On the Complexity of Iterated Weak Dominance in Constant-Sum Games

  • Felix Brandt
  • Markus Brill
  • Felix Fischer
  • Paul Harrenstein
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5814)

Abstract

In game theory, a player’s action is said to be weakly dominated if there exists another action that, with respect to what the other players do, is never worse and sometimes strictly better. We investigate the computational complexity of the process of iteratively eliminating weakly dominated actions (IWD) in two-player constant-sum games, i.e., games in which the interests of both players are diametrically opposed. It turns out that deciding whether an action is eliminable via IWD is feasible in polynomial time whereas deciding whether a given subgame is reachable via IWD is NP-complete. The latter result is quite surprising as we are not aware of other natural computational problems that are intractable in constant-sum games. Furthermore, we slightly improve a result by Conitzer and Sandholm [6] by showing that typical problems associated with IWD in win-lose games with at most one winner are NP-complete.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Apt, K.R.: Uniform proofs of order independence for various strategy elimination procedures. Contributions to Theoretical Economics 4(1) (2004)Google Scholar
  2. 2.
    Bernheim, B.: Rationalizable strategic behavior. Econometrica 52(4), 1007–1028 (1984)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brandenburger, A., Friedenberg, A., Keisler, H.J.: Admissibility in games. Econometrica 76(2), 307–352 (2008)MATHMathSciNetGoogle Scholar
  4. 4.
    Brandt, F., Brill, M., Fischer, F., Harrenstein, P.: Computational aspects of Shapley’s saddles. In: Proc. of 8th AAMAS Conference, pp. 209–216 (2009)Google Scholar
  5. 5.
    Brandt, F., Fischer, F., Harrenstein, P., Shoham, Y.: Ranking games. Artificial Intelligence 173(2), 221–239 (2009)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Conitzer, V., Sandholm, T.: Complexity of (iterated) dominance. In: Proc. of 6th ACM-EC Conference, pp. 88–97. ACM Press, New York (2005)Google Scholar
  7. 7.
    Ewerhart, C.: Iterated weak dominance in strictly competitive games of perfect information. Journal of Economic Theory 107, 474–482 (2002)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gilboa, I., Kalai, E., Zemel, E.: The complexity of eliminating dominated strategies. Mathematics of Operations Research 18(3), 553–565 (1993)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Knuth, D.E., Papadimitriou, C.H., Tsitsiklis, J.N.: A note on strategy elimination in bimatrix games. Operations Research Letters 7, 103–107 (1988)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kohlberg, E., Mertens, J.-F.: On the strategic stability of equilibria. Econometrica 54, 1003–1037 (1986)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Koller, D., Megiddo, N.: The complexity of two-person zero-sum games in extensive from. Games and Economic Behavior 4(4), 528–552 (1992)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Luce, R.D., Raiffa, H.: Games and Decisions: Introduction and Critical Survey. John Wiley & Sons Inc., Chichester (1957)MATHGoogle Scholar
  13. 13.
    Marx, L.M., Swinkels, J.M.: Order independence for iterated weak dominance. Games and Economic Behavior 18, 219–245 (1997)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Myerson, R.B.: Game Theory: Analysis of Conflict. Harvard University Press, Cambridge (1991)MATHGoogle Scholar
  15. 15.
    Osborne, M.: An Introduction to Game Theory. Oxford University Press, Oxford (2004)Google Scholar
  16. 16.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  17. 17.
    Pearce, D.: Rationalizable strategic behavior and the problem of perfection. Econometrica 52(4), 1029–1050 (1984)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Samuelson, L.: Dominated strategies and common knowledge. Games and Economic Behavior 4, 284–313 (1992)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Shimoji, M.: On the equivalence of weak dominance and sequential best response. Games and Economic Behavior 48, 385–402 (2004)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Shoham, Y., Leyton-Brown, L.: Multiagent Systems – Algorithmic, Game-Theoretic, and Logical Foundations. Cambridge University Press, Cambridge (2009)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Felix Brandt
    • 1
  • Markus Brill
    • 1
  • Felix Fischer
    • 1
  • Paul Harrenstein
    • 1
  1. 1.Institut für InformatikLudwig-Maximilians-Universität MünchenMünchenGermany

Personalised recommendations