  In coalition formation games self-organized coalitions are created as a
result of the strategic interactions of independent agents. 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 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.
