Communications of the ACM
On nash-equilibria of approximation-stable games
SAGT'10 Proceedings of the Third international conference on Algorithmic game theory
Rationality and Strongly Polynomial Solvability of Eisenberg-Gale Markets with Two Agents
SIAM Journal on Discrete Mathematics
Market equilibrium under separable, piecewise-linear, concave utilities
Journal of the ACM (JACM)
Complexity and economics: computational constraints may not matter empirically
ACM SIGecom Exchanges
A revealed preference approach to computational complexity in economics
Proceedings of the 12th ACM conference on Electronic commerce
The complexity of nash equilibria in limit-average games
CONCUR'11 Proceedings of the 22nd international conference on Concurrency theory
Complexity of rational and irrational Nash equilibria
SAGT'11 Proceedings of the 4th international conference on Algorithmic game theory
Approximation algorithm for security games with costly resources
WINE'11 Proceedings of the 7th international conference on Internet and Network Economics
Polynomial time algorithms for multi-type branching processesand stochastic context-free grammars
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
A complementary pivot algorithm for markets under separable, piecewise-linear concave utilities
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
On the complexity of computing probabilistic bisimilarity
FOSSACS'12 Proceedings of the 15th international conference on Foundations of Software Science and Computational Structures
Computing game metrics on markov decision processes
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part II
On the Computational Complexity of Stochastic Controller Optimization in POMDPs
ACM Transactions on Computation Theory (TOCT)
WINE'12 Proceedings of the 8th international conference on Internet and Network Economics
The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions
ACM Transactions on Economics and Computation - Special Issue on Algorithmic Game Theory
The empirical implications of rank in Bimatrix games
Proceedings of the fourteenth ACM conference on Electronic commerce
The complexity of non-monotone markets
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Complexity of Rational and Irrational Nash Equilibria
Theory of Computing Systems
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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.