Anonymous and Separable Hedonic Coalition Formation Games: Nash Stability Under Different Membership Rights

We consider hedonic coalition formation games. A hedonic coalition formation game is a pair which consists of a finite set of agents and a list of agents’ preferences such that each agent has preferences over all coalitions containing her. We study the existence of a Nash stable partition under different membership rights for anonymous and separable hedonic coalition formation games. We prove that for anonymous and separable hedonic games, the existence of a Nash stable partition is always guaranteed when the membership rights are Free Exit-Approved Entry or Approved Exit-Free Entry, but the existence of a Nash stable partition is not guaranteed when the membership rights are Free Exit-Free Entry. We also analyze the relation of the anonymity and separability with the other sufficient conditions which guarantee the existence of a Nash stable partition under different membership rights.