We study approximation algorithms for revenue maximization based on static item pricing, where a seller chooses prices for various goods in the market, and then the buyers purchase utility-maximizing bundles at these given prices. We formulate two somewhat general techniques for designing good pricing algorithms for this setting: Price Doubling and Item Halving. Using these techniques, we unify many of the existing results in the item pricing literature under a common framework, as well as provide several new item pricing algorithms for approximating both revenue and social welfare. More specifically, for a variety of settings with item pricing, we show that it is possible to deterministically obtain a log-approximation for revenue and a constant-approximation for social welfare simultaneously: thus one need not sacrifice revenue if the goal is to still have decent welfare guarantees. In addition, we provide a new black-box reduction from revenue to welfare based on item pricing, which immediately gives us new revenue-approximation algorithms (e.g., for gross substitutes valuations). The main technical contribution of this paper is a @math -approximation algorithm for revenue maximization based on the Item Halving technique, for settings where buyers have XoS valuations, where @math is the number of goods and @math is the average supply. Surprisingly, ours is the first known item pricing algorithm with polylogarithmic approximations for such general classes of valuations, and partially resolves an important open question from the algorithmic pricing literature about the existence of item pricing algorithms with logarithmic factors for general valuations. We also use the Item Halving framework to form envy-free item pricing mechanisms for the popular setting of multi-unit markets, providing a log-approximation to revenue in this case.
Despite the tremendous body of work on revenue maximization, ours is the first known poly-logarithmic approximation algorithm based on static, item pricing for complex valuations such as submodular and XoS functions. Partial exceptions include the @math -approximation for simple submodular functions' in @cite_4 and the @math -approximation in @cite_16 for settings consisting of a large number of buyers having the exact same valuation function. Although our central result improves upon the upper bound of @math from @cite_4 for XoS functions, it is pertinent to mention that their result holds for a more general information model where the seller is not aware of the buyer valuations. For such a general model, it is reasonable to expect that the only possible strategy would be to price items uniformly (common price), as is done in @cite_4 @cite_16 . The presence of a matching lower bound of @math @cite_16 for uniform pricing motivates the need for pricing different goods differently. For such a scheme, we argue that knowledge of buyer valuations is necessary since it allows us to quantify each good's relative value in the market.
@cite_16 @cite_4
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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. We investigate nonparametric multiproduct pricing problems, in which we want to find revenue maximizing prices for products @math based on a set of customer samples @math . We mostly focus on the unit-demand case, in which products constitute strict substitutes and each customer aims to purchase a single product. In this setting a customer sample consists of a number of nonzero values for different products and possibly an additional product ranking. Once prices are fixed, each customer chooses to buy one of the products she can afford based on some predefined selection rule. We distinguish between the min-buying, max-buying, and rank-buying models. Some of our results also extend to single-minded pricing, in which case products are strict complements and every customer seeks to buy a single set of products, which she purchases if the sum of prices is below her valuation for that set. For the min-buying model we show that the revenue maximization problem is not approximable within factor @math for some constant @math , unless @math , thereby almost closing the gap between the known algorithmic results and previous lower bounds. We also prove inapproximability within @math , @math being an upper bound on the number of nonzero values per customer, and @math under slightly stronger assumptions and provide matching upper bounds. Surprisingly, these hardness results hold even if a price ladder constraint, i.e., a predefined order on the prices of all products, is given. Without the price ladder constraint we obtain similar hardness results for the special case of uniform valuations, i.e., the case that every customer has identical values for all the products she is interested in, assuming specific hardness of the balanced bipartite independent set problem in constant degree graphs or hardness of refuting random 3CNF formulas. Introducing a slightly more general problem definition in which customers are given as an explicit probability distribution, we obtain inapproximability within @math assuming @math . These results apply to single-minded pricing as well. For the max-buying model a polynomial-time approximation scheme exists if a price ladder is given. We give a matching lower bound by proving strong NP-hardness. Assuming limited product supply, we analyze a generic local search algorithm and prove that it is 2-approximate. Finally, we discuss implications for the rank-buying model. In the design and analysis of revenue-maximizing auctions, auction performance is typically measured with respect to a prior distribution over inputs. The most obvious source for such a distribution is past data. The goal of this paper is to understand how much data is necessary and sufficient to guarantee near-optimal expected revenue. Our basic model is a single-item auction in which bidders' valuations are drawn independently from unknown and nonidentical distributions. The seller is given m samples from each of these distributions "for free" and chooses an auction to run on a fresh sample. How large does m need to be, as a function of the number k of bidders and e 0, so that a (1 -- e)-approximation of the optimal revenue is achievable? We prove that, under standard tail conditions on the underlying distributions, m = poly(k, 1 e) samples are necessary and sufficient. Our lower bound stands in contrast to many recent results on simple and prior-independent auctions and fundamentally involves the interplay between bidder competition, non-identical distributions, and a very close (but still constant) approximation of the optimal revenue. It effectively shows that the only way to achieve a sufficiently good constant approximation of the optimal revenue is through a detailed understanding of bidders' valuation distributions. Our upper bound is constructive and applies in particular to a variant of the empirical Myerson auction, the natural auction that runs the revenue-maximizing auction with respect to the empirical distributions of the samples. To capture how our sample complexity upper bound depends on the set of allowable distributions, we introduce α-strongly regular distributions, which interpolate between the well-studied classes of regular (α = 0) and MHR (α = 1) distributions. We give evidence that this definition is of independent interest. We consider the item pricing problem for revenue maximization, where a single seller with multiple distinct items caters to multiple buyers with unknown subadditive valuation functions who arrive in a sequence. The seller sets the prices on individual items, and we design randomized pricing strategies to maximize expected revenue. We consider dynamic uniform strategies, which can change the price upon the arrival of each buyer but the price on all unsold items is the same at all times, and static nonuniform strategies, which can assign different prices to different items but can never change it after setting it initially. We design pricing strategies that guarantee poly-logarithmic (in number of items) approximation to maximum possible social welfare, which is an upper bound on revenue. We also show that any static uniform pricing strategy cannot yield such approximation, thus highlighting a large gap between the powers of dynamic and static pricing. Finally, our pricing strategies imply poly-logarithmic ...