On Hedonic Games with Common Ranking Property

Abstract

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 [21], where output is shared equally among affiliates, is called 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 [2]. Nonetheless, a core stable solution is not necessarily a socially desirable (Pareto optimal) outcome. We improve upon existing results by proving that every instance of HGCRP has a solution that is both Pareto optimal and core stable. We establish that finding such a solution is, however, $\mathsf {NP}$-$\mathsf {HARD}$, by proving the stronger statement that finding any Pareto optimal solution is $\mathsf {NP}$-$\mathsf {HARD}$. We show that the gap between the total utility of a core stable solution and that of the socially optimal solution (OPT) is bounded by |N|, where N is the set of agents, and that this bound is tight. Our investigations reveal that finding a solution, whose total utility is within a constant factor of that of OPT, is intractable.

This work is supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK) through grant 118E126.