Biform game consensus analysis of group decision making with unconnected social network

In today's network era, people's decisions are susceptibly influenced by others, especially the ones they trust. This study confines to studying social network group decision making (SNGDM). Due to the mutual influence of consensus level and consensus adjustment among decision makers (DMs), this study utilizes biform game theory to propose an innovative consensus mechanism for facilitating group decision making with unconnected social networks. Specifically, in the context of a DM social network with multiple trust relationship-based connected components, we construct a multi-objective programming model to determine the consensus adjustment. Within each connected component, we employ the digraph game theory to study DMs' consensus adjustments, leveraging the directional and asymmetrical characteristics of trust-relationships. We then analyze the consensus adjustments of feasible DM coalitions using built optimization models and define the di-Myerson value. Additionally, we construct several axiomatic systems to show the rationality of consensus allocation results. We identify partial trust-relationships that increase the consensus adjustments of DMs as irrational, and design an algorithm to address them, thereby reducing the cost of consensus. Finally, we present a case study that showcases the real-world application of our new theoretical results. This is the first bi-form game consensus mechanism based on trust relationship for SNGDM.