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

TY - GEN

T1 - On the Computational Complexity of Decision Problems About Multi-player Nash Equilibria

AU - Berthelsen, Marie Louisa Tølbøll

AU - Hansen, Kristoffer Arnsfelt

N1 - Conference code: 12

PY - 2019

Y1 - 2019

N2 - 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, (Formula Presented) -complete, when (Formula Presented). We show that, unless they turn into trivial problems, they are (Formula Presented) -hard even for 3-player zero-sum games. We also obtain new results about several other decision problems. We show that when (Formula Presented) the problems of deciding if a game has a Pareto optimal Nash equilibrium or deciding if a game has a strong Nash equilibrium are (Formula Presented) -complete. The latter result rectifies a previous claim of (Formula Presented) -completeness in the literature. We show that deciding if a game has an irrational valued Nash equilibrium is (Formula Presented) -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 (Formula Presented) -complete when (Formula Presented), answering a question of Garg, Mehta, Vazirani, and Yazdanbod.

AB - 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, (Formula Presented) -complete, when (Formula Presented). We show that, unless they turn into trivial problems, they are (Formula Presented) -hard even for 3-player zero-sum games. We also obtain new results about several other decision problems. We show that when (Formula Presented) the problems of deciding if a game has a Pareto optimal Nash equilibrium or deciding if a game has a strong Nash equilibrium are (Formula Presented) -complete. The latter result rectifies a previous claim of (Formula Presented) -completeness in the literature. We show that deciding if a game has an irrational valued Nash equilibrium is (Formula Presented) -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 (Formula Presented) -complete when (Formula Presented), answering a question of Garg, Mehta, Vazirani, and Yazdanbod.

UR - https://www.scopus.com/pages/publications/85075234435

U2 - 10.1007/978-3-030-30473-7_11

DO - 10.1007/978-3-030-30473-7_11

M3 - Article in proceedings

SN - 978-3-030-30472-0

T3 - Lecture Notes in Computer Science

SP - 153

EP - 167

BT - Algorithmic Game Theory - 12th International Symposium, SAGT 2019, Proceedings

A2 - Fotakis, Dimitris

A2 - Markakis, Evangelos

PB - Springer VS

CY - Cham

T2 - 12th International Symposium: International Symposium on Algorithmic Game Theory

Y2 - 30 September 2019 through 3 October 2019

ER -