Equilibrium Learning in Combinatorial Auctions: Computing Approximate Bayesian Nash Equilibria via Pseudogradient Dynamics

Abstract

Applications of combinatorial auctions (CA) as market mechanisms are prevalent in practice, yet their Bayesian Nash equilibria (BNE) remain poorly understood. Analytical solutions are known only for a few cases where the problem can be reformulated as a tractable partial differential equation (PDE). In the general case, finding BNE is known to be computationally hard. Previous work on numerical computation of BNE in auctions has relied either on solving such PDEs explicitly, calculating pointwise best-responses in strategy space, or iteratively solving restricted subgames. In this study, we present a generic yet scalable alternative multi-agent equilibrium learning method that represents strategies as neural networks and applies policy iteration based on gradient dynamics in self-play. Most auctions are ex-post nondifferentiable, so gradients may be unavailable or misleading, and we rely on suitable pseudogradient estimates instead. Although it is well-known that gradient dynamics cannot guarantee convergence to NE in general, we observe fast and robust convergence to approximate BNE in a wide variety of auctions and present a sufficient condition for convergence.

Equilibrium Learning in Auction Markets

“…We presented our findings in several workshop papers. The latest of these (Heidekrüger et al 2021b), also contains a first theoretical result: In monotonic auction games, NPGA provably converges to anapproximation of the unique BNE. This generalizes known results about gradient dynamics in complete-information games (Mertikopoulos and Zhou 2019) and differentiable…”

Learning Revenue-Maximizing Auctions With Differentiable Matching

“…Others have taken a similar approach but applied to different mechanism design problems, including finding welfare-maximizing auctions [49] and facility location [21]. Other applications of gradient-based methods to problems in mechanism design include [25,53].…”