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Proportional Fairness in Obnoxious Facility Location

Alexander Lam, Haris Aziz, Bo Li, Fahimeh Ramezani, Toby Walsh

TL;DR

This work considers the obnoxious facility location problem and proposes a hierarchy of distance-based proportional fairness concepts that ensure that groups of agents at the same location are guaranteed to be a distance from the facility proportional to their group size.

Abstract

We consider the obnoxious facility location problem (in which agents prefer the facility location to be far from them) and propose a hierarchy of distance-based proportional fairness concepts for the problem. These fairness axioms ensure that groups of agents at the same location are guaranteed to be a distance from the facility proportional to their group size. We consider deterministic and randomized mechanisms, and compute tight bounds on the price of proportional fairness. In the deterministic setting, we show that our proportional fairness axioms are incompatible with strategyproofness, and prove asymptotically tight $ε$-price of anarchy and stability bounds for proportionally fair welfare-optimal mechanisms. In the randomized setting, we identify proportionally fair and strategyproof mechanisms that give an expected welfare within a constant factor of the optimal welfare. Finally, we prove existence results for two extensions to our model.

Proportional Fairness in Obnoxious Facility Location

TL;DR

This work considers the obnoxious facility location problem and proposes a hierarchy of distance-based proportional fairness concepts that ensure that groups of agents at the same location are guaranteed to be a distance from the facility proportional to their group size.

Abstract

We consider the obnoxious facility location problem (in which agents prefer the facility location to be far from them) and propose a hierarchy of distance-based proportional fairness concepts for the problem. These fairness axioms ensure that groups of agents at the same location are guaranteed to be a distance from the facility proportional to their group size. We consider deterministic and randomized mechanisms, and compute tight bounds on the price of proportional fairness. In the deterministic setting, we show that our proportional fairness axioms are incompatible with strategyproofness, and prove asymptotically tight -price of anarchy and stability bounds for proportionally fair welfare-optimal mechanisms. In the randomized setting, we identify proportionally fair and strategyproof mechanisms that give an expected welfare within a constant factor of the optimal welfare. Finally, we prove existence results for two extensions to our model.
Paper Structure (35 sections, 41 theorems, 51 equations, 3 figures, 1 table)

This paper contains 35 sections, 41 theorems, 51 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Model
  5. Proportional Fairness Axioms
  6. Individual Fair Share
  7. Unanimous Fair Share
  8. Deterministic Setting
  9. Price of Fairness
  10. Utilitarian Welfare
  11. Egalitarian Welfare
  12. Incompatibility with Strategyproofness
  13. $\epsilon$-Price of Anarchy and $\epsilon$-Price of Stability
  14. Randomized Mechanisms
  15. Strategyproofness
  16. ...and 20 more sections

Key Result

Proposition 1

The lowest value of $\alpha$ for which an $\alpha$-IFS solution always exists is $\alpha=2$.

Figures (3)

  • Figure 1: OFLP with agent location profile $(0.1,0.1,0.8,0.8, 0.8,0.8)$ represented by x. The facility locations (represented by •) correspond to a utilitarian outcome, $f^*_{UW}=0$; a proportionally fair outcome, $\textmd{2-UFS}=0.3$; and an egalitarian outcome, $f^*_{EW}=0.45$.
  • Figure 2: The lower bound instance in the proof of Theorem \ref{['thm: 2-IFS UW']} for $n=4$. $f^*_{UW}$ represents the utilitarian welfare maximizing facility placement, whilst $f^*_{2IFS}$ maximizes utilitarian welfare under the constraints of 2-IFS. The red intervals denote locations that are infeasible under 2-IFS.
  • Figure 3: The instance in the proof of Theorem \ref{['thm: 2-UFS EW']}. $f^*_{EW}$ represents the egalitarian welfare maximizing facility placement, whilst $2UFS(x)$ represents the interval of facility placements satisfying 2-UFS. The red intervals denote locations that are infeasible under 2-UFS.

Theorems & Definitions (81)

  • Definition 1: $\alpha$-Individual Fair Share (IFS)
  • Proposition 1
  • proof
  • Definition 2: $\alpha$-Unanimous Fair Share (UFS)
  • Proposition 2
  • proof
  • Definition 3: Price of Fairness for Utilitarian/Egalitarian Welfare
  • Theorem 1
  • proof : Lower Bound Proof
  • Theorem 2
  • ...and 71 more