  In a social network, individuals express their opinions on several
interdependent topics, and therefore the evolution of their opinions on these
topics is also mutually dependent. In this work, we propose a differential game
model for the multi-dimensional opinion formation of a social network whose
population of agents interacts according to a communication graph. Each
individual's opinion evolves according to an aggregation of disagreements
between the agent's opinions and its graph neighbors on multiple interdependent
topics exposed to an unknown extraneous disturbance. For a social network with
strategist agents the opinions evolve over time with respect to the
minimization of a quadratic cost function that solely represents each
individual's motives against the disturbance. We find the unique
Nash/worst-case equilibrium solution for the proposed differential game model
of coupled multi-dimensional opinions under an open-loop information structure.
Moreover, we propose a distributed implementation of the Nash/worst-case
equilibrium solution. We examine the non-distributed and proposed distributed
open-loop Nash/worst-case strategies on a small social network with strategist
agents in a two-dimensional opinion space. Then we compare the opinions evolved
based on the Nash/worst-case strategy with the opinions corresponding to social
optimality actions for non-strategist agents.
