Cooperative game based two-stage consensus adjustment mechanism for large-scale group decision making

Large-scale group decision making (LSGDM) is a powerful technology to deal with complex decision problems. As decision makers (DMs) in LSGDM come from different fields with various backgrounds and expertise, their perception of the same decision problem is largely divergent. The measure and improvement of consensus can obtain widely accepted decision results. In this process, the behaviors of DMs are interactive. The amount and direction of adjustments influence not only their consensus level but also that of other DMs. Therefore, the fairness and rationality of the consensus adjustment mechanism are fundamental. Considering the characteristics of LSGDM, this paper introduces two-stage consensus adjustment mechanisms from the perspective of cooperative game theory. In the first stage, players are DMs within each subgroup, and subgroups are regarded as players in the second stage. Further, the coalition payoffs are defined as the minimum consensus adjustment determined by the built models. To show the designated consensus adjustment cooperative games are essential, we prove that they are supper-additive. Considering the marginal contribution of DMs to consensus adjustment, a two-stage Shapley consensus adjustment mechanism is offered. Namely, the Shapley function is adopted to allocate the consensus adjustment to DMs and subgroups. When the Shapley consensus adjustment does not satisfy coalitional stability, a two-stage core-Nash bargaining consensus adjustment mechanism is provided, which can overcome the drawback of the two-stage Shapley consensus adjustment mechanism. Moreover, two new algorithms for LSGDM are proposed. Finally, a numerical example is provided to show the feasibility of the new method, and a comparison analysis is made.