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

2026-04-29

AI Summary

This work investigates the computational complexity of approximating competitive equilibria in Fisher markets with piecewise-linear concave utility functions, where each buyer receives a $(1-\delta)$-approximately optimal bundle and the market clears up to an $\varepsilon$-approximation. By leveraging a reduction from the PPAD complexity class and assuming the PCP-for-PPAD conjecture, the paper establishes for the first time that this problem is PPAD-hard for any constant $\delta > 0$ and $\varepsilon < 1/9$. Notably, this constitutes the first natural economic equilibrium problem whose hardness proof inherently relies on the PCP-for-PPAD conjecture, thereby providing a theoretical lower bound for computing such equilibria and highlighting the conjecture’s essential role in characterizing the complexity of approximate market equilibria.

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 $(1-δ)$-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 $δ> 0$, 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 $δ$: 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.

Problem

• Fisher markets
• competitive equilibrium
• approximate optimality
• PPAD-hardness
• SPLC utilities

Innovation

• Fisher markets
• PPAD-hardness
• approximate competitive equilibrium
• PCP-for-PPAD conjecture
• SPLC utilities