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A Catalog of EXISTS-R-Complete Decision Problems About Nash Equilibria in Multi-Player Games

Authors Vittorio Bilò, Marios Mavronicolas

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Vittorio Bilò
Marios Mavronicolas

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Vittorio Bilò and Marios Mavronicolas. A Catalog of EXISTS-R-Complete Decision Problems About Nash Equilibria in Multi-Player Games. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 17:1-17:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.STACS.2016.17

Abstract

[Schaefer and Stefankovic, Theory of Computing Systems, 2015] provided an explicit formulation of EXISTS-R as the class capturing the complexity of deciding the Existential Theory of the Reals, and established that deciding, given a 3-player game, whether or not it has a Nash equilibrium with no probability exceeding a given rational is EXISTS-R-complete. Four more decision problems about Nash equilibria for 3-player games were very recently shown EXISTS-R-complete via a chain of individual, problem-specific reductions in [Garg et al., Proceedings of ICALP 2015]; determining more such EXISTS-R-complete problems was posed there as an open problem. In this work, we deliver an extensive catalog of EXISTS-R-complete decision problems about Nash equilibria in 3-player games, thus resolving completely the open problem from [Garg et al., Proceedings of ICALP 2015]. Towards this end, we present a single and very simple, unifying reduction from the EXISTS-R-complete decision problem from [Schaefer and Stefankovic, Theory of Computing Systems, 2015] to (almost) all the decision problems about Nash equilibria that were before shown NP-complete for 2-player games in [Bilo and Mavronicolas, Proceedings of SAGT 2012; Conitzer and Sandholm, Games and Economic Behavior, 2008; Gilboa and Zemel, Games and Economic Behavior, 1989]. Encompassed in the catalog are the four decision problems shown EXISTS-R-complete in [Garg et al., Proceedings of ICALP 2015].

Subject Classification

Keywords
  • Nash equilibrium
  • complexity of equilibria
  • EXISTS-R-completeness

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References

  1. V. Bilò and M. Mavronicolas. The complexity of decision problems about Nash equilibria in win-lose games. In Proc. of the 5th Inte'l Symposium on Algorithmic Game Theory, volume 7615 of LNCS, pages 37-48, 2012. Google Scholar
  2. V. Bilò and M. Mavronicolas. Complexity of rational and irrational Nash equilibria. Theory of Computing Systems, 54(3):491-527, 2014. Google Scholar
  3. V. Bonifaci, U. Di Orio, and L. Laura. The complexity of uniform Nash equilibria and related subgraph problems. Theoretical Computer Science, 401(1-3):144-152, 2008. Google Scholar
  4. J. Canny. Some algebraic and geometric computations in PSPACE. In Proc. of the 20th Annual ACM Symp. on Theory of Computing, pages 460-467, 1988. Google Scholar
  5. X. Chen, X. Deng, and S.-H. Teng. Settling the complexity of computing two-player Nash equilibria. Journal of the ACM, 56(3), 2009. Google Scholar
  6. V. Conitzer and T. Sandholm. New complexity results about Nash equilibria. Games and Economic Behavior, 63(2):621-641, 2008. Google Scholar
  7. C. Daskalakis, P. W. Goldberg, and C. H. Papadimitriou. The complexity of computing a Nash equilibrium. SIAM Journal on Computing, 39(1):195-259, 2009. Google Scholar
  8. M. J. Garey and D. J. Johnson. Computers and Intractability - A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  9. J. Garg, R. Mehta, V. V. Vazirani, and S. Yazdanbod. ETR-completeness for decision versions of multi-player (symmetric) Nash equilibria. In Proc. of the 42nd Int'l Colloquium on Automata, Languages and Programming, volume 9134 of LNCS, pages 554-566, 2015. Google Scholar
  10. I. Gilboa and E. Zemel. Nash and correlated equilibria: Some complexity considerations. Games and Economic Behavior, 1(1):80-93, 1989. Google Scholar
  11. N. E. Mnev. The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In Topology and Geometry - Rohlin Seminar, number 1346 in LNM, pages 527-543, 1988. Google Scholar
  12. J. F. Nash. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences of the United States of America, 36:48-49, 1950. Google Scholar
  13. J. F. Nash. Non-cooperative games. Annals of Mathematics, 54(2):286-295, 1951. Google Scholar
  14. C. H. Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. Journal of Computer and System Sciences, 48(3):498-532, 1994. Google Scholar
  15. M. Schaefer. Complexity of some geometric and topological problems. In Proc. of the 17th Int'l Symp. on Graph Drawing, volume 5849 of LNCS, pages 334-344, 2010. Google Scholar
  16. M. Schaefer. Realizability of graphs and linkages. In Thirty Essays on Geometric Graph Theory, 2013. Google Scholar
  17. M. Schaefer and D. Štefankovič. Fixed points, Nash equilibria and the existential theory of the reals. Theory of Computing Systems, First online: 04 November 2015. Google Scholar
  18. A. Tarski. A decision method for elementary algebra and geometry. RAND Corporation, 1948. Google Scholar
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