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Discounted Mean-Field Game model of a dense static crowd with variable information crossed by an intruder
by Matteo Butano, Cécile Appert-Rolland, Denis Ullmo
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Submission summary
| Authors (as registered SciPost users): | Matteo Butano · Denis Ullmo |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2302.08945v2 (pdf) |
| Code repository: | https://github.com/matteobutano/mfg_ergodic_intruder |
| Date submitted: | 2024-01-09 14:43 |
| Submitted by: | Butano, Matteo |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
It was demonstrated in [Bonnemain et al., Phys. Rev. E 107, 024612 (2023)] that the anticipation pattern displayed by a dense crowd crossed by an intruder can be successfully described by a minimal Mean-Field Games model. However, experiments show that changes in the pedestrian knowledge significantly modify the dynamics of the crowd. Here, we show that the addition of a single parameter, the discount factor $\gamma$, which gives a lower weight to events distant in time, is sufficient to observe the whole variety of behaviors observed in the experiments. We present a comparison between the discounted MFG and the experimental data, also providing new analytic results and insight about how the introduction of $\gamma$ modifies the model.
Author comments upon resubmission
Dear SciPost Editor,
We thank you for forwarding the reports of the referees concerning our submission "Discounted Mean-Field Game model of a dense static crowd with variable information crossed by an intruder". We understand that the referees found positive qualities to our work, referee I in particular stating that it was "clear and well written", and has "convincing results". Both, however, considered that it had flaws that had to be corrected before publication. Both referees expressed the concern that we derive time independent equations, (Eqs.\ (10) to (14) which are then transposed in the moving frame as Eqs.\ (15) to (20)), which are not used in practice when we turn to the numerical implementation of the model. This furthermore had two consequences, the first one being that one may wonder why they are introduced, and the second being that the existence of an "ergodic/time-independent" regime is postulated, but in fact never demonstrated.
We admit that the reason we were not including numerics for the time-independent equations was that we did not know how to do this at the time of our first submission. The referees' comments on this point gave us the incentive to investigate that question again, and it turns out that we were able to solve that problem. The approach that we have used is quite non-trivial (we had to use a mixed representation between the (u,m) variable far from the cylinder and the (Phi,Gamma) close to it) and is briefly discussed in an appendix C.
With this tool now at hands, we have significantly modified the content of our paper, and in particular: - All the simulations in the core of the paper are now made in the "ergodic/time-independent" framework. - We have checked in appendix C that the solution found in this way indeed corresponded to what was obtained at t=T/2 for the time dependent calculation, demonstrating in this way the correctness of our hypothesis about the ergodic solution. - As suggested by referee 1, we have added a discussion on how the discount factor modifies the four-quadrant repartition discussed in our previous paper [Bonnemain et al., Phys. Rev. E 107, 024612 (2023)]
We feel that with these new results, as well as the significant rewriting of the manuscript, we have answered all comments and suggestions of both referees, and we hope our paper is now ready for publication in SciPost.
List of changes
- A brief discussion of the role of $U_0$ has been introduced in the paragraph following Eq. (4).
- Section 2.2 has been completely redrafted and is now split into two subsections (2.2.1 & 2.2.2).
- A new section 3 has been introduced to discuss the parameter space of the model, with three new figures (Figs. 2,3 &4).
- The first paragraph of section 4 has been modified to stress the fact that we now do numerical computations in the time-independent framework.
- Details of the derivation of the discounted HJB equation have been moved from section 2.1 to a new appendix A.
- Old appendix A is now appendix B
- Old appendix B has been suppressed.
- Appendix C, which briefly describes the numerics, has been completely rewritten, and contains now two figures (Figs.8 & 9) comparing the density profiles obtained with our new numerical scheme with the ones obtained from the time-dependent equations.
- Appendix D introduces the scaled version of the MFG equations and identifies the relevant parameter of the model.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 3) on 2024-1-17 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2302.08945v2, delivered 2024-01-17, doi: 10.21468/SciPost.Report.8421
Report
I thank the authors for the efforts they put into revising their manuscript. They took into account my remarks, and responded in great details. I appreciate hte inclusion of the four-quadrants comparison and their efforts in discussing the impact of the discount factor on it, as well as the numerical derivation of the time-independent calculations. Although the latter does not change the conclusions about the comparison to experiments, it justifies the theoretical development presented in Section 2.
Although I am happy to support publication of this manuscript, I have two additional remarks I would like the authors to seriously consider
- I don't find the "discount term" particularly intuitive, as it is an inverse time. Would it be meaningful to use its inverse, as an anticipatory time scale? One could then argue that the classical MFG theory is for infinite anticipation, and that this work looks into finite anticipation times (such as when pedestrians have their back to the obstacles). I would find such a description simpler than arguing for a cost of actions, more intuitive, and probably capable of reaching a larger community of scientists interested in crowds dynamics
- I reiterate that, in my opinion, claims such as "We strongly believe in the validity of our approach" are inadequate and do not serve the manuscript well. Instead, saying that "The strong agreement between our model and experimental data supports the idea that, although people do not solve non-linear systems while walking, their basic anticipation mechanisms allows them to, etc" would be both more rigorous and convincing.
Requested changes
See above
Author: Matteo Butano on 2024-02-08 [id 4303]
(in reply to Report 1 on 2024-01-17)We thank the referee for taking the time to read and review the latest version of our manuscript. We welcome his/her appreciation for our modifications to it, and we acknowledge his/her remarks. We have modified the manuscript accordingly to the best of our ability and highlighted such modifications in blue in the attached text. In the following we answer in detail the referee's remarks.
"I don't find the "discount term" particularly intuitive, as it is an inverse time. Would it be meaningful to use its inverse, as an anticipatory time scale? One could then argue that the classical MFG theory is for infinite anticipation, and that this work looks into finite anticipation times (such as when pedestrians have their back to the obstacles). I would find such a description simpler than arguing for a cost of actions, more intuitive, and probably capable of reaching a larger community of scientists interested in crowds dynamics"
"I reiterate that, in my opinion, claims such as "We strongly believe in the validity of our approach" are inadequate and do not serve the manuscript well. Instead, saying that "The strong agreement between our model and experimental data supports the idea that, although people do not solve non-linear systems while walking, their basic anticipation mechanisms allows them to, etc. " would be both more rigorous and convincing. "
Attachment:
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