When do Social Learners Affect Collective Performance Negatively? The Predictions of a Dynamical-System Model

The paper formulates the 2D system for individual and social learners, derives the Jacobian at the symmetric fixed point for m = 0.5, and proves the critical threshold s_c = 1/f′(0.5), with bistability for s > s_c and the necessity of hyperconformity (f′(0.5) > 1) for bistability. These claims appear in the main text and are proven in SM Sec. 4 (including Eq. 8), using the model defined in Eqs. (3)–(6) and the symmetric, increasing conformity function f (Eq. 2) . The candidate solution arrives at the same conclusions via a clean 1D reduction for x(t): x′ = (1−s)m + s f(x) − x, then linearization at x* = 1/2 and symmetry arguments. It also correctly identifies f′(1/2) = α for the canonical family and the regimes α ≤ 1 (no bistability) vs α > 1 (pitchfork/bistability) consistent with the paper’s figures and discussion . Minor nuance: at the exact boundary (e.g., α = 1 with s = 1), stability is marginal rather than asymptotically stable—neither the paper nor the model dwells on this edge case. Overall, both are correct and aligned; the proofs differ in presentation (2D Jacobian vs 1D reduced flow).