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

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