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Bilinear Games: Polynomial Time Algorithms for Rank Based Subclasses

  • Jugal Garg
  • Albert Xin Jiang
  • Ruta Mehta
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7090)

Abstract

Motivated by the sequence form formulation of Koller et al. [16], this paper considers bilinear games, represented by two payoff matrices (A,B) and compact polytopal strategy sets. Bilinear games are very general and capture many interesting classes of games including bimatrix games, two player Bayesian games, polymatrix games, and two-player extensive form games with perfect recall as special cases, and hence are hard to solve in general. For a bilinear game, we define its best response polytopes (BRPs) and characterize its Nash equilibria as the fully-labeled pairs of the BRPs. Rank of a game (A,B) is defined as rank(A + B). In this paper, we give polynomial-time algorithms for computing Nash equilibria of (i) rank-1 games, (ii) FPTAS for constant-rank games, and (iii) when rank(A) or rank(B) is constant.

Keywords

Nash Equilibrium Polynomial Time Algorithm Payoff Matrice Fully Polynomial Time Approximation Scheme Time Poly 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Adsul, B., Garg, J., Mehta, R., Sohoni, M.: Rank-1 bimatrix games: A homeomorphism and a polynomial time algorithm. In: STOC, pp. 195–204 (2011)Google Scholar
  2. 2.
    Avis, D., Rosenberg, G.D., Savani, R., von Stengel, B.: Enumeration of Nash equilibria for two-player games. Economic Theory 42, 9–37 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Charnes, A.: Constrained games and linear programming. Proceedings of the National Academy of Sciences of the USA 39, 639–641 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, X., Deng, X.: Settling the complexity of 2-player Nash-equilibrium. In: FOCS, pp. 261–272 (2006)Google Scholar
  5. 5.
    Dantzig, G.B.: Linear Programming and Extensions. Princeton Univ. Press (1963)Google Scholar
  6. 6.
    Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. In: STOC, pp. 71–78 (2006)Google Scholar
  7. 7.
    Daskalakis, C., Papadimitriou, C.H.: On a Network Generalization of the Minmax Theorem. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 423–434. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Fudenberg, D., Tirole, J.: Game Theory. MIT Press (1991)Google Scholar
  9. 9.
    Garg, J., Jiang, A.X., Mehta, R.: Bilinear games: Polynomial time algorithms for rank based subclasses. arXiv:1109.6182 (2011)Google Scholar
  10. 10.
    Glicksberg, I.L.: A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points, vol. 3(1), pp. 170–174. AMS (1952)Google Scholar
  11. 11.
    Howson Jr., J.: Equilibria of polymatrix games. Management Science (1972)Google Scholar
  12. 12.
    Howson Jr, J., Rosenthal, R.: Bayesian equilibria of finite two-person games with incomplete information. Management Science, 313–315 (1974)Google Scholar
  13. 13.
    Immorlica, N., Kalai, A.T., Lucier, B., Moitra, A., Postlewaite, A., Tennenholtz, M.: Dueling algorithms. In: STOC (2011)Google Scholar
  14. 14.
    Kannan, R., Theobald, T.: Games of fixed rank: A hierarchy of bimatrix games. Economic Theory, 1–17 (2009)Google Scholar
  15. 15.
    Koller, D., Megiddo, N., von Stengel, B.: Fast algorithms for finding randomized strategies in game trees. In: STOC, pp. 750–759 (1994)Google Scholar
  16. 16.
    Koller, D., Megiddo, N., von Stengel, B.: Efficient computation of equilibria for extensive two-person games. Games and Economic Behavior 14, 247–259 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kontogiannis, S., Spirakis, P.: Exploiting Concavity in Bimatrix Games: New Polynomially Tractable Subclasses. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX 2010, LNCS, vol. 6302, pp. 312–325. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  18. 18.
    Lipton, R., Markakis, E., Mehta, A.: Playing large games using simple strategies. In: EC, pp. 36–41. ACM Press, New York (2003)Google Scholar
  19. 19.
    Ponssard, J.P., Sorin, S.: The LP formulation of finite zero-sum games with incomplete information. International Journal of Game Theory 9, 99–105 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    von Stengel, B.: Equilibrium computation for two-player games in strategic and extensive form. In: Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V. (eds.) Algorithmic Game Theory, ch. 3, pp. 53–78 (2007)Google Scholar
  21. 21.
    Vavasis, S.: Approximation algorithms for indefinite quadratic programming. Mathematical Programming 57, 279–311 (1992)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Jugal Garg
    • 1
  • Albert Xin Jiang
    • 2
  • Ruta Mehta
    • 1
  1. 1.Indian Institute of TechnologyBombayIndia
  2. 2.University of British ColumbiaCanada

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