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Properties of Path-Independent Choice Correspondences and Their Applications to Efficient and Stable Matchings

Keisuke Bando, Kenzo Imamura, Yasushi Kawase

TL;DR

The paper introduces path-independence (PI) for choice correspondences, proving that PI entails rationalizability and endows the choice set with a generalized-matroid (g-matroid) structure; it further links PI to ordinal concavity, showing that correspondences rationalized by ordinally concave (and, with size-restriction, by ordinal-concave plus size-restricted-concave) utilities satisfy PI and LAD. Building on these foundations, it develops a theory of constrained efficient matching under PI and LAD, providing a cycle-based characterization (PSIC-free maximal stable matchings) and a polynomial-time procedure to compute constrained-efficient outcomes. The results apply broadly to realistic market-design settings with ties, such as diversity constraints, quotas, and reserves, by representing school- and market-constraints via M${}^ atural$-concave or laminar-concave utilities. Overall, the work offers a cohesive framework linking discrete convexity, matroidal structures, and stable/efficient matching in markets with indifferences.

Abstract

Choice correspondences are crucial in decision-making, especially when faced with indifferences or ties. While tie-breaking can transform a choice correspondence into a choice function, it often introduces inefficiencies. This paper introduces a novel notion of path-independence (PI) for choice correspondences, extending the existing concept of PI for choice functions. Intuitively, a choice correspondence is PI if any consistent tie-breaking produces a PI choice function. This new notion yields several important properties. First, PI choice correspondences are rationalizabile, meaning they can be represented as the maximization of a utility function. This extends a core feature of PI in choice functions. Second, we demonstrate that the set of choices selected by a PI choice correspondence for any subset forms a generalized matroid. This property reveals that PI choice correspondences exhibit a nice structural property. Third, we establish that choice correspondences rationalized by ordinally concave functions inherently satisfy the PI condition. This aligns with recent findings that a choice function satisfies PI if and only if it can be rationalized by an ordinally concave function. Building on these theoretical foundations, we explore stable and efficient matchings under PI choice correspondences. Specifically, we investigate constrained efficient matchings, which are efficient (for one side of the market) within the set of stable matchings. Under responsive choice correspondences, such matchings are characterized by cycles. However, this cycle-based characterization fails in more general settings. We demonstrate that when the choice correspondence of each school satisfies both PI and monotonicity conditions, a similar cycle-based characterization is restored. These findings provide new insights into the matching theory and its practical applications.

Properties of Path-Independent Choice Correspondences and Their Applications to Efficient and Stable Matchings

TL;DR

The paper introduces path-independence (PI) for choice correspondences, proving that PI entails rationalizability and endows the choice set with a generalized-matroid (g-matroid) structure; it further links PI to ordinal concavity, showing that correspondences rationalized by ordinally concave (and, with size-restriction, by ordinal-concave plus size-restricted-concave) utilities satisfy PI and LAD. Building on these foundations, it develops a theory of constrained efficient matching under PI and LAD, providing a cycle-based characterization (PSIC-free maximal stable matchings) and a polynomial-time procedure to compute constrained-efficient outcomes. The results apply broadly to realistic market-design settings with ties, such as diversity constraints, quotas, and reserves, by representing school- and market-constraints via M-concave or laminar-concave utilities. Overall, the work offers a cohesive framework linking discrete convexity, matroidal structures, and stable/efficient matching in markets with indifferences.

Abstract

Choice correspondences are crucial in decision-making, especially when faced with indifferences or ties. While tie-breaking can transform a choice correspondence into a choice function, it often introduces inefficiencies. This paper introduces a novel notion of path-independence (PI) for choice correspondences, extending the existing concept of PI for choice functions. Intuitively, a choice correspondence is PI if any consistent tie-breaking produces a PI choice function. This new notion yields several important properties. First, PI choice correspondences are rationalizabile, meaning they can be represented as the maximization of a utility function. This extends a core feature of PI in choice functions. Second, we demonstrate that the set of choices selected by a PI choice correspondence for any subset forms a generalized matroid. This property reveals that PI choice correspondences exhibit a nice structural property. Third, we establish that choice correspondences rationalized by ordinally concave functions inherently satisfy the PI condition. This aligns with recent findings that a choice function satisfies PI if and only if it can be rationalized by an ordinally concave function. Building on these theoretical foundations, we explore stable and efficient matchings under PI choice correspondences. Specifically, we investigate constrained efficient matchings, which are efficient (for one side of the market) within the set of stable matchings. Under responsive choice correspondences, such matchings are characterized by cycles. However, this cycle-based characterization fails in more general settings. We demonstrate that when the choice correspondence of each school satisfies both PI and monotonicity conditions, a similar cycle-based characterization is restored. These findings provide new insights into the matching theory and its practical applications.

Paper Structure

This paper contains 26 sections, 30 theorems, 83 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Organization of the Paper
  4. Preliminaries
  5. Matroids and Generalized Matroids
  6. Properties of Choice Functions
  7. Properties of Choice Correspondences
  8. PI Choice Correspondences
  9. Rationalizability
  10. Proof of \ref{['thm:rationalize']}
  11. G-matroid
  12. PI and Ordinal concavity
  13. Constrained Efficient Matching
  14. Matching Model
  15. Main Result
  16. ...and 11 more sections

Key Result

Proposition 1

For any g-matroid $\mathcal{F}\subseteq 2^I$ and $X\in\mathcal{F}$, if $|X|<\max\{|Y|: Y\in\mathcal{F}\}$, then there is an element $i\in I$ such that $X+i\in\mathcal{F}$.

Figures (6)

  • Figure 1: Relations of $X$, $S$, $T$, $Z$, and $S^*$
  • Figure 2: Case (i)
  • Figure 3: Case (ii)
  • Figure 4: Classes of choice correspondences
  • Figure 5: Illustration of the bridging property. The green region represent the set $(B\setminus A)\cup((X\setminus Y)\setminus A)$, which contains the element $j$ required to satisfy the condition.
  • ...and 1 more figures

Theorems & Definitions (69)

  • Proposition 1
  • proof
  • Theorem 1: yokote2024rationalizing
  • Definition 1
  • Definition 2
  • Example 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 59 more