Consensus methods with Nash and Kalai–Smorodinsky bargaining game for large-scale group decision-making

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.