On the many-to-one strongly stable fractional matching set

TL;DR

This paper characterizes the set of strongly stable fractional matchings in many-to-one markets with q-responsive preferences, showing they can be expressed as convex combinations of stable matchings ordered by common preferences.

Contribution

It provides a novel characterization of strongly stable fractional matchings as convex hulls of connected stable matchings sets in many-to-one markets.

Findings

Strongly stable fractional matchings are unions of convex hulls of connected stable matchings.

Such matchings can be represented as convex combinations of stable matchings ordered by common preferences.

The results extend understanding of fractional stability in markets with q-responsive preferences.

Abstract

For a many-to-one matching market where firms have strict and $\boldsymbol{q}$-responsive preferences, we give a characterization of the set of strongly stable fractional matchings as the union of the convex hull of all connected sets of stable matchings. Also, we prove that a strongly stable fractional matching is represented as a convex combination of stable matchings that are ordered in the common preferences of all firms.