Truthful Randomized Mechanisms for Combinatorial Auctions

Transcription

1 Truthful Randoized Mechaniss for Cobinatorial Auctions Shahar Dobzinski Noa Nisan Michael Schapira March 20, 2006 Abstract We design two coputationally-efficient incentive-copatible echaniss for cobinatorial auctions with general bidder preferences. Both echaniss are randoized, and are incentivecopatible in the universal sense. This is in contrast to recent previous work that only addresses the weaker notion of incentive copatibility in expectation. The first echanis obtains an O( )-approxiation of the optial social welfare for arbitrary bidder valuations this is the best approxiation possible in polynoial tie. The second one obtains an O(log 2 )- approxiation for a subclass of bidder valuations that includes all subodular bidders. This iproves over the best previously obtained incentive-copatible echanis for this class which only provides an O( )-approxiation. 1 Introduction 1.1 Background The field of Algorithic Mechanis Design attepts to design efficient echaniss for decentralized coputerized settings. These echaniss ust take into account both the strategic behavior of the different participants and the usual algorithic efficiency considerations. Target applications include any types of protocols for Internet environent that necessitate looking at both issues strategic and algorithic together. For an introduction see [20]. The basic strategic notions are taken fro the field of echanis design a subfield of econoic theory (see [18, 23]), and in ost of the work in coputational settings, as in this one, the very robust notion of equilibriu in doinant strategies is used. It is well known ([18], see [20]) that without loss of generality, we can liit ourselves to looking at incentive copatible echaniss, also known as truthful echaniss or strategy-proof echaniss. In such echaniss participants are always rationally otivated to correctly report their private inforation. The ain difficulty in this field is the fact that the basic technique of echanis design naely VCG echaniss [25, 4, 11] can only be applied in cases where the exact optial outcoe is achieved. However, in ost interesting coputational applications, exact optiization is NP-hard, and coputationally-speaking we ust settle for approxiations or heuristics. As was observed in [20, 17], the VCG technique cannot be applied in such cases, and in fact [21] showed that this inapplicability was essentially universal. Thus, the challenge is to design alternative incentivecopatible echaniss for interesting applications. The proble of cobinatorial auctions has gained the status of the paradigatic proble of this field. For a thorough overview see [5]. In a cobinatorial auction, ites are auctioned to n players. Each player i has a valuation function v i that describes his value v i (S) for each subset The School of Coputer Science and Engineering, The Hebrew University of Jerusale, {shahard, noa, ikesch}@cs.huji.ac.il. 1

2 S of ites. The basic question is how to construct the auction echanis that allocates all the ites in a way that axiizes the social welfare Σ i v i (S i )wheres i is the set of ites allocated to bidder i. This proble indeed exhibits the basic issues of algorithic echanis design: finding the exact optiu is coputationally hard, even for the ost interesting special cases, but several approxiation algoriths, with varying approxiation ratios, are known for the general case as well as for various interesting special cases [16, 6, 7, 8]. However, these approxiation algoriths do not yield incentive copatible echaniss. In a landark paper, Lehann et al [17] were able to design an incentive-copatible, efficientlycoputable, approxiation echanis which achieves an approxiation ratio that is as good as coputationally possible Θ( ) [24] for the special case of single-inded bidders. This is the case in which each bidder is only interested in a single bundle of goods. For this special case, as well as soe other single-paraeter scenarios a host of incentive copatible echaniss have been designed in the last few years (e.g., [19, 1, 10, 9]). However, alost nothing is known for ore general cases in which bidders have coplex ulti-diensional preferences. Only two results are known in ulti-diensional settings 1 : the first is a pair of algoriths that copletely optiize over a very restricted range of allocations and then use the usual VCG echanis. These get a barely better than trivial approxiation ratio of O(/ log ) for the general case [12] and a weak O( ) for the copleent-free case [6] both ratios being quite far fro what is coputationally possible. The second result is the echanis of [2] that applies only to the special case of auctions with any duplicates of each good and indeed is not a VCG echanis. Soe evidence showing that obtaining a non-vcg incentive-copatible echanis for cobinatorial auctions and related probles would be difficult was given in [14]. 1.2 Randoized Mechaniss It was observed in [20] that randoized echaniss can soeties provide better approxiation ratios than deterinistic ones. There are two possible definitions for incentive copatibility of a randoized echanis. The first and stronger one, defines an incentive-copatible randoized echanis as a probability distribution over incentive copatible deterinistic echaniss. Thus, this definition requires that for any fixed outcoe of the rando choices ade by the echanis, players still axiize their utility by reporting their true valuations. This definition was used in [20, 10, 9], and will be called incentive copatible in the universal sense. The weaker definition only requires that players axiize their expected utility, where the expectation is over the rando choices of the echanis (but still for every behavior of the other players). This was used in [15, 8] (see below), and will be called incentive copatibility in expectation. There are two ajor iplications of the difference between these two notions: 1. Attitude towards risk: randoized echaniss that are incentive copatible in expectation only otivate risk-neutral bidders to act truthfully. Risk-averse bidders ay benefit fro strategic behavior. In contrast, the universal sense of incentive copatibility applies to any attitude towards risk, as it applies to every possible realization of the rando coins. 2. Knowledge of the randoization results: randoized echaniss that are incentive copatible in expectation induce truthful behavior only as long as players have no inforation about the outcoes of the rando coin flips before they need to act. Thus, in order to ensure truthful behavior the echanis ust utilize cryptography-grade randoness, and keep it secret fro the players. In contrast, any randoization that is effective algorithically suffices to ensure truthful behavior in the universal case. (In a siilar vein, technically speaking, 1 This is true not only for cobinatorial auctions but also for any other coputationally-hard proble. 2

3 using a pseudorando generator will destroy the foral incentive copatibility properties of randoized echaniss that are incentive copatible in expectation, due to the slight sub-polynoial change in probabilities of outcoes.) In the recent [15] a rather general technique was developed for converting approxiation algoriths into randoized echaniss that are incentive copatible in expectation. The technique is based on randoized rounding of the LP relaxation, and relies on a clever representation of the LP solution as a scaled convex cobination of integer solutions. In particular, they design a randoized echanis for general cobinatorial auctions that is incentive copatible in expectation and obtains the coputationally-optial approxiation ratio of O( ). Very recently, [8] used a different but soewhat related randoized rounding procedure to obtain another randoized echanis for the case of cobinatorial auctions with copleent-free bidders. This echanis is, again, incentive-copatible in expectation, and achieves an approxiation ratio of O( log log log ), which is worse than what he obtains algorithically a ratio of Our Results We present the first randoized echanis for cobinatorial auctions that is incentive copatible is the universal sense. This is another step towards the holy grail of obtaining a deterinistic one. Theore: There exists a polynoial-tie coputable randoized echanis for cobinatorial auctions with general bidders that is incentive copatible in the universal sense and obtains a O( ) approxiation ratio. 2 The algorith runs in tie that is polynoial in the natural paraeters of the proble: the nuber of players n and the nuber of ites. Access to the (exponentially long) valuation functions of the players is done using the usual deand queries [3, 6, 7], in which bidders are presented with a vector of ite prices p 1...p and reply with the set of ites S that axiizes their utility under these prices v(s) j S p j. The approxiation factor entioned in the theore is in expectation, however, our result is technically stronger: for any fixed >0 we provide a echanis that obtains poly() -approxiation with probability of at least 1. Our techniques are quite siple, copletely different than the ethods of [15, 8], and do not rely on the LP-relaxation of the proble. They are ore in line with the rando sapling ethods that were used for auctioning digital goods [10, 9]. These techniques can be viewed as providing a general fraework for obtaining randoized incentive copatible echaniss in the universal sense. In particular, a significant property of this fraework is that it provides, for any >0, a echanis that achieves an approxiation ratio not just in expectation, but with probability 1. We stress that this cannot be achieved by the usual techniques of aplification, since repetition can destroy the incentive properties. Using the sae fraework, we are also able to design iproved echaniss for the iportant special case of subodular valuations, and actually even for a ore general class of valuations tered XOS in [16] and fractionally-subadditive in [8] 3. This iproves over the truthful deterinistic O( )-approxiation achieved in [6]. Theore: There exists a polynoial-tie coputable randoized echanis for cobinatorial auctions with subodular bidders that is incentive copatible in the universal sense and obtains a O(log 2 ) approxiation ratio. 2 Soewhat unusually, the equilibriu obtained is in doinant strategies even for the adaptive query odel which usually only supports ex-post equilibria. 3 For the XOS class, the bidders ust also be able to answer, so called, XOS queries [6]. 3

4 Beyond the use of randoization, this theore is sub-optial in two other senses, which reain e as open probles: first, the approxiation ratio achieved is worse than the ratio of e 1 that is coputationally possible [7]; second, our echanis does not apply to the soewhat wider class of copleent-free valuations that is handled in [8, 6]. The ajor open proble left is that of finding deterinistic O( )-approxiation efficientlycoputable incentive-copatible echaniss for cobinatorial auctions. 2 Preliinaries In a cobinatorial auction, a set M of ites, M = {1,..., }, issoldton bidders. Every bidder values bundles of ites, rather than only assigning values to single ites. The value that bidder i assigns to bundle S is defined by a valuation function v i :2 M R +. Two standard assuptions regarding each bidder i, arethatv i is noralized (v i ( ) = 0), and onotone (for every S T M, v i (S) v i (T )). The allocation proble is to partition the ites between the bidders in a way that axiizes the total social welfare. I.e., to find a partition S 1,..., S n of M, that axiizes Σ i v i (S i ). Even though the size of the input is exponential in (each v i is described by 2 real nubers) we require algoriths to run in tie polynoial in the natural paraeters of the proble, and n. An iportant issue is how the input can be accessed. In this paper we follow the black box approach: we assue that we are given an oracle for each valuation function. The oracle is liited to soe predefined type of queries. A coon type of query is the deand query (e.g., [6, 7, 3]). A deand query to a valuation v i specifies a vector p =(p 1...p ) of ite prices. The answer to the query is a set that would be deanded by the queried bidder under these ite prices. I.e., a subset S that axiizes v i (S) j S p j. In this paper we seek algoriths that are incentive copatible (a.k.a. truthful). That is, algoriths which ensure that it is in the best interest of each of the bidders to always reveal his true preferences when asked. In the case of randoized echaniss this translates to being incentive copatible in the universal sense randoized echaniss that are a probability distribution over incentive copatible deterinistic echaniss. In other words, telling the truth is the doinant strategy of each bidder, regardless of the coins tossed by the echanis. This is a uch stronger requireent than incentive copatibility in expectation (see [15]). Soe special cases of cobinatorial auctions have recently received great attention. In particular, cobinatorial auctions in which all bidders are known to have subodular valuations are the subject of extensive research (e.g., [16, 6, 13, 7]). A valuation v is subodularif v(s T )+v(s T ) v(s)+v(t ) for all S, T M. All subodular valuations are known to be strictly contained in the ore general class of valuations tered XOS in [16], and fractionally-subadditive in [8]. A valuation v is said to be XOS if there are additive valuations {a 1,..., a t }, such that v(s) =ax k {a k (S)} for all S M 4. See [7] for a ore thorough explanation. For every XOS valuation v =ax k {a k }, and bundle S, we call an additive valuation a such that a(s) = arg ax k {a k (S)} a axiizing clause for S in v i. We require XOS bidders to be able to answer XOS queries. In this type of queries the question is in the for of a bundle and the answer is a axiizing clause for that bundle. 3 A Fraework For Designing Incentive-Copatible Mechaniss The design of a randoized approxiation algorith coprises two basic steps: first, we are interested in aking sure that the expected value of the solution produced by the algorith is not 4 A valuation a is additive if for every S M, a(s) =Σ j Sa({j}) 4

5 far fro the optiu. Second, we wish to be able to find a solution with a value close to the expectation with high probability. Usually, the ain difficulty is in achieving the first goal and proving that a solution close to the expectation can be obtained with soe (perhaps polynoially low) probability. Aplification of the probability of success is then easily attainable by running the algorith a polynoial nuber of ties and choosing the best solution. In contrast, the design of a randoized echanis is inherently different: in general, running a echanis ultiple ties and choosing the best output does not preserve the truthfulness of the echanis. In addition, it is well known that in order to ensure truthfulness, the price a bidder pays for the bundle he is allocated cannot depend on inforation he provides. The fraework we introduce here helps us overcoe these probles. The fraework relies on the exaination of two distinct possible cases: either there is one bidder such that allocating all ites to hi is a good approxiation to the welfare, or there is no such bidder. I.e., there is no sall group of bidders that contributes a lot to the optial solution. In the first case, achieving a good approxiation is easy - allocate all ites to that bidder. In the second and ore coplicated case, we will perfor a fixed-price auction, and will have to prove that we get a good approxiation. The key observation used in handling the second case is that two randoly chosen groups that consist (in expectation) of a constant fraction of the bidders have any properties in coon (e.g., both hold a constant fraction of the total welfare.) This idea is siilar to the ain principle in rando-sapling auctions for digital goods [10, 9]. However, our situation is uch ore coplex due to the ulti-paraeter setting of cobinatorial auctions, in contrast to the single-paraeter setting of [10, 9]. In addition, our goal is to optiize the welfare, and not axiize revenue. Moreover, we do not assue that all the ites are identical and that there is an unliited supply of ites, as in the case of digital goods. Fro a coputational point of view, another difference is that the probles we consider are NP-hard to approxiate. The fraework allows us, with high probability, to distinguish between the two cases, and provides us with the tools for finding the price used in the fixed-price auction. The ain difficulty in tailoring the fraework to a specific setting is showing that the fixed-price auction guarantees a good approxiation. Indeed, in the two echaniss we are about to present in this paper the price used in the fixed-price auction is deterined in a copletely different anner. The Fraework: Phase I: Partitioning the Bidders We assign each bidder to exactly one of the following three sets: SEC-PRICE with probability 1, FIXED with probability 2, and STAT with probability 2. Only bidders fro SEC-PRICE will be allowed to participate in the second-price auction. Bidders in STAT will never get any ites, so we can safely use this group to gather the necessary statistics (see next phase). The bidders in FIXED will be the only bidders who participate in the fixed-price auction. Phase II: Gathering Statistics The goal in this phase is to use the bidders in STAT in order to find prices for the secondprice auction with reserve price, and the fixed-price auction. Both auctions will be conducted in the next phases. To ensure incentive copatibility, a bidder should have no influence on the price of the bundle he is offered. This is why the prices of bundles offered to bidders in SEC-PRICE and FIXED in the following phases will be deterined using bidders in STAT only. The bidders in STAT never get any ites and so have no incentive to isreport their preferences. Finding the price of the fixed-price auction is echanis specific. However, the reserve price for the second price auction is generally deterined by applying an approxiation algorith 5

6 to bidders fro STAT. If no sall groups of bidders contributes a large fraction of the optial solution (the first case), we can prove that with high probability the reserve price we obtain is a good approxiation to the optial welfare. On the other hand, If there is one bidder with very high value for the bundle of all ites (the second case), we will see that this reserve price has no effect on the result of the second-price auction. Phase III: A Second-Price Auction We now conduct a second-price auction with a reserve price for selling the bundle of all ites to one of the bidders. Intuitively, one can think of this phase as handling the first case, where there is one bidder with a very high value for the bundle of all ites. A second-price auction will allocate the bundle of all ites to the bidder that values it the ost. If there is one bidder with a very high value for this bundle, he will be placed in SEC-PRICE with probability 1. We then get a good approxiation to the welfare, and the algorith terinates. The purpose of the reserve price is to handle the second case, where no sall group of bidders contributes a lot to the optial solution 5. If this is the situation, allocating all ites to one bidder ay provide a bad approxiation. Fortunately, in the previous phase we obtained a reserve price which is a good approxiation to the optial welfare. Therefore, if there is a winning bidder, we know that we have a good approxiation because the revenue obtained in the second-price auction (which is at least the reserve price) is a lower bound on the welfare. If we do not have a winning bidder, we continue to the next phase. Phase IV: A Fixed-Price Auction We go over the bidders in FIXED one by one, in soe arbitrary order, asking each one for his deand under a fixed price per ite, obtained earlier fro the bidders in STAT. We allocate each bidder his ost deanded set, and charge hi the appropriate price. We reove the set allocated to hi fro the set of ites that are offered to the next bidders. This phase is eant to handle the second case, where no sall group of bidders contributes a lot to the optial solution. Indeed, it can be shown that since FIXED is a randoly chosen group that consists of a constant fraction of all bidders, it also holds, with high probability, a constant fraction of the optial welfare. In addition, we show that in the second case the bidders in STAT aid us in choosing a fixed-price that leads to a good approxiation. The way this price is chosen is echanis-specific, and is not the sae in our two echaniss. For every possible tosses of coins the fraework produces a truthful deterinistic echanis. First, bidders who are in STAT never get any ites, and thus have no incentive to isreport their preferences. A bidder can get ites in exactly one of the following ways: by participating in the second-price auction with the reserve price, or by participating in the fixed-price auction. It is well known that second-price auctions with a reserve price are incentive copatible. The fixed-price auction is also clearly incentive copatible, as each bidder gets the bundle that axiizes his deand, given prices which he does not affect. 4 Cobinatorial Auctions with General Valuations In this section we exhibit an incentive-copatible echanis for approxiating cobinatorial auctions with general valuations. The incentive copatibility of the echanis is guaranteed by its use of the fraework. As in all echaniss built using the fraework, the ain difficulty is to 5 Of course, both a second-price auction and a second-price auction with a reserve price are incentive-copatible. 6

7 analyze the case in which no sall group of bidders contributes a lot to the optial solution. In the case of general valuations this is translated to the case where no bidder assigns a value to M that is higher than the -fraction of the value of the optial fractional solution. In this case, our echanis uses the bidders of STAT to approxiate the value of the optial fractional solution. We set the ite price for the fixed-price auction to be (approxiately) the value of the approxiation we obtained, divided by the nuber of ites. The iportant technical observation is that for each ite we anage to sell at this price, we lose a value of at ost O( ) ties this price (copared to the optial fractional solution). The revenue we get in this case sets a lower bound on the welfare we achieve. Although the echanis does use the LP relaxation of the proble, LP does play a relatively inor role, and we ainly use it for the analysis. This is in contrast to previous related work [15, 8], where the technique itself is LP based. The reader is referred to the appendix for the standard LP relaxation of the proble. The Algorith: Input: n bidders 6, each with a general valuation v i that is represented by a deand oracle, a rational nuber 0 <<1. Output: An allocation of the ites, which is a O( )-approxiation to the optial allocation. 3 The Algorith: Phase I: Partitioning the Bidders 1. Assign each bidder to exactly one of the following three sets: SEC-PRICE with probability 1, FIXED with probability 2, and STAT with probability 2. Phase II: Gathering Statistics 2. Calculate the value of the optial fractional solution in the cobinatorial auction with all ites, but only with the bidders in STAT. Denote this value by OPT STAT. Phase III: A Second-Price Auction 3. Conduct a second-price auction with a reserve price of OPT STAT, in which the bundle M of all ites is sold to the bidders in SEC-PRICE. If there is a winning bidder, allocate all the ites to that bidder and output this allocation. Otherwise, proceed to the next step. Phase IV: A Fixed-Price Auction 4. Let R = M. Letp = OP T STAT For each bidder i FIXED, in soe arbitrary order: (a) Let S i be the deand of bidder i given the following prices: p for each ite in R, and for each ite in M R. (b) Allocate S i to bidder i, and set his price to be p S i. (c) Let R = R \ S i. Theore 4.1 For any constant >0, there exists a randoized and truthful polynoial-tie echanis that achieves an O( )-approxiation with probability Both in this echanis and in the XOS echanis, we assue that n is not constant. If n is constant, one can easily get a truthful 1 -approxiation by bundling all ites together and perforing a second price auction. n 7

8 Proof: The algorith produces a feasible allocation. In addition, the algorith is clearly incentive copatible, since it was designed using the fraework. It is left to prove that it obtains the desired approxiation ratio with probability 1. Denote by OPT the optial fractional solution. There are two possible cases: 1. There is a bidder i such that v i (M) OPT. 2. For each bidder i, v i (M) < OPT. We start by handling the first case. Let i be soe bidder such that v i (M) OPT. Observe that with probability 1 bidder i is in SEC-PRICE. If there is no such bidder is in SEC-PRICE, then the algorith fail to guarantee any approxiation ratio. This happens with probability of at ost. The next proposition shows that if bidder i is in SEC-PRICE the algorith obtains an O( )-approxiation. Proposition 4.2 If there exists a bidder i in SEC-PRICE such that v i (M) OPT, then the allocation generated by the algorith is an O( )-approxiation to the optial allocation. Proof: Let i be the bidder in SEC-PRICE with the highest value for M. By the conditions of the lea, v i (M) OPT. Clearly, since STAT N, wehavethat: OPT STAT OPT v i (M) Hence, due to the properties of second-price auctions with a reserve price, all ites will be sold to i, and the algorith will terinate in Step 3. Thus, we get an allocation that is an O( )- approxiation to the optial one. The second case is ore involved. For each bidder i, v i (M) < OPT. We will take advantage of the fact that no bidder contributes a lot to the optial fractional solution, and see that, with high probability, OPTSTAT is a good approxiation to the optial fractional solution. We will see that the sae holds for OPTFIXED, which is the value of the optial fractional solutions in the cobinatorial auctions with all ites and with bidders fro FIXED only. Lea 4.3 If for each bidder i, v i (M) < OPT,thenwithprobability1 o(1): 1. 4 OPT OPT STAT 2. 4 OPT OPT FIXED Proof: We will start by proving that the probability that the first event does not occur is o(1). The proof for the second is alost identical. The lea will then follow, by applying the union bound. Let A be the rando variable that receives the value of OPTSTAT. For every bidder i we denote by A i the rando variable that receives the value of bidder i in OPTSTAT. Let {x i,s} 1 i n,s M be the set of variables in the fractional solution, OPT. Since every bidder is placed in STAT with probability 2,andSTAT N, wehavethate[a] =Σ i 2 E[A i] Σ i 2 Σ Sx i,s v i (S) = 2 OPT. If the conditions of the lea hold, we also have that for each i, A i < OPT. We can use this fact to set an upper bound on the probability that A gets a value that is substantially saller than its expectation. We ake use of the following corollary fro Chebyshev s inequality: 8

9 Clai 4.4 Let X be the su of independent rando variables, each of which lies in [0,t]. Then, for any α>0, Pr[ X E[X] α] te[x] α 2. Since for each i, A i [0, OPT ], we have that Pr[A < 4 OPT ] Pr[ A 2 OPT 4 OPT ] OPT 2 OPT ( 4 OPT ) 2 8 With probability of 1 o(1) we have that the values of the optial fractional solutions for FIXED and STAT are close to OPT. If this is the case, we will show that we anage to achieve an O( ) approxiation factor. With probability of at ost o(1) this is not the case, and the 2 algorith fails to provide any approxiation ratio. Although the second-price auction was designed to handle the first case, when there is one bidder that contributes a lot to the welfare, it is still possible that soe bidder i in SEC-PRICE will be allocated the bundle M in Step 3. However, notice that bidder i was forced to pay at least OPT STAT. Therefore, that bidder s value for the bundle M is greater than OPT STAT, which by Lea 4.3 is at least OP T 4. Hence, allocating the bundle M to bidder i provides an O( ) approxiation to the optial solution. If no bidder in SEC-PRICE got the bundle M then the algorith attepts to sell ites to the bidders in FIXED (Step 5). As before, we clai that the revenue is a lower bound to the social welfare. The next lea shows that in this case the revenue will be Ω( 3 OPT ). Hence, Step 5 will result in an allocation that is a Ω( )-approxiation to the optial allocation. 3 Lea 4.5 If the following conditions hold: 1. The algorith reaches Step 5 2. For each bidder i, v i (M) < OPT 3. For the ite-price p it holds that: 2 OPT OPT FIXED 4 OPT p OPT 8 then the revenue of the algorith is Ω( 3 OPT ). Proof: Let {y i,s } i FIXED,S M be the variables in the fractional solution OPTFIXED. We will restrict our attention to bundles in OPTFIXED that are profitable when setting a price of p for each ite. That is, let T be the set of pairs (i, S) such that y i,s > 0, and v i (S) p S > 0. The next clai shows that we do not lose too uch by ignoring all other bundles in OPTFIXED. Clai 4.6 Σ (i,s) T y i,s v i (S) 1 2 OPT FIXED Proof: Define T to be the copleent set of T. Forally, T consists of all pairs (i, S) such that y i,s > 0inOPT FIXED, but v i(s) p S 0. It is easy to see that OPT FIXED = 9

10 Σ (i,s) T y i,s v i (S) +Σ (i,s) T y i,s v i (S). Since OPTFIXED 4 OPT it is enough to bound fro above the contribution of T to OPTFIXED to prove the clai. Σ (i,s) T y i,s v i (S) Σ (i,s) T y i,s p S p OPT 8 OPT FIXED 2 where the first inequality is because of the definition of T and the second inequality is due to the LP constraints. Let us now calculate the revenue we get in Step 5. Without loss of generality, assue the bidders in FIXED are 1,..., 2n. In the first iteration of Step 5, bidder 1 is asked for his ost deanded set. The key observation is that if there is soe S such that x 1,S > 0and(1,S) T then bidder 1 s deand is not epty. Recall that for each ite in S 1 we gain a revenue of p. We will now upper bound what we lose by assigning S 1 to bidder 1 in coparison to OPTFIXED. Notice, that by assigning S 1 to bidder 1 we lose both the value of all the fractional bundles assigned to bidder 1 in OPTFIXED, and of all the bundles in OPT FIXED that contain soe ite fro S 1. The value of all the fractional bundles assigned to bidder 1 in OPTFIXED OPT is at ost : Σ (1,S) T y 1,S v 1 (S) OPT because v 1 (M) < OPT and Σ (1,S) y 1,S 1, due to the constraints of the LP forulation. We will now bound the value of all the bundles in OPTFIXED that contain soe ite fro S 1. Fix soe ite j S 1. Again, using the constraints of the LP and v i (M) < OPT, Σ (i,s) T j S y i,s v i (S) OPT To conclude, for every ite we sell to bidder 1 at price p 2 OPT 16, we lose bundles in T that are together worth at ost 2 OPT. The analysis continues by reoving fro OPTFIXED all pairs (i, S) which can not be assigned now (either i =1,orj S i and j S), and applying siilar arguents to the rest of the bidders in FIXED. The revenue achieved by the algorith is an O( )-approxiation to the value of OPT 2 FIXED. Since OPTFIXED 4 OPT we have that it is a O( ) approxiation to OPT, and the theore 3 follows. 5 Cobinatorial Auctions with XOS Valuations Like the echanis for approxiating cobinatorial auctions with general valuations, the echanis for XOS valuations is also based on the general fraework. Again, the ain challenge involved in designing this echanis is analyzing the case in which no sall group of bidders contributes a lot to the optial solution. The way this is achieved for XOS valuations is entirely different fro the way it is done with general valuations. Suppose we assign a bundle S to a bidder with an XOS valuation v i. By the definition of XOS valuations, v(s) is deterined by the value S gets under soe additive valuation a (we will also refer to a as the axiizing clause). We can look at the whole process as iplicitly assigning a 10

11 price to each ite in S (the price that a assigns to each ite in S.) We will use this property for finding a price for the fixed-price auction. To see how this is done we ust first introduce the following definition: Definition 5.1 We say that an allocation of the ites T =(T 1,..., T n ) is supported by a price p, if for each bidder i and each possible bundle S i T i, it holds that v i (S i ) S i p. WecallΣ i T i p the supported value of T. We now show that for every allocation it is possible to find a contained allocation and a price that supports it, and holds a considerable part of the welfare of the original allocation. Lea 5.2 For every allocation T =(T 1,..., T n ) it is possible to find in polynoial tie an allocation (S 1,..., S n ) and a price p that supports it, such that for each i, S i T i,andσ i S i p Ω( Σ iv i (T i ) log ). Proof: GivenanallocationT, we query each bidder i s XOS oracle for the axiizing XOS clause for T i. We refer to the value of an ite in T i as the ite s value in the axiizing clause of T i.letw =Σ i v i (T i ) (i.e., the welfare value of T.) Define the set P = { W 2, W,..., W 2,W}. Notice that P = O(log ). Round down each ite s value in the axiizing clauses to the nearest value in P.Letp P be the (rounded down) ite value that contributes the ost to the welfare. Notice that we ignore ites with value lower than W 2 our loss is not too high since the su of these ites values is less than W 2. We can now define (S 1,..., S n ) to be the allocation in which S i T i and the (rounded down) value of every ite in T i is at least p. There is still the atter of finding such a price that would enable us to get a good approxiation in the fixed-price auction. We prove that one can use the bidders in STAT to find such a price for the bidders in FIXED with high probability. We also note that if a valuation is known to be subodular, an XOS oracle for it can be siulated using a deand oracle [6]. Thus, if all bidders are known to be subodular our echanis can be ipleented using deand oracles only. The Algorith: Input: n bidders, v 1,..., v n, each represented by a deand and a XOS oracle, a rational nuber 0 << 1 2. Output: An allocation of the ites, which is an O( log2 )-approxiation to the optial allocation. 3 The Algorith: Phase I: Partitioning the Bidders 1. Assign each bidder to exactly one of the following three sets: SEC-PRICE with probability 1, FIXED with probability 2, and STAT with probability 2. Phase II: Gathering Statistics 2. Find an allocation that is an O(1) approxiation to the value of the optial solution in the cobinatorial auction with all ites, but only with the bidders in STAT (e.g., using the algoriths of [6, 7]). Denote this value by OPT STAT. 3. Using the allocation obtained in the previous step, find a price p and an allocation T = (T 1,..., T STAT ), such that T is supported by p (rounded down to the nearest power of 2), and Σ i STAT T i p Ω( OPT STAT log ). 11

12 Phase III: A Second-Price Auction 4. Conduct a second-price auction with a reserve price of 2 OPT STAT 100 log 2, in which the bundle M of all ites is sold to the bidders in SEC-PRICE. If there is a winning bidder, allocate all the ites to that bidder and output this allocation. Otherwise, proceed to the next step. Phase IV: A Fixed-Price Auction 5. Let R = M. Letp = p /2. 6. For each bidder i FIXED, in soe arbitrary order: (a) Let S i be the deand of bidder i given the following prices: p for each ite in R, and for each ite in M R. (b) Allocate S i to bidder i, and set his price to be p S i. (c) Let R = R \ S i. Theore 5.3 For any constant 0 << 1 2, there exists a randoized and truthful algorith that achieves an O( log2 )-approxiation with probability 1. 3 Proof: The algorith produces a feasible allocation. Incentive copatibility of the algorith is guaranteed since it was built using the fraework. It is left to prove that it obtains the desired approxiation ratio with probability 1. We will now prove that the the algorith provides the approxiation ratio. Let R = 2 OPT 100 log 2. There are two possible cases: 1. There is a bidder i such that v i (M) R. 2. For each bidder i, v i (M) <R. We handle the first case in a way siilar to the way we handled the first case in the correctness proof for the algorith of Section 4. Let i be soe bidder such that v i (M) R. Observe that with probability 1 bidder i is in SEC-PRICE. If there is no such bidder is in SEC-PRICE, then the algorith fails to guarantee any approxiation ratio. This happens with probability of at ost. If bidder i is in SEC-PRICE the algorith obtains a R-approxiation. The next proof is siilar to the proof of Lea 4.2. Proposition 5.4 If there exists a bidder i in SEC-PRICE such that v i (M) R, then the allocation generated by the algorith is a O(R)-approxiation to the optial allocation. Let us now exaine the second case, where for each bidder i, v i (M) <R. The basic idea is to use the bidders in STAT to find a price that, with high probability, will obtain an allocation of the ites to the bidders in FIXED in Step 6. To show this we need to prove that OPT FIXED, the value of the optial solution consisting of the bidders in FIXED only, has a value that is close to the value of the total welfare. This will be done in a siilar way to the previous algorith. However, unlike the previous algorith, we have to prove that if a price is good in OPT FIXED (i.e. supports an allocation that holds a substantial part of the welfare), then it can be found using the bidders in STAT. As in Lea 5.2, we restrict our attention to prices which are greater than OPT 2 log. Lea 5.5 If for each bidder i, v i (M) <R, then with probability higher than : 12

13 1. 4 OPT OPT STAT 2. 4 OPT OPT FIXED OPT 3. Let P = {p p is a power of 2, and 2 log p OPT log, and there exists an allocation T that is supported by p, andthesupportedvalueoft is at least 4 OPT log }. Then, for every p k P there exists an allocation T k of the ites to the bidders in FIXED only such that T k is supported by p k,andthesupportedvalueoft k is at least 2 16 OPT log. Proof: The proof that the probability that one of the first two events does not occur is o(1) is identical to that of Lea 4.3. We now bound fro above the probability that the third event occurs and use the union bound to coplete the proof. Let T =(T 1,..., T n ) be an allocation, and p k P a price such that the supported value of T is at least 4 OPT log,andt is supported by p k. We now turn our attention to the bidders in FIXED. Observe that for each bidder i FIXED, v i (T i ) T i p t. Therefore, we will prove that there exists a T k with the desired value by looking at the expected value of T, restricted only to bidders in FIXED. Let A i be the rando variable that gets the value of p T i with probability 2,and0with probability 1 2. Let A =Σ ia i. Since every bidder i is placed in FIXED with probability 2 we have that E[A] =Σ i E[A i ]= 2 Σ ip T i 4 OPT log }. Using Clai 4.4, and since for each i, A i [0,R], we have that Pr[A < 2 OPT 16 log ] Pr[ A 2 OPT 8log 2 OPT 16 log ] R 2 OPT 8log ( 2 OPT 16 log 32R log )2 2 OPT Since there are less than log possible choices of p k, we can apply the union bound to verify 32R log log that the fourth event does not occur with probability 2 OPT. By our choice of R, wegetthat they all hold siultaneously with probability of at least Given that the conditions of Lea 5.5 hold, we will show that we anage to achieve an O( log ) 3 approxiation factor. With probability of at ost 2 2 this is not the case, and the algorith fails to provide any approxiation ratio. If soe bidder i in SEC-PRICE was allocated M in Step 4, then he was forced to pay at least 2 OPT STAT 2 OPT 100. Therefore, that bidder s value for M is greater than STAT log 2 100, which by Lea log is at least O( 3 OPT log 2 ). Hence, allocating M to bidder i provides a a O( log2 ) approxiation to 3 the optial solution. If no bidder in SEC-PRICE got the bundle M then the algorith attepts to sell ites to the bidders in FIXED (Step 6). The next two leas show that in this case we will get an allocation that is an O( log )-approxiation to the optial allocation. 3 Lea 5.6 Let T p =(T 1,..., T n ) be an allocation that axiizes Σ i v i (T i ) such that 1. T p is supported by p. 2. For each bidder i/ FIXED, T i =. Then, if the algorith reaches Step 6 the approxiation ratio achieved is O(Σ i T i p). 13

14 Proof: We first note that by assigning T i to each bidder i and charging a price of T i p, we gain a revenue of Σ i T i p, while all bidders are profitable. We will use this revenue as a lower bound to the welfare that can be achieved. Notice that we do not guarantee that the actual revenue the echanis gets is a constant factor away fro Σ i T i p. We will now upper bound the revenue we lose by assigning S 1 to bidder 1, coparing to the allocation considered earlier. Without loss of generality, assue the bidders in FIXED are nubered 1,..., 2n. In the first iteration of Step 5, bidder 1 is asked for his ost deanded set. First, we could have assigned T 1 to bidder 1 and gain a revenue of T 1 p 2. (Recall that the the price for ite is p 2.) However, we did not lose too uch because the value of T 1 is at ost twice the value of S 1. The last stateent is true since bidder 1 could gain a profit of at least T 1 p 2 by choosing T 1,and S 1 has at least that value being bidder 1 s ost deanded set. We note again that the revenue we achieve in this case (but not the welfare) ight be very sall coparing to v i (T i ). The second possible lose occurs when there is an ite j S 1, and there exists another bidder i with j T i. Because T p is supported by p, wehavethatv i (T i \{j}) ( T i 1) p. Suing overallsuchites,wehavethatweloseavalueofatost S 1 p 2 v 1(S 1 ). The inequality holds since S 1 is profitable to bidder 1 under a price per ite of p 2. To conclude, by assigning T 1 to bidder 1 we lose a revenue of O(T 1 ). The analysis continues by reoving fro T 2,..., T all ites which can not be assigned now, and using induction to apply 2 siilar arguents to the rest of the bidders in FIXED. Lea 5.7 If the following conditions hold: 1. The algorith reaches Step OPT OPT STAT 3. 4 OPT OPT FIXED OPT 4. Let P = {p p is a power of 2, and 2 log p OPT log, and there exists an allocation T that is supported by p, andthesupportedvalueoft is at least 4 OPT log }. Then, for every p k P there exists an allocation T k of the ites to the bidders in FIXED only such that T k is supported by p k,andthesupportedvalueoft k is at least 2 16 OPT log. Then the algorith produces an allocation that is an O( log )-approxiation to the welfare. 3 Proof: Observe that in Step 3 we have found an allocation that is supported by p and worth ore than OPT STAT log OP T 4log. Obviously, an allocation restricted to bidders in STAT only is also an allocation for all bidders with the sae value. We can therefore deduce that there exists an allocation T p of the ites to bidders in FIXED such that T p =(T 1,..., T n ) is supported by p, and worth at least 2 16 OPT log. Clearly, all conditions of Lea 5.6 hold. Therefore, the algorith is an O( log )-approxiation 2 to the value of OPT FIXED. Since OPT FIXED log 4 OPT we have that it is an O( ) approxiation to 3 OPT. Acknowledgents We thank Moshe Babaioff, Liad Blurosen, Uri Feige, Ron Lavi, Ahuva Mu ale, and Chaitanya Sway for helpful discussions and coents. The authors are supported by grants fro the Israel Science Foundation and the USA-Israel Bi-national Science Foundation. 14

15 References [1] A. Archer, C. Papadiitriou, K. Talwar, and E. Tardos. An approxiate truthful echanis for cobinatorial auctions with single paraeter agent. In Proceedings of the 14th Annual ACM Syposiu on Discrete Algoriths (SODA), [2] Yair Bartal, Rica Gonen, and Noa Nisan. Incentive copatible ulti unit cobinatorial auctions. In TARK 03, [3] Liad Blurosen and Noa Nisan. On the coputational power of iterative auctions I: Deand queries. Working Paper. Preliinary version in EC [4] E. H. Clarke. Multipart pricing of public goods. Public Choice, 2:19 33, [5] P. Craton, Y. Shoha, and R. Steinberg (Editors). Cobinatorial Auctions. MIT Press. Forthcoing., [6] Shahar Dobzinski, Noa Nisan, and Michael Schapira. Approxiation algoriths for cobinatorial auctions with copleent-free bidders. In STOC 05: Proceedings of the thirty-seventh annual ACM syposiu on Theory of coputing, pages , New York, NY, USA, ACM Press. [7] Shahar Dobzinski and Michael Schapira. An iproved approxiation algorith for cobinatorial auctions with subodular bidders. In SODA [8] Uriel Feige. On axiizing welfare where the utility functions are subadditive. To appear in STOC [9] Aos Fiat, Andrew V. Goldberg, Jason D. Hartline, and Anna R. Karlin. Copetitive generalized auctions. In STOC, pages 72 81, [10] Andrew Goldberg, Jason Hartline, Anna Karlin, Mike Saks, and Andrew Wright. Copetitive auctions. Subitted. [11] T. Groves. Incentives in teas. Econoetrica, pages , [12] Ron Holzan, Noa Kfir-Dahav, Dov Monderer, and Moshe Tennenholtz. Bundling equilibriu in cobinatrial auctions. Gaes and Econoic Behavior, 47: , [13] Subhash Khot, Richard Lipton, Evangelos Markakis, and Aranyak Mehta. Inapproxiability results for cobinatorial auctions with subodular utility functions. To appear in WINE [14] Ron Lavi, Ahuva Mu ale, and Noa Nisan. Towards a characterization of truthful cobinatorial auctions. In FOCS 03: Proceedings of the 44th Annual IEEE Syposiu on Foundations of Coputer Science, page 574, Washington, DC, USA, IEEE Coputer Society. [15] Ron Lavi and Chaitanya Sway. Truthful and near-optial echanis design via linear prograing. In FOCS [16] Benny Lehann, Daniel Lehann, and Noa Nisan. Cobinatorial auctions with decreasing arginal utilities. In ACM conference on electronic coerce, [17] Daniel Lehann, Liadan Ita O Callaghan, and Yoav Shoha. Truth revelation in approxiately efficient cobinatorial auctions. In JACM 49(5), pages , Sept

16 [18] A. Mas-Collel, W. Whinston, and J. Green. Microeconoic Theory. Oxford university press, [19] Ahuva Mua le and Noa Nisa. Truthful approxiation echaniss for restricted cobinatorial auctions. In AAAI-02, [20] Noa Nisan and Air Ronen. Algorithic echanis design. In STOC, [21] Noa Nisan and Air Ronen. Coputationally feasible vcg-based echaniss. In ACM Conference on Electronic Coerce, [22] Noa Nisan and Ilya Segal. The counication requireents of efficient allocations and supporting prices. To appear in Journal of Econoic Theory. [23] M. J. Osborne and A. Rubistein. A Course in Gae Theory. MIT press, [24] Tuoas Sandhol. Algorith for optial winner deterination in cobinatorial auctions. Artificial Intelligence, 135(1-2):1 54, [25] Willia Vickrey. Counterspeculation, auctions and copetitive sealed tenders. Journal of Finance, 16(1):8 37, March A The Standard LP Forulation of a Cobinatorial Auction Maxiize: Σ i,s x i,s v i (S) Subject to: For each ite j: Σ i,s j S x i,s 1 for each bidder i: Σ S x i,s 1 for each i, S: x i,s 0 We reark that the LP relaxation can be solved using deand oracles only [22]. 16