SciPost Submission Page
Schrödinger approach to Mean Field Games with negative coordination
by Thibault Bonnemain, Thierry Gobron, Denis Ullmo
 Published as SciPost Phys. 9, 059 (2020)
Submission summary
As Contributors:  Denis Ullmo 
Arxiv Link:  https://arxiv.org/abs/2006.01221v2 (pdf) 
Date accepted:  20201005 
Date submitted:  20200729 09:59 
Submitted by:  Ullmo, Denis 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Mean Field Games provide a powerful framework to analyze the dynamics of a large number of controlled agents in interaction. Here we consider such systems when the interactions between agents result in a negative coordination and analyze the behavior of the associated system of coupled PDEs using the now well established correspondence with the non linear Schr\"odinger equation. We focus on the long optimization time limit and on configurations such that the game we consider goes through different regimes in which the relative importance of disorder, interactions between agents and external potential varies, which makes possible to get insights on the role of the forwardbackward structure of the Mean Field Game equations in relation with the way these various regimes are connected.
Published as SciPost Phys. 9, 059 (2020)
Author comments upon resubmission
please find enclosed a revised version of our manuscript.
We thanks both referees for their very positive appreciation of our work, and for their detailed reading of our manuscript. Referee 1 in particular made a list of rather precise suggestions of changes, that we have integrally implemented. We detail the list of change below.
Best regards
Denis Ullmo (for the authors)
List of changes
i) As pointed out by referee 1, there was some ambiguity in our notations as whether our results apply for any dimensionality or only for d=1. We have changed the notations so that everything that apply to an arbitrary d use boldface fonts for the coordinates (essentially everything up to the end of section III) and that results restricted to d=1 use normal font for the coordinates (essentially everything from section IV onward).
ii) We have clarified the sentence in section 3.2 to make clear that the scale that emerges is indeed the healing length $\nu$.
ii) We have corrected misprints in Eq. (8) and in the expression of X between Eqs.(28) and (29), and have clarified the notations in Eqs. (26) and (32).
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 1 on 2020827 (Invited Report)
Report
I did not see any changes following my suggestion but this is not a reason not to publish this ^paper as is
Denis Ullmo on 20200918
We thanks again the referee for his/her support of our paper, and we apologize if he/she had difficulties locating the changes we have made following the suggestions in his/her first report. Indeed, because the suggestions where mainly about relatively minor point, we thought the rather compact description we gave in our first answer would be sufficient. Although we understand the referee does not explicitly requires these details, we provide them below anyhow.
List of changes made following referee 1 first report :
We have modified Eq. (10), the text below Eq. (16), Eqs.(17) and (23) to (27), as well as (32) to (36) to clarify what is valid for an arbitrary d.
It should indeed. The misprint has been corrected.
The new scale is $\eta$, not $L$. We have modified the text around this sentence to avoid that confusion.
The change has been made.
we have corrected that misprint.
We have included these modifications in Eqs (32) to (36).
Final remarks
We take advantage of this second communication with the referee to thank him/her again for his/her very thorough reading of our manuscript.
As a last comment, we stress that we did not introduce modifications of our manuscript after the report of referee 2. Indeed this report suggested "acceptance of this paper essentially as it", and the optional changes mentioned (generalization to non quadratic MFG or time dependent U_0) would be a research program in itself, presumably better adapted for future publication(s).
With these clarifications, we hope that our paper can be accepted in SciPost without further delays.
Best regards
Denis Ullmo (for the authors)