Constant Inapproximability for Fisher Markets

TL;DR

This paper proves that computing constant-factor approximate market equilibria in Fisher markets with SPLC utilities is PPAD-hard, establishing inapproximability bounds and extending results to Arrow-Debreu markets.

Contribution

It demonstrates PPAD-hardness for constant approximations in Fisher markets with SPLC utilities, strengthening previous inapproximability results.

Findings

Computing any approximation better than 1/11 is PPAD-complete.

The inapproximability result applies to both Fisher and Arrow-Debreu markets with SPLC utilities.

No polynomial-time scheme can achieve constant approximation unless PPAD=P.

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.