Fisher Markets with Approximately Optimal Bundles and the Need for a PCP Theorem for PPAD

TL;DR

This paper investigates the computational complexity of finding approximately optimal bundles in Fisher markets with SPLC utilities, establishing PPAD-hardness under a key conjecture and highlighting the conjecture's necessity for such hardness results.

Contribution

It proves PPAD-hardness for computing approximate market equilibria with constant accuracy, assuming the PCP-for-PPAD conjecture, and shows this conjecture is essential for such hardness proofs.

Findings

PPAD-hardness holds for some constant δ > 0 under the PCP-for-PPAD conjecture.

Hardness persists even with identical budgets and linear capped utilities.

The PCP-for-PPAD conjecture is necessary to prove hardness for these market problems.

Abstract

We study the problem of computing a competitive equilibrium with approximately optimal bundles in Fisher markets with separable piecewise-linear concave (SPLC) utility functions, meaning that every buyer receives a $(1-\delta)$-optimal bundle, instead of a perfectly optimal one. We establish the first intractability result for the problem by showing that it is PPAD-hard for some constant $\delta > 0$, assuming the PCP-for-PPAD conjecture. This hardness result holds even if all buyers have identical budgets (competitive equilibrium with equal incomes), linear capped utilities, and even if we also allow $\varepsilon$-approximate clearing instead of perfect clearing, for any constant $\varepsilon < 1/9$. Importantly, we show that the PCP-for-PPAD conjecture is in fact required to show hardness for constant $\delta$: showing PPAD-hardness for finding such approximate market equilibria in a broad class of markets encompassing those generated by our hardness result would prove the conjecture. This is the first natural problem where the conjecture is provably required to establish hardness for it.