A rational logit dynamic for decision-making under uncertainty: well-posedness, vanishing-noise limit, and numerical approximation

The paper proves the vanishing-noise limit for the κ-exponential logit dynamic (6) under bounded/Lipschitz U and a uniform positivity assumption U≥U0>0, culminating in Proposition 2: sup_{t∈[0,T]}||μ_t^η−μ_t^0||→0 (with μ^0 solving the limit dynamic (9)) via η-uniform estimates and a Grönwall argument, as detailed in the Appendix (key bounds (23)–(29), (31)–(32)) . The candidate solution proves the same limit using a different, tighter route: normalize the weights, derive a uniform asymptotic for e_κ(U/η) on [U0,Ū] exploiting the explicit form and derivative identity e'_κ(z)=e_κ(z)/√(1+κ^2 z^2) (cf. (3) in the paper), obtain a uniform-in-η Lipschitz bound for the vector fields, and apply Duhamel + Grönwall to get uniform-in-time convergence; this matches the result and assumptions of Proposition 2 and uses the same positivity assumption to drop the max{·,0} term . The model’s proof also gives a sharper rate (O(η^2) for the field discrepancy) than is stated in the paper’s text, but that strengthens rather than conflicts with the paper, whose numerical section reports near-first-order behavior without contradicting theory (32) . A minor omission common to both write-ups in the presented excerpts is that well-posedness of the limit equation (9) is not proved explicitly (though it is standard under U≥U0 and Lipschitz assumptions and is implicitly used in Proposition 2).