# 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 accepted: 2020-10-05 Date submitted: 2020-07-29 09:59 Submitted by: Ullmo, Denis Submitted to: SciPost Physics Academic field: Physics Specialties: 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.

Published as SciPost Phys. 9, 059 (2020)

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).

### Submission & Refereeing History

Resubmission 2006.01221v2 on 29 July 2020
Submission 2006.01221v1 on 5 June 2020

## Reports on this Submission

### Report

I did not see any changes following my suggestion but this is not a reason not to publish this ^paper as is

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Denis Ullmo  on 2020-09-18

Category:

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 :

1) Notation issue. It is not clear always clear to me when the development is valid for players evolving in R d for d > 1 or the evolution is 1restricted to scalar situations d = 1. For example in section 3, Eq.(31) is written for d = 1 whereas Eq.(35) which follows seems to hold for ≥ 1 (and so is also the case for Eqs.(32) and (33)). In section 4 , one is limited to d = 1, etc...

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.

2) Misprint ? In Eq.(8) on page 5. Does the term (U 0 + gΓφ)Φ should read (U 0 + gΓΦ)Φ ?

It should indeed. The misprint has been corrected.

3) In section 3.2. [...] by looking at the expression for the energy (15), one can note that a natural length scale appears [....]. Could the authors give a hint to see how one spontaneously sees this L.

The new scale is $\eta$, not $L$. We have modified the text around this sentence to avoid that confusion.

4) In Eq.(26), m_er =\frac{\lmabda+U_0}{|g|} would it be perhaps be more clear if one writes m_er(x) = \frac{\lmabda+U_0(x)}{|g|}.

5) Misprint ? Between Eqs.(28) and (29). Does X = \sqrt{2\lambda/\mu \omega} should perhaps read as X = \sqrt{2\lambda/\mu \omega_0}

we have corrected that misprint.

6) After Eq.(32). The perturbations are infinitesimal fields so one means that δm = δm(x, t) and δv = δv(x, t) and δm 0 stands for δm 0 (x) and hence:

∂ t δm(x, t) = −∇ [m er (x)δv(x, t)] , ∂ t δv(x, t) = − μ g ∇δm(x, t)

and then the same remark as expressed in point 1) holds again. Indeed in Eq.(37) and (38) one has δm(x, t) := δm(x)e ±ωt and so x → x in these expressions.

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)