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.
Clearly, the current paper is closely associated with the growing body of work on item pricing and more general envy-free schemes @cite_2 @cite_13 @cite_1 for social welfare and revenue maximization. Item pricing for maximizing only welfare has traditionally been a sought after area of research owing to its ties to Walrasian equilibrium; since our focus is primarily on revenue, we refer the reader to @cite_25 @cite_8 @cite_7 for more recent algorithmic perspectives on the subject. On the other hand, the problem of item pricing for revenue maximization has recently gained traction in computer science with respect to both sequential and simultaneous mechanisms. A steady stream of research has yielded near-optimal approximation algorithms for a variety of settings including but not limited to unit demand @cite_29 @cite_5 @cite_3 , single minded @cite_12 @cite_33 @cite_6 , graph minded @cite_23 , multi-unit markets @cite_2 @cite_13 , and unlimited supply settings @cite_26 @cite_29 . One of the main contributions of this paper is a general framework that captures many of the above results, and provides a recipe for converting them into bicriteria approximations.
@cite_26 @cite_33 @cite_7 @cite_8 @cite_29 @cite_1 @cite_3 @cite_6 @cite_23 @cite_2 @cite_5 @cite_13 @cite_25 @cite_12
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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 consider prior-free auctions for revenue and welfare maximization when agents have a common budget. The abstract environments we consider are ones where there is a downward-closed and symmetric feasibility constraint on the probabilities of service of the agents. These environments include position auctions where slots with decreasing click-through rates are auctioned to advertisers. We generalize and characterize the envy-free benchmark from Hartline and Yan (2011) to settings with budgets and characterize the optimal envy-free outcomes for both welfare and revenue. We give prior-free mechanisms that approximate these benchmarks. A building block in our mechanism is a clinching auction for position auction environments. This auction is a generalization of the multi-unit clinching auction of (2008) and a special case of the polyhedral clinching auction of (2012). For welfare maximization, we show that this clinching auction is a good approximation to the envy-free optimal welfare for position auction environments. For profit maximization, we generalize the random sampling profit extraction auction from (2002) for digital goods to give a 10.0-approximation to the envy-free optimal revenue in symmetric, downward-closed environments. The profit maximization question is of interest even without budgets and our mechanism is a 7.5-approximation which improving on the 30.4 bound of Ha and Hartline (2012). We study the revenue maximization problem of a seller with n heterogeneous items for sale to a single buyer whose valuation function for sets of items is unknown and drawn from some distribution D. We show that if D is a distribution over subadditive valuations with independent items, then the better of pricing each item separately or pricing only the grand bundle achieves a constant-factor approximation to the revenue of the optimal mechanism. This includes buyers who are k-demand, additive up to a matroid constraint, or additive up to constraints of any downwards-closed set system (and whose values for the individual items are sampled independently), as well as buyers who are fractionally subadditive with item multipliers drawn independently. Our proof makes use of the core-tail decomposition framework developed in prior work showing similar results for the significantly simpler class of additive buyers [LY13, BILW14]. In the second part of the paper, we develop a connection between approximately optimal simple mechanisms and approximate revenue monotonicity with respect to buyers' valuations. Revenue non-monotonicity is the phenomenon that sometimes strictly increasing buyers' values for every set can strictly decrease the revenue of the optimal mechanism [HR12]. Using our main result, we derive a bound on how bad this degradation can be (and dub such a bound a proof of approximate revenue monotonicity); we further show that better bounds on approximate monotonicity imply a better analysis of our simple mechanisms. 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 ...