AUCTION LEARNING AS A TWO-PLAYER GAME

Abstract

Designing an incentive compatible auction that maximizes expected revenue is a central problem in Auction Design. While theoretical approaches to the problem have hit some limits, a recent research direction initiated by Duetting et al. (2019) consists in building neural network architectures to find optimal auctions. We propose two conceptual deviations from their approach which result in enhanced performance. First, we use recent results in theoretical auction design to introduce a time-independent Lagrangian. This not only circumvents the need for an expensive hyper-parameter search (as in prior work), but also provides a single metric to compare the performance of two auctions (absent from prior work). Second,the optimization procedure in previous work uses an inner maximization loop to compute optimal misreports. We amortize this process through the introduction of an additional neural network. We demonstrate the effectiveness of our approach by learning competitive or strictly improved auctions compared to prior work. Both results together further imply a novel formulation of Auction Design as a two-player game with stationary utility functions.

1. INTRODUCTION

Efficiently designing truthful auctions is a core problem in Mathematical Economics. Concrete examples include the sponsored search auctions conducted by companies as Google or auctions run on platforms as eBay. Following seminal work of Vickrey (Vickrey, 1961) and Myerson (Myerson, 1981) , auctions are typically studied in the independent private valuations model: each bidder has a valuation function over items, and their payoff depends only on the items they receive. Moreover, the auctioneer knows aggregate information about the population that each bidder comes from, modeled as a distribution over valuation functions, but does not know precisely each bidder's valuation (outside of any information in this Bayesian prior). A major difficulty in designing auctions is that valuations are private and bidders need to be incentivized to report their valuations truthfully. The goal of the auctioneer is to design an incentive compatible auction which maximizes expected revenue. Auction Design has existed as a rigorous mathematical field for several decades and yet, complete characterizations of the optimal auction only exist for a few settings. While Myerson's Nobel prizewinning work provides a clean characterization of the single-item optimum (Myerson, 1981) , optimal multi-item auctions provably suffer from numerous formal measures of intractability (including computational intractability, high description complexity, non-monotonicity, and others) (Daskalakis et al., 2014; Chen et al., 2014; 2015; 2018; Hart & Reny, 2015; Thanassoulis, 2004 ). An orthogonal line of work instead develops deep learning architectures to find the optimal auction. Duetting et al. (2019) initiated this direction by proposing RegretNet, a feed-forward architecture. They frame the auction design problem as a constrained learning problem and lift the constraints into the objective via the augmented Lagrangian method. Training RegretNet involves optimizing this Lagrangian-penalized objective, while simultaneously updating network parameters and the Lagrangian multipliers themselves. This architecture produces impressive results: recovering nearoptimal auctions in several known multi-item settings, and discovering new mechanisms when a theoretical optimum is unknown. Yet, this approach presents several limitations. On the conceptual front, our main insight is a connection to an exciting line of recent works (Hartline & Lucier, 2010; Hartline et al., 2011; Bei & Huang, 2011; Daskalakis & Weinberg, 2012; Rubinstein & Weinberg, 2018; Dughmi et al., 2017; Cai et al., 2019) on ε-truthful-to-truthful reductions.foot_0 On the technical front, we identify three areas for improvement. First, their architecture is difficult to train in practice as the objective is nonstationary. Specifically, the Lagrangian multipliers are time-dependent and they increase following a pre-defined schedule, which requires careful hyperparameter tuning (see §3.1 for experiments illustrating this). Leveraging the aforementioned works in Auction Theory, we propose a stationary Lagrangian objective. Second, all prior work inevitably finds auctions which are not precisely incentive compatible, and does not provide a metric to compare, say, an auction with revenue 1.01 which is 0.002-truthful, or one with revenue 1 which is 0.001-truthful. We argue that our stationary Lagrangian objective serves as a good metric (and that the second auction of our short example is "better" for our metric). Finally, their training procedure requires an inner-loop optimization (essentially, this inner loop is the bidders trying to maximize utility in the current auction), which is itself computationally expensive. We use amortized optimization to make this process more efficient.

CONTRIBUTIONS

This paper leverages recent work in Auction Theory to formulate the learning of revenue-optimal auctions as a two-player game. We develop a new algorithm ALGnet (Auction Learning Game network) that produces competitive or better results compared to Duetting et al. ( 2019)'s RegretNet. In addition to the conceptual contributions, our approach yields the following improvements (as RegretNet is already learning near-optimal auctions, our improvement over RegretNet is not due to significantly higher optimal revenues). -Easier hyper-parameter tuning: By constructing a time-independent loss function, we circumvent the need to search for an adequate parameter scheduling. Our formulation also involves less hyperparameters, which makes it more robust. Furthermore, these technical contributions together now imply a novel formulation of auction learning as a two-player game (not zero-sum) between an auctioneer and a misreporter. The auctioneer is trying to design an incentive compatible auction that maximizes revenue while the misreporter is trying to identify breaches in the truthfulness of these auctions. The paper decomposes as follows. Section 2 introduces the standard notions of auction design. Section 3 presents our game formulation for auction learning. Section 4 provides a description of ALGnet and its training procedure. Finally, Section 5 presents numerical evidence for the effectiveness of our approach.

RELATED WORK

Auction design and machine learning. Machine learning and computational learning theory have been used in several ways to design auctions from samples of bidder valuations. Machine learning has been used to analyze the sample complexity of designing optimal revenue-maximizing auctions. 



By ε-truthful, we mean the expected total regret R is bounded by ε. See Prop. 1 for a definition of R.



-A metric to compare auctions: We propose a metric to compare the quality of two auctions which are not incentive compatible. -More efficient training: We replace the inner-loop optimization of prior work with a neural network, which makes training more efficient. -Online auctions: Since the learning formulation is time-invariant, ALGnet is able to quickly adapt in auctions where the bidders' valuation distributions varies over time. Such setting appears for instance in the online posted pricing problem studied in Bubeck et al. (2017).

Dütting et al., 2015). Our approach is different as we build a deep learning architecture for auction design. Auction design and deep learning. While Duetting et al. (2019) is the first paper to design auctions through deep learning, several other paper followed-up this work. Feng et al. (2018) extended it to budget constrained bidders, Golowich et al. (2018) to the facility location problem. Tacchetti et al. (2019) built architectures based on the Vickrey-Clarke-Groves mechanism. Rahme et al. (2021) used permutation-equivariant networks to design symmetric auctions. Shen et al. (2019) and Duetting

