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. The proposed utility functions also account for how the relative resources within the social group affect both an agent's stubbornness and social influence. 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：我们提出了一种连续时间非线性观点动力学模型，其中效用最大化的智能体通过社会影响网络相互连接。该模型的一个显著特征是在效用函数中引入了一个依赖于观点的资源惩罚项，从而限制智能体持有过大的观点幅度。所提出的效用函数还考虑了社会群体内相对资源如何影响智能体的固执程度及社会影响力。每个近视型智能体通过沿其效用函数的梯度上升方向修正观点来最大化自身效用，进而形成了所提出的观点动力学机制。我们证明：对于任意社会影响网络，观点最终有界；对于弱对抗关系网络，利用收缩理论证明存在全局指数稳定的均衡点。我们建立了共识均衡存在的条件，并分析了智能体在共识状态下的相对主导性。同时从博弈论视角分析了潜在的观点形成博弈，包括纳什均衡以及以满意度比率衡量的无政府代价。此外，我们还探讨了双智能体场景中观点的振荡行为。最后，仿真实验验证了我们的结论。