With the significant advancements in communication technology, group decision-making (GDM) can now be implemented online, allowing a large number of decision-makers (DMs) to participate concurrently. However, current methods for large-scale group decision-making (LSGDM) are primarily suitable for 20 to 50 DMs, and their effectiveness in scenarios involving thousands or even tens of thousands of participants has yet to be fully validated. Furthermore, as the number of participants increases, the evaluation information becomes increasingly diverse and complex. At the same time, the social networks associated with the DMs typically become sparse, making information sharing and consensus building more challenging. In light of these challenges, we develop two new methods based on cooperative games to effectively address the challenges in super LSGDM. First, we propose a two-stage semi-supervised fuzzy C-means (FCM) clustering method with trust constraints, which aims to address the issue of sparsity in relationships within large-scale social networks. This method utilizes trust relationships as reliable resources and prior knowledge to guide and supervise the clustering process. On this basis, we discuss three scenarios from the perspective of cooperative games: (i) subgroup optimal consensus adjustments in non-cooperative situations, (ii) group optimal consensus adjustments in cooperative situations, and (iii) subgroup optimal consensus adjustments in cooperative situations. Subsequently, we view the consensus adjustment allocation as a cost cooperative game problem and propose two new LSGDM consensus methods based on Nash Bargaining (NB) and Kalai–Smorodinsky Bargaining (KSB). Finally, experiments on real datasets demonstrate the superiority and reliability of our proposed LSGDM methods.

中文翻译：

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用于大规模群体决策的 Nash 和 Kalai-Smorodinsky 讨价还价游戏的共识方法


随着通信技术的重大进步，集团决策 （GDM） 现在可以在线实施，允许大量决策者 （DM） 同时参与。然而，目前的大规模群体决策 （LSGDM） 方法主要适用于 20 到 50 个 DM，它们在涉及数千甚至数万参与者的场景中的有效性尚未得到充分验证。此外，随着参与者数量的增加，评估信息变得越来越多样化和复杂。与此同时，与 DM 相关的社交网络通常变得稀疏，这使得信息共享和建立共识更具挑战性。鉴于这些挑战，我们开发了两种基于合作博弈的新方法，以有效应对超级 LSGDM 中的挑战。首先，我们提出了一种具有信任约束的两阶段半监督模糊 C 均值 （FCM） 聚类方法，旨在解决大规模社交网络中关系稀疏的问题。这种方法利用信任关系作为可靠的资源和先验知识来指导和监督集群过程。在此基础上，我们从合作博弈的角度讨论了三种情景：（i） 非合作情况下的子组最优共识调整，（ii） 合作情况下的组最优共识调整，以及 （iii） 合作情况下的子组最优共识调整。随后，我们将共识调整分配视为成本合作博弈问题，并提出了两种基于 Nash Bargaining （NB） 和 Kalai-Smorodinsky Bargaining （KSB） 的新 LSGDM 共识方法。 最后，在真实数据集上的实验证明了我们提出的 LSGDM 方法的优越性和可靠性。