research-article

∃ℝ-complete Decision Problems about (Symmetric) Nash Equilibria in (Symmetric) Multi-player Games

Authors:

M. MavronicolasAuthors Info & Claims

ACM Transactions on Economics and Computation (TEAC), Volume 9, Issue 3

Article No.: 14, Pages 1 - 25

https://doi.org/10.1145/3456758

Published: 29 May 2021 Publication History

Abstract

A decision problem about Nash equilibria is concerned with whether a given game has a Nash equilibrium with certain natural properties. We settle the complexity of such decision problems over multi-player games, establishing that (nearly) all decision problems that were before shown NP-complete over 2-player games in References [5, 12, 18] become ∃ℝ-complete over multi-player games. ∃ℝ [27] is the class capturing the complexity of deciding the Existential Theory of the Reals. Specifically, we present a simple, unifying, parametric polynomial time reduction from the problem of deciding, given a 3-player (symmetric) game, whether there is a (symmetric) Nash equilibrium where all strategies played with non-zero probability come from a given set, which was shown ∃ℝ-complete in Reference [17]. By suitably working on the tuning parameters, our reduction delivers two extensive catalogs of ∃ℝ-complete decision problems in multi-player games. The first addresses Nash equilibria in general games, while the second encompasses symmetric Nash equilibria in symmetric games. These results resolve completely the main open problem from Reference [17] to enlarge the class of ∃ℝ-complete decision problems about (symmetric) Nash equilibria in multi-player (symmetric) games.

References

[1]

Per Austrin, Mark Braverman, and Eden Chlamtac. 2013. Inapproximability of -complete variants of Nash equilibrium. Theor. Comput. 9, 3 (Jan. 2013), 117–142.

[2]

Vittorio Bilò and Marios Mavronicolas. 2014. Complexity of rational and irrational Nash equilibria. Theor. Comput. Syst. 54, 3 (Apr. 2014), 491–527.

Digital Library

[3]

Vittorio Bilò and Marios Mavronicolas. 2016. A catalog of -complete decision problems about Nash equilibria in multi-player games. In Proceedings of the 33rd International Symposium on Theoretical Aspects of Computer Science (LIPIcs), Vol. 47. Schloss Dagstuhl, Leibniz Zentrum fuer Informatik, 17:1–17:13.

[4]

Vittorio Bilò and Marios Mavronicolas. 2017. -complete decision problems about symmetric Nash equilibria in symmetric multi-player games. In Proceedings of the 34th International Symposium on Theoretical Aspects of Computer Science (LIPIcs), Vol. 66. Schloss Dagstuhl, Leibniz Zentrum fuer Informatik, 13:1–13:14.

[5]

Vittorio Bilò and Marios Mavronicolas. 2020. The complexity of computational problems about Nash equilibria in symmetric win-lose games. Algorithmica (Sept. 2020).

[6]

Vincenzo Bonifaci, Ugo Di Orio, and Luigi Laura. 2008. The complexity of uniform Nash equilibria and related subgraph problems. Theor. Comput. Sci. 401, 1–3 (July 2008), 144–152.

Digital Library

[7]

Mark Braverman, Young Kun-Ko, and Omri Weinstein. 2015. Approximating the best Nash equilibrium in O(no(lgn)) time breaks the exponential time hypothesis. In Proceedings of the 26th ACM-SIAM Symposium on Discrete Algorithms. ACM Press, 970–982.

[8]

George W. Brown and John von Neumann. 1950. Solutions of games by differential equations. Contrib. Theor. Games, Ann. Math. Stud.24 (1950), 73–79.

[9]

John Canny. 1988. Some algebraic and geometric computations in . In Proceedings of the 20th ACM Symposium on Theory of Computing. ACM Press, 460–467.

[10]

Xi Chen, Xiaotie Deng, and Shang-Hua Teng. 2009. Settling the complexity of computing two-player Nash equilibria. J. ACM 56, 3 (2009), 14:1–14:57.

Digital Library

[11]

Bruno Codenotti and Daniel Štefankovič. 2005. On the computational complexity of Nash equilibria for (0, 1) bimatrix games. Inform. Proc. Lett. 94, 3 (May 2005), 145–150.

[12]

Vincent Conitzer and Tuomas Sandholm. 2008. New complexity results about Nash equilibria. Games Econ. Behav. 63, 2 (July 2008), 621–641.

[13]

Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou. 2009. The complexity of computing a Nash equilibrium. SIAM J. Comput. 39, 1 (2009), 195–259.

Digital Library

[14]

Argyrios Deligkas, John Fearnley, and Rahul Savani. 2018. Inapproximability results for constrained approximate Nash equilibria. Inf. Comput. 262 (2018), 40–56.

[15]

Kousha Etessami and Mihalis Yannakakis. 2010. On the complexity of Nash equilibria and other fixed points. SIAM J. Comput. 39, 6 (2010), 2531–2597.

Digital Library

[16]

Michael J. Garey and David S. Johnson. 1979. Computers and Intractability — A Guide to the Theory of -Completeness. W. H. Freeman.

Digital Library

[17]

Jugal Garg, Ruta Mehta, Vijay V. Vazirani, and Sadra Yazdanbod. 2018. -completeness for decision versions of multi-player (symmetric) Nash equilibria. ACM Trans. Econ. Comput. 6, 1 (2018), 1:1–1:23.

Digital Library

[18]

Itzhak Gilboa and Eitan Zemel. 1989. Nash and correlated equilibria: Somel. Games Econ. Behav. 1, 1 (Mar. 1989), 80–93.

[19]

Kristoffer A. Hansen. 2017. The real computational complexity of minmax value and equilibrium refinements in multi-player games. In Proceedings of the 10th International Symposium on Algorithmic Game Theory (Lecture Notes in Computer Science), Vol. 10504. Springer, 119–130.

Digital Library

[20]

Elad Hazan and Robert Kraugthgamer. 2011. How hard is it to approximate the best Nash equilibrium? SIAM J. Comput. 40, 1 (2011), 79–91.

Digital Library

[21]

Nicolai E. Mnëv. 1988. The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In Topology and Geometry – Rohlin Seminar (Lecture Notes in Mathematics), Vol. 1346. Springer, 527–543.

[22]

John F. Nash. 1950. Equilibrium points in -person games. Proc. Nat. Acad. Sci. Unit. States Amer. 36 (1950), 48–49.

[23]

John F. Nash. 1951. Non-cooperative games. Ann. Math. 54, 2 (1951), 286–295.

[24]

Christos H. Papadimitriou. 1994. On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci. 48, 3 (Aug. 1994), 498–532.

Digital Library

[25]

Marcus Schaefer. 2010. Complexity of some geometric and topological problems. In Proceedings of the 17th International Symposium on Graph Drawing (Lecture Notes in Computer Science), Vol. 5849. Springer-Verlag, 334–344.

Digital Library

[26]

Marcus Schaefer. 2013. Realizability of graphs and linkages. In Thirty Essays on Geometric Graph Theory. Springer, 461–482.

[27]

Marcus Schaefer and Daniel Štefankovič. 2017. Fixed points, Nash equilibria and the existential theory of the reals. Theor. Comput. Syst. 60, 2 (Feb. 2017), 172–193.

Digital Library

[28]

Alfred Tarski. 1948. A Decision Method for Elementary Algebra and Geometry. RAND Corporation.

Cited By

Kapron BSamieefar K(2025)The Computational Complexity of Equilibria with Strategic ConstraintsSOFSEM 2025: Theory and Practice of Computer Science10.1007/978-3-031-82697-9_9(112-127)Online publication date: 21-Jan-2025
https://dl.acm.org/doi/10.1007/978-3-031-82697-9_9
Bilò VHansen KMavronicolas M(2023)Computational Complexity of Decision Problems About Nash Equilibria in Win-Lose Multi-player GamesAlgorithmic Game Theory10.1007/978-3-031-43254-5_3(40-57)Online publication date: 4-Sep-2023
https://dl.acm.org/doi/10.1007/978-3-031-43254-5_3
Berthelsen MHansen K(2022)On the Computational Complexity of Decision Problems About Multi-player Nash EquilibriaTheory of Computing Systems10.1007/s00224-022-10080-166:3(519-545)Online publication date: 1-Jun-2022
https://dl.acm.org/doi/10.1007/s00224-022-10080-1

Index Terms

∃ℝ-complete Decision Problems about (Symmetric) Nash Equilibria in (Symmetric) Multi-player Games
1. Theory of computation
  1. Computational complexity and cryptography
    1. Complexity classes
    2. Problems, reductions and completeness

Recommendations

∃R-Completeness for Decision Versions of Multi-Player (Symmetric) Nash Equilibria

As a result of a series of important works [7--9, 15, 23], the complexity of two-player Nash equilibrium is by now well understood, even when equilibria with special properties are desired and when the game is symmetric. However, for multi-player games, ...
Multi-player Approximate Nash Equilibria
AAMAS '17: Proceedings of the 16th Conference on Autonomous Agents and MultiAgent Systems

In this paper we study the complexity of finding approximate Nash equilibria in multi-player normal-form games. First, for any constant number m, we present a polynomial-time algorithm for computing a relative (1 - 1(1+(m-1)^m))-Nash equilibrium in ...
On the Computational Complexity of Decision Problems About Multi-player Nash Equilibria
Abstract
We study the computational complexity of decision problems about Nash equilibria in m-player games. Several such problems have recently been shown to be computationally equivalent to the decision problem for the existential theory of the reals, or ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Economics and Computation

ACM Transactions on Economics and Computation Volume 9, Issue 3

September 2021

181 pages

ISSN:2167-8375

EISSN:2167-8383

DOI:10.1145/3468852

Editors:
David Pennock
Microsoft, USA
,
Ilya Segal
Stanford University, USA

Issue’s Table of Contents

Copyright © 2021 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 29 May 2021

Accepted: 01 December 2020

Revised: 01 August 2020

Received: 01 January 2018

Published in TEAC Volume 9, Issue 3

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

3
Total Citations
View Citations
102
Total Downloads

Downloads (Last 12 months)15
Downloads (Last 6 weeks)1

Reflects downloads up to 05 Mar 2025

Other Metrics

View Author Metrics

Citations

Cited By

Kapron BSamieefar K(2025)The Computational Complexity of Equilibria with Strategic ConstraintsSOFSEM 2025: Theory and Practice of Computer Science10.1007/978-3-031-82697-9_9(112-127)Online publication date: 21-Jan-2025
https://dl.acm.org/doi/10.1007/978-3-031-82697-9_9
Bilò VHansen KMavronicolas M(2023)Computational Complexity of Decision Problems About Nash Equilibria in Win-Lose Multi-player GamesAlgorithmic Game Theory10.1007/978-3-031-43254-5_3(40-57)Online publication date: 4-Sep-2023
https://dl.acm.org/doi/10.1007/978-3-031-43254-5_3
Berthelsen MHansen K(2022)On the Computational Complexity of Decision Problems About Multi-player Nash EquilibriaTheory of Computing Systems10.1007/s00224-022-10080-166:3(519-545)Online publication date: 1-Jun-2022
https://dl.acm.org/doi/10.1007/s00224-022-10080-1

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

HTML Format

View this article in HTML Format.

Figures

Tables

Media

View Issue’s Table of Contents