Harmonious Equilibria in Roommate Problems

We study the roommate problem with self-matching agents and preferences with indifferences. Following the approach of Richter and Rubinstein (2024a, 2024b) and Herings (2024), we de_ne equilibria as constraints shaped by social norms, combined with outcomes where agents make optimal choices given those constraints. We propose four new equilibrium notions, conceptually distinct from taboo equilibrium of Richter and Rubinstein (2024a, 2024b) and expectational equilibrium of Herings (2024). Each equilibrium allows an agent a personalized set of contracts, with additional assumptions on minimality of constraints or maximality of permissions. Motivated by outcome-equivalence, we focus on two out of four of these equilibria, the minimal aggregate constraint equilibrium (MACE) and the maximal aggregate permission equilibrium (MAPE), both guaranteed to exist. MACE and MAPE outcomes are not nested, but with strict preferences, an MAPE is an MACE. We further show that taboo equilibrium outcomes, MACE outcomes, and individually rational Pareto-optimal outcomes coincide, while there is an equivalence between the expectational equilibrium outcomes, stable outcomes, and MAPE outcomes, if the former two exist. Finally, we argue that MAPE is logically independent of all concepts that are proposed in the literature to address the non-existence of stable outcomes in the classical roommate problem.