Revisiting stability in one-to-one matching problems

This paper studies the stability of a status quo matching by considering the set of matching locations as a primitive of a one-to-one matching problem, alongside the agents and their preferences. As such we generalize the approach of Morrill (J Econ Theory 145:1739-1756, 2010) who was the first to study matching problems with location restrictions. We develop two novel stability concepts, direct and (coalition-) trade stability, akin to Gale-Shapley stability and Alcalde's (Econ Des 1:275-287, 1995) concept of exchange stability, respectively, and derive connections with existing stability concepts. We show that coalition-trade stability is a refinement of direct stability. We then demonstrate that when there are no matching restrictions, direct stability is equivalent to Gale-Shapley stability and coalition-trade stability is equivalent to requiring both exchange stability and Gale-Shapley stability. In addition, we reveal a link between trade dominance and indirect dominance, Harsanyi's farsighted dominance concept. For the class of individually rational matching problems, we show that indirect dominance is a refinement of trade dominance. However, these two dominance notions do not always generate the same stable (set of) matchings.