We consider the problem of computing a (pure) Bayes–Nash equilibrium in the first-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 \(\varepsilon\) -equilibrium of the auction is PPAD-complete, and computing an exact equilibrium is FIXP-complete. We also provide an efficient algorithm for solving a special case of the problem for a fixed number of bidders and available bids.


  1. first-price auctions
  2. Bayes–Nash equilibria
  3. approximate equilibria
  4. subjective priors
  5. PPAD
  6. FIXP

MSC codes

  1. 68Q17
  2. 91A68
  3. 68Q25
  4. 91B26

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Information & Authors


Published In

cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 80 - 131
ISSN (online): 1095-7111


Submitted: 23 July 2021
Accepted: 14 September 2022
Published online: 14 February 2023


  1. first-price auctions
  2. Bayes–Nash equilibria
  3. approximate equilibria
  4. subjective priors
  5. PPAD
  6. FIXP

MSC codes

  1. 68Q17
  2. 91A68
  3. 68Q25
  4. 91B26



School of Informatics, Informatics Forum, University of Edinburgh, Edinburgh EH8 9AB, United Kingdom.
Department of Data Science, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen 91058, Germany.
Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom.
LASIGE, Faculdade de Ciências, Universidade de Lisboa, Lisbon 1749-016, Portugal.

Funding Information

FCT via Lasige Research Unit: UIDB/00408/2020
Funding: Part of this work was done while the second and fifth authors were members of the Operations Research group at Technical University of Munich, School of Management, supported by the Alexander von Humboldt Foundation with funds from the German Federal Ministry of Education and Research (BMBF). The third author was supported by an EPSRC doctoral studentship (reference 1892947). The fourth author was partially supported by the ERC Advanced Grant 788893 AMDROMA, “Algorithmic and Mechanism Design Research in Online Markets,” and MIUR PRIN project ALGADIMAR, “Algorithms, Games, and Digital Markets.” The fifth author was supported by FCT via LASIGE Research Unit, reference UIDB/00408/2020.

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