Equilibria, Fixed Points, and Complexity Classes

TL;DR

This paper reviews the computational complexity of finding equilibria and fixed points across various models, highlighting the roles of classes like PLS, PPAD, and FIXP in understanding these problems.

Contribution

It provides a comprehensive overview of the computational principles and complexity classes related to equilibrium and fixed point problems in diverse models.

Findings

Identifies common computational principles underlying equilibrium problems

Classifies key problems into complexity classes PLS, PPAD, and FIXP

Summarizes the computational difficulty of various equilibrium computations

Abstract

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.