Additively separable hedonic games (ASHGs) are a prominent model of coalition formation where agents' preferences are derived from their individual valuations of peers. While social welfare maximization in ASHGs has traditionally focused mostly on utilitarian welfare, Nash welfare -- a well-established metric in economics which balances fairness with efficiency and offers scale invariance -- has been entirely overlooked. In this paper, we initiate the study of Nash welfare in ASHGs. We point out desirable properties fulfilled by partitions with high Nash welfare. This includes guaranteed contractual Nash stability in symmetric games, even for any approximation of Nash welfare. This is particularly appealing since, as for other welfare notions, Nash welfare turns out to be NP-hard to maximize, even for the ASHG subclass of symmetric aversion to enemies games (AEGs). A main focus of our study is on approximation algorithms for the Nash welfare objective. We present packing-based algorithms with approximation ratios for well-established subclasses of ASHGs: $n-1$ for AEGs and $2n$ for appreciation of friends games. This is complemented by a strict inapproximability result showing it is NP-hard to approximate Nash welfare within a factor of $1.0000759$ in general ASHGs. Further, we investigate the restricted settings with an upper bound on the coalition size or number of coalitions, and draw the boundary between the cases admitting efficient algorithms and those yielding NP-hardness: bounding the allowed size or number of coalitions by $2$ admits polynomial-time solvability, whereas bounds of $3$ or more yield NP-hardness or unbounded inapproximability.

翻译：可加可分离享乐博弈（ASHGs）是一种重要的联盟形成模型，其中代理人的偏好源于其对同伴的个体估值。尽管ASHGs中的社会福利最大化传统上主要关注功利主义福利，但纳什福利——经济学中一种兼顾公平与效率、具有尺度不变性的成熟度量指标——却完全被忽视。本文首次在ASHGs中研究纳什福利。我们指出，具有高纳什福利的划分满足理想性质。这包括在对称博弈中保证契约纳什稳定性，即使对于纳什福利的任何近似也成立。这一性质尤为吸引人，因为与其他福利概念类似，纳什福利的最大化被证明是NP难的，即使对于ASHGs的子类——对称厌恶敌人博弈（AEGs）也是如此。我们研究的主要焦点是纳什福利目标的近似算法。针对ASHGs的典型子类，我们提出了基于打包的算法，其近似比分别为：AEGs为$n-1$，朋友欣赏博弈为$2n$。此外，我们给出了一个严格不可近似性结果，表明在一般ASHGs中，将纳什福利近似到因子$1.0000759$以内是NP难的。进一步，我们探讨了联盟规模或联盟数量存在上界受限情形，并划定了存在高效算法与导致NP难问题之间的边界：将允许的联盟规模或数量限制为$2$时可获得多项式时间可解性，而限制为$3$或更大时则导致NP难或不可有界近似。