On the Complexity of Equilibrium Computation in First-Price Auctions

Abstract

We consider the problem of computing a (pure) Bayes-Nash equilibrium in the rst-price auction with continuous value distributions and discrete bidding space. We prove that when bidders have independent subjective prior beliefs about the value distributions of the other bidders, computing an 𝜀-equilibrium of the auction is PPAD-complete, and computing an exact equilibrium is FIXP-complete. * Y. Giannakopoulos was supported by the Alexander von Humboldt Foundation with funds from the German Federal Ministry of Education and Research (BMBF). He was also an associated researcher with the Research Training Group GRK 2201 "Advanced Optimization in a Networked Economy", funded by the German Research Foundation (DFG). A. Hollender was supported by an EPSRC doctoral studentship (Reference 1892947). D. Poc ¸as was supported by FCT via LASIGE Research Unit, ref. UIDB/00408/2020.1 For the o cial Nobel Prize announcements see here, here and here.

Strong Approximations and Irrationality in Financial Networks with Financial Derivatives

“…Our results here are not conclusive in that there is a gap between our necessary and sufficient conditions, the cycles which involve both switched on and switched off nodes being not fully understood. 10 We conjecture that for any network with a weakly switched cycle there exist rational values for assets and liabilities that lead to irrational solution (Appendix C contains one such example); however, our arguments and scheme cannot be easily generalised to those instances. We leave the full characterisation as an open problem.…”

“…A complexity class regarding fixed point computations of total search problems, is the FIXP class defined by Etessami and Yannakakis in [9], where the authors define and introduce the FIXP complexity class and prove that the problem of computing a 3-player (or more) Nash equilibrium is complete for this class. Not many FIXP-complete problems are known in the literature, but some example we are aware of can be found in [13,10,11].…”

A heuristic for estimating Nash equilibria in first-price auctions with correlated values

“…Research on first-price auctions received renewed impetus in 2019 when Google switched its display advertising market place to first-price [16,17]. This led to a surge of interest in new topics such as computational complexity [18], numerical approximation [19] or more recently, the use of neural networks to compute the auction's equilibrium [20].…”