Pedestrians in static crowds are not grains, but game players

Thibault Bonnemain, Matteo Butano, Théophile Bonnet, Iñaki Echeverría-Huarte, Antoine Seguin, Alexandre Nicolas, Cécile Appert-Rolland, and Denis Ullmo

Phys. Rev. E 107, 024612 – Published 28 February 2023

Abstract

The local navigation of pedestrians is assumed to involve no anticipation beyond the most imminent collisions, in most models. These typically fail to reproduce some key features experimentally evidenced in dense crowds crossed by an intruder, namely, transverse displacements toward regions of higher density due to the anticipation of the intruder's crossing. We introduce a minimal model based on mean-field games, emulating agents planning out a global strategy that minimizes their overall discomfort. By solving the problem in the permanent regime thanks to an elegant analogy with the nonlinear Schrödinger's equation, we are able to identify the two main variables governing the model's behavior and to exhaustively investigate its phase diagram. We find that, compared to some prominent microscopic approaches, the model is remarkably successful in replicating the experimental observations associated with the intruder experiment. In addition, the model can capture other daily-life situations such as partial metro boarding.

Received 13 September 2022
Accepted 1 February 2023

DOI:https://doi.org/10.1103/PhysRevE.107.024612

Physics Subject Headings (PhySH)

Collective dynamics Complex systems Optimization problems Transportation research

Mean field theory Schroedinger equation

Interdisciplinary PhysicsNonlinear DynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Thibault Bonnemain ^1,2,*, Matteo Butano ³, Théophile Bonnet^4,3,†, Iñaki Echeverría-Huarte ⁵, Antoine Seguin ⁶, Alexandre Nicolas ⁷, Cécile Appert-Rolland ⁴, and Denis Ullmo ³

¹Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne, NE1 8ST, United Kingdom
²Department of Mathematics, King's College, London KCL WC2R 2LS, United Kingdom
³Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
⁴Université Paris-Saclay, CNRS, IJCLab, 91405 Orsay, France
⁵Laboratorio de Medios Granulares, Departamento de Física y Matemática Aplicada, Univ. Navarra, 31080 Pamplona, Spain
⁶Université Paris-Saclay, CNRS, FAST, 91405 Orsay, France
⁷Institut Lumière Matière, CNRS & Université Claude Bernard Lyon 1, 69622 Villeurbanne, France

^*Corresponding author: thibault.bonnemain@kcl.ac.uk
^†Current address: Université Paris-Saclay, CEA, Service d'Etudes des Réacteurs et de Mathématiques Appliquées, 91191 Gif-sur-Yvette, France.

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Issue

Vol. 107, Iss. 2 — February 2023

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Images

Figure 1
Data (middle row) and velocity (bottom row) fields induced in a static crowd by a cylindrical intruder that crosses it; the transparency of the velocity arrows is linearly related to the local density. Column 1: Simulations of a monolayer of vibrated disks. Column 2: Simulations of an agent-based model wherein agents may anticipate the most imminent collision. All fields have been averaged over many realizations. Column 3: Results of the mean-field game introduced in this paper. Column 4: Controlled experiments of Ref. [15]. Note the relatively symmetrical density dip in front and behind the intruder, as well as the transverse moves. Columns 1–3: The crowd's density and intruder's size have been adjusted to match the experimental data (average density of 2.5 ped/ $m^{2}$ ). Details of simulations and videos showcasing time evolution can be found in the Appendix.
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Figure 2
Typical density and velocity fields induced by the crossing intruder in the permanent regime, as predicted by the MFG model in different regions of the parameter space. Parameters taken in the small $c_{s} / v$ and small $ξ / R$ quadrant display good visual agreement with the experimental data.
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Figure 3
Boarding a crowded metro coach at rush hour. Left: Morning rush hour of November 18, 2021, on the platform of Metro A in Lyon, France. The doors are about to close and the gap between boarding passengers and those who preferred to wait for the next metro is clearly visible. Right: Snapshot from a MFG simulation at $t = 0.9 T$ . Players start uniformly distributed on the platform and would like to get on the coach before the doors close, at $t = T$ . Just before that moment, the players closest to the doors choose to rush toward the coach and cram themselves in it despite the high density. Others prefer to stay on the platform (see Movie S4 [27] for the whole process). Simulations have been performed in a box of dimensions 15×15 over a time $T = 10$ , with an initial density on the platform $m_{0} = 0.2$ . Parameters are chosen to have healing length $ξ = 1.1$ , and speed of sound $c_{s} = 0.45$ , while $c_{coach} = 0$ and $c_{platform} = 6.21$ .
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Figure 4
Variation of the agent speed term for the TTC model, while keeping the other parameters equal to those of the picture displayed in the letter.
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Figure 5
Variation of the inertial term for the TTC model, while keeping the other parameters equal to those of the picture displayed in the letter.
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Figure 6
Variation of the TTC amplitude term for the TTC model, while keeping the other parameters equal to those of the picture displayed in the paper.
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