Consensus dynamics, network interaction, and Shapley indices in the Choquet framework

We consider a set \(N = \{ 1,\ldots ,n \}\) of interacting agents whose individual opinions are denoted by \( x_{i}\), \(i \in N \) in some domain \({\mathbb {D}}\subseteq {\mathbb {R}}\). The interaction among the agents is expressed by a symmetric interaction matrix with null diagonal and off-diagonal coefficients in the open unit interval. The interacting network structure is thus that of a complete graph with edge values in (0, 1). In the Choquet framework, the interacting network structure is the basis for the construction of a consensus capacity \(\mu \), where the capacity value \(\mu (S)\) of a coalition of agents \(S \subseteq N\) is defined to be proportional to the sum of the edge interaction values contained in the subgraph associated with S. The capacity \(\mu \) is obtained in terms of its 2-additive Mobius transform \(m_{\mu }\), and the corresponding Shapley power and interaction indices are identified. We then discuss two types of consensus dynamics, both of which refer significantly to the notion of context opinion. The second type converges simply the plain mean, whereas the first type produces the Shapley mean as the asymptotic consensual opinion. In this way, it provides a dynamical realization of Shapley aggregation.