Bundle Pricing with Comparable Items

2006

We present approximation and online algorithms for a number of problems of pricing items for sale so as to maximize seller's revenue in an unlimited supply setting. Our first result is an O(k)-approximation algorithm for pricing items to single-minded bidders who each want at most k items. This improves over recent independent work of Briest and Krysta [6] who achieve an O(k 2 ) bound. For the case k = 2, where we obtain a 4-approximation, this can be viewed as the following graph vertex pricing problem: given a (multi) graph G with valuations w e on the edges, find prices p i ≥ 0 for the vertices to maximize

Lecture Notes in Computer Science, 2015

We study the problem of selling n items to a single buyer with an additive valuation function. We consider the valuation of the items to be correlated, i.e., desirabilities of the buyer for the items are not drawn independently. Ideally, the goal is to design a mechanism to maximize the revenue. However, it has been shown that a revenue optimal mechanism might be very complicated and as a result inapplicable to real-world auctions. Therefore, our focus is on designing a simple mechanism that achieves a constant fraction of the optimal revenue. Babaioff et al. propose a simple mechanism that achieves a constant fraction of the optimal revenue for independent setting with a single additive buyer. However, they leave the following problem as an open question: "Is there a simple, approximately optimal mechanism for a single additive buyer whose value for n items is sampled from a common base-value distribution?" Babaioff et al. show a constant approximation factor of the optimal revenue can be achieved by either selling the items separately or as a whole bundle in the independent setting. We show a similar result for the correlated setting when the desirabilities of the buyer are drawn from a common base-value distribution. It is worth mentioning that the core decomposition lemma which is mainly the heart of the proofs for efficiency of the mechanisms does not hold for correlated settings. Therefore we propose a modified version of this lemma which is applicable to the correlated settings as well. Although we apply this technique to show the proposed mechanism can guarantee a constant fraction of the optimal revenue in a very weak correlation, this method alone can not directly show the efficiency of the mechanism in stronger correlations. Therefore, via a combinatorial approach we reduce the problem to an auction with a weak correlation to which the core decomposition technique is applicable. In addition, we introduce a generalized model of correlation for items and show the proposed mechanism achieves an O(log k) approximation factor of the optimal revenue in that setting.

Australian Economic Papers, 2011

We examine when a revenue-maximising auctioneer prefers to auction a homogenous product in one bundle (a single-object auction) than to sell the item in two or more shares. When the items are super-additive the auctioneer always prefers a single-object auction. When the product valuations are sub-additive, the auctioneer is more likely to choose a share auction when there are a large number of potential bidders. When there are a small number of bidders the auctioneer will tend to prefer a single-object auction.

Proceedings of the 41st Annual Hawaii International Conference on System Sciences (HICSS 2008), 2008

Current combinatorial auctions require bidders to specify the valuations of bundles at the start of the auction. We propose an alternative mechanism called RevalSlot, based on the common value model, which allows a bidder to participate in the auction even if she can only identify a range within which her valuations lie. This will increase bidder participation, and at the same time maximize revenue. As the auction progresses, bidders get information which helps them to converge to a value for each bundle. RevalSlot is a combination of two processes. One helps bidders to zero in on a value for each bundle, and the other is an ascending proxy auction. We present theoretical and experimental results which confirm the efficacy of our mechanism.

Lecture Notes in Computer Science, 2010

We investigate the extent to which price updates can increase the revenue of a seller with little prior information on demand. We study prior-free revenue maximization for a seller with unlimited supply of n item types facing m myopic buyers present for k < log n days. For the static (k = 1) case, Balcan et al. show that one random item price (the same on each item) yields revenue within a Θ(log m+log n) factor of optimum and this factor is tight.

European Journal of Operational Research, 2004

Single-item auctions have many desirable properties. Mechanisms exist to ensure optimality, incentive compatibility and market-clearing prices. When multiple items are offered through individual auctions, a bidder wanting a bundle of items faces an exposure problem if the bidder places a high value on a combination of goods but a low value on strict subsets of the desired collection. To remedy this, combinatorial auctions permit bids on bundles of goods. However, combinatorial auctions are hard to optimize and may not have incentive compatible mechanisms or market-clearing individual item prices. Several papers give approaches to provide incentive compatibility and imputed, individual prices. We find the relationships between these approaches and analyze their advantages and disadvantages.

Operations Research, 2006

recommends vehicles to consumers based on their requirements and budget constraints. Through the website, GM has access to large quantities of data that reflect consumer preferences. Motivated by the availability of such data, we formulate a nonparametric approach to multiproduct pricing. We consider a class of models of consumer purchasing behavior, each of which relates observed data on a consumer's requirements and budget constraint to subsequent purchasing tendencies. To price products, we aim at optimizing prices with respect to a sample of consumer data. We offer a bound on the sample size required for the resulting prices to be near-optimal with respect to the true distribution of consumers. The bound exhibits a dependence of O n log n on the number n of products being priced, showing that-in terms of sample complexity-the approach is scalable to large numbers of products. With regards to computational complexity, we establish that computing optimal prices with respect to a sample of consumer data is NP-complete in the strong sense. However, when prices are constrained by a price ladder-an ordering of prices defined prior to price determination-the problem becomes one of maximizing a supermodular function with real-valued variables. It is not yet known whether this problem is NP-hard. We provide a heuristic for our price-ladderconstrained problem, together with encouraging computational results. Finally, we apply our approach to a data set from the Auto Choice Advisor website. Our analysis provides insights into the current pricing policy at GM and suggests enhancements that may lead to a more effective pricing strategy.

Networks, 2012

In Stackelberg pricing a leader sets prices for items to maximize revenue from a follower purchasing a feasible subset of items. We consider computationally bounded followers who cannot optimize exactly over the range of all feasible subsets, but who apply publicly known algorithms to determine the items to purchase. This corresponds to general multidimensional pricing when customers cannot optimize their valuation functions efficiently but still aim to act rationally to the best of their ability. We consider two versions of this novel type of pricing problem. In the MIN-KNAPSACK variant items are weighted objects and the follower seeks to purchase a min-cost selection of objects of some bounded weight. When he uses a greedy 2-approximation algorithm, we provide a polynomial-time (2 + ε)-approximation algorithm for the leader's revenue maximization problem based on so-called near-uniform price assignments. We also prove the problem to be strongly NP-hard. In the SET-COVER variant items are subsets of some ground set which the follower seeks to cover. When he uses a standard primal-dual approach, we prove that exact revenue maximization is possible in polynomial time when

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 2012

We consider profit maximization pricing problems, where we are given a set of m customers and a set of n items. Each customer c is associated with a subset Sc ⊆ [n] of items of interest, together with a budget Bc, and we assume that there is an unlimited supply of each item. Once the prices are fixed for all items, each customer c buys a subset of items in Sc, according to its buying rule. The goal is to set the item prices so as to maximize the total profit. We study the unit-demand min-buying pricing (UDP MIN) and the single-minded pricing (SMP) problems. In the former problem, each customer c buys the cheapest item i ∈ Sc, if its price is no higher than the budget Bc, and buys nothing otherwise. In the latter problem, each customer c buys the whole set Sc if its total price is at most Bc, and buys nothing otherwise. Both problems are known to admit O(min {log(m + n), n})-approximation algorithms. We prove that they are log 1− (m + n) hard to approximate for any constant , unless NP ⊆ DTIME(n log δ n), where δ is a constant depending on. Restricting our attention to approximation factors depending only on n, we show that these problems are 2 log 1−δ n-hard to approximate for any δ > 0 unless NP ⊆ ZPTIME(n log δ n), where δ is some constant depending on δ. We also prove that restricted versions of UDP MIN and SMP, where the sizes of the sets Sc are bounded by k, are k 1/2−-hard to approximate for any constant. We then turn to the Tollbooth Pricing problem, a special case of SMP, where each item corresponds to an edge in the input graph, and each set

European Journal of Operational Research, 2017

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights • We model the problem to determine the optimal composition and pricing of multiple bundles. • We take advantage of the problem's mathematical structure to develop a two-phase solution approach. • The optimal price of each bundle depends on the composition of all the other bundles. • We demonstrated that the marginal utility of composing an additional bundle is always non-negative.