A Stackelberg Game Framework for Double-Incentive Consensus Mechanism With Cost Budget in Group Decision Making

The opinions of experts often exhibit initial variance in group decision-making process due to the differences in background, knowledge, stance, and other influential factors. Thus, an incentive mechanism is critical to motivate experts to adjust individuals’ opinions and achieve a consensus solution. The incentive mechanism is usually costly and included in the consensus reaching process (CRP), and its effect relies on the behavior interaction between the moderator and experts. Within the Stackelberg game framework, we present the maximum-experts and minimum-cost consensus models with multiple incentives and cost budget. First, from a two-side perspective of the moderator and experts, a consensus model with maximum-return modification and maximum-experts feedback (MRMECM) is built to pursue the maximum number of consensus experts under an established cost budget, where the incentive mechanism is realized with the allocation of unit return to each return-driven expert. Then, a double incentive mechanism (DIM) is designed with the modification and shared return incentives. Subsequently, the MRMECM with DIM is constructed to improve the utilization efficiency of cost budget. Finally, the DIM module is integrated into the maximum-return modification and minimum-cost feedback consensus model (MRMCCM), which aims to minimize the consensus cost in the feedback mechanism. All the proposed consensus models are built as bi-level programming models in a unified Stackelberg game framework. We propose a hybrid approach that combines the best-response update strategy with the differential evolution (DE) algorithm to solve the equilibrium solutions. Consequently, several experimental studies are performed to validate the effectiveness of the proposed models.