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Fairness in Limited Resources Settings

Eitan Bachmat, Inbal Livni Navon

TL;DR

An adaptation of proportional fairness is introduced and it is shown that it has a bounded price of fairness, indicating greater robustness, and a variant of equal opportunity is proposed that also has a bounded price of fairness.

Abstract

In recent years many important societal decisions are made by machine-learning algorithms, and many such important decisions have strict capacity limits, allowing resources to be allocated only to the highest utility individuals. For example, allocating physician appointments to the patients most likely to have some medical condition, or choosing which children will attend a special program. When performing such decisions, we consider both the prediction aspect of the decision and the resource allocation aspect. In this work we focus on the fairness of the decisions in such settings. The fairness aspect here is critical as the resources are limited, and allocating the resources to one individual leaves less resources for others. When the decision involves prediction together with the resource allocation, there is a risk that information gaps between different populations will lead to a very unbalanced allocation of resources. We address settings by adapting definitions from resource allocation schemes, identifying connections between the algorithmic fairness definitions and resource allocation ones, and examining the trade-offs between fairness and utility. We analyze the price of enforcing the different fairness definitions compared to a strictly utility-based optimization of the predictor, and show that it can be unbounded. We introduce an adaptation of proportional fairness and show that it has a bounded price of fairness, indicating greater robustness, and propose a variant of equal opportunity that also has a bounded price of fairness.

Fairness in Limited Resources Settings

TL;DR

An adaptation of proportional fairness is introduced and it is shown that it has a bounded price of fairness, indicating greater robustness, and a variant of equal opportunity is proposed that also has a bounded price of fairness.

Abstract

In recent years many important societal decisions are made by machine-learning algorithms, and many such important decisions have strict capacity limits, allowing resources to be allocated only to the highest utility individuals. For example, allocating physician appointments to the patients most likely to have some medical condition, or choosing which children will attend a special program. When performing such decisions, we consider both the prediction aspect of the decision and the resource allocation aspect. In this work we focus on the fairness of the decisions in such settings. The fairness aspect here is critical as the resources are limited, and allocating the resources to one individual leaves less resources for others. When the decision involves prediction together with the resource allocation, there is a risk that information gaps between different populations will lead to a very unbalanced allocation of resources. We address settings by adapting definitions from resource allocation schemes, identifying connections between the algorithmic fairness definitions and resource allocation ones, and examining the trade-offs between fairness and utility. We analyze the price of enforcing the different fairness definitions compared to a strictly utility-based optimization of the predictor, and show that it can be unbounded. We introduce an adaptation of proportional fairness and show that it has a bounded price of fairness, indicating greater robustness, and propose a variant of equal opportunity that also has a bounded price of fairness.
Paper Structure (29 sections, 6 theorems, 97 equations, 3 figures)

This paper contains 29 sections, 6 theorems, 97 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Adapting resource allocation definitions
  4. Connecting group fairness definitions to resource allocation fairness
  5. Finding the price of fairness
  6. Suggesting new fairness definitions
  7. Related works
  8. Open questions and discussion
  9. Preliminaries and notations
  10. Loss minimization
  11. Group fairness definitions
  12. Resource allocation fairness definitions in the prediction setting
  13. Connections between fairness definitions
  14. Equal opportunity and max-min fairness
  15. Non-equal base rates
  16. ...and 14 more sections

Key Result

Lemma 4.4

For $c<0.5$, let $\mathcal{D}$ be a not degenerate distribution with respect to groups $S_1,\ldots,S_m$, capacity $2c$ and function family $\mathcal{F}$ that is closed under group post-processing. Let $\mathcal{F}_{EO}\subseteq\mathcal{F}$ be the set of functions in $\mathcal{F}$ satisfying equal op

Figures (3)

  • Figure 1: Simulating the output distribution of a predictor $p$ with a larger variance on $S_1$ (in blue) compared to $S_2$. The red line is a capacity limit of $1\%$ of the population, and it contains a large majority of individuals from $S_1$. See Appendix \ref{['app:simulations']} for more information.
  • Figure 2: Simulating the output distribution of a predictor $p$ with a large gap between the distribution over $S_1$ and $S_2$. In such cases the true positive count with fairness requirements is significantly larger than without. See Appendix \ref{['app:simulations']} for more information.
  • Figure 3: Simulating the output distribution for higher capacity under different type of tail dominance. See Appendix \ref{['app:simulations']}.

Theorems & Definitions (58)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Claim 2.4
  • Definition 2.5
  • Definition 2.6
  • Claim 2.7
  • Definition 2.8: Demographic parity
  • Definition 2.9: Equal opportunity hardt2016equality
  • Definition 2.10: Group calibration
  • ...and 48 more