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Fairness and Efficiency in Online Class Matching

MohammadTaghi Hajiaghayi, Shayan Chashm Jahan, Mohammad Sharifi, Suho Shin, Max Springer

TL;DR

This work defines the ``price of fairness,'' which represents the trade-off between optimal and fair matching, and demonstrates that increasing the level of fairness in the approximation of the solution leads to a decrease in the objective of maximizing USW, following an inverse proportionality relationship.

Abstract

The online bipartite matching problem, extensively studied in the literature, deals with the allocation of online arriving vertices (items) to a predetermined set of offline vertices (agents). However, little attention has been given to the concept of class fairness, where agents are categorized into different classes, and the matching algorithm must ensure equitable distribution across these classes. We here focus on randomized algorithms for the fair matching of indivisible items, subject to various definitions of fairness. Our main contribution is the first (randomized) non-wasteful algorithm that simultaneously achieves a $1/2$ approximation to class envy-freeness (CEF) while simultaneously ensuring an equivalent approximation to the class proportionality (CPROP) and utilitarian social welfare (USW) objectives. We supplement this result by demonstrating that no non-wasteful algorithm can achieve an $α$-CEF guarantee for $α> 0.761$. In a similar vein, we provide a novel input instance for deterministic divisible matching that demonstrates a nearly tight CEF approximation. Lastly, we define the ``price of fairness,'' which represents the trade-off between optimal and fair matching. We demonstrate that increasing the level of fairness in the approximation of the solution leads to a decrease in the objective of maximizing USW, following an inverse proportionality relationship.

Fairness and Efficiency in Online Class Matching

TL;DR

This work defines the ``price of fairness,'' which represents the trade-off between optimal and fair matching, and demonstrates that increasing the level of fairness in the approximation of the solution leads to a decrease in the objective of maximizing USW, following an inverse proportionality relationship.

Abstract

The online bipartite matching problem, extensively studied in the literature, deals with the allocation of online arriving vertices (items) to a predetermined set of offline vertices (agents). However, little attention has been given to the concept of class fairness, where agents are categorized into different classes, and the matching algorithm must ensure equitable distribution across these classes. We here focus on randomized algorithms for the fair matching of indivisible items, subject to various definitions of fairness. Our main contribution is the first (randomized) non-wasteful algorithm that simultaneously achieves a approximation to class envy-freeness (CEF) while simultaneously ensuring an equivalent approximation to the class proportionality (CPROP) and utilitarian social welfare (USW) objectives. We supplement this result by demonstrating that no non-wasteful algorithm can achieve an -CEF guarantee for . In a similar vein, we provide a novel input instance for deterministic divisible matching that demonstrates a nearly tight CEF approximation. Lastly, we define the ``price of fairness,'' which represents the trade-off between optimal and fair matching. We demonstrate that increasing the level of fairness in the approximation of the solution leads to a decrease in the objective of maximizing USW, following an inverse proportionality relationship.

Paper Structure

This paper contains 27 sections, 14 theorems, 45 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Work
  4. Online Matching.
  5. Fair Division.
  6. Fair Online Matching.
  7. Price of Fairness
  8. Model
  9. Matching.
  10. Class Valuations.
  11. Definitions of Fairness
  12. Definitions of Efficiency.
  13. Online Model
  14. Randomized Algorithms
  15. Improved CEF Upper Bounds
  16. ...and 12 more sections

Key Result

Theorem 1.1

For randomized matching of indivisible items, the Random algorithm satisfies non-wastefulness, $\frac{1}{2}$-CEF, $\frac{1}{2}$-CPROP, and $\frac{1}{2}$-USW.

Figures (3)

  • Figure 1: Examples of class envy-free (CEF) and non-wasteful (NW) matchings where bolded lines indicate a matching. Red nodes indicate agents in the first class, blue nodes indicate agents in the second class, and white nodes indicate items.
  • Figure 2: Impossibility constructions for the upper bound results of Theorems \ref{['thm:cef_upper']} and \ref{['thm:divisible']}. (a) the indivisible setting construction for an at most $\left(\frac{e^2-1}{e^2+1}\right)$-CEF approximation, (b) the divisible setting construction for an at most 0.677-CEF approximation.
  • Figure 3: Hardness instance for Theorem \ref{['thm:cef_not_nsw']}.

Theorems & Definitions (49)

  • Theorem 1.1: Randomized algorithm; informal
  • Theorem 1.2: Indivisible CEF upper bound; Informal
  • Theorem 1.3: Divisible CEF upper bound; Informal
  • Theorem 1.4: Price of fairness; Informal
  • Definition 2.1: Class Envy-Freeness
  • Definition 2.2: Class Envy-Freeness Up to One Item
  • Definition 2.3: Class Proportional Fairness
  • Definition 2.4: Non-Wastefulness
  • Definition 2.5: Utilitarian Social Welfare
  • Proposition 2.6
  • ...and 39 more