October  2021, 8(4): 359-380. doi: 10.3934/jdg.2021007

Origin-to-destination network flow with path preferences and velocity controls: A mean field game-like approach

1. 

Department of Mathematics, Università di Trento, Via Sommarive, 14, I-38123 Povo, Trento, Italy

2. 

Department of Economics, Business, Mathematics and Statistics, Università di Trieste, Via dell'Università, 1, I-34127 Trieste, Italy

3. 

Department of Management, Università Ca' Foscari Venezia, Fondamenta S. Giobbe, 873, I-30121 Cannaregio, Venezia, Italy

* Corresponding author: Rosario Maggistro

Received  June 2020 Revised  November 2020 Published  October 2021 Early access  February 2021

In this paper we consider a mean field approach to modeling the agents flow over a transportation network. In particular, beside a standard framework of mean field games, with controlled dynamics by the agents and costs mass-distribution dependent, we also consider a path preferences dynamics obtained as a generalization of the so-called noisy best response dynamics. We introduce this last dynamics to model the fact that the agents choose their path on the basis of both the network congestion state and the observation of the agents' decision that have preceded them. We prove the existence of a mean field equilibrium obtained as a fixed point of a map over a suitable set of time-varying mass-distributions, defined edge by edge in the network. We also address the case where the admissible set of controls is suitably bounded depending on the mass-distribution on the edge itself.

Citation: Fabio Bagagiolo, Rosario Maggistro, Raffaele Pesenti. Origin-to-destination network flow with path preferences and velocity controls: A mean field game-like approach. Journal of Dynamics & Games, 2021, 8 (4) : 359-380. doi: 10.3934/jdg.2021007
References:
[1]

Y. AchdouF. Camilli and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. Control Optim., 50 (2012), 77-109.  doi: 10.1137/100790069.  Google Scholar

[2]

F. BagagioloD. BausoR. Maggistro and M. Zoppello, Game theoretic decentralized feedback controls in Markov jump processes, J Optim. Theory Appl., 173 (2017), 704-726.  doi: 10.1007/s10957-017-1078-3.  Google Scholar

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F. Bagagiolo, S. Faggian, R. Maggistro and R. Pesenti, Optimal control of the mean field equilibrium for a pedestrian tourists' flow model, Netw. Spat. Econ., (2019). doi: 10.1007/s11067-019-09475-4.  Google Scholar

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F. Bagagiolo, R. Maggistro and R. Pesenti, A mean field approach to model flows of agents with path preferences over a network, 2019 IEEE 58th Conference on Decision and Control (CDC), Nice, France, 2019. doi: 10.1109/CDC40024.2019.9029794.  Google Scholar

[5]

F. Bagagiolo and R. Pesenti, Non-memoryless pedestrian flow in a crowded environment with target sets, in Advances in Dynamic and Mean Field Games, Ann. Internat. Soc. Dynam. Games, 15, Birkhäuser/Springer, Cham, 2017, 3-25. doi: 10.1007/978-3-319-70619-1_1.  Google Scholar

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G. Barles and B. Perthame, Discontinuous solutions of deterministic optimal stopping time problems, RAIRO Modél. Math. Anal. Numér., 21 (1987), 557-579.  doi: 10.1051/m2an/1987210405571.  Google Scholar

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F. CamilliR. De Maio and A. Tosin, Transport of measures on networks, Netw. Heterog. Media, 12 (2017), 191-215.  doi: 10.3934/nhm.2017008.  Google Scholar

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G. ComoK. SavlaD. AcemogluM. A. Dahleh and E. Frazzoli, Stability analysis of transportation networks with multiscale driver decisions, SIAM J. Control Optim., 51 (2013), 230-252.  doi: 10.1137/110820804.  Google Scholar

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R. Y. Guo and H. J. Huang, A mobile lattice gas model for simulating pedestrian evacuation, Phys. A, 387 (2008), 580-586.  doi: 10.1016/j.physa.2007.10.001.  Google Scholar

[19]

D. HelbingP. MolnárI. J. Farkas and K. Bolay, Self-organizing pedestrian movement, Environ. Plan. B Plan. Des., 28 (2001), 361-383.  doi: 10.1068/b2697.  Google Scholar

[20]

S. P. Hoogendoorn and P. H. L. Bovy, Pedestrian route-choice and activity scheduling theory and models, Transp. Res. B, 38 (2004), 169-190.  doi: 10.1016/S0191-2615(03)00007-9.  Google Scholar

[21]

S. P. Hoogendoorn and P. H. L. Bovy, Pedestrian travel behavior modeling, Netw. Spatial Econ., 5 (2005), 193-216.  doi: 10.1007/s11067-005-2629-y.  Google Scholar

[22]

M. HuangR. P. Malhamé and P. E. Caines, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainly equivalence principle, Commun. Inf. Syst., 6 (2006), 221-251.  doi: 10.4310/CIS.2006.v6.n3.a5.  Google Scholar

[23]

R. L. Hughes, A continuum theory for the flow of pedestrians, Transp. Res. B, 36 (2002), 507-535.  doi: 10.1016/S0191-2615(01)00015-7.  Google Scholar

[24]

H. Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Engrg. Chuo Univ., 28 (1985), 33-77.   Google Scholar

[25]

A. LachapelleJ. Salomon and G. Turinici, Computation of mean field equilibria in economics, Math. Models Methods Appl. Sci., 20 (2010), 567-588.  doi: 10.1142/S0218202510004349.  Google Scholar

[26]

A. Lachapelle and M.-T. Wolfram, On a mean field game approach modeling congestion and aversion in pedestrian crowds, Transp. Res. B, 45 (2011), 1572-1589.  doi: 10.1016/j.trb.2011.07.011.  Google Scholar

[27]

J. M. Lasry and P. L. Lions, Juex à champ moyen. II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[28]

R. Maggistro and G. Como, Stability and optimality of multi-scale transportation networks with distributed dynamic tolls, 2018 IEEE Conference on Decision and Control (CDC), Miami Beach, FL, 2018. doi: 10.1109/CDC.2018.8619804.  Google Scholar

[29]

B. Piccoli and F. Rossi, Measure-theoretic models for crowd dynamics, in Crowd Dynamics. Vol. 1, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 2018,137-165.  Google Scholar

[30]

B. Piccoli and F. Rossi, Transport equation with nonlocal velocity in Wasserstein spaces: Convergence of numerical schemes, Acta Appl. Math., 124 (2013), 73-105.  doi: 10.1007/s10440-012-9771-6.  Google Scholar

[31]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738.  doi: 10.1007/s00205-010-0366-y.  Google Scholar

[32]

R. W. Rosenthal, A class of games possessing pure-strategy Nash equilibria, Internat. J. Game Theory, 2 (1973), 65-67.  doi: 10.1007/BF01737559.  Google Scholar

show all references

References:
[1]

Y. AchdouF. Camilli and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. Control Optim., 50 (2012), 77-109.  doi: 10.1137/100790069.  Google Scholar

[2]

F. BagagioloD. BausoR. Maggistro and M. Zoppello, Game theoretic decentralized feedback controls in Markov jump processes, J Optim. Theory Appl., 173 (2017), 704-726.  doi: 10.1007/s10957-017-1078-3.  Google Scholar

[3]

F. Bagagiolo, S. Faggian, R. Maggistro and R. Pesenti, Optimal control of the mean field equilibrium for a pedestrian tourists' flow model, Netw. Spat. Econ., (2019). doi: 10.1007/s11067-019-09475-4.  Google Scholar

[4]

F. Bagagiolo, R. Maggistro and R. Pesenti, A mean field approach to model flows of agents with path preferences over a network, 2019 IEEE 58th Conference on Decision and Control (CDC), Nice, France, 2019. doi: 10.1109/CDC40024.2019.9029794.  Google Scholar

[5]

F. Bagagiolo and R. Pesenti, Non-memoryless pedestrian flow in a crowded environment with target sets, in Advances in Dynamic and Mean Field Games, Ann. Internat. Soc. Dynam. Games, 15, Birkhäuser/Springer, Cham, 2017, 3-25. doi: 10.1007/978-3-319-70619-1_1.  Google Scholar

[6]

G. Barles and B. Perthame, Discontinuous solutions of deterministic optimal stopping time problems, RAIRO Modél. Math. Anal. Numér., 21 (1987), 557-579.  doi: 10.1051/m2an/1987210405571.  Google Scholar

[7] M. BeckmannC. B. McGuire and C. B. Winsten, Studies in the Economics of Transportation, Yale University Press, New Haven, CT, 1956.   Google Scholar
[8]

N. Bellomo and C. Dogbé, On the modelling crowd dynamics from scaling to hyperbolic macroscopic models, Math. Models Methods Appl. Sci., 18 (2008), 1317-1345.  doi: 10.1142/S0218202508003054.  Google Scholar

[9]

C. Burstedde, A. Kirchner, K. Klauck, A. Schadschneider and J. Zittartz, Cellular automaton approach to pedestrian dynamics-applications, in Pedestrian and Evacuation Dynamics, Springer, 2001, 87-98 Google Scholar

[10]

F. CamilliE. Carlini and C. Marchi, A model problem for mean field games on networks, Discrete Contin. Dyn. Syst., 35 (2015), 4173-4192.  doi: 10.3934/dcds.2015.35.4173.  Google Scholar

[11]

F. CamilliR. De Maio and A. Tosin, Transport of measures on networks, Netw. Heterog. Media, 12 (2017), 191-215.  doi: 10.3934/nhm.2017008.  Google Scholar

[12]

R. M. Colombo and M. D. Rosini, Pedestrian flows and non-classical shocks, Math. Methods Appl. Sci., 28 (2005), 1553-1567.  doi: 10.1002/mma.624.  Google Scholar

[13]

G. ComoK. SavlaD. AcemogluM. A. Dahleh and E. Frazzoli, Stability analysis of transportation networks with multiscale driver decisions, SIAM J. Control Optim., 51 (2013), 230-252.  doi: 10.1137/110820804.  Google Scholar

[14]

E. CristianiB. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9 (2011), 155-182.  doi: 10.1137/100797515.  Google Scholar

[15]

E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, MS & A (Modeling, Simulation & Applications), 12, Springer, Cham, 2014. doi: 10.1007/978-3-319-06620-2.  Google Scholar

[16]

E. CristianiF. S. Priuli and A. Tosin, Modeling rationality to control self-organization of crowds: An environmental approach, SIAM J. Appl. Math., 75 (2015), 605-629.  doi: 10.1137/140962413.  Google Scholar

[17]

O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Math., 2003, Springer, Berlin, 2011,205-266. doi: 10.1007/978-3-642-14660-2_3.  Google Scholar

[18]

R. Y. Guo and H. J. Huang, A mobile lattice gas model for simulating pedestrian evacuation, Phys. A, 387 (2008), 580-586.  doi: 10.1016/j.physa.2007.10.001.  Google Scholar

[19]

D. HelbingP. MolnárI. J. Farkas and K. Bolay, Self-organizing pedestrian movement, Environ. Plan. B Plan. Des., 28 (2001), 361-383.  doi: 10.1068/b2697.  Google Scholar

[20]

S. P. Hoogendoorn and P. H. L. Bovy, Pedestrian route-choice and activity scheduling theory and models, Transp. Res. B, 38 (2004), 169-190.  doi: 10.1016/S0191-2615(03)00007-9.  Google Scholar

[21]

S. P. Hoogendoorn and P. H. L. Bovy, Pedestrian travel behavior modeling, Netw. Spatial Econ., 5 (2005), 193-216.  doi: 10.1007/s11067-005-2629-y.  Google Scholar

[22]

M. HuangR. P. Malhamé and P. E. Caines, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainly equivalence principle, Commun. Inf. Syst., 6 (2006), 221-251.  doi: 10.4310/CIS.2006.v6.n3.a5.  Google Scholar

[23]

R. L. Hughes, A continuum theory for the flow of pedestrians, Transp. Res. B, 36 (2002), 507-535.  doi: 10.1016/S0191-2615(01)00015-7.  Google Scholar

[24]

H. Ishii, Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Engrg. Chuo Univ., 28 (1985), 33-77.   Google Scholar

[25]

A. LachapelleJ. Salomon and G. Turinici, Computation of mean field equilibria in economics, Math. Models Methods Appl. Sci., 20 (2010), 567-588.  doi: 10.1142/S0218202510004349.  Google Scholar

[26]

A. Lachapelle and M.-T. Wolfram, On a mean field game approach modeling congestion and aversion in pedestrian crowds, Transp. Res. B, 45 (2011), 1572-1589.  doi: 10.1016/j.trb.2011.07.011.  Google Scholar

[27]

J. M. Lasry and P. L. Lions, Juex à champ moyen. II. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[28]

R. Maggistro and G. Como, Stability and optimality of multi-scale transportation networks with distributed dynamic tolls, 2018 IEEE Conference on Decision and Control (CDC), Miami Beach, FL, 2018. doi: 10.1109/CDC.2018.8619804.  Google Scholar

[29]

B. Piccoli and F. Rossi, Measure-theoretic models for crowd dynamics, in Crowd Dynamics. Vol. 1, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 2018,137-165.  Google Scholar

[30]

B. Piccoli and F. Rossi, Transport equation with nonlocal velocity in Wasserstein spaces: Convergence of numerical schemes, Acta Appl. Math., 124 (2013), 73-105.  doi: 10.1007/s10440-012-9771-6.  Google Scholar

[31]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738.  doi: 10.1007/s00205-010-0366-y.  Google Scholar

[32]

R. W. Rosenthal, A class of games possessing pure-strategy Nash equilibria, Internat. J. Game Theory, 2 (1973), 65-67.  doi: 10.1007/BF01737559.  Google Scholar

Figure 1.  The graph topology used in the paper
Figure 2.  Fixed point scheme
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