Equitable Mechanism Design for Facility Location

TL;DR

This paper explores strategy-proof mechanisms for facility location that aim to maximize equitability among agents, introducing new approximation approaches for the Gini index and analyzing Nash welfare as an alternative.

Contribution

It presents a fundamental impossibility result for bounding Gini index approximation and proposes new methods to approximate the complemented Gini index and Nash welfare.

Findings

No strategy proof mechanism can bound the optimal Gini index approximation.

Proposed approximation ratios for complemented Gini index with deterministic and randomized mechanisms.

Analysis of how mechanisms approximate Nash welfare.

Abstract

We consider strategy proof mechanisms for facility location which maximize equitability between agents. As is common in the literature, we measure equitability with the Gini index. We first prove a simple but fundamental impossibility result that no strategy proof mechanism can bound the approximation ratio of the optimal Gini index of utilities for one or more facilities. We propose instead computing approximation ratios of the complemented Gini index of utilities, and consider how well both deterministic and randomized mechanisms approximate this. In addition, as Nash welfare is often put forwards as an equitable compromise between egalitarian and utilitarian outcomes, we consider how well mechanisms approximate the Nash welfare.