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

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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 Biló 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 games has a non-symmetric Nash equilibrium is ∃R-complete when m≥3, answering a question of Garg, Mehta, Vazirani, and Yazdanbod.
Original languageEnglish
Title of host publicationAlgorithmic Game Theory - 12th International Symposium, SAGT 2019, Proceedings
EditorsDimitris Fotakis, Evangelos Markakis
Number of pages15
Place of publicationCham
PublisherSpringer VS
Publication year2019
ISBN (print)978-3-030-30472-0
ISBN (Electronic)978-3-030-30473-7
Publication statusPublished - 2019
Event12th International Symposium: International Symposium on Algorithmic Game Theory - Athen, Greece
Duration: 30 Sep 20193 Oct 2019
Conference number: 12


Conference12th International Symposium: International Symposium on Algorithmic Game Theory
SeriesLecture Notes in Computer Science

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