Rationality and Reciprocity of Opinion Dynamics in Games

The paper proves a Z2-symmetric pitchfork in the 2-agent, 2-strategy opinion dynamics and derives the Lyapunov–Schmidt (LS) reduced normal form with explicit parameter dependence; see Theorem 1 and its reduction dzc/dτ = [(η^{-1}p + (α+γ)ũ)/(2(α^2−αγ+γ^2))]zc − zc^3 + [(p+2p⊥)/(2(α^2−αγ+γ^2))] + h.o.t. (their Eq. (14), Appendix Eqs. (18)–(19)) . The candidate solution independently reconstructs the same normal form by restricting to the symmetric invariant subspace, expanding tanh to cubic order, and identifying the unfolding directions “tilt” (∝ p zc) and “bias” (∝ p+2p⊥). Small discrepancies appear only in intermediate coefficients (a factor-of-two in the p zc slope and the omission of d in the time rescaling), but after the standard normal-form/time rescalings the candidate’s final equation coincides with the paper’s stated normal form and parameter dependence. The paper’s derivation via LS partial derivatives (Appendix: Jacobian singularity at u=2/(α+γ), cubic coefficient −2(α^2−αγ+γ^2), and the mixed p, p⊥ derivatives) is correct and complete , and the model solution reaches the same conclusion by a different (but consistent) route.