Differential Game Strategies for Social Networks with Self-Interested Individuals

A social network population engages in collective actions as a direct result of forming a particular opinion. The strategic interactions among the individuals acting independently and selfishly naturally portray a noncooperative game. Nash equilibrium allows for self-enforcing strategic interactions between selfish and self-interested individuals. This paper presents a differential game approach to the opinion formation problem in social networks to investigate the evolution of opinions as a result of a Nash equilibrium. The opinion of each individual is described by a differential equation, which is the continuous-time Hegselmann-Krause model for opinion dynamics with a time delay in input. The objective of each individual is to seek optimal strategies for her own opinion evolution by minimizing an individual cost function. Two differential game problems emerge, one for a population that is not stubborn and another for a population that is stubborn. The open-loop Nash equilibrium actions and their associated opinion trajectories are derived for both differential games using Pontryagin's principle. Additionally, the receding horizon control scheme is used to practice feedback strategies where the information flow is restricted by fixed and complete social graphs as well as the second neighborhood concept. The game strategies were executed on the well-known Zachary's Karate Club social network. The resulting opinion trajectories associated with the game strategies showed consensus, polarization, and disagreement in final opinions.