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Pedestrian Models Based on Rational Behaviour

  • Rafael BailoEmail author
  • José A. Carrillo
  • Pierre Degond
Chapter
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

Abstract

Following the paradigm set by attraction-repulsion-alignment schemes, a myriad of individual-based models have been proposed to calculate the evolution of abstract agents. While the emergent features of many agent systems have been described astonishingly well with force-based models, this is not the case for pedestrians. Many of the classical schemes have failed to capture the fine detail of crowd dynamics, and it is unlikely that a purely mechanical model will succeed. As a response to the mechanistic literature, we will consider a model for pedestrian dynamics that attempts to reproduce the rational behaviour of individual agents through the means of anticipation. Each pedestrian undergoes a two-step time evolution based on a perception stage and a decision stage. We will discuss the validity of this game theoretical-based model in regimes with varying degrees of congestion, ultimately presenting a correction to the mechanistic model in order to achieve realistic high-density dynamics.

Notes

Acknowledgements

JAC acknowledges support by the EPSRC grant no. EP/P031587/1. PD acknowledges support by the EPSRC grant no. EP/M006883/1, by the Royal Society and the Wolfson Foundation through a Royal Society Wolfson Research Merit Award no. WM130048. PD is on leave from CNRS, Institut de Mathmatiques de Toulouse, France. JAC and PD acknowledge support by the National Science Foundation (NSF) under Grant no. RNMS11-07444(KI-Net).

References

  1. 1.
    C. Appert-Rolland, A. Jelić, P. Degond, J. Fehrenbach, J. Hua, A. Cretual, R. Kulpa, A. Marin, A.-H. Olivier, S. Lemercier, and J. Pettré. Experimental Study of the Following Dynamics of Pedestrians. In Pedestr. Evacuation Dyn. 2012, pages 305–315. Springer International Publishing, Cham, 2014.Google Scholar
  2. 2.
    I. L. Bajec, M. Mraz, and N. Zimic. Boids with a fuzzy way of thinking. Proc. ASC, 25:58–62, 2003.zbMATHGoogle Scholar
  3. 3.
    M. Batty. Predicting where we walk. Nature, 388(6637):19–20, jul 1997.Google Scholar
  4. 4.
    N. Bellomo and A. Bellouquid. On the modelling of vehicular traffic and crowds by kinetic theory of active particles. In Math. Model. Collect. Behav. Socio-Economic Life Sci., pages 273–296. Birkhäuser Boston, Boston, 2010.Google Scholar
  5. 5.
    N. Bellomo and A. Bellouquid. On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms. Networks Heterog. Media, 6(3):383–399, aug 2011.Google Scholar
  6. 6.
    N. Bellomo, C. Bianca, and V. Coscia. On the modeling of crowd dynamics: An overview and research perspectives. SeMA J., 54(1):25–46, apr 2011.MathSciNetzbMATHGoogle Scholar
  7. 7.
    N. Bellomo and C. Dogbe. On the Modeling of Traffic and Crowds: A Survey of Models, Speculations, and Perspectives. SIAM Rev., 53(3):409–463, jan 2011.MathSciNetzbMATHGoogle Scholar
  8. 8.
    A. Borzì and S. Wongkaew. Modeling and control through leadership of a refined flocking system. Math. Model. Methods Appl. Sci., 25(02):255–282, feb 2015.MathSciNetzbMATHGoogle Scholar
  9. 9.
    V. Braitenberg. Vehicles: Experiments in Synthetic Psychology. MIT Press, Cambridge, Massachusetts, 1984.Google Scholar
  10. 10.
    M. Caponigro, M. Fornasier, B. Piccoli, and E. Trélat. Sparse stabilization and optimal control of the Cucker-Smale model. Math. Control Relat. Fields, 3(4):447–466, sep 2013.MathSciNetzbMATHGoogle Scholar
  11. 11.
    J. A. Carrillo, M. Fornasier, J. Rosado, and G. Toscani. Asymptotic Flocking Dynamics for the Kinetic CuckerSmale Model. SIAM J. Math. Anal., 42(1):218–236, jan 2010.MathSciNetzbMATHGoogle Scholar
  12. 12.
    J. A. Carrillo, M. Fornasier, G. Toscani, and F. Vecil. Particle, kinetic, and hydrodynamic models of swarming. In Math. Model. Collect. Behav. Socio-Economic Life Sci., pages 297–336. Birkhäuser Boston, Boston, 2010.Google Scholar
  13. 13.
    J. A. Carrillo, S. Martin, and M.-T. Wolfram. An improved version of the Hughes model for pedestrian flow. Math. Model. Methods Appl. Sci., 26(04):671–697, apr 2016.MathSciNetzbMATHGoogle Scholar
  14. 14.
    E. Cristiani, B. Piccoli, and A. Tosin. Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints. In Math. Model. Collect. Behav. Socio-Economic Life Sci., pages 337–364. Birkhäuser Boston, Boston, 2010.Google Scholar
  15. 15.
    F. Cucker and S. Smale. Emergent Behavior in Flocks. IEEE Trans. Automat. Contr., 52(5):852–862, may 2007.MathSciNetzbMATHGoogle Scholar
  16. 16.
    F. Cucker and S. Smale. On the mathematics of emergence. Japanese J. Math., 2(1):197–227, 2007.MathSciNetzbMATHGoogle Scholar
  17. 17.
    J. E. Cutting, P. M. Vishton, and P. A. Braren. How we avoid collisions with stationary and moving objects. Psychol. Rev., 102(4):627–651, 1995.Google Scholar
  18. 18.
    W. Daamen and S. P. Hoogendoorn. Controlled Experiments to derive Walking Behaviour. Eur. J. Transp. Infrastruct. Res., 3(1):39–59, 2003.Google Scholar
  19. 19.
    W. Daamen and S. P. Hoogendoorn. Experimental Research of Pedestrian Walking Behavior. Transp. Res. Rec. J. Transp. Res. Board, 1828(January):20–30, jan 2003.Google Scholar
  20. 20.
    P. Degond, C. Appert-Rolland, M. Moussaïd, J. Pettré, and G. Theraulaz. A Hierarchy of Heuristic-Based Models of Crowd Dynamics. J. Stat. Phys., 152(6):1033–1068, sep 2013.MathSciNetzbMATHGoogle Scholar
  21. 21.
    P. Degond, C. Appert-Rolland, J. Pettré, and G. Theraulaz. Vision-based macroscopic pedestrian models. Kinet. Relat. Model., 6(4):809–839, nov 2013.MathSciNetzbMATHGoogle Scholar
  22. 22.
    M. R. D’Orsogna, Y. L. Chuang, A. L. Bertozzi, and L. S. Chayes. Self-Propelled Particles with Soft-Core Interactions: Patterns, Stability, and Collapse. Phys. Rev. Lett., 96(10):104302, mar 2006.Google Scholar
  23. 23.
    J. J. Fruin. Pedestrian Planning and Design. Metropolitan Association of Urban Designers and Environmental Planners, New York, 1971.Google Scholar
  24. 24.
    Q. Gibson. Social Forces. J. Philos., 55(11):441, may 1958.Google Scholar
  25. 25.
    G. Gigerenzer. Why Heuristics Work. Perspect. Psychol. Sci., 3(1):20–29, jan 2008.Google Scholar
  26. 26.
    J. R. Gill and K. Landi. Traumatic Asphyxial Deaths Due to an Uncontrolled Crowd. Am. J. Forensic Med. Pathol., 25(4):358–361, dec 2004.Google Scholar
  27. 27.
    S.-Y. Ha, T. Ha, and J.-H. Kim. Emergent Behavior of a Cucker-Smale Type Particle Model With Nonlinear Velocity Couplings. IEEE Trans. Automat. Contr., 55(7):1679–1683, jul 2010.Google Scholar
  28. 28.
    B. D. Hankin and R. A. Wright. Passenger Flow in Subways. Oper. Res. Q., 9(2):81, jun 1958.Google Scholar
  29. 29.
    D. Helbing. A mathematical model for the behavior of pedestrians. Behav. Sci., 36(4):298–310, oct 1991.Google Scholar
  30. 30.
    D. Helbing. A Fluid Dynamic Model for the Movement of Pedestrians. Complex Syst., 6:391–415, may 1992.Google Scholar
  31. 31.
    D. Helbing. Self-organization in Pedestrian Crowds. In Soc. Self-Organization, pages 71–99. 2012.Google Scholar
  32. 32.
    D. Helbing, L. Buzna, A. Johansson, and T. Werner. Self-Organized Pedestrian Crowd Dynamics: Experiments, Simulations, and Design Solutions. Transp. Sci., 39(1):1–24, feb 2005.Google Scholar
  33. 33.
    D. Helbing, I. J. Farkas, and T. Vicsek. Simulating dynamical features of escape panic. Nature, 407(6803):487–490, 2000.Google Scholar
  34. 34.
    D. Helbing, A. Johansson, and H. Z. Al-Abideen. Crowd turbulence: the physics of crowd disasters. Fifth Int. Conf. Nonlinear Mech., (June):967–969, aug 2007.Google Scholar
  35. 35.
    D. Helbing, A. Johansson, and H. Z. Al-Abideen. Dynamics of crowd disasters: An empirical study. Phys. Rev. E, 75(4):046109, apr 2007.Google Scholar
  36. 36.
    D. Helbing and P. Molnár. Social force model for pedestrian dynamics. Phys. Rev. E, 51(5):4282–4286, may 1995.Google Scholar
  37. 37.
    D. Helbing, P. Molnár, I. J. Farkas, and K. Bolay. Self-organizing pedestrian movement. Environ. Plan. B Plan. Des., 28(3):361–383, 2001.Google Scholar
  38. 38.
    D. Helbing and T. Vicsek. Optimal self-organization. New J. Phys., 1:13.1–13.7=17, 1999.zbMATHGoogle Scholar
  39. 39.
    L. F. Henderson. The Statistics of Crowd Fluids. Nature, 229(5284):381–383, feb 1971.Google Scholar
  40. 40.
    L. F. Henderson. On the fluid mechanics of human crowd motion. Transp. Res., 8(6):509–515, dec 1974.Google Scholar
  41. 41.
    S. P. Hoogendoorn and W. Daamen. Pedestrian Behavior at Bottlenecks. Transp. Sci., 39(2):147–159, may 2005.Google Scholar
  42. 42.
    B. Hopkins, A. Churchill, S. Vogt, and L. Rönnqvist. Braking Reaching Movements: A Test of the Constant Tau-Dot Strategy Under Different Viewing Conditions. J. Mot. Behav., 36(1):3–12, may 2004.Google Scholar
  43. 43.
    R. L. Hughes. A continuum theory for the flow of pedestrians. Transp. Res. Part B Methodol., 36(6):507–535, jul 2002.Google Scholar
  44. 44.
    R. L. Hughes. The Flow of Human Crowds. Annu. Rev. Fluid Mech., 35(1):169–182, jan 2003.Google Scholar
  45. 45.
    A. Jelić, C. Appert-Rolland, S. Lemercier, and J. Pettré. Properties of pedestrians walking in line: Fundamental diagrams. Phys. Rev. E, 85(3):036111, mar 2012.Google Scholar
  46. 46.
    Y.-q. Jiang, P. Zhang, S. Wong, and R.-x. Liu. A higher-order macroscopic model for pedestrian flows. Phys. A Stat. Mech. its Appl., 389(21):4623–4635, nov 2010.Google Scholar
  47. 47.
    A. Johansson and D. Helbing. Analysis of Empirical Trajectory Data of Pedestrians. In Pedestr. Evacuation Dyn. 2008, pages 203–214. Springer Berlin Heidelberg, Berlin, Heidelberg, 2010.Google Scholar
  48. 48.
    N. R. Johnson. Panic at the ‘Who Concert Stampede’: An Empirical Assessment. Soc. Probl., 34(4):362–373, 1987.Google Scholar
  49. 49.
    T. Kretz, A. Grünebohm, M. Kaufman, F. Mazur, and M. Schreckenberg. Experimental study of pedestrian counterflow in a corridor. J. Stat. Mech. Theory Exp., 2006(10):P10001–P10001, oct 2006.Google Scholar
  50. 50.
    T. Kretz, A. Grünebohm, and M. Schreckenberg. Experimental study of pedestrian flow through a bottleneck. J. Stat. Mech. Theory Exp., (10), 2006.Google Scholar
  51. 51.
    S. Lemercier, A. Jelić, R. Kulpa, J. Hua, J. Fehrenbach, P. Degond, C. Appert-Rolland, S. Donikian, and J. Pettré. Realistic following behaviors for crowd simulation. Comput. Graph. Forum, 31(2pt2):489–498, may 2012.zbMATHGoogle Scholar
  52. 52.
    K. Lewin. Field Theory in Social Science. Harper, 1951.Google Scholar
  53. 53.
    M. J. Lighthill and G. B. Whitham. On Kinematic Waves I - Flood Movement in Long Rivers. Proc. R. Soc. A Math. Phys. Eng. Sci., 229(1178):281–316, may 1955.Google Scholar
  54. 54.
    M. J. Lighthill and G. B. Whitham. On Kinematic Waves II - A Theory of Traffic Flow on Long Crowded Roads. Proc. R. Soc. A Math. Phys. Eng. Sci., 229(1178):317–345, may 1955.Google Scholar
  55. 55.
    L. Luo, Z. Fu, X. Zhou, K. Zhu, H. Yang, and L. Yang. Fatigue effect on phase transition of pedestrian movement: experiment and simulation study. J. Stat. Mech. Theory Exp., 2016(10):103401, oct 2016.MathSciNetGoogle Scholar
  56. 56.
    M. Moussaïd, E. G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettré, C. Appert-Rolland, P. Degond, and G. Theraulaz. Traffic instabilities in self-organized pedestrian crowds. PLoS Comput. Biol., 8(3), 2012.Google Scholar
  57. 57.
    M. Moussaïd, D. Helbing, S. Garnier, A. Johansson, M. Combe, and G. Theraulaz. Experimental study of the behavioural mechanisms underlying self-organization in human crowds. Proc. R. Soc. B Biol. Sci., 276(1668):2755–2762, 2009.Google Scholar
  58. 58.
    M. Moussaïd, D. Helbing, and G. Theraulaz. How simple rules determine pedestrian behavior and crowd disasters. Proc. Natl. Acad. Sci., 108(17):6884–6888, 2011.Google Scholar
  59. 59.
    M. Moussaïd, N. Perozo, S. Garnier, D. Helbing, and G. Theraulaz. The walking behaviour of pedestrian social groups and its impact on crowd dynamics. PLoS One, 5(4):1–7, 2010.Google Scholar
  60. 60.
    M. Mri and H. Tsukaguchi. A new method for evaluation of level of service in pedestrian facilities. Transp. Res. Part A Gen., 21(3):223–234, may 1987.Google Scholar
  61. 61.
    K. M. Ngai, F. M. Burkle, A. Hsu, and E. B. Hsu. Human Stampedes: A Systematic Review of Historical and Peer-Reviewed Sources. Disaster Med. Public Health Prep., 3(04):191–195, dec 2009.Google Scholar
  62. 62.
    S. J. Older. Movement of Pedestrians on Footways in Shopping Streets. Traffic Eng. Control, 10(4):160–163, 1968.Google Scholar
  63. 63.
    A. Polus, J. L. Schofer, and A. Ushpiz. Pedestrian Flow and Level of Service. J. Transp. Eng., 109(1):46–56, jan 1983.Google Scholar
  64. 64.
    C. W. Reynolds. Flocks, herds and schools: A distributed behavioral model. ACM SIGGRAPH Comput. Graph., 21(4):25–34, aug 1987.Google Scholar
  65. 65.
    C. W. Reynolds. Steering behaviors for autonomous characters. Game Dev. Conf., pages 763–782, 1999.Google Scholar
  66. 66.
    P. R. Schrater, D. C. Knill, and E. P. Simoncelli. Mechanisms of visual motion detection. Nat. Neurosci., 3(1):64–68, jan 2000.Google Scholar
  67. 67.
    A. Seyfried, B. Steffen, W. Klingsch, and M. Boltes. The fundamental diagram of pedestrian movement revisited. J. Stat. Mech. Theory Exp., (10):41–53, 2005.Google Scholar
  68. 68.
    D. Strömbom. Collective motion from local attraction. J. Theor. Biol., 283(1):145–151, 2011.MathSciNetzbMATHGoogle Scholar
  69. 69.
    D. Strömbom, R. P. Mann, A. M. Wilson, S. Hailes, A. J. Morton, D. J. T. Sumpter, and A. J. King. Solving the shepherding problem: heuristics for herding autonomous, interacting agents. J. R. Soc. Interface, 11(100):20140719–20140719, aug 2014.Google Scholar
  70. 70.
    Transportation Research Board. Highway Capacity Manual: Special Report 209. U.S. Dept. of Transportation, Federal Highway Administration, Washington, D.C., 1985.Google Scholar
  71. 71.
    Transportation Research Board. Highway Capacity Manual 2000. U.S. Dept. of Transportation, Federal Highway Administration, Washington, D.C., 2000.Google Scholar
  72. 72.
    H. M. Traquair. Clinical perimetry. Kimpton, London, 1876.Google Scholar
  73. 73.
    T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet. Novel Type of Phase Transition in a System of Self-Driven Particles. Phys. Rev. Lett., 75(6):1226–1229, aug 1995.MathSciNetGoogle Scholar
  74. 74.
    W. H. Warren and B. R. Fajen. From Optic Flow to Laws of Control. In Opt. Flow Beyond, pages 307–337. Springer Netherlands, Dordrecht, 2004.Google Scholar
  75. 75.
    U. Weidmann. Transporttechnik der Fussgänger, Transporttechnische Eigenschaften des Fussgängerverkehrs (Literturauswertung), volume 90. Institut für Verkehrsplanung, Transporttechnik, Strassen- und Eisenbahnbau (IVT), ETH Zürich, 1993.Google Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Rafael Bailo
    • 1
    Email author
  • José A. Carrillo
    • 1
  • Pierre Degond
    • 1
  1. 1.Department of MathematicsImperial College LondonLondonUK

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