Equilibria, fixed points, and complexity classes

Mihalis Yannakakis
<span title="">2009</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/x3ibtg2oifh47c73tgo66gjrw4" style="color: black;">Computer Science Review</a> </i> &nbsp;
A B S T R A C T 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
more &raquo; ... 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 two-player normal form games, and (mixed) Nash equilibria in normal form games with three (or more) players. This paper reviews the underlying computational principles and the corresponding classes. . games); stable configurations of neural networks; analysis of 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. Most of these models and problems have been studied mathematically for a long time, leading to the development of rich theories. Yet, some of their most basic algorithmic questions are still not resolved; in particular, it is not known whether they can be solved in polynomial time. Despite the broad diversity of these problems, there are certain common computational principles that underlie many of these different types of problems, which are captured by the complexity classes PLS, PPAD, and FIXP. In this paper we will review these principles, the corresponding classes, and the types of problems they contain. 1574-0137/$ -see front matter c
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