Large roommate problem with non-transferable random utility

We analyze a large roommate problem (i.e., marriage matching in which the marriage is not restricted solely to matchings between men and women) with non-transferable utility. It is well known that while a roommate problem may not have a stable proper matching, each roommate problem does have an stable improper matching. In a random utility model with types from Dagsvik (2000) and Menzel (2015), we show that all improper stable matchings are asymptotically close to being a proper stable matching. Moreover, the distribution of types in stable matchings (proper or not) converges to the unique maximizer of an expression that is a sum of two terms: the average “welfare” of the matching and the Shannon entropy of the distribution. In the noiseless limit, when the random component of the utility is reduced to zero, the distribution of types of matched pairs converges to the outcome of the transferable utility model.