Strategyproof and Proportionally Fair Facility Location

TL;DR

This paper investigates strategyproof and proportionally fair mechanisms for a one-dimensional facility location problem, characterizing their properties, uniqueness, and approximation guarantees for social welfare and costs.

Contribution

It introduces a hierarchy of fairness axioms, characterizes strategyproof mechanisms satisfying these axioms, and identifies the unique Uniform Phantom mechanism with optimal approximation guarantees.

Findings

Many fairness axioms become equivalent under strategyproofness.

The Uniform Phantom mechanism is unique and strategyproof, satisfying multiple fairness axioms.

It achieves the best approximation of social welfare among strategyproof, fair mechanisms.

Abstract

We focus on a simple, one-dimensional collective decision problem (often referred to as the facility location problem) and explore issues of strategyproofness and proportionality-based fairness. We introduce and analyze a hierarchy of proportionality-based fairness axioms of varying strength: Individual Fair Share (IFS), Unanimous Fair Share (UFS), Proportionality (as in Freeman et al, 2021), and Proportional Fairness (PF). For each axiom, we characterize the family of mechanisms that satisfy the axiom and strategyproofness. We show that imposing strategyproofness renders many of the axioms to be equivalent: the family of mechanisms that satisfy proportionality, unanimity, and strategyproofness is equivalent to the family of mechanisms that satisfy UFS and strategyproofness, which, in turn, is equivalent to the family of mechanisms that satisfy PF and strategyproofness. Furthermore, there is a unique such mechanism: the Uniform Phantom mechanism, which is studied in Freeman et al. (2021). We also characterize the outcomes of the Uniform Phantom mechanism as the unique (pure) equilibrium outcome for any mechanism that satisfies continuity, strict monotonicity, and UFS. Finally, we analyze the approximation guarantees, in terms of optimal social welfare and minimum total cost, obtained by mechanisms that are strategyproof and satisfy each proportionality-based fairness axiom. We show that the Uniform Phantom mechanism provides the best approximation of the optimal social welfare (and also minimum total cost) among all mechanisms that satisfy UFS.