Auction Algorithms for Market Equilibrium with Weak Gross Substitute Demands

TL;DR

This paper introduces a simple auction algorithm for approximate market equilibrium in Arrow-Debreu markets with weak gross substitutes demands, leveraging a price update oracle, and applies it to approximate Nash social welfare solutions.

Contribution

It presents a novel auction algorithm for WGS demands using a price oracle, and provides the first polynomial-time approximation for a relaxed Nash social welfare problem.

Findings

Efficient auction algorithm for WGS demands with a price oracle.

Polynomial-time approximation for Nash social welfare with capped utilities.

Implementation of specific oracles for bounded elasticities and Gale demand systems.

Abstract

We consider the Arrow--Debreu exchange market model under the assumption that the agents' demands satisfy the weak gross substitutes (WGS) property. We present a simple auction algorithm that obtains an approximate market equilibrium for WGS demands assuming the availability of a price update oracle. We exhibit specific implementations of such an oracle for WGS demands with bounded price elasticities and for Gale demand systems. As an application of our result, we obtain an efficient algorithm to find an approximate spending-restricted market equilibrium for WGS demands, a model that has been recently introduced as a continuous relaxation of the Nash social welfare (NSW) problem. This leads to a polynomial-time constant factor approximation algorithm for the NSW problem with capped additive separable piecewise linear utility functions; only a pseudopolynomial approximation algorithm was known for this setting previously.