The Computational Complexity of Weak Saddles
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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.
KeywordsGame theory Solution concepts Shapley’s saddles Computational complexity
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- 2.Baumeister, D., Brandt, F., Fischer, F., Hoffmann, J., Rothe, J.: The complexity of computing minimal unidirectional covering sets. In: Proceedings of the 7th International Conference on Algorithms and Complexity (CIAC). Lecture Notes in Computer Science (LNCS), vol. 6078, pp. 299–310. Springer, Berlin (2010) Google Scholar
- 5.Brandt, F., Brill, M., Fischer, F., Harrenstein, P.: Computational aspects of Shapley’s saddles. In: Proceedings of the 8th International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), pp. 209–216 (2009) Google Scholar
- 7.Chen, X., Deng, X., Teng, S.-H.: Settling the complexity of computing two-player Nash equilibria. J. ACM 56(3) (2009) Google Scholar
- 21.Shapley, L.: Order matrices, I. Technical Report RM-1142, The RAND Corporation (1953a) Google Scholar
- 22.Shapley, L.: Order matrices, II. Technical Report RM-1145, The RAND Corporation (1953b) Google Scholar
- 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