On the Computational Complexity of Decision Problems About Multi-player Nash Equilibria

We study the computational complexity of decision problems about Nash equilibria in m -player games. Several such problems have recently been shown to be computationally equivalent to the decision problem for the existential theory of the reals, or stated in terms of complexity classes, ∃ ℝ -complet...

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Veröffentlicht in:Theory of computing systems Jg. 66; H. 3; S. 519 - 545
Hauptverfasser: Berthelsen, Marie Louisa Tølbøll, Hansen, Kristoffer Arnsfelt
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.06.2022
Springer Nature B.V
Schlagworte:
Complexity
Computer Science
Decision theory
Equilibrium
Game theory
Games
Pareto optimum
Questions
Special Issue on Algorithmic Game Theory (SAGT 2019)
Theory of Computation
Zero sum games
Nash equilibrium
Computational complexity
Existential theory of the reals
ISSN:1432-4350, 1433-0490
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Zusammenfassung:We study the computational complexity of decision problems about Nash equilibria in m -player games. Several such problems have recently been shown to be computationally equivalent to the decision problem for the existential theory of the reals, or stated in terms of complexity classes, ∃ ℝ -complete, when m ≥ 3. We show that, unless they turn into trivial problems, they are ∃ ℝ -hard even for 3-player zero-sum games. We also obtain new results about several other decision problems. We show that when m ≥ 3 the problems of deciding if a game has a Pareto optimal Nash equilibrium or deciding if a game has a strong Nash equilibrium are ∃ ℝ -complete. The latter result rectifies a previous claim of NP-completeness in the literature. We show that deciding if a game has an irrational valued Nash equilibrium is ∃ ℝ -hard, answering a question of Bilò and Mavronicolas, and address also the computational complexity of deciding if a game has a rational valued Nash equilibrium. These results also hold for 3-player zero-sum games. Our proof methodology applies to corresponding decision problems about symmetric Nash equilibria in symmetric games as well, and in particular our new results carry over to the symmetric setting. Finally we show that deciding whether a symmetric m -player game has a non-symmetric Nash equilibrium is ∃ ℝ -complete when m ≥ 3, answering a question of Garg, Mehta, Vazirani, and Yazdanbod.
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ISSN:1432-4350
1433-0490
DOI:10.1007/s00224-022-10080-1