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 and 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.

[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.

[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.

[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.

[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.

[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.

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

[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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

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

[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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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