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.

中文翻译：

---

改进的分数享乐游戏的稳定结果

在联盟形成游戏中，由于独立代理人的战略互动而形成了自组织联盟。在本文中，我们假设对于每对特工（i，  j），权重\（w_ {i，j} = w_ {j，i} \）反映出i和j受益于同一个联盟。我们考虑（对称）修正分数享乐游戏，即联盟形成博弈，其中代理的效用使得代理i的总收益属于联盟（由\（w_ {i，j} \ ）超过所有其他代理j属于同一个联盟）在该联盟的所有其他成员（即不包括她自己）上平均。修饰的分数享乐游戏构成了一类简洁可表示的享乐游戏。我们对以下情况感兴趣：代理人单独或共同选择组建新的联盟或加入现有的联盟，直到达成稳定的结果。为此，我们考虑导致强大的Nash稳定结局，Nash稳定结局或核心稳定结局的常见稳定性概念：在一般权重和0–1权重的情况下，我们研究它们的存在，复杂性和性能。特别是，