The Computational Complexity of Weak Saddles

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


We continue the recently initiated study of the computational aspects of weak saddles, an ordinal set-valued solution concept proposed by Shapley. Brandt et al. gave a polynomial-time algorithm for computing weak saddles in a subclass of matrix games, and showed that certain problems associated with weak saddles of bimatrix games are NP-complete. The important question of whether weak saddles can be found efficiently was left open. We answer this question in the negative by showing that finding weak saddles of bimatrix games is NP-hard, under polynomial-time Turing reductions. We moreover prove that recognizing weak saddles is coNP-complete, and that deciding whether a given action is contained in some weak saddle is hard for parallel access to NP and thus not even in NP unless the polynomial hierarchy collapses. Our hardness results are finally shown to carry over to a natural weakening of weak saddles.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Basu, K., Weibull, J.: Strategy subsets closed under rational behavior. Economics Letters 36, 141–146 (1991)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baumeister, D., Brandt, F., Fischer, F., Hoffmann, J., Rothe, J.: The complexity of computing minimal unidirectional covering sets. Technical report (2009),
  3. 3.
    Bernheim, B.: Rationalizable strategic behavior. Econometrica 52(4), 1007–1028 (1984)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brandt, F., Fischer, F.: Computing the minimal covering set. Mathematical Social Sciences 56(2), 254–268 (2008)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    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
  6. 6.
    Brandt, F., Brill, M., Fischer, F., Harrenstein, P.: On the complexity of iterated weak dominance in constant-sum games. In: Mavronicolas, M., Papadopoulou, V.G. (eds.) SAGT 2009. LNCS, vol. 5814, pp. 287–298. Springer, Heidelberg (2009)Google Scholar
  7. 7.
    Chen, X., Deng, X.: Settling the complexity of 2-player Nash-equilibrium. In: Proc. of 47th FOCS Symposium, pp. 261–272. IEEE Press, Los Alamitos (2006)Google Scholar
  8. 8.
    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
  9. 9.
    Daskalakis, C., Goldberg, P., Papadimitriou, C.: The complexity of computing a Nash equilibrium. In: Proc. of 38th STOC, pp. 71–78. ACM Press, New York (2006)Google Scholar
  10. 10.
    Duggan, J., Le Breton, M.: Dutta’s minimal covering set and Shapley’s saddles. Journal of Economic Theory 70, 257–265 (1996)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dutta, B.: Covering sets and a new Condorcet choice correspondence. Journal of Economic Theory 44, 63–80 (1988)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gilboa, I., Kalai, E., Zemel, E.: The complexity of eliminating dominated strategies. Mathematics of Operations Research 18(3), 553–565 (1993)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hemaspaandra, E., Hemaspaandra, L., Rothe, J.: Exact analysis of Dodgson elections: Lewis Carroll’s 1876 voting system is complete for parallel access to NP. Journal of the ACM 44(6), 806–825 (1997)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Luce, R.D., Raiffa, H.: Games and Decisions: Introduction and Critical Survey. John Wiley & Sons Inc., Chichester (1957)MATHGoogle Scholar
  15. 15.
    McKelvey, R.D., Ordeshook, P.C.: Symmetric spatial games without majority rule equilibria. The American Political Science Review 70(4), 1172–1184 (1976)CrossRefGoogle Scholar
  16. 16.
    Myerson, R.B.: Game Theory: Analysis of Conflict. Harvard University Press, Cambridge (1991)MATHGoogle Scholar
  17. 17.
    Nash, J.F.: Non-cooperative games. Annals of Mathematics 54(2), 286–295 (1951)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  19. 19.
    Pearce, D.: Rationalizable strategic behavior and the problem of perfection. Econometrica 52(4), 1029–1050 (1984)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Samuelson, L.: Dominated strategies and common knowledge. Games and Economic Behavior 4, 284–313 (1992)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Shapley, L.: Order matrices. I. Technical Report RM-1142, The RAND Corporation (1953)Google Scholar
  22. 22.
    Shapley, L.: Order matrices. II. Technical Report RM-1145, The RAND Corporation (1953)Google Scholar
  23. 23.
    Shapley, L.: Some topics in two-person games. In: Dresher, M., Shapley, L.S., Tucker, A.W. (eds.) Advances in Game Theory. Annals of Mathematics Studies, vol. 52, pp. 1–29. Princeton University Press, Princeton (1964)Google Scholar
  24. 24.
    von Neumann, J.: Zur Theorie der Gesellschaftspiele. Mathematische Annalen 100, 295–320 (1928)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)MATHGoogle Scholar
  26. 26.
    Wagner, K.: More complicated questions about maxima and minima, and some closures of NP. Theoretical Computer Science 51, 53–80 (1987)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Felix Brandt
    • 1
  • Markus Brill
    • 1
  • Felix Fischer
    • 1
  • Jan Hoffmann
    • 1
  1. 1.Institut für InformatikLudwig-Maximilians-Universität MünchenMünchenGermany

Personalised recommendations