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

A Scalable Neural Network for DSIC Affine Maximizer Auction Design

Automated auction design aims to find empirically high-revenue mechanisms through machine learning. Existing works on multi item auction scenarios can be roughly divided into RegretNet-like and affine maximizer auctions (AMAs) approaches. However, the former cannot strictly ensure dominant strategy incentive compatibility (DSIC), while the latter faces scalability issue due to the large number of allocation candidates. To address these limitations, we propose AMenuNet, a scalable neural network that constructs the AMA parameters (even including the allocation menu) from bidder and item representations. AMenuNet is always DSIC and individually rational (IR) due to the properties of AMAs, and it enhances scalability by generating candidate allocations through a neural network. Additionally, AMenuNet is permutation equivariant, and its number of parameters is independent of auction scale. We conduct extensive experiments to demonstrate that AMenuNet outperforms strong baselines in both contextual and non-contextual multi-item auctions, scales well to larger auctions, generalizes well to different settings, and identifies useful deterministic allocations. Overall, our proposed approach offers an effective solution to automated DSIC auction design, with improved scalability and strong revenue performance in various settings.Comment: NeurIPS 2023 (spotlight

Learning and Robustness With Applications To Mechanism Design

The design of economic mechanisms, especially auctions, is an increasingly important part of the modern economy. A particularly important property for a mechanism is strategyproofness -- the mechanism must be robust to strategic manipulations so that the participants in the mechanism have no incentive to lie. Yet in the important case when the mechanism designer's goal is to maximize their own revenue, the design of optimal strategyproof mechanisms has proved immensely difficult, with very little progress after decades of research. Recently, to escape this impasse, a number of works have parameterized auction mechanisms as deep neural networks, and used gradient descent to successfully learn approximately optimal and approximately strategyproof mechanisms. We present several improvements on these techniques. When an auction mechanism is represented as a neural network mapping bids from outcomes, strategyproofness can be thought of as a type of adversarial robustness. Making this connection explicit, we design a modified architecture for learning auctions which is amenable to integer-programming-based certification techniques from the adversarial robustness literature. Existing baselines are empirically strategyproof, but with no way to be certain how strong that guarantee really is. By contrast, we are able to provide perfectly tight bounds on the degree to which strategyproofness is violated at any given point. Existing neural networks for auctions learn to maximize revenue subject to strategyproofness. Yet in many auctions, fairness is also an important concern -- in particular, fairness with respect to the items in the auction, which may represent, for instance, ad impressions for different protected demographic groups. With our new architecture, ProportionNet, we impose fairness constraints in addition to the strategyproofness constraints, and find approximately fair, approximately optimal mechanisms which outperform baselines. With PreferenceNet, we extend this approach to notions of fairness that are learned from possibly vague human preferences. Existing network architectures can represent additive and unit-demand auctions, but are unable to imposing more complex exactly-k constraints on the allocations made to the bidders. By using the Sinkhorn algorithm to add differentiable matching constraints, we produce a network which can represent valid allocations in such settings. Finally, we present a new auction architecture which is a differentiable version of affine maximizer auctions, modified to offer lotteries in order to potentially increase revenue. This architecture is always perfectly strategyproof (avoiding the Lagrangian-based constrained optimization of RegretNet) -- to achieve this goal, however, we need to accept that we cannot in general represent the optimal auction