  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.
