Differential games for crowd dynamics and applications

Julian Barreiro-Gomez, Nader Masmoudi

Research output: Contribution to journalArticlepeer-review


We study crowd dynamics by means of both non-atomic and atomic differential games, which are also known as macroscopic and microscopic models, respectively; and we consider few crowd-related applications and experiments. Mainly, we study the modeling for Hajj motion (Tawaf), the most important event for Muslim pilgrims taking place in Mecca. We formally show that the proposed game-theoretical model is a potential-cost-dependent version of the well-known Hughes model, and the existing connection with the Lighthill–Whitham–Richards traffic model. We show that for a particular value in the potential cost, one obtains the same Hughes crowd model. Hence, we introduce a mild approximation that allows the computation of semi-explicit/explicit solutions for the proposed game problems. We study four main components that may be used by a central planner: (i) the effect of the clusters (delegations) over the motion, (ii) the inflow control to optimize the flux, (iii) how organized crowds evolve in comparison to disorganized ones, and (iv) how the creation of corrals with time-delays over their motion can be beneficial for the crowd evolution efficiency. We show that a faster performance is exhibited when pilgrims do Tawaf individually than when clustered groups are made, suggesting a possible direction for the policies design. Hence, we also present a simple ON/OFF control over the inflow that can be implemented by a central planner such that the flux is maximized. Related to this last mentioned policy, we have shown that delaying a group to integrate into the crowd can be beneficial for their efficient and faster motion. Also, we show that organized crowds evolve faster than disorganized ones, suggesting that education programs to perform Tawaf may potentially improve the flow. Finally, we show that other type of behaviors could be captured by means of the appropriate design of the cost functional. For example, the consideration of stress during an evacuation can be captured by means of the suitable modification of the cost functional, and as another example, we model the drafting tactics in bicycling where small densities can be beneficial for the performance. We present numerical and simulation results for all the applications using both the macroscopic and microscopic models, which certify the suitability of these models to capture the main features of the real behavior.

Original languageEnglish (US)
Pages (from-to)2703-2742
Number of pages40
JournalMathematical Models and Methods in Applied Sciences
Issue number13
StatePublished - Dec 15 2023


  • Differential games
  • Tawaf motion
  • atomic games
  • crowd dynamics
  • crowd marathon
  • drafting
  • non-atomic games

ASJC Scopus subject areas

  • Modeling and Simulation
  • Applied Mathematics


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