Opinions that move closer together in an online network are often considered a sign that people have genuinely reached consensus; whether this narrowing carries a cost is a question existing methods cannot answer. Coevolutionary opinion formation games measure the social cost of equilibria through the Price of Anarchy (PoA), but their agents follow numerical update rules, while opinion simulations with large language model (LLM) agents capture language-based reasoning and report only descriptive indices. We introduce the Hybrid Coevolutionary Opinion Game (H-COG), in which analytical and LLM-driven agents share one network that is rewired by opinion similarity and initialized from real Reddit data. To our knowledge, no prior framework lets both kinds of agents coevolve with the network inside one formal game, and none has measured the social cost or PoA of LLM-driven populations in opinion formation. We prove that on any fixed network the opinion stage has a unique equilibrium and a closed-form social optimum, and we show that the hybrid game satisfies the assumptions of a convergence theorem for coevolutionary opinion games, so its dynamics reach an approximate Nash equilibrium. On this foundation, all simulation runs converge. Echo chambers form under every composition, and the initial topology has no detectable effect on them, which points to the rewiring rule as their source. LLM-driven populations show lower polarization, that is, a narrower spread of opinions, with about five times the PoA of analytical populations. Decomposing social cost shows why: about half of the gap is disagreement with neighbors, and the other half comes from agents being pulled away from their own prior positions. In these populations, opinions draw closer largely because agents give up their own positions, at a cost in efficiency.

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