Nash Stability in Fractional Hedonic Games with Bounded Size Coalitions

Gianpiero Monaco, Luca Moscardelli

Abstract: We consider fractional hedonic games, a natural and succinct subclass of hedonic games able to model many real-world settings in which agents have to organize themselves in groups called coalitions. An outcome of the game, also called coalition structure, is a partition of all agents into coalitions. Previous work assumed that coalitions can be of any size. However, in many real-world situations, the size of the coalitions is bounded: vehicles, offices, classrooms and project teams are some examples of possible coalitions of bounded size. In this paper, we initiate the study k-fractional hedonic games (\(\mathrm {{{k}\!-\!FHG}}\)), in which all coalitions have size at most k, by considering Nash stable coalition structures, i.e., outcomes in which no agent can improve her utility by unilaterally changing her own coalition; in particular, we study existence of, convergence to, complexity and efficiency of Nash stable outcomes, and we also provide results about the complexity and approximation of optimal outcomes. We perform a thoroughgoing analysis of \(\mathrm {{{k}\!-\!FHG}}\) for \(k=2,3,4\). We remark that, on the one hand, considering these values of k is interesting in itself as many real world scenarios (as some of the aforementioned ones) deal with coalitions of small size; on the other hand, studying \(\mathrm {{{k}\!-\!FHG}}\) for small values of k both represents a necessary step for understanding the cases with higher values of k and already constitutes a challenging task.