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Equilibria, Fixed Points, and Complexity Classes Yannakakis, Mihalis Equilibria Fixed points Computational Complexity Game Theory Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind. Examples include Nash equilibria in games; market equilibria; computing optimal strategies and the values of competitive games (stochastic and other games); stable configurations of neural networks; analysing basic stochastic models for evolution like branching processes and for language like stochastic context-free grammars; and models that incorporate the basic primitives of probability and recursion like recursive Markov chains. It is not known whether these problems can be solved in polynomial time. There are certain common computational principles underlying different types of equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP. Representative complete problems for these classes are respectively, pure Nash equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria in 2-player normal form games, and (mixed) Nash equilibria in normal form games with 3 (or more) players. This paper reviews the underlying computational principles and the corresponding classes. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Mihalis Yannakakis 2008 Is Part Of LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2008.1311 urn:nbn:de:0030-drops-13110 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1311 eng https://creativecommons.org/licenses/by-nd/3.0/legalcode