Acyclic roommates

Abstract

In the context of the stable roommates problem, this paper provides an alternative characterization of acyclic instances with n roommates; one that requires checking n - 1 fewer equations than symmetry of the utility functions (Rodrigues-Neto, 2007). We introduce the concepts of agent-cycles and cycle equations and prove that an instance is acyclic if and only if there exists a representation of preferences such that all cycle equations of agent-cycles of length 3 containing an agent i hold. In this case, there is a unique stable matching.