The Complexity of Non-Monotone Markets

TL;DR

This paper proves that computing approximate market equilibria in non-monotone utility markets, including those with CES functions and certain parameters, is computationally PPAD-hard or PPAD-complete, highlighting fundamental complexity barriers.

Contribution

It introduces non-monotone utilities and establishes PPAD-hardness and completeness results for computing market equilibria with these utilities, resolving open problems in economic computation.

Findings

PPAD-hardness for approximate equilibria with linear non-monotone utilities

PPAD-completeness for CES utilities with elasticity parameter less than -1

Resolution of a long-standing open problem in market equilibrium computation

Abstract

We introduce the notion of non-monotone utilities, which covers a wide variety of utility functions in economic theory. We then prove that it is PPAD-hard to compute an approximate Arrow-Debreu market equilibrium in markets with linear and non-monotone utilities. Building on this result, we settle the long-standing open problem regarding the computation of an approximate Arrow-Debreu market equilibrium in markets with CES utility functions, by proving that it is PPAD-complete when the Constant Elasticity of Substitution parameter \rho is any constant less than -1.