 Abstract:

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 epsilon > 0, compute a vector x' (a mixed strategy profile) that is within distance \epsilon (say, in l_infinity) of some (exact) Nash equilibrium.
We show that approximation of an (actual) Nash equilibrium for games with 3 players, even to within any nontrivial constant additive factor epsilon < 1/2 in just one desired coordinate, is at least as hard as the long standing squareroot 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 contextfree 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 squareroot sum and arithmetic circuit decision problems.
 Copyright:
 2007
 Links To Paper
 Link to full version of tech report
 Bibtex format
 @InProceedings{EDIINFRR1164,
 author = {
Kousha Etessami
and Mihalis Yannakakis
},
 title = {On the Complexity of Nash Equilibria and Other Fixed Points},
 book title = {Proceedings of 48th IEEE Symp. on Foundations of Computer Science (FOCS), 2007.},
 publisher = {IEEE},
 year = 2007,
 month = {Oct},
 doi = {10.1109/FOCS.2007.39},
 url = {http://homepages.inf.ed.ac.uk/kousha/nash_focs07_full_j_spec_issue_sub.pdf},
 }
