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Computational Complexity of Decision Problems About Nash Equilibria in Win-Lose Multi-player Games

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Algorithmic Game Theory (SAGT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14238))

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Abstract

We revisit the computational complexity of decision problems about existence of Nash equilibria in multi-player games satisfying certain natural properties. Such problems have generally been shown to be complete for the complexity class \(\exists \mathbb {R}\), that captures the complexity of the decision problem for the Existential Theory of the Reals. For most of these problems, we show that their complexity remains unchanged even when restricted to win-lose games, where all utilities are either 0 or 1.

The second author is supported by the Independent Research Fund Denmark under grant no. 9040-00433B. The third author is supported by funds for the promotion of research at University of Cyprus.

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Notes

  1. 1.

    To keep notation simple, we use \(\textsf{U}_i\) to denote both the utility of player i when some certain strategy profile is realized and the expected utility of player i when some mixed profile is played.

  2. 2.

    The relative interior of \(\varDelta ^n\) is the set \(\textsf{relint}(\varDelta ^n) = \{ x \in \varDelta ^n \mid \textsf{N}_{\epsilon }(x) \cap \textsf{aff}(\varDelta ^n \subseteq \varDelta ^n\ \ \text{ for } \text{ some } \epsilon > 0 \}\), where \(\textsf{aff}(\varDelta ^n)\) is the affine hull of \(\varDelta ^n\) and \(\textsf{N}_{\epsilon }(x)\) is a ball of radius \(\epsilon \) centered at x.

References

  1. Abbott, T.G., Kane, D.M., Valiant, P.: On the complexity of two-player win-lose games. In: FOCS, pp. 113–122 (2005). https://doi.org/10.1109/SFCS.2005.59

  2. Aumann, R.J.: Acceptable points in games of perfect information. Pacific J. Math. 10(2), 381–417 (1960). https://doi.org/10.2140/pjm.1960.10.381

    Article  MathSciNet  MATH  Google Scholar 

  3. Austrin, P., Braverman, M., Chlamtac, E.: Inapproximability of NP-complete variants of Nash equilibrium. Theory Comput. 9, 117–142 (2013). https://doi.org/10.4086/toc.2013.v009a003

  4. Berthelsen, M.L.T., Hansen, K.A.: On the computational complexity of decision problems about multi-player Nash equilibria. Theory Comput. Syst. 66(3), 519–545 (2022). https://doi.org/10.1007/s00224-022-10080-1

    Article  MathSciNet  MATH  Google Scholar 

  5. Bilò, V., Mavronicolas, M.: The complexity of computational problems about Nash equilibria in symmetric win-lose games. Algorithmica 83(2), 447–530 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bilò, V., Mavronicolas, M.: \(\exists \mathbb{R} \)-complete decision problems about (symmetric) Nash equilibria in (symmetric) multi-player games. ACM Trans. Econ. Comput. 9(3), 14:1-14:25 (2021). https://doi.org/10.1145/3456758

    Article  MathSciNet  MATH  Google Scholar 

  7. Blanc, M., Hansen, K.A.: Computational complexity of multi-player evolutionarily stable strategies. In: Santhanam, R., Musatov, D. (eds.) CSR 2021. LNCS, vol. 12730, pp. 1–17. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-79416-3_1

    Chapter  Google Scholar 

  8. Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull. Am. Math. Soc. 21(1), 1–46 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bürgisser, P., Cucker, F.: Exotic quantifiers, complexity classes, and complete problems. Found. Comput. Math. 9(2), 135–170 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Buss, J.F., Frandsen, G.S., Shallit, J.O.: The computational complexity of some problems of linear algebra. J. Comput. Syst. Sci. 58(3), 572–596 (1999). https://doi.org/10.1006/jcss.1998.1608

    Article  MathSciNet  MATH  Google Scholar 

  11. Canny, J.F.: Some algebraic and geometric computations in PSPACE. In: STOC, pp. 460–467. ACM (1988). https://doi.org/10.1145/62212.62257

  12. Chen, X., Deng, X., Teng, S.H.: Settling the complexity of two-player Nash equilibrium. J. ACM 56(3), 14:-14:57 (2009)

    Article  MATH  Google Scholar 

  13. Conitzer, V., Sandholm, T.: New complexity results about Nash equilibria. Games Econom. Behav. 63(2), 621–641 (2008). https://doi.org/10.1016/j.geb.2008.02.015

    Article  MathSciNet  MATH  Google Scholar 

  14. Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. SIAM J. Comput. 39(1), 195–259 (2009). https://doi.org/10.1137/070699652

    Article  MathSciNet  MATH  Google Scholar 

  15. Deligkas, A., Fearnley, J., Melissourgos, T., Spirakis, P.: Approximating the existential theory of the reals. J. Comput. Syst. Sci. 125, 106–128 (2022). https://doi.org/10.10616/j.jcss.2021.11002

    Article  MathSciNet  MATH  Google Scholar 

  16. Deligkas, A., Fearnley, J., Savani, R.: Inapproximability results for constrained approximate Nash equilibria. Inf. Comput. 262, 40–56 (2018). https://doi.org/10.1016/j.ic2018.06.001

    Article  MathSciNet  MATH  Google Scholar 

  17. Erickson, J., van der Hoog, I., Miltzow, T.: Smoothing the gap between NP and \(\exists \mathbb{R} \). In: FOCS, pp. 1022–1033 (2020). https://doi.org/10.1109/FOCS46700.2020.00099

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Garg, J., Mehta, R., Vazirani, V.V., Yazdanbod, S.: \(\exists \mathbb{R} \)-completeness for decision versions of multi-player (symmetric) Nash equilibria. ACM Trans. Econ. Comput. 6(1), 1:1–1:23 (2018). https://doi.org/10.1145/3175494

  20. Gilboa, I., Zemel, E.: Nash and correlated equilibria: some complexity considerations. Games Econom. Behav. 1(1), 80–93 (1989). https://doi.org/10.1016/0899-8256(89)90006-7

    Article  MathSciNet  MATH  Google Scholar 

  21. Hansen, K.A.: The real computational complexity of minmax value and equilibrium refinements in multi-player games. Theory Comput. Syst. 63(7), 1554–1571 (2019). https://doi.org/10.1007/s00224-018-9887-9

    Article  MathSciNet  MATH  Google Scholar 

  22. Maschler, M., Solan, E., Zamir, S.: Game Theory. Cambridge University Press, Cambridge (2013)

    Google Scholar 

  23. Nash, J.: Non-cooperative games. Ann. Math. 2(54), 286–295 (1951). https://doi.org/10.2307/1969529

    Article  MathSciNet  MATH  Google Scholar 

  24. Schaefer, M., Štefankovič, D.: Fixed points, Nash equilibria, and the existential theory of the reals. Theory Comput. Syst. 60, 172–193 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. University of Salento, Lecce, Italy

    Vittorio Bilò

  2. Aarhus University, Aarhus, Denmark

    Kristoffer Arnsfelt Hansen

  3. University of Cyprus, Nicosia, Cyprus

    Marios Mavronicolas

Authors
  1. Vittorio Bilò
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  2. Kristoffer Arnsfelt Hansen
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  3. Marios Mavronicolas
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Corresponding author

Correspondence to Marios Mavronicolas .

Editor information

Editors and Affiliations

  1. Royal Holloway University of London, Egham, UK

    Argyrios Deligkas

  2. University of Edinburgh, Edinburgh, UK

    Aris Filos-Ratsikas

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Bilò, V., Hansen, K.A., Mavronicolas, M. (2023). Computational Complexity of Decision Problems About Nash Equilibria in Win-Lose Multi-player Games. In: Deligkas, A., Filos-Ratsikas, A. (eds) Algorithmic Game Theory. SAGT 2023. Lecture Notes in Computer Science, vol 14238. Springer, Cham. https://doi.org/10.1007/978-3-031-43254-5_3

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  • DOI: https://doi.org/10.1007/978-3-031-43254-5_3

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  • Print ISBN: 978-3-031-43253-8

  • Online ISBN: 978-3-031-43254-5

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