Population games on dynamic community networks

Both the paper’s Theorem 2 proof and the model’s Phase-2 solution hinge on a positivity claim for the inter-action encounter factor x_1^T W x_2 when y is interior. In the paper, the step “since W is connected, non-negative, and has strictly positive diagonal entries, sgn(x_i' W x_j') = sgn(y_i y_j)” is asserted inside the Lyapunov argument (equations (12)–(13)) but is not generally valid; connectedness and positive diagonal do not ensure that x_i' W x_j' > 0 whenever y_i y_j > 0, because the supports of x_i and x_j can be segregated on communities that are not adjacent in W. The candidate solution repeats this gap in Step 2 by claiming κ(x)=x_1^T W x_2>0 for all interior y from irreducibility and positive diagonal. Thus, as written, both proofs are missing a needed hypothesis (or a different argument) ensuring strictly positive cross-encounter rates whenever both actions are present at the population level. The remaining parts of each argument (pairwise-comparison reduction, Lyapunov/KL calculations, and 2×2 ESS algebra) are otherwise consistent with the model.