: in addition to the above restrictions, eachagent i’s valuation takes the form ui(S) = φ(|S|) for all S ⊂ N , whereφ : {0...L} → R. That is, agents care only about the number of itemsthey receive.

For the purpose of providing a lower bound, we restrict attention to thecase of N = 2. (The bound extends to N > 2, by considering the case whereall agents other than two have valuations that are identically zero.) In mostcases, we will apply Proposition 1 in the following way: for each valuationu ∈ U1, define a dual valuation u∗ by

u∗(S) = u(L)− u(L\S).The idea is that if agent 1 has valuation u and agent 2 has valuation u∗,the total surplus for any allocation of objects between the agents is u(L).Thus, by Proposition 1, the set of states F = {(u, u∗) : (u, u∗) ∈ U1 × U2}constitutes a fooling set, and the dimensionality of the message space in thecontinuous case is at least dimF, while the number of bits in the discretecase is at least log2 |F|.

5.1 Unrestricted Valuations

Here U1 = U2 = U , where U is the set of valuations satisfying monotonicityand normalization. (In addition, ui(S) ∈ {0...R} in the discrete case.) Onemay easily verify that the dual of any valuation in U is also in U , thus thefooling set is {(u, u∗) : u ∈ U}. Communicating the fooling set is equivalentto communicating a valuation from U . Thus, by Proposition 1, the lowerbound is dimU and log2 |U | in the continuous and discrete case respectively.Note that this lower bound is essentially tight for N = 2, since it is achievedby the protocol in which agent 1 announces his valuation and then agent 1announces an efficient allocation.In the continuous case, dimU = 2L − 1. In the discrete case, |U | is the

number of monotone functions on subsets of L elements with range {0...R}.This number can be bounded below by (R+ 1)(

LL/2), by considering valuations

with u(S) = 0 for |S| < L/2 and u(S) = R for |S| > L/2. Thus, we have

16

Corollary 1 With continuous unrestricted valuations, the dimensionality ofthe message space in an efficient protocol is at least 2L − 1. With discreteunrestricted valuations, the number of bits communicated by an efficient pro-tocol is at least

¡LL/2

¢log2 (R+ 1).

Therefore, the communication burden is exponential in the number of ob-jects, both in the continuous and discrete case.9 It an interesting open prob-lem how the required communication grows with the number N of agents.10

A trivial upper bound on communication is obtained by having all but oneagent announce their valuations, and the remaining agent announce an effi-cient allocation. This yields the amount of communication (N − 1)(2L − 1)/ (N − 1) log2 |U |+ L log2 |N | in the continuous/discrete case respectively.

5.2 Submodular Valuations

Here U1 = U2 = U , where U is the class of submodular valuations. Weare unable to use the approach of the previous subsection directly, sincethe dual of a submodular valuation is not submodular (unless both areidentically constant). We will modify the approach, by defining duality insuch a way that the social surplus is constant only on a subset of outcomeseK = {k ∈ K : |k−1(1)| = |k−1(2)| = L/2}.Consider the set eU of valuations u ∈ R2L such that• u(S) = 4|S|d for |S| < L/2,• u(S) = 2Ld for |S| > L/2,• u(S) ∈ [(2L− 1)d, 2Ld] for |S| = L/2,• u({1, ..., L/2}) + u({L/2 + 1, ..., L}) = (4L− 1) d,

where d = 1 in the continuous case and d =£R2L

¤in the discrete case ([·]

stands for the integer part). One can easily verify that in both cases, eU ⊂ U .9More precisely, in the discrete case, Stirling’s formula implies that¡LL/2

¢ ∼p2/ (πL) · 2L.10That such an increase might be required is suggested by the finding of Bikhchandani

and Ostroy (2001) that personalized prices for subsets of objects are needed to ensure theexistence of a competitive equilibrium.

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Note that in any state (u1, u2) ∈ eU × eU , the set of efficient outcomesis exactly K. (Indeed, any allocation from K brings a total surplus of atleast (4L− 2) d, while any other allocation brings a total surplus of at most(4L− 4) d.) Thus, we can apply Proposition 1 with the two agents’ valua-tions restricted to eU and the outcomes restricted to eK (the last bullet aboveensures that thus constructed valuation classes on eK are normalized) Forthis purpose, for each u ∈ eU , define its “quasi-dual” bu ∈ eU as follows:• bu(S) = (4L− 1) d− u(L\S) for |S| = L/2,• bu(S) = 4|S|d for |S| < L/2.• bu(S) = 2Ld for |S| > L/2,By construction, the set of efficient outcomes in any state (u, bu) ∈ eU× eU is

exactly eK, hence by Proposition 1 these outcomes constitute a fooling set. Inthe continuous case, the dimensionality of this set is dim eU = ¯ eK ¯− 1, whilein the discrete case, its cardinality is

¯ eU ¯ = (d+ 1)| eK|−1 (where ¯ eK ¯ = ¡ LL/2

¢).

Thus, Proposition 1 implies

Corollary 2 With continuous submodular valuations, the dimensionality ofthe message space in an efficient protocol is at least

¡LL/2

¢− 1. With discreteunrestricted valuations, the number of bits communicated by an efficient pro-tocol is at least

h¡LL/2

¢− 1i log2 ¡£ R2L¤+ 1¢.Note that the discrete lower bound is non-trivial only when R ≥ 2L. (If

R < 2L, a lower bound of¡R/2R/4

¢− 1 bits is obtained by restricting attentionto R items.)

5.3 Homogeneous Valuations

Here U1 = U2 = U , where U is the class of homogeneous valuations.Since the dual of a homogeneous valuation is homogeneous, the fooling setcan be constructed here in the same way as with unrestricted valuations:{(u, u∗) : u ∈ U}. Communicating the fooling set is equivalent to communi-cating a valuation from U . Thus, by Proposition 1, the lower bound is dimUand log2 |U | in the continuous and discrete case respectively.

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In the continuous case, dimU = L. In the discrete case, |U | is the numberof monotone functions with domain {1..L} and range {0..R}, which is exactlythe number of ways that R indistinguishable balls (corresponding to thelocations of jumps) can be partitioned into L+ 1 bins:

¡R+LR

¢. This yields

Corollary 3 With continuous homogeneous valuations, the dimensionalityof the message space in an efficient protocol is at least L. With discrete un-restricted valuations, the number of bits communicated by an efficient protocolis at least log2

¡R+LR

¢.

Just as with unrestricted valuations, this lower bound is essentially tightfor N = 2, since it is achieved by the protocol in which agent 1 announces hisvaluation and then agent 1 announces an efficient allocation. The problemof how the required communication grows with N is again open.The result for the continuous case can be linked to a result obtained by

Calsamiglia (1977). In his model, instead of L indivisible goods there existsa single infinitely divisible good, i.e., L = [0, 1]. In this case, U is the spaceof nondecreasing functions on [0, 1], which is infinitely-dimensional under areasonable topology. The same argument as that in Corollary 3 establishesthat the dimensionality of the message space must be at least dimU = ∞.Therefore, finite-dimensional communication cannot realize efficiency, whichis exactly the result obtained by Calsamiglia (1977).11

These results should be contrasted with the case where both agents’ val-uations are known to be concave. In this case, a Walrasian equilibrium witha single price exists, and thus efficiency is achieved with a one-dimensionalmessage space (regardless of whether the objects are divisible or not).

6 ApproximationThis section considers the case where an outcome is not required to maximizethe social surplus exactly, but only to realize a given fraction of the maximumsurplus. Formally, we replace the efficient choice correspondence with therelaxed correspondence

11Calsamiglia (1977) restricts the valuation of agent 1 to be concave and that of agent2 to be convex. Since the dual of a concave valuation is convex, the analysis goes throughwithout modification. Similarly, the agents’ valuations can be restricted to be arbitrarilysmooth (even analytical), without changing the argument.

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K∗c (v) =

(k ∈ K :

Xi∈N

vik ≥1

cmaxj∈K

Xi∈N

vij

),

where c > 1 is a given approximation factor (the inverse of the fraction ofmaximum surplus to be achieved).12 The approximation factor c may be aconstant or may depend on the parameters of the problem, such as N andL.Note that arbitrarily close approximation can be realized with finite com-

munication, in which agents’s valuations are announced rounded off with asufficiently fine precision.For this reason, in this section we restrict attentionto discrete communication, measuring it with the number of bits transmitted.Also, note that approximation in the discrete case for a given input precisionR provides a lower bound on approximation in the discrete case for a higherinput precision R0 > R, as well as on approximation in the continuous case.Finally, note that in the discrete case with N agents and input precisionR, approximation to within a factor c < NR

NR−1 = 1 + 1NR−1 is equivalent

to precise efficiency (since inefficiency loses at least one unit of surplus, andthe maximum social surplus is at most NR). These observations allow usto derive approximation results from the results on exact efficiency in thediscrete case.

6.1 Unrestricted Valuations

Using Corollary 1 for the case N = 2 and R = 1, we see that approximationwithin a factor c < 1 + 1

NR−1 = 2 requires the communication of at least¡LL/2

¢bits. This implies

Corollary 4 With unrestricted valuations (either continuous or discrete),approximation within a factor c < 2 requires the communication of at least¡LL/2

¢bits.

Thus, exponential communication is required to realize an approximationfactor below 2. This result can be strengthened for N > 2: Nisan (2001)

12Note that this is a “worst-case” definition, since it requires uniform approximationacross all states. We consider an even weaker requirement of “average-case” approximationin subsection 6.5 below.

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shows that approximation to a factor c > N requires exponential communi-cation in L as long as N ≤ L1/3−ε, with ε > 0.13An upper bound on communication can be obtained by considering the

auction in which all objects are sold together as a bundle. The social surplusrealized by this auction is

maxiui(L) ≥ 1

N

Xi

ui(L) ≥ 1

Nmaxk

Xi

vik.

Thus, the bundled auction realizes the approximation factor N . Puttingthe results together, we see that any improvement on the bundled auctionrequires exponential communication.

6.2 Submodular Valuations

Using Corollary 2 for the case N = 2 and R = 2L, we see that approximationwithin a factor c < 1 + 1

NR−1 = 1 +1

4L−1 requires the communication of atleast

¡LL/2

¢− 1 bits. This impliesCorollary 5 With submodular valuations, either continuous or discrete withprecision R ≥ 2L, approximation within a factor c < 1 + 1

4L−1 requires thecommunication of at least

¡LL/2

¢− 1 bits.This result is substantially weaker than that for unrestricted valuations.

For example, it does not rule out the possibility that any given approxima-tion factor greater than 1 can be realized with polynomial communication.Nevertheless, Corollary 5 does rule out the possibility of a so-called “FullyPolynomial Approximation Scheme” - i.e., the realization of a 1 + ² approx-imation with communication that is polynomial in L,R, ²−1. Corollary 5

13Nisan (2001) establishes this lower bound by considering the following “packing” prob-lem: each of the N agents holds a collection of subsets of L, and the aim is to approximatethe maximum number of subsets in the union of their collections that can be packed to-gether, i.e., that are pairwise disjoint. This problem is clearly seen to be a special case ofthe combinatorial allocation problem (where the range of valuations is {0, 1} and ui(S) = 1if the subset S is in the collection of agent i).In the case where N ≥ L1/3, this lower bound only implies that a L1/3−²-approximation

requires exponential communication in L. Nisan (2001) shows that this is close to beingtight, by suggesting a O(

√L) approximation with polynomial communication using the

mechanisms of Lovasz (1975).

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rules out the possibility of such a scheme, since it implies that approxima-tion within ε(L) = 1/ (4L) requires exponential communication in L. Tohave an economic example, consider an ascending-bid auction with per-itembids and bid increment ε. The number of bits transmitted (whether to bid orabstain on a given object at a given price) is O(NLε−1). Corollary 5 impliesthat such an auction cannot realize an O (ε)-approximation of efficiency, forotherwise it would be a FPAS.The only upper bound known for the submodular case is the 2-approximation

of Lehmann et al. (2001), which is achieved by allocating the objects sequen-tially to the agents with the highest current marginal benefit for them.

6.3 Homogeneous Valuations

In this case, approximation within a factor of 1+² is possible using communi-cation that is polynomial in logR, logL,N, ²−1. In this communication, eachagent i rounds his valuation to integer multiples of ²ui(L). Each roundedvaluation can be fully specified using log2R+ ²

−1 log (L+ 1) bits (the valueof u(L) and the location of L+ 1 jump points.This setting offers an interesting example where efficiency can be ap-

proximated arbitrarily closely using O(logL) bits, despite the fact that byCorollary 3, exact efficiency requires communicating L real variables, in asense, exponentially longer communication. This approximation can be ex-tended to the problem of allocating an infinitely divisible good consideredby Calsamiglia (1977), provided that the agents’ valuations are restrictedto be Lipschitz continuous with some constant Amaxui(L). Indeed, in thiscase splitting the good into L = A/ε indivisible units and having each agentreveal rounded-off valuations for such discretized allocations as describedabove realizes the approximation factor 1 + 2ε while communicating onlylog2R+ ε

−1 log (Aε−1 + 1) bits.

6.4 The Procurement Problem

The procurement problem is the combinatorial allocation problem in whichthe “items” are obligations to supply objects, thus the valuations are the neg-ative costs of producing combinations of objects. This reverses the monotonic-ity of valuations - instead of “free disposal,” we have “free addition” of them.The analysis of exact efficiency for the procurement problem is exactly thesame way as for the “selling” problem, with the same results. However, the

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analysis of approximation of cost-minimization in the procurement problemturns out to differ from the approximation value-maximization in the “sell-ing” problem.To illustrate the difference, consider the protocol in which all items are

auctioned together as a bundle, and thus go to the agent with the lowestcost of producing the whole bundle. If the agents’ costs exhibit very largediseconomies of scale/scope, the cost of this outcome might be very highrelative to the minimal cost achieved by a more even allocation of items, andso the approximation factor may be arbitrarily large. This is in contrast tothe “selling” problem, where we have shown that the bundled auction ensuresfactor N -approximation.Nisan (2001) proves a lower bound on approximation in procurement

auctions using a reduction to the “set covering” problem. In this problem,there are two agents, each of whom holds a collection of subsets of L. Theaim is to approximate the cover number of the combined collection, i.e.,the minimal number of sets in the combined collection whose union is L.A lower bound on communication that is exponential in L is proven therefor a c logL-approximation for any constant c < 1/2. The reduction to theprocurement problem obtains by defining each agent’s cost of a subset S tobe the minimum number of subsets in her collection that together cover S (orinfinity if such covering is impossible). Thus, the exponential lower boundobtained for the covering problem applies also to the procurement problem.In particular, when N << logL, approximation in the procurement problemis much harder than in the selling problem, where N-approximation is trivial.A matching upper bound is constructed by Nisan (2001) for the case

where the agents’ costs are subadditive. This bound follows from classicalgorithms for approximate set covering using the following iterative proce-dure: repeatedly procure the set of items S with the minimal average costuntil all items are procured.

6.5 Average-case approximation

One may relax the notion of approximation further, by requiring only thatthe expected surplus be close to optimal. This, of course, depends on thedistribution on valuations. In the terminology of communication complex-ity, this concept is called “distributional complexity.” This concept is alsostronger less demanding than worst-case complexity in that it counts theexpected rather than worst-case number of bits transmitted.

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Proposition 2 There exists a distribution D over valuations in the combi-natorial allocation problem and a constant c > 1 such that every communica-tion protocol that realizes an expected social surplus within a factor of c fromoptimal when all agents’ valuations are drawn independently from D requiresthe transmission, on expectation, of an exponential number of bits in L.

Proof. Uses the known lower bounds on the distributional complexity ofthe “disjointness problem” (Babai, Frankl, and Simon 1986). Details to beadded.

7 Deterministic Protocols using Linear Pro-gramming

The communication lower bounds shown still leaves room for protocols thathandle restricted classes of valuations. In this section we sketch some possi-bility results along these lines. These protocols are not “pretty” in any sense,rather they only suggest that, for restricted classes of valuations, communi-cation is not a bottleneck.It is well known that the allocation problem of combinatorial auctions

may be phrased as an integer programming problem (see Vohra and de Vries(2000) for a survey). This integer programming problem is commonly re-laxed to a linear programming problem, and in some cases it is known thatthis linear program will indeed return integer allocations, solving the originalproblem as well. In particular it is known that this is the case if all valuationssatisfy the “gross substitutes” property (Kelso and Crawford 1982, Gul andStacchetti 1999). What we wish to point here is that in all such cases, effi-ciency may be realized with a protocol that only requires polynomial amountof communication.The basic correspondence between many auction protocols and primal-

dual methods of linear programming was pointed out in Bikhchandani et al.(2001). However, to prove a theorem about communication we will need touse separation-based LP algorithms. A separation-based linear programmingalgorithm may be used in cases of linear programs that have an exponentialnumber of constraints (inequalities) that are given implicitly. Specifically,such an algorithm does not receive the inequalities as an explicit input, butrather is provided with a “separation oracle” as its input: whenever an infea-sible solution is presented to this oracle, it must be able to produce a violated

24

inequality. Algorithms of this type can solve linear programs in polynomialtime, given just this type of an oracle. The reader is referred to any textbookon linear programming (e.g., Karlin 1991) for more information.The allocation problem itself is usually phrased as having an exponential

number of variables (xiS specifying the allocation of the bundle S of goodsto bidder i), but only a polynomial number of significant inequalities (thateach good is sold only once, and that each bidder is allocated only one set):

Maximize:P

i,S xiSu

i(S)Subject to:

• For all goods j: Pi,S|j∈S xiS ≤ 1.

• For all bidders i: PS xiS ≤ 1.

• For all i, S: xiS ≥ 0.

In order to use a separation-based linear programming algorithm, we moveto the dual that has just a polynomial number of variables:

Minimize:P

j pj +P

iwiSubject to:

• For all i, S: wi +P

j∈S pj ≥ ui(S).• For all j: pj ≥ 0.• For all i: wi ≥ 0.

It is important to note that in the dual, each inequality specifies a con-dition that depends on the valuation of a single player! Now the separation-based LP is used, where the protocol runs the separation-based LP algorithm,and whenever an oracle query needs to me made (i.e. a violated inequalityis desired), the current solution is presented to all players, and each of themresponds with a violated inequality, if any14. Since each inequality is heldby some single player, if a violated equality exists it is found, and the proto-col can continue. The communication that takes place at each such stage is

14A violated inequality is exactly the selection of a subset S of items that would giveplayer i at least utility wi under the prices pj . Finding this set may be a computationallydifficult problem depending on the representation of ui, but this does not concern us here.

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polynomial, and since the algorithm is known to terminate within a polyno-mial number of steps, we have that the whole protocol requires polynomialcommunication.In order to ensure truthfulness on the part of all the bidders, the protocol

should simply impose Vickrey-Groves-Clarke payments on allocated bundles.The calculation of these payments will require running n different linearprograms in the same way. This is clearly not very elegant, but still insuch a mechanism truth remains a dominant strategy. A delicate detail toensure is that this new protocol does not enrich the set of strategies of theplayers beyond those that correspond to valuations (since Vickrey-Groves-Clarke payments only ensure domination if the set of strategies is the setof valuations). This enriching may happen, e.g., if a player may reply withinconsistent inequalities during different stages of the n + 1 LP algorithms— i.e. those that correspond to no possible valuation. This however may beverified by the mechanism — making sure that the set of inequalities producedby any player is satisfied by some valuation (being an LP problem in itself),in which case the mechanism may ignore any offending inequality.

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