Automata, Probability, and Recursion
CIAA '08 Proceedings of the 13th international conference on Implementation and Applications of Automata
The complexity of computing a Nash equilibrium
Communications of the ACM - Inspiring Women in Computing
Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations
Journal of the ACM (JACM)
Settling the complexity of computing two-player Nash equilibria
Journal of the ACM (JACM)
The Complexity of Nash Equilibria in Simple Stochastic Multiplayer Games
ICALP '09 Proceedings of the 36th Internatilonal Collogquium on Automata, Languages and Programming: Part II
On the complexity of constrained Nash equilibria in graphical games
Theoretical Computer Science
New algorithms for approximate Nash equilibria in bimatrix games
Theoretical Computer Science
Computational Aspects of Equilibria
SAGT '09 Proceedings of the 2nd International Symposium on Algorithmic Game Theory
Computing equilibria by incorporating qualitative models?
Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems: volume 1 - Volume 1
A direct reduction from k-player to 2-player approximate nash equilibrium
SAGT'10 Proceedings of the Third international conference on Algorithmic game theory
The computational complexity of trembling hand perfection and other equilibrium refinements
SAGT'10 Proceedings of the Third international conference on Algorithmic game theory
2-Player nash and nonsymmetric bargaining games: algorithms and structural properties
SAGT'10 Proceedings of the Third international conference on Algorithmic game theory
On the Complexity of Nash Equilibria and Other Fixed Points
SIAM Journal on Computing
Approximate nash equilibria in bimatrix games
ICCCI'11 Proceedings of the Third international conference on Computational collective intelligence: technologies and applications - Volume Part II
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
On the complexity of approximating a Nash equilibrium
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Survey: Equilibria, fixed points, and complexity classes
Computer Science Review
Survey: Nash equilibria: Complexity, symmetries, and approximation
Computer Science Review
New differential evolution selective mutation operator for the nash equilibria problem
ICCCI'12 Proceedings of the 4th international conference on Computational Collective Intelligence: technologies and applications - Volume Part II
On the Complexity of Approximating a Nash Equilibrium
ACM Transactions on Algorithms (TALG) - Special Issue on SODA'11
Differential evolution as a new method of computing nash equilibria
Transactions on Computational Collective Intelligence IX
Hi-index | 0.00 |
We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, \Gamma, with 3 or more players, and given\in \le 0, compute a vector {x'} (a mixed strategy profile) that is within distance\in (say, in l\infty) of some (exact) Nash equilibrium. We show that approximation of an (actual) Nash equilibrium for games with 3 players, even to within any non-trivial constant additive factor \in \le 1/2 in just one desired coordinate, is at least as hard as the long standing square-root sum problem, as well as more general arithmetic circuit decision problems, and thus that even placing the approximation problem in NP would resolve a major open problem in the complexity of numerical computation. Furthermore, we show that the (exact or approximate) computation of Nash equilibria for 3 or more players is complete for the class of search problems, which we call FIXP, that can be cast as fixed point computation problems for functions represented by algebraic circuits (straight line programs) over basis {+, *,-, /,max, min}, with rational constants. We show that the linear fragment of FIXP equals PPAD.Many problems in game theory, economics, and probability theory, can be cast as fixed point problems for such algebraic functions. We discuss several important such problems: computing the value of Shapley's stochastic games, and the simpler games of Condon, extinction probabilities of branching processes, termination probabilities of stochastic context-free grammars, and of Recursive Markov Chains. We show that for some of them, the approximation, or even exact computation, problem can be placed in PPAD, while for others, they are at least as hard as the square-root sum and arithmetic circuit decision problems.