NDLI: On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract)

Content Provider	IEEE Xplore Digital Library
Author	Etessami, K. Yannakakis, M.
Copyright Year	2007
Description	Author affiliation: Edinburgh Univ., Edinburgh (Etessami, K.)
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 epsiv > 0, compute a vector x' (a mixed strategy profile) that is within distance e (say in t^) 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 epsiv < 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.
Starting Page	113
Ending Page	123
File Size	347066
Page Count	11
File Format	PDF
ISBN	9780769530109
ISSN	02725428
DOI	10.1109/FOCS.2007.39
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2007-10-21
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Game theory Computer science Nash equilibrium Circuits Stochastic processes Informatics Search problems Fixed-point arithmetic Equations History
Content Type	Text
Resource Type	Article

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