Optimal Pricing Is Hard (original) (raw)

The Complexity of Optimal Multidimensional Pricing

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, 2013

We resolve the complexity of revenue-optimal deterministic auctions in the unit-demand single-buyer Bayesian setting, i.e., the optimal item pricing problem, when the buyer's values for the items are independent. We show that the problem of computing a revenue-optimal pricing can be solved in polynomial time for distributions of support size 2, and its decision version is NP-complete for distributions of support size 3. We also show that the problem remains NP-complete for the case of identical distributions. * Columbia University.

Optimal auctions with correlated bidders are easy

Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11, 2011

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 ( 3 2 + ǫ)-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.

Bayesian Combinatorial Auctions

Lecture Notes in Computer Science, 2008

We study the following Bayesian setting: m items are sold to n selfish bidders in m independent second-price auctions. Each bidder has a private valuation function that expresses complex preferences over all subsets of items. Bidders only have beliefs about the valuation functions of the other bidders, in the form of probability distributions. The objective is to allocate the items to the bidders in a way that provides a good approximation to the optimal social welfare value. We show that if bidders have submodular valuation functions, then every Bayesian Nash equilibrium of the resulting game provides a 2-approximation to the optimal social welfare. Moreover, we show that in the full-information game a pure Nash always exists and can be found in time that is polynomial in both m and n.

Duality and optimality of auctions for uniform distributions

Proceedings of the fifteenth ACM conference on Economics and computation - EC '14, 2014

We develop a general duality-theory framework for revenue maximization in additive Bayesian auctions. The framework extends linear programming duality and complementarity to constraints with partial derivatives. The dual system reveals the geometric nature of the problem and highlights its connection with the theory of bipartite graph matchings. We demonstrate the power of the framework by applying it to a multiple-good monopoly setting where the buyer has uniformly distributed valuations for the items, the canonical long-standing open problem in the area. We propose a deterministic selling mechanism called Straight-Jacket Auction (SJA) which we prove to be exactly optimal for up to 6 items, and conjecture its optimality for any number of goods. The duality framework is used not only for proving optimality, but perhaps more importantly, for deriving the optimal mechanism itself; as a result, SJA is defined by natural geometric constraints. satisfies all three properties. Indeed, for Property 1, we have two cases: If j ∈ J then, by using Property 1, we get

Pricing combinatorial auctions

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.

Lower Bounds on Revenue of Approximately Optimal Auctions

arXiv preprint arXiv:1210.0275, 2012

We obtain revenue guarantees for the simple pricing mechanism of a single posted price, in terms of a natural parameter of the distribution of buyers' valuations. Our revenue guarantee applies to the single item n buyers setting, with values drawn from an arbitrary joint distribution. Specifically, we show that a single price drawn from the distribution of the maximum valuation Vmax = max{V1, V2, . . . , Vn} achieves a revenue of at least a 1 e fraction of the geometric expecation of Vmax. This generic bound is a measure of how revenue improves/degrades as a function of the concentration/spread of Vmax. We further show that in absence of buyers' valuation distributions, recruiting an additional set of identical bidders will yield a similar guarantee on revenue. Finally, our bound also gives a measure of the extent to which one can simultaneously approximate welfare and revenue in terms of the concentration/spread of Vmax.

Optimal Bundle Pricing for Homogeneous Items

2006

We consider a revenue maximization problem where we are selling a set of m items, each of which available in a certain quantity (possibly unlimited) to a set of n bidders. Bidders are single minded, that is, each bidder requests exactly one subset, or bundle of items. Each bidder has a valuation for the requested bundle that we assume to be known to the seller. The task is to find an envy-free pricing such as to maximize the revenue of the seller. We derive several complexity results and algorithms for several variants of this pricing problem. In fact, the settings that we consider address problems where the different items are 'homogeneous' in some sense. First, we introduce the notion of affine price functions that can be used to model situations much more general than the usual combinatorial pricing model that is mostly addressed in the literature. We derive fixed-parameter polynomial time algorithms as well as inapproximability results. Second, we consider the special case of combinatorial pricing, and introduce a monotonicity constraint that can also be seen as 'global' envy-freeness condition. We show that the problem remains strongly NP-hard, and we derive a PTAS -thus breaking the inapproximability barrier known for the general case. As a special case, we finally address the notorious highway pricing problem under the global envy-freeness condition.

Approximation algorithms and online mechanisms for item pricing

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

Optimal and Efficient Parametric Auctions

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, 2013

Consider a seller who seeks to provide service to a collection of interested parties, subject to feasibility constraints on which parties may be simultaneously served. Assuming that a distribution is known on the value of each party for service-arguably a strong assumption-Myerson's seminal work provides revenue optimizing auctions [12]. We show instead that, for very general feasibility constraints, only knowledge of the median of each party's value distribution, or any other quantile of these distributions, or approximations thereof, suffice for designing simple auctions that simultaneously approximate both the optimal revenue and the optimal welfare. Our results apply to all downward-closed feasibility constraints under the assumption that the underlying, unknown value distributions are monotone hazard rate, and to all matroid feasibility constraints under the weaker assumption of regularity of the underlying distributions. Our results jointly generalize the single-item results obtained by Azar and Micali [2] on parametric auctions, and Daskalakis and Pierrakos [6] for simultaneously approximately optimal and efficient auctions.

Implementing the Optimal Auction

2003

In a general framework with independent private values of the bidders, we propose a game, with a simple economic interpretation, that allows implementing the optimal auction outcome when the seller ignores the distributions of the different bidders' valuations. In this robust or detail-free implementation procedure, a second-price auction is organized and the winner volunteers a payment to the seller; this payment can then be challenged by another bidder who knows the distribution of the winner's valuation. Dans un cadre du modèle d'enchères avec des valeurs privées indépendantes, nous proposons un jeu, ayant une interprétation économique simple, qui permet de mettre en oeuvre les enchères optimales même quand le vendeur ignore les distributions des volontés à payer des différents soumissionnaires. Dans cette procédure robuste (detail-free), une enchère au deuxième prix est organisée et le gagnant de cette enchère propose un paiement au vendeur; ce paiement peut alors...