Measuring satisfaction in societies with opinion leaders and mediators

An opinion leader-follower model (OLF) is a two-action collective decision-making model for societies, in which three kinds of actors are considered: opinion leaders, followers, and independent actors. In OLF the initial decision of the followers can be modified by the influence of the leaders. Once the final decision is set, a collective decision is taken applying the simple majority rule [22]. We consider a generalization of OLF, the gOLF models which allow collective decision taken by rules different from the single majority rule. Inspired in this model we define two new families of collective decision-making models associated with influence games [15]. We define the oblivious and non-oblivious influence models. We show that gOLF models are nonoblivious influence models played on a two layered bipartite influence graph. Together with OLF models, the satisfaction measure was introduced and studied. We analyze the computational complexity of the satisfaction measure for gOLF models and the other collective decision-making models introduced in the paper. We show that computing the satisfaction measure is #P-hard in all the considered models. On the other hand, we provide two subfamilies of decision models in which the satisfaction measure can be computed in polynomial time. Exploiting the relationship with influence games, we can relate the satisfaction measure with the Rae index of an associated simple game. The Rae index is closely related to the Banzhaf. Thus, our results also extend the families of simple games for which computing the Rae index and the Banzhaf value is computationally hard.