Pairwise Stable Matching in Large Economies

Michael Greinecker,Christopher Kah

doi:10.3982/ecta16228

Abstract

We formulate a stability notion for two‐sided pairwise matching problems with individually insignificant agents in distributional form. Matchings are formulated as joint distributions over the characteristics of the populations to be matched. Spaces of characteristics can be high‐dimensional and need not be compact. Stable matchings exist with and without transfers, and stable matchings correspond precisely to limits of stable matchings for finite‐agent models. We can embed existing continuum matching models and stability notions with transferable utility as special cases of our model and stability notion. In contrast to finite‐agent matching models, stable matchings exist under a general class of externalities.

Highlights

If we look at only two measurable spaces (X X ) and (Y Y), we write X ⊗ Y for the product σ-algebra
Let f, VW, and VM be any matching; we show it is not stable
We show in Proposition E1 below that we can find a continuous version of VW when W is locally compact and M compact

Summary

Introduction

Under the assumption that W and M are compact, that preferences are negatively transitive, and that type spaces and preferences are compatible with small indifference curves in a measure-theoretic sense, we obtain for a topologically large set of matching problems the existence of extremal matchings in Theorem G1 and a version of the lone wolf theorem in Theorem G2.3 Let φ : M(W ) × M(M) → 2M(W∅×M∅) be the correspondence in the marriage model that maps a matching problem to the corresponding set of stable matchings and assume that W and M are both compact.

Results

Conclusion