Consensus modelling in group decision making: a dynamical approach based on fuzzy preferences

The notion of consensus plays an important role in group decision making, particularly when the collective preference structure is generated by a dynamical aggregation process of the single individual preference structures. In this dynamical process of aggregation each single decision maker gradually transforms his/her preference structure by combining it, through iterative weighted averaging, with the preference structures of the remaining decision makers. In this way, the collective decision emerges dynamically as a result of the consensual interaction among the various decision makers in the group. From the point of view of applied mathematics, the models of consensual dynamics stand in the context of multi-agent complex systems, with interactive and nonlinear dynamics. The consensual interaction among the various agents (decision makers) acts on their state variables (the preferences) in order to optimize an appropriate measure of consensus, which can be of type ‘hard’ (unanimous agreement within the group of decision makers) or ‘soft’ (partial agreement within the group of decision makers). In this paper, we study the modelling of consensus reaching when the individual testimonies are assumed to be expressed as fuzzy preference relations. Here consensus is meant as the degree to which most of the experts agree on the preferences associated to the most relevant alternatives. First of all we derive a degree of dissensus based on linguistic quantifiers and then we introduce a form of network dynamics in which the quantifiers are represented by scaling functions. Finally, assuming that the decision makers can express their preferences in a more flexible way, i.e. by using triangular fuzzy numbers, we describe the iterative process of opinion transformation towards consensus via the gradient dynamics of a cost function expressed as a linear combination of a dissensus cost function and an inertial cost function.