On hedonic games with common ranking property

Hedonic games are a prominent model of coalition formation, in which each agent’s utility only depends on the coalition she resides. The subclass of hedonic games that models the formation of general partnerships (Larson 2018), where all affiliates receive the same utility, is referred to as hedonic games with common ranking property (HGCRP). Aside from their economic motivation, HGCRP came into prominence since they are guaranteed to have core stable solutions that can be found efficiently (Farrell and Scotchmer Q. J. Econ. 103(2), 279–297 1988). We improve upon existing results by proving that every instance of HGCRP has a solution that is Pareto optimal, core stable, and individually stable. The economic significance of this result is that efficiency is not to be totally sacrificed for the sake of stability in HGCRP. We establish that finding such a solution is NP-hard even if the sizes of the coalitions are bounded above by 3; however, it is polynomial time solvable if the sizes of the coalitions are bounded above by 2. We show that the gap between the total utility of a core stable solution and that of the socially-optimal solution (OPT) is bounded above by n, where n is the number of agents, and that this bound is tight. Our investigations reveal that computing OPT is inapproximable within better than \(O(n^{1-\epsilon })\) for any fixed \(\epsilon > 0\), and that this inapproximability lower bound is polynomially tight. However, OPT can be computed in polynomial time if the sizes of the coalitions are bounded above by 2.