Mean-field games for high-density crowds: The discount factor reproduces experiments but raises numerical issues

Appert-Rolland, Cécile; Butano, Matteo; Ullmo, Denis · 2025 · Crossref

DOI: 10.1051/epjconf/202533404022

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Summary

This paper addresses the challenge of modeling high-density crowd dynamics, specifically focusing on pedestrian anticipation behaviors that occur on time scales longer than immediate collision avoidance. While classical agent-based models typically handle only short-term tactical decisions, Mean-Field Games (MFG) offer a framework to capture strategic anticipation, such as pedestrians detouring to avoid future congestion or moving away from an approaching obstacle before physical contact. The authors highlight that while MFGs can reproduce experimental observations, introducing a "discount factor" to tune the level of anticipation creates significant numerical instabilities near solid boundaries, such as walls or obstacles. The study utilizes a quadratic Mean-Field Game model where agents optimize an individual cost function involving a running cost dependent on crowd density and a terminal cost. The discount factor, denoted by $\gamma$, weights future events exponentially, reflecting limited information or partial blindness. The model is formulated as a system of coupled partial differential equations: a forward-in-time density equation and a backward-in-time Hamilton-Jacobi-Bellman (HJB) equation for the value function. The authors analyze the permanent (steady-state) regime, which is relevant for scenarios like a cylinder moving through a static crowd. In this regime, the value function exhibits a logarithmic divergence near impenetrable obstacles due to infinite potential costs, causing standard finite difference numerical schemes to fail. For the case where the discount factor is zero ($\gamma = 0$), the authors demonstrate that a Cole-Hopf transformation can linearize the HJB equation, mapping the system to a form resembling the stationary Non-Linear Schrödinger equation. This transformation eliminates the logarithmic divergence, allowing for stable numerical solutions. However, when $\gamma > 0$, this transformation is no longer valid globally because the discount term prevents the linearization. To resolve this, the authors propose a mixed numerical approach. They divide the spatial domain into two overlapping regions: a region very close to the obstacle where the discount factor's effect is negligible, allowing the use of the transformed variables, and a region further away where the original variables can be used directly without encountering divergence issues. The significance of this work lies in providing a robust numerical method for solving Mean-Field Game equations with non-zero discount factors. By circumventing the instability near boundaries, this approach enables the accurate simulation of crowd behaviors involving long-term anticipation. This facilitates the application of MFG models to a wider range of realistic scenarios, such as evacuations and obstacle avoidance, where pedestrians adjust their strategies based on predicted future conditions rather than just immediate surroundings.

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