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Fast, Fair and Truthful Distributed Stable Matching for Common Preferences

Juho Hirvonen, Sara Ranjbaran

TL;DR

This work studies a natural special case of the stable matching problem where all agents on one side share common preferences and shows how algorithms for sampling random colorings can be used to break ties fairly and extend the results to fractional stable matching.

Abstract

Stable matching is a fundamental problem studied both in economics and computer science. The task is to find a matching between two sides of agents that have preferences over who they want to be matched with. A matching is stable if no pair of agents prefer each other over their current matches. The deferred acceptance algorithm of Gale and Shapley solves this problem in polynomial time. Further, it is a mechanism: the proposing side in the algorithm is always incentivised to report their preferences truthfully. The deferred acceptance algorithm has a natural interpretation as a distributed algorithm (and thus a distributed mechanism). However, the algorithm is slow in the worst case and it is known that the stable matching problem cannot be solved efficiently in the distributed setting. In this work we study a natural special case of the stable matching problem where all agents on one side share common preferences. We show that in this case the deferred acceptance algorithm does yield a fast and truthful distributed mechanism for finding a stable matching. We show how algorithms for sampling random colorings can be used to break ties fairly and extend the results to fractional stable matching.

Fast, Fair and Truthful Distributed Stable Matching for Common Preferences

TL;DR

This work studies a natural special case of the stable matching problem where all agents on one side share common preferences and shows how algorithms for sampling random colorings can be used to break ties fairly and extend the results to fractional stable matching.

Abstract

Stable matching is a fundamental problem studied both in economics and computer science. The task is to find a matching between two sides of agents that have preferences over who they want to be matched with. A matching is stable if no pair of agents prefer each other over their current matches. The deferred acceptance algorithm of Gale and Shapley solves this problem in polynomial time. Further, it is a mechanism: the proposing side in the algorithm is always incentivised to report their preferences truthfully. The deferred acceptance algorithm has a natural interpretation as a distributed algorithm (and thus a distributed mechanism). However, the algorithm is slow in the worst case and it is known that the stable matching problem cannot be solved efficiently in the distributed setting. In this work we study a natural special case of the stable matching problem where all agents on one side share common preferences. We show that in this case the deferred acceptance algorithm does yield a fast and truthful distributed mechanism for finding a stable matching. We show how algorithms for sampling random colorings can be used to break ties fairly and extend the results to fractional stable matching.
Paper Structure (18 sections, 18 theorems, 2 figures)

This paper contains 18 sections, 18 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Related work
  4. Background and definitions
  5. Distributed algorithms
  6. Mechanism design
  7. Stable matching and deferred acceptance
  8. Variants of deferred acceptance.
  9. Almost stable matching.
  10. Fast and fair deferred acceptance algorithm
  11. Our basic algorithm
  12. Proof of Lemma \ref{['lem:mechanism-main']}
  13. Coloring in the CONGEST model
  14. Generalisation to fractional loads and capacities
  15. Modified deferred acceptance algorithm
  16. ...and 3 more sections

Key Result

Theorem 1

There is an incentive-compatible distributed implementation of the deferred acceptance algorithm in the CONGEST model for one-sided common preferences, with running time $O(\Delta S + \log^* n)$.

Figures (2)

  • Figure 1: A matching instance and a stable matching in bold. The preferences are given as a list for the neighbours from left to right, with 1 indicating the highest preference. The matching is stable: there is no pair of unmatched agents connected by an edge such that they would prefer each other over their current match. Since there are more agents on one side, some agents must remain unmatched.
  • Figure 2: Two instances of stable matching that only differ in the preferences of agent $v$ on the left. This causes the unique stable matching to change, and therefore agent $u$ on the right has to learn the preferences of $v$ in order to compute a stable matching.

Theorems & Definitions (18)

  • Theorem 1: Informal, Theorem \ref{['thm:da-arbitrary-tb']}
  • Theorem 2: Informal, Theorem \ref{['thm:fractional-det-da']}
  • Theorem 3: Informal, Theorem \ref{['thm:fair-almost-uniform']}
  • Theorem 4: Informal, Theorem \ref{['thm:fast-fair-da']}
  • Theorem 5: Floreen et al. floreen10stable
  • Lemma 1
  • Theorem 6
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 8 more