Equilibrium Computation in First-Price Auctions with Correlated Priors

TL;DR

This paper investigates the computational complexity of finding Bayes-Nash equilibria in first-price auctions with correlated priors, establishing NP-hardness in discrete cases and proposing approximation algorithms for certain settings.

Contribution

It provides the first NP-hardness proof for pure equilibrium existence without prior subjectivity assumptions and introduces algorithms for approximate equilibrium computation.

Findings

NP-hardness of pure equilibrium existence in discrete settings

Polynomial-time approximation algorithms for symmetric and fixed-bidder auctions

Two approaches: bid sparsification and bid densification

Abstract

We consider the computational complexity of computing Bayes-Nash equilibria in first-price auctions, where the bidders' values for the item are drawn from a general (possibly correlated) joint distribution. We show that when the values and the bidding space are discrete, determining the existence of a pure Bayes-Nash equilibrium is NP-hard. This is the first hardness result in the literature of the problem that does not rely on assumptions of subjectivity of the priors, or convoluted tie-breaking rules. We then present two main approaches for achieving positive results, via bid sparsification and via bid densification. The former is more combinatorial and is based on enumeration techniques, whereas the latter makes use of the continuous theory of the problem developed in the economics literature. Using these approaches, we develop polynomial-time approximation algorithms for computing equilibria in symmetric settings or settings with a fixed number of bidders, for different (discrete or continuous) variants of the auction.