A Stackelberg game model for large-scale group decision making based on cooperative incentives

With the rapid development of novel technological paradigms such as e-democracy, e-government, and collective intelligence, large-scale group decision making (LSGDM) has become an emerging topic. In LSGDM, a clustering algorithm is usually adopted to cluster experts into multiple subgroups with the aim of reducing the dimension of the problem. Because of the clustering process, a large group spans three hierarchies of a moderator, multiple subgroup spokesmen, and corresponding experts in each subgroup. The moderator first proposes an incentive to encourage subgroups to modify their opinions to reach an expected degree of consensus. Then, in each subgroup, the spokesman invokes sub-games with involved experts to negotiate the number of changed opinions through allocating incentives. Considering such a hierarchical decision-making structure, this study introduces a hierarchical Stackelberg game model to address the interactions between different players. A critical value is proposed to define the reward or punishment according to the degree of cooperativeness of experts and subgroups. The existence and uniqueness of the Stackelberg equilibrium is verified. We also put forward a consensus model for group decision making based on the Stackelberg equilibrium. A numerical example is provided to demonstrate the applicability of the model, and nu-merical studies are given to investigate the influence of some parameters.