Hedonic Games (HGs) are a classical framework modeling coalition formation of strategic agents guided by their individual preferences. According to these preferences, it is desirable that a coalition structure (i.e. a partition of agents into coalitions) satisfies some form of stability. The most well-known and natural of such notions is arguably core-stability. Informally, a partition is core-stable if no subset of agents would like to deviate by regrouping in a so-called core-blocking coalition. Unfortunately, core-stable partitions seldom exist and even when they do, it is often computationally intractable to find one. To circumvent these problems, we propose the notion of $\varepsilon$-fractional core-stability, where at most an $\varepsilon$-fraction of all possible coalitions is allowed to core-block. It turns out that such a relaxation may guarantee both existence and polynomial-time computation. Specifically, we design efficient algorithms returning an $\varepsilon$-fractional core-stable partition, with $\varepsilon$ exponentially decreasing in the number of agents, for two fundamental classes of HGs: Simple Fractional and Anonymous. From a probabilistic point of view, being the definition of $\varepsilon$-fractional core equivalent to requiring that uniformly sampled coalitions core-block with probability lower than $\varepsilon$, we further extend the definition to handle more complex sampling distributions. Along this line, when valuations have to be learned from samples in a PAC-learning fashion, we give positive and negative results on which distributions allow the efficient computation of outcomes that are $\varepsilon$-fractional core-stable with arbitrarily high confidence.

翻译：享乐博弈（HGs）是一个经典框架，用于刻画由个体偏好引导的智能体联盟形成过程。根据这些偏好，理想情况下联盟结构（即智能体划分为多个联盟的划分方式）应满足某种形式的稳定性。其中最著名且最自然的概念当属核心稳定性。非正式地说，一个划分是核心稳定的，如果不存在任何智能体子集希望通过重组形成所谓的核心阻塞联盟来偏离当前状态。然而，核心稳定划分很少存在，即便存在，其求解也常常是计算上不可行的。为规避这些问题，我们提出了$\varepsilon$-分数核心稳定性的概念，其中最多允许$\varepsilon$比例的所有可能联盟构成核心阻塞。事实证明，这种松弛策略既能保证存在性，又能实现多项式时间计算。具体而言，我们针对享乐博弈的两个基础类别——简单分数博弈和匿名博弈——设计了高效算法，可返回$\varepsilon$指数级随智能体数量递减的$\varepsilon$-分数核心稳定划分。从概率视角来看，由于$\varepsilon$-分数核心的定义等价于要求均匀采样的联盟以低于$\varepsilon$的概率形成核心阻塞，我们进一步将定义扩展至更复杂的采样分布。沿着这一方向，当需要以PAC学习方式从样本中学习估值函数时，我们将给出关于哪些分布允许以任意高置信度高效计算$\varepsilon$-分数核心稳定结果的正反两方面结论。