Abstract
The king of refinements of Nash equilibrium is trembling hand perfection. We show that it is NP-hard and Sqrt-Sum-hard to decide if a given pure strategy Nash equilibrium of a given three-player game in strategic form with integer payoffs is trembling hand perfect. Analogous results are shown for a number of other solution concepts, including proper equilibrium, (the strategy part of) sequential equilibrium, quasi-perfect equilibrium and CURB.
The proofs all use a reduction from the problem of comparing the minmax value of a three-player game in strategic form to a given rational number. This problem was previously shown to be NP-hard by Borgs et al., while a Sqrt-Sum hardness result is given in this paper. The latter proof yields bounds on the algebraic degree of the minmax value of a three-player game that may be of independent interest.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Basu, K., Weibull, J.W.: Strategy subsets closed under rational behavior. Economics Letters 2(36), 141–146 (1991)
Basu, S., Pollack, R., Roy, M.F.: On the combinatorial and algebraic complexity of quantifier elimination. Journal of the ACM 43(6), 1002–1045 (1996)
Benisch, M., Davis, G.B., Sandholm, T.: Algorithms for closed under rational behavior (CURB) sets. Journal of Artificial Intelligence Research (Forthcoming 2010)
Blume, L., Brandenburger, A., Dekel, E.: Lexicographic probabilities and equilibrium refinements. Econometrica 59, 81–98 (1991)
Borgs, C., Chayes, J.T., Immorlica, N., Kalai, A.T., Mirrokni, V.S., Papadimitriou, C.H.: The myth of the folk theorem. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 365–372. ACM, New York (2008)
Chen, X., Deng, X.: Settling the complexity of two-player Nash equilibrium. In: Proceedings of 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), pp. 261–272 (2006)
Coste, M., Roy, M.: Thom’s lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebraic sets. Journal of Symbolic Computation 5(1-2), 121–129 (1988)
van Damme, E.: A relation between perfect equilibria in extensive form games and proper equilibria in normal form games. International Journal of Game Theory 13, 1–13 (1984)
van Damme, E.: Stability and Perfection of Nash Equlibria, 2nd edn. Springer, Heidelberg (1991)
Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. In: Procedings of the 38th Annual ACM Symposium on the Theory of Computing (STOC 2006), pp. 71–78 (2006)
Etessami, K., Lochbihler, A.: The computational complexity of evolutionarily stable strategies. International Journal of Game Theory 31(1), 93–113 (2008)
Etessami, K., Yannakakis, M.: Recursive markov decision processes and recursive stochastic games. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 891–903. Springer, Heidelberg (2005)
Etessami, K., Yannakakis, M.: On the complexity of Nash equilibria and other fixed points (extended abstract). In: Proc. 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007), pp. 113–123 (2007)
Garey, M.R., Graham, R.L., Johnson, D.S.: Some NP-complete geometric problems. In: Proceedings of the 8th Annual ACM Symposium on Theory of Computing, STOC 1976, Hershey, PA, May 3-5, pp. 10–22. ACM Press, New York (1976)
Govindan, S., Klumpp, T.: Perfect equilibrium and lexicographic beliefs. International Journal of Game Theory 31(2), 229–243 (2003)
Graham, R.L.: 10 - Problems and Solutions. In: P73: Euclidian Minimum Spanning Trees. Bulletin of the EATCS, vol. 24, pp. 205–206. EATCS (October 1984)
Hansen, K.A., Hansen, T.D., Miltersen, P.B., Sørensen, T.B.: Approximability and parameterized complexity of minmax values. In: Papadimitriou, C., Zhang, S. (eds.) WINE 2008. LNCS, vol. 5385, pp. 684–695. Springer, Heidelberg (2008)
Hillas, J., Kohlberg, E.: Foundations of strategic equilibria. In: Aumann, R.J., Hart, S. (eds.) Handbook of Game Theory, ch. 42, vol. 3, pp. 1597–1663. Elsevier Science, Amsterdam (2002)
Kohlberg, E., Reny, P.J.: Independence on relative probability spaces and consistent assessments in game trees. Journal of Economic Theory 75(2), 280–313 (1997)
Koller, D., Megiddo, N., von Stengel, B.: Efficient computation of equilibria for extensive form games. Games and Economic Behavior 14, 247–259 (1996)
Kreps, D.M., Wilson, R.: Sequential equilibria. Econometrica 50(4), 863–894 (1982)
Mertens, J.F.: Two examples of strategic equilibrium. Games and Economic Behavior 8, 378–388 (1995)
Morandi, P.: Field and Galois Theory. Graduate Texts in Mathematics, vol. 167. Springer, Heidelberg (1996)
Myerson, R.B.: Refinements of the Nash equilibrium concept. International Journal of Game Theory 15, 133–154 (1978)
Selten, R.: A reexamination of the perfectness concept for equilibrium points in extensive games. International Journal of Game Theory 4, 25–55 (1975)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hansen, K.A., Miltersen, P.B., Sørensen, T.B. (2010). The Computational Complexity of Trembling Hand Perfection and Other Equilibrium Refinements. In: Kontogiannis, S., Koutsoupias, E., Spirakis, P.G. (eds) Algorithmic Game Theory. SAGT 2010. Lecture Notes in Computer Science, vol 6386. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16170-4_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-16170-4_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-16169-8
Online ISBN: 978-3-642-16170-4
eBook Packages: Computer ScienceComputer Science (R0)