Opinion Dynamics with Median Aggregation

Paper in proceeding, 2025

Understanding the formation and evolution of opinions is of broad interdisciplinary interest. Many classical models for opinion formation focus on the impact of different notions of locality, e.g., locality due to network effects among agents or the role of the proximity of opinions. In practice, however, opinion formation is often governed by the interplay of local and global influences. In this paper, we study an asynchronous opinion dynamics in a social network. Each agent has a static intrinsic opinion as well as a public opinion that is updated asynchronously over time. Moreover, agents have access to a global aggregate (e.g., the outcome of a vote) of all public opinions. We focus on the popular median voting rule and show that pure Nash equilibria always exist. For every initial state of the dynamics, a pure equilibrium can be reached. The set of reachable equilibria forms a complete lattice, and extremal equilibria can be computed in polynomial time. Indeed, there are instances and initial states from which the number of reachable equilibria is exponentially large. The global median in these equilibria can be any of the initial opinions. We show that by uniformly increasing the influence of the aggregate median we can enforce that the median opinion is the same in every reachable equilibrium. We can compute the increase scheme that achieves this property in polynomial time. Furthermore, we show that finding the k most influential agents is NP-complete.