Opinion Dynamics under Social Pressure

The paper’s Theorem 4 states that on a complete graph with honesty parameter γ>1, the estimator g_n(H_n) = [Ψ̂_n(γ−1)+1]/(γ+1) converges almost surely, equals Φ when its limit lies strictly between 1/(γ+1) and γ/(γ+1), and otherwise saturates at the endpoints, thereby only certifying that Φ lies beyond the corresponding threshold. This is explicitly stated in the paper (Theorem 4 and surrounding discussion) , with the model and goals set up in Section II and the complete-graph analysis in Section IV . The candidate solution proves the same claims via a different, concise 1D stochastic-approximation argument on r_n=Ψ̂_n, deriving a mean-field drift r↦G(r) and showing sign(G(r)−r)=sign(Φ−g(r)) with g(r)=[(γ−1)r+1]/(γ+1), hence g_∞=clamp(Φ,[1/(γ+1),γ/(γ+1)]). The paper’s proof works through a 3D ODE (X_n,Y_n,Z_n) and convergence of (p^0_n,p^1_n) to a deterministic limit before translating to Ψ̂_n and g_n(H_n) . Both arrive at the same conclusion; the proofs are substantively different but compatible.