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, \(\exists \mathbb {R}\)-complete, when \(m\ge 3\). We show that, unless they turn into trivial problems, they are \(\exists \mathbb {R}\)-hard even for 3-player zero-sum games.

[1]  Vittorio Bilò,et al.  Existential-R-Complete Decision Problems about Symmetric Nash Equilibria in Symmetric Multi-Player Games , 2017, STACS.

[2]  Ruta Mehta,et al.  ∃R-Completeness for Decision Versions of Multi-Player (Symmetric) Nash Equilibria , 2018, ACM Trans. Economics and Comput..

[3]  Vittorio Bilò,et al.  Complexity of Rational and Irrational Nash Equilibria , 2013, Theory of Computing Systems.

[4]  S. Smale,et al.  On a theory of computation and complexity over the real numbers; np-completeness , 1989 .

[5]  R. Aumann.  Acceptable points in games of perfect information. , 1960 .

[6]  Jeffrey Shallit,et al.  The Computational Complexity of Some Problems of Linear Algebra , 1996 .

[7]  Tuomas Sandholm,et al.  On the verification and computation of strong nash equilibrium , 2013, AAMAS.

[8]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I: Introduction. Preliminaries. The Geometry of Semi-Algebraic Sets. The Decision Problem for the Existential Theory of the Reals , 1992, J. Symb. Comput..

[9]  Kousha Etessami,et al.  On the Complexity of Nash Equilibria and Other Fixed Points , 2010, SIAM J. Comput..

[10]  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).

[11]  Vittorio Bilò,et al.  A Catalog of EXISTS-R-Complete Decision Problems About Nash Equilibria in Multi-Player Games , 2016, STACS.

[12]  J. Nash.  NON-COOPERATIVE GAMES , 1951, Classics in Game Theory.

[13]  John F. Canny,et al.  Some algebraic and geometric computations in PSPACE , 1988, STOC '88.

[14]  Vincent Conitzer,et al.  New complexity results about Nash equilibria , 2008, Games Econ. Behav..

[15]  Ruta Mehta,et al.  Settling Some Open Problems on 2-Player Symmetric Nash Equilibria , 2015, SAGT.

[16]  J. Koenigsmann.  Defining Z in Q , 2010 .

[17]  Kristoffer Arnsfelt Hansen.  The Real Computational Complexity of Minmax Value and Equilibrium Refinements in Multi-player Games , 2018, Theory of Computing Systems.

[18]  Marcus Schaefer,et al.  Complexity of Some Geometric and Topological Problems , 2009, GD.

[19]  Marcus Schaefer,et al.  Fixed Points, Nash Equilibria, and the Existential Theory of the Reals , 2017, Theory of Computing Systems.

[20]  J. Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I , 1989 .

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

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

[23]  Eitan Zemel,et al.  Nash and correlated equilibria: Some complexity considerations , 1989 .

[24]  S. Basu,et al.  Algorithms in real algebraic geometry , 2003 .

[25]  Vittorio Bilò,et al.  A Catalog of ∃ R-Complete Decision Problems About Nash Equilibria in Multi-Player Games , 2016 .

[26]  Felipe Cucker,et al.  Exotic Quantifiers, Complexity Classes, and Complete Problems , 2007, ICALP.