Generalized replicator dynamics based on mean-field pairwise comparison dynamic

The paper states and justifies well-posedness of the GRD (12) under assumptions (2)–(3) on U and the structural/Lipschitz conditions on C, by invoking a Banach-space Picard–Lindelöf theorem (Zeidler [52]) and Cheung’s existence theory for pairwise comparison dynamics; it explicitly checks the needed boundedness/Lipschitz bounds (56)–(59) for the induced vector field, and concludes a unique strong solution with mass conservation and invariance of the probability simplex (via the pairwise-comparison structure) . The candidate solution mirrors the same program (define the vector field, prove local Lipschitz on M2(Ω), use a contraction mapping, then argue invariance of P(Ω)), but the key step proving positivity is circular: it treats μt as a nonnegative measure to bound inflow/outflow terms (e.g., I_A(t) ≥ 0 and O_A(t) ≤ C* μt(A)) when positivity is precisely what must be established. With signed measures, those inequalities do not hold a priori. Hence the candidate’s argument does not actually establish invariance of P(Ω), while the paper’s appeal to the established framework (Cheung [11] + Zeidler [52]) does. Apart from this gap, the model’s Lipschitz and fixed-point steps are consistent with the paper’s bounds.