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 & 

