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Harm Ratio: A Novel and Versatile Fairness Criterion

Soroush Ebadian, Rupert Freeman, Nisarg Shah

Abstract

Envy-freeness has become the cornerstone of fair division research. In settings where each individual is allocated a disjoint share of collective resources, it is a compelling fairness axiom which demands that no individual strictly prefer the allocation of another individual to their own. Unfortunately, in many real-life collective decision-making problems, the goal is to choose a (common) public outcome that is equally applicable to all individuals, and the notion of envy becomes vacuous. Consequently, this literature has avoided studying fairness criteria that focus on individuals feeling a sense of jealousy or resentment towards other individuals (rather than towards the system), missing out on a key aspect of fairness. In this work, we propose a novel fairness criterion, individual harm ratio, which is inspired by envy-freeness but applies to a broad range of collective decision-making settings. Theoretically, we identify minimal conditions under which this criterion and its groupwise extensions can be guaranteed, and study the computational complexity of related problems. Empirically, we conduct experiments with real data to show that our fairness criterion is powerful enough to differentiate between prominent decision-making algorithms for a range of tasks from voting and fair division to participatory budgeting and peer review.

Harm Ratio: A Novel and Versatile Fairness Criterion

Abstract

Envy-freeness has become the cornerstone of fair division research. In settings where each individual is allocated a disjoint share of collective resources, it is a compelling fairness axiom which demands that no individual strictly prefer the allocation of another individual to their own. Unfortunately, in many real-life collective decision-making problems, the goal is to choose a (common) public outcome that is equally applicable to all individuals, and the notion of envy becomes vacuous. Consequently, this literature has avoided studying fairness criteria that focus on individuals feeling a sense of jealousy or resentment towards other individuals (rather than towards the system), missing out on a key aspect of fairness. In this work, we propose a novel fairness criterion, individual harm ratio, which is inspired by envy-freeness but applies to a broad range of collective decision-making settings. Theoretically, we identify minimal conditions under which this criterion and its groupwise extensions can be guaranteed, and study the computational complexity of related problems. Empirically, we conduct experiments with real data to show that our fairness criterion is powerful enough to differentiate between prominent decision-making algorithms for a range of tasks from voting and fair division to participatory budgeting and peer review.
Paper Structure (22 sections, 9 theorems, 13 equations, 7 figures, 1 table)

This paper contains 22 sections, 9 theorems, 13 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Summary of Significance and Implications
  4. Model
  5. Fairness
  6. Private Goods Division
  7. Harm Ratio: A Novel Fairness Criterion
  8. Comparison to Envy-Freeness and Proportionality
  9. Groupwise Extension
  10. Maximum Nash Welfare Solution
  11. Computation
  12. Experiments
  13. Private Goods
  14. Peer Review
  15. Participatory Budgeting
  16. ...and 7 more sections

Key Result

Proposition 1

Any proportionally fair outcome is also a maximum Nash welfare outcome and lies in the core. Any outcome in the core satisfies proportionality and Pareto optimality.

Figures (7)

  • Figure 1: Instance for \ref{['thm:pef-ef']}. Blue and red rectangles show two different allocations; the rectangle in each good's column covers the agents among which the good is equally divided. The blue allocation is EF, but $(n/2)$-IHR, as witnessed by the existence of the red allocation.
  • Figure 2: The figure depicts a hierarchy of fairness notions for private goods division (already known) and the public outcomes model (based on our novel fairness definitions and results). For private goods division, the implications marked with (⁎) hold for cake-cutting CFSW19FSV20. For the public outcomes model, we show the implications marked with (+) when the utility set $\mathcal{U}$ is compact and upper convex (Theorem \ref{['thm:mnw-pf']}).
  • Figure 3: Scatter plots showing instance-wise individual harm ratio and private envy ratio, and bar charts displaying their averages as well as the fraction of instances with infinite ratios. In the scatter plots, the inset plot shows the full set of instances, while the main plot is zoomed into its lower left region that contains approximately 90% of the instances.
  • Figure 4: Scatter plots showing instance-wise public and shuffle envy ratios and bar charts displaying average public and shuffle envy ratios. In the scatter plots, the main plot is zoomed into the lower left region of the graph that contains more than 90% of all data points -- except for CVPR'17 where CoBRA and TPMS lie far from the others -- while the inset plot shows the full set of instances.
  • Figure 5: Plots show normalized public group envy ratio as a function of the minimum size of the envious group of agents $S$.
  • ...and 2 more figures

Theorems & Definitions (29)

  • Definition 1: Proportional Fairness (PF)
  • Definition 2: Maximum Nash Welfare (MNW)
  • Definition 3: Core
  • Definition 4: Proportionality (Prop)
  • Definition 5: Pareto Optimality (PO)
  • Proposition 1: PF $\Rightarrow$ MNW, PF $\Rightarrow$ Core $\Rightarrow$ (Prop+PO)
  • proof
  • Remark 1
  • Definition 6: Envy-Freeness (EF)
  • Definition 7: Group Fairness (GF)
  • ...and 19 more