The Roommate Problem: Is More Stable than You Think

Abstract

Stable matchings may fail to exist in the roommate matching problem, both when utility is transferable and when it is not. We show that when utility is transferable, the existence of a stable matching is restored when there is an even number of individuals of indistinguishable characteristics and tastes (types). As a consequence, when the number of individuals of any given type is large enough there always exist "quasi-stable" matchings: a stable matching can be restored with minimal policy intervention. Our results build on an analogy with an associated bipartite problem; it follows that the tools crafted in empirical studies of the marriage problem can easily be adapted to the roommate problem.

On the partnership formation problem

“…Assignment games associated with the partnership formation problem are also considered in Biró et al (2012) and Chiappori et al (2014), which are similar to ours, but different in the use of pairs {(i, i ) | i ∈ N}. In the following, we discuss the difference among the three assignment games.…”

“…We finally note that our approach by using an associated assignment game is not totally new, and similar approach is already used in Biró et al (2012) and Chiappori et al (2014). Indeed, Biró et al (2012) and Chiappori et al (2014) associate certain assignment games with a given partnership formation problem, and use the maximum weight of a matching in the associated assignment games to characterize the existence of an equilibrium in the partnership formation problem.…”

“…Indeed, Biró et al (2012) and Chiappori et al (2014) associate certain assignment games with a given partnership formation problem, and use the maximum weight of a matching in the associated assignment games to characterize the existence of an equilibrium in the partnership formation problem. Based on the characterization, Biró et al (2012) and Chiappori et al (2014) propose algorithms for checking the existence of an equilibrium in partnership formation problem. A drawback of their algorithms is that the information about the joint values of agents pairs are needed to compute the maximum weight of a matching.…”

“…In contrast, the associated assignment game used in this paper is similar to but different from the ones in Biró et al (2012) and Chiappori et al (2014) (see Section 3.1 for more discussion on the difference of the associated assignment games). This difference makes it possible to obtain a characterization for the existence of an equilibrium in terms of demand correspondences of agents.…”

“…An equilibrium in the partnership formation problem may not exist (see, e.g., Chiappori et al (2014); Talman & Yang (2011)). For example, it is easy to see that the partnership formation problem with N = {1, 2, 3}, E = {(1, 2), (2, 3), (1, 3)}, and v(i, j) = 1 for every (i, j) ∈ E has no equilibrium.…”