Nash Welfare and Facility Location

TL;DR

This paper studies the problem of locating a facility along a line to maximize Nash welfare, proposing a polynomial-time approximation algorithm and a strategy-proof mechanism that balance fairness and efficiency.

Contribution

It introduces the first polynomial-time approximation algorithm for maximizing Nash welfare in facility location and designs a strategy-proof mechanism with bounded approximation ratio.

Findings

The approximation algorithm effectively balances fairness and efficiency.

The strategy-proof mechanism ensures truthful reporting with bounded approximation.

The approach provides a new perspective on fair resource allocation in location problems.

Abstract

We consider the problem of locating a facility to serve a set of agents located along a line. The Nash welfare objective function, defined as the product of the agents' utilities, is known to provide a compromise between fairness and efficiency in resource allocation problems. We apply this welfare notion to the facility location problem, converting individual costs to utilities and analyzing the facility placement that maximizes the Nash welfare. We give a polynomial-time approximation algorithm to compute this facility location, and prove results suggesting that it achieves a good balance of fairness and efficiency. Finally, we take a mechanism design perspective and propose a strategy-proof mechanism with a bounded approximation ratio for Nash welfare.