We design an expected polynomial-time, truthful-in-expectation, (1-1 e)-approximation mechanism for welfare maximization in a fundamental class of combinatorial auctions. Our results apply to bidders with valuations that are m matroid rank sums (MRS), which encompass most concrete examples of submodular functions studied in this context, including coverage functions, matroid weighted-rank functions, and convex combinations thereof. Our approximation factor is the best possible, even for known and explicitly given coverage valuations, assuming P != NP. Ours is the first truthful-in-expectation and polynomial-time mechanism to achieve a constant-factor approximation for an NP-hard welfare maximization problem in combinatorial auctions with heterogeneous goods and restricted valuations. Our mechanism is an instantiation of a new framework for designing approximation mechanisms based on randomized rounding algorithms. A typical such algorithm first optimizes over a fractional relaxation of the original problem, and then randomly rounds the fractional solution to an integral one. With rare exceptions, such algorithms cannot be converted into truthful mechanisms. The high-level idea of our mechanism design framework is to optimize directly over the (random) output of the rounding algorithm, rather than over the input to the rounding algorithm. This approach leads to truthful-in-expectation mechanisms, and these mechanisms can be implemented efficiently when the corresponding objective function is concave. For bidders with MRS valuations, we give a novel randomized rounding algorithm that leads to both a concave objective function and a (1-1 e)-approximation of the optimal welfare.
Without incentive-compatibility constraints, the welfare maximization problem with submodular bidder valuations is completely solved. Vondr 'a k @cite_20 gave a @math -approximation algorithm for the problem, improving over the @math -approximation given in @cite_3 . The algorithm in @cite_20 works in the value oracle model, where each valuation @math is modeled as a black box'' that returns the value @math of a queried set @math in a single operation. The approximation factor of @math is unconditionally optimal in the value-oracle model (for polynomial communication) @cite_12 , and is also optimal (for polynomial time) for certain succinctly represented submodular valuations, assuming @math @cite_11 . The result of @cite_11 implies that @math is the optimal approximation factor in our model as well, assuming @math . We show in Appendix that our oracle model is no more powerful than polynomial-time computation in the special case of explicitly represented coverage functions, for which @math is optimal assuming @math @cite_11 . In contrast, the work of @cite_23 improves on the approximation factor of @math by using , which can not be simulated in polynomial time for explicit coverage functions.
@cite_3 @cite_23 @cite_20 @cite_12 @cite_11
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We provide tight information-theoretic lower bounds for the welfare maximization problem in combinatorial auctions. In this problem, the goal is to partition m items among k bidders in a way that maximizes the sum of bidders' values for their allocated items. Bidders have complex preferences over items expressed by valuation functions that assign values to all subsets of items. We study the "black box" setting in which the auctioneer has oracle access to the valuation functions of the bidders. In particular, we explore the well-known value query model in which the permitted query to a valuation function is in the form of a subset of items, and the reply is the value assigned to that subset of items by the valuation function. We consider different classes of valuation functions: submodular,subadditive, and superadditive. For these classes, it has been shown that one can achieve approximation ratios of 1 -- 1 e, 1 √m, and √ m m, respectively, via a polynomial (in k and m) number of value queries. We prove that these approximation factors are essentially the best possible: For any fixed e > 0, a (1--1 e + e)-approximation for submodular valuations or an 1 m1 2-e-approximation for subadditive valuations would require exponentially many value queries, and a log1+e m m-approximation for superadditive valuations would require a superpolynomial number of value queries. In the Submodular Welfare Problem, m items are to be distributed among n players with utility functions wi: 2[m] → R+. The utility functions are assumed to be monotone and submodular. Assuming that player i receives a set of items Si, we wish to maximize the total utility ∑i=1n wi(Si). In this paper, we work in the value oracle model where the only access to the utility functions is through a black box returning wi(S) for a given set S. Submodular Welfare is in fact a special case of the more general problem of submodular maximization subject to a matroid constraint: max f(S): S ∈ I , where f is monotone submodular and I is the collection of independent sets in some matroid. For both problems, a greedy algorithm is known to yield a 1 2-approximation [21, 16]. In special cases where the matroid is uniform (I = S: |S| ≤ k) [20] or the submodular function is of a special type [4, 2], a (1-1 e)-approximation has been achieved and this is optimal for these problems in the value oracle model [22, 6, 15]. A (1-1 e)-approximation for the general Submodular Welfare Problem has been known only in a stronger demand oracle model [4], where in fact 1-1 e can be improved [9]. In this paper, we develop a randomized continuous greedy algorithm which achieves a (1-1 e)-approximation for the Submodular Welfare Problem in the value oracle model. We also show that the special case of n equal players is approximation resistant, in the sense that the optimal (1-1 e)-approximation is achieved by a uniformly random solution. Using the pipage rounding technique [1, 2], we obtain a (1-1 e)-approximation for submodular maximization subject to any matroid constraint. The continuous greedy algorithm has a potential of wider applicability, which we demonstrate on the examples of the Generalized Assignment Problem and the AdWords Assignment Problem. We design an expected polynomial time, truthful in expectation, (1-1 e)-approximation mechanism for welfare maximization in a fundamental class of combinatorial auctions. Our results apply to bidders with valuations that are matroid rank sums (MRS), which encompass most concrete examples of submodular functions studied in this context, including coverage functions and matroid weighted-rank functions. Our approximation factor is the best possible, even for known and explicitly given coverage valuations, assuming P ≠ NP. Ours is the first truthful-in-expectation and polynomial-time mechanism to achieve a constant-factor approximation for an NP-hard welfare maximization problem in combinatorial auctions with heterogeneous goods and restricted valuations. Our mechanism is an instantiation of a new framework for designing approximation mechanisms based on randomized rounding algorithms. A typical such algorithm first optimizes over a fractional relaxation of the original problem, and then randomly rounds the fractional solution to an integral one. With rare exceptions, such algorithms cannot be converted into truthful mechanisms. The high-level idea of our mechanism design framework is to optimize directly over the (random) output of the rounding algorithm, rather than over the input to the rounding algorithm. This approach leads to truthful-in-expectation mechanisms, and these mechanisms can be implemented efficiently when the corresponding objective function is concave. For bidders with MRS valuations, we give a novel randomized rounding algorithm that leads to both a concave objective function and a (1-1 e)-approximation of the optimal welfare. We study the problem of fair allocation for indivisible goods. We use the the maxmin share paradigm introduced by Budish as a measure for fairness. Procacciafirst (EC'14) were first to investigate this fundamental problem in the additive setting. In contrast to what real-world experiments suggest, they show that a maxmin guarantee (1- @math allocation) is not always possible even when the number of agents is limited to 3. While the existence of an approximation solution (e.g. a @math - @math allocation) is quite straightforward, improving the guarantee becomes subtler for larger constants. Procaccia provide a proof for existence of a @math - @math allocation and leave the question open for better guarantees. Our main contribution is an answer to the above question. We improve the result of ! to a @math factor in the additive setting. The main idea for our @math - @math allocation method is clustering the agents. To this end, we introduce three notions and techniques, namely reducibility, matching allocation, and cycle-envy-freeness, and prove the approximation guarantee of our algorithm via non-trivial applications of these techniques. Our analysis involves coloring and double counting arguments that might be of independent interest. One major shortcoming of the current studies on fair allocation is the additivity assumption on the valuations. We alleviate this by extending our results to the case of submodular, fractionally subadditive, and subadditive settings. More precisely, we give constant approximation guarantees for submodular and XOS agents, and a logarithmic approximation for the case of subadditive agents. Furthermore, we complement our results by providing close upper bounds for each class of valuation functions. Finally, we present algorithms to find such allocations for additive, submodular, and XOS settings in polynomial time. 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 @math [STOC'06], where @math is the number of items. This was subsequently improved to @math [Dobzinski, APPROX'07] and then to @math [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 @math . 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. We consider the problem of maximizing welfare when allocating m items to n players with subadditive utility functions. Our main result is a way of rounding any fractional solution to a linear programming relaxation to this problem so as to give a feasible solution of welfare at least half that of the value of the fractional solution. This approximation ratio of 1 2 improves over an Ω(1 log m) ratio of Dobzinski, Nisan and Schapira [STOC 2005]. We also show an approximation ratio of 1 - 1 e when utility functions are fractionally subadditive. A result similar to this last result was previously obtained by Dobzinski and Schapira [Soda 2006], but via a different rounding technique that requires the use of a so called "XOS oracle".The randomized rounding techniques that we use are oblivious in the sense that they only use the primal solution to the linear program relaxation, but have no access to the actual utility functions of the players. This allows us to suggest new incentive compatible mechanisms for combinatorial auctions, extending previous work of Lavi and Swamy [FOCS 2005]. 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−e. 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 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 @math 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. We consider the problem of maximizing a nonnegative (possibly non-monotone) submodular set function with or without constraints. [9] showed a 2 5-approximation for the unconstrained problem and also proved that no approximation better than 1 2 is possible in the value oracle model. Constant-factor approximation has been also known for submodular maximization subject to a matroid independence constraint (a factor of 0.309 [33]) and for submodular maximization subject to a matroid base constraint, provided that the fractional base packing number ν is bounded away from 1 (a 1 4-approximation assuming that ν ≥ 2 [33]). In this paper, we propose a new algorithm for submodular maximization which is based on the idea of simulated annealing. We prove that this algorithm achieves improved approximation for two problems: a 0.41-approximation for unconstrained submodular maximization, and a 0.325-approximation for submodular maximization subject to a matroid independence constraint. On the hardness side, we show that in the value oracle model it is impossible to achieve a 0.478-approximation for submodular maximization subject to a matroid independence constraint, or a 0.394-approximation subject to a matroid base constraint in matroids with two disjoint bases. Even for the special case of cardinality constraint, we prove it is impossible to achieve a 0.491-approximation. (Previously it was conceivable that a 1 2-approximation exists for these problems.) It is still an open question whether a 1 2-approximation is possible for unconstrained submodular maximization. An extensive body of recent work studies the welfare guarantees of simple and prevalent combinatorial auction formats, such as selling m items via simultaneous second price auctions (SiSPAs) [1], [2], [3]. These guarantees hold even when the auctions are repeatedly executed and the players use no-regret learning algorithms to choose their actions. Unfortunately, off-the-shelf no-regret learning algorithms for these auctions are computationally inefficient as the number of actions available to the players becomes exponential. We show that this obstacle is inevitable: there are no polynomial-time no-regret learning algorithms for SiSPAs, unless RP ⊆ NP, even when the bidders are unit-demand. Our lower bound raises the question of how good outcomes polynomially-bounded bidders may discover in such auctions. To answer this question, we propose a novel concept of learning in auctions, termed "no-envy learning." This notion is founded upon Walrasian equilibrium, and we show that it is both efficiently implementable and results in approximately optimal welfare, even when the bidders have valuations from the broad class of fractionally subadditive (XOS) valuations (assuming demand oracle access to the valuations) or coverage valuations (even without demand oracles). No-envy learning outcomes are a relaxation of no-regret learning outcomes, which maintain their approximate welfare optimality while endowing them with computational tractability. Our positive and negative results extend to several auction formats that have been studied in the literature via the smoothness paradigm. Our positive results for XOS valuations are enabled by a novel Follow-The-Perturbed-Leader algorithm for settings where the number of experts and states of nature are both infinite, and the payoff function of the learner is non-linear. We show that this algorithm has applications outside of auction settings, establishing significant gains in a recent application of no-regret learning in security games. Our efficient learning result for coverage valuations is based on a novel use of convex rounding schemes and a reduction to online convex optimization. We consider the problem of dividing indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin share, that is defined to be the maximum value that an agent can ensure for herself, if she were to partition the goods into n bundles, and then receive a minimum valued bundle. Since maximin fair allocations (i.e., allocations in which each agent gets at least her maximin share) do not always exist, prior work has focussed on approximation results that aim to find allocations in which the value of the bundle allocated to each agent is (multiplicatively) as close to her maximin share as possible. In particular, Procaccia and Wang (2014) along with (2015) have shown that under additive valuations a 2 3-approximate maximin fair allocation always exists and can be found in polynomial time. We complement these results by developing a simple and efficient algorithm that achieves the same approximation guarantee. Furthermore, we initiate the study of approximate maximin fair division under submodular valuations. Specifically, we show that when the valuations of the agents are nonnegative, monotone, and submodular, then a 1 10-approximate maximin fair allocation is guaranteed to exist. In fact, we show that such an allocation can be efficiently found by using a simple round-robin algorithm. A technical contribution of the paper is to analyze the performance of this combinatorial algorithm by employing the concept of multilinear extensions. Let @math be a monotone submodular set function, and let @math be a matroid. We consider the problem @math . It is known that the greedy algorithm yields a @math -approximation [M. L. Fisher, G. L. Nemhauser, and L. A. Wolsey, Math. Programming Stud., no. 8 (1978), pp. 73-87] for this problem. For certain special cases, e.g., @math , the greedy algorithm yields a @math -approximation. It is known that this is optimal both in the value oracle model (where the only access to @math is through a black box returning @math for a given set @math ) [G. L. Nemhauser and L. A. Wolsey, Math. Oper. Res., 3 (1978), pp. 177-188] and for explicitly posed instances assuming @math [U. Feige, J. ACM, 45 (1998), pp. 634-652]. In this paper, we provide a randomized @math -approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [J. Combin. Optim., 8 (2004), pp. 307-328], and a continuous greedy process that may be of independent interest. As a special case, our algorithm implies an optimal approximation for the submodular welfare problem in the value oracle model [J. Vondrak, Proceedings of the @math th ACM Symposium on Theory of Computing, 2008, pp. 67-74]. As a second application, we show that the generalized assignment problem (GAP) is also a special case; although the reduction requires @math to be exponential in the original problem size, we are able to achieve a @math -approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constraints. We consider the maximization problem in the value oracle model of functions defined on k-tuples of sets that are submodular in every orthant and r-wise monotone, where k g 2 and 1 l r l k. We give an analysis of a deterministic greedy algorithm that shows that any such function can be approximated to a factor of 1 (1 + r). For r e k, we give an analysis of a randomized greedy algorithm that shows that any such function can be approximated to a factor of 1 (1+sk 2. In the case of k e r e 2, the considered functions correspond precisely to bisubmodular functions, in which case we obtain an approximation guarantee of 1 2. We show that, as in the case of submodular functions, this result is the best possible both in the value query model and under the assumption that NP ≠ RP. Extending a result of , we show that for any k g 3, submodularity in every orthant and pairwise monotonicity (i.e., r e 2) precisely characterize k-submodular functions. Consequently, we obtain an approximation guarantee of 1 3 (and thus independent of k) for the maximization problem of k-submodular functions. We provide a computationally efficient black-box reduction from mechanism design to algorithm design in very general settings. Specifically, we give an approximation-preserving reduction from truthfully maximizing any objective under arbitrary feasibility constraints with arbitrary bidder types to (not necessarily truthfully) maximizing the same objective plus virtual welfare (under the same feasibility constraints). Our reduction is based on a fundamentally new approach: we describe a mechanism's behavior indirectly only in terms of the expected value it awards bidders for certain behavior, and never directly access the allocation rule at all. Applying our new approach to revenue, we exhibit settings where our reduction holds both ways. That is, we also provide an approximation-sensitive reduction from (non-truthfully) maximizing virtual welfare to (truthfully) maximizing revenue, and therefore the two problems are computationally equivalent. With this equivalence in hand, we show that both problems are NP-hard to approximate within any polynomial factor, even for a single monotone sub modular bidder. We further demonstrate the applicability of our reduction by providing a truthful mechanism maximizing fractional max-min fairness. We study anonymous posted price mechanisms for combinatorial auctions in a Bayesian framework. In a posted price mechanism, item prices are posted, then the consumers approach the seller sequentially in an arbitrary order, each purchasing her favorite bundle from among the unsold items at the posted prices. These mechanisms are simple, transparent and trivially dominant strategy incentive compatible (DSIC). We show that when agent preferences are fractionally subadditive (which includes all submodular functions), there always exist prices that, in expectation, obtain at least half of the optimal welfare. Our result is constructive: given black-box access to a combinatorial auction algorithm A, sample access to the prior distribution, and appropriate query access to the sampled valuations, one can compute, in polytime, prices that guarantee at least half of the expected welfare of A. As a corollary, we obtain the first polytime (in n and m) constant-factor DSIC mechanism for Bayesian submodular combinatorial auctions, given access to demand query oracles. Our results also extend to valuations with complements, where the approximation factor degrades linearly with the level of complementarity. We study anonymous posted price mechanisms for combinatorial auctions in a Bayesian framework. In a posted price mechanism, item prices are posted, then the consumers approach the seller sequentially in an arbitrary order, each purchasing her favorite bundle from among the unsold items at the posted prices. These mechanisms are simple, transparent and trivially dominant strategy incentive compatible (DSIC). We show that when agent preferences are fractionally subadditive (which includes all submodular functions), there always exist prices that, in expectation, obtain at least half of the optimal welfare. Our result is constructive: given black-box access to a combinatorial auction algorithm A, sample access to the prior distribution, and appropriate query access to the sampled valuations, one can compute, in polytime, prices that guarantee at least half of the expected welfare of A. As a corollary, we obtain the first polytime (in n and m) constant-factor DSIC mechanism for Bayesian submodular combinatorial auctions, given access to demand query oracles. Our results also extend to valuations with complements, where the approximation factor degrades linearly with the level of complementarity. A line of recent work provides welfare guarantees of simple combinatorial auction formats, such as selling m items via simultaneous second price auctions (SiSPAs) ( 2008, Bhawalkar and Roughgarden 2011, 2013). These guarantees hold even when the auctions are repeatedly executed and players use no-regret learning algorithms. Unfortunately, off-the-shelf no-regret algorithms for these auctions are computationally inefficient as the number of actions is exponential. We show that this obstacle is insurmountable: there are no polynomial-time no-regret algorithms for SiSPAs, unless RP @math NP, even when the bidders are unit-demand. Our lower bound raises the question of how good outcomes polynomially-bounded bidders may discover in such auctions. To answer this question, we propose a novel concept of learning in auctions, termed "no-envy learning." This notion is founded upon Walrasian equilibrium, and we show that it is both efficiently implementable and results in approximately optimal welfare, even when the bidders have fractionally subadditive (XOS) valuations (assuming demand oracles) or coverage valuations (without demand oracles). No-envy learning outcomes are a relaxation of no-regret outcomes, which maintain their approximate welfare optimality while endowing them with computational tractability. Our results extend to other auction formats that have been studied in the literature via the smoothness paradigm. Our results for XOS valuations are enabled by a novel Follow-The-Perturbed-Leader algorithm for settings where the number of experts is infinite, and the payoff function of the learner is non-linear. This algorithm has applications outside of auction settings, such as in security games. Our result for coverage valuations is based on a novel use of convex rounding schemes and a reduction to online convex optimization. In this article, we study the problem of online market clearing where there is one commodity in the market being bought and sold by multiple buyers and sellers whose bids arrive and expire at different times. The auctioneer is faced with an online clearing problem of deciding which buy and sell bids to match without knowing what bids will arrive in the future. For maximizing profit, we present a (randomized) online algorithm with a competitive ratio of ln(pmax − pmin) p 1, when bids are in a range [pmin, pmax], which we show is the best possible. A simpler algorithm has a ratio twice this, and can be used even if expiration times are not known. For maximizing the number of trades, we present a simple greedy algorithm that achieves a factor of 2 competitive ratio if no money-losing trades are allowed. We also show that if the online algorithm is allowed to subsidize matches---match money-losing pairs if it has already collected enough money from previous pairs to pay for them---then it can actually be 1-competitive with respect to the optimal offline algorithm that is not allowed subsidy. That is, for maximizing the number of trades, the ability to subsidize is at least as valuable as knowing the future. We also consider objectives of maximizing buy or sell volume and social welfare. We present all of these results as corollaries of theorems on online matching in an incomplete interval graph.We also consider the issue of incentive compatibility, and develop a nearly optimal incentive-compatible algorithm for maximizing social welfare. For maximizing profit, we show that no incentive-compatible algorithm can achieve a sublinear competitive ratio, even if only one buy bid and one sell bid are alive at a time. However, we provide an algorithm that, under certain mild assumptions on the bids, performs nearly as well as the best fixed pair of buy and sell prices, a weaker but still natural performance measure. This latter result uses online learning methods, and we also show how such methods can be used to improve our “optimal” algorithms to a broader notion of optimality. Finally, we show how some of our results can be generalized to settings in which the buyers and sellers themselves have online bidding strategies, rather than just each having individual bids. A number of recent results on optimization problems involving submodular functions have made use of the "multilinear relaxation" of the problem. We present a general approach to deriving inapproximability results in the value oracle model, based on the notion of "symmetry gap". Our main result is that for any fixed instance that exhibits a certain "symmetry gap" in its multilinear relaxation, there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries. This unifies several known hardness results for submodular maximization, e.g. the optimality of (1-1 e)-approximation for monotone submodular maximization under a cardinality constraint, and the impossibility of (1 2+epsilon)-approximation for unconstrained (non-monotone) submodular maximization. It follows from our result that (1 2+epsilon)-approximation is also impossible for non-monotone submodular maximization subject to a (non-trivial) matroid constraint. On the algorithmic side, we present a 0.309-approximation for this problem, improving the previously known factor of 1 4-o(1).As another application, we consider the problem of maximizing a non-monotone submodular function over the bases of a matroid. A (1 6-o(1))-approximation has been developed for this problem, assuming that the matroid contains two disjoint bases. We show that the best approximation one can achieve is indeed related to packings of bases in the matroid. Specifically, for any k≫=2, there is a class of matroids of fractional base packing number nu = k (k-1), such that any algorithm achieving a better than (1-1 nu)-approximation for this class would require exponentially many value queries. On the positive side, we present a 1 2 (1-1 nu-o(1))-approximation algorithm for the same problem. Our hardness results hold in fact for very special symmetric instances. For such symmetric instances, we show that the approximation factors of 1 2 (for submodular maximization subject to a matroid constraint) and 1-1 nu (for a matroid base constraint) can be achieved algorithmically and hence are optimal. We consider the problem of pricing n items to maximize revenue when faced with a series of unknown buyers with complex preferences, and show that a simple pricing scheme achieves surprisingly strong guarantees. We show that in the unlimited supply setting, a random single price achieves expected revenue within a logarithmic factor of the total social welfare for customers with general valuation functions, which may not even necessarily be monotone. This generalizes work of Guruswami et. al [18], who show a logarithmic factor for only the special cases of single-minded and unit-demand customers. In the limited supply setting, we show that for subadditive valuations, a random single price achieves revenue within a factor of 2O(√(log n loglog n) of the total social welfare, i.e., the optimal revenue the seller could hope to extract even if the seller could price each bundle differently for every buyer. This is the best approximation known for any item pricing scheme for subadditive (or even submodular) valuations, even using multiple prices. We complement this result with a lower bound showing a sequence of subadditive (in fact, XOS) buyers for which any single price has approximation ratio 2Ω(log1 4 n), thus showing that single price schemes cannot achieve a polylogarithmic ratio. This lower bound demonstrates a clear distinction between revenue maximization and social welfare maximization in this setting, for which [12,10] show that a fixed price achieves a logarithmic approximation in the case of XOS [12], and more generally subadditive [10], customers. We also consider the multi-unit case examined by [1111] in the context of social welfare, and show that so long as no buyer requires more than a 1 -- e fraction of the items, a random single price now does in fact achieve revenue within an O(log n) factor of the maximum social welfare.