Approximating Equilibrium under Constrained Piecewise Linear Concave Utilities with Applications to Matching Markets

TL;DR

This paper develops a fixed parameter approximation scheme for computing approximate equilibria in Fisher markets with constrained piecewise linear concave utilities, addressing computational hardness and extending to matching markets and Arrow-Debreu models.

Contribution

It introduces a novel fixed parameter approximation scheme for equilibrium computation in PLC utility markets, solving an open problem and simplifying existing algorithms.

Findings

Provides a polynomial-time approximation scheme parameterized by agents and accuracy.

Extends results to markets with satiation and Arrow-Debreu exchange markets.

Offers a faster, simpler algorithm for matching markets.

Abstract

We study the equilibrium computation problem in the Fisher market model with constrained piecewise linear concave (PLC) utilities. This general class captures many well-studied special cases, including markets with PLC utilities, markets with satiation, and matching markets. For the special case of PLC utilities, although the problem is PPAD-hard, Devanur and Kannan (FOCS 2008) gave a polynomial-time algorithm when the number of items is constant. Our main result is a fixed parameter approximation scheme for computing an approximate equilibrium, where the parameters are the number of agents and the approximation accuracy. This provides an answer to an open question by Devanur and Kannan for PLC utilities, and gives a simpler and faster algorithm for matching markets as the one by Alaei, Jalaly and Tardos (EC 2017). The main technical idea is to work with the stronger concept of thrifty equilibria, and approximating the input utility functions by `robust' utilities that have favorable marginal properties. With some restrictions, the results also extend to the Arrow--Debreu exchange market model.