Game-Theoretic Analysis of the Hegselmann-Krause Model for Opinion Dynamics in Finite Dimensions

TL;DR

This paper analyzes the Hegselmann-Krause opinion dynamics model in finite dimensions, providing new bounds on termination time and a game-theoretic perspective on asynchronous updates, advancing understanding of both homogeneous and heterogeneous cases.

Contribution

It offers the sharpest known bound on termination time independent of dimension and introduces a game-theoretic framework for asynchronous dynamics analysis.

Findings

Termination time depends only on the number of agents

Asynchronous dynamics are equivalent to potential game best responses

Polynomial bound on convergence time for random updates

Abstract

We consider the Hegselmann-Krause model for opinion dynamics and study the evolution of the system under various settings. We first analyze the termination time of the synchronous Hegselmann-Krause dynamics in arbitrary finite dimensions and show that the termination time in general only depends on the number of agents involved in the dynamics. To the best of our knowledge, that is the sharpest bound for the termination time of such dynamics that removes dependency of the termination time from the dimension of the ambient space. This answers an open question in [1] on how to obtain a tighter upper bound for the termination time. Furthermore, we study the asynchronous Hegselmann-Krause model from a novel game-theoretic approach and show that the evolution of an asynchronous Hegselmann-Krause model is equivalent to a sequence of best response updates in a well-designed potential game. We then provide a polynomial upper bound for the expected time and expected number of switching topologies until the dynamic reaches an arbitrarily small neighborhood of its equilibrium points, provided that the agents update uniformly at random. This is a step toward analysis of heterogeneous Hegselmann-Krause dynamics. Finally, we consider the heterogeneous Hegselmann-Krause dynamics and provide a necessary condition for the finite termination time of such dynamics. In particular, we sketch some future directions toward more detailed analysis of the heterogeneous Hegselmann-Krause model.