Computational complexity of mixed Bayes–Nash equilibria in independent-private-value first-price auctions

Determine the computational complexity of computing mixed Bayes–Nash equilibria in discrete first-price auctions with independent private values.

Background

The paper studies the computational complexity of approximate mixed Bayes–Nash equilibria in discrete first-price auctions when bidders’ values are drawn from a correlated joint prior. Its main result establishes PPAD-completeness for that correlated-prior setting. The independent-private-value (IPV) setting is identified as a more fundamental case in which each bidder’s value is privately drawn independently, but the cited literature had not determined the complexity of computing equilibria.

This problem remains unresolved in the paper: the authors use the IPV case as motivation and context for studying correlated priors, without proving a complexity classification for IPV auctions. The open issue concerns the computational problem of finding mixed Bayes–Nash equilibria, rather than the existence or characterization of equilibria in the auction-theoretic sense.