Core Matching Mechanisms

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Consider a house matching problem with at least as many houses as agents.  Fix a material characteristic function that summarizes all submatchings achievable by any given coalition of agents.  Together with a profile of agent preferences over houses, the characteristic function defines a cooperative game.  If the core of any such game is nonempty and unique, then the characteristic function defines a core mechanism that maps each profile of preferences to the unique element of the core of the game induced by the profile of preferences.  I show that the set of core mechanisms coincides with the set of all trading cycle mechanisms. Such mechanisms use cycles, similar to Gale's, where agents point to their preferred houses, but allow for more complex control rights  (e.g., ownership of multiple houses).  The set of trading cycle mechanisms is strictly nested between Papai's (2000) hierarchical exchange mechanisms and the set of trading and braiding mechanisms characterized by Bade (2014) and Pycia and Unver (2017).