FIXP

In computer science, FIXP is a complexity class introduced by Kousha Etessami and Mihalis Yannakakis at 2010.^[1] It represents problems that can be solved by computing a fixed point of a function that satisfies the conditions of Brouwer's fixed point theorem. More formally, FIXP contains search problems that can be cast as fixed point computation problems for functions represented by algebraic circuits over basis {+,*,-,/,max,min} with rational constants.

They prove that some fundamental problems in economics and game theory are complete for FIXP, in particular:

• Nash equilibrium computation - computing an exact or approximate mixed-strategy Nash equilibrium in a game with three or more players.
• Market equilibrium computation - computing competitive-equilibrium prices for exchange economies with algebraic demand functions.

Relations to other classes

The piecewise-linear fragment of FIXP equals PPAD.

Computing an approximate Nash equilibrium to any factor smaller than 1/2 is at least as hard as the square-root sum problem.