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Speeding up deferred acceptance

Gregory Z. Gutin, Daniel Karapetyan, Philip R. Neary, Alexander Vickery, Anders Yeo

TL;DR

The paper addresses the computational cost of finding stable matchings in one-to-one markets by introducing accelerated deferred acceptance (ADA), which integrates the idea of iterated deletion of unattractive alternatives (IDUA) to prune sure-to-be rejected proposals. ADA produces the same stable matching as the classic DA (specifically the man-optimal outcome when men propose), and is strategy-proof for the proposing side, while reducing the number of proposals and rounds and achieving earlier final matches. The authors prove the core properties and demonstrate, via a substantial set of simulations, that efficiency gains can be substantial and often strict, especially as market size grows and preferences become more heterogeneous. These findings have practical relevance for market design (e.g., labor, school choice) and suggest promising extensions to more general DA variants and richer environments.

Abstract

A run of the deferred acceptance (DA) algorithm may contain proposals that are sure to be rejected. We introduce the accelerated deferred acceptance algorithm that proceeds in a similar manner to DA but with sure-to-be rejected proposals ruled out. Accelerated deferred acceptance outputs the same stable matching as DA but does so more efficiently: it terminates in weakly fewer rounds, requires weakly fewer proposals, and final pairs match no later. Computational experiments show that these efficiency savings can be strict.

Speeding up deferred acceptance

TL;DR

The paper addresses the computational cost of finding stable matchings in one-to-one markets by introducing accelerated deferred acceptance (ADA), which integrates the idea of iterated deletion of unattractive alternatives (IDUA) to prune sure-to-be rejected proposals. ADA produces the same stable matching as the classic DA (specifically the man-optimal outcome when men propose), and is strategy-proof for the proposing side, while reducing the number of proposals and rounds and achieving earlier final matches. The authors prove the core properties and demonstrate, via a substantial set of simulations, that efficiency gains can be substantial and often strict, especially as market size grows and preferences become more heterogeneous. These findings have practical relevance for market design (e.g., labor, school choice) and suggest promising extensions to more general DA variants and richer environments.

Abstract

A run of the deferred acceptance (DA) algorithm may contain proposals that are sure to be rejected. We introduce the accelerated deferred acceptance algorithm that proceeds in a similar manner to DA but with sure-to-be rejected proposals ruled out. Accelerated deferred acceptance outputs the same stable matching as DA but does so more efficiently: it terminates in weakly fewer rounds, requires weakly fewer proposals, and final pairs match no later. Computational experiments show that these efficiency savings can be strict.
Paper Structure (18 sections, 12 theorems, 8 equations, 10 figures, 2 tables, 3 algorithms)

This paper contains 18 sections, 12 theorems, 8 equations, 10 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. The environment
  3. Matching problems
  4. DA and IDUA
  5. Deferred acceptance
  6. Iterated deletion of unattractive alternatives
  7. Accelerated deferred acceptance
  8. The algorithm
  9. Fewer proposals, fewer rounds, and earlier pairings
  10. Computational experiments
  11. Instance generator
  12. Number of rounds
  13. Number of proposals
  14. Proportion of final pairs matched by round
  15. Execution time
  16. ...and 3 more sections

Key Result

Theorem 1

Every instance of the stable matching problem possesses at least one stable matching.

Figures (10)

  • Figure 1: The proportion of final pairs matched by round.
  • Figure 2: Number of rounds by $n$. DA: dashed lines; ADA: solid lines.
  • Figure 3: Number of rounds as a function of $c$.
  • Figure 4: The distributions of the number of rounds for $c = 0.0$, $c = 0.5$ and $c = 0.9$.
  • Figure 5: The relation between the number of rounds for DA and ADA.
  • ...and 5 more figures

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem : GaleShapley:1962:AMM
  • Theorem : GaleShapley:1962:AMM
  • Definition 4: Unattractive alternatives
  • Definition 5: The iterated deletion of unattractive alternatives (IDUA)
  • Lemma 1: BalinskiRatier:1997:GutinNeary:2023:GEB
  • Lemma 2: BalinskiRatier:1997:GutinNeary:2023:GEB
  • Theorem 1
  • ...and 14 more