We propose a continuous-time nonlinear model of opinion dynamics with utility-maximizing agents connected via a social influence network. A distinguishing feature of the proposed model is the inclusion of an opinion-dependent resource-penalty term in the utilities, which limits the agents from holding opinions of large magnitude. This model is applicable in scenarios where the opinions pertain to the usage of resources, such as money, time, computational resources etc. Each agent myopically seeks to maximize its utility by revising its opinion in the gradient ascent direction of its utility function, thus leading to the proposed opinion dynamics. We show that, for any arbitrary social influence network, opinions are ultimately bounded. For networks with weak antagonistic relations, we show that there exists a globally exponentially stable equilibrium using contraction theory. We establish conditions for the existence of consensus equilibrium and analyze the relative dominance of the agents at consensus. We also conduct a game-theoretic analysis of the underlying opinion formation game, including on Nash equilibria and on prices of anarchy in terms of satisfaction ratios. Additionally, we also investigate the oscillatory behavior of opinions in a two-agent scenario. Finally, simulations illustrate our findings.

翻译：本文提出了一种连续时间非线性观点动力学模型，其中通过社会影响网络连接的个体以效用最大化为目标。该模型的一个显著特点是在效用函数中引入了依赖于观点的资源惩罚项，该惩罚项限制了个体持有过大强度观点的行为。该模型适用于观点涉及资源使用（如金钱、时间、计算资源等）的场景。每个个体短视地寻求通过沿其效用函数梯度上升方向修正其观点来最大化自身效用，从而导出了所提出的观点动力学。我们证明，对于任意社会影响网络，观点最终是有界的。对于具有弱对抗关系的网络，我们利用收缩理论证明了全局指数稳定平衡点的存在。我们建立了共识平衡点存在的条件，并分析了在共识状态下个体的相对主导性。我们还对潜在的观点形成博弈进行了博弈论分析，包括纳什均衡以及基于满意度比率定义的无政府价格。此外，我们还研究了两主体场景下观点的振荡行为。最后，仿真结果验证了我们的发现。