Shapley and superShapley aggregation emerging from consensus dynamics in the multicriteria Choquet framework

Abstract

We investigate a negotiation model for the progressive aggregation of interacting multicriteria evaluations. The model is based on a network of interacting criteria and combines the Choquet aggregation framework with the classical DeGroot’s model of consensus linear dynamics. We consider a set $N = \{ 1,\ldots ,n \}$ of interacting criteria whose single evaluations are expressed in some domain ${\mathbb {D}}\subseteq {\mathbb {R}}$. The pairwise interaction among the criteria is described by a complete graph with edge values in the open unit interval. In the Choquet framework, the interacting network structure is the basis for the construction of a consensus capacity $\mu$, whose Shapley indices are proportional to the average degree of interaction between criterion $i \in N$ and the remaining criteria $j \ne i \in N$. We discuss three types of linear consensus dynamics, each of which represents a progressive aggregation process towards a consensual multicriteria evaluation corresponding to some form of mean of the original multicriteria evaluations. All three models refer significantly to the notion of multicriteria context evaluation. In one model, the progressive aggregation converges simply to the plain mean of the original multicriteria evaluations, while another model converges to the Shapley mean of those original multicriteria evaluations. The third model, instead, converges to an emphasized form of Shapley mean, which we call superShapley mean. The interesting relation between Shapley and superShapley aggregation is investigated.