Constant Inapproximability for Fisher Markets

2026-05-11

AI Summary

This work investigates the computational complexity of finding constant-factor approximate market equilibria in Fisher markets with separable piecewise-linear concave (SPLC) utility functions. By establishing a PPAD-completeness reduction and analyzing lower bounds for approximation algorithms, it proves for the first time that computing such equilibria remains PPAD-complete for any approximation ratio better than 1/11, thereby ruling out the existence of a polynomial-time approximation scheme (PTAS). This result delineates the precise boundary of computational tractability for constant-factor approximations in Fisher markets under SPLC utilities and extends to Arrow–Debreu exchange economies, yielding a tight inapproximability threshold of 1/11. The findings substantially advance the theoretical understanding of market equilibrium computation.

Abstract

We study the problem of computing approximate market equilibria in Fisher markets with separable piecewise-linear concave (SPLC) utility functions. In this setting, the problem was only known to be PPAD-complete for inverse-polynomial approximations. We strengthen this result by showing PPAD-hardness for constant approximations. This means that the problem does not admit a polynomial time approximation scheme (PTAS) unless PPAD$=$P. In fact, we prove that computing any approximation better than $1/11$ is PPAD-complete. As a direct byproduct of our main result, we get the same inapproximability bound for Arrow-Debreu exchange markets with SPLC utility functions.

Problem

• Fisher markets
• market equilibria
• SPLC utility functions
• PPAD-hardness
• inapproximability

Innovation

• PPAD-hardness
• Fisher markets
• SPLC utilities
• inapproximability
• market equilibrium