Designing an incentive compatible auction that maximizes expected revenue is an intricate task. The single-item case was resolved in a seminal piece of work by Myerson in 1981. Even after 30--40 years of intense research, the problem remains unsolved for settings with two or more items. We overview recent research results that show how tools from deep learning are shaping up to become a powerful tool for the automated design of near-optimal auctions auctions. In this approach, an auction is modeled as a multilayer neural network, with optimal auction design framed as a constrained learning problem that can be addressed with standard machine learning pipelines. Through this approach, it is possible to recover to a high degree of accuracy essentially all known analytically derived solutions for multi-item settings and obtain novel mechanisms for settings in which the optimal mechanism is unknown.

We consider the classic principal-agent model of contract theory, in which a principal designs an outcome-dependent compensation scheme to incentivize an agent to take a costly and unobservable action. When all of the model parameters-including the full distribution over principal rewards resulting from each agent action-are known to the designer, an optimal contract can in principle be computed by linear programming. In addition to their demanding informational requirements, such optimal contracts are often complex and unintuitive, and do not resemble contracts used in practice.This paper examines contract theory through the theoretical computer science lens, with the goal of developing novel theory to explain and justify the prevalence of relatively simple contracts, such as linear (pure commission) contracts. First, we consider the case where the principal knows only the first moment of each action's reward distribution, and we prove that linear contracts are guaranteed to be worst-case optimal, ranging over all reward distributions consistent with the given moments. Second, we study linear contracts from a worst-case approximation perspective, and prove several tight parameterized approximation bounds.

A central object in optimal stopping theory is the single-choice prophet inequality for independent, identically distributed random variables: Given a sequence of random variables X 1 , . . . , X n drawn independently from a distribution F, the goal is to choose a stopping time τ so as to maximize α such that for all distributions F we haveWhat makes this problem challenging is that the decision whether τ = t may only depend on the values of the random variables X 1 , . . . , X t and on the distribution F. For quite some time the best known bound for the problem was α ≥ 1 − 1/e ≈ 0.632 [Hill and Kertz, 1982]. Only recently this bound was improved by Abolhassani et al. [2017], and a tight bound of α ≈ 0.745 was obtained by Correa et al. [2017].The case where F is unknown, such that the decision whether τ = t may depend only on the values of the first t random variables but not on F, is equally well motivated (e.g., [Azar et al., 2014]) but has received much less attention. A straightforward guarantee for this case of α ≥ 1/e ≈ 0.368 can be derived from the solution to the secretary problem. Our main result is that this bound is tight. Motivated by this impossibility result we investigate the case where the stopping time may additionally depend on a limited number of samples from F. An extension of our main result shows that even with o(n) samples α ≤ 1/e, so that the interesting case is the one with Ω(n) samples. Here we show that n samples allow for a significant improvement over the secretary problem, while O(n 2 ) samples are equivalent to knowledge of the distribution: specifically, with n samples α ≥ 1 − 1/e ≈ 0.632 and α ≤ ln(2) ≈ 0.693, and with O(n 2 ) samples α ≥ 0.745 − ε for any ε > 0.

Designing double auctions is a complex problem, especially when there are restrictions on the sets of buyers and sellers that may trade with one another. The goal of this paper is to develop "black-box reductions" from doubleauction design to the exhaustively-studied problem of designing single-sided mechanisms.We consider several desirable properties of a double auction: feasibility, dominant-strategy incentive-compatibility, the still stronger incentive constraints offered by a deferred-acceptance implementation, exact and approximate welfare maximization, and budget-balance. For each of these properties, we identify sufficient conditions on the two one-sided mechanisms-one for the buyers, one for the sellers-and on the method of composition, that guarantee the desired property of the double auction.Our framework also offers new insights into classic double-auction designs, such as the VCG and McAfee auctions with unit-demand buyers and unit-supply sellers.

Consider a gambler and a prophet who observe a sequence of independent, non-negative numbers. The gambler sees the numbers one-by-one whereas the prophet sees the entire sequence at once. The goal of both is to decide on fractions of each number they want to keep so as to maximize the weighted fractional sum of the numbers chosen.The classic result of Krengel and Sucheston (1977-78) asserts that if both the gambler and the prophet can pick one number, then the gambler can do at least half as well as the prophet. Recently, Kleinberg and Weinberg (2012) have generalized this result to settings where the numbers that can be chosen are subject to a matroid constraint.In this note we go one step further and show that the bound carries over to settings where the fractions that can be chosen are subject to a polymatroid constraint. This bound is tight as it is already tight for the simple setting where the gambler and the prophet can pick only one number. An interesting application of our result is in mechanism design, where it leads to improved results for various problems.

Deferred-acceptance auctions are mechanisms whose allocation rule can be implemented using an adaptive reverse greedy algorithm. Milgrom and Segal recently introduced these auctions and proved that they satisfy remarkable incentive guarantees: in addition to being dominant strategy and incentive compatible, they are weakly group-strategyproof and can be implemented by ascending-clock auctions. Neither forward greedy mechanisms nor the VCG mechanism generally possess any of these additional incentive properties. The goal of this paper is to initiate the study of deferred-acceptance auctions from an approximation standpoint. We study what fraction of the optimal social welfare can be guaranteed by these auctions in two canonical problems, knapsack auctions and combinatorial auctions with single-minded bidders. For knapsack auctions, we prove a separation between deferred-acceptance auctions and arbitrary dominant-strategy incentive-compatible mechanisms. For combinatorial auctions with single-minded bidders, we design novel polynomial-time mechanisms that achieve the best of both worlds: the incentive guarantees of a deferred-acceptance auction, and approximation guarantees close to the best possible.

We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms and is used to derive new and improved results for combinatorial markets (with and without complements), multidimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees.

Designing an incentive compatible auction that maximizes expected revenue is an intricate task. The single-item case was resolved in a seminal piece of work by Myerson in 1981. Even after 30-40 years of intense research the problem remains unsolved for seemingly simple multibidder, multi-item settings. In this work, we initiate the exploration of the use of tools from deep learning for the automated design of optimal auctions. We model an auction as a multi-layer neural network, frame optimal auction design as a constrained learning problem, and show how it can be solved using standard pipelines. We prove generalization bounds and present extensive experiments, recovering essentially all known analytical solutions for multi-item settings, and obtaining novel mechanisms for settings in which the optimal mechanism is unknown.