The Complexity of Optimal Multidimensional Pricing

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.

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.

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

4OR, 2011

We consider the problem of pricing items in order to maximize the revenue obtainable from a set of single minded customers. We relate the tractability of the problem to structural properties of customers' valuations: the problem admits an efficient approximation algorithm, parameterized along the inhomogeneity of the valuations.

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.

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

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

2013

We study the optimal strategy for selling multiple items in a setting where bidders can bid for individual items and for subsets of items. There is a large literature examining search problems with a single good, but few papers that generalize the problem to vector offers for subsets of items. One of the challenges in extending to multiple goods is that, as the state space expands, the problem can become computationally intractable. We show how to solve the dynamic optimization problem so that it becomes computationally feasible when the number of items is not too large. We also consider special cases of the model, including an “additive” case and a “single item” case, and present several intuitive structural results about the optimal policy and value functions for these specialized cases. Finally, we consider extensions to a stopping rule problem. JEL Classification Numbers: C61, D83.

2002

2011

Consider the problem of pricing n items under an unlimited supply with m buyers. Each buyer is interested in a bundle of at most k of the items. These buyers are single minded, which means each of them has a budget and they will either buy all the items if the total price is within their budget or they will buy none of the items. The goal is to price each item with profit margin p 1 , p 2 , ..., p n so as to maximize the overall profit. When k = 2, such a problem is called the graph-vertex-pricing problem. Another special case of the problem is the highwaypricing problem when the items (toll-booths) are arranged linearly on a line and each buyer (as a driver) is interested in paying for a path that consists of consecutive items. The goal again is to price the items (tolls) so as to maximize the total profits.