Takas istikrarı: Oda kısıtı bulunan oda arkadaşı problemi üzerine bir analiz

This paper analyzes exchange stability for the roommate problem under the restriction that the rooms are in a scarce supply. While the central axioms in matching problems are Gale-Shapley stability (Gale and Shapley, 1962) and Pareto optimality, recently a new property, exchange stability is proposed. We demonstrate that the set of exchange stable matchings, which can be empty, is a subset of Pareto optimal matchings. We also prove an impossibility result that there is no well-defined solution which coincides with the set of exchange stable matchings whenever it is non-empty and satisfies consistency. Moreover, we show that Gale-Shapley stability and exchange stability are independent concepts, thus, algorithms to find Gale-Shapley stable solutions cannot be used for identifying exchange stable solutions. In addition, a necessary condition for a matching to satisfy both Gale-Shapley stability and exchange stability is found for the marriage problem, which is a special case of the roommate problem.