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 evolved opinions based on the Nash/worst-case strategy with the opinions corresponding to social optimality actions for non-strategist agents.

在社交网络中，个人对几个相互依赖的话题表达自己的观点，因此他们对这些话题的观点的演变也是相互依赖的。在这项工作中，我们提出了一种微分博弈模型，用于社交网络的多维意见形成，该网络的代理群体根据通信图进行交互。每个人的观点都是根据智能体与其图邻居之间关于多个相互依赖的主题（暴露于未知的外部干扰）的观点之间的分歧的聚合而演变的。对于具有策略代理的社交网络，随着时间的推移，关于二次成本函数的最小化的意见会不断演变，二次成本函数仅代表每个人反对干扰的动机。我们为所提出的开环信息结构下耦合多维意见的微分博弈模型找到了独特的纳什/最坏情况均衡解。此外，我们提出了纳什/最坏情况均衡解决方案的分布式实现。我们研究了在二维意见空间中具有策略代理的小型社交网络上的非分布式和提出的分布式开环纳什/最坏情况策略。然后，我们将基于纳什/最坏情况策略的演化意见与对应于非策略代理的社会最优行动的意见进行比较。