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）是一种经典框架，用于模拟由个体偏好引导的战略智能体联盟形成过程。基于这些偏好，理想的联盟结构（即智能体被划分为若干联盟的划分）应满足某种形式的稳定性。其中最为著名且自然的稳定性概念当属核心稳定性。非正式地，若不存在任何智能体子集通过重组为所谓的“核心阻塞联盟”进行偏离，则该划分是核心稳定的。遗憾的是，核心稳定划分很少存在，即便存在，其计算也往往极其困难。为解决这些问题，我们提出$\varepsilon$-分数核心稳定性的概念，允许最多有$\varepsilon$比例的联盟构成核心阻塞。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，我们针对两类基础享乐博弈——简单分数型博弈和匿名型博弈——设计了高效算法，可返回一个$\varepsilon$-分数核心稳定划分，其中$\varepsilon$随智能体数量呈指数衰减。从概率视角看，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$\varepsilon$的概率构成核心阻塞，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学习方式从样本中获取估值时，我们给出了关于哪些分布允许高效计算具有任意高置信度的$\varepsilon$-分数核心稳定结果的正反面结论。