The Real Computational Complexity of Minmax Value and Equilibrium Refinements in Multi-player Games

We show that for several solution concepts for finite n -player games, where n ≥ 3, the task of simply verifying its conditions is computationally equivalent to the decision problem of the existential theory of the reals. This holds for trembling hand perfect equilibrium, proper equilibrium, and CUR...

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Bibliographic Details
Published in:Theory of computing systems Vol. 63; no. 7; pp. 1554 - 1571
Main Author: Hansen, Kristoffer Arnsfelt
Format: Journal Article
Language:English
Published: New York Springer US 01.10.2019
Springer Nature B.V
Subjects:
Completeness
Complexity
Computer Science
Decision theory
Equilibrium
Equivalence
Game theory
Special Issue on Algorithmic Game Theory (SAGT 2017)
Theory of Computation
Minmax value
Existential theory of the reals
Nash equilibrium refinements
Computational complexity
ISSN:1432-4350, 1433-0490
Online Access:Get full text
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Summary:We show that for several solution concepts for finite n -player games, where n ≥ 3, the task of simply verifying its conditions is computationally equivalent to the decision problem of the existential theory of the reals. This holds for trembling hand perfect equilibrium, proper equilibrium, and CURB sets in strategic form games and for (the strategy part of) sequential equilibrium, trembling hand perfect equilibrium, and quasi-perfect equilibrium in extensive form games of perfect recall. For obtaining these results we first show that the decision problem for the minmax value in n -player games, where n ≥ 3, is also equivalent to the decision problem for the existential theory of the reals. Our results thus improve previous results of NP- hardness as well as Sqrt-Sum - hardness of the decision problems to completeness for ∃ ℝ , the complexity class corresponding to the decision problem of the existential theory of the reals. As a byproduct we also obtain a simpler proof of a result by Schaefer and Štefankovič giving ∃ ℝ -completeness for the problem of deciding existence of a probability constrained Nash equilibrium.
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ISSN:1432-4350
1433-0490
DOI:10.1007/s00224-018-9887-9