Agent-based Modeling: Equilibrium, Echo Chambers, and Efficiency in Hybrid Coevolutionary Opinion Games

Abstract

Online discussion of political and gender-related issues is often heated, and when opinions in a network draw closer, the convergence is readily taken as genuine consensus. Whether it carries a cost is a question existing methods cannot answer: coevolutionary opinion formation games measure the Price of Anarchy (PoA) of agents that update by numerical rules, while simulations with large language model (LLM) agents report only descriptive indices. We introduce the Hybrid Coevolutionary Opinion Game (H-COG), in which analytical and LLM-driven agents share one network, choose their neighbors by opinion similarity in every round, and hold stances drawn from real Reddit comments on gun control and abortion. To our knowledge, H-COG is the first framework to place Friedkin-Johnsen best-response agents and LLM agents in one coevolutionary game and to measure the social cost and PoA of LLM-driven populations. We prove that on any fixed network, given the LLM agents'opinions, the analytical agents'opinion stage has a unique equilibrium and the social optimum has a closed form, and that the convergence guarantee of Chen et al. for optimistic gradient ascent carries over to H-COG. All runs converge structurally. LLM-driven populations are less polarized yet have about five times the PoA of analytical ones; half of the gap comes from agents being pulled away from their own prior positions, a distance we prove must carry a cost whenever expressed opinions are more concentrated than intrinsic ones. Echo chambers form under every composition and grow out of the rewiring rule rather than the initial topology. Opinions in these populations draw closer largely because agents give up their own positions.