The Computational Complexity of Trembling Hand Perfection and Other Equilibrium Refinements

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.

[1]  Fan Chung Graham,et al.  Internet and Network Economics, Third International Workshop, WINE 2007, San Diego, CA, USA, December 12-14, 2007, Proceedings , 2007, WINE.

[2]  Tuomas Sandholm,et al.  Algorithms for Closed Under Rational Behavior (CURB) Sets , 2010, J. Artif. Intell. Res..

[3]  Roger B. Myerson,et al.  Acceptable and predominant correlated equilibria , 1986 .

[4]  Ronald L. Graham,et al.  Some NP-complete geometric problems , 1976, STOC '76.

[5]  E. Damme.  Refinements of the Nash Equilibrium Concept , 1983 .

[6]  Paul W. Goldberg,et al.  The Complexity of Computing a Nash Equilibrium , 2009, SIAM J. Comput..

[7]  Marie-Françoise Roy,et al.  On the combinatorial and algebraic complexity of quantifier elimination , 1996, JACM.

[8]  Tuomas Sandholm,et al.  Algorithms for Rationalizability and CURB Sets , 2006, AAAI.

[9]  E. Kohlberg,et al.  Foundations of Strategic Equilibrium , 1996 .

[10]  Michel Coste,et al.  Thom's Lemma, the Coding of Real Algebraic Numbers and the Computation of the Topology of Semi-Algebraic Sets , 1988, J. Symb. Comput..

[11]  E. Vandamme.  Stability and perfection of nash equilibria , 1987 .

[12]  Kousha Etessami,et al.  Recursive Markov Decision Processes and Recursive Stochastic Games , 2005, ICALP.

[13]  Kristoffer Arnsfelt Hansen,et al.  Approximability and Parameterized Complexity of Minmax Values , 2008, WINE.

[14]  Kousha Etessami,et al.  On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract) , 2010, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[15]  Patrick J. Morandi.  Field and Galois theory , 1996 .

[16]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[17]  E. Damme.  Stability and perfection of Nash equilibria , 1987 .

[18]  Jean-François Mertens.  Two examples of strategic equilibrium , 1995 .

[19]  J. Weibull,et al.  Strategy subsets closed under rational behavior , 1991 .

[20]  Xiaotie Deng,et al.  Settling the Complexity of Two-Player Nash Equilibrium , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[21]  Eddie Dekel,et al.  Lexicographic Probabilities and Equilibrium Refinements , 1991 .

[22]  Philip J. Reny,et al.  Independence on Relative Probability Spaces and Consistent Assessments in Game Trees , 1997 .

[23]  Kousha Etessami,et al.  The computational complexity of evolutionarily stable strategies , 2008, Int. J. Game Theory.

[24]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[25]  Anna Gál,et al.  Lower Bounds on Streaming Algorithms for Approximating the Length of the Longest Increasing Subsequence , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[26]  Cynthia Dwork,et al.  Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008 , 2008, STOC.

[27]  E. Damme.  A relation between perfect equilibria in extensive form games and proper equilibria in normal form games , 1984 .

[28]  D. Koller,et al.  Efficient Computation of Equilibria for Extensive Two-Person Games , 1996 .

[29]  R. Selten.  Reexamination of the perfectness concept for equilibrium points in extensive games , 1975, Classics in Game Theory.

[30]  Tilman Klumpp,et al.  Perfect equilibrium and lexicographic beliefs , 2003, Int. J. Game Theory.

[31]  David M. Kreps,et al.  Sequential Equilibria Author ( s ) : , 1982 .

[32]  The myth of the Folk Theorem , 2010 .