Core-selecting combinatorial auctions are popular auction designs that constrain prices to eliminate the incentive for any group of bidders---with the seller---to renegotiate for a better deal. They help overcome the low-revenue issues of classical combinatorial auctions. We introduce a new class of core-selecting combinatorial auctions that leverage bidder information available to the auction designer. We model such information through constraints on the joint type space of the bidders---these are constraints on bidders' private valuations that are known to hold by the auction designer before bids are elicited. First, we show that type space information can overcome the well-known impossibility of incentive-compatible core-selecting combinatorial auctions. We present a revised and generalized version of that impossibility result that depends on how much information is conveyed by the type spaces. We then devise a new family of core-selecting combinatorial auctions and show that they minimize the sum of bidders' incentives to deviate from truthful bidding. We develop new constraint generation techniques---and build upon existing quadratic programming techniques---to compute core prices, and conduct experiments to evaluate the incentive, revenue, fairness, and computational merits of our new auctions. Our new core-selecting auctions directly improve upon existing designs that have been used in many high-stakes auctions around the world. We envision that they will be a useful addition to any auction designer's toolkit.

The design of multi-item, multi-bidder auctions involves a delicate balancing act of economic objectives, bidder incentives, and real-world complexities. Efficient auctions, that is, auctions that allocate items to maximize total bidder value, are practically desirable since they promote the most economically beneficial use of resources. Arguably the biggest drawback of efficient auctions, however, is their potential to generate very low revenue. In this work, we show how the auction designer can artificially inject competition into the auction to boost revenue while striving to maintain efficiency. First, we invent a new auction family that enables the auction designer to specify competition in a precise, expressive, and interpretable way. We then introduce a new model of bidder behavior and individual rationality to understand how bidders act when prices are too competitive. Next, under our bidder behavior model, we use our new competitive auction class to derive the globally revenue-optimal efficient auction under two different knowledge models for the auction designer: knowledge of full bidder value distributions and knowledge of bidder value quantiles. Finally, we study a third knowledge model for the auction designer: knowledge of historical bidder valuation data. In this setting we present sample and computationally efficient learning algorithms that find high-revenue probably-efficient competitive auctions from bidder data. Our learning algorithms are instance adaptive and can be run in parallel across bidders, unlike most prior approaches to data-driven auction design.

Sequence-independent lifting is a procedure for strengthening valid inequalities of an integer program. We generalize the sequence-independent lifting method of Gu, Nemhauser, and Savelsbergh (GNS lifting) for cover inequalities and correct an error in their proposed generalization. We obtain a new sequence-independent lifting technique---piecewise-constant (PC) lifting---with a number of important properties. We derive a broad set of sufficient conditions under which PC lifting yields facets---the first characterization of facet-defining sequence-independent liftings that are efficiently computable from the underlying cover. Finally, we demonstrate via experiments that PC lifting can be a useful alternative to GNS lifting. We test PC lifting atop a number of novel cover inequality generation routines, which prove to be effective in experiments with CPLEX. PC lifting delivers strong numerical properties making it practically relevant for integer programming solvers.

We develop a versatile new methodology for multidimensional mechanism design that incorporates side information about agent types to generate high social welfare and high revenue simultaneously. Prominent sources of side information in practice include predictions from a machine-learning model trained on historical agent data, advice from domain experts, and even the mechanism designer's own gut instinct. In this paper we adopt a prior-free perspective that makes no assumptions on the correctness, accuracy, or source of the side information. First, we design a meta-mechanism that integrates input side information with an improvement of the classical VCG mechanism. The welfare, revenue, and incentive properties of our meta-mechanism are characterized by novel constructions we introduce based on the notion of a weakest competitor, which is an agent that has the smallest impact on welfare. We show that our meta-mechanism, when carefully instantiated, simultaneously achieves strong welfare and revenue guarantees parameterized by errors in the side information. When the side information is highly informative and accurate, our mechanism achieves welfare and revenue competitive with the total social surplus, and its performance decays continuously and gradually as the quality of the side information decreases. Finally, we apply our meta-mechanism to a setting where each agent's type is determined by a constant number of parameters. Specifically, agent types lie on constant-dimensional subspaces (of the potentially high-dimensional ambient type space) that are known to the mechanism designer. We use our meta-mechanism to obtain the first known welfare and revenue guarantees in this setting.

Cutting plane methods play a significant role in modern solvers for tackling mixed-integer programming (MIP) problems. Proper selection of cuts would remove infeasible solutions in the early stage, thus largely reducing the computational burden without hurting the solution accuracy. However, the major cut selection approaches heavily rely on heuristics, which strongly depend on the specific problem at hand and thus limit their generalization capability. In this paper, we propose a data-driven and generalizable cut selection approach, named Cut Ranking, in the settings of multiple instance learning. To measure the quality of the candidate cuts, a scoring function, which takes the instance-specific cut features as inputs, is trained and applied in cut ranking and selection. In order to evaluate our method, we conduct extensive experiments on both synthetic datasets and real-world datasets. Compared with commonly used heuristics for cut selection, the learning-based policy has shown to be more effective, and is capable of generalizing over multiple problems with different properties. Cut Ranking has been deployed in an industrial solver for large-scale MIPs. In the online A/B testing of the product planning problems with more than $10^7$ variables and constraints daily, Cut Ranking has achieved the average speedup ratio of 12.42% over the production solver without any accuracy loss of solution.

A shrinking market is a ubiquitous challenge faced by various industries. In this paper we formulate the first formal model of shrinking markets in multi-item settings, and study how mechanism design and machine learning can help preserve revenue in an uncertain, shrinking market. Via a sample-based learning mechanism, we prove the first guarantees on how much revenue can be preserved by truthful multi-item, multi-bidder auctions (for limited supply) when only a random unknown fraction of the population participates in the market. We first present a general reduction that converts any sufficiently rich auction class into a randomized auction robust to market shrinkage. Our main technique is a novel combinatorial construction called a winner diagram that concisely represents all possible executions of an auction on an uncertain set of bidders. Via a probabilistic analysis of winner diagrams, we derive a general possibility result: a sufficiently rich class of auctions always contains an auction that is robust to market shrinkage and market uncertainty. Our result has applications to important practically-constrained settings such as auctions with a limited number of winners. We then show how to efficiently learn an auction that is robust to market shrinkage by leveraging practically-efficient routines for solving the winner determination problem.

The incorporation of cutting planes within the branch-and-bound algorithm, known as branch-and-cut, forms the backbone of modern integer programming solvers. These solvers are the foremost method for solving discrete optimization problems and thus have a vast array of applications in machine learning, operations research, and many other fields. Choosing cutting planes effectively is a major research topic in the theory and practice of integer programming. We conduct a novel structural analysis of branch-and-cut that pins down how every step of the algorithm is affected by changes in the parameters defining the cutting planes added to the input integer program. Our main application of this analysis is to derive sample complexity guarantees for using machine learning to determine which cutting planes to apply during branch-and-cut. These guarantees apply to infinite families of cutting planes, such as the family of Gomory mixed integer cuts, which are responsible for the main breakthrough speedups of integer programming solvers. We exploit geometric and combinatorial structure of branch-and-cut in our analysis, which provides a key missing piece for the recent generalization theory of branch-and-cut.

We develop a new framework for designing truthful, high-revenue (combinatorial) auctions for limited supply. Our mechanism learns within an instance. It generalizes and improves over previously-studied random-sampling mechanisms. It first samples a participatory group of bidders, then samples several learning groups of bidders from the remaining pool of bidders, learns a high-revenue auction from the learning groups, and finally runs that auction on the participatory group. Previous work on random-sampling mechanisms focused primarily on unlimited supply. Limited supply poses additional significant technical challenges, since allocations of items to bidders must be feasible. We prove guarantees on the performance of our mechanism based on a market-shrinkage term and a new complexity measure we coin partition discrepancy. Partition discrepancy simultaneously measures the intrinsic complexity of the mechanism class and the uniformity of the set of bidders. We then introduce new auction classes that can be parameterized in a way that does not depend on the number of bidders participating, and prove strong guarantees for these classes. We show how our mechanism can be implemented efficiently by leveraging practically-efficient routines for solving winner determination. Finally, we show how to use structural revenue maximization to decide what auction class to use with our framework when there is a constraint on the number of learning groups.

Cutting-plane methods have enabled remarkable successes in integer programming over the last few decades. State-of-the-art solvers integrate a myriad of cutting-plane techniques to speed up the underlying tree-search algorithm used to find optimal solutions. In this paper we provide sample complexity bounds for cut-selection in branch-and-cut (B&C). Given a training set of integer programs sampled from an application-specific input distribution and a family of cut selection policies, these guarantees bound the number of samples sufficient to ensure that using any policy in the family, the size of the tree B&C builds on average over the training set is close to the expected size of the tree B&C builds. We first bound the sample complexity of learning cutting planes from the canonical family of Chvátal-Gomory cuts. Our bounds handle any number of waves of any number of cuts and are fine tuned to the magnitudes of the constraint coefficients. Next, we prove sample complexity bounds for more sophisticated cut selection policies that use a combination of scoring rules to choose from a family of cuts. Finally, beyond the realm of cutting planes for integer programming, we develop a general abstraction of tree search that captures key components such as node selection and variable selection. For this abstraction, we bound the sample complexity of learning a good policy for building the search tree.

A two-part tariff is a pricing scheme that consists of an up-front lump sum fee and a per unit fee. Various products in the real world are sold via a menu, or list, of two-part tariffs---for example gym memberships, cell phone data plans, etc. We study learning high-revenue menus of two-part tariffs from buyer valuation data, in the setting where the mechanism designer has access to samples from the distribution over buyers' values rather than an explicit description thereof. Our algorithms have clear direct uses, and provide the missing piece for the recent generalization theory of two-part tariffs. We present a polynomial time algorithm for optimizing one two-part tariff. We also present an algorithm for optimizing a length-L menu of two-part tariffs with run time exponential in L but polynomial in all other problem parameters. We then generalize the problem to multiple markets. We prove how many samples suffice to guarantee that a two-part tariff scheme that is feasible on the samples is also feasible on a new problem instance with high probability. We then show that computing revenue-maximizing feasible prices is hard even for buyers with additive valuations. Then, for buyers with identical valuation distributions, we present a condition that is sufficient for the two-part tariff scheme from the unsegmented setting to be optimal for the market-segmented setting. Finally, we prove a generalization result that states how many samples suffice so that we can compute the unsegmented solution on the samples and still be guaranteed that we get a near-optimal solution for the market-segmented setting with high probability.

We consider active learning under incentive compatibility constraints. The main application of our results is to economic experiments, in which a learner seeks to infer the parameters of a subject’s preferences: for example their attitudes towards risk, or their beliefs over uncertain events. By cleverly adapting the experimental design, one can save on the time spent by subjects in the laboratory, or maximize the information obtained from each subject in a given laboratory session; but the resulting adaptive design raises complications due to incentive compatibility. A subject in the lab may answer questions strategically, and not truthfully, so as to steer subsequent questions in a profitable direction. We analyze two standard economic problems: inference of preferences over risk from multiple price lists, and belief elicitation in experiments on choice over uncertainty. In the first setting, we tune a simple and fast learning algorithm to retain certain incentive compatibility properties. In the second setting, we provide an incentive compatible learning algorithm based on scoring rules with query complexity that differs from obvious methods of achieving fast learning rates only by subpolynomial factors. Thus, for these areas of application, incentive compatibility may be achieved without paying a large sample complexity price.

We explore questions dealing with the learnability of models of choice over time. We present a large class of preference models defined by a structural criterion for which we are able to obtain an exponential improvement over previously known learning bounds for more general preference models. This in particular implies that the three most important discounted utility models of intertemporal choice - exponential, hyperbolic, and quasi-hyperbolic discounting - are learnable in the PAC setting with VC dimension that grows logarithmically in the number of time periods. We also examine these models in the framework of active learning. We find that the commonly studied stream-based setting is in general difficult to analyze for preference models, but we provide a redeeming situation in which the learner can indeed improve upon the guarantees provided by PAC learning. In contrast to the stream-based setting, we show that if the learner is given full power over the data he learns from - in the form of learning via membership queries - even very naive algorithms significantly outperform the guarantees provided by higher level active learning algorithms.