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A Gale-Shapley View of Unique Stable Marriages

Kartik Gokhale, Amit Kumar Mallik, Ankit Kumar Misra, Swaprava Nath

TL;DR

This work studies unique stable matching (USM) in the classic stable marriage problem through an algorithmic lens centered on the deferred acceptance (DA) process. It introduces two novel sufficient conditions, MaxProp and MaxRou, and their proposer-specific variants, showing that MaxRou implies MaxProp and that both imply USM; moreover, these conditions are largely disjoint from the established SPC and NCC criteria when $n\ge 3$. A complete, efficiently verifiable characterization of MaxProp is provided, based on three structural properties and an acyclicity condition, which clarifies the subspace of USM that MaxProp captures. The paper also analyzes the relationship between MaxProp/MaxRou and existing conditions across cases $n\ge 3$ and $n=2$, revealing sharp separations and notable special-case coincidences. Overall, the results illuminate the internal structure of USM by linking DA dynamics (proposals and rounds) to stability outcomes, with potential applicability to partial preference domains and many-to-one extensions.

Abstract

Stable marriage of a two-sided market with unit demand is a classic problem that arises in many real-world scenarios. In addition, a unique stable marriage in this market simplifies a host of downstream desiderata. In this paper, we explore a new set of sufficient conditions for unique stable matching (USM) under this setup. Unlike other approaches that also address this question using the structure of preference profiles, we use an algorithmic viewpoint and investigate if this question can be answered using the lens of the deferred acceptance (DA) algorithm (Gale and Shapley, 1962). Our results yield a set of sufficient conditions for USM (viz., MaxProp and MaxRou) and show that these are disjoint from the previously known sufficiency conditions like sequential preference and no crossing. We also provide a characterization of MaxProp that makes it efficiently verifiable, and shows the gap between MaxProp and the entire USM class. These results give a more detailed view of the sub-structures of the USM class.

A Gale-Shapley View of Unique Stable Marriages

TL;DR

This work studies unique stable matching (USM) in the classic stable marriage problem through an algorithmic lens centered on the deferred acceptance (DA) process. It introduces two novel sufficient conditions, MaxProp and MaxRou, and their proposer-specific variants, showing that MaxRou implies MaxProp and that both imply USM; moreover, these conditions are largely disjoint from the established SPC and NCC criteria when . A complete, efficiently verifiable characterization of MaxProp is provided, based on three structural properties and an acyclicity condition, which clarifies the subspace of USM that MaxProp captures. The paper also analyzes the relationship between MaxProp/MaxRou and existing conditions across cases and , revealing sharp separations and notable special-case coincidences. Overall, the results illuminate the internal structure of USM by linking DA dynamics (proposals and rounds) to stability outcomes, with potential applicability to partial preference domains and many-to-one extensions.

Abstract

Stable marriage of a two-sided market with unit demand is a classic problem that arises in many real-world scenarios. In addition, a unique stable marriage in this market simplifies a host of downstream desiderata. In this paper, we explore a new set of sufficient conditions for unique stable matching (USM) under this setup. Unlike other approaches that also address this question using the structure of preference profiles, we use an algorithmic viewpoint and investigate if this question can be answered using the lens of the deferred acceptance (DA) algorithm (Gale and Shapley, 1962). Our results yield a set of sufficient conditions for USM (viz., MaxProp and MaxRou) and show that these are disjoint from the previously known sufficiency conditions like sequential preference and no crossing. We also provide a characterization of MaxProp that makes it efficiently verifiable, and shows the gap between MaxProp and the entire USM class. These results give a more detailed view of the sub-structures of the USM class.
Paper Structure (20 sections, 19 theorems, 8 equations, 1 figure, 1 algorithm)

This paper contains 20 sections, 19 theorems, 8 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Related works
  4. Preliminaries
  5. Remark.
  6. Current State-of-the-art Sufficient Conditions
  7. Our Results
  8. MaxProposals and MaxRounds
  9. Remark.
  10. MaxProp implies USM
  11. A Characterization of MaxProp
  12. Discussions.
  13. Position of MaxProp in the USM space
  14. MaxProp is disjoint from SPC for $n \geqslant 3$
  15. Discussions.
  16. ...and 5 more sections

Key Result

Theorem 1

If for any two distinct stable matchings $\mu_1$ and $\mu_2$ in $\succ$, if each man finds $\mu_1$ at least as preferred as $\mu_2$, then every woman will find $\mu_2$ at least as preferred as $\mu_1$.

Figures (1)

  • Figure 1: The above two figures illustrate the sub-structures of the USM class for $n \geqslant 3$ and $n=2$ respectively. However, the gap between MaxProp and MaxRou is empty for $n=3$ (\ref{['lemma:mp-mr-n=3']}) and non-empty for $n\geqslant 4$. The dashed lines and the shaded regions denote the new sub-structures of USM that are contributions of this paper. We also characterize the class MaxProp and provide the complexity of verification. In \ref{['fig:nequal2']}, the fact USM = SPC was known from eeckhout2000uniqueness. We provide a direct proof of this fact.

Theorems & Definitions (50)

  • Definition 1: Matching
  • Definition 2: Blocking Pair
  • Definition 3: Stable Matching
  • proof
  • proof
  • Theorem 1: gale1985some
  • Definition 4: Sequential Preference Condition
  • Example 1: USM but not SPC
  • Definition 5: No Crossing Condition
  • Definition 6: MaxProp and MaxRou
  • ...and 40 more