Learning Competitive Equilibria in Noisy Combinatorial Markets
LLearning Competitive Equilibria in Noisy CombinatorialMarkets
Enrique Areyan Viqueira
Brown UniversityProvidence, [email protected]
Cyrus Cousins
Brown UniversityProvidence, [email protected]
Amy Greenwald
Brown UniversityProvidence, [email protected]
ABSTRACT
We present a methodology to robustly estimate the competitiveequilibria (CE) of combinatorial markets under the assumption thatbuyers do not know their precise valuations for bundles of goods,but instead can only provide noisy estimates. We first show tightlower- and upper-bounds on the buyers’ utility loss, and hencethe set of CE, given a uniform approximation of one market byanother. We then develop a learning framework for our setup, andpresent two probably-approximately-correct algorithms for learn-ing CE, i.e., producing uniform approximations that preserve CE,with finite-sample guarantees. The first is a baseline that uses Ho-effding’s inequality to produce a uniform approximation of buyers’valuations with high probability. The second leverages a connec-tion between the first welfare theorem of economics and uniformapproximations to adaptively prune value queries when it deter-mines that they are provably not part of a CE. We experiment withour algorithms and find that the pruning algorithm achieves betterestimates than the baseline with far fewer samples.
KEYWORDS
Competitive Equilibria Learning, Noisy Combinatorial Markets,PAC Algorithms for Combinatorial Markets
Combinatorial Markets (CMs) are a class of markets in which buy-ers are interested in acquiring bundles of goods, and their valuesfor these bundles can be arbitrary. Real-world examples of CMsinclude: spectrum auctions [13] allocation of landing and take-offslots at airports [6]; internet ad placement [14]; and procurementof bus routes [10]. An outcome of a CM is an assignment of bun-dles to buyers together with prices for the goods. A competitiveequilibrium (CE) is an outcome of particular interest in CMs andother well-studied economic models [8, 35]. In a CE, buyers areutility-maximizing (i.e., they maximize their utilities among all fea-sible allocations at the posted prices) and the seller maximizes itsrevenue (again, over all allocations at the posted prices).While CEs are a static equilibrium concept, they can some-times arise as the outcome of a dynamic price adjustment process(e.g., [11]). In such a process, prices might be adjusted by an imagi-nary Walrasian auctioneer, who poses demand queries to buyers:i.e., asks them their demands at given prices. Similarly, we imagine
An earlier version of this paper [2] appeared in Proceedings of the 2nd Games, Agents,and Incentives Workshop (GAIW@ AAMAS 2020). An extended abstract of this paper toappear in Proc.of the 20th International Conference on Autonomous Agents and MultiagentSystems (AAMAS 2021), U. Endriss, A. Nowé, F. Dignum, A. Lomuscio (eds.), May 3–7,2021, London, UK . 2021. that prices in a CM are set by a market maker, who poses valuequeries to buyers: i.e., asks them their values on select bundles.One of the defining features of CMs is that they afford buyersthe flexibility to express complex preferences, which in turn hasthe potential to increase market efficiency. However, the extensiveexpressivity of these markets presents challenges for both the mar-ket maker and the buyers. With an exponential number of bundlesin general, it is infeasible for a buyer to evaluate them all. We thuspresent a model of noisy buyer valuations: e.g., buyers might useapproximate or heuristic methods to obtain value estimates [15].In turn, the market maker chooses an outcome in the face of un-certainty about the buyers’ valuations. We call the objects of studyin this work noisy combinatorial markets (NCM) to emphasize thatbuyers do not have direct access to their values for bundles, butinstead can only noisily estimate them.In this work, we formulate a mathematical model of NCMs. Ourgoal is then to design learning algorithms with rigorous finite-sample guarantees that approximate the competitive equilibria ofNCMs. Our first result is to show tight lower- and upper-boundson the set of CE, given uniform approximations of buyers’ valu-ations. We then present two learning algorithms. The first one—Elicitation Algorithm; EA—serves as a baseline. It uses Hoeffding’sinequality [19] to produce said uniform approximations. Our secondalgorithm—Elicitation Algorithm with Pruning; EAP—leveragesthe first welfare theorem of economics to adaptively prune valuequeries when it determines that they are provably not part of a CE.After establishing the correctness of our algorithms, we evaluatetheir empirical performance using both synthetic unit-demandvaluations and two spectrum auction value models. The formerare a class of valuations central to the literature on economicsand computation [27], for which there are efficient algorithms tocompute CE [17]. In the spectrum auction value models, the buyers’valuations are characterized by complements, which complicate thequestions of existence and computability of CE. In all three models,we measure the average quality of learned CE via our algorithms,compared to the CE of the corresponding certain market (i.e., here,“certain” means lacking uncertainty), as a function of the numberof samples. We find that EAP often yields better error guaranteesthan EA using far fewer samples, because it successfully prunesbuyers’ valuations (i.e., it ceases querying for values on bundles ofgoods that a CE provably does not comprise), even without any apriori knowledge of the market’s combinatorial structure.As the market size grows, an interesting tradeoff arises betweencomputational and sample efficiency. To prune a value query andretain rigorous guarantees on the quality of the learned CE, wemust solve a welfare-maximizing problem whose complexity growswith the market’s size. Consequently, at each iteration of EAP, for a r X i v : . [ c s . G T ] J a n ach value query, we are faced with a choice. Either solve saidwelfare-maximizing problem and potentially prune the value query(thereby saving on future samples), or defer attempts to prune thevalue query, until more is known about the market. To combat thissituation, we show that an upper bound on the optimal welfare’svalue (rather than the precise value) suffices to obtain rigorousguarantees on the learned CE’s quality. Such upper bounds can befound easily, by solving a relaxation of the welfare-maximizationproblem. Reminiscent of designing admissible heuristics in classicalsearch problems, this methodology applies to any combinatorialmarket, but at the same time allows for the application of domain-dependent knowledge to compute these upper bounds, when avail-able. Empirically, we show that a computationally cheap relaxationof the welfare-maximization problem yields substantial sample andcomputational savings in a large market. Related Work.
The idea for this paper stemmed from the workon abstraction in Fisher markets by Kroer et al. [22]. There, theauthors tackle the problem of computing equilibria in large marketsby creating an abstraction of the market, computing equilibria inthe abstraction, and lifting those equilibria back to the originalmarket. Likewise, we develop a pruning criterion which in effectbuilds an abstraction of any CM, where then compute a CE, whichis provably also an approximate CE in the original market.The mathematical formalism we adopt follows that of AreyanViqueira et al. [1]. There, the authors propose a mathematical frame-work for empirical game-theoretic analysis [37], and algorithmsthat learn the Nash equilibria of simulation-based games [33, 34].In this paper, we extend this methodology to market equilibria, andprovide analogous results in the case of CMs. Whereas intuitively, abasic pruning criterion for games is arguably more straightforward—simply prune dominated strategies—the challenge in this work wasto discover a pruning criterion that would likewise prune valuationsthat are provably not part of a CE.Jha and Zick [20] have also tackled the problem of learning CE inCM. Whereas our approach is to accurately learn only those com-ponents of the buyers’ valuations that determine a CE (up to PACguarantees), their approach bypasses the learning of agent pref-erences altogether, going straight for learning a solution concept,such as a CE. It is an open question as to whether one approachdominates the other, in the context of noisy CMs.Another related line of research is concerned with learning valua-tion functions from data [4, 5, 26]. In contrast, our work is concernedwith learning buyers’ valuations only in so much as it facilitateslearning CE. Indeed, our main conclusion is that CE often can belearned from just a subset of the buyers’ valuations.There is also a long line of work on preference elicitation incombinatorial auctions (e.g., [12]), where an auctioneer aims topose value queries in an intelligent order so as to minimize the com-putational burden on the bidders, while still clearing the auction.Finally, our pruning criterion relies on a novel application ofthe first welfare theorem of economics. While prior work has con-nected economic theory with algorithmic complexity [30], thiswork connects economic theory with statistical learning theory. The market structure they investigate is not identical to the structure studied here.Thus, at present, our results are not directly comparable.
We write X + to denote the set of positive values in a numerical set X including zero. Given an integer 𝑘 ∈ Z , we write [ 𝑘 ] to denotethe first 𝑘 integers, inclusive: i.e., [ 𝑘 ] = { , , . . . , 𝑘 } . Given a finiteset of integers 𝑍 ⊂ Z , we write 2 𝑍 to denote the power set of 𝑍 .A combinatorial market is defined by a set of goods and a setof buyers. We denote the set of goods by 𝐺 = [ 𝑚 ] , and the set ofbuyers by 𝑁 = [ 𝑛 ] . We index an arbitrary good by 𝑗 ∈ 𝐺 , and anarbitrary buyer by 𝑖 ∈ 𝑁 . A bundle of goods is a set of goods 𝑆 ⊆ 𝐺 .Each buyer 𝑖 is characterized by their preferences over bundles,represented as a valuation function 𝑣 𝑖 : 2 𝐺 ↦→ R + , where 𝑣 𝑖 ( 𝑆 ) ∈ R + is buyer 𝑖 ’s value for bundle 𝑆 . We assume valuations are normalizedso that 𝑣 𝑖 (∅) =
0, for all 𝑖 ∈ 𝑁 . Using this notation, a combinatorialmarket—market, hereafter—is a tuple 𝑀 = ( 𝐺, 𝑁, { 𝑣 𝑖 } 𝑖 ∈ 𝑁 ) .Given a market 𝑀 , an allocation S = ( 𝑆 , . . . , 𝑆 𝑛 ) denotes an as-signment of goods to buyers, where 𝑆 𝑖 ⊆ 𝐺 is the bundle assigned tobuyer 𝑖 . We consider only feasible allocations. An allocation S is fea-sible if 𝑆 𝑖 ∩ 𝑆 𝑘 = ∅ for all 𝑖, 𝑘 ∈ 𝑁 such that 𝑖 ≠ 𝑘 . We denote the setof all feasible allocations of market 𝑀 by F ( 𝑀 ) . The welfare of allo-cation S is defined as 𝑤 (S) = (cid:205) 𝑖 ∈ 𝑁 𝑣 𝑖 ( 𝑆 𝑖 ) . A welfare-maximizingallocation S ∗ is a feasible allocation that yields maximum welfareamong all feasible allocations, i.e., S ∗ ∈ arg max S∈F( 𝑀 ) 𝑤 ( 𝑀 ) . Wedenote by 𝑤 ∗ ( 𝑀 ) the welfare of any welfare-maximizing allocation S ∗ , i.e., 𝑤 ∗ ( 𝑀 ) = 𝑤 (S ∗ ) = (cid:205) 𝑖 ∈ 𝑁 𝑣 𝑖 ( 𝑆 ∗ 𝑖 ) .A pricing profile P = ( 𝑃 , . . . , 𝑃 𝑛 ) is a vector of 𝑛 pricing func-tions, one function 𝑃 𝑖 : 2 𝐺 ↦→ R + for each buyer, each mappingbundles to prices, 𝑃 𝑖 ( 𝑆 ) ∈ R + . The seller’s revenue of allocation S given a pricing P is (cid:205) 𝑖 ∈ 𝑁 𝑃 𝑖 ( 𝑆 𝑖 ) . We refer to pair (S , P) as a market outcome —outcome, for short. Given an outcome, buyer 𝑖 ’sutility is difference between its attained value and its payment, 𝑣 𝑖 ( 𝑆 𝑖 ) − 𝑃 𝑖 ( 𝑆 𝑖 ) , and the seller’s utility is equal to its revenue.In this paper, we are interested in approximations of one marketby another. We now define a mathematical framework in whichto formalize such approximations. In what follows, whenever wedecorate a market 𝑀 , e.g., 𝑀 ′ , what we mean is that we decorateeach of its components: i.e., 𝑀 ′ = ( 𝐺 ′ , 𝑁 ′ , { 𝑣 ′ 𝑖 } 𝑖 ∈ 𝑁 ′ ) .It will be convenient to refer to a subset of buyer–bundle pairs.We use the notation I ⊆ 𝑁 × 𝐺 for this purpose.Markets 𝑀 and 𝑀 ′ are compatible if 𝐺 = 𝐺 ′ and 𝑁 = 𝑁 ′ .Whenever a market 𝑀 is compatible with a market 𝑀 ′ , an out-come of 𝑀 is also an outcome of 𝑀 ′ . Given two compatible mar-kets 𝑀 and 𝑀 ′ , we measure the difference between them at I as ∥ 𝑀 − 𝑀 ′ ∥ I = max ( 𝑖,𝑆 ) ∈I | 𝑣 𝑖 ( 𝑆 ) − 𝑣 ′ 𝑖 ( 𝑆 )| . When I = 𝑁 × 𝐺 , thisdifference is precisely the infinity norm. Given 𝜀 > 𝑀 and 𝑀 ′ arecalled 𝜀 -approximations of one another if ∥ 𝑀 − 𝑀 ′ ∥ ∞ ≤ 𝜀 .The solution concept of interest in this paper is competitive equi-librium . A competitive equilibrium consists of two conditions: theutility-maximization (UM) condition and the revenue-maximization(RM) condition. UM ensures that the allocation maximizes buyers’utilities given the pricings, while RM ensures that the seller maxi-mizes its utility. Together, both conditions constitute an equilibriumof the market, i.e., an outcome where no agent has an incentiveto deviate by, for example, relinquishing its allocation. We nowformalize this solution concept, followed by its relaxation, centralwhen working with approximate markets. A competitive equilibrium is always guaranteed to exists [9]. efinition 1 (Competitive Eqilibrium).
Given a market 𝑀 ,an outcome (S , P) is a competitive equilibrium (CE) if: (UM) ∀ 𝑖 ∈ 𝑁,𝑇 ⊆ 𝐺 : 𝑣 𝑖 ( 𝑆 𝑖 ) − 𝑃 𝑖 ( 𝑆 𝑖 ) ≥ 𝑣 𝑖 ( 𝑇 ) − 𝑃 𝑖 ( 𝑇 ) (RM) ∀S ′ ∈ F ( 𝑀 ) : (cid:205) 𝑖 ∈ 𝑁 𝑃 𝑖 ( 𝑆 𝑖 ) ≥ (cid:205) 𝑖 ∈ 𝑁 𝑃 𝑖 ( 𝑆 ′ 𝑖 ) Definition 2 (Approximate Competitive Eqilibria).
Let 𝜀 > . An outcome (S , P) is a 𝜀 -competitive equilibrium ( 𝜀 -CE) if it is aCE in which UM holds up to 𝜀 : 𝜀 -(UM) ∀ 𝑖 ∈ 𝑁,𝑇 ⊆ 𝐺 : 𝑣 𝑖 ( 𝑆 𝑖 ) − 𝑃 𝑖 ( 𝑆 𝑖 ) + 𝜀 ≥ 𝑣 𝑖 ( 𝑇 ) − 𝑃 𝑖 ( 𝑇 ) For 𝛼 ≥
0, we denote by CE 𝛼 ( 𝑀 ) the set of all 𝛼 -approximateCE of 𝑀 , i.e., CE 𝛼 ( 𝑀 ) = {(S , P) : (S , P) is a 𝛼 -approximate CE of 𝑀 } . Note that CE ( 𝑀 ) is the set of (exact) CE of market 𝑀 , whichwe denote CE( 𝑀 ) .Theorem 1 (Competitive Eqilibrium Approximation). Let 𝜀 > . If 𝑀 and 𝑀 ′ are compatible markets such that ∥ 𝑀 − 𝑀 ′ ∥ ∞ ≤ 𝜀 ,then CE( 𝑀 ) ⊆ CE 𝜀 ( 𝑀 ′ ) ⊆ CE 𝜀 ( 𝑀 ) . Proof. We prove that: CE 𝛼 ( 𝑀 ) ⊆ CE 𝛼 + 𝜀 ( 𝑀 ′ ) , for 𝛼 ≥
0. Thisresult then implies
CE( 𝑀 ) ⊆ CE 𝜀 ( 𝑀 ′ ) when 𝛼 =
0; likewise, it(symmetrically) implies CE 𝜀 ( 𝑀 ′ ) ⊆ CE 𝜀 ( 𝑀 ) when 𝛼 = 𝜀 .Let 𝑀 and 𝑀 ′ be compatible markets s.t. ∥ 𝑀 − 𝑀 ′ ∥ ∞ ≤ 𝜀 . Sup-pose (S , P) is a 𝛼 -competitive equilibrium of 𝑀 . Our task is toshow that (S , P) , interpreted as an outcome of 𝑀 ′ , is a ( 𝛼 + 𝜀 ) -competitive equilibrium of 𝑀 ′ .First, note that the RM condition is immediately satisfied, because S and P do not change when interpreting (S , P) as an outcome of 𝑀 ′ . Thus, we need only show that the approximation holds for theUM condition: 𝑣 ′ 𝑖 ( 𝑆 𝑖 ) − 𝑃 𝑖 ( 𝑆 𝑖 ) ≥ 𝑣 𝑖 ( 𝑆 𝑖 ) − 𝑃 𝑖 ( 𝑆 𝑖 ) − 𝜀, ∀ 𝑖, 𝑆 𝑖 (1) ≥ 𝑣 𝑖 ( 𝑇 ) − 𝑃 𝑖 ( 𝑇 ) − 𝛼 − 𝜀, ∀ 𝑇 ⊆ 𝐺 (2) ≥ 𝑣 ′ 𝑖 ( 𝑇 ) − 𝑃 𝑖 ( 𝑇 ) − 𝛼 − 𝜀, ∀ 𝑇 ⊆ 𝐺 (3)where (1) and (3) follow because ∥ 𝑀 − 𝑀 ′ ∥ ∞ ≤ 𝜀 , and (2) followsbecause (S , P) is a 𝛼 -approximate CE of 𝑀 . □ We now present a formalism in which to model noisy combinato-rial markets. Intuitively, a noisy market is one in which buyers’valuations over bundles are not known precisely; rather, only noisysamples are available.Definition 3 (Conditional Combinatorial Markets). A con-ditional comb. market 𝑀 X = (X , 𝐺, 𝑁, { 𝑣 𝑖 } 𝑖 ∈ 𝑁 ) consists of a set ofconditions X , a set of goods 𝐺 , a set of buyers 𝑁 , and a set of condi-tional valuation functions { 𝑣 𝑖 } 𝑖 ∈ 𝑁 , where 𝑣 𝑖 : 2 𝐺 × X ↦→ R + . Givena condition 𝑥 ∈ X , the value 𝑣 𝑖 ( 𝑆, 𝑥 ) is 𝑖 ’s value for bundle 𝑆 ⊆ 𝐺 . Definition 4 (Expected Combinatorial Market).
Let 𝑀 X = (X , 𝐺, 𝑁, { 𝑣 𝑖 } 𝑖 ∈ 𝑁 ) be a conditional combinatorial market and let D be a distribution over X . For all 𝑖 ∈ 𝑁 , define the expected valuationfunction 𝑣 𝑖 : 2 𝐺 ↦→ R + by 𝑣 𝑖 ( 𝑆, D) = E 𝑥 ∼D [ 𝑣 𝑖 ( 𝑆, 𝑥 )] , and the corre-sponding expected combinatorial market as 𝑀 D = ( 𝐺, 𝑁, { 𝑣 𝑖 } 𝑖 ∈ 𝑁 ) . The goal of this work is to design algorithms that learn theapproximate CE of expected combinatorial markets. We will learntheir equilibria given access only to their empirical counterparts,which we define next. Definition 5 (Empirical Combinatorial Market).
Let 𝑀 X = (X , 𝐺, 𝑁, { 𝑣 𝑖 } 𝑖 ∈ 𝑁 ) be a conditional combinatorial market and let D be a distribution over X . Denote by 𝒙 = ( 𝑥 , . . . , 𝑥 𝑡 ) ∼ D a vec-tor of 𝑡 samples drawn from X according to distribution D . For all 𝑖 ∈ 𝑁 , we define the empirical valuation function ˆ 𝑣 𝑖 : 2 𝐺 ↦→ R + by ˆ 𝑣 𝑖 ( 𝑆 ) = 𝑡 (cid:205) 𝑡𝑙 = 𝑣 𝑖 ( 𝑆, 𝑥 𝑙 ) , and the corresponding empirical combina-torial market as ˆ 𝑀 𝒙 = ( 𝐺, 𝑁, { ˆ 𝑣 𝑖 } 𝑖 ∈ 𝑁 ) . Observation 1 (Learnability).
Let 𝑀 X be a conditional combi-natorial market and let D be a distribution over X . Let 𝑀 D and ˆ 𝑀 𝒙 be the corresponding expected and empirical combinatorial markets.If, for some 𝜀, 𝛿 > , it holds that P (cid:16)(cid:13)(cid:13) 𝑀 D − ˆ 𝑀 𝒙 (cid:13)(cid:13) ≤ 𝜀 (cid:17) ≥ − 𝛿 , then thecompetitive equilibria of 𝑀 D are learnable: i.e, any competitive equi-librium of 𝑀 D is a 𝜀 -competitive equilibrium of ˆ 𝑀 𝒙 with probabilityat least − 𝛿 . Theorem 1 implies that CE are approximable to within any de-sired 𝜀 > 𝛿 > Let 𝑀 X be a con-ditional combinatorial market, D a distribution over X , and I ⊆ 𝑁 × 𝐺 an index set. Suppose that for all 𝑥 ∈ X and ( 𝑖, 𝑆 ) ∈ I ,it holds that 𝑣 𝑖 ( 𝑆, 𝑥 ) ∈ [ , 𝑐 ] where 𝑐 ∈ R + . Then, with probabil-ity at least − 𝛿 over samples 𝒙 = ( 𝑥 , . . . , 𝑥 𝑡 ) ∼ D , it holds that (cid:13)(cid:13) 𝑀 D − ˆ 𝑀 𝒙 (cid:13)(cid:13) I ≤ 𝑐 √︁ ln ( |I| / 𝛿 ) / 𝑡 , where 𝛿 > . (Proof in the Appendix) Hoeffding’s inequality is a convenient and simple bound, whereonly knowledge of the range of values is required. However, theunion bound can be inefficient in large combinatorial markets. Thisshortcoming can be addressed via uniform convergence boundsand Rademacher averages [3, 7, 21]. Furthermore, sharper empiri-cal variance sensitive bounds have been shown to improve samplecomplexity in learning the Nash equilibria of black-box games [1].In particular, to obtain a confidence interval of radius 𝜀 in a combi-natorial market with index set I = 𝑁 × 𝐺 , Hoeffding’s inequalityrequires 𝑡 ∈ O( 𝑐 | 𝐺 | / 𝜀 ln | 𝑁 | / 𝛿 ) samples. Uniform convergencebounds can improve the | 𝐺 | term arising from the union bound , andvariance-sensitive bounds can largely replace dependence on 𝑐 with variances . Nonetheless, even without these augmentations,our methods are statistically efficient in | 𝐺 | , requiring only polyno-mial sample complexity to learn exponentially large combinatorialmarkets. EA (Algorithm 2) is a preference elicitation algorithm for combi-natorial markets. The algorithm places value queries, but is onlyassumed to elicit noisy values for bundles. The following guaranteefollows immediately from Lemma 1.Theorem 2 (Elicitation Algorithm Guarantees).
Let 𝑀 X be a conditional market, D be a distribution over X , I an index set, 𝑡 ∈ N > a number of samples, 𝛿 > , and 𝑐 ∈ R + . Suppose that forall 𝑥 ∈ X and ( 𝑖, 𝑆 ) ∈ I , it holds that 𝑣 𝑖 ( 𝑆, 𝑥 ) ∈ [ , 𝑐 ] . If EA outputs ({ ˆ 𝑣 𝑖 } ( 𝑖,𝑆 ) ∈I , ˆ 𝜀 ) on input ( 𝑀 X , D , I , 𝑡, 𝛿, 𝑐 ) , then, with probability atleast − 𝛿 , it holds that (cid:13)(cid:13) 𝑀 D − ˆ 𝑀 𝒙 (cid:13)(cid:13) I ≤ 𝑐 √︁ ln ( |I| / 𝛿 ) / 𝑡 . Proof. The result follows from Lemma 1. □ lgorithm 1 Elicitation Algorithm with Pruning (EAP)
Input : 𝑀 X , D , 𝒕 , 𝜹 , 𝑐, 𝜀 .A conditional combinatorial market 𝑀 X , a distribution D over X ,a sampling schedule 𝒕 , a failure probability schedule 𝜹 , a pruningbudget schedule 𝝅 , a valuation range 𝑐 , and a target approx. error 𝜀 . Output : Valuation estimates ˆ 𝑣 𝑖 ( 𝑆 ) , for all ( 𝑖, 𝑆 ) , approximationerrors ˆ 𝜀 𝑖,𝑆 , failure probability ˆ 𝛿 , and CE error ˆ 𝜀 . I ← 𝑁 × 𝐺 {Initialize index set} ( ˆ 𝑣 𝑖 ( 𝑆 ) , ˆ 𝜀 𝑖,𝑆 ) ← ( , 𝑐 / ) , ∀( 𝑖, 𝑆 ) ∈ I {Initialize outputs} for 𝑘 ∈ , . . . , | 𝒕 | do ({ ˆ 𝑣 𝑖 } ( 𝑖,𝑆 ) ∈I , ˆ 𝜀 ) ← EA ( 𝑀 X , D , I , 𝑡 𝑘 , 𝛿 𝑘 , 𝑐 ) {Call Alg. 2} ˆ 𝜀 𝑖,𝑆 ← ˆ 𝜀, ∀( 𝑖, 𝑆 ) ∈ I {Update error rates} if ˆ 𝜀 ≤ 𝜀 or 𝑘 = | 𝒕 | or I = ∅ then return ({ ˆ 𝑣 𝑖 } 𝑖 ∈ 𝑁 , { ˆ 𝜀 𝑖,𝑆 } ( 𝑖,𝑆 ) ∈ 𝑁 × 𝐺 , (cid:205) 𝑘𝑙 = 𝛿 𝑙 , ˆ 𝜀 ) end if Let ˆ 𝑀 be the market with valuations { ˆ 𝑣 𝑖 } ( 𝑖,𝑆 ) ∈I I prune ← ∅ {Initialize set of indices to prune} I candidates ← a subset of I of size at most 𝜋 𝑘 {Select some active pairs as candidates for pruning} for ( 𝑖, 𝑆 ) ∈ I candidates do Let ˆ 𝑀 −( 𝑖,𝑆 ) be the ( 𝑖, 𝑆 ) -submarket of ˆ 𝑀 . Let 𝑤 ⋄( 𝑖,𝑆 ) an upper bound of 𝑤 ∗ ( ˆ 𝑀 −( 𝑖,𝑆 ) ) . if ˆ 𝑣 𝑖 ( 𝑆 ) + 𝑤 ⋄( 𝑖,𝑆 ) + 𝜀𝑛 < 𝑤 ∗ ( ˆ 𝑀 ) then I prune ← I prune ∪ ( 𝑖, 𝑆 ) end if end for I ← I \ I prune end for
EA elicits buyers’ valuations for all bundles, but in certain situ-ations, some buyer valuations are not relevant for computing aCE—although bounds on all of them are necessary to guaranteestrong bounds on the set of CE (Theorem 1). For example, in afirst-price auction for one good, it is enough to accurately learn thehighest bid, but is not necessary to accurately learn all other bids,if it is known that they are lower than the highest. Since our goalis to learn CE, we present EAP (Algorithm 1), an algorithm thatdoes not sample uniformly, but instead adaptively decides whichvalue queries to prune so that, with provable guarantees, EAP’sestimated market satisfies the conditions of Theorem 1.EAP (Algorithm 1) takes as input a sampling schedule 𝒕 , a fail-ure probability schedule 𝜹 , and a pruning budget schedule 𝝅 . Thesampling schedule 𝒕 is a sequence of | 𝒕 | strictly decreasing inte-gers 𝑡 > 𝑡 > · · · > 𝑡 | 𝒕 | , where 𝑡 𝑘 is the total number of samplesto take for each ( 𝑖, 𝑆 ) pair during EAP’s 𝑘 -th iteration. The fail-ure probability schedule 𝜹 is a sequence of the same length as 𝒕 , where 𝛿 𝑘 ∈ ( , ) is the 𝑘 -th iteration’s failure probability and (cid:205) 𝑘 𝛿 𝑘 ∈ ( , ) is the total failure probability. The pruning budgetschedule 𝝅 is a sequence of integers also of the same length as 𝒕 , where 𝜋 𝑘 is the maximum number of ( 𝑖, 𝑆 ) pruning candidatepairs. The algorithm progressively elicits buyers’ valuations viarepeated calls to EA. However, between calls to EA, EAP searchesfor value queries that are provably not part of a CE; the size ofthis search is dictated by the pruning schedule. All such queries Algorithm 2
Elicitation Algorithm (EA)
Input : 𝑀 X , D , I , 𝑡, 𝛿, 𝑐 .A conditional combinatorial market 𝑀 X , a distribution D over X ,an index set I , sample size 𝑡 , failure prob. 𝛿 , and valuation range 𝑐 . Output : Valuation estimates ˆ 𝑣 𝑖 ( 𝑆 ) , for all ( 𝑖, 𝑆 ) ∈ I , and anapproximation error ˆ 𝜀 . ( 𝑥 , . . . , 𝑥 𝑡 ) ∼ D {Draw 𝑡 samples from D } for ( 𝑖, 𝑆 ) ∈ I do ˆ 𝑣 𝑖 ( 𝑆 ) ← 𝑡 (cid:205) 𝑡𝑙 = 𝑣 𝑖 ( 𝑆, 𝑥 𝑙 ) end for ˆ 𝜀 ← 𝑐 √︁ ln ( |I| / 𝛿 ) / 𝑡 {Compute error} return ({ ˆ 𝑣 𝑖 } ( 𝑖,𝑆 ) ∈I , ˆ 𝜀 ) (i.e., buyer–bundle pairs) then cease to be part of the index set withwhich EA is called in future iterations.In what follows, we prove several intermediate results, whichenable us to prove the main result of this section, Theorem 4, whichestablishes EAP’s correctness. Specifically, the market learned byEAP—with potentially different numbers of samples for different ( 𝑖, 𝑆 ) pairs—is enough to provably recover any CE of the underlyingmarket.Lemma 2 (Optimal Welfare Approximations). Let 𝑀 and 𝑀 ′ be compatible markets such that they 𝜀 -approximate one another.Then | 𝑤 ∗ ( 𝑀 ) − 𝑤 ∗ ( 𝑀 ′ )| ≤ 𝜀𝑛 . Proof. Let S ∗ be a welfare-maximizing allocation for 𝑀 and U ∗ be a welfare-maximizing allocation for 𝑀 ′ . Let 𝑤 ∗ ( 𝑀 ) be themaximum achievable welfare in market 𝑀 . Then, 𝑤 ∗ ( 𝑀 ) = ∑︁ 𝑖 ∈ 𝑁 𝑣 𝑖 (S ∗ 𝑖 ) ≥ ∑︁ 𝑖 ∈ 𝑁 𝑣 𝑖 (U ∗ 𝑖 ) ≥ ∑︁ 𝑖 ∈ 𝑁 𝑣 ′ 𝑖 (U ∗ 𝑖 ) − 𝜀𝑛 = 𝑤 ∗ ( 𝑀 ′ ) − 𝜀𝑛 The first inequality follows from the optimality of S ∗ in 𝑀 , and thesecond from the 𝜀 -approximation assumption. Likewise, 𝑤 ∗ ( 𝑀 ′ ) ≥ 𝑤 ∗ ( 𝑀 ) − 𝜀𝑛 , so the result holds. □ The key to this work was the discovery of a pruning criterionthat removes ( 𝑖, 𝑆 ) pairs from consideration if they are provablynot part of any CE. Our check relies on computing the welfare ofthe market without the pair: i.e., in submarkets.Definition 6. Given a market 𝑀 and buyer–bundle pair ( 𝑖, 𝑆 ) ,the ( 𝑖, 𝑆 ) -submarket of 𝑀 , denoted by 𝑀 −( 𝑖,𝑆 ) , is the market obtainedby removing all goods in 𝑆 and buyer 𝑖 from market 𝑀 . That is, 𝑀 −( 𝑖,𝑆 ) = ( 𝐺 \ 𝑆, 𝑁 \ { 𝑖 } , { 𝑣 𝑘 } 𝑘 ∈ 𝑁 \{ 𝑖 } ) . Lemma 3 (Pruning Criteria).
Let 𝑀 and 𝑀 ′ be compatible mar-kets such that ∥ 𝑀 − 𝑀 ′ ∥ ∞ ≤ 𝜀 . In addition, let ( 𝑖, 𝑆 ) be a buyer,bundle pair, and 𝑀 ′−( 𝑖,𝑆 ) be the ( 𝑖, 𝑆 ) -submarket of 𝑀 ′ . Finally, let 𝑤 ⋄( 𝑖,𝑆 ) ∈ R + upper bound 𝑤 ∗ ( 𝑀 ′−( 𝑖,𝑆 ) ) , i.e., 𝑤 ∗ ( 𝑀 ′−( 𝑖,𝑆 ) ) ≤ 𝑤 ⋄( 𝑖,𝑆 ) . Ifthe following pruning criterion holds, then 𝑆 is not allocated to 𝑖 inany welfare-maximizing allocation of 𝑀 : 𝑣 ′ 𝑖 ( 𝑆 ) + 𝑤 ⋄( 𝑖,𝑆 ) + 𝜀𝑛 < 𝑤 ∗ ( 𝑀 ′ ) . (4)roof. Let S ∗ , U ∗ , and U ∗−( 𝑖,𝑆 ) be welfare-maximizing alloca-tions of markets 𝑀, 𝑀 ′ , and 𝑀 ′−( 𝑖,𝑆 ) , respectively. Then, 𝑤 ∗ ( 𝑀 ) ≥ 𝑤 ∗ ( 𝑀 ′ ) − 𝜀𝑛 (5) > 𝑣 ′ 𝑖 ( 𝑆 ) + 𝑤 ⋄( 𝑖,𝑆 ) + 𝜀𝑛 (6) ≥ 𝑣 ′ 𝑖 ( 𝑆 ) + 𝑤 ∗ ( 𝑀 ′−( 𝑖,𝑆 ) ) + 𝜀𝑛 (7) ≥ 𝑣 𝑖 ( 𝑆 ) − 𝜀 + 𝑤 ∗ ( 𝑀 −( 𝑖,𝑆 ) ) − 𝜀 ( 𝑛 − ) + 𝜀𝑛 (8) = 𝑣 𝑖 ( 𝑆 ) + 𝑤 ∗ ( 𝑀 −( 𝑖,𝑆 ) ) (9)The first inequality follows from Lemma 2. The second followsfrom Equation (4) and the third because 𝑤 ⋄( 𝑖,𝑆 ) is an upper boundof 𝑤 ∗ ( 𝑀 ′−( 𝑖,𝑆 ) ) . The fourth inequality follows from the assump-tion that ∥ 𝑀 − 𝑀 ′ ∥ ∞ ≤ 𝜀 , and by Lemma 2 applied to submarket 𝑀 −( 𝑖,𝑆 ) . Therefore, the allocation where 𝑖 gets 𝑆 cannot be welfare-maximizing in market 𝑀 . □ Lemma 3 provides a family of pruning criteria parameterizedby the upper bound 𝑤 ⋄( 𝑖,𝑆 ) . The closer 𝑤 ⋄( 𝑖,𝑆 ) is to 𝑤 ∗ ( 𝑀 ′−( 𝑖,𝑆 ) ) , thesharper the pruning criterion, with the best pruning criterion being 𝑤 ⋄( 𝑖,𝑆 ) = 𝑤 ∗ ( 𝑀 ′−( 𝑖,𝑆 ) ) . However, solving for 𝑤 ∗ ( 𝑀 ′−( 𝑖,𝑆 ) ) exactly caneasily become a bottleneck, as the pruning loop requires a solutionto many such instances, one per ( 𝑖, 𝑆 ) pair (Line 12 of Algorithm 1).Alternatively, one could compute looser upper bounds, and therebytrade off computation time for opportunities to prune more ( 𝑖, 𝑆 ) pairs, when the upper bound is not tight enough. In our experiments,we show that even relatively loose but cheap-to-compute upperbounds result in significant pruning and, thus, savings along bothdimensions—computational and sample complexity.To conclude this section, we establish the correctness of EAP.For our proof we rely on the following generalization of the firstwelfare theorem of economics, which handles additive errors.Theorem 3 (First Welfare Theorem [29]). For 𝜀 > , let (S , P) be an 𝜀 -competitive equilibrium of 𝑀 . Then, S is a welfare-maximizing allocation of 𝑀 , up to additive error 𝜀𝑛 . Theorem 4 (Elicitation Algorithm with Pruning Guaran-tees).
Let 𝑀 X be a conditional market, let D be a distribution over X , and let 𝑐 ∈ R + . Suppose that for all 𝑥 ∈ X and ( 𝑖, 𝑆 ) ∈ I , it holdsthat 𝑣 𝑖 ( 𝑆, 𝑥 ) ∈ [ , 𝑐 ] , where 𝑐 ∈ R . Let 𝒕 be a sequence of strictlyincreasing integers, and 𝜹 a sequence of the same length as 𝒕 suchthat 𝛿 𝑘 ∈ ( , ) and (cid:205) 𝑘 𝛿 𝑘 ∈ ( , ) . If EAP outputs ({ ˆ 𝑣 𝑖 } 𝑖 ∈ 𝑁 , { ˆ 𝜀 𝑖,𝑆 } ( 𝑖,𝑆 ) ∈ 𝑁 × 𝐺 , − (cid:205) 𝑘 𝛿 𝑘 , ˆ 𝜀 ) on input ( 𝑀 X , D , 𝒕 , 𝜹 , 𝑐, 𝜀 ) ,then the following holds with probability at least − (cid:205) 𝑘 𝛿 𝑘 :1. (cid:13)(cid:13) 𝑀 D − ˆ 𝑀 (cid:13)(cid:13) I ≤ ˆ 𝜀 𝑖,𝑆 CE( 𝑀 D ) ⊆ CE 𝜀 ( ˆ 𝑀 ) ⊆ CE 𝜀 ( 𝑀 D ) Here ˆ 𝑀 is the empirical market obtained via EAP , i.e., the marketwith valuation functions given by { ˆ 𝑣 𝑖 } 𝑖 ∈ 𝑁 . Proof. To show part 1, note that at each iteration 𝑘 of EAP,Line 5 updates the error estimates for each ( 𝑖, 𝑆 ) after a call to EA(Line 4 of EAP) with input failure probability 𝛿 𝑘 . Theorem 2 impliesthat each call to EA returns estimated values that are within ˆ 𝜀 oftheir expected value with probability at least 1 − 𝛿 𝑘 . By union bound-ing all calls to EA within EAP, part 1 then holds with probabilityat least 1 − (cid:205) 𝑘 𝛿 𝑘 .To show part 2, note that only pairs ( 𝑖, 𝑆 ) for which Equation (4)holds are removed from index set I (Line 14 of EAP). By Lemma 3, no such pair can be part of any approximate welfare-maximizingallocation of the expected market, 𝑀 D . By Theorem 3, no such paircan be a part of any CE. Consequently, ˆ 𝑀 contains accurate enoughestimates (up to 𝜀 ) of all ( 𝑖, 𝑆 ) pairs that may participate in any CE.Part 2 then follows from Theorem 1. □ The goal of our experiments is to robustly evaluate the empiricalperformance of our algorithms. To this end, we experiment with avariety of qualitatively different inputs. In particular, we evaluateour algorithms on both unit-demand valuations, the Global Syn-ergy Value Model (GSVM) [16], and the Local Synergy Value Model(LSVM) [32]. Unit-demand valuations are a class of valuations cen-tral to the literature on economics and computation [27] for whichefficient algorithms exist to compute CE [17]. GSVM and LSVMmodel situations in which buyers’ valuations encode complements;CE are not known be efficiently computable, or even representable,in these markets.While CE are always guaranteed to exist (e.g., [9]), in the worstcase, they might require personalized bundle prices. These pricesare computationally complex, not to mention out of favor [18]. Apricing P = ( 𝑃 , . . . , 𝑃 𝑛 ) is anonymous if it charges every buyer thesame price, i.e., 𝑃 𝑖 = 𝑃 𝑘 = 𝑃 for all 𝑖 ≠ 𝑘 ∈ 𝑁 . Moreover, an anony-mous pricing is linear if there exists a set of prices { 𝑝 , . . . , 𝑝 𝑚 } ,where 𝑝 𝑗 is good 𝑗 ’s price, such that 𝑃 ( 𝑆 ) = (cid:205) 𝑗 ∈ 𝑆 𝑝 𝑗 . In whatfollows, we refer to linear, anonymous pricings as linear prices.Where possible, it is preferable to work with linear prices, as theyare simpler, e.g., when bidding in an auction [24]. In our presentstudy—one of the first empirical studies on learning CE—we thusfocus on linear prices, leaving as future research the empirical effect of more complex pricings. To our knowledge, there have been no analogous attempts atlearning CE; hence, we do not reference any baseline algorithmsfrom the literature. Rather, we compare the performance of EAP,our pruning algorithm, to EA, investigating the quality of the CElearned by both, as well as their sample efficiencies.
We first explain our experimental setup, and then present results.We let 𝑈 [ 𝑎, 𝑏 ] denote the continuous uniform distribution overrange [ 𝑎, 𝑏 ] , and 𝑈 { 𝑘, 𝑙 } , the discrete uniform distribution over set { 𝑘, 𝑘 + , . . . , 𝑙 } , for 𝑘 ≤ 𝑙 ∈ N . Simulation of Noisy Combinatorial Markets.
We start by drawingmarkets from experimental market distributions. Then, fixing amarket, we simulate noisy value elicitation by adding noise drawnfrom experimental noise distributions to buyers’ valuations in themarket. We refer to a market realization 𝑀 drawn from an ex-perimental market distribution as the ground-truth market. Ourexperiments then measure how well we can approximate the CE ofa ground-truth market 𝑀 given access only to noisy samples of it.Fix a market 𝑀 and a condition set X = [ 𝑎, 𝑏 ] , where 𝑎 < 𝑏 .Define the conditional market 𝑀 X , where 𝑣 𝑖 ( 𝑆, 𝑥 𝑖𝑆 ) = 𝑣 𝑖 ( 𝑆 ) + 𝑥 𝑖𝑆 ,for 𝑥 𝑖𝑆 ∈ X . In words, when eliciting 𝑖 ’s valuation for 𝑆 , we assume Note that all our theoretical results hold for any pricing profile. Lahaie and Lubin [25], for example, search for prices in between linear and bundle. dditive noise, namely 𝑥 𝑖𝑆 . The market 𝑀 X together with distribu-tion D over X is the model from which our algorithms elicit noisyvaluations from buyers. Then, given samples 𝒙 of 𝑀 X , the empiricalmarket ˆ 𝑀 𝒙 is the market estimated from the samples. Note that ˆ 𝑀 𝒙 is the only market we get to observe in practice.We consider only zero-centered noise distributions. In this case,the expected combinatorial market 𝑀 D is the same as the ground-truth market 𝑀 since, for every 𝑖, 𝑆 ∈ 𝑁 × 𝐺 it holds that 𝑣 𝑖 ( 𝑆, D) = E D [ 𝑣 𝑖 ( 𝑆, 𝑥 𝑖𝑆 )] = E D [ 𝑣 𝑖 ( 𝑆 ) + 𝑥 𝑖𝑆 ] = 𝑣 𝑖 ( 𝑆 ) . While this noise struc-ture is admittedly simple, we robustly evaluate our algorithms alonganother dimension, as we study several rich market structures (unit-demand, GSVM, and LSVM). An interesting future direction wouldbe to also study richer noise structures, e.g., letting noise vary witha bundle’s size, or other market characteristics. Utility-Maximization (UM) Loss.
To measure the quality of a CE (S ′ , P ′ ) computed for a market 𝑀 ′ in another market 𝑀 , we firstdefine the per-buyer metric UM-Loss
𝑀,𝑖 as follows,
UM-Loss
𝑀,𝑖 (S ′ , P ′ ) = max 𝑆 ⊆ 𝐺 ( 𝑣 𝑖 ( 𝑆 ) − 𝑃 ′ ( 𝑆 )) − ( 𝑣 𝑖 ( 𝑆 ′ 𝑖 ) − 𝑃 ′ ( 𝑆 ′ 𝑖 )) , i.e., the difference between the maximum utility 𝑖 could have at-tained at prices P ′ and the utility 𝑖 attains at the outcome (S ′ , P ′ ) .Our metric of interest is then UM-Loss 𝑀 defined as, UM-Loss 𝑀 (S ′ , P ′ ) = max 𝑖 ∈ 𝑁 UM-Loss
𝑀,𝑖 (S ′ , P ′ ) , which is a worst-case measure of utility loss over all buyers in themarket. Note that it is not useful to incorporate the SR conditioninto a loss metric, because it is always satisfied.In our experiments, we measure the UM loss that a CE of an em-pirical market obtains, evaluated in the corresponding ground-truthmarket. Thus, given an empirical estimate ˆ 𝑀 𝒙 of 𝑀 , and a CE ( ˆ S , ˆ P) in ˆ 𝑀 𝒙 , we measure UM-Loss 𝑀 ( ˆ S , ˆ P) , i.e., the loss in 𝑀 at prices ˆ P of CE ( ˆ S , ˆ P) . Theorem 1 implies that if ˆ 𝑀 𝒙 is an 𝜀 -approximationof 𝑀 , then UM-Loss 𝑀 ( ˆ S , ˆ P) ≤ 𝜀 . Moreover, Theorem 2 yields thesame guarantees, but with probability at least 1 − 𝛿 , provided the 𝜀 -approximation holds with probability at least 1 − 𝛿 . Sample Efficiency of
EAP . We say that algorithm 𝐴 has better sample efficiency than algorithm 𝐵 if 𝐴 requires fewer samples than 𝐵 to achieve at least the same 𝜀 accuracy.Fixing a condition set X , a distribution D over X , and a con-ditional market 𝑀 X , we use the following experimental design toevaluate EAP’s sample efficiency relative to that of EA. Given adesired error guarantee 𝜀 >
0, we compute the number of samples 𝑡 ( 𝜀 ) that would be required for EA to achieve accuracy 𝜀 . We thenuse the following doubling strategy as a sampling schedule forEAP, 𝒕 ( 𝑡 ( 𝜀 )) = [ 𝑡 ( 𝜀 ) / , 𝑡 ( 𝜀 ) / , 𝑡 ( 𝜀 ) , 𝑡 ( 𝜀 )] , rounding to the nearestinteger as necessary, and the following failure probability schedule 𝜹 = [ . , . , . , . ] , which sums to 0 . unconstrained pruning budget schedule by 𝝅 = [∞ , ∞ , ∞ , ∞] , which by convention means that at every iteration,all active pairs are candidates for pruning. Using these schedules,we run EAP with a desired accuracy of zero. We denote by 𝜀 EAP ( 𝜀 ) the approximation guarantee achieved by EAP upon termination. 𝜀 = . 𝜀 = . P min ˆ P max ˆ P min ˆ P max Uniform 0.0018 0.0020 0.0074 0.0082Preferred-Good 0.0019 0.0023 0.0080 0.0094Preferred-Good-Distinct 0.0000 0.0020 0.0000 0.0086Preferred-Subset 0.0019 0.0022 0.0076 0.0090
Table 1: Average
UM-Loss for 𝜀 ∈ { . , . } . A buyer 𝑖 is endowed with unit-demand valuations if, for all 𝑆 ⊆ 𝐺 , 𝑣 𝑖 ( 𝑆 ) = max 𝑗 ∈ 𝑆 𝑣 𝑖 ({ 𝑗 }) . In a unit-demand market, all buyers haveunit-demand valuations. A unit-demand market can be compactlyrepresented by matrix V , where entry 𝑣 𝑖 𝑗 ∈ R + is 𝑖 ’s value for 𝑗 , i.e., 𝑣 𝑖 𝑗 = 𝑣 𝑖 ({ 𝑗 }) . In what follows, we denote by V a random variableover unit-demand valuations.We construct four different distributions over unit-demand mar-kets: Uniform, Preferred-Good, Preferred-Good-Distinct, andPreferred-Subset. All distributions are parameterized by 𝑛 and 𝑚 , the number of buyers and goods, respectively. A uniform unit-demand market V ∼
Uniform is such that for all 𝑖, 𝑗, 𝑣 𝑖 𝑗 ∼ 𝑈 [ , ] .When V ∼
Preferred-Good, each buyer 𝑖 has a preferred good 𝑗 𝑖 , with 𝑗 𝑖 ∼ 𝑈 { , . . . , 𝑚 } and 𝑣 𝑖 𝑗 𝑖 ∼ 𝑈 [ , ] . Conditioned on 𝑣 𝑖 𝑗 𝑖 , 𝑖 ’s value for good 𝑘 ≠ 𝑗 𝑖 is given by 𝑣 𝑖𝑘 = 𝑣 𝑖𝑗𝑖 / 𝑘 . DistributionPreferred-Good-Distinct is similar to Preferred-Good, exceptthat no two buyers have the same preferred good. (Note that thePreferred-Good-Distinct distribution is only well defined when 𝑛 ≤ 𝑚 .) Finally, when V ∼
Preferred-Subset, each buyer 𝑖 is in-terested in a subset of goods 𝑖 𝐺 ⊆ 𝐺 , where 𝑖 𝐺 is drawn uniformlyat random from the set of all bundles. Then, the value 𝑖 has for 𝑗 isgiven by 𝑣 𝑖 𝑗 ∼ 𝑈 [ , ] , if 𝑗 ∈ 𝑖 𝐺 ; and 0, otherwise.In unit-demand markets, we experiment with three noise mod-els, low, medium, and high, by adding noise drawn from 𝑈 [− . , . ] , 𝑈 [− , ] , and 𝑈 [− , ] , respectively. We choose 𝑛, 𝑚 ∈ { , , , } . Unit-demand Empirical UM Loss of EA . As a learned CE is a CEof a learned market, we require a means of computing the CE ofa market—specifically, a unit-demand market V . To do so, we firstsolve for the welfare-maximizing allocation S ∗ V of V , by solvingfor the maximum weight matching using Hungarian algorithm [23]in the bipartite graph whose weight matrix is given by V . Fixing S ∗ V , we then solve for prices via linear programming [9]. In general,there might be many prices that couple with S ∗ V to form a CE of V . For simplicity, we solve for two pricings given S ∗ V , the revenue-maximizing P max and revenue-minimizing P min , where revenue isdefined as the sum of the prices.For each distribution, we draw 50 markets, and for each suchmarket V , we run EA four times, each time to achieve guarantee 𝜀 ∈ { . , . , . , . } . EA then outputs an empirical estimate ˆ V for each V . We compute outcomes (S ∗ ˆ V , ˆ P max ) and (S ∗ ˆ V , ˆ P min ) , andmeasure UM-Loss V (S ∗ ˆ V , ˆ P max ) and UM-Loss V (S ∗ ˆ V , ˆ P min ) . We thenaverage across all market draws, for both the minimum and themaximum pricings. Table 1 summarizes a subset of these results.The error guarantees are consistently met across the board, indeedby one or two orders of magnitude, and they degrade as expected: Since in all our experiments, we draw values from continuous distributions, weassume that the set of markets with multiple welfare-maximizing allocations is ofnegligible size. Therefore, we can ignore ties. igure 1: Mean
EAP sample efficiency relative to EA , 𝜀 = . . Each ( 𝑖, 𝑗 ) pair is annotated with the corresponding % saving. i.e., with higher values of 𝜀 . We note that the quality of the learnedCE is roughly the same for all distributions, except in the caseof ˆ P min and Preferred-Good-Distinct, where learning is moreaccurate. For this distribution, it is enough to learn the preferredgood of each buyer. Then, one possible CE is to allocate each buyerits preferred good and price all goods at zero which yields near no UM-Loss . Note that, in general, pricing all goods at zero is not aCE, unless the market has some special structure, like the marketsdrawn from Preferred-Good-Distinct.
Unit-demand Sample Efficiency.
We use pruning schedule 𝝅 = [∞ , ∞ , ∞ , ∞] and for each ( 𝑖, 𝑗 ) pair, we use the Hungarian algo-rithm [23] to compute the optimal welfare of the market without ( 𝑖, 𝑗 ) . In other words, in each iteration, we consider all active ( 𝑖, 𝑗 ) pairs as pruning candidates (Algorithm 1, Line 11), and for each wecompute the optimal welfare (Algorithm 1, Line 14).For each market distribution, we compute the average of thenumber of samples used by EAP across 50 independent marketdraws. We report samples used by EAP as a percentage of thenumber of samples used by EA to achieve the same guarantee,namely, 𝜀 EAP ( 𝜀 ) , for each initial value of 𝜀 . Figure 1 depicts theresults of these experiments as heat maps, for all distributions andfor 𝜀 = .
05, where darker colors indicate more savings, and thusbetter EAP sample efficiency.A few trends arise, which we note are similar for other values of 𝜀 . For a fixed number of buyers, EAP’s sample efficiency improvesas the number of goods increases, because fewer goods can beallocated, which means that there are more candidate values toprune, resulting in more savings. On the other hand, the sampleefficiency usually decreases as the number of buyers increases; thisis to be expected, as the pruning criterion degrades with the numberof buyers (Lemma 3). While savings exceed 30% across the board,we note that Uniform, the market with the least structure, achievesthe least savings, while Preferred-Subset and Preferred-Good-Distinct achieve the most. This finding shows that EAP is capableof exploiting the structure present in these distributions, despitenot knowing anything about them a priori .Finally, we note that sample efficiency quickly degrades forhigher values of 𝜀 . In fact, for high enough values of 𝜀 (in ourexperiments, 𝜀 = . ( 𝑖, 𝑗 ) pairs early enough:i.e., during the first few iterations of EAP. When 𝜀 is large, however,our sampling schedule does not allocate enough samples early on.When designing sampling schedules for EAP, one must allocateenough (but not too many) samples at the beginning of the schedule. Precisely how to determine this schedule is an empirical question,likely dependent on the particular application at hand. In this next set of experiments, we test the empirical performance ofour algorithms in more complex markets, where buyers valuationscontain synergies. Synergies are a common feature of many high-stakes combinatorial markets. For example, telecommunicationservice providers might value different bundles of radio spectrumlicenses differently, depending on whether the licenses in the bundlecomplement one another. For example, a bundle including NewJersey and Connecticut might not be very valuable unless it alsocontains New York City.Specifically, we study the Global Synergy Value Model (GSVM) [16]and the Local Synergy Value Model (LSVM) [32]. These models ormarkets capture buyers’ synergies as a function of buyers’ types andtheir (abstract) geographical locations. In both GSVM and LSVM,there are 18 licenses, with buyers of two types: national or regional.A national buyer is interested in larger packages than regional buy-ers, whose interests are limited to certain regions. GSVM has sixregional bidders and one national bidder and models geographicalregions as two circles. LSVM has five regional bidders and one na-tional bidder and uses a rectangular model. The models differ in theexact ways buyers’ values are drawn, but in any case, synergies aremodeled by suitable distance metrics. In our experiments, we drawinstances of both GSVM and LSVM using SATS, a universal spec-trum auction test suite developed by researchers to test algorithmsfor combinatorial markets [36].
Experimental Setup.
On average, the value a buyer has for anarbitrary bundle in either GSVM or LSVM markets is approximately80. We introduce noise i.i.d. noise from distribution 𝑈 [− , ] whoserange is 2, or 2.5% of the expected buyer’s value for a bundle. AsGSVM’s buyers’ values are at most 400, and LSVM’s are at most 500,we use valuation ranges 𝑐 =
402 and 𝑐 =
502 for GSVM and LSVM,respectively. We note that a larger noise range yields qualitativelysimilar results with errors scaling accordingly.For the GSVM markets, we use the pruning budget schedule 𝝅 = [∞ , ∞ , ∞ , ∞] . For each ( 𝑖, 𝑆 ) pair, we solve the welfare maxi-mization problem using an off-the-shelf solver. In an LSVM market,the national bidder demands all 18 licenses. The welfare optimiza-tion problem in an LSVM market is solvable in a few seconds. Still, the many submarkets (in the hundreds of thousands) call for afinite pruning budget schedule and a cheaper-to-compute welfare We include ILP formulations and further technical details in the appendix. Approximately 20 seconds in our experiments, details appear in the appendix.
SVM LSVM 𝜀 EA EAP 𝜀 EAP
UM Loss EA EAP 𝜀 EAP
UM Loss1.25 2 , ± . ± .
01 0 . ± . , ± , ± , . ± .
00 0 . ± . ± . ± .
02 0 . ± . , ± , ± , . ± .
00 0 . ± . ± . ± .
03 0 . ± . , ± , ±
933 3 . ± .
01 0 . ± . ± . ± .
04 0 . ± . , ± , ±
211 7 . ± .
01 0 . ± . Table 2: GSVM (left group) and LSVM (right group) results. Each group reports sample efficiency and UM loss. Each row ofthe table reports results for a fixed value of 𝜀 . Results are 95% confidence intervals over 40 GSVM market draws and 50 LSVMmarket draws, except for EA ’s number of samples in the case of GSVM which is a deterministic quantity (a GSVM market isof size 4,480). The values in bold indicate the more sample efficient algorithm. Numbers of samples are reported in millions. upper bound. In fact, to address LSVM’s size complexity, we slightlymodify EAP, as explained next. A two-pass strategy for LSVM.
Because of the complexity ofLSVM markets, we developed a heuristic pruning strategy, in whichwe perform two pruning passes during each iteration of EAP. Theidea is to compute a computationally cheap upper bound on wel-fare with pruning budget schedule 𝝅 = [∞ , ∞ , ∞ , ∞] in the firstpass, use this bound instead of the optimal for each active ( 𝑖, 𝑆 ) .We compute this bound using the classic relaxation technique tocreate admissible heuristics. Concretely, given a candidate ( 𝑖, 𝑆 ) pair, we compute the maximum welfare in the absence of pair ( 𝑖, 𝑆 ) ,ignoring feasibility constraints: 𝑤 ⋄( 𝑖,𝑆 ) = ∑︁ 𝑘 ∈ 𝑁 \{ 𝑖 } max { 𝑣 𝑘 ( 𝑇 ) | 𝑇 ∈ 𝐺 and 𝑆 ∩ 𝑇 = ∅} After this first pass, we undertake a second pass over all re-maining active pairs. For each active pair, we compute the optimalwelfare without pair ( 𝑖, 𝑆 ) , but using the following finite pruningbudget schedule 𝝅 = [ , , , ] . In other words, we carry outthis computation for just a few of the remaining candidate pairs.We chose this pruning budget schedule so that one iteration of EAPwould take approximately two hours.One choice remains undefined for the second pruning pass:which ( 𝑖, 𝑆 ) candidate pairs to select out of those not pruned inthe first pass? For each iteration 𝑘 , we sort the ( 𝑖, 𝑆 ) in descendingorder according to the upper bound on welfare computed in thefirst pass, and then we select the bottom 𝝅 𝑘 pairs (180 during thefirst iteration, 90 during the second, etc.). The intuition for thischoice is that pairs with lower upper bounds might be more likelyto satisfy Lemma 3’s pruning criteria than pairs with higher upperbounds. Note that the way candidate pairs are selected for the sec-ond pruning pass uses no information about the underlying market,and is thus widely applicable. We will have more to say about thelack a priori information used by EAP in what follows. Results.
Table 2 summarizes the results of our experiments withGSVM and LSVM markets. The table shows 95% confidence inter-vals around the mean number of samples needed by EA and EAPto achieve the indicated accuracy ( 𝜀 ) guarantee for each row ofthe table. The table also shows confidence intervals around themean 𝜀 guarantees achieved by EAP, denoted 𝜀 EAP , and confidenceintervals over the UM loss metric. Several observations follow.Although ultimately a heuristic method, on average EAP usesfar fewer samples than EA and produces significantly better 𝜀 guar-antees. We emphasize that EAP is capable of producing these re-sults without any a priori knowledge about the underlying market. Instead, EAP autonomously samples those quantities that can prov-ably be part of an optimal solution. The EAP guarantees are slightlyworse in the LSVM market than for GSVM, where we prune all eligi-ble ( 𝑖, 𝑆 ) pairs. In general, there is a tradeoff between computationaland sample efficiency: at the cost of more computation, to find morepairs to prune up front, one can save on future samples. Still, evenwith a rather restricted pruning budget 𝝅 = [ , , , ] (com-pared to hundreds of thousands potentially active ( 𝑖, 𝑆 ) pairs), EAPachieves substantial savings compared to EA in the LSVM market.Finally, the UM loss metric follows a trend similar to those ob-served for unit-demand markets, i.e., the error guarantees are con-sistently met and degrade as expected (worst guarantees for highervalues of 𝜀 ). Note that in our experiments, all 40 GSVM marketinstances have equilibria with linear and anonymous prices. Incontrast, only 18 out of 32 LSVM market do, so the table reportsUM loss over this set. For the remaining 32 markets, we reporthere a UM loss of approximately 12 ± regardless of the value of 𝜀 .This high UM loss is due to the lack of CE in linear pricings whichdominates any UM loss attributable to the estimation of values. In this paper, we define noisy combinatorial markets as a model ofcombinatorial markets in which buyers’ valuations are not knownwith complete certainty, but noisy samples can be obtained, forexample, by using approximate methods, heuristics, or truncatingthe run-time of a complete algorithm. For this model, we tacklethe problem of learning CE. We first show tight lower- and upper-bounds on the buyers’ utility loss, and hence the set of CE, given auniform approximation of one market by another. We then developlearning algorithms that, with high probability, learn said uniformapproximations using only finitely many samples. Leveraging thefirst welfare theorem of economics, we define a pruning criterionunder which an algorithm can provably stop learning about buy-ers’ valuations for bundles, without affecting the quality of the setof learned CE. We embed these conditions in an algorithm thatwe show experimentally is capable of learning CE with far fewersamples than a baseline. Crucially, the algorithm need not knowanything about this structure a priori ; our algorithm is generalenough to work in any combinatorial market. Moreover, we expectsubstantial improvement with sharper sample complexity bounds;in particular, variance-sensitive bounds can be vastly more effi-cient when the variance is small, whereas Hoeffding’s inequalityessentially assumes the worst-case variance.
Acknowledgements.
This work was supported by NSF AwardCMMI-1761546 and by DARPA grant FA8750.
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PPENDIXTheoretical Proofs
Proof. (Lemma 1 of the main paper)Let 𝑀 X be a conditional combinatorial market, D a distributionover X , and I ⊆ 𝑁 × 𝐺 an index set. Let 𝒙 = ( 𝑥 , . . . , 𝑥 𝑡 ) ∼ D bea vector of 𝑡 samples drawn from D . Suppose that for all 𝑥 ∈ X and ( 𝑖, 𝑆 ) ∈ I , it holds that 𝑣 𝑖 ( 𝑆, 𝑥 ) ∈ [ , 𝑐 ] where 𝑐 ∈ R + . Let 𝛿 > 𝜀 >
0. Then, by Hoeffding’s inequality [19], 𝑃𝑟 (| 𝑣 𝑖 ( 𝑆 ) − ˆ 𝑣 𝑖 ( 𝑆 )| ≥ 𝜀 ) ≤ 𝑒 − 𝑡 ( 𝜀𝑐 ) (10)Now, applying union bound over all events | 𝑣 𝑖 ( 𝑆 ) − ˆ 𝑣 𝑖 ( 𝑆 )| ≥ 𝜀 where ( 𝑖, 𝑆 ) ∈ I , 𝑃𝑟 (cid:169)(cid:173)(cid:171) (cid:216) ( 𝑖,𝑆 ) ∈I | 𝑣 𝑖 ( 𝑆 ) − ˆ 𝑣 𝑖 ( 𝑆 )| ≥ 𝜀 (cid:170)(cid:174)(cid:172) ≤ ∑︁ ( 𝑖,𝑆 ) ∈I 𝑃𝑟 (| 𝑣 𝑖 ( 𝑆 ) − ˆ 𝑣 𝑖 ( 𝑆 )| ≥ 𝜀 ) (11)Using bound (10) in the right-hand side of (11), 𝑃𝑟 (cid:169)(cid:173)(cid:171) (cid:216) ( 𝑖,𝑆 ) ∈I | 𝑣 𝑖 ( 𝑆 ) − ˆ 𝑣 𝑖 ( 𝑆 )| ≥ 𝜀 (cid:170)(cid:174)(cid:172) ≤ ∑︁ ( 𝑖,𝑆 ) ∈I 𝑒 − 𝑡 ( 𝜀𝑐 ) = |I| 𝑒 − 𝑡 ( 𝜀𝑐 ) (12)Where the last equality follows because the summands on theright-hand size of eq. (12) do not depend on the summation index.Now, note that eq. (12) implies a lower bound for the event thatcomplements (cid:208) ( 𝑖,𝑆 ) ∈I | 𝑣 𝑖 ( 𝑆 ) − ˆ 𝑣 𝑖 ( 𝑆 )| ≥ 𝜀 , 𝑃𝑟 (cid:169)(cid:173)(cid:171) (cid:217) ( 𝑖,𝑆 ) ∈I | 𝑣 𝑖 ( 𝑆 ) − ˆ 𝑣 𝑖 ( 𝑆 )| ≤ 𝜀 (cid:170)(cid:174)(cid:172) ≥ − |I| 𝑒 − 𝑡 ( 𝜀𝑐 ) (13)The event (cid:209) ( 𝑖,𝑆 ) ∈I | 𝑣 𝑖 ( 𝑆 ) − ˆ 𝑣 𝑖 ( 𝑆 )| ≤ 𝜀 is equivalent to the eventmax ( 𝑖,𝑆 ) ∈I | 𝑣 𝑖 ( 𝑆 ) − ˆ 𝑣 𝑖 ( 𝑆 )| ≤ 𝜀 . Setting 𝛿 = |I| 𝑒 − 𝑡 ( 𝜀𝑐 ) and solvingfor 𝜀 yields 𝜀 = 𝑐 √︁ ln ( |I| / 𝛿 ) / 𝑡 .The results follows by substituting 𝜀 in eq. (13). □ Mathematical Programs
For our experiments, we solve for CE in linear prices. To computeCE in linear prices, we first solve for a welfare-maximizing allo-cation S ∗ and then, fixing S ∗ , we solve for CE linear prices. Notethat, if a CE in linear prices exists, then it is supported by anywelfare-maximizing allocation [30]. Moreover, since valuations inour experiments are drawn from continuous distributions, we as-sume that the set of welfare-maximizing allocations for a givenmarket is of negligible size.Next, we present the mathematical programs we used to com-pute welfare-maximizing allocations and find linear prices. Givena combinatorial market 𝑀 , the following integer linear program,(14), computes a welfare-maximizing allocation S ∗ . Note this for-mulation is standard in the literature [28].maximize ∑︁ 𝑖 ∈ 𝑁,𝑆 ⊆ 𝐺 𝑣 𝑖 ( 𝑆 ) 𝑥 𝑖𝑆 subject to ∑︁ 𝑖 ∈ 𝑁,𝑆 | 𝑗 ∈ 𝑆 𝑥 𝑖𝑆 ≤ , 𝑗 = , . . . , 𝑚 ∑︁ 𝑆 ⊆ 𝐺 𝑥 𝑖𝑆 ≤ , 𝑖 = , . . . , 𝑛𝑥 𝑖𝑆 ∈ { , } , 𝑖 ∈ 𝑁, 𝑆 ⊆ 𝐺 (14) Given a market 𝑀 and a solution S ∗ to (14), the following setof linear inequalities, (15), define all linear prices that couple withallocation S ∗ to form a CE in 𝑀 . The inequalities are defined overvariables 𝑃 , . . . , 𝑃 𝑚 where 𝑃 𝑗 is good 𝑗 ’s price. The price of bundle 𝑆 is then (cid:205) 𝑗 ∈ 𝑆 𝑃 𝑗 . 𝑣 𝑖 ( 𝑆 ) − (cid:205) 𝑗 ∈ 𝑆 𝑃 𝑗 ≤ 𝑣 𝑖 ( 𝑆 ∗ 𝑖 ) − (cid:205) 𝑗 ∈ 𝑆 ∗ 𝑖 𝑃 𝑗 , 𝑖 ∈ 𝑁, 𝑆 ⊆ 𝐺 If 𝑗 ∉ ∪ 𝑖 ∈ 𝑁 𝑆 ∗ 𝑖 , then 𝑃 𝑗 = , 𝑗 = , . . . , 𝑚𝑃 𝑗 ≥ , 𝑗 ∈ 𝐺 (15)The first set of inequalities of (15) enforce the UM conditions.The second set of inequalities states that the price of goods notallocated to any buyer in S ∗ must be zero. In the case of linearpricing, this condition is equivalent to the RM condition. In practice,a market might not have CE in linear pricings, i.e., the set of feasiblesolutions of (15) might be empty. In our experiments, we solve thefollowing linear program, (16), which is a relaxation of (15). Inlinear program (16), we introduce slack variables 𝛼 𝑖𝑆 to relax theUM constraints. We define as objective function the sum of all slackvariables, (cid:205) 𝑖 ∈ 𝑁,𝑆 ⊆ 𝐺 𝛼 𝑖𝑆 , which we wish to minimize.minimize ∑︁ 𝑖 ∈ 𝑁,𝑆 ⊆ 𝐺 𝛼 𝑖𝑆 subject to 𝑣 𝑖 ( 𝑆 ) − (cid:205) 𝑗 ∈ 𝑆 𝑃 𝑗 − 𝛼 𝑖𝑆 ≤ 𝑣 𝑖 ( 𝑆 ∗ 𝑖 ) − (cid:205) 𝑗 ∈ 𝑆 ∗ 𝑖 𝑃 𝑗 , 𝑖 ∈ 𝑁, 𝑆 ⊆ 𝐺 If 𝑗 ∉ ∪ 𝑖 ∈ 𝑁 𝑆 ∗ 𝑖 , then 𝑃 𝑗 = , 𝑗 = , . . . , 𝑚𝑃 𝑗 ≥ , 𝑗 ∈ 𝐺𝛼 𝑖𝑆 ≥ , 𝑖 ∈ 𝑁, 𝑆 ⊆ 𝐺 (16)As reported in the main paper, for each GSVM market we found thatthe optimal solution of (16) was such that (cid:205) 𝑖 ∈ 𝑁,𝑆 ⊆ 𝐺 𝛼 𝑖𝑆 =
0, whichmeans that an exact CE in linear prices was found. In contrast,for LSVM markets only 18 out of 50 markets had linear prices( (cid:205) 𝑖 ∈ 𝑁,𝑆 ⊆ 𝐺 𝛼 𝑖𝑆 =
0) whereas 32 did not ( (cid:205) 𝑖 ∈ 𝑁,𝑆 ⊆ 𝐺 𝛼 𝑖𝑆 > ) . Experiments’ Technical Details
We used the COIN-OR [31] library, through Python’s PuLP (https://pypi.org/project/PuLP/) interface, to solve all mathematical pro-grams. We wrote all our experiments in Python, and once thedouble-blind review period finalizes, we will release all code pub-licly. We ran our experiments in a cluster of 2 Google’s GCloud c2-standard-4c2-standard-4