Learning Equilibria in Asymmetric Auction Games

Abstract

Computing Bayesian Nash equilibrium strategies in auction games is a challenging problem that is not well-understood. Such equilibria can be modeled as systems of nonlinear partial differential equations. It was recently shown that neural pseudogradient ascent (NPGA), an implementation of simultaneous gradient ascent via neural networks, converges to a Bayesian Nash equilibrium for a wide variety of symmetric auction games. Whereas symmetric auction models are widespread in the theoretical literature, in most auction markets in the field, one can observe different classes of bidders having different valuation distributions and strategies. Asymmetry of this sort is almost always an issue in real-world multiobject auctions, in which different bidders are interested in different packages of items. Such environments require a different implementation of NPGA with multiple interacting neural networks having multiple outputs for the different allocations in which the bidders are interested. In this paper, we analyze a wide variety of asymmetric auction models. Interestingly, our results show that we closely approximate Bayesian Nash equilibria in all models in which the analytical Bayes–Nash equilibrium is known. Additionally, we analyze new and larger environments for which no analytical solution is known and verify that the solution found approximates equilibrium closely. The results provide a foundation for generic equilibrium solvers that can be used in a wide range of auction games. History: Accepted by Ram Ramesh, Area Editor for Data Science & Machine Learning. Funding: This work was supported by Deutsche Forschungsgemeinschaft [Grant BI-1056/I-9]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.1281 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0151 ) at ( http://dx.doi.org/10.5281/zenodo.7407158 ).