Regarding the roommate problem, the paper below shows that when utility is transferable, the existence of a stable match in it is restored, as long as there is an even number of agents with distinguishable characteristics and preferences. Furthermore, with a sufficiently large group of individuals, even in the absence of a stable match, there can be a “quasi-stable” match, which can be obtained with minimal policy intervention.
This study begins by considering that the difference between a marriage problem and the roommate problem is the bipartite requirement. Additionally, it says that since Gale and Shapley, it is well-thought-out that a stable matching may not exist for a roommate problem under non-transferable utility. Nevertheless, this work finds out that cloning (duplicating the economy by “cloning” each agent) is a simple modification that restores its existence. Moreover, the cloning operation is also a solution for the symmetry constraint which is analyzed as a difficulty for finding a stable match under the roommate problem.
In the end, the authors say that in general full cloning may not be needed, because in any roommate matching problem with some categories in which there is an odd number of individuals, by adding one individual in each of them, the result will be a stable matching.
Also, the paper proofs that the existence of a stable matching can be restored when a bounded number of individuals from a population is removed.
This reading concludes by acknowledging that the usefulness of their proposed “cloning technique” to restore stability, when there is a non-transferable utility version of the roommate problem, is still unknown.
Chiappori, Pierre-André and Galichon, Alfred and Salanié, Bernard, The Roommate Problem is More Stable than You Think (March 29, 2012). Available at: http://www.columbia.edu/~pc2167/RoommatePbm-29Mars2012.pdf