Popularity on the roommate diversity problem

Abstract

A recently introduced restricted variant of the multidimensional stable roommates problem is the roommate diversity problem: each agent belongs to one of two types (e.g., red and blue), and the agents' preferences over the rooms solely depend on the fraction of agents of their own type among their roommates. We study this variant with the notion of popularity.

We show that in the roommate diversity problem with the room size fixed to 2, the problem becomes tractable. Particularly, a popular partitioning of agents is guaranteed to exist and can be computed in polynomial time. Additionally, a mixed popular partitioning of agents is always guaranteed to exist in any roommate diversity game. By contrast, when there are no restrictions on the room size of a roommate diversity game, a popular partitioning may fail to exist and the problem becomes intractable.