Relaxed core stability in hedonic games

The core is a well-known and fundamental notion of stability in games intended to model coalition formation such as hedonic games: an outcome is core stable if there exists no blocking coalition, i.e., no set of agents that may profit by forming a coalition together. The fact that the cardinality of a blocking coalition, i.e., the number of deviating agents that have to coordinate themselves, can be arbitrarily high, and the fact that agents may benefit only by a tiny amount from their deviation, while they could incur in a higher cost for deviating, suggest that the core is not able to suitably model practical scenarios in large and highly distributed multi-agent systems. For this reason, we consider relaxed core stable outcomes where the notion of permissible deviations is modified along two orthogonal directions: the former takes into account the size q of the deviating coalition, and the latter the amount of utility gain, in terms of a multiplicative factor k, for each member of the deviating coalition. These changes result in two different notions of stability, namely, the q-size core and k-improvement core. We consider fractional hedonic games, that is a well-known subclass of hedonic games for which core stable outcomes are not guaranteed to exist and it is computationally hard to decide non-emptiness of the core; we investigate these relaxed concepts of stability with respect to their existence, computability and performance in terms of price of anarchy and price of stability, by providing in many cases tight or almost tight bounds. Interestingly, the considered relaxed notions of core also possess the appealing property of recovering, in some notable cases, the convergence, the existence and the possibility of computing stable solutions in polynomial time.