  In this paper, I study optimizing the opinion formation of a social network
of a population of individuals on a graph whose opinion evolves according to
the Hegselmann-Krause model for opinion dynamics. I propose an optimization
problem based on a differential game for a population of individuals who are
not stubborn. The objective of each individual is to seek an optimal control
policy for her own opinion evolution by optimizing a personal performance
index. The Nash equilibrium actions and the associated opinion trajectory with
the equilibrium actions are derived for the opinion optimization model using
Pontryagin's principle. 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 that in non-stubborn Zachary's
network, the opinions moved toward the average opinion of the network, but a
consensus of final opinions did not necessarily emerge.
