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Fairness in Repeated Matching: A Maximin Perspective

Eugene Lim, Tzeh Yuan Neoh, Nicholas Teh

TL;DR

The computational challenges associated with finding (anytime) optimal outcomes are investigated and it is demonstrated that these problems are generally computationally intractable but also several special cases whereby the problem(s) can be solved efficiently.

Abstract

We study a sequential decision-making model where a set of items is repeatedly matched to the same set of agents over multiple rounds. The objective is to determine a sequence of matchings that either maximizes the utility of the least advantaged agent at the end of all rounds (optimal) or at the end of every individual round (anytime optimal). We investigate the computational challenges associated with finding (anytime) optimal outcomes and demonstrate that these problems are generally computationally intractable. However, we provide approximation algorithms, fixed-parameter tractable algorithms, and identify several special cases whereby the problem(s) can be solved efficiently. Along the way, we also establish characterizations of Pareto-optimal/maximum matchings, which may be of independent interest to works in matching theory and house allocation.

Fairness in Repeated Matching: A Maximin Perspective

TL;DR

The computational challenges associated with finding (anytime) optimal outcomes are investigated and it is demonstrated that these problems are generally computationally intractable but also several special cases whereby the problem(s) can be solved efficiently.

Abstract

We study a sequential decision-making model where a set of items is repeatedly matched to the same set of agents over multiple rounds. The objective is to determine a sequence of matchings that either maximizes the utility of the least advantaged agent at the end of all rounds (optimal) or at the end of every individual round (anytime optimal). We investigate the computational challenges associated with finding (anytime) optimal outcomes and demonstrate that these problems are generally computationally intractable. However, we provide approximation algorithms, fixed-parameter tractable algorithms, and identify several special cases whereby the problem(s) can be solved efficiently. Along the way, we also establish characterizations of Pareto-optimal/maximum matchings, which may be of independent interest to works in matching theory and house allocation.

Paper Structure

This paper contains 33 sections, 34 theorems, 79 equations, 1 figure, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Work
  4. Repeated matching.
  5. Repeated fair division.
  6. Multi-agent sequential decision-making.
  7. Santa Claus problem.
  8. Other sequential decision-making models.
  9. Preliminaries
  10. Matchings.
  11. Valuations.
  12. Instances.
  13. Objective.
  14. Efficiency.
  15. Allocations and Bistochastic Matrices
  16. ...and 18 more sections

Key Result

Lemma 2.0

Suppose $A\in\mathbb{R}^{n\times m}$ is an allocation with Then, there exist a sequence of matchings $S$ consisting of $d\leq m^{2}-m+1$ unique matchings that satisfy $v_{i}^{T}(S)\geq v_{i}(A)$. This can be computed in polynomial time.

Figures (1)

  • Figure :

Theorems & Definitions (58)

  • Lemma 2.0
  • Theorem 3.1
  • Proposition 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • Lemma 3.3
  • Theorem 4.1
  • Proposition 4.1
  • proof
  • ...and 48 more