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In a social network, individuals express their opinions on severalinterdependent topics, and therefore the evolution of their opinions on thesetopics is also mutually dependent. In this work, we propose a differential gamemodel for the multi-dimensional opinion formation of a social network whosepopulation of agents interacts according to a communication graph. Eachindividual's opinion evolves according to an aggregation of disagreementsbetween the agent's opinions and its graph neighbors on multiple interdependenttopics exposed to an unknown extraneous disturbance. For a social network withstrategist agents the opinions evolve over time with respect to theminimization of a quadratic cost function that solely represents eachindividual's motives against the disturbance. We find the uniqueNash/worst-case equilibrium solution for the proposed differential game modelof coupled multi-dimensional opinions under an open-loop information structure.Moreover, we propose a distributed implementation of the Nash/worst-caseequilibrium solution. We examine the non-distributed and proposed distributedopen-loop Nash/worst-case strategies on a small social network with strategistagents in a two-dimensional opinion space. Then we compare the opinions evolvedbased on the Nash/worst-case strategy with the opinions corresponding to socialoptimality actions for non-strategist agents.

A Game Approach to Multi-dimensional Opinion Dynamics in Social Networks with Stubborn Strategist Agents