On Welfare Approximation and Stable Pricing
aa r X i v : . [ c s . G T ] N ov On Welfare Approximation and Stable Pricing
Michal Feldman , Nick Gravin , and Brendan Lucier Tel-Aviv University; [email protected] Microsoft Research; [email protected] , [email protected] Abstract.
We study the power of item-pricing as a tool for approxi-mately optimizing social welfare in a combinatorial market. We considermarkets with m indivisible items and n buyers. The goal is to set pricesto the items so that, when agents purchase their most demanded setssimultaneously, no conflicts arise and the obtained allocation has nearlyoptimal welfare. For gross substitutes valuations, it is well known thatit is possible to achieve optimal welfare in this manner. We ask: can oneachieve approximately efficient outcomes for valuations beyond gross sub-stitutes? We show that even for submodular valuations, and even withonly two buyers, one cannot guarantee an approximation better thanΩ( √ m ). The same lower bound holds for the class of single-minded buy-ers as well. Beyond the negative results on welfare approximation, ourresults have daunting implications on revenue approximation for thesevaluation classes: in order to obtain good approximation to the collectedrevenue, one would necessarily need to abandon the common approachof comparing the revenue to the optimal welfare; a fundamentally newapproach would be required. Combinatorial markets are a canonical setting in which economic and computa-tional theory collide. In a combinatorial market scenario, a variety of indivisibleand heterogeneous goods are to be allocated among buyers with idiosyncraticand potentially complex preferences over subsets. A natural economic goal is tofind (and implement) allocations that are efficient, meaning that the total valuegenerated by the allocation is maximized. Over the past decade, the algorithmicmechanism design community has made a concerted effort to understand theinterplay between the requirement of aligning buyer incentives (from economics)and the tool of approximation (from computer science) in the context of thiswelfare maximization problem.An important observation from economic theory is that, in many scenarios,markets can be resolved using a very simple instrument: item prices. Ratherthan execute an auction or other mechanism, one sets an appropriate price foreach good and allow buyers to select their favorite bundles. Indeed, this is thestrategy employed by most retail outlets in the western world, even in scenarioswhere items are truly heterogeneous (e.g., antique stores). This prevelance canbe attributed to the natural and distributed manner in which buyers can respondto prices posted by a seller.n markets with limited supply, a desirable property of item prices is that each buyer can simultaneously obtain her most demanded set. That is, no item isoverdemanded at the specified prices. This is a natural market fairness criterion,and (as argued elsewhere [10]) is important for maintaining buyer satisfaction.Following [6], we will say that a choice of prices is stable if it satisfies thisfairness condition. When each buyer wants at most a single item, this propertyis analogous to envy-freeness (no buyer prefers another’s outcome to her own).Hence, for unit-demand buyers, stable prices are equivalent to envy-free prices[10].It is well known that when each buyer wants at most a single item (i.e., unit-demand preferences), or when the value a buyer derives from an item is indepen-dent of what other items she is allocated (i.e., additive preferences), there alwaysexist stable prices for which the resulting market outcome achieves optimal effi-ciency [18]. This is referred to as a Walrasian (or competitive) equilibrium. Suchprices exist more generally for the class of gross substitutes preferences [12]. Thisclass includes all additive and unit-demand valuations, but is strictly containedin the class of submodular valuations. To summarize, when buyers have grosssubstitutes valuations, item pricing can efficiently resolve a market.In this work we study the power of stable item prices to approximate welfarewhen valuations do not satisfy the gross substitutes condition. It is known thatfor any class of valuations beyond gross substitutes, there exist instances forwhich no stable prices can clear the market in an optimally efficient assignment[9]. We ask: for what classes of buyer preferences do there exist prices that gener-ate approximately efficient outcomes, when buyers select their utility-maximizingsets? Is it possible to recover a constant fraction of the optimal welfare whenbuyer valuations are, for example, submodular?
It might not come at a surprise that item prices are not sufficient to obtainhigh welfare if items are complementary. After all, if items have value only whensold together, then it seems intuitive to bundle (rather than setting individualprices). Perhaps more surprisingly, we also show that a polynomial gap may existeven if all valuations are submodular. This is despite the existential nature ofthe question, which carries the assumption that buyer types are publicly known.Even for two buyers with submodular valuations, there may not exist prices thatattain any reasonable approximation to the optimal welfare. Crucially, theselower bounds do not stem from computational considerations, but rather areexistential. That is, stable item prices necessarily provide a poor approximationfor welfare.
Main Result:
There exists an instance with two buyers with submodular valu-ations, such that there does not exist a stable outcome that obtains better thanan Ω( √ m ) -approximation to the optimal social welfare. The result is formulated for the objective of welfare maximization. A differ-ent, and also very natural, objective is seller revenue, which has been extensivelytudied in the literature on the design of revenue-maximizing envy-free prices[2,4,5,11]. Although our results do not have a direct implication for the revenueobjective, as far as we are aware, all existing methods for approximating envy-free revenue proceed by comparing to the benchmark of optimal welfare. Animplication of our result is that such an approach cannot generalize to submod-ular valuations and beyond. Since the welfare generated via stable prices cannotapproximate optimal welfare, certainly the revenue from stable prices cannot ap-proximate this benchmark either. Any approximation analysis for the revenue ofstable prices for submodular valuations would therefore require a fundamentallynew approach.These negative results may lead one to suspect that a super-constant lowerbound is unavoidable for any valuation class beyond gross substitutes, even fortwo buyers (similar in spirit to the result of [9], showing that the class of grosssubstitutes valuations is maximal with respect to Walrasian equilibrium exis-tence). In Appendix B we show that this is not the case: for two budget-additivebuyers, there always exists a set of item prices that achieves a quarter of theoptimal welfare.We also consider single-minded valuations, for which we give a lower boundof Ω( √ m ). For this case, a matching upper bound of O ( √ m ) follows from theresults in [5]. These results are obtained via an analysis of the configuration LPand its dual. Techniques.
Our lower bound construction is based upon a connection betweenthe existence of Walrasian prices and the integrality gap of the configurationLP for a market, but requires new ideas. In particular, a simple configurationLP argument presented in [6] showing a lower bound of Ω( m ) for the broaderclass of fractionally subadditive valuations completely fails for the class of sub-modular valuations. One reason is that the example in [6] works by exploitingthe fact that complement-free valuations can have “hidden complementarities:”a marginal valuation function of a complement-free valuation (i.e., the extravalue derived from additional items on top of a fixed set of items) is not neces-sarily complement-free. One might therefore guess that submodular valuations,which do not exhibit such hidden complementarities [13], might overcome thislower bound and yield a constant approximation. Another reason is that the ex-ample from [6] uses symmetric valuation functions, but symmetric submodularfunctions are gross substitutes [9], and thus lower bounds based on symmetricconstructions cannot apply. Our work is related to the literature on algorithmic pricing [10], which studies theproblem of setting envy-free item prices in full-information settings. This work Formally, given a valuation v on a set of items M and a set S ⊆ M , the marginalvaluation of a set T ⊆ M \ S is defined by v ( T | S ) = v ( T ∪ S ) − v ( S ). ocuses mainly on algorithms for finding prices that approximately maximize theseller’s revenue, assuming that buyers are unit-demand [2,4,5,11].One motivation for our work is the non-existence of Walrasian equilibriumin general combinatorial markets. Characterizations of existence of Walrasianequilibria were studied in, for example, [1,12,15,3,9]. Extensions of envy-freepricing for valuation classes beyond gross substitutes were studied in [17,7], butwhereas we focus on item prices those works consider more general paymentrules that assign prices to bundles of goods.Our model can be viewed as a relaxation of the notion of Walrasian equilib-rium, where we do not insist that all items are sold. A similar relaxation wasrecently considered in [6], showing that if, in addition to our relaxation, onecan also bundle items into packages prior to setting prices, then at least half ofthe optimal social welfare can be obtained for arbitrary valuations. Our nega-tive result provides additional motivation for such a bundling operation. Indeed,our results illustrate that insisting on outcomes that simultaneously satisfy thedemands of all agents, using item prices, can prevent a good approximation tosocial welfare. The results of [6] show that this difficulty can also be circumventedvia bundling.A different relaxation of Walrasian equilibrium was presented in [8]. In theirequilibrium notion, no buyer wishes to add additional items to his allocation(but might prefer a non-superset of their current allocation). This can be viewedas a partial relaxation of the buyer-side equilibrium constraints in a Walrasianequilibrium. In contrast, our approach is to relax the seller-side constraint thatall items must be sold. The setting considered in this work consists of a set M of m indivisible objectsand a set of n buyers. Each buyer has a valuation function v i ( · ) : 2 M → R ≥ that indicates his value for every set of objects. As standard, we assume thatvaluations are monotone non-decreasing (i.e., v i ( S ) ≤ v i ( T ) for every S ⊆ T ⊆ M ) and are normalized so that v i ( ∅ ) = 0. The profile of buyer valuations isdenoted by v = ( v , . . . , v n ).An allocation of M is a vector of sets X = ( X , X , . . . , X n ), where X i denotesthe bundle assigned to buyer i , for i ∈ [ n ], and X is the set of unallocatedobjects; i.e., X = M \ ∪ i ∈ [ n ] X i . It is required that X i ∩ X k = ∅ for every i = k .The social welfare of an allocation X is SW ( X ) = P ni =1 v i ( X i ), and the optimalwelfare is denoted by OPT. An allocation X gives an α -approximation for thesocial welfare if SW ( X ) ≥ (1 /α ) · OPT. A price vector p = ( p , . . . , p m ) consistsof a price p j for each object j ∈ M . As standard, we assume that each buyer hasa quasi-linear utility function; i.e., the utility of buyer i being allocated bundle X i under prices p is u i ( X i , p ) = v i ( X i ) − P j ∈ X i p j . Note that, for unit-demand buyers, envy-freeness is equivalent to our notion of sta-bility. iven prices p = ( p , . . . , p m ), the demand correspondence D i ( p ) of buyer i contains the sets of objects that maximize buyer i ’s utility: D i ( p ) = (cid:26) S ∗ : S ∗ ∈ argmax S ⊆ M { u i ( S, p ) } (cid:27) . An allocation X i is optimal for buyer i with respect to prices p if X i ∈ D i ( p ).A tuple ( X , p ) is said to be stable if for every buyer i , X i is optimal for i withrespect to p . We will sometimes refer to such a tuple as a stable outcome . Werefer to a pricing p as being stable if there exists a supporting allocation X suchthat ( X , p ) is stable. In this section, we present a characterization of stable outcomes. This character-ization makes use of the configuration linear program (LP), which encodes theproblem of maximizing social welfare over all fractional allocations of a combi-natorial market. The configuration LP is given by the following linear program:max X i,S v i ( S ) · x i,S s.t. X S ⊆ M x i,S ≤ i ∈ N X i,S ∋ j x i,S ≤ j ∈ Mx i,S ∈ [0 ,
1] for every i ∈ N, S ⊆ M (1)It was shown in [16] that a Walrasian Equilibrium exists if and only if theintegrality gap of the configuration LP is 1. Since, by definition, an outcomeis stable if and only if it is a WE over the set of items that it allocates, Astraightforward corollary of that result is as follows: Corollary 1.
For any M ′ ⊆ M , there exists a stable tuple ( X , p ) with S i X i = M ′ iff the configuration LP restricted to the items in M ′ has integrality gap . The following hierarchy of complement-free valuation classes have been studiedextensively in the literature. – v is additive if v ( S ) = P j ∈ S v ( { j } ) for all S ⊂ M . – v is XOS if there exists a collection of additive functions A ( · ) , . . . , A k ( · )such that for every set S ⊆ M , v ( S ) = max ≤ i ≤ k A i ( S ). – v is submodular if for every two sets S ⊆ T ⊆ M and item j ∈ M , v ( j | T ) ≤ v ( j | S ) (where v ( j | S ) := v ( S ∪ { j } ) − v ( S ) for every set S ⊆ M ). v is budget additive if there is a value B ≥ v ( S ) = min { P j ∈ S v ( { j } ) , B } for every set S ⊆ M .These valuations exhibit the following hierarchy: additive ⊂ budget additive ⊂ submodular ⊂ XOS, where all containments are strict. See [13] for a detaileddiscussion.
The following theorem, which is the main result of the paper, shows that evenfor two submodular buyers, a large gap in welfare might be unavoidable in stableoutcomes.
Theorem 1.
There exists an instance with two buyers with submodular valua-tions, such that there does not exist a stable outcome that obtains better than Ω( √ m ) -approximation to the optimal social welfare.Proof. Before going into the details of the proofs, let us provide a high-leveldescription of our approach. By Corollary 1, we wish to construct a pair ofsubmodular valuations for which the configuration LP has integrality gap strictlygreater than 1, even as items are removed from the market, unless almost allitems are removed. We begin by considering an instance that is not submodular,but that has this desired property. We will use an example from [6], in whichagent 1 is XOS and agent 2 is unit-demand. A key property of this example isthat the XOS agent always wants all items and the unit-demand agent alwayswants exactly one item. Our approach will be to massage these valuations intoa pair of submodular functions, while retaining the lower bounds.Call the original two valuations v and v . Our first idea is to make thefunctions submodular by adding a very large, strictly concave function h to eachof v and v . This will make the contribution of v and v to the social welfarenegligible by comparison, but recall that a stable outcome exists only if theintegrality gap is exactly one, so it is enough for even a tiny component of thewelfare to be improvable via fractional allocation.Unfortunately, this modification ruins the properties of the original exam-ple. Specifically, because of the concavity of h , the optimal assignment dividesall items equally between the players, and this does not align with the origi-nal example. We therefore apply an additional trick: we divide the items into √ m equal-sized buckets, and implement the original example on each bucketseparately.We now begin with the formal proof, which requires some preparation. Define h ( z ) = z + P zi =1 /i . Function h is symmetric, submodular, and satisfies thefollowing property. Claim.
For every set function f ( · ), there is a sufficiently small ε > v ( S ) = h ( | S | ) + ǫ · f ( S ) is submodular for any ε ≥ ǫ ≥ roof. In order to ensure that v ( · ) is submodular we show below that marginalvalues are decreasing. Let S ( S . We need to ensure that v ( S ∪ { i } ) − v ( S ) ≥ v ( S ∪ { i } ) − v ( S ) . This inequality for a fixed ǫ is equivalent to the following1 | S | +1 − | S | +1 ≥ ǫ · ( f ( S ∪ { i } ) − f ( S ) − f ( S ∪ { i } ) + f ( S ))We observe that the left hand side of the last inequality has a lower bound of n +1)( n +2) and the right hand side has an upper bound of 2 ǫ · max S ⊆ [ n ] f ( S ).Therefore, the inequality holds true for any ǫ < ε = n +1)( n +2) max S ⊆ [ n ] f ( S ) . ⊓⊔ We are now ready to construct our valuations. Consider an instance with 2buyers and m = k items for some integer k >
0. Suppose that the k items arepartitioned into k buckets, B , . . . , B k , where | B i | = k for every i = 1 , . . . , k .The valuation functions of the buyers are as follows. Define XOS ( S ) = | S | for every S such that | S | >
1, and
XOS ( S ) = 2 if | S | = 1. The valuation functionof the first buyer is given by v ( S ) = h ( | S | ) + ε · f ( S ) , where f ( S ) = max j =1 ,...,k XOS ( S ∩ B j ) . We now turn to the second buyer. Let
U nit ( S ) = 1 − k for every S such that | S |≥ S = ∅ ). Note U nit ( S ) is a symmetric unit-demand function.The valuation function of the second buyer is given by v ( S ) = h ( | S | ) + ε · f ( S ) , where f ( S ) = k X j =1 U nit ( S ∩ B j ) . By Claim 3, we can take sufficiently small ε such that both valuation functions v ( · ) and v ( · ) are submodular.Recall that our solution concept allows for some items to be unsold. Ournext argument is similar in spirit to the argument for XOS valuations given in[6] (see Appendix A). Let K denote the set of items sold. The maximum of h ( S ) + h ( K \ S ) is attained when | S | is as close to | K | / ε any integral allocation that maximizes v ( S ) + v ( K \ S )must divide items equally among two buyers (up to a single item). Among these,the optimal integral allocation is one that maximizes f ( S ) + f ( K \ S ). We cannow show that if K is not too small, then a stable outcome cannot exist. Claim. If | K |≥ k , then there exists a fractional allocation that does strictlybetter than the optimal integral allocation. Proof.
Suppose that | K |≥ k . Without loss of generality let B be the bucketwith the largest number of sold items, say t . We note that the number of items t in B is at least 4.ne can easily verify that the allocation that maximizes f ( S ) + f ( K \ S )is one that assigns all items in B to the first buyer, and a single item fromevery remaining non-empty bucket to the second buyer. Indeed, this allocationmaximizes the value of f ( · ) and almost matches the maximum of f ( · ), andthese two cannot be maximized simultaneously. Let S and S be sets of itemsassigned respectively to the first and second buyer in this allocation. We note that | S |≤ k and | S |≤ k −
1. Therefore, the remaining at least 4 k − (2 k −
1) = 2 k + 1unallocated items can be divided among the two buyers into two sets J and J such that buyers receive the same (up to one) number of items, as required.Thus in the optimal integral allocation the first buyer receives S ∪ J and thesecond buyer receives S ∪ J .By the characterization given in [16] (see also Section 2.1), this allocationadmits a stable pricing if and only if this is also an optimal fractional allocation.We next construct a fractional allocation with a higher social welfare.We observe that | J | > k , so we can choose T ⊂ J such that | T | = | S | = t. We recall that t ≥ . Let π : S → T be a bijection between items in S and T .We consider the following fractional solution { y i,S } : y ,S ∪ J = t − t − y , { j }∪ J ∪ T \{ π ( j ) } = 1 t ( t − , for each j ∈ S y , { j }∪ S ∪ J \{ π ( j ) } = 1 t , for each j ∈ S Let ℓ be the number of nonempty buckets in K . One can easily verify that thisis a feasible solution, and the welfare obtained by { y i,S } is given by SW ( y ) = h ( | S ∪ J | ) + h ( | S ∪ J | ) + t − t − · f ( S ∪ J )+ X j ∈ S t ( t − · f ( { j } ∪ J ∪ T \ { π ( j ) } )+ X j ∈ S t · f ( { j } ∪ S ∪ J \ { π ( j ) } ) ≥ h ( | S ∪ J | ) + h ( | S ∪ J | ) + t − t − · t + | S |· t ( t − · t · X j ∈ S f ( j ∪ S )= h ( | S ∪ J | ) + h ( | S ∪ J | ) + t − t − · t + 2 t − ℓ · (cid:18) − k (cid:19) . The welfare in the optimal integral allocation x is given by SW ( x ) = h ( | S ∪ J | ) + h ( | S ∪ J | ) + t + ( ℓ − · (cid:18) − k (cid:19) . t follows that SW ( y ) − SW ( x ) ≥ − k − t − t − > . ⊓⊔ We conclude that no allocation of at least 4 k items admits a stable outcome.Now since k = √ m and v ( S ) = Θ( | S | ), v ( S ) = Θ( | S | ), this gives a gap ofΩ( √ m ) in the social welfare, completing the proof of Theorem 1. ⊓⊔ We now consider buyers with single-minded valuations. A single minded buyeris interested in a single set and derives no utility from any strict subset of it.Single minded buyers exhibit strong complementarity between items.We first present an instance such that there does not exist a stable outcomethat obtains better than Ω( √ m ) approximation to the optimal social welfare.Readers who are familiar with the NP-hardness of welfare approximation withina factor Ω( √ m ) for single minded bidders (e.g., [14]) might assume that ourΩ( √ m ) lower bound follows similarly. We reiterate here that our lower boundis purely existential and is completely independent of computational considera-tions, and is therefore unrelated to the computational result. Example 1.
Consider a combinatorial auction with n single-minded buyers and m = n ( n − items. Each item is represented by a pair of integers ( a, b ) such that M = { ( a, b ) | a + b ≤ n, a, b ∈ [ n ] } . Each buyer i for i ∈ [ n −
1] desires a set S ∗ i = { ( a, b ) | a = i or b = n − i, ( a, b ) ∈ M } of size n −
1, for a value of n + 1.Note that | S ∗ i ∩ S ∗ k | = 1 for all i = k , and every object a ∈ M is contained in atmost 2 sets. Buyer n desires the set of all items, M , for a value of m .The optimal outcome allocates all objects to buyer n , for a total value of m . Suppose this optimal outcome can be supported by item prices p . Then P a ∈ M p a ≤ m , but a simple counting argument below demonstrates that P a ∈ S ∗ ℓ p a ≤ n < n + 1 for some ℓ . n ( n −
1) = 2 v n ( M ) ≥ X a ∈ M p a ≥ n − X i =1 X a ∈ S ∗ i p a . Buyer ℓ would demand set S ∗ ℓ at these prices, a contradiction. Thus, at anypricing equilibrium, some set S ∗ ℓ must be allocated to a buyer ℓ ; but this impliesthat no other set S ∗ k can be allocated, as S ∗ k ∩ S ∗ i = ∅ for all k = i . Thusthe total social welfare at any equilibrium outcome with item prices is at most n + 1 = O ( √ m ). ⊓⊔ Finally, we note that the last example is essentially the worst possible case.That is, for every instance of single-minded buyers, there exists a stable outcomethat achieves at least an O ( √ m ) approximation to the optimal social welfare.The latter statement is implied by [5], i.e., the agrument goes through a care-ful analysis of the configuration LP along with its dual program. In particular,one can show how to construct a price vector for which (i) no buyer can gettrictly positive utility, and (ii) there exists a fractionally optimal solution inwhich all buyers obtain zero utility. Then, a standard greedy algorithm appliedto the buyers in the support of the fractional solution will yield an O ( √ m ) ap-proximation. Since it is not possible for any buyer to obtain strictly positiveutility, this outcome is necessarily stable. This paper studies the problem of resolving combinatorial markets using itemprices and utility-maximizing allocations, with the objective of maximizing eco-nomic efficiency. We approach the problem through the lens of approximation,and ask how much social welfare can be obtained using item prices. We establishnegative results, showing that no stable pricing can provide reasonable approxi-mation to social welfare in several cases of interest. These negative results holdeven for simple complement-free valuations, and even with only two buyers.
Open problems
Our model and results leave a number of directions for futureresearch.First, we show that even with only two submodular buyers, there is a lowerbound of Ω( √ m ) on the approximation ratio of social welfare. On the positiveside, for gross-substitute buyers there always exists a competitive equilibrium,i.e., efficient stable pricing. It is left open whether there are subclasses of sub-modular valuations (such as budget additive or coverage valuations) that admitconstant approximation to optimal welfare. For the case of two consumers withbudget additive valuations we answer this question in the affirmative. However,for larger instances of budget additive valuations or other valuation classes thisquestion is left open.Second, throughout the paper we assume that items are indivisible and het-erogeneous. It would be an interesting research direction to relax partially theseassumption. For example, one could assume that every item in the market hasa few identical copies and that every buyer does not want more than one copyof each item. It would be interesting to see how the efficiency of item pricingdepends on the minimal number of item copies. Given our mostly negative re-sults for valuations with complements, we would like to understand under whatrelaxations one can obtain positive results (e.g., constant approximation of theoptimal social welfare) for single-minded buyers. Acknowledgments
The work of Michal Feldman was partially supported by the European Re-search Council under the European Union’s Seventh Framework Programme(FP7/2007-2013) / ERC grant agreement number 337122. eferences
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APPENDIXA A lower bound for XOS valuations
For completeness, we present an example from [6], demonstrating a linear gapin welfare for XOS valuations.
Theorem 2 ([6]).
There exists an instance with two buyers with XOS valua-tions, such that there does not exist a stable outcome that obtains better than Ω( m ) ( ≥ m/ ) approximation to the optimal social welfare.Proof. Consider an auction with m items and two buyers with the followingsymmetric XOS valuations. Buyer 1 is unit-demand and values every subsetat 1 / − δ , for a sufficiently small δ (that will be determined soon). Buyer 2values any subset of size k at max(1 , k/ m/ m >
2, and allocates one item to each of thebuyers when m = 2 for a total value of 3 / − δ . We claim that there is no stablepricing that sells more than two items. For every m >
2, an optimal integralsolution obtains a value of m/ m/ m/ δ < m − . Consider the fractionalsolution in which the allocation of the first (unit demand) buyer is given by x , { j } = 1 /m for every j ∈ [ m ], and the allocation of the second (XOS) buyer isgiven by x , { j } = m ( m − for every j ∈ [ m ], and x , [ m ] = m − m − . One can easilyverify that this is a feasible solution, and the welfare obtained by { x i,S } is givenby SW ( x ) = m + m − − δ , which is greater than m for every δ < m − ,as required. We conclude that a stable outcome can have at most two allocatedobject, and thus the highest welfare that can be obtained in a stable allocationis 3 / − δ , resulting in a linear gap of m/ B Two budget additive valuations: an upper bound
It has been shown in the last two sections that for submodular and XOS valu-ations, one cannot hope for a constant approximation, even for instances withonly two buyers. One may wonder whether this is an unavoidable property ofany valuation beyond gross substitutes. In this section we show that this is stilltoo early to jump into this conclusion. In particular, for the case of two budget-additive buyers, constant approximation can be achieved. budget-additive valuation of buyer i is specified by budget B and item val-ues v ij for every j ∈ M . The value of some set S is then v i ( S ) = min { B, P j ∈ S v ij } .Without loss of generality we may assume that for each item j neither v j ≤ B ,nor v j ≤ B . Theorem 3.
For every instance of two budget-additive buyers, there exists astable outcome, which is -approximation to the optimal social welfare.Proof. Let B and B denote the budgets of buyers 1 and 2, respectively, andsuppose without loss of generality that B ≥ B . We distinguish between twocases. Case 1: P j v j ≥ B (i.e., the value of buyer 1 for the grand bundle exhaustsher budget). In this case, set a price of p j = v j for every item j . Thus, everybundle S (including S = ∅ ) such that P j ∈ S v j ≥ B is in buyer 1’s demandset (and gives her utility 0). Since v j ≤ B for each j and P j v j ≥ B we canfind a set S , such that B ≥ P j ∈ S v j ≥ B /
2. Any subset of S is a demandset of buyer 1. Let D be a bundle in buyer 2’s demand set. We let D be theallocation of buyer 2. We note that v ( D ) ≥ X j ∈ D p j = X j ∈ D v j . We let D = S \ D be the allocation of buyer 1. We can estimate the socialwelfare of this allocation as follows. SW = v ( D ) + v ( D ) = X j ∈ S \ D v j + v ( D ) ≥ X j ∈ S \ D v j + X j ∈ D v j ≥ X j ∈ S v j ≥ B . The optimal social welfare is at most B + B , which cannot exceed our SW by more than a factor of 4. Case 2: P j v j < B . Let S = { j : v j ≥ v j } , and S = { j : v j > v j } .Consider the pricing where p j = v j for every item j ∈ S and p j = v j for everyitem j ∈ S . Since buyer 2 is indifferent about taking any item from S , thereexists a set in buyer 2’s demand set that is contained in S , we call it D . Welet buyer 2 to be allocated the items in D and buyer 1 be allocated the itemsin S ∪ ( S \ D ). The social welfare SW for this allocation is SW = X j ∈ S v j + X j ∈ S \ D v j + min X j ∈ D v j , B . We assume that D = ∅ , as otherwise we would have the optimal socialwelfare. If v ( D ) + v ( S \ D ) < B , then u ( S ) > u ( D ) and we arrive at aontradiction with the fact that D was a demand set of buyer 2. Indeed, u ( D ) = v ( D ) − X j ∈ D v j = v ( D ) + v ( S \ D ) − X j ∈ S v j < min X j ∈ S v j , B − X j ∈ S v j = v ( S ) − X j ∈ S p j = u ( S )Therefore, v ( D ) + v ( S \ D ) ≥ B , then it also holds that SW = v ( D ) + v ( D ) + v ( S \ D ) ≥ B . (2)In addition to that, since v ( D ) ≥ P j ∈ D p j = P j ∈ D v j , it follows that SW = v ( D ) + v ( D ) + v ( S \ D ) ≥ X j ∈ M v j . (3)Consider the sum of Equations (2) and (3). The left hand side is twice thesocial welfare obtained by the suggested allocation, and the right hand side is B + P j ∈ M v j , which is clearly an upper bound on OPT. It follows that we getat least 1 //