Welfare maximization in bilateral trade has been extensively studied in recent years. Previous literature obtained incentive-compatible approximation mechanisms only for the private values case. In this paper, we study welfare maximization in bilateral trade with interdependent values. Designing mechanisms for interdependent settings is much more challenging because the values of the players depend on the private information of the others, requiring complex belief updates and strategic inference.

We present a constant-factor approximation algorithm for the Nash Social Welfare (NSW) maximization problem with subadditive valuations accessible via demand queries. More generally, we propose a framework for NSW optimization which assumes two subroutines which (1) solve a configuration-type LP under certain additional conditions, and (2) round the fractional solution with respect to utilitarian social welfare. In particular, a constant-factor approximation for submodular valuations with value queries can also be derived from our framework.

We study the bilateral trade problem where a seller owns a single indivisible item, and a potential buyer seeks to purchase it. Previous mechanisms for this problem only considered the case where the values of the buyer and the seller are drawn from independent distributions. In contrast, this paper studies bilateral trade mechanisms when the values are drawn from a joint distribution. We prove that the buyer-offering mechanism guarantees an approximation ratio of e/e−1 ≈ 1.582 to the social welfare even if the values are drawn from a joint distribution. The buyer-offering mechanism is Bayesian incentive compatible, but the seller has a dominant strategy. We prove the buyer-offering mechanism is optimal in the sense that no Bayesian mechanism where one of the players has a dominant strategy can obtain an approximation ratio better than e/e−1. We also show that no mechanism in which both sides have a dominant strategy can provide any constant approximation to the social welfare when the values are drawn from a joint distribution. Finally, we prove some impossibility results on the power of general Bayesian incentive compatible mechanisms. In particular, we show that no deterministic Bayesian incentive-compatible mechanism can provide an approximation ratio better than 1+ln2/2≈ 1.346.

In this paper we revisit the notion of simplicity in mechanisms. We consider a seller of m heterogeneous items, facing a single buyer with valuation v. We observe that previous attempts to define complexity measures often fail to classify mechanisms that are intuitively considered simple (e.g., the "selling separately" mechanism) as such. We suggest to view a menu as simple if a bundle that maximizes the buyer's profit can be found by conducting a few primitive operations that are considered simple. The primitive complexity of a menu is the number of primitive operations needed to (adaptively) find a profit-maximizing entry in the menu. In this paper, the primitive operation that we study is essentially computing the outcome of the "selling separately" mechanism.

We explore the performance of polynomial-time incentive-compatible mechanisms in single-crossing domains. Single-crossing domains were extensively studied in the economics literature. Roughly speaking, a domain is single crossing if monotonicity characterizes incentive compatibility (intuitively, an algorithm is monotone if a bidder that "improves" his valuation is allocated a better outcome). That is, single-crossing domains are the standard mathematical formulation of domains that are informally known as "single parameter". In all major single-crossing domains studied so far (e.g., welfare maximization in various auctions with single-minded bidders, makespan minimization on related machines), the performance of the best polynomial-time incentive-compatible mechanisms matches the performance of the best polynomial-time non-incentive-compatible algorithms. Our two main results make progress in understanding the power of incentive-compatible polynomial-time mechanisms in single-crossing domains:

We study incentive-compatible mechanisms that maximize the Nash Social Welfare. Since traditional incentive-compatible mechanisms cannot maximize the Nash Social Welfare even approximately, we propose changing the traditional model. Inspired by a widely used charging method (e.g., royalties, a lawyer that charges some percentage of possible future compensation), we suggest charging the players some percentage of their value of the outcome. We call this model the percentage fee model. We show that there is a mechanism that maximizes exactly the Nash Social Welfare in every setting with non-negative valuations. Moreover, we prove an analog of Roberts theorem that essentially says that if the valuations are non-negative, then the only implementable social choice functions are those that maximize weighted variants of the Nash Social Welfare. We develop polynomial time incentive compatible approximation algorithms for the Nash Social Welfare with subadditive valuations and prove some hardness results. 26 pages. This is the TheoretiCS journal version

A rapidly growing literature on lying in behavioral economics and psychology shows that individuals often do not lie even when lying maximizes their utility. In this work, we attempt to incorporate these findings into the theory of mechanism design. We consider players that have a preference for truth-telling and will only lie if their benefit from lying is sufficiently larger than the loss of the others. To accommodate such players, we introduce α-moral mechanisms, in which the gain of a player from misreporting his true value, comparing to truth-telling, is at most α times the loss that the others incur due to misreporting. Note that a 0-moral mechanism is a truthful mechanism. We develop a theory of moral mechanisms in the canonical setting of single-item auctions within the "reasonable" range of α, 0 ≤ α ≤ 1. We identify similarities and disparities to the standard theory of truthful mechanisms. In particular, we show that the allocation function does not uniquely determine the payments and is unlikely to admit a simple characterization. In contrast, recall that monotonicity characterizes the allocation function of truthful mechanisms. Our main technical effort is invested in determining whether the auctioneer can exploit the preference for truth-telling of the players to extract more revenue comparing to truthful mechanisms. We show that the auctioneer can indeed extract more revenue when the values of the players are correlated, even when there are only two players. However, we show that truthful mechanisms are revenue-maximizing even among moral ones when the values of the players are independently drawn from certain identical distributions (e.g., the uniform and exponential distributions). A by-product of our proof that optimal moral mechanisms are truthful is an alternative proof to Myerson’s optimal truthful mechanism characterization in the settings that we consider. We flesh out this approach by providing an alternative proof that does not involve moral mechanisms to Myerson’s characterization of optimal truthful mechanisms to all settings in which the values are independently drawn from regular distributions (not necessarily identical).

We study the communication complexity of dominant strategy implementations of combinatorial auctions. We start with two domains that are generally considered “easy”: multi-unit auctions with decreasing marginal values and combinatorial auctions with gross substitutes valuations. For both domains we have fast algorithms that find the welfare-maximizing allocation with communication complexity that is poly-logarithmic in the input size. This immediately implies that welfare maximization can be achieved in ex-post equilibrium with no significant communication cost, by using VCG payments. In contrast, we show that in both domains the communication complexity of any dominant strategy implementation that achieves the optimal welfare is polynomial in the input size.

Consider a seller that intends to auction some item. The seller can invest money and effort in advertising in different market segments in order to recruit n bidders to the auction. Alternatively, the seller can have a much cheaper and focused marketing operation and recruit the same number of bidders from a single market segment. Which marketing operation should the seller choose?

The class of gross substitutes (GS) set functions plays a central role in Economics and Computer Science. GS belongs to the hierarchy of complement free valuations introduced by Lehmann, Lehmann and Nisan, along with other prominent classes: GS ⊊ Submodular ⊊ XOS ⊊ Subadditive$. The GS class has always been more enigmatic than its counterpart classes, both in its definition and in its relation to the other classes. For example, while it is well understood how closely the Submodular, XOS and Subadditive classes (point-wise) approximate one another, approximability of these classes by GS remained wide open. In particular, the largest gap known between Submodular and GS valuations was some constant ratio smaller than 2. Our main result is the existence of a submodular valuation (one that is also budget additive) that cannot be approximated by GS within a ratio better than $Ømega(łog m/łogłog m), where m is the number of items. En route, we uncover a new symmetrization operation that preserves GS, which may be of independent interest. We show that our main result is tight with respect to budget additive valuations. However, whether GS approximates general submodular valuations within a poly-logarithmic factor remains open, even in the special case of concave of GS valuations (a subclass of Submodular containing budget additive). For concave of Rado valuations (Rado is a significant subclass of GS, containing, e.g., weighted matroid rank functions and OXS), we show approximability by GS within an O(łog2m) factor.

Let (f,P) be an incentive compatible mechanism where f is the social choice function and P is the payment function. In many important settings, f uniquely determines P (up to a constant) and therefore a common approach is to focus on the design of f and neglect the role of the payment function. Fadel and Segal [JET, 2009] question this approach by taking the lenses of communication complexity: can it be that the communication complexity of an incentive compatible mechanism that implements f (that is, computes both the output and the payments) is much larger than the communication complexity of computing the output? I.e., can it be that ccIC(f)>>cc(f)? Fadel and Segal show that for every f, ccIC(f)≤ exp(cc(f)). They also show that fully computing the incentive compatible mechanism is strictly harder than computing only the output: there exists a social choice function f such that ccIC(f)=cc(f)+1. In a follow-up work, Babaioff, Blumrosen, Naor, and Schapira [EC’08] provide a social choice function f such that ccIC(f)=Θ(n· cc(f)), where n is the number of players. The question of whether the exponential upper bound of Fadel and Segal is tight remained wide open. In this paper we solve this question by explicitly providing a function f such that ccIC(f)= exp(cc(f)). In fact, we establish this via two very different proofs. In contrast, we show that if the players are risk-neutral and we can compromise on a randomized truthful-in-expectation implementation (and not on deterministic ex-post implementation) gives that ccTIE(f)=poly(n,cc(f)) for every function f, as long as the domain of f is single parameter or a convex multi-parameter domain. We also provide efficient algorithms for deterministic computation of payments in several important domains.

We study a central problem in algorithmic mechanism design: constructing truthful mechanisms for welfare maximization in combinatorial auctions with submodular bidders. Dobzinski, Nisan, and Schapira provided the first mechanism that guarantees a nontrivial approximation ratio of $O(\log^2 m)$ [STOC'06, ACM, New York, 2006, pp. 644--652], where $m$ is the number of items. This approximation ratio was subsequently improved to $O(\log m\log \log m)$ [S. Dobzinski, APPROX'07, Springer, Berlin, 2007, pp. 89--103] and then to $O(\log m)$ [P. Krysta and B. Vöcking, ICALP'12, Springer, Heidelberg, 2012, pp. 636--647]. In this paper we develop the first mechanism that breaks the logarithmic barrier. Specifically, the mechanism provides an approximation ratio of $O(\sqrt {\log m})$. Similarly to previous constructions, our mechanism uses polynomially many value and demand queries and, in fact, provides the same approximation ratio for the larger class of XOS (also known as fractionally subadditive) valuations. We also develop a computationally efficient implementation of the mechanism for combinatorial auctions with budget additive bidders. Although, in general, computing a demand query is NP-hard for budget additive valuations, we observe that the specific form of demand queries that our mechanism uses can be efficiently computed when bidders are budget additive.

We study a communication variant of local search. There is some fixed, commonly known graph G. Alice holds fA and Bob holds fB, both are functions that specify a value for each vertex. The goal is to find a local maximum of fA+fB with respect to G, i.e., a vertex v for which (fA+fB)(v)≥ (fA+fB)(u) for each neighbor u of v.

We introduce a combinatorial variant of the cost sharing problem: several services can be provided to each player and each player values every combination of services differently. A publicly known cost function specifies the cost of providing every possible combination of services. A combinatorial cost sharing mechanism is a protocol that decides which services each player gets and at what price. We look for dominant strategy mechanisms that are (economically) efficient and cover the cost, ideally without overcharging (i.e., budget balanced). Note that unlike the standard cost sharing setting, combinatorial cost sharing is a multi-parameter domain. This makes designing dominant strategy mechanisms with good guarantees a challenging task. We present the Potential Mechanism -- a combination of the VCG mechanism and a well-known tool from the theory of cooperative games: Hart and Mas-Colell's potential function. The potential mechanism is a dominant strategy mechanism that always covers the incurred cost. When the cost function is subadditive the same mechanism is also approximately efficient. Our main technical contribution shows that when the cost function is submodular the potential mechanism is approximately budget balanced in three settings: supermodular valuations, symmetric cost function and general symmetric valuations, and two players with general valuations.

We study combinatorial auctions with bidders that exhibit endowment effect. In most of the previous work on cognitive biases in algorithmic game theory (e.g., [Kleinberg and Oren, EC'14] and its follow-ups) the focus was on analyzing the implications and mitigating their negative consequences. In contrast, in this paper we show how in some cases cognitive biases can be harnessed to obtain better outcomes. Specifically, we study Walrasian equilibria in combinatorial markets. It is well known that Walrasian equilibria exist only in limited settings, e.g., when all valuations are gross substitutes, but fails to exist in more general settings, e.g., when the valuations are submodular. We consider combinatorial settings in which bidders exhibit the endowment effect, that is, their value for items increases with ownership. Our main result shows that when the valuations are submodular, even a mild degree of endowment effect is sufficient to guarantee the existence of Walrasian equilibria. In fact, we show that in contrast to Walrasian equilibria with standard utility maximizing bidders -- in which the equilibrium allocation must be efficient -- when bidders exhibit endowment effect any local optimum can be an equilibrium allocation. Our techniques reveal interesting connections between the LP relaxation of combinatorial auctions and local maxima. We also provide lower bounds on the intensity of the endowment effect that the bidders must have in order to guarantee the existence of a Walrasian equilibrium in various settings.

We analyze the revenue loss due to market shrinkage. Specifically, consider a simple market with one item for sale and n bidders whose values are drawn from some joint distribution. Suppose that the market shrinks as a single bidder retires from the market. Suppose furthermore that the value of this retiring bidder is fixed and always strictly smaller than the values of the other bidders. We show that even this slight decrease in competition might cause a significant fall of a multiplicative factor of 1/e+1 ~0.268 in the revenue that can be obtained by a dominant strategy ex-post individually rational mechanism. In particular, our results imply a solution to an open question that was posed by Dobzinski, Fu, and Kleinberg [STOC'11].

We introduce a combinatorial variant of the cost sharing problem: several services can be provided to each player and each player values every combination of services differently. A publicly known cost function specifies the cost of providing every possible combination of services. A combinatorial cost sharing mechanism is a protocol that decides which services each player gets and at what price. We look for dominant strategy mechanisms that are (economically) efficient and cover the cost, ideally without overcharging (i.e., budget balanced). Note that unlike the standard cost sharing setting, combinatorial cost sharing is a multi-parameter domain. This makes designing dominant strategy mechanisms with good guarantees a challenging task.

We present fast algorithms for sketching valuation functions. Let N (| N | = n ) be some ground set and v :2 N → R be a function. We say that v˜:2 N → R is an α-sketch of v if for every set S we have that v ( S )/α ≤ v˜( S ) ≤ v ( S ) and v˜ can be described in poly ( n ) bits. Goemans et al. [SODA’09] showed that if v is submodular then there exists an õ (√ n )-sketch that can be constructed using polynomially many value queries (this is essentially the best possible, as Balcan and Harvey [STOC’11] show that no submodular function admits an n 1/3 - ϵ -sketch). Based on their work, Balcan et al. [COLT’12] and Badanidiyuru et al. [SODA’12] show that if v is subadditive, then there exists an õ (√ n )-sketch that can be constructed using polynomially many demand queries. All previous sketches are based on complicated geometric constructions. The first step in their constructions is proving the existence of a good sketch by finding an ellipsoid that “approximates” v well (this is done by applying John’s theorem to ensure the existence of an ellipsoid that is “close” to the polymatroid that is associated with v ). The second step is to show that this ellipsoid can be found efficiently, and this is done by repeatedly solving a certain convex program to obtain better approximations of John’s ellipsoid. In this article, we give a significantly simpler, nongeometric proof for the existence of good sketches and utilize the proof to obtain much faster algorithms that match the previously obtained approximation bounds. Specifically, we provide an algorithm that finds õ (√ n )-sketch of a submodular function with only õ ( n 3/2 value queries, and we provide an algorithm that finds õ (√ n )-sketch of a subadditive function with O ( n ) demand and value queries.

We characterize the communication complexity of truthful mechanisms. Our departure point is the well known taxation principle. The taxation principle asserts that every truthful mechanism can be interpreted as follows: every player is presented with a menu that consists of a price for each bundle (the prices depend only on the valuations of the other players). Each player is allocated a bundle that maximizes his profit according to this menu. We define the taxation complexity of a truthful mechanism to be the logarithm of the maximum number of menus that may be presented to a player. Our main finding is that in general the taxation complexity essentially equals the communication complexity. The proof consists of two main steps. First, we prove that for rich enough domains the taxation complexity is at most the communication complexity. We then show that the taxation complexity is much smaller than the communication complexity only in "pathological" cases and provide a formal description of these extreme cases. Next, we study mechanisms that access the valuations via value queries only. In this setting we establish that the menu complexity - a notion that was already studied in several different contexts - characterizes the number of value queries that the mechanism makes in exactly the same way that the taxation complexity characterizes the communication complexity. Our approach yields several applications, including strengthening the solution concept with low communication overhead, fast computation of prices, and hardness of approximation by computationally efficient truthful mechanisms.

We study a central problem in Algorithmic Mechanism Design: constructing truthful mechanisms for welfare maximization in combinatorial auctions with submodular bidders. Dobzinski, Nisan, and Schapira provided the first mechanism that guarantees a non-trivial approximation ratio of O(log^2 m) [STOC'06], where m is the number of items. This was subsequently improved to O( log m log log m) [Dobzinski, APPROX'07] and then to O(m) [Krysta and Vocking, ICALP'12]. In this paper we develop the first mechanism that breaks the logarithmic barrier. Specifically, the mechanism provides an approximation ratio of O( m). Similarly to previous constructions, our mechanism uses polynomially many value and demand queries, and in fact provides the same approximation ratio for the larger class of XOS (a.k.a. fractionally subadditive) valuations. We also develop a computationally efficient implementation of the mechanism for combinatorial auctions with budget additive bidders. Although in general computing a demand query is NP-hard for budget additive valuations, we observe that the specific form of demand queries that our mechanism uses can be efficiently computed when bidders are budget additive.

A long-standing open question in algorithmic mechanism design is whether there exist computationally efficient truthful mechanisms for combinatorial auctions, with performance guarantees close to those possible without considerations of truthfulness. In this article, we answer this question negatively: the requirement of truthfulness can impact dramatically the ability of a mechanism to achieve a good approximation ratio for combinatorial auctions. More precisely, we show that every universally truthful randomized mechanism for combinatorial auctions with submodular valuations that approximates optimal social welfare within a factor of m 1/2−ϵ must use exponentially many value queries, where m is the number of items. Furthermore, we show that there exists a class of succinctly represented submodular valuation functions, for which the existence of a universally truthful polynomial-time mechanism that provides an m 1/2−ϵ -approximation would imply NP = RP . In contrast, ignoring truthfulness, there exist constant-factor approximation algorithms for this problem, and ignoring computational efficiency, the VCG mechanism is truthful and provides optimal social welfare. These are the first hardness results for truthful polynomial-time mechanisms for any type of combinatorial auctions, even for deterministic mechanisms. Our approach is based on a novel direct hardness technique that completely skips the notoriously hard step of characterizing truthful mechanisms. The characterization step was the main obstacle for proving impossibility results in algorithmic mechanism design so far.

We study combinatorial auctions where each item is sold separately but simultaneously via a second price auction. We ask whether it is possible to efficiently compute in this game a pure Nash equilibrium with social welfare close to the optimal one. We show that when the valuations of the bidders are submodular, in many interesting settings (e.g., constant number of bidders, budget additive bidders) computing an equilibrium with good welfare is essentially as easy as computing, completely ignoring incentives issues, an allocation with good welfare. On the other hand, for subadditive valuations, we show that computing an equilibrium requires exponential communication. Finally, for XOS (a.k.a. fractionally subadditive) valuations, we show that if there exists an efficient algorithm that finds an equilibrium, it must use techniques that are very different from the ones currently known.

Competitive equilibrium, the central equilibrium notion in markets with indivisible goods, is based on pricing each good such that the demand for goods equals their supply and the market clears. This equilibrium notion is not guaranteed to exist beyond the narrow case of substitute goods, might result in zero revenue even when consumers value the goods highly, and overlooks the widespread practice of pricing bundles rather than individual goods. Alternative equilibrium notions proposed to address these shortcomings have either made a strong assumption on the ability to withhold supply in equilibrium, or have allowed an exponential number of prices. In this paper we study the notion of competitive bundling equilibrium – a competitive equilibrium over the market induced by partitioning the goods into bundles. Such an equilibrium is guaranteed to exist, is succinct, and satisfies the fundamental economic condition of market clearance. We establish positive welfare and revenue guarantees for this solution concept: For welfare we show that in markets with homogeneous goods, there always exists a competitive bundling equilibrium that achieves a logarithmic fraction of the optimal welfare. We also extend this result to establish nontrivial welfare guarantees for markets with heterogeneous goods. For revenue we show that in a natural class of markets for which competitive equilibrium does not guarantee positive revenue, there always exists a competitive bundling equilibrium that extracts as revenue a logarithmic fraction of the optimal welfare. Both results are tight. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We consider reallocation problems in settings where the initial endowment of each agent consists of a subset of the resources. The private information of the players is their value for every possible subset of the resources. The goal is to redistribute resources among agents to maximize efficiency. Monetary transfers are allowed, but participation is voluntary.

We study the necessity of interaction between individuals for obtaining approximately efficient economic allocations. We view this as a formalization of Hayek's classic point of view that focuses on the information transfer advantages that markets have relative to centralized planning. We study two settings: combinatorial auctions with unit demand bidders (bipartite matching) and combinatorial auctions with subadditive bidders. In both settings we prove that non-interactive protocols require exponentially larger communication costs than interactive ones, even those that only use a modest amount of interaction.

We prove the first communication complexity lower bound for constant-factor approximation of the submodular welfare problem. More precisely, we show that a -approximation (≃ 0.816) for welfare maximization in combinatorial auctions with submodular valuations would require exponential communication. We also show NP-hardness of -approximation in a computational model where each valuation is given explicitly by a table of constant size. Both results rule out better than (1 − )-approximations in every oracle model with a separate oracle for each player, such as the demand oracle model. Our main tool is a new construction of monotone submodular functions that we call multi-peak submodular functions. Roughly speaking, given a family of sets , we construct a monotone submodular function f with a high value f(S) for every set S ∊ (a “peak”), and a low value on every set that does not intersect significantly any set in . We also study two other related problems: max-min allocation (for which we also get hardness of -approximation, in both models), and combinatorial public projects (for which we prove hardness of -approximation in the communication model, and hardness of -approximation in the computational model, using constant size valuations).

In many settings the power of truthful mechanisms is severely bounded. In this paper we use randomization to overcome this problem in the multi-unit auction setting. In particular, we construct a fully polynomial-time approximation scheme (FPTAS) for multi-unit auctions that is truthful in expectation, whereas there is evidence that no polynomial-time truthful deterministic mechanism provides an approximation ratio better than 2. We leverage the FPTAS to show for the first time that truthful in expectation polynomial-time mechanisms are provably stronger than polynomial-time universally truthful mechanisms. Specifically, we show that there is a setting, related to multi-unit auctions, in which (1) there is a nonpolynomial time truthful mechanism that always outputs the optimal solution, and that (2) no universally truthful randomized mechanism can provide an approximation ratio better than 2 in polynomial time, but (3) an FPTAS that is truthful in expectation exists.

Many large decentralized systems rely on information propagation to ensure their proper function. We examine a common scenario in which only participants that are aware of the information can compete for some reward, and thus informed participants have an incentive not to propagate information to others. One recent example in which such tension arises is the 2009 DARPA Network Challenge (finding red balloons). We focus on another prominent example: Bitcoin, a decentralized electronic currency system.

We study combinatorial procurement auctions, where a buyer with a valuation function v and budget B wishes to buy a set of items. Each item i has a cost ci and the buyer is interested in a set S that maximizes v(S) subject to ∑i∈Sci ≤ β. Special cases of combinatorial procurement auctions are well-studied problems from submodular optimization. In particular, when the costs are all equal (cardinality constraint), a classic result by Nemhauser et al shows that the greedy algorithm provides an e/e-1 approximation.

One of the fundamental questions of Algorithmic Mechanism Design is whether there exists an inherent clash between truthfulness and computational tractability: in particular, whether polynomial-time truthful mechanisms for combinatorial auctions are provably weaker in terms of approximation ratio than non-truthful ones. This question was very recently answered for universally truthful mechanisms for combinatorial auctions [4], and even for truthful-in-expectation mechanisms [12]. However, both of these results are based on information-theoretic arguments for valuations given by a value oracle, and leave open the possibility of polynomial-time truthful mechanisms for succinctly described classes of valuations.

Motivated by the problem of querying and communicating bidders' valuations in combinatorial auctions, we study how well different classes of set functions can be sketched. More formally let f be a function mapping subsets of some ground set [n] to the non-negative real numbers. We say that f′ is an α-sketch of f if for every set S, the value f′(S) lies between f(S)/α and f(S), and f′ can be specified by poly(n) bits. We show that for every subadditive function f there exists an α-sketch where α = n1/2 · O(polylog(n)). Furthermore, we provide an algorithm that finds these sketches with a polynomial number of demand queries. This is essentially the best we can hope for since: 1. We show that there exist subadditive functions (in fact, XOS functions) that do not admit an o(n1/2) sketch. (Balcan and Harvey [3] previously showed that there exist functions belonging to the class of substitutes valuations that do not admit an O(n1/3) sketch.) 2. We prove that every deterministic algorithm that accesses the function via value queries only cannot guarantee a sketching ratio better than n1−ε. We also show that coverage functions, an interesting subclass of submodular functions, admit arbitrarily good sketches. Finally, we show an interesting connection between sketching and learning. We show that for every class of valuations, if the class admits an α-sketch, then it can be α-approximately learned in the PMAC model of Balcan and Harvey. The bounds we prove are only information-theoretic and do not imply the existence of computationally efficient learning algorithms in general.

We consider submodular optimization problems, and provide a general way of translating oracle inapproximability results arising from the symmetry gap technique to computational complexity inapproximability results, where the submodular function is given explicitly (under the assumption that NP ≠ RP). Applications of our technique include an optimal computational hardness of (1/2 + ε)-approximation for maximizing a symmetric nonnegative submodular function, an optimal hardness of (1-(1-1/k)k + ε)-approximation for welfare maximization in combinatorial auctions with k submodular bidders (for constant k), super-constant hardness for maximizing a nonnegative submodular function over matroid bases, and tighter bounds for maximizing a monotone submodular function subject to a cardinality constraint. Unlike the vast majority of computational inapproximability results, our approach does not use the PCP machinery or the Unique Games Conjecture, but relies instead on a direct reduction from Unique-SAT using list-decodable codes.

We present a new framework for the design of computationally-efficient and incentive-compatible mechanisms for combinatorial auctions. The mechanisms obtained via this framework are randomized, and obtain incentive compatibility in the universal sense (in contrast to the substantially weaker notion of incentive compatibility in expectation). We demonstrate the usefulness of our techniques by exhibiting two mechanisms for combinatorial auctions with general bidder preferences. The first mechanism obtains an optimal O(m)-approximation to the optimal social welfare for arbitrary bidder valuations. The second mechanism obtains an O(log2m)-approximation for a class of bidder valuations that contains the important class of submodular bidders. These approximation ratios greatly improve over the best (known) deterministic incentive-compatible mechanisms for these classes.

We exhibit incentive compatible multi-unit auctions that are not affine maximizers (i.e., are not of the VCG family) and yet approximate the social welfare to within a factor of 1+ɛ. For the case of two-item two-bidder auctions we show that these auctions, termed Triage auctions, are the only scalable ones that give an approximation factor better than 2. “Scalable ” means that the allocation does not depend on the units in which the valuations are measured. We deduce from this that any scalable computationally-efficient incentive-compatible auction for m items and n ≥ 2 bidders cannot approximate the social welfare to within a factor better than 2. This is in contrast to arbitrarily good approximations that can be reached under computational constraints alone, and in contrast to the existence of incentivecompatible mechanisms that achieve the optimal allocation.

We study procurement auctions when the buyer has complement-free (subadditive) objectives in the budget feasibility model (Singer 2010). For general subadditive functions we give a randomized universally truthful mechanism which is an O(log2 n) approximation, and an O(log3 n) deterministic truthful approximation mechanism; both mechanisms are in the demand oracle model. For cut functions, an interesting case of nonincreasing objectives, we give both randomized and deterministic truthful and budget feasible approximation mechanisms that achieve a constant approximation factor.

We show that every universally truthful randomized mechanism for combinatorial auctions with submodular valuations that provides an approximation ratio of m1/ 2 -ε must use exponentially many value queries, where m is the number of items. In contrast, ignoring incentives there exist constant ratio approximation algorithms for this problem. Our approach is based on a novel direct hardness technique that completely skips the notoriously hard step of characterizing truthful mechanisms. The characterization step was the main obstacle for proving impossibility results in algorithmic mechanism design so far. We demonstrate two additional applications of our new technique: (1) an impossibility result for universally-truthful polynomial time flexible combinatorial public projects and (2) an impossibility result for truthful-in-expectation mechanisms for exact combinatorial public projects. The latter is the first result that bounds the power of polynomial-time truthful in expectation mechanisms in any setting.

We consider the problem of designing a revenue-maximizing auction for a single item, when the values of the bidders are drawn from a correlated distribution. We observe that there exists an algorithm that finds the optimal randomized mechanism that runs in time polynomial in the size of the support. We leverage this result to show that in the oracle model introduced by Ronen and Saberi [FOCS'02], there exists a polynomial time truthful in expectation mechanism that provides a (1.5+ε)-approximation to the revenue achievable by an optimal truthful-in-expectation mechanism, and a polynomial time deterministic truthful mechanism that guarantees 5/3 approximation to the revenue achievable by an optimal deterministic truthful mechanism.

We present the first monotone randomized polynomial-time approximation scheme (PTAS) for minimizing the makespan of parallel related machines ($Q||C_{\max}$), the paradigmatic problem in single-parameter algorithmic mechanism design. This result immediately gives a polynomial-time, truthful (in expectation) mechanism whose approximation guarantee attains the best-possible one for all polynomial-time algorithms (assuming $P\neq NP$). Our algorithmic techniques are flexible and also yield a monotone deterministic quasi-PTAS for $Q||C_{\max}$ and a monotone randomized PTAS for max-min scheduling on related machines.

We present an incentive-compatible polynomial-time approximation scheme for multi-unit auctions with general k-minded player valuations. The mechanism fully optimizes over an appropriately chosen sub-range of possible allocations and then uses VCG payments over this sub-range. We show that obtaining a fully polynomial-time incentive-compatible approximation scheme, at least using VCG payments, is NP-hard. For the case of valuations given by black boxes, we give a polynomial-time incentive-compatible 2-approximation mechanism and show that no better is possible, at least using VCG payments.

In many settings the power of truthful mechanisms is severely bounded. In this paper we use randomization to overcome this problem. In particular, we construct an FPTAS for multi-unit auctions that is truthful in expectation, whereas there is evidence that no polynomial-time truthful deterministic mechanism provides an approximation ratio better than 2. We also show for the first time that truthful in expectation polynomial-time mechanisms are provably stronger than polynomial-time universally truthful mechanisms. Specifically, we show that there is a setting in which: (1) there is a non-polynomial time truthful mechanism that always outputs the optimal solution, and that (2) no universally truthful randomized mechanism can provide an approximation ratio better than 2 in polynomial time, but (3) an FPTAS that is truthful in expectation exists.

We consider the problem of designing truthful mechanisms to minimize the makespan on m unrelated machines. In their seminal paper, Nisan and Ronen [14] showed a lower bound of 2, and an upper bound of m, thus leaving a large gap. They conjectured that their upper bound is tight, but were unable to prove it. Despite many attempts that yield positive results for several special cases, the conjecture is far from being solved: the lower bound was only recently slightly increased to 2.61 [5,10], while the best upper bound remained unchanged.

We present the first monotone randomized polynomial-time approximation scheme (PTAS) for minimizing the makespan of parallel related machines (Q||Cmax), the paradigmatic problem in single-parameter algorithmic mechanism design. This result immediately gives a polynomial-time, truthful (in expectation) mechanism whose approximation guarantee attains the best-possible one for all polynomial-time algorithms (assuming P not equal to NP). Our algorithmic techniques are flexible and also yield, among other results, a monotone deterministic quasi-PTAS for Q||Cmaxand a monotone randomized PTAS for max-min scheduling on related machines.

We study multi-unit auctions where the bidders have a budget constraint, a situation very common in practice that has received very little attention in the auction theory literature. Our main result is an impossibility: there are no incentive-compatible auctions that always produce a Pareto-optimal allocation. We also obtain some surprising positive results for certain special cases.

We characterize truthful mechanisms in two multi-parameter domains. The first characterization shows that every mechanism for combinatorial auctions with two subadditive bidders that always allocates all items is an affine maximizer. The second result shows that every truthful machine scheduling mechanism for 2 unrelated machines that yields a finite approximation of the minimum makespan, must be task independent. That is, the mechanism must determine the allocation of each job separately.

We present an incentive-compatible polynomial-time approximation scheme for multi-unit auctions with general k-minded playervaluations. The mechanism fully optimizes over an appropriately chosen sub-range of possible allocations and then uses VCG payments over this sub-range. We show that obtaining a fully polynomial-time incentive-compatible approximation scheme, at least using VCG payments, is NP-hard. For the case of valuations given by black boxes, we give a polynomial-time incentive-compatible 2-approximation mechanism and show that no better is possible, at least using VCG payments.

We consider computationally-efficient incentive-compatiblemechanisms that use the VCG payment scheme, and study how well theycan approximate the social welfare in auction settings. We present anovel technique for setting lower bounds on the approximation ratioof this type of mechanisms. Specifically, for combinatorial auctionsamong submodular (and thus also subadditive) bidders we prove an Ω(m1/6) lower bound, which is close to the knownupper bound of O(m1/2), and qualitatively higher than theconstant factor approximation possible from a purely computationalpoint of view.

Congestion games are non-cooperative games where the utility of a player from using a certain resource depends on the total number of players that are using the same resource. While most work so far took a distributed game-theoretic approach to this problem, this paper studies centralized solutions for congestion games. The first part of the paper analyzes the problem from a computational perspective. We analyze the computational complexity of the welfare-maximization problem, for which we provide both approximation algorithms and lower bounds. We study this optimization problem under different kinds of congestion effects (externalities) among the players: positive, negative, and unrestricted. Our main algorithmic result is a constant approximation algorithm for congestion games with unrestricted externalities. In the second part of the paper, we also take the strategic behavior of the players into account, and present centralized truthful mechanisms for congestion-game environments. Our main result in this part is an incentive- compatible mechanism for m-resource n-player congestion games that achieves an O(vm log n) approximation to the optimal welfare. We also describe an important and useful connection between congestion games and combinatorial auctions. This connection allows us to use insights and methods from the combinatorial-auction literature for solving congestion-game problems.

Congestion games are non-cooperative games where the utility of a player from using a certain resource depends on the total number of players that are using the same resource. While most work so far took a game-theoretic approach to this problem, we study centralized solutions for congestion games from a computational point of view. We analyze the computational complexity of the welfare-maximization problem, and provide both approximation algorithms and lower bounds. Throughout the paper, different kinds of congestion effects (externalities) among the players are considered: positive, negative, and unrestricted. Our main algorithmic result is a constant approximation algorithm for congestion games with unrestricted externalities. We describe an important and useful connection between congestion games and combinatorial auctions. This connection allows us to use insights and methods from the combinatorial-auction literature for solving congestion games. Finally, we initiate the study of strategic centralized mechanisms in congestion-game environments.

We exhibit three approximation algorithms for the allocation problem in combinatorial auctions with complement free bidders. The running time of these algorithms is polynomial in the number of items $m$ and in the number of bidders n, even though the "input size" is exponential in m. The first algorithm provides an O(log m) approximation. The second algorithm provides an O(√ m) approximation in the weaker model of value oracles. This algorithm is also incentive compatible. The third algorithm provides an improved 2-approximation for the more restricted case of "XOS bidders", a class which strictly contains submodular bidders. We also prove lower bounds on the possible approximations achievable for these classes of bidders. These bounds are not tight and we leave the gaps as open problems.