Two Randomized Mechanisms for Combinatorial Auctions Shahar Dobzinski The School of Computer Science and Engineering, The Hebrew University of Jerusalem

Abstract. This paper discusses two advancements in the theory of designing truthful randomized mechanisms. Our first contribution is a new framework for developing truthful randomized mechanisms. The framework enables the construction of mechanisms with polynomially small failure probability. This is in contrast to previous mechanisms that fail with constant probability. Another appealing feature of the new framework is that bidding truthfully is a strongly dominant strat√ egy. The power of the framework is demonstrated by an O( m)-mechanism for combinatorial √ m ). auctions that succeeds with probability 1 − O( log m The other major result of this paper is an O(log m log log m) randomized truthful mechanism for combinatorial auction with subadditive bidders. The best previously-known truthful √ mechanism for this setting guaranteed an approximation ratio of O( m). En route, the new mechanism also provides the best approximation ratio for combinatorial auctions with submodular bidders currently achieved by truthful mechanisms.

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Introduction

1.1

Background

The field of Algorithmic Mechanism Design [15] has received much attention in recent years. The main goal of research in this field is to design efficient algorithms for computationally-hard problems, in a way that handles the strategic behavior of the participants. In particular, we are interested in mechanisms that are truthful, i.e., where the dominant strategy of each bidder is to report his true preferences1 . For a recent excellent introduction to the basics of mechanism design the reader is referred to [14]. Arguably the most important problem in algorithmic mechanism design is the design of truthful efficient mechanisms for combinatorial auctions. In a combinatorial auction we have n bidders and a set M of items, |M | = m. Each bidder i has a valuation function vi . The common assumptions are that each valuation function is normalized (vi (∅) = 0) and monotone (for every S ⊆ T ⊆ M, vi (T ) ≥ vi (S)). The goal is to find a partition S1 , ..., Sn of the items, such that the total social welfare, Σi vi (Si ), is maximized. Notice that a naive representation of each valuation function requires exponential number of bits in n and m, the parameters we require our algorithms to be polynomial in. This paper therefore assumes that the valuations are represented as black boxes which can answer a specific natural type of queries, demand queries 2 . 1

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One could also consider algorithms that lead to another arbitrary equilibrium, but the revelation principle tells us that the restriction to truthful mechanisms is essentially without loss of generality. In a demand query the bidder is presented with a price of pj for each item, and the answer is the bundle that maximizes the profit, v(S) − Σj∈S pj .

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Combinatorial auctions demonstrate the clash between the computational and economic aspects that stands in the heart of algorithmic mechanism design: on one hand, obtaining a truthful mechanism is easy, ignoring computational limitations: find the optimal solution and use VCG payments, probably the main technique of mechanism design. Unfortunately, combinatorial auctions are hard 1 to exactly solve and even to approximate: in the general case, an approximation ratio of m 2 −² cannot be obtained in polynomial time, for any constant ² > 0 [13]. It is known that in general using VCG together with an approximation algorithm (in contrast to the optimal algorithm) does not result in a truthful mechanism [16], hence other techniques are required. Recently, a series of papers [11, 4, 5] used randomization to construct truthful mechanisms with good approximation ratios. Referring to randomization seems necessary, as we have some evidence that deterministic mechanisms do not have much power [10, 2]. Two types of randomized mechanisms were considered: mechanisms that are truthful in expectation, and mechanisms that are truthful in the universal sense. Mechanisms that are truthful in expectation are truthful only with respect to bidders that maximize their expected utility (risk-neutral bidders), and do not know the outcome of the random coin flips. Mechanisms that are truthful in the universal sense are stronger, as they are truthful for every type of bidders, and even if the outcome of the random coins is known in advance. For a more thorough discussion of the differences the reader is referred to [4]; This paper considers only mechanisms that are truthful in the universal sense. 1.2

Our Results

An Improved Framework for Designing Truthful Mechanisms In [4] a general framework for designing truthful mechanisms was presented. The framework uses random-sampling methods (introduced in [7]), and was quite successful in enabling the design of mechanisms that provide good approximation ratios and are truthful in the universal sense (rather than mechanisms that are truthful in expectation [11, 5]). However, the framework of [4] suffers from two major drawbacks: – The probability of success: In general, given a randomized mechanism, running the mechanism again (in order to increase the success probability) makes the mechanism no longer truthful. Thus, the main goal of the framework of [4] was to achieve good approximation ratios with high probability. Unfortunately, this was achieved by trading success probability with approximation ratio. For example, in its application to combinatorial auction with general bidders, √ the framework provided an approximation ratio of O( ²3m ) with probability 1 − ², for any ² > 0. In particular, this means that in order to obtain a probability of success that is better than a constant, we have to compromise on a non-optimal approximation guarantee (by more than a constant factor). – The motivation of bidders to participate: The framework of [4] uses a randomly selected group of bidders that cannot win any items at all. Yet, bidders in it are still required to provide information about their valuations. In other words, bidding truthfully is only a weakly dominant strategy for these bidders, as their utility will always be 0 regardless of the information they provide. One contribution of the current paper is an improved framework that overcomes both limitations. The improved framework enables the design of mechanisms that provide good approximation ratios with high probability, without compromising on the approximation ratio. In addition, for every outcome of the random coins bidding truthfully can sometime strictly improve the utility of each bidder. The main application of the improved framework is the following theorem, that guarantees

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the optimal approximation ratio possible for combinatorial auctions with general bidders with high success probability: √ Theorem: There exists a truthful randomized O( m)-approximation mechanism for combinatorial auctions with general bidders. The approximation ratio is achieved with probability of at least √ m ). Bidding truthfully is a strongly dominant strategy of each bidder. 1 − O( log m The analysis of the framework of [4] was quite straightforward. The improved framework presented in this paper is more involved, and its analysis requires some effort.

A Mechanism for Combinatorial Auctions with Subadditive Bidders Another line of research that has gained popularity recently is determining the approximation ratios possible when the bidders’ valuations are restricted (e.g., [12, 3, 5, 6, 9]). Most of the research concentrated in the case when the bidders’ valuations are known to be either subadditive (for every S, T ⊆ M , v(S) + v(T ) ≥ v(S ∪ T )) or submodular (for every S, T ⊆ M , v(S) + v(T ) ≥ v(S ∪ T ) + v(S ∩ T )). Between these two classes lies the syntactic class of XOS valuations (see [4] for a definition), that strictly contains the class of submodular valuations, but is strictly contained in the class of subadditive valuations. 2 In [4] another application of the framework is presented, an O( log²3 m ) mechanism for combinatorial auctions with XOS bidders that succeeds with probability of 1 − ². The improved framework presented in this paper can be applied to construct an O(log2 m) mechanism for the same setting that succeeds with probability 1 − O( log1 m ). Yet, for the case of combinatorial auctions with subad√ ditive bidders, the best mechanism until now guaranteed an approximation ratio of only O( m) [3]. log m Feige [5] presents an O( log log m )-mechanism that is truthful in expectation for this case3 . Ignoring incentives issues, a ratio of 2 can be achieved [5]. This paper presents the first truthful mechanism that provides a poly-logarithmic approximation ratio for combinatorial auctions with subadditive bidders. En route, this mechanism also improves upon the best mechanism known for combinatorial auctions with submodular bidders [4]. Another important improvement over [4] is this paper’s mechanism uses the simple and natural demand queries, and not, as in [4], the syntactically defined XOS queries. Theorem: There exists a truthful O(log m log log m)-mechanism for combinatorial auctions with subadditive bidders, that succeeds with probability of 1 − O( log1 m ). We point out that unlike the improved framework that this paper describes, where our main contribution is game-theoretic, the main improvement of the mechanism for subadditive bidders is of algorithmic nature: we define a new combinatorial property of valuation functions, namely, αsupporting prices, and show that if a class of valuations exhibit this property, then we can essentially derive an O(α)-approximation algorithm. Finally, we prove that every subadditive valuation has an O(log m)-supporting prices, and show how to use the framework to derive a truthful mechanism that achieves almost this approximation ratio. 3

In fact the mechanism of [5] uses an even weaker notion than truthfulness in expectation, as truthfulness maximizes the expected utility only approximately.

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1.3

Open Questions

The main question we leave open is to determine the exact approximation ratio that truthful mechanisms can guarantee for combinatorial auctions with subadditive bidders. Obtaining mechanisms with constant approximation ratios is of particular interest. It will be also very interesting to understand the power of randomization; √ For example, for combinatorial auctions with general valuations this paper presents an optimal O( m)-randomized mechanism. Yet, the best deterministic mechm anism achieves a ratio of O( √log ) [8]. For combinatorial auctions with subadditive bidders, the m √ gap is also large: the best deterministic mechanism achieves a ratio of O( m) [3], while the best randomized mechanism is this paper’s O(log m log log m) mechanism. A recent result [2] suggests that this gap cannot be bridged, at least with the current techniques. A second open question is to develop mechanisms that fail not only with polynomially small probability, but with exponentially small probability, as common in traditional algorithm design. This seems beyond the power of the framework, and any other random-sampling based techniques, due to the need to “gather statistics”, which fails with polynomially small probability. Developing such mechanisms seems very challenging.

2

An Improved Framework for Truthful Mechanisms

The new framework presented in this section uses two types of auctions as subroutines: a secondprice auction with non-anonymous reserve prices, and a fixed price auction. A second-price auction with non-anonymous reserve prices is a variation of the well-known second-price auction. Here each bidder is presented with a (different) reserve price. If the winner (the bidder with the highest bid) bids above his reserve price, he wins the item and pays the maximum price of the second-highest bid and his reserve price. If the winner bids below his reserve price, then he pays nothing and the item is not allocated. The auction is truthful if each bidder’s reserve price is independent of his bid. In a fixed-price auction with price p the bidders are ordered arbitrarily, and we assign the first bidder his most demanded set when the price of each item is p. Then, we assign the second bidder his most demanded set from the set of remaining items (the items that were not allocated to the first bidder), again, when the price of each remaining item is p and so on. If p does not depend on the valuations of the bidders, then a fixed-price auction is truthful. Let us now present the framework itself. To create a specific mechanism, we have to specify four parameters. These parameters are marked in bold. The Framework 1. Add each bidder to exactly one of the following groups of bidders: with probability 12 to STAT, and with probability 12 to FIXED. 2. For each set of bidders S, we denote by OP T S an estimation of the optimal solution with the participation of bidders in S only. 3. Conduct a second-price auction with non-anonymous reserve prices for selling the bundle of all items with the participation of bidders in STAT only. Set the reserve price of the ith bidder OP T

ST AT \{i} that is in STAT to be , for some α. α If there was a winner in the second-price auction, continue to Case 1, otherwise continue to Case 2.

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Case 1: There was a winner. 4. Conduct a second-price auction with non-anonymous reserve prices for selling the bundle of all items with the participation of all bidders. Set the reserve price of each bidder i ∈ ST AT OP T

ST AT \{i} to . Set the reserve price of each bidder i ∈ F IXED to 0. Let w be the winning α bidder. Assign w the bundle of all items, and let w pay max(maxw6=i∈N vi (M ), rw ), where rw denotes the reserve price of bidder w. Finish the mechanism. Case 2: There was no winner. 5. Use the bidders in STAT to calculate a price per item of p to be used in a fixed-price auction among the bidders in FIXED. Conduct a fixed-price auction with this price. Let Si denote the bundle bidder i is assigned in this auction. 6. Select, uniformly at random, a price r from the set [ OP TαST AT , OP TαST AT , · · · , OP TβST AT ], for some β. Conduct a second-price auction with a reserve price of r for selling the bundle of all items among the bidders of FIXED. Let j denote the winner in auction (if one exists). If there are at least two bids above r, allocate the bundle to the bidder with the highest bid, let him pay the second-most highest bid, and finish the mechanism. Otherwise, continue to the next step. 7. If there is no bid above r, then assign each bidder i the bundle Si and let him pay |Si | · p. If there is exactly one bid above r, let this bidder j choose his maximum-profit allocation of the following two, and assign accordingly: – Assign bidder j all items, and let him pay r. – Assign bidder j the bundle Sj and let him pay |Sj | · p. Assign the rest of the bidders no items at all.

We will first see that mechanisms constructed using the framework are truthful. As an application, we will see an improved mechanism for combinatorial auctions with general bidders. Theorem 1. Mechanisms constructed using the framework are truthful. Moreover, for each bidder i bidding truthfully is a strongly dominant strategy. Proof. We first prove that the (strongly) dominant strategy of each bidder in STAT is to bid truthfully. Then, we show that this is true for bidders in FIXED too. Look at some bidder in STAT. This bidder can only win the bundle of all items, by participating a second-price auction with non-anonymous reserve prices. Observe that the reserve price does not depend on the declaration of the bidder. Clearly, such an auction is truthful, hence the truthfulness of the mechanism with respect to bidders in STAT. Also observe that for every input there is a declaration that makes the bidder win and gain a positive utility, thus truthfully reporting his valuation is a strongly dominant strategy. We now handle bidders in FIXED. If there is a winner in the second-price auction, then bidders in FIXED essentially participate in a second-price auction, which is truthful. Otherwise, there was no winner in the second-price auction. Observe that if the value of all items for some bidder is below r, then, independently of his declaration, he either wins some items via the fixed-price auction, that is truthful, or does not win any items at all. If his value of all items is above r and there is another bidder with a value above r, then this bidder participates in a second-price auction which is truthful. Finally, if this bidder is the only one that bids above r then he chooses between getting the bundle of all items with a price of r, or getting the bundle he won in the fixed-price auction in the corresponding price. In particular, he cannot be hurt by declaring above r. Notice

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that in any case no other bidder gets any items at all, thus all other bidders have no incentive to bid untruthfully in the fixed-price auction4 . u t We do note that in the two applications of the framework presented in the paper we use iterative methods to compute an estimation of the optimal solution (namely, solving the corresponding linear program). Thus it might be the case that a bidder will have no incentive to truthfully answer some queries (but will have no incentive to answer untruthfully too). This problem do not arise if the valuations can be succinctly described, or in any other situation where the estimation of the optimal solution can be done in a non-iterative way. 2.1

The Solution Concept: Dominant Strategies vs. Ex-Post Nash

It is well known in the economic literature that in iterative mechanisms the revelation principle do not hold anymore and the solution concept becomes ex-post Nash5 . Consider an iterative variant of a second-price auction where the first bidder bids, and the second bidder bids after him. If the strategy of the first bidder is: “if the first bidder bids 5 then I bid 3, otherwise I bid 4”, then bidding truthfully is no longer a dominant strategy of the first bidder. As this example demonstrate, in order to prove that in an iterative mechanism is bidding truthfully is a dominant strategy, we have to make sure, roughly speaking, that bidders do not get “extra information” from the previous rounds (i.e., the bid of the first player in the example). See, e.g., [11] for a discussion. Fortunately, this is not the case with mechanisms constructed using the framework, but showing it is subtle. Let us now explicitly explain why bidding truthfully is a dominant strategy in mechanisms constructed using the framework. We will have to be careful about the implementation details, though. In the first stage, we ask each bidder i to declare vi (M ). We hide bidder i’s bid from the rest of the bidders. After determining the winner, we use the rest of the bidders in STAT to calculate the winner’s reserve price. The crucial point is that the only new information that bidders in STAT gained is the identity of the bidder. Thus, the only way the winner’s bid can influence this calculation is by bidding lower and losing, but then the winner’s profit can only decrease to zero. If the winner bids above his reserve price, we use the bids of the bidders in FIXED to determine the identity of the bidder that will be allocated the bundle of all items. Otherwise, we calculate, using bidders in STAT, the price per item p for the fixed price auction (again, the bids of the bidders in FIXED are hidden from bidders in STAT), and perform the fixed price auction. We now allocate the items to bidders in FIXED according to the framework. Notice that when the bidders participate in the fixed-price auction, they do not know the bids on the bundle of all items, and thus they are not influenced by these bids. The mechanism now have all the information it needs to decide the resulting allocation. 2.2

Combinatorial Auctions with General Bidders

√ We now present the main application of the improved framework, an O( m) approximation mech√ m ). To make the mechanism concrete, we need to anism that succeeds with probability 1 − O( log m specify four parameters: 4

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Interestingly enough, in this case even if the other bidders do lie regarding their preferences in the fixed-price auction, this will not hurt the approximation ratio! In an ex-post Nash equilibrium bidding truthfully is a (weakly) dominant strategy of each bidder if all other bidders bid truthfully.

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– – – –

√ The value of α: we let α = 100 m. The value of β: we let β = 100 log m. T ST AT The value of p: we let the price per item in the fixed-price auction to be p = OP100m . The way we estimate the optimal solution restricted to a set of bidders: we use the value of the optimal fractional allocation (details in the appendix).

Let us start the proof of the approximation ratio, and in the same time provide some intuition regarding the framework. Some notation first: we denote by OP TS∗ the optimal fractional solution ∗ √T , with the participation of bidders in S only. Bidder i is called a dominant bidder if vi (M ) ≥ OP m ∗

OP T and super dominant if vi (M ) ≥ 10 log m . The analysis involves considering three different cases; the first two are similar in all mechanisms developed using the framework, while the third one is much more algorithmically challenging. The main difficulty (but not the only one) in this case is to show that a fixed-price auction provides a good approximation ratio. See the mechanism for subadditive bidders for a much more complicated example than the one presented in this section.

Case 1: There is a Super-Dominant Bidder We first prove that if there is at least one superdominant bidder then the mechanism always ends with a good approximation ratio. There are two possibilities: there might be a super-dominant bidder in STAT, or all super-dominant bidders are in FIXED. In the first case we observe that the reserve price offered to each of the bidders is at OP√ T∗ most 100 , since OP TS∗ ≤ OP T ∗ for any set S. Thus, there must be a winner in the second-price m auction. Allocating the bundle of all item to the bidder that values it the most, which must be a super-dominant bidder, is a good approximation to the optimal welfare, as required. Assume that all super-dominant bidders are in FIXED. If there was a winner in the second-price auction of Step 4, then all bidders essentially participate in a second-price auction. As in the previous case, we are guaranteed to get a good approximation ratio. If there was no winner, then the analysis is divided into cases once again, depending on the number of super-dominant bidders. If there are at least two super-dominant bidders, we will have a second-price auction for selling the bundle of all items, and we are guaranteed to get a good approximation. The case where there is exactly one super-dominant bidder is a bit more tricky. This super-dominant bidder i has to choose between getting the bundle of all items with a price of r, or getting the bundle Si he won in the fixed-price and pay |Si | · p. If i takes M we get a good approximation ratio. If i takes the bundle Si , observe OP T ∗ OP T ∗ OP T ∗ that the profit of this bidder from taking M is at least vi (M ) − r ≥ 10 log m − 100 log m ≥ 2 log m , thus the profit from taking Si must be bigger, and in particular vi (Si ) ≥ approximation ratio even if i takes Si .

OP T ∗ 2 log m ,

which gives a good

Case 2: Many Dominant Bidders If there are more than log m dominant bidders, then with 1 probability of at least 1 − 2log1 m = 1 − m at least one dominant bidder will be in STAT, since each bidder is selected to STAT with a probability of 21 . If this happens, then using similar arguments to the previous claim, we are guaranteed that the bundle of all items will be sold to some dominant bidder. Thus, we get a good approximation ratio with high probability. Case 3: A Few Dominant Bidders The last case we need to handle is the case where there are at most log m dominant bidders, and no super-dominant bidders. For most mechanisms developed using the framework this is the non-standard part that requires a deeper combinatorial understanding of the problem. For the mechanism presented in this section, we borrow ideas from [4]. The

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second mechanism we present (for subadditive valuations) requires much more effort to handle this third case. OP T ∗ OP T ∗ First notice that dominant bidders contribute together a value of at most 10 log m · log m = 10 to the optimal welfare. Let N D be the set of bidders that are not dominant. Clearly, OP TN∗ D ≥ ∗ 9OP T 10 . The next lemma is a corollary of Lemma 7 in the appendix: Lemma 1. With probability of at least O( √1m ), we have that both events hold together: OP TN∗ D∩ST AT ≥ OP T ∗ 8 ,

and OP TN∗ D∩F IXED ≥

OP T ∗ 8 .

With probability of at most O( √1m ) the conclusion of the lemma does not hold, and we assume that the approximation ratio is 0. Otherwise, if there was a winner in the second-price auction of OP√ T∗ Step 4, we get a good approximation ratio, since the reserve price is at least 400 . If there was no m winner, and there are at least two bidders that value the bundle of all items above r, we also get ∗ √ T ). The next lemma a good approximation since one of them will get the bundle, and r ≥ Ω( OP m handles the case where no bidder bids above r (see the appendix for a proof): Lemma 2. Let A = F IXED ∩ N D. Then, a fixed-price auction with the participation of bidders OP T ∗ in FIXED with a price per item p = c·mA , for some c > 2, returns an allocation that has a value ∗ OP T of O( c√mA ). We have to handle an additional single case where there is only one bidder i that bids above OP√T r. Our concern is that the profit from taking M , vi (M ) − r, is smaller than 100 , and the profit m from taking the bundle Si that i won in the fixed-price auction is larger, but vi (Si ) is low. We will see that this case occurs with very small probability. First, observe that this problem does not arise OP√T for the next (if possible) value of r, r0 = r + 100 , since vi (M ) < r0 . Also notice that for smaller m values of r, like r00 = r −

OP√T , 100 m

we have that vi (M ) − r0 >

OP√T . 100 m

Thus, if bidder i chooses to

OP√T . 100 m

take Si in this case, it holds that vi (Si ) ≥ (For smaller values we may have more than one 0 bidder that bids above r , but this is a “good” case, similarly to before.) We conclude that there √ exists at most one value of r that gives a bad approximation ratio. However, there are O( logmm ) possible values, and we choose one uniformly at random. Hence, we are not guaranteed to get a √ m ). In conclusion, we have the following good approximation ratio with probability of at most O( log m theorem: √ Theorem 2. There exists an O( m)-approximation mechanism for combinatorial auction with √ m) general bidders that succeeds with probability 1 − O( log m

3

A Truthful Mechanism for Subadditive Valuations

This section introduces a new truthful mechanism for combinatorial auctions with subadditive bidders. The mechanism uses the framework of the previous section. However, the main novelty of the mechanism is combinatorial: roughly speaking, we prove that for every subadditive valuation and bundle S, one can set a price for each item such that every subset of S is profitable in these prices, yet the sum of prices is high (comparing to the value of S). We use this property to show that if all valuations are subadditive, then a fixed-price auction, with a well-chosen price per item, returns an allocation that is a good approximation to the optimal welfare. This will enable the construction of the mechanism.

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3.1

On A Combinatorial Property of Subadditive Valuations

− Definition 1. A vector of non-negative prices → p = (p1 , ..., pm ) α-supports a set of items S in a valuation v if the following two conditions hold together: – S is strongly profitable: for every T ⊆ S we have that v(T ) ≥ Σj∈T pj . – The prices are high: Σj∈S pj ≥ v(S) α . In [3] it is proved that the class of XOS valuations is exactly the class of valuations for which every subset of items has 1-supporting prices. The main combinatorial property we prove is that if v is subadditive, then for every set of items S there are O(log m)-supporting prices. − Lemma 3. Let v be a subadditive valuation. Then, for every S ⊆ M there exists a vector → p that log m -supports it. Furthermore, the price of each item is either 0 or p, for some p > 0. These prices 2e can be found by using polynomially many demand queries. Proof. The lemma will be proved by constructing a series of sets S = S0 ⊇ S1 ⊇ · · · ⊇ Sk = T . Each i−1 ) set Si is defined to be the most demanded set when the price of each item j ∈ Si−1 is |Sv(S , i−1 | log m and the price of the rest of the items is ∞ (that is, Si is the answer to the demand query with these | prices). The construction stops at the first step k where |Sk | ≥ |Sk−1 . Clearly, k ≤ log m. 2 v(S) i Let us first prove that v(T ) ≥ e . Denote by pj the price of item j in the i’th demand query. i−1 ) i In each step i we have that Σj∈Si−1 pij ≤ v(S log m . Thus the profit from the set Si−1 in prices {pj }j 1 is at least v(Si−1 )(1 − log m ). The most demanded set Si must be at least as profitable, hence at least as valuable: v(Si ) ≥ v(Si−1 )(1 − log1 m ). After k ≤ log m steps we have that v(T ) = v(Sk ) ≥

v(S)(1 − log1 m )log m ≥ v(S) e . We now prove that T is strongly profitable in prices {pij }j . Proposition 1. Let v be a subadditive valuation. Let S be the most demanded set (the most profitable set) when the price of each item j is pj . Then S is strongly profitable in these prices. Proof. We use the following fact: for each two subsets, S and T , S ⊆ T , and a subadditive valuation v, we have that v(S) ≥ v(T ) − v(T \ S). In other words, the value of S is at least as large as the marginal value of S given T . Given this fact, assume, for a contradiction, that the most demanded set T in prices (p1 , ..., pm ) in v is not strongly profitable. Thus, there exists a set S ⊆ T such that v(S) < Σj∈S pj . But then it follows by the fact that v(T \ S) − Σj∈T \S pj > v(T ) − Σj∈T pj , a contradiction to our assumption that S is the most demanded set. All that is left is to prove the correctness of the fact. Since v is subadditive we have that v(S) + v(T \ S) ≥ v(T ). By subtracting v(T \ S) from both sides of the equation, we get v(S) ≥ v(T ) − v(T \ S), as needed. u t To finish the lemma, we explicitly describe the log2em -supporting prices of S: we set the price of each item j ∈ T to pkj , and the price of each item j ∈ / T to 0. u t It will be interesting to determine if subadditive valuations exhibit c-supporting prices for every bundle, for some constant c. The following example rules out the possibility that c < 2: let the set of items M consist of m homogeneous items. The subadditive valuation v is defined as follows: v(M ) = 2, and v(S)=1, for all S 6= M . We will show that M does not have 2-supporting prices. By

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the definition of a strongly profitable set for each set S, |S| = m−1, we have that Σj∈S pj ≤ 1. Thus 1 1 . But then Σj∈M pj = Σj∈M \{j 0 } pj + pj 0 ≤ 1 + m−1 , there must exist some item j 0 with pj ≤ m−1 while v(M ) = 2. Towards developing the mechanism itself, we require a generalization of the α-supporting prices property: Definition 2. Let v1 , ..., vn be subadditive valuations, and let A = (S1 , ..., Sn ) be an allocation. Let − → → p be a vector of non-negative prices. We say the − p α-supports an allocation A if both the following conditions hold: – For each i, and for each T ⊆ Ti we have that vi (T ) ≥ Σj∈T pj . Σ p – Σi vi (Si ) ≥ jα j → We will say that − p supports A if the second condition does not necessarily hold. We will sometime abuse the notation and say that an allocation A is α-supported by a price p → if there exists a vector − p ∈ {0, p}m that α-supports an allocation A. Lemma 4. Let A = (S1 , ..., Sn ) be an allocation. If all valuations are subadditive, then there exists → a vector of non-negative prices − p ∈ {0, p}m that log2em -supports A. These prices can be found using polynomially many demand queries. Proof. We reduce the problem of finding α-supporting prices for an allocation to the problem of finding α-supporting prices for a single valuation and a set of items (Lemma 3). For the allocation A, define m0 = Σi |Si | · n new items, M 0 = {u11 , ..., u1|S1 | , ..., un1 , ..., un|Sn | }. Let v be a new subadditive valuation: for each T ⊆ m0 , v(T ) = Σi vi (∪j|uij ∈T {j}). v is the sum of subadditive valuations, and thus is subadditive. In addition, observe that if one can answer demand queries for each valuation vi , one can answer demand queries for v too: query vi , using the prices of items ui1 , ..., ui|S1 | . The answer of the demand query for v is clearly the union of the answers of the queries for the vi ’s. The lemma now follows by applying Lemma 3. u t The next lemma turns this combinatorial property into an algorithmic result: → Lemma 5. Let A = (S1 , ..., Sn ) be an allocation, and let − p ∈ {0, p}m support it, where p > 0. If all valuations are subadditive, then a fixed-price auction with a price of p2 returns an allocation that has a value of Ω(Σi Σj∈Si pj ). Proof. Denote the allocation produced by the fixed-price auction by (ALG1 , ..., ALGn ). Let LBi denote the sum of prices of items held in A by bidders i+1, ..., n immediately after bidder i is queried n and allocated a set of items in the fixed-price auction. I.e., LBi = Σt=i+1 Σj∈St \(∪it=1 ALGt ) pj . Let LB0 = Ω(Σi Σj∈Si pj ). To prove the lemma it suffices to show that for every i ≥ 1, vi (ALGi ) = Ω(LBi−1 − LBi ). We can therefore assume that for all i and each item j ∈ Si , pj = p (otherwise we remove each item j with pj = 0 from the Si ’s). First observe that LBi−1 − LBi is the sum of two terms: the sum of prices of items in Si that are available but were not allocated to bidder i (i.e., Σj∈Si \(∪it=1 ALGt ) pj ), and the sum of prices of items allocated in the fixed-price auction to bidder i but belong to bidders i + 1, .., n in (S1 , ..., Sn ) (i.e., Σj∈ALGi ∩(∪nt=i+1 St ) pj ). To bound the first term, let T = Si \ ∪it=1 ALGt . Notice that T ⊆ Si and Si is strongly profitable − in a price of p per item (recall that (S1 , ..., Sn ) is supported by → p ). Hence, if the price of each item

11 ) is p, then the profit from T is at least v(T 2 . We get that the profit from the most demanded set in v(T ) a price of p per item, ALGi , is at least 2 . However, ALGi is the most demanded set, so it must ) be at least as profitable, and in particular vi (ALGi ) ≥ v(T 2 . As for the second term, for any set U ∈ ALGi , by Proposition 1 we have that v(U ) ≥ Σj∈U pj /2. u t In particular, the last inequality is true for ALGi ∩ (∪nt=i+1 St ). The lemma follows.

As evident from the last two lemmas, for every allocation there exists a price p such that a fixedprice auction with a price of p per item returns an allocation that is an O(log m) approximation to the welfare of the allocation. 3.2

The O(log m log log m)-Truthful Mechanism for Subadditive Valuations

Let us now specify the four parameters of the framework needed for the mechanism: let α = 500 log m log log m, and β = 100 log log m. To estimate the value of the optimal solution restricted to a set of bidders we use the 2-approximation algorithm of Feige [5]. To find the price to be used in the fixed-price auction, we obtain an allocation A that is a 2 approximation to OP TST AT [5]. We then find a vector {0, p}m that O(log m)-supports A (Lemma 4), and let p be the price used in the fixed-price auction. T OP T A bidder i is called super dominant if vi (M ) ≥ 100 OP log log m , and dominant if vi (M ) ≥ 500 log m log log m . As in the mechanism for general bidders, if there is a super-dominant bidder, then the mechanism will surely end with a good approximation ratio. If there are more than log log m dominant bidders, then with probability log1 m one of them will be selected to the STAT group, and we are guaranteed to get a good approximation in this case too. As usual, the hard case is when there are at most log log m dominant bidders, and no superdominant bidders. The main effort is to show that we can get a good price p for the fixed-price auction using bidders in STAT. Observe that in this case dominant bidders contribute a value of T OP T 99OP T at most log log m · 100 OP log log m = 100 . Hence we have that OP TN D ≥ 100 , where N D is the set of bidders that are not dominant. The next lemma proves that if there is an price that O(log m)supports an allocation with a good value, then this price also supports an allocation with a good value restricted to bidders in FIXED: Lemma 6. Let A = (A1 , ..., An ) be an allocation where Σi vi (Ai ) ≥ OPt T . Let p be a price that 10 log m-supports it6 . Let S be a set of bidders where each bidder is selected independently at random T 1 to A with probability 21 . If vi (M ) ≤ 500·t·logOP m log log m then, with probability 1 − log 1.5 m , there is an T allocation B restricted to bidders in S only, where p supports B, and Σi |Bi | · p ≥ 2·tOP log m . Proof. Let us start with partitioning A into up to log m + 1 sets U1 , ..., Ulog m+1 , where the set Uk includes all Ai ’s such that 2k−1 ≤ |Ai | < 2k . Let L denote the set of “large” Uk ’s. I.e., L = {Uk | |Uk | ≥ 5 · log log m}. We claim that the set L contributes most of the “supported” welfare of A: T ΣUk ∈L Σi|Ai ∈Uk |Ai | · p ≥ 5· 1OP . To see this, first assume that |Ulog m+1 | < 5 · log log m. By the ·t·log m 2

conditions of the lemma we have that vi (M ) ≤ OP T log log m· 1000· 1 ·t·log m log log m 2

OP T . 200· 12 ·t·log m

OP T . 1000· 12 ·t·log m log log m

Thus, Σi|Ai ∈Ulog m+1 |Ai | · p ≤ 5 ·

≤ As for the next set, Ulog m , if |Ulog m | < 5·log log m The welfare supported by it is clearly at most the same. However, if |Ulog m−1 | < 5·log log m, then we have T that Σi|Ai ∈Ulog m−1 |Ai | · p ≤ 400·OP . The latter statement is true since for each Ai ∈ Ulog m+1 , 1 ·t·log m 2

6

Notice that the existence of such a price is guaranteed by Lemma 4.

12

and Aj ∈ Ulog m−1 we have that |Ai | > |Aj |/2. Summing all Uk with |Ulog k | < 5 · log log m we get T T . Hence, ΣUk ∈L Σi|Ai ∈Uk |Ai | · p ≥ 10· 1OP . a geometric series with a sum of at most 50· 1OP ·t·log m ·t log m 2

2

We now show that with high probability, at least 14 of the Ai ’s in each Uk ∈ L are selected to the set of bidders S. This will finish the proof since the contribution to the supported welfare of two Ai ’s in a specific Uk is the same, up to a factor of 2. Fix a set Uk ∈ L. Since Uk ≥ 5 log log m and by the Chernoff bounds the probability that S 3 contains less than 18 sets from Uk is at most 2−5· 4 ·log log m ≤ log13 m . Using the union bound for all possible O(log m) choices of Uk ∈ L, the lemma follows. u t By itself the lemma is not enough to finish this case: the allocation A from the lemma involves all bidders, while we only have information regarding allocations to bidders in STAT. Moreover, we do not know a-priori the price p from the lemma, and the conclusion of the lemma must hold for each OP T one of the many possibilities. To this end, observe that only prices that are larger than 500m log m can O(log m) support an allocation with a good value, since otherwise the sum of prices of the supported OP T allocation (S1 , ..., Sn ) is not high enough: Σi |Si | · p ≤ m · p ≤ 500 log m . We therefore restrict our OP T OP T OP T attention to the set of prices P = { 1000 log m , 500 log m , 250 log m , · · · }. Observe that |P | = O(log m). We claim that with probability of at least 1 − O( log1 m ) the following conditions hold together (again, otherwise we assume the algorithm provides an approximation ratio of 0): 1. OP TST AT ≥ OP T /8. 2. OP TF IXED ≥ OP T /8. 3. For each price p ∈ P , if there is an allocation that hold a constant fraction of the welfare, and is O(log m)-supported by p, then there is an allocation to bidders in F IXED that holds a constant fraction of the welfare and is O(log m)-supported by p. By Lemma 7, the first two conditions do not hold with probability of O( log1 m ). The third condition alone does not hold with probability of at most O( log12 m ), as there are O(log m) “subconditions”, one for each price, each sub-condition does not hold with probability of at most O( log12 m ) (by Lemma 6). The claim follows by using the union bound. With high probability and by the first condition, if there was a winner in the second-price auction, then we get a good approximation ratio. Next we assume that there was no winner in the second-price auction. Recall that bidders in FIXED face a randomly selected price r. As in the previous mechanism, if two bidders bid above r then we get a good approximation ratio. The problematic case is when there is one bidder that bids above r and chooses to take the bundle he won in the fixed-price auction. Again, there is exactly one value of r that causes this. We choose a value of r uniformly at random from a set of O(log m) possible values, hence the mechanism fails to provide a good approximation ratio in this case with probability O( log1 m ). We now claim that the price p that O(log m)-supports the allocation to bidders in STAT also supports some allocation with a good value restricted to bidders in FIXED: an allocation with a good value restricted to bidders in STAT is also an allocation with the same value to all bidders. This allocation is O(log m)-supported by some price p ∈ P . By the third condition, there must be an allocation with a good value to bidders in FIXED that is O(log m) supported by this price. Finally, by Lemma 5 we get that the fixed-price auction returns an allocation that is an O(log m)approximation to the optimal welfare. We have the following theorem:

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Theorem 3. There exists a truthful randomized mechanism for combinatorial auctions with subadditive bidders that obtains an approximation ratio of O(log m log log m) with probability of at least 1 − O( log1 m ). Acknowledgements I thank Noam Nisan, Ariel Procaccia, Michael Schapira, and the anonymous reviewers for helpful discussions and comments. I thank Liad Blumrosen for a discussion that lead to this paper. This research was supported by grants from the Israel Science Foundation and the USA-Israel Bi-national Science Foundation.

References 1. Liad Blumrosen and Noam Nisan. On the computational power of iterative auctions I: demand queries. Discussion paper no. 381, The Center for the Study of Rationality, The Hebrew University. An extended abstract in EC’05 contained preliminary results. 2. Shahar Dobzinski and Noam Nisan. Limitations of vcg-based mechanisms. In STOC’07. 3. Shahar Dobzinski, Noam Nisan, and Michael Schapira. Approximation algorithms for combinatorial auctionss with complement-free bidders. In STOC’05. 4. Shahar Dobzinski, Noam Nisan, and Michael Schapira. Truthful randomized mechanisms for combinatorial auctions. In STOC’06. 5. Uriel Feige. On maximizing welfare where the utility functions are subadditive. In STOC’06. 6. Uriel Feige and Jan Vondrak. Approximation algorithms for allocation problems: Improving the factor of 1-1/e. In FOCS’06. 7. Andrew Goldberg, Jason Hartline, Anna Karlin, Mike Saks, and Andrew Wright. Competitive auctions. Games and Economic Behaviour, 2006. 8. Ron Holzman, Noa Kfir-Dahav, Dov Monderer, and Moshe Tennenholtz. Bundling equilibrium in combinatrial auctions. Games and Economic Behavior, 47:104–123, 2004. 9. Subhash Khot, Richard J. Lipton, Evangelos Markakis, and Aranyak Mehta. Inapproximability results for combinatorial auctions with submodular utility functions. In WINE’05, 2005. 10. Ron Lavi, Ahuva Mu’alem, and Noam Nisan. Towards a characterization of truthful combinatorial auctions. In The 44th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2003. 11. Ron Lavi and Chaitanya Swamy. Truthful and near-optimal mechanism design via linear programming. In FOCS 2005. 12. Benny Lehmann, Daniel Lehmann, and Noam Nisan. Combinatorial auctions with decreasing marginal utilities. In ACM conference on electronic commerce, 2001. 13. Noam Nisan. The communication complexity of approximate set packing and covering. In ICALP 2002. 14. Noam Nisan. 2007. Introduction to Mechanism Design (for Computer Scientists). In “Algorithmic Game Theory”, N. Nisan, T. Roughgarden, E. Tardos and V. Vazirani, editors. 15. Noam Nisan and Amir Ronen. Algorithmic mechanism design. In STOC, 1999. 16. Noam Nisan and Amir Ronen. Computationally feasible vcg-based mechanisms. In ACM Conference on Electronic Commerce, 2000. 17. Noam Nisan and Ilya Segal. The communication requirements of efficient allocations and supporting prices, 2006. In the Journal of Economic Theory.

A A.1

Appendix Lemmas and Missing Proofs

Lemma 7. Let T be a set of bidders where each bidder is selected independently at random to T with probability 21 . Denote by OP T ∗ the optimal fractional solution, and by OP TT∗ the value of the

14

optimal solution restricted only to bidders in T only. If for each bidder i, vi (M ) < ∗ probability 1 − 8t we have that OP TT∗ ≥ · OP4T .

OP T ∗ , t

then with

We note that an analogous statement can be obtained by replacing the optimal fractional solution OP T ∗ by the optimal integral solution OP T . Proof. Let {yS,i }S,i be the set of items allocated to bidder i in the optimal fractional solution. Let A be the random variable that receives the value of OP TT∗ . For every bidder i denote by Ai the random variable that receives the value ΣS yS,i vi (S) if i ∈ T , and 0 otherwise. Observe that for every set T , Σi∈T ΣS yS,i vi (S) ≤ OP TT∗ . Every bidder is placed in T with probability 21 , and thus we have that E[A] = Σi∈T OP2 T . Let us use the fact that for each i, Ai < OPt T to set an upper bound on the probability that A gets a value that is substantially smaller than its expectation. We make use of the following corollary from Chebyshev’s inequality: Claim. Let X be the sum of independent random variables, each of which lies in [0, r]. Then, for any α > 0, Pr[|X − E[X]| ≥ α] ≤ rE[X] α2 . Since for each i, Ai ∈ [0, OPc T ], we have that 1 1 1 Pr[A < · OP T ∗ ] ≤ Pr[|A − · OP T ∗ | ≥ · OP T ∗ ] ≤ 4 2 4

OP T ∗ 1 · 2 · OP T ∗ t ( 14 · OP T ∗ )2

≤

8 t u t

Proof of Lemma 2 Proof. Let {yi,S }i∈A,S⊆M be the variables in the fractional solution OP TA∗ . We will restrict our attention to bundles in OP TA∗ that are profitable when setting a price of p for each item. That is, let T be the set of pairs (i, S) such that yi,S > 0, and vi (S) − p · |S| > 0. The next claim shows that we do not lose too much by ignoring all other bundles in OP TA∗ . Claim. Σ(i,S)∈T yi,S vi (S) ≥

1 2

· OP TA∗

Proof. Define T to be the “complement” set of T . Formally, T consists of all pairs (i, S) such that yi,S > 0 in OP TA∗ , but vi (S) − p · |S| ≤ 0. By definition, OP TA∗ = Σ(i,S)∈T yi,S vi (S) + Σ(i,S)∈T yi,S vi (S). We bound from above the contribution of T to OP TA∗ to prove the claim. Σ(i,S)∈T yi,S vi (S) ≤ Σ(i,S)∈T yi,S p · |S| ≤ m · p ≤ m ·

OP TA∗ ·OP TA∗ ≤ c·m c

where the first inequality is because of the definition of T and the second inequality is due to the LP constraints. The lemma follows for c ≥ 2. u t Let us now calculate the revenue we get in the fixed-price auction. Without loss of generality, let bidder 1 be the bidder that is queried first in the fixed-price auction, 2 the bidder that is queried next, and so on. Bidder 1 is asked for his most demanded set. The key observation is that if there is some S such that x1,S > 0 and (1, S) ∈ T then bidder 1’s demand set is not empty. Recall that for each item in S1 we gain a revenue of p.

15

We now upper bound the “loss” incurred by allocating S1 to bidder 1 in comparison to OP TA∗ . Notice, that by allocating S1 to bidder 1 we lose both the value of all the fractional bundles assigned to bidder 1 in OP TA∗ , and of all the bundles in OP TA∗ that contain some item from S1 . The value OP√ T∗ of all the fractional bundles assigned to bidder 1 in OP TA∗ is at most 100 (if bidder 1 is not a m dominant bidder, otherwise the loss is 0): Σ(1,S)∈T y1,S v1 (S) ≤

OP TA∗ √ 100 m

∗

OP√ T because v1 (M ) < 100 and Σ(1,S) y1,S ≤ 1, due to the constraints of the LP formulation. m We now bound the value of all the bundles in OP TA∗ that contain some item from S1 . Fix some OP√ T∗ item j ∈ S1 . Again, using the constraints of the LP, and vi (M ) < 100 which holds for bidders in m A, OP TA∗ √ Σ(i,S)∈T |j∈S yi,S vi (S) ≤ 100 m OP T ∗

To conclude, for every item we sell to bidder 1 at price p = c·mA , we lose bundles in T that are OP√ T∗ together worth at most 2 · 100 . The analysis continues by removing from OP TA∗ all pairs (i, S) m which can not be assigned now (either i = 1, or j ∈ Si and j ∈ S), and applying similar arguments √ to the rest of the bidders in FIXED. In total we get that the algorithm sells at least 25 m items OP T ∗ (taking into account only profitable bundles under price p), each with a revenue of c·mA , hence our 25OP T ∗ total revenue (which is also a lower bound to the welfare of the allocation) is at least c·√mA . u t A.2

The Standard LP Formulation of a Combinatorial Auction

Maximize: Σi,S xi,S vi (S) Subject to: – For each item j: Σi,S|j∈S xi,S ≤ 1 – for each bidder i: ΣS xi,S ≤ 1 – for each i, S: xi,S ≥ 0 We remark that the LP relaxation can be solved using demand oracles only [17, 1].