# Schrödinger approach to Mean Field Games with negative coordination

### Submission summary

 As Contributors: Denis Ullmo Arxiv Link: https://arxiv.org/abs/2006.01221v2 (pdf) Date submitted: 2020-07-29 09:59 Submitted by: Ullmo, Denis Submitted to: SciPost Physics Discipline: Physics Subject area: Statistical and Soft Matter Physics 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 forward-backward structure of the Mean Field Game equations in relation with the way these various regimes are connected.

###### Current status:
Editor-in-charge assigned

Dear SciPost editors,

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).