On Revenue Maximization with Sharp Multi-Unit Demands

We consider markets consisting of a set of indivisible items, and buyers that have {\em sharp} multi-unit demand. This means that each buyer $i$ wants a specific number $d_i$ of items; a bundle of size less than $d_i$ has no value, while a bundle of size greater than $d_i$ is worth no more than the most valued $d_i$ items (valuations being additive). We consider the objective of setting prices and allocations in order to maximize the total revenue of the market maker. The pricing problem with sharp multi-unit demand buyers has a number of properties that the unit-demand model does not possess, and is an important question in algorithmic pricing. We consider the problem of computing a revenue maximizing solution for two solution concepts: competitive equilibrium and envy-free pricing. For unrestricted valuations, these problems are NP-complete; we focus on a realistic special case of "correlated values" where each buyer $i$ has a valuation v_i\qual_j for item $j$, where $v_i$ and \qual_j are positive quantities associated with buyer $i$ and item $j$ respectively. We present a polynomial time algorithm to solve the revenue-maximizing competitive equilibrium problem. For envy-free pricing, if the demand of each buyer is bounded by a constant, a revenue maximizing solution can be found efficiently; the general demand case is shown to be NP-hard.Comment: page2

Quadratic Regularization of Unit-Demand Envy-Free Pricing Problems and Application to Electricity Markets

We consider a profit-maximizing model for pricing contracts as an extension of the unit-demand envy-free pricing problem: customers aim to choose a contract maximizing their utility based on a reservation bill and multiple price coefficients (attributes). A classical approach supposes that the customers have deterministic utilities; then, the response of each customer is highly sensitive to price since it concentrates on the best offer. A second approach is to consider logit model to add a probabilistic behavior in the customers' choices. To circumvent the intrinsic instability of the former and the resolution difficulties of the latter, we introduce a quadratically regularized model of customer's response, which leads to a quadratic program under complementarity constraints (QPCC). This allows to robustify the deterministic model, while keeping a strong geometrical structure. In particular, we show that the customer's response is governed by a polyhedral complex, in which every polyhedral cell determines a set of contracts which is effectively chosen. Moreover, the deterministic model is recovered as a limit case of the regularized one. We exploit these geometrical properties to develop an efficient pivoting heuristic, which we compare with implicit or non-linear methods from bilevel programming. These results are illustrated by an application to the optimal pricing of electricity contracts on the French market.Comment: 37 pages, 9 figures; adding a section on the pricing of electricity contract

Revenue maximizing envy-free fixed-price auctions with budgets

Traditional incentive-compatible auctions [6,16] for selling multiple goods to unconstrained and budgeted bidders can discriminate between bidders by selling identical goods at different prices. For this reason, Feldman et al. [7] dropped incentive compatibility and turned the attention to revenue maximizing envy-free item-pricing allocations for budgeted bidders. Envy-free allocations were suggested by classical papers [9,15]. The key property of such allocations is that no one envies the allocation and the price charged to anyone else. In this paper we consider this classical notion of envy-freeness and study fixed-price mechanisms which use nondiscriminatory uniform prices for all goods. Feldman et al. [7] gave an item-pricing mechanism that obtains 1/2 of the revenue obtained from any envy-free fixed-price mechanism for identical goods. We improve over this result by presenting an FPTAS for the problem that returns an (1 − ε)-approximation of the revenue obtained by any envy-free fixed-price mechanism for any ε > 0 and runs in polynomial time in the number of bidders n and 1/ ε even for exponential supply of goods m. Next, we consider the case of budgeted bidders with matching-type preferences on the set of goods, i.e., the valuation of each bidder for each item is either v i or 0. In this more general case, we prove that it is impossible to approximate the optimum revenue within O( min (n,m)1/2 − ε ) for any ε > 0 unless P = NP. On the positive side, we are able to extend the FPTAS for identical goods to budgeted bidders in the case of constant number of different types of goods. Our FPTAS gives also a constant approximation with respect to the general envy-free auction

Lottery pricing equilibria

We extend the notion of Combinatorial Walrasian Equilibrium, as defined by Feldman et al. [2013], to settings with budgets. When agents have budgets, the maximum social welfare as traditionally defined is not a suitable benchmark since it is overly optimistic. This motivated the liquid welfare of [Dobzinski and Paes Leme 2014] as an alternative. Observing that no combinatorial Walrasian equilibrium guarantees a non-zero fraction of the maximum liquid welfare in the absence of randomization, we instead work with randomized allocations and extend the notions of liquid welfare and Combinatorial Walrasian Equilibrium accordingly. Our generalization of the Combinatorial Walrasian Equilibrium prices lotteries over bundles of items rather than bundles, and we term it a lottery pricing equilibrium. Our results are two-fold. First, we exhibit an efficient algorithm which turns a randomized allocation with liquid expected welfare W into a lottery pricing equilibrium with liquid expected welfare 3-√5/2 W (≈ 0.3819-W). Next, given access to a demand oracle and an α-approximate oblivious rounding algorithm for the configuration linear program for the welfare maximization problem, we show how to efficiently compute a randomized allocation which is (a) supported on polynomially-many deterministic allocations and (b) obtains [nearly] an α fraction of the optimal liquid expected welfare. In the case of subadditive valuations, combining both results yields an efficient algorithm which computes a lottery pricing equilibrium obtaining a constant fraction of the optimal liquid expected welfare. © Copyright 2016 ACM

Approximation Algorithms for the Max-Buying Problem with Limited Supply

We consider the Max-Buying Problem with Limited Supply, in which there are $n$ items, with $C_i$ copies of each item $i$, and $m$ bidders such that every bidder $b$ has valuation $v_{ib}$ for item $i$. The goal is to find a pricing $p$ and an allocation of items to bidders that maximizes the profit, where every item is allocated to at most $C_i$ bidders, every bidder receives at most one item and if a bidder $b$ receives item $i$ then $p_i \leq v_{ib}$. Briest and Krysta presented a 2-approximation for this problem and Aggarwal et al. presented a 4-approximation for the Price Ladder variant where the pricing must be non-increasing (that is, $p_1 \geq p_2 \geq \cdots \geq p_n$). We present an $e/(e-1)$-approximation for the Max-Buying Problem with Limited Supply and, for every $\varepsilon > 0$, a $(2+\varepsilon)$-approximation for the Price Ladder variant

Stackelberg Network Pricing is Hard to Approximate

In the Stackelberg Network Pricing problem, one has to assign tariffs to a certain subset of the arcs of a given transportation network. The aim is to maximize the amount paid by the user of the network, knowing that the user will take a shortest st-path once the tariffs are fixed. Roch, Savard, and Marcotte (Networks, Vol. 46(1), 57-67, 2005) proved that this problem is NP-hard, and gave an O(log m)-approximation algorithm, where m denote the number of arcs to be priced. In this note, we show that the problem is also APX-hard