Envy-freeness in 3D Hedonic Games

We study the problem of partitioning a set of agents into coalitions based on the agents' additively separable preferences, which can also be viewed as a hedonic game. We apply three successively weaker solution concepts, namely envy-freeness, weakly justified envy-freeness, and justified envy-freeness. In a model in which coalitions may have any size, trivial solutions exist for these concepts, which provides a strong motivation for placing restrictions on coalition size. In this paper, we require feasible coalitions to have size three. We study the existence of partitions that are envy-free, weakly justified envy-free, and justified envy-free, and the computational complexity of finding such partitions, if they exist. We present a comprehensive complexity classification, in terms of the restrictions placed on the agents' preferences. From this, we identify a general trend that for the three successively weaker solution concepts, existence and polynomial-time solvability hold under successively weaker restrictions.