Item Pricing for Revenue Maximization in Combinatorial Auctions

Consider the problem of a retailer with various goods for sale, attempting to set prices to maximize revenue. If customers have separate valuations over the different goods, and these are known to the retailer, then the goods can be priced separately and the problem is not so difficult. However, when customers have valuations over sets of items, this becomes a combinatorial auction problem, and the problem becomes computationally hard even when valuations are fully known in advance. In this talk we present some simple randomized algorithms and mechanisms for a number of interesting cases of this problem, both in the limited and unlimited supply setting. This talk is based on joint work with Avrim Blum and Yishay Mansour

Approximately Optimal Mechanism Design: Motivation, Examples, and Lessons Learned

Optimal mechanism design enjoys a beautiful and well-developed theory, and also a number of killer applications. Rules of thumb produced by the field influence everything from how governments sell wireless spectrum licenses to how the major search engines auction off online advertising. There are, however, some basic problems for which the traditional optimal mechanism design approach is ill-suited --- either because it makes overly strong assumptions, or because it advocates overly complex designs. The thesis of this paper is that approximately optimal mechanisms allow us to reason about fundamental questions that seem out of reach of the traditional theory. This survey has three main parts. The first part describes the approximately optimal mechanism design paradigm --- how it works, and what we aim to learn by applying it. The second and third parts of the survey cover two case studies, where we instantiate the general design paradigm to investigate two basic questions. In the first example, we consider revenue maximization in a single-item auction with heterogeneous bidders. Our goal is to understand if complexity --- in the sense of detailed distributional knowledge --- is an essential feature of good auctions for this problem, or alternatively if there are simpler auctions that are near-optimal. The second example considers welfare maximization with multiple items. Our goal here is similar in spirit: when is complexity --- in the form of high-dimensional bid spaces --- an essential feature of every auction that guarantees reasonable welfare? Are there interesting cases where low-dimensional bid spaces suffice?Comment: Based on a talk given by the author at the 15th ACM Conference on Economics and Computation (EC), June 201

Welfare and Revenue Guarantees for Competitive Bundling Equilibrium

We study equilibria of markets with $m$ heterogeneous indivisible goods and $n$ consumers with combinatorial preferences. It is well known that a competitive equilibrium is not guaranteed to exist when valuations are not gross substitutes. Given the widespread use of bundling in real-life markets, we study its role as a stabilizing and coordinating device by considering the notion of \emph{competitive bundling equilibrium}: a competitive equilibrium over the market induced by partitioning the goods for sale into fixed bundles. Compared to other equilibrium concepts involving bundles, this notion has the advantage of simulatneous succinctness ($O(m)$ prices) and market clearance. Our first set of results concern welfare guarantees. We show that in markets where consumers care only about the number of goods they receive (known as multi-unit or homogeneous markets), even in the presence of complementarities, there always exists a competitive bundling equilibrium that guarantees a logarithmic fraction of the optimal welfare, and this guarantee is tight. We also establish non-trivial welfare guarantees for general markets, two-consumer markets, and markets where the consumer valuations are additive up to a fixed budget (budget-additive). Our second set of results concern revenue guarantees. Motivated by the fact that the revenue extracted in a standard competitive equilibrium may be zero (even with simple unit-demand consumers), we show that for natural subclasses of gross substitutes valuations, there always exists a competitive bundling equilibrium that extracts a logarithmic fraction of the optimal welfare, and this guarantee is tight. The notion of competitive bundling equilibrium can thus be useful even in markets which possess a standard competitive equilibrium

A General Theory of Sample Complexity for Multi-Item Profit Maximization

The design of profit-maximizing multi-item mechanisms is a notoriously challenging problem with tremendous real-world impact. The mechanism designer's goal is to field a mechanism with high expected profit on the distribution over buyers' values. Unfortunately, if the set of mechanisms he optimizes over is complex, a mechanism may have high empirical profit over a small set of samples but low expected profit. This raises the question, how many samples are sufficient to ensure that the empirically optimal mechanism is nearly optimal in expectation? We uncover structure shared by a myriad of pricing, auction, and lottery mechanisms that allows us to prove strong sample complexity bounds: for any set of buyers' values, profit is a piecewise linear function of the mechanism's parameters. We prove new bounds for mechanism classes not yet studied in the sample-based mechanism design literature and match or improve over the best known guarantees for many classes. The profit functions we study are significantly different from well-understood functions in machine learning, so our analysis requires a sharp understanding of the interplay between mechanism parameters and buyer values. We strengthen our main results with data-dependent bounds when the distribution over buyers' values is "well-behaved." Finally, we investigate a fundamental tradeoff in sample-based mechanism design: complex mechanisms often have higher profit than simple mechanisms, but more samples are required to ensure that empirical and expected profit are close. We provide techniques for optimizing this tradeoff

Pricing Ad Slots with Consecutive Multi-unit Demand

We consider the optimal pricing problem for a model of the rich media advertisement market, as well as other related applications. In this market, there are multiple buyers (advertisers), and items (slots) that are arranged in a line such as a banner on a website. Each buyer desires a particular number of {\em consecutive} slots and has a per-unit-quality value $v_i$ (dependent on the ad only) while each slot $j$ has a quality $q_j$ (dependent on the position only such as click-through rate in position auctions). Hence, the valuation of the buyer $i$ for item $j$ is $v_iq_j$. We want to decide the allocations and the prices in order to maximize the total revenue of the market maker. A key difference from the traditional position auction is the advertiser's requirement of a fixed number of consecutive slots. Consecutive slots may be needed for a large size rich media ad. We study three major pricing mechanisms, the Bayesian pricing model, the maximum revenue market equilibrium model and an envy-free solution model. Under the Bayesian model, we design a polynomial time computable truthful mechanism which is optimum in revenue. For the market equilibrium paradigm, we find a polynomial time algorithm to obtain the maximum revenue market equilibrium solution. In envy-free settings, an optimal solution is presented when the buyers have the same demand for the number of consecutive slots. We conduct a simulation that compares the revenues from the above schemes and gives convincing results.Comment: 27page

Fast Iterative Combinatorial Auctions via Bayesian Learning

Iterative combinatorial auctions (CAs) are often used in multi-billion dollar domains like spectrum auctions, and speed of convergence is one of the crucial factors behind the choice of a specific design for practical applications. To achieve fast convergence, current CAs require careful tuning of the price update rule to balance convergence speed and allocative efficiency. Brero and Lahaie (2018) recently introduced a Bayesian iterative auction design for settings with single-minded bidders. The Bayesian approach allowed them to incorporate prior knowledge into the price update algorithm, reducing the number of rounds to convergence with minimal parameter tuning. In this paper, we generalize their work to settings with no restrictions on bidder valuations. We introduce a new Bayesian CA design for this general setting which uses Monte Carlo Expectation Maximization to update prices at each round of the auction. We evaluate our approach via simulations on CATS instances. Our results show that our Bayesian CA outperforms even a highly optimized benchmark in terms of clearing percentage and convergence speed.Comment: 9 pages, 2 figures, AAAI-1