Tight Information-Theoretic Lower Bounds for Welfare Maximization in Combinatorial Auctions

Transcription

1 Tight Inforation-Theoretic Lower Bounds for Welfare Maxiization in Cobinatorial Auctions Vahab Mirrokni Jan Vondrák Theory Group, Microsoft Dept of Matheatics Research Princeton University Redond, WA 9805 Princeton, NJ Michael Schapira School of Engineering and Coputer Science Hebrew University of Jerusale, Israel ABSTRACT We provide tight inforation-theoretic lower bounds for the welfare axiization proble in cobinatorial auctions In this proble, the goal is to partition ites aong k bidders in a way that axiizes the su of bidders values for their allocated ites Bidders have coplex preferences over ites expressed by valuation functions that assign values to all subsets of ites We study the black box setting in which the auctioneer has oracle access to the valuation functions of the bidders In particular, we explore the well-known value query odel in which the peritted query to a valuation function is in the for of a subset of ites, and the reply is the value assigned to that subset of ites by the valuation function We consider different classes of valuation functions: subodular, subadditive, and superadditive For these classes, it has been shown that one can achieve approxiation ratios of, e, and log, respectively, via a polynoial (in k and ) nuber of value queries We prove that these approxiation factors are essentially the best possible: For any fixed ɛ > 0, a ( /e + ɛ)-approxiation for subodular valuations or an -approxiation for subadditive valuations would require / ɛ exponentially any value queries, and a log+ɛ -approxiation for superadditive valuations would require a superpolynoial nuber of value queries INTRODUCTION Cobinatorial auctions are a central research area at the intersection of econoics, gae theory, and coputer science The welfare axiization proble in cobinatorial auctions is an abstraction of any coputational and econoic resource-allocation probles In this proble, an auctioneer sells a set M of ites to a set of k bidders The value of bidder i for any subset (bundle) of ites is Supported by grants fro the Israel Science Foundation EC 08, July 8, 008, Chicago, Illinois, USA given by a valuation set function v i : M R +, where v i(s) represents i s axiu willingness to pay for the bundle S The two standard assuptions on each v i are that if S T then v i(s) v i(t ) (onotonicity), and that v i( ) = 0 (noralization) The objective is to partition M into k disjoint subsets S, S,, S k in a way that axiizes the expression P i k vi(si) (ie, the social welfare) Algoriths for axiizing welfare in cobinatorial auctions are required to be polynoial in the natural paraeters of the proble, and k However, since the input (the valuation functions) is of exponential size one ust specify how it can be accessed Most works in this field have taken a black box approach in which bidders valuation functions are accessed via oracles that can answer specific type of queries Three types of queries have been considered [,, 5]: Value queries: The query to a valuation function v i is in the for of a bundle S M, and the response is v i(s) Deand queries: The query to a valuation function v i is in the for of a price vector p = (p,, p ) and the response is the bundle T ost deanded by v i given these prices That is, T = argax S M v i(s) P j S pj General queries: We allow any type of query (to each valuation function alone) This odel captures the counication coplexity (between the bidders) of the proble Due to its strength it is ostly interesting for proving lower bounds Value queries are strictly less powerful than deand queries, which, in turn, are strictly less powerful than general queries [,, 4] In this paper we focus our attention on the value query odel It has been shown that coputing an optial solution for social-welfare axiization requires an exponential nuber of queries even in the general queries odel [3] Hence, we are interested in the approxiability of this proble For the general case, the approxiability of the proble is wellunderstood in all odels [, 3] Researchers have also studied the approxiability of the proble for restricted classes of valuation functions Two failies of such classes, that

2 have natural econoic interpretations [, 5], have been considered: Subadditive functions, and superadditive functions A set function f : M R + is subadditive iff for any two sets S and T, f(s) + f(t ) f(s T ), and is superadditive iff for any two disjoint sets S and T, f(s)+f(t ) f(s T ) An iportant special case of subadditive functions are subodular functions A set function f is subodular iff for any two sets S and T, f(s) + f(t ) f(s T ) + f(s T ) Subodular functions (that are onotone and noralized) are also reasonable to consider fro an econoic perspective as they characterize functions with decreasing arginal utilities [] We present tight inforation-theoretic lower bounds for subodular, subadditive, and superadditive valuation functions in the value query odel We prove the following theores: Theore: For any fixed ɛ > 0, achieving an approxiation ratio of + ɛ for welfare-axiization with subodular e functions requires an exponential nuber of value queries This atches the ( )-approxiation (achieved with a e polynoial nuber of value queries) recently shown by Vondrák [7], who iproved over the -approxiation shown by Lehann et al [] (We note that this proble can be foralized as the proble of axiizing a subodular function subject to a atroid constraint Hence, the greedy algorith developed by Fisher et al [6] provides a -approxiation for this proble) The only previously known inforation-theoretic lower bound for this proble was O( ) (see [3], in the general queries odel Our lower bound strengthens the + ɛ lower bound dependent on P NP proven by Khot et al [0] We stress e that our lower bound is independent of any coputational coplexity assuptions and holds even for algoriths of unbounded coputational power, that are bounded only in ters of the nuber of value queries they can ake Also, we reark that the sae inapproxiability result does not hold in stronger query odels it is known that ( /e+ɛ)- approxiation is possible with polynoially any deand queries [7] Theore: For any fixed ɛ > 0, achieving an approxiation ratio of for welfare-axiization with subadditive ɛ functions requires an exponential nuber of value queries This atches the upper bound of presented by Dobzinski et al [5] (achieved using a polynoial nuber of value queries) The previously known lower bound in the value query odel was [4] In fact, our lower bound holds 4 even for the ore restricted subclass of fractionally subadditive valuations [6], introduced in [] under the nae of XOS Theore: For any fixed ɛ > 0, achieving an approxiation ratio of log+ɛ for welfare-axiization with superadditive functions requires a super-polynoial nuber of value queries This nearly atches the upper bound of log presented by Holzan et al [9] (achieved via a polynoial nuber of value queries) A siilar lower bound was known for general valuation functions [, ] We extend this lower bound to the restricted class of superadditive functions In fact, the lower bound holds for a superadditive analogue of fractionally subadditive functions (that is strictly contained in the class of superadditive valuation functions) VALUE-QUERY COMPLEXITY OF SUB- MODULAR WELFARE MAXIMIZATION In this section, we construct an exaple showing that it is ipossible to achieve an approxiation factor better than /e for subodular utility functions in the value oracle odel We consider algoriths whose running tie is potentially unbounded, we only count the nuber of value queries posed by the algorith More precisely, we prove the following Theore For any fixed β > 0 and k, any (possibly randoized) ( ( /k) k + β)-approxiation algorith for ites and k players with subodular valuation functions requires e Ω() value queries, otherwise it fails with high probability Since ( /k) k is arbitrarily close to /e for large enough k, this iplies the following Corollary For any fixed ɛ > 0, there is no ( /e + ɛ)-approxiation for an arbitrary nuber of players, using a subexponential nuber of queries We note that our exaples use the sae subodular valuation function for all players, just like in the NP-hardness result of [0] Thus the proble is hard to approxiate even in the special case where all utility functions are equal Our construction is different fro [0], however While the hardness reduction of [0] uses explicit coverage-type subodular functions, our valuation functions are not exactly of the coverage type Our construction is inspired by a lower bound developed by Feige et al [8] for the proble of axiizing non-onotone subodular functions Overview of the Proof Consider a k-unifor hypergraph H = (X, E) and a function f : X R + where f(s) is the nuber of hyperedges incident with the set of vertices S This is a coverage-type subodular function The idea is that it is hard to distinguish instances where H is a coplete k-partite k-unifor hypergraph (and allocating one part X i to each player results in a perfect solution ), and instances where H is a coplete k-unifor hypergraph (and then there is no perfect solution ) Since vertices of the hypergraph could be labeled arbitrarily on the input, it s hard for any algorith to find a set of vertices significantly overlapping with any X i and hence it cannot distinguish these two cases In order to ake the exaple work, we have to odify the coverage-type functions slightly We consider a ground set

3 X partitioned into X X X k The functions f(s) that we define depend only on the fractions of X i that S contains: x i = S X i / X i To siplify the notation, we work with continuous functions f(x,, x k ) The following lea states the properties that we need f(x,, x k ) to satisfy Lea 3 Let X = X X X k as above and let f : [0, ] k R be a function with continuous first partial derivatives, and second partial derivatives alost everywhere Define a discrete function f : X R so that f(s) = f S X,, S X «k X X k If 0 everywhere for each i, then the function f is onotone If f 0 alost everywhere for any i, j, then the function f is subodular Proof For onotonicity, it s sufficient to observe that if f 0, then f is non-decreasing in each coordinate Hence, adding eleents cannot decrease the value of f For the subodularity condition, fix an eleent in a X i and consider a set S paraeterized by x i = S X i / X i The arginal value of a added to S is equal to f S(a) = f(x,, x i + X,, x k) f(x,, x i,, x k ) i = Z / Xi 0 (x,, x i + t,, x k )dt We want to prove that f S(a) cannot increase by adding eleents to S, ie by increasing any coordinate x j Because is continuous and its derivative along x j, f, is nonpositive except at finitely any points, f is non-increasing in x j By shifting the entire integral to a higher value of x j, the arginal value cannot increase Hence, we need our continuous functions to satisfy f 0 and f 0 for all i, j [k], which iplies onotonicity and subodularity in the discrete case We call such functions sooth subodular To shorten notation, we write f(x) = f(x,, x k ) In each instance, all players have the sae valuation function We find two functions f, g such that we have f(x) = g(x) whenever ax i,j x i x j ɛ As we show later, this will iply that f and g are indistinguishable by a subexponential nuber of queries We construct these two functions as follows Lea 4 For any β > 0 and integer k, there is ɛ > 0 and two sooth subodular functions f, g : [0, ] k R + such that To be ore precise, on any axis-parallel line there are only finitely any points where f is not defined If ax i,j x i x j ɛ, then f(x) = g(x) and the function value depends only on x = k P k i= xi ax{ P k i= f(xi,, x ik) x ij 0 & j; P k } ( β)k i= xij = ax{ P k i= g(xi,, x ik) x ij 0 & j; P k i= xij = } ( ( /k) k + β)k Proof We start by considering two sooth subodular functions, otivated by the exaples of k-unifor hypergraphs that we discussed above f(x) = Q k i= ( xi) g(x) = ( x) k, where x = k P k i= xi The optial solution with valuation function f is x ii =, x ij = 0 for i j This way, each player gets value and P i f(xi,, x ik) = k For g, on the other hand, the value depends only on the average of the coordinates x By the concavity of g, the optiu solution is to set x ij = /k for all i, j, which gives total value P i g(xi,, x ik) = k( ( /k) k ) It reains to perturb the functions so that f(x) = g(x) for vectors satisfying ax i,j x i x j ɛ Let h(x) denote the difference of the two functions, h(x) = f(x) g(x) = ( x) k Q k i= ( xi) P Again, we denote x = k k i= xi Also, let δ = axi,j xi x j First, we estiate h(x) and its first derivatives in ters of x and δ We use very crude bounds, to siplify the analysis Clai h(x) kδ( x) k h(x) k 4 δ ( x) k 3 kδ( x) k, ie k 3 ( x) k/ p h(x) We have h(x) = ( x) k Q k i= ( xi) If kδ x, we get iediately h(x) ( x) k kδ( x) k So let s assue kδ < x Then, since x i x + δ for all i, we get h(x) ( x) k ( x δ) k = «k ( x) k δ ( x) k kδ x x

4 For a lower bound on h(x), suppose that δ = x x For k =, we are done iediately since h(x) = ( x) ( x δ/)( x + δ/) = 4 δ Hence, we assue k > and define η = ( x (x +x)) k Ie, x = x (k )η δ/, x = x (k )η + δ/, and the average of the reaining coordinates is x + η By the arithetic-geoetric ean inequality, Q i, ( xi) is axiized when these variables are all equal: h(x) ( x) k ( x η) k ( x + (k )η δ)( x + (k )η + δ) = ( x) k ( x η) k ( x + (k )η) + 4 δ ( x η) k Again by the arithetic-geoetric ean inequality, ( x) k ( x η) k ( x+(k )η) If η ( x), we are k done because then the last ter is at least δ ( x) k 4e So we can assue η > ( x) In this case, we throw away k the last ter and write h(x) ( x) k ( x + (k )η) ( x η) k «= ( x) k η + (k ) η «! k x x ( x) k + η «(k ) η «! (k ) x x = ( x) k «η (k )! ( x) ( x) k k «! (k ) ( x)k k using k > and η > ( x) We observe it always holds k that δ k( x): If the iniu coordinate is x in, we have x k xin +, hence x k k in k x (k ) and δ x in k( x) Consequently, h(x) k ( x) k k 4 δ ( x) k 3 Let δ = ax i,j x i x j We estiate the partial derivative = Y ( x i) ( x) k i j Define η = (xj x) Ie, xj = x + (k )η and the average of the reaining coordinates is x η By the arithetic- k geoetric ean inequality, ( x + η) k ( x) k = ( x) k ( + η x k «Since η = (xj x) ( x), we can estiate ( + k k η x )k + k η Also, we know that all coordinates x differ fro x η by at ost δ, in particular x j = x + (k )η x η + δ, hence kη δ and ( x) k k η x δ( x)k For a lower bound, it s enough to observe that each coordinate is at ost x + δ, and so ( x δ) k ( x) k = ( x) k δ «k! x «( x) (k k δ ) x = (k )δ( x) k assuing that (k ) δ x bound directly fro proof of the clai ; otherwise we get the sae ( x) k This finishes the We return to our construction We define f(x) = f(x) φ(h(x)) where φ : R R is defined so that φ(t) = t for sall t 0, then φ(t) is increasing and concave with a controlled second derivative and finally φ(t) is bounded by a sall constant everywhere More precisely, For t [0, ɛ ], we set φ(t) = t We choose ɛ = kɛ Ie, for ax i,j x i x j ɛ, we have h(x) ɛ by Clai and then f(x) = g(x) For t [ɛ, ɛ ], the first derivative of φ is continuous at t = ɛ and its second derivative is φ (t) = α/t for t [ɛ, ɛ ] Hence, φ (t) = Z t ɛ α τ dτ = α ln t ɛ We choose α = / ln ɛ and ɛ = ɛ, so that φ (ɛ ) = 0 Since 0 φ (t) everywhere, we have 0 φ(ɛ ) ɛ For t > ɛ, we set φ(t) = φ(ɛ ) Hence, we have 0 φ(t) ɛ everywhere and f(x) = f(x) φ(h(x)) f(x) ɛ Next, we want to show that we didn t corrupt the onotonicity and subodularity of f too badly We have = f φ (h) We have 0 φ (h), and f, So, f = = ( φ (h)) f + φ (h) g g are both nonnegative 0 For the second partial derivatives, we get f φ h (h) φ (h) = ( φ f (h)) + φ g (h) φ (h) The first two ters for a convex cobination of non-positive values To control the third ter, we have φ (h) α/h

5 In Clai 3, we showed that k3 ( x) k/ p h(x) We can conclude that f x φ (h) j αk6 ( x) k We need to ake the second partial derivatives non-positive Since g = k ( k x)k, it is enough to add a suitable ultiple of g to both functions: ˆf = f + αk 6 g, ĝ = ( + αk 6 )g Then ˆf, ĝ are sooth subodular Recall that we have α = / ln For a given β > 0, we kɛ choose ɛ = k e 4k6 /β, so that β = αk 6 and we increase g only by a factor of + β We also get ɛ = kɛ β, and therefore ˆf(x) f(x) f(x) ɛ f(x) β Thus ˆf and ĝ satisfy the conditions of the lea Now we are ready to prove Theore Proof Consider a large set of eleents X, partitioned into equal parts X,, X k Lea 4 defines two sooth subodular functions By Lea 3, we define discrete utility functions f, g : X R + We consider two instances, where all utility functions are equal to either f or g We present one of these two instances to a (possibly randoized) algorith The labeling of the eleents of X is arbitrary and unknown to the algorith; we can assue that it is uniforly rando Let us assue that an algorith queries a set S of size s Let x i = S X i / X i Since the partition is uniforly rando, each x i is a rando variable of expectation s/k and variance O(s) We consider k fixed here, while the nuber of eleents = X is very large We can assue that s ɛ, otherwise the deviation of x i fro its expectation can never be ore than ɛ Otherwise, by standard bounds, the probability of x i deviating fro its expectation by ore than ɛ decays as e Ω(ɛ /s) = e Ω(ɛ) Hence, it happens only with exponentially sall probability that x i x j > ɛ for any i j Let s call such a query unbalanced Consider any fixed sequence of q queries Unless the nuber of queries q is exponentially large in, it still happens only with an exponentially sall probability that any query is unbalanced (by the union bound) Therefore, with high probability, no query is unbalanced Now consider any (possibly randoized) algorith, using a subexponential nuber of value queries In the randoized case, let us condition on the rando bits on the algorith Given this, the sequence of queries can depend only on the obtained answers Note that for balanced queries, we have f(x) = g(x) and the function value depends only P on x = k k i= xi, ie on the size of the queried set, which is the algorith s own choice Hence with high probability, the algorith always follows the sae sequence of (balanced) queries and the answers obtained are the sae for f(x) and g(x) Intuitively, the algorith never learns any inforation about the partition (X, X,, X k ) In the case of a randoized algorith, we can now average over the choices of its rando bits Still, with high probability it never asks any unbalanced query and cannot distinguish between f(x) and g(x) If the underlying instance corresponds to f(x), the algorith will never find any set whose value differs fro that of g(x), and hence any solution obtained is at ost ( ( /k) k + β) of the optiu 3 VALUE-QUERY COMPLEXITY OF SUB- ADDITIVE WELFARE MAXIMIZATION In this section, we construct an exaple showing that it is ipossible to achieve an approxiation factor significantly better than for the subadditive welfare proble, using a polynoial nuber of value queries In fact, we prove our result for the ore restricted class of fractionally subadditive valuation functions [, 6] that is known to strictly contain all subodular valuation functions A fractionally subadditive function is the pointwise axiu over a set of linear valuation functions Definition 3 A linear valuation function (also known as additive) is a set function a : M R + that assigns a non-negative value to every singleton {j M}, and for all S M it holds that a(s) = P j S a({j}) Definition 3 A fractionally subadditive function is a set function f : S M R +, for which there is a finite set of linear valuation functions A = {a,, a l } such that f(s) = ax ai A a i(s), for every S M We prove the following theore Theore 3 For any fixed ɛ > 0, a -approxiation ɛ algorith for fractionally subadditive valuation functions requires exponentially any value queries We note that the result of Theore 3 can be shown to hold even for the case that bidders have the sae valuation function (as will be explained later) Proof We shall use probabilistic arguents (siilar to those in Section ) to show that any algorith that obtains an -approxiation to the social-welfare requires an exponential nuber of value queries For siplicity, we shall ɛ start by proving the theore for the case that bidders have different valuation functions We shall later discuss how the proof can be extended to the ore restricted case that all bidders have the sae valuation function Fix a sall constant δ > 0 (to be deterined later) We shall construct a cobinatorial auction with ites and k = bidders For every S let a S be the linear valuation function that assigns a value of to each ite j S, and 0 to each ite j / S Let ā be the additive valuation that assigns every ite j [] a value of +δ δ

6 Let v,, v k be an k-tuple of (equal) valuation functions defined as follows: v i = ax{a S: S (+δ) δ, ā} That is, v i is the pointwise axiu over the set of additive valuation functions that contains a S for all S of a certain size, and ā Choose, uniforly at rando, a partition of the ites into disjoint bundles of ites T,, T k such that for each i, T i = Let v,, v k be the k-tuple of valuation functions defined as follows: v i = ax{v i, a Ti } We shall prove that for every player i, it takes an exponential nuber of value queries to distinguish between the case that i s valuation function is v i and the case that i s valuation function is v i It is easy to see that the axiu social-welfare attainable if the valuation functions are v, v k is O( +δ ), while the optial social-welfare if the valuation functions are v, v k is Hence, the fact that it requires an exponential nuber of value queries to distinguish between the valuation-functions profiles v, v k and v, v k iplies that one cannot get an approxiation ratio better than Ω( ) in less than an exponential nuber δ of value queries Consider a specific player i Fix a bundle S of size saller or equal to +δ It holds that v i(s) = ax{ S, ( + δ) δ } v i ight assign a value higher than v i to S but only if S T i > ( + δ) δ Using standard probabilistic arguents, and relying on the Chernoff bounds, it can be shown that P r[ S T i > ( + δ) δ ] is exponentially sall (see Section ) Now, consider a bundle S of size greater than +δ v i will S assign to S the value of ( + δ) δ v i ight assign S a higher value, but only if ax{v i, a T,, a Tk } It is still true that distinguishing between v,, v k and v,, v k requires an exponential nuber of value queries (using the sae probabilistic arguents as before and the union bound) It is also still true that the ratio between the social welfare if the valuation functions are v,, v k, and the social welfare if the valuation functions are v,, v k is O( δ )(the ratio between O( +δ ) and, respectively) Hence, in order to obtain an approxiationratio better than Ω( ), an exponential nuber of value δ queries in required 4 VALUE-QUERY COMPLEXITY OF SU- PERADDITIVE WELFARE MAXIMIZA- TION In this section, we construct an exaple showing that it is ipossible to achieve an approxiation factor significantly better than log for the superadditive welfare proble, using a polynoial nuber of value queries The construction of the exaple will be done in two steps First, we shall define a subclass of superadditive valuation functions we ter in-linear functions This is a superadditive analogue of fractionally subadditive functions [, 6] We shall then prove our lower bound for this ore restricted class Definition 4 A in-linear function is a set function f : M R + such that there is a finite set of linear valuation functions A = {a,, a l } such that for every S M f(s) = in ai A{a i(s)} It is easy to show (and analogous to the proofs in [, ]) that in-linear functions are contained in the class of superadditive valuation functions, and are a superclass of superodular valuation functions For copleteness we present the siple proofs below Siple exaples deonstrate that these containents are strict S S T i > ( + δ) δ Again, using standard probabilistic arguents it can be shown S that P r[ S T i > ( + δ) ] is exponentially sall δ We conclude that for every bundle S, only with exponentially sall probability does one gather sufficient inforation to distinguish between the case that i s valuation is v i and the case that it is v i Hence, it requires an exponential nuber of value queries to distinguish between v i and v i in the worst case This concludes the proof of the theore We note that this proof can be extended to the case that all bidders have the sae valuation function Observe, that all the v i functions are identical We shall show how it is possible to ake all the v i identical as well Inforally, in the construction of v i, we have associated every bidder i with a bundle T i However, it is possible to define v i in a way that associates every bidder i with the entire partition T,, T k This is done by defining each v i to be Clai 4 Any in-linear function is superadditive Proof Let f = in a A a be a in-linear function Let S and T be two disjoint subsets of ites By definition there are linear functions a S, a T, and a S T in A for which the value of S, T, and S T, is iniized Therefore, it ust hold that a S(S) a S T (S) and a T (T ) a S T (T ) Hence: f(s T ) = a S T (S T ) = a S T (S) + a S T (T ) a S(S) + a T (T ) = f(s) + f(t ) Definition 4 A set function f is superodular iff for any two sets S and T, f(s) + f(t ) f(s T ) + f(s T ) Clai 4 Any superodular valuation function is a inlinear function

7 Proof Let f be a superodular valuation function Fix an order on the ites, wlog,,, For every set S we define a linear function a S as follows: For every j S a S({j}) = f({,, j}) f({,, j }) For every j / S a S({j}) =, where represents a very large nuber (in particular f(m) << ) It is easy to see that a S(S) = f(s) for any S We want to show that f is in-linear and the finite set of linear functions is A = {a T } T M For this, we need to show that a S(S) = in at A a T (S) Observe that for any S, T such that S is not contained in T, it is ipossible that the iniu for S is achieved by a T (because for soe j S a T ({j}) = ) So, we are left with the case that S T Here we exploit the well-known fact that onotone superodular functions have increasing arginal values That is, if U V M, and j is in neither U nor V, then f(u {j}) f(u) f(v {j}) f(v ) It is easy to see that this iplies that for S T, for any j S a S({j}) a T ({j}) Hence, a S is indeed the linear function for which the iniu is achieved (for bundle S) This iplies that f = in A We are now ready to prove the following theore: Theore 4 For any ɛ > 0, a log+ɛ -approxiation algorith for in-linear valuation functions requires a superpolynoial nuber of value queries We note that the result of Theore 4 can be shown to hold even for the case that all bidders have the sae valuation function (as will be discussed later) Proof For siplicity, we shall start by proving the theore for the case that bidders have different valuation functions We shall later discuss how the proof can be extended to the ore restricted case that all bidders have the sae valuation function We use probabilistic arguents siilar to those in Section Fix ɛ > 0 We construct an auction with M = + log +ɛ ites and N = k = M consists of two disjoint sets log +ɛ M and M, such that M = and M = Each log +ɛ bidder i [k] is associated with a unique ite d i in M We partition M, uniforly at rando, into k = bundles T,, T k of equal size (ie, of size log +ɛ ) log +ɛ For each bidder i, let b i be the linear valuation function that assigns a value of to d i and 0 to all other ites By a S, we denote a linear function that assigns a value of to each ite in S and 0 to all other ites Let v,, v k be the k-tuple of valuation functions defined as follows: v i = in{b i, a S:S M, S = } This function has a very siple interpretation: It assigns a value of to every set that contains d i and ore than ites in M It assigns 0 to all other bundles Let v,, v k be the k-tuple of valuation functions defined as follows: v i = in{b i, a S:S M, S = & S T i } This function too has a siple interpretation: It assigns a value of to every set that contains d i and T i, or d i and ore than ites in M It assigns 0 to all other sets We shall show that it would take a super-polynoial nuber of value queries to distinguish between the case that i s valuation function is v i and the case that it is v i It is easy to see that if the valuation functions of the bidders are v,, v k then the optial social welfare is On the other hand, if the valuation functions of the players are v,, v k log +ɛ then the optial social welfare value is (assign every bidder i the bundle that contains d i and T i) Hence, it follows that achieving an approxiation of log+ɛ requires a super-polynoial nuber of value queries (required to distinguish between v,, v k and v,, v k) Observe that, for each i, v i and v i assign exactly the sae value to all bundles, except for bundles that contain d i and at ost ites in M Also observe, that the difference is that v i assigns a value of 0 to all these bundles, while v i assigns a value of to such bundles that contain T i (and 0 to all other such sets) What is the probability that a set of size at ost in M contains Ti? Let S be a bundle in M of size at ost Recall that Ti is uniforly distributed over all sets of size log +ɛ For every ite in T i, the probability that it is contained in S is at ost Therefore, P r[t i S] log+ɛ This iplies that log+ɛ, ie, a super-polynoial nuber of value queries, ay be required to distinguish between v i and v i The theore follows We note that this proof can be extended to the case that all bidders have the sae valuation function Observe that all the v i functions are identical So, it suffices to show that the v is can be converted to be identical, while still aintaining the following properties: It ust require a superpolynoial nuber of value queries to distinguish between v,, v k and v,, v k The ratio between the social welfare if the valuation functions are v,, v k, and the social welfare log +ɛ if the valuation functions are v,, v k ust reain (the ratio between and, respectively) This can log +ɛ be done in a way analogous to that shown in the proof of Theore 3 Inforally, in the construction of v i, we have associated every bidder i with a bundle T i However, it is possible to define v i in a way that associates every bidder i with the entire partition T,, T k 5 OPEN QUESTIONS We conclude by exhibiting the following two open questions: For the case of subodular valuation functions, the only inforation-theoretic lower bound in the odels of general and deand queries is O(/) There are indications that suggest the existence of a constant

8 ( ɛ) lower bound (APX-hardness results in the deand query and general query odels [3, 7], and an integrality gap [4]) Proving such a lower bound in any of these two odels is a very interesting open question and sees to require non-trivial cobinatorial constructions We have presented tight lower bounds in the value query odel for subodular and subadditive valuation functions There is still a gap between log log and in the case of superadditive valuations We have not considered the class of superodular valuation functions, for which no inforation-theoretic lower bound is known in any of the odels Proving inforationtheoretic lower bounds for this class is an open proble 6 REFERENCES [] L Blurosen and N Nisan On the coputational power of iterative auctions I: Deand queries In Proc of the 6th ACM Conference on Electronic Coerce (EC), 005 [] L Blurosen and N Nisan On the coputational power of iterative auctions I: Ascending auctions In Proc of the 6th ACM Conference on Electronic Coerce (EC), 005 [3] S Dobzinski Private counication, 006 [4] S Dobzinski and M Schapira An iproved approxiation algorith for cobinatorial auctions with subodular bidders In Proc of the 8rd Annual ACM Syposiu on Discrete Algoriths (SODA), 006, [5] S Dobzinski, N Nisan and M Schapira Approxiation algoriths for cobinatorial auctions with copleent-free bidders In Proc of the 37th Annual ACM Syposiu on Theory of Coputing (STOC), 005, [6] U Feige On axiizing welfare when utility functions are subadditive In Proc of the 38th Annual ACM Syposiu on Theory of Coputing (STOC), 006, 4 50 [7] U Feige and J Vondrák Approxiation algoriths for cobinatorial allocation probles: Iproving the factor of /e In Proc of the 47th Annual IEEE Syposiu on the Foundations of Coputer Science (FOCS), 006, [8] U Feige, V Mirrokni and J Vondrák Maxiizing non-onotone subodular functions In Proc of the 47th Annual IEEE Syposiu on the Foundations of Coputer Science (FOCS), 007, [9] R Holzan, N Kfir-Dahav, D Monderer and M Tennenholtz Bundling equilibriu in cobinatorial auctions Gaes and Econoic Behavior 47, 004, 04 3 [0] S Khot, R Lipton, E Markakis and A Mehta Inapproxiability results for cobinatorial auctions with subodular utility functions In Proc of WINE 005 [] B Lehann, D Lehann and N Nisan Cobinatorial auctions with decreasing arginal utilities In Proc of the 3rd ACM Conference on Electronic Coerce (EC), 00 [] N Nisan Bidding and allocation in cobinatorial auctions In Proc of the nd ACM Conference on Electronic Coerce (EC), 000 [3] N Nisan and I Segal The counication requireents of efficient allocations and supporting prices Journal of Econoic Theory, 9:, 006, 9 4 [4] N Nisan and I Segal Exponential counication inefficiency of deand queries In Theoretical Aspects of Rationality and Knowledge X, June 005 [5] G Nehauser, L Wolsey and M Fisher An analysis of the approxiations for axiizing subodular set functions Matheatical Prograing 4, 978, [6] M Fisher, G Nehauser and L Wolsey An analysis of approxiations for axiizing subodular set functions II Math Prograing Study 8, 978, [7] J Vondrák Optial approxiation for the Subodular Welfare Proble in the value oracle odel To appear in Proc of the 40th Annual ACM Syposiu on Theory of Coputing (STOC), 008