Opinion Dynamics for Utility Maximizing Agents: Exploring the Impact of the Resource Penalty

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. In addition, we also investigate the oscillatory behavior of opinions in a two-agent scenario. Finally, simulations illustrate our findings.