# Stable outcomes in modified fractional hedonic games

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## Abstract

In *coalition formation games* self-organized coalitions are created as a result of the strategic interactions of independent agents. In this paper we assume that for each couple of agents (*i*, *j*), weight \(w_{i,j}=w_{j,i}\) reflects how much agents *i* and *j* benefit from belonging to the same coalition. We consider the (symmetric) *modified fractional hedonic game*, that is a coalition formation game in which agents’ utilities are such that the total benefit of agent *i* belonging to a coalition (given by the sum of \(w_{i,j}\) over all other agents *j* belonging to the same coalition) is averaged over all the other members of that coalition, i.e., excluding herself. Modified fractional hedonic games constitute a class of succinctly representable hedonic games. We are interested in the scenario in which agents, individually or jointly, choose to form a new coalition or to join an existing one, until a stable outcome is reached. To this aim, we consider common stability notions leading to strong Nash stable outcomes, Nash stable outcomes or core stable outcomes: we study their existence, complexity and performance, both in the case of general weights and in the case of 0–1 weights. In particular, we completely characterize the existence of the considered stable outcomes and show many tight or asymptotically tight results on the performance of these natural stable outcomes for modified fractional hedonic games, also highlighting the differences with respect to the model of fractional hedonic games, in which the total benefit of an agent in a coalition is averaged over all members of that coalition, i.e., including herself.

## Keywords

Coalition formation games Hedonic games Nash equilibrium Core Price of anarchy Price of stability## Notes

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