The computational complexity of trembling hand perfection and other equilibrium refinements

Authors:
Kristoffer Arnsfelt Hansen;Peter Bro Miltersen;Troels Bjerre Sørensen
Affiliations:
Department of Computer Science, Aarhus University, Århus N, Denmark;Department of Computer Science, Aarhus University, Århus N, Denmark;Department of Computer Science, University of Warwick, Coventry, United Kingdom
Venue:
SAGT'10 Proceedings of the Third international conference on Algorithmic game theory
Year:
2010

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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.