Advertisement

Computing Constrained Approximate Equilibria in Polymatrix Games

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10504)

Abstract

This paper studies constrained approximate Nash equilibria in polymatrix games. We show that is \(\mathtt {NP}\)-hard to decide if a polymatrix game has a constrained approximate equilibrium for 9 natural constraints and any non-trivial \(\epsilon \). We then provide a QPTAS for polymatrix games with bounded treewidth and logarithmically many actions per player that finds constrained approximate equilibria for a wide family of constraints.

References

  1. 1.
    Austrin, P., Braverman, M., Chlamtac, E.: Inapproximability of NP-complete variants of Nash equilibrium. Theory Comput. 9, 117–142 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Babichenko, Y., Barman, S., Peretz, R.: Simple approximate equilibria in large games. In: Proceedings of the EC, pp. 753–770 (2014)Google Scholar
  3. 3.
    Barman, S., Ligett, K.: Finding any nontrivial coarse correlated equilibrium is hard. In: Proceedings of EC, pp. 815–816 (2015)Google Scholar
  4. 4.
    Barman, S., Ligett, K., Piliouras, G.: Approximating Nash equilibria in tree polymatrix games. In: Hoefer, M. (ed.) SAGT 2015. LNCS, vol. 9347, pp. 285–296. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-48433-3_22CrossRefGoogle Scholar
  5. 5.
    Bilò, V., Mavronicolas, M.: The complexity of decision problems about Nash equilibria in win-lose games. In: Serna, M. (ed.) SAGT 2012. LNCS, pp. 37–48. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-33996-7_4CrossRefGoogle Scholar
  6. 6.
    Bilò, V., Mavronicolas, M.: A catalog of \(\exists \mathbb{R}\)-complete decision problems about Nash equilibria in multi-player games. In: Proceedings of STACS, pp. 17:1–17:13 (2016)Google Scholar
  7. 7.
    Bilò, V., Mavronicolas, M.: \(\exists \mathbb{R}\)-complete decision problems about symmetric Nash equilibria in symmetric multi-player games. In: Proceedings of STACS, pp. 13:1–13:14 (2017)Google Scholar
  8. 8.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bonifaci, V., Di Iorio, U., Laura, L.: The complexity of uniform Nash equilibria and related regular subgraph problems. Theor. Comput. Sci. 401(1–3), 144–152 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bosse, H., Byrka, J., Markakis, E.: New algorithms for approximate Nash equilibria in bimatrix games. Theor. Comput. Sci. 411(1), 164–173 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Braverman, M., Kun-Ko, Y., Weinstein, O.: Approximating the best Nash equilibrium in n\({}^{\text{o}}\)\({}^{\text{(log } \text{ n) }}\)-time breaks the exponential time hypothesis. In: Proceedings of SODA, pp. 970–982 (2015)Google Scholar
  12. 12.
    Chen, X., Deng, X., Teng, S.-H.: Settling the complexity of computing two-player Nash equilibria. J. ACM 56(3), 57 (2009). Article No. 14MathSciNetCrossRefGoogle Scholar
  13. 13.
    Conitzer, V., Sandholm, T.: New complexity results about Nash equilibria. Games Econ. Behav. 63(2), 621–641 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Czumaj, A., Deligkas, A., Fasoulakis, M., Fearnley, J., Jurdziński, M., Savani, R.: Distributed methods for computing approximate equilibria. In: Cai, Y., Vetta, A. (eds.) WINE 2016. LNCS, vol. 10123, pp. 15–28. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-54110-4_2CrossRefGoogle Scholar
  15. 15.
    Czumaj, A., Fasoulakis, M., Jurdziński, M.: Approximate Nash equilibria with near optimal social welfare. In: Proceedings of IJCAI, pp. 504–510 (2015)Google Scholar
  16. 16.
    Czumaj, A., Fasoulakis M., Jurdziński, M.: Approximate plutocratic and egalitarian Nash equilibria. In: Proceedings of AAMAS, pp. 1409–1410 (2016)Google Scholar
  17. 17.
    Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. SIAM J. Comput. 39(1), 195–259 (2009)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Daskalakis, C., Mehta, A., Papadimitriou, C.H.: Progress in approximate Nash equilibria. In: Proceedings of EC, pp. 355–358 (2007)Google Scholar
  19. 19.
    Daskalakis, C., Mehta, A., Papadimitriou, C.H.: A note on approximate Nash equilibria. Theor. Comput. Sci. 410(17), 1581–1588 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Deligkas, A., Fearnley, J., Igwe, T.P., Savani, R.: An empirical study on computing equilibria in polymatrix games. In: Proceedings of AAMAS, pp. 186–195 (2016)Google Scholar
  21. 21.
    Deligkas, A., Fearnley, J., Savani, R.: Inapproximability results for approximate Nash equilibria. In: Cai, Y., Vetta, A. (eds.) WINE 2016. LNCS, vol. 10123, pp. 29–43. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-54110-4_3CrossRefGoogle Scholar
  22. 22.
    Deligkas, A., Fearnley, J., Savani, R., Spirakis, P.G.: Computing approximate Nash equilibria in polymatrix games. Algorithmica 77(2), 487–514 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Elkind, E., Goldberg, L.A., Goldberg, P.W.: Nash equilibria in graphical games on trees revisited. In: Proceedings of EC, pp. 100–109 (2006)Google Scholar
  24. 24.
    Elkind, E., Goldberg, L.A., Goldberg, P.W.: Computing good Nash equilibria in graphical games. In: Proceedings of EC, pp. 162–171 (2007)Google Scholar
  25. 25.
    Fearnley, J., Goldberg, P.W., Savani, R., Sørensen, T.B.: Approximate well-supported Nash equilibria below two-thirds. In: Serna, M. (ed.) SAGT 2012. LNCS, pp. 108–119. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-33996-7_10CrossRefGoogle Scholar
  26. 26.
    Garg, J., Mehta, R., Vazirani, V.V., Yazdanbod, S.: ETR-completeness for decision versions of multi-player (symmetric) Nash equilibria. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 554–566. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-47672-7_45CrossRefGoogle Scholar
  27. 27.
    Gilboa, I., Zemel, E.: Nash and correlated equilibria: some complexity considerations. Games Econ. Behav. 1(1), 80–93 (1989)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Greco, G., Scarcello, F.: On the complexity of constrained Nash equilibria in graphical games. Theor. Comput. Sci. 410(38–40), 3901–3924 (2009)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Hazan, E., Krauthgamer, R.: How hard is it to approximate the best Nash equilibrium? SIAM J. Comput. 40(1), 79–91 (2011)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Kontogiannis, S.C., Spirakis, P.G.: Well supported approximate equilibria in bimatrix games. Algorithmica 57(4), 653–667 (2010)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Lipton, R.J., Markakis, E., Mehta, A.: Playing large games using simple strategies. In: Proceedings of EC, pp. 36–41 (2003)Google Scholar
  32. 32.
    Moore, C., Robson, J.M.: Hard tiling problems with simple tiles. Discrete Comput. Geom. 26(4), 573–590 (2001)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Ortiz, L.E., Irfan, M.T.: FPTAS for mixed-strategy Nash equilibria in tree graphical games and their generalizations. CoRR, abs/1602.05237 (2016)Google Scholar
  34. 34.
    Ortiz, L.E., Irfan, M.T.: Tractable algorithms for approximate Nash equilibria in generalized graphical games with tree structure. In: Proceedings of AAAI, pp. 635–641 (2017)Google Scholar
  35. 35.
    Rubinstein, A.: Inapproximability of Nash equilibrium. In: Proceedings of STOC, pp. 409–418 (2015)Google Scholar
  36. 36.
    Tsaknakis, H., Spirakis, P.G.: An optimization approach for approximate Nash equilibria. Internet Math. 5(4), 365–382 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.TechnionHaifaIsrael
  2. 2.University of LiverpoolLiverpoolUK

Personalised recommendations