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Consistent Conjectures in Dynamic Matching Markets

Laura Doval, Pablo Schenone

Abstract

We provide a framework to study stability notions for two-sided dynamic matching markets in which matching is one-to-one and irreversible. The framework gives center stage to the set of matchings an agent anticipates would ensue should they remain unmatched, which we refer to as the agent's conjectures. A collection of conjectures, together with a pairwise stability and individual rationality requirement given the conjectures, defines a solution concept for the economy. We identify a sufficient condition--consistency--for a family of conjectures to lead to a nonempty solution (cf. Hafalir, 2008). As an application, we introduce two families of consistent conjectures and their corresponding solution concepts: continuation-value-respecting dynamic stability, and the extension to dynamic markets of the solution concept in Hafalir (2008), sophisticated dynamic stability.

Consistent Conjectures in Dynamic Matching Markets

Abstract

We provide a framework to study stability notions for two-sided dynamic matching markets in which matching is one-to-one and irreversible. The framework gives center stage to the set of matchings an agent anticipates would ensue should they remain unmatched, which we refer to as the agent's conjectures. A collection of conjectures, together with a pairwise stability and individual rationality requirement given the conjectures, defines a solution concept for the economy. We identify a sufficient condition--consistency--for a family of conjectures to lead to a nonempty solution (cf. Hafalir, 2008). As an application, we introduce two families of consistent conjectures and their corresponding solution concepts: continuation-value-respecting dynamic stability, and the extension to dynamic markets of the solution concept in Hafalir (2008), sophisticated dynamic stability.
Paper Structure (28 sections, 6 theorems, 37 equations)

This paper contains 28 sections, 6 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Model
  4. Two-sided economy
  5. Arrivals
  6. Dynamic Matching
  7. Available agents and induced economies
  8. Preferences
  9. A solution concept given conjectures
  10. Consistent conjectures
  11. Candidate matchings
  12. Toward consistent conjectures
  13. Why consistent conjectures?
  14. Not all solution concepts have consistent conjectures
  15. Two solution concepts with consistent conjectures
  16. ...and 13 more sections

Key Result

Theorem 4.1

Suppose $\Sigma_{T-1}^\Phi$ is a nonempty-valued correspondence. Fix $E^T$ and suppose the conjectures $\{\varphi_1(\emptyset,k):k\in A_1\cup B_1\}$ are consistent. Then, $\Sigma_T^\Phi(E^T)$ is nonempty. In particular, $M^\star(\varphi_1,\Sigma_{T-1}^\Phi)\subset\Sigma_T^\Phi(E^T)$.

Theorems & Definitions (29)

  • Definition 1: Period-$t$ matching
  • Definition 2: Irreversible dynamic matching
  • Definition 3: Individual rationality
  • Definition 4: Static notion of stability
  • Remark 2.1: Notation
  • Remark 2.2: Simplifying assumptions
  • Definition 5: Conjectures
  • Definition 6: Solutions induced by $\Phi$
  • Definition 7: Recursive
  • Definition 8: One-period economy induced by $\varphi_1$
  • ...and 19 more