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

Abstract

We consider a set N = {1, ..., n} of interacting agents whose individual opinions are denoted by x i , i ∈ N in some domain D ⊆ 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 µ, where the capacity value µ(S) of a coalition of agents S ⊆ N is defined to be proportional to the sum of the edge interaction values contained in the subgraph associated to S. The capacity µ is obtained in terms of its 2-additive Möbius transform m µ , 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.

A bibliometric analysis and basic model introduction of opinion dynamics

“…As the most productive country, China has coworked with 27 countries/regions 220 times, approaching 50% cooperation has been deployed with the USA (55 times and accounting for 25%), Australia (30 times and accounting for 13.64%), and Spain (20 times and accounting for 9.09%). Among the 6 countries/regions, the USA is the country with the highest citations (7,443 times), the highest links (37), and the highest TLS with 268 times of cooperation with 37 countries/regions in total. 55 collaborative papers make China and the USA the first main cooperator with each other while the other countries which have coworked with the USA for not less than 15 papers are Italy (22), England (17), and Australia (15).…”

“…The most striking and interesting finding was that the final electoral outcome depended on how strongly the individuals in the society connected, besides the initial opinion of each individual. Bortot et al (2020a) focused on the classical DeGroot approach to consensus dynamics and network interaction and described how the Shapley aggregation of individual opinions emerges naturally in the non-additive framework of Choquet integration [37]. dynamics [43], and financial crisis.…”

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

“…Fundamental contributions in this research field have been made by several authors, following the seminal papers by: Shapley (1953) on cooperative game theory; French (1956) and Harary (1959) on social power theory; DeGroot (1974), Chatterjee (1975), Chatterjee and Seneta (1977), Berger (1981), Kelly (1982) and French (1981) on DeGroot's consensus formation model. An extended bibliography can be found in Bortot et al (2019).…”