Matrix analysis
Decreasing the nesting depth of expressions involving square roots
Journal of Symbolic Computation
Theory of linear and integer programming
Theory of linear and integer programming
Journal of Computer and System Sciences - 26th IEEE Conference on Foundations of Computer Science, October 21-23, 1985
Some algebraic and geometric computations in PSPACE
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Exponential lower bounds for finding Brouwer fixed points
Journal of Complexity
On total functions, existence theorems and computational complexity
Theoretical Computer Science
A bound on the proportion of pure strategy equilibria in generic games
Mathematics of Operations Research
A problem that is easier to solve on the unit-cost algebraic RAM
Journal of Complexity
The complexity of stochastic games
Information and Computation
On the complexity of the parity argument and other inefficient proofs of existence
Journal of Computer and System Sciences - Special issue: 31st IEEE conference on foundations of computer science, Oct. 22–24, 1990
Computational complexity of fixed points and intersection points
Journal of Complexity
Complexity and real computation
Complexity and real computation
Foundations of statistical natural language processing
Foundations of statistical natural language processing
Optimal solution of nonlinear equations
Optimal solution of nonlinear equations
On the Power of Random Access Machines
Proceedings of the 6th Colloquium, on Automata, Languages and Programming
The Analysis of Local Search Problems and Their Heuristics
STACS '90 Proceedings of the 7th Annual Symposium on Theoretical Aspects of Computer Science
Playing large games using simple strategies
Proceedings of the 4th ACM conference on Electronic commerce
A characterization of the class of functions computable in polynomial time on Random Access Machines
STOC '81 Proceedings of the thirteenth annual ACM symposium on Theory of computing
Some NP-complete geometric problems
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
Universality of Nash Equilibria
Mathematics of Operations Research
Derandomizing polynomial identity tests means proving circuit lower bounds
Computational Complexity
Reducibility among equilibrium problems
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Settling the Complexity of Two-Player Nash Equilibrium
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Computing Nash Equilibria: Approximation and Smoothed Complexity
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract)
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
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)
On the Complexity of Numerical Analysis
SIAM Journal on Computing
The Complexity of Computing a Nash Equilibrium
SIAM Journal on Computing
Recursive concurrent stochastic games
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part II
The game world is flat: the complexity of nash equilibria in succinct games
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
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 $\epsilon0$, compute an approximation within $\epsilon$ of some (actual) Nash equilibrium. We show that approximation of an actual Nash equilibrium, even to within any nontrivial constant additive factor $\epsilon We show similar results for market equilibria: it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium, where the economy is given by explicit algebraic formulas for the excess demand functions. We define a class, FIXP, which captures search problems 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 (exact or approximate) computation of Nash equilibria for 3 or more players is complete for FIXP. The price equilibrium problem for exchange economies with algebraic demand functions is another FIXP-complete problem. We show that the piecewise linear fragment of FIXP equals PPAD. Many other 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, probabilities of stochastic context-free grammars, and termination probabilities 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.