Theory of Computing Systems

, Volume 49, Issue 1, pp 139–161 | Cite as

The Computational Complexity of Weak Saddles

  • Felix Brandt
  • Markus Brill
  • Felix Fischer
  • Jan Hoffmann


We study the computational aspects of weak saddles, an ordinal set-valued solution concept proposed by Shapley. F. Brandt et al. recently 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-hard. 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. Most of our hardness results are shown to carry over to a natural weakening of weak saddles.


Game theory Solution concepts Shapley’s saddles Computational complexity 


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Felix Brandt
    • 1
  • Markus Brill
    • 1
  • Felix Fischer
    • 2
  • Jan Hoffmann
    • 3
  1. 1.Institut für InformatikTechnische Universität MünchenGarchingGermany
  2. 2.Harvard School of Engineering and Applied SciencesCambridgeUSA
  3. 3.Institut für InformatikLudwig-Maximilians-Universität MünchenMünchenGermany

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