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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.
Wankhede et al. (Sun,) studied this question.