Linear quadratic evacuation mean-field game models with negative definite state cost matrices

Modeling and understanding crowd evacuation dynamics has been a long-standing problem. Most realistic models involve nonlinear effects to capture individual velocity decrease with crowd density increase. This leads to useful but essentially intractable partial differential equation-based models. We consider here for tractability purposes, a class of linear quadratic large-scale evacuation games where velocity can be improved through crowd avoidance. This is simulated in the agent cost functions through negative costs which accrue when agents drift away from variously defined population mean trajectories, in a multi-exit situation. The presence of negative cost components generically induces a finite escape time phenomenon if the time horizon is not adequately bounded. We formulate two types of models for which we provide sufficient time horizon upper bounds for agent cost convergence and establish existence of limiting mean field game equilibria as well as their ε-Nash property. This is joint work with Noureddine Toumi and Jérôme Le Ny.