# American Institute of Mathematical Sciences

April  2012, 8(2): 507-520. doi: 10.3934/jimo.2012.8.507

## A Stackelberg game management model of the urban public transport

 1 School of Mathematical Sciences, Qufu Normal University, Qufu Shandong, 273165, China 2 School of Traffic and Transportation, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China 3 School of Management Science, Qufu Normal University, Rizhao Shandong, 276800

Received  January 2011 Revised  January 2012 Published  April 2012

For the urban public transport management problem, based on analysis of the competition and cooperation relationship among operators on a common network, we establish a generalized Nash equilibrium game first; and then by analyzing the dynamic interaction between manager and operators, we propose a non-cooperative Stackelberg game model in which the manager is the leader and the operators are the followers. To solve the model, we transform it into a variational inequality problem, and the gap function method and the augmented Lagrange algorithm are applied. The given numerical experiments show the efficiency of the proposed model and algorithms.
Citation: Lianju Sun, Ziyou Gao, Yiju Wang. A Stackelberg game management model of the urban public transport. Journal of Industrial and Management Optimization, 2012, 8 (2) : 507-520. doi: 10.3934/jimo.2012.8.507
##### References:
 [1] M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Second edition, "Nonlinear Programming: Theory and Algorithms," Wiley, New York, 1993. [2] A. Bensoussan, Points de Nash dans le cas de fonctionnelles quadratiques et jeux différentials linéaires à $N$ personnes, SIAM Journal Control, 12 (1974), 460-499. doi: 10.1137/0312037. [3] D. P. Bertsekas, "Constrained Optimization and Lagrange Multiplier Methods," Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. [4] T. Boulogne, E. Altman, H. Kameda and O. Pourtallier, Mixed equilibrium (ME) for multi-class routing games, IEEE Transactions on Automatic Control, 47 (2002), 903-916. doi: 10.1109/TAC.2002.1008357. [5] W.-K. Ching, S.-M. Choi and M. Huang, Optimal service capacity in a multiple-server queueing system: A game theory approach, Journal of Industrial and Management Optimization, 6 (2010), 73-102. [6] M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Mathematical Programming, 53 (1992), 99-110. doi: 10.1007/BF01585696. [7] P. T. Harker, Generalized Nash games and quasi-variational inequalities, European Journal of Operational Research, 54 (1991), 81-94. doi: 10.1016/0377-2217(91)90325-P. [8] P. T. Harker, Multiple equilibrium behaviors on networks, Transportation Science, 22 (1988), 39-46. doi: 10.1287/trsc.22.1.39. [9] B.-S. He, H. Yang and C.-S. Zhang, A modified augmented Lagrangian method for a class of monotone variational inequalities, European Journal of Operational Research, 159 (2004), 35-51. doi: 10.1016/S0377-2217(03)00385-0. [10] D. W. Hearn and M. V. Ramana, Solving congestion toll pricing models, in "Equilibrium and Advanced Transportation Modeling" (eds. P. Marcotte and S. Nguyen), Kluwer Academic Publishers, (1998), 109-124. [11] O. Kofi and I. Ugboro, Effective strategic planning in public transit systems, Transportation Research Part E, 44 (2008), 420-439. doi: 10.1016/j.tre.2006.10.008. [12] R. J. La and V. Anantharam, Optimal routing control: Repeated game approach, IEEE Transactions on Automatic Control, 47 (2002), 437-450. doi: 10.1109/9.989076. [13] G. Laporte, J. A. Mesa and P. Federico, A game theoretic frame work for the robust railway transit network design problem, Transportation Research Part B, 44 (2010), 447-459. doi: 10.1016/j.trb.2009.08.004. [14] Z.-Q. Luo, J.-S. Pang and D. Ralph, "Mathematical Programs with Equilibrium Constraints," Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511983658. [15] Q. Meng, H. Yang and M. G. H. Bell, An equivalent continuously differentiable model and a locally convergent algorithm for the continuous network design problem, Transportation Research Part B, 35 (2001), 83-105. doi: 10.1016/S0191-2615(00)00016-3. [16] Y. Nie, H. M. Zhang and D. H. Lee, Models and algorithms for the traffic assignment problem with link capacity constraints, Transportation Research Part B, 38 (2004), 285-312. doi: 10.1016/S0191-2615(03)00010-9. [17] J. S. Pang, The generalized quasi-variational inequality problem, Mathematics of Operations Research, 7 (1982), 211-222. doi: 10.1287/moor.7.2.211. [18] M. Patriksson and R. T. Rockafellar, A mathematical model and descent algorithm for bilevel traffic management, Transportation Science, 36 (2002), 271-291. doi: 10.1287/trsc.36.3.271.7826. [19] K. Shimizu, Y. Ishizuka and J. F. Bard, "Non-Differentiable and Two-Level Mathematical Programming," Kluwer Academic Publishers, Boston, MA, 1997. [20] L. J. Sun and Z. Y. Gao, An equilibrium model for urban transit assignment based on game theory, European Journal of Operational Research, 181 (2007), 305-314. doi: 10.1016/j.ejor.2006.05.028. [21] T. Van Vuren, D. Van Vliet and M. Smith, Combined equilibrium in a network with partial route guidance, in "Traffic Control Methods" (eds. S. Yagar and S. E. Rowen), Engineering Foundation Press, New York, (1990), 375-387. [22] T. Van Vuren and D. Watling, A multiple user class assignment model for route guidance, Transportation Research Record, 1306 (1991), 22-31. [23] Q.-Y. Yan, J.-B. Li and J.-L. Zhang, Licensing schemes in Stackelberg model under asymmetric information of product costs, Journal of Industrial and Management Optimization, 3 (2007), 763-774. doi: 10.3934/jimo.2007.3.763. [24] H. Yang, X. N. Zhang and Q. Meng, Modeling private highways in networks with entry-exit based toll charges, Transportation Research Part B, 38 (2004), 191-213. doi: 10.1016/S0191-2615(03)00050-X. [25] H. Yang, X. N. Zhang and Q. Meng, Stackelberg games and multiple equilibrium behaviors on networks, Transportation Research Part B, 41 (2007), 841-861. doi: 10.1016/j.trb.2007.03.002. [26] J. Zhou, W. H. K. Lam and B. Heydecker, The generalized Nash equilibrium model for oligopolistic transit market with elastic demand, Transportation Research Part B, 39 (2005), 519-544. doi: 10.1016/j.trb.2004.07.003. [27] D. Zhu and P. Marcotte, Modified descent methods for solving the monotone variational inequality problem, Operations Research Letters, 14 (1993), 111-120. doi: 10.1016/0167-6377(93)90103-N.

show all references

##### References:
 [1] M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Second edition, "Nonlinear Programming: Theory and Algorithms," Wiley, New York, 1993. [2] A. Bensoussan, Points de Nash dans le cas de fonctionnelles quadratiques et jeux différentials linéaires à $N$ personnes, SIAM Journal Control, 12 (1974), 460-499. doi: 10.1137/0312037. [3] D. P. Bertsekas, "Constrained Optimization and Lagrange Multiplier Methods," Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. [4] T. Boulogne, E. Altman, H. Kameda and O. Pourtallier, Mixed equilibrium (ME) for multi-class routing games, IEEE Transactions on Automatic Control, 47 (2002), 903-916. doi: 10.1109/TAC.2002.1008357. [5] W.-K. Ching, S.-M. Choi and M. Huang, Optimal service capacity in a multiple-server queueing system: A game theory approach, Journal of Industrial and Management Optimization, 6 (2010), 73-102. [6] M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Mathematical Programming, 53 (1992), 99-110. doi: 10.1007/BF01585696. [7] P. T. Harker, Generalized Nash games and quasi-variational inequalities, European Journal of Operational Research, 54 (1991), 81-94. doi: 10.1016/0377-2217(91)90325-P. [8] P. T. Harker, Multiple equilibrium behaviors on networks, Transportation Science, 22 (1988), 39-46. doi: 10.1287/trsc.22.1.39. [9] B.-S. He, H. Yang and C.-S. Zhang, A modified augmented Lagrangian method for a class of monotone variational inequalities, European Journal of Operational Research, 159 (2004), 35-51. doi: 10.1016/S0377-2217(03)00385-0. [10] D. W. Hearn and M. V. Ramana, Solving congestion toll pricing models, in "Equilibrium and Advanced Transportation Modeling" (eds. P. Marcotte and S. Nguyen), Kluwer Academic Publishers, (1998), 109-124. [11] O. Kofi and I. Ugboro, Effective strategic planning in public transit systems, Transportation Research Part E, 44 (2008), 420-439. doi: 10.1016/j.tre.2006.10.008. [12] R. J. La and V. Anantharam, Optimal routing control: Repeated game approach, IEEE Transactions on Automatic Control, 47 (2002), 437-450. doi: 10.1109/9.989076. [13] G. Laporte, J. A. Mesa and P. Federico, A game theoretic frame work for the robust railway transit network design problem, Transportation Research Part B, 44 (2010), 447-459. doi: 10.1016/j.trb.2009.08.004. [14] Z.-Q. Luo, J.-S. Pang and D. Ralph, "Mathematical Programs with Equilibrium Constraints," Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511983658. [15] Q. Meng, H. Yang and M. G. H. Bell, An equivalent continuously differentiable model and a locally convergent algorithm for the continuous network design problem, Transportation Research Part B, 35 (2001), 83-105. doi: 10.1016/S0191-2615(00)00016-3. [16] Y. Nie, H. M. Zhang and D. H. Lee, Models and algorithms for the traffic assignment problem with link capacity constraints, Transportation Research Part B, 38 (2004), 285-312. doi: 10.1016/S0191-2615(03)00010-9. [17] J. S. Pang, The generalized quasi-variational inequality problem, Mathematics of Operations Research, 7 (1982), 211-222. doi: 10.1287/moor.7.2.211. [18] M. Patriksson and R. T. Rockafellar, A mathematical model and descent algorithm for bilevel traffic management, Transportation Science, 36 (2002), 271-291. doi: 10.1287/trsc.36.3.271.7826. [19] K. Shimizu, Y. Ishizuka and J. F. Bard, "Non-Differentiable and Two-Level Mathematical Programming," Kluwer Academic Publishers, Boston, MA, 1997. [20] L. J. Sun and Z. Y. Gao, An equilibrium model for urban transit assignment based on game theory, European Journal of Operational Research, 181 (2007), 305-314. doi: 10.1016/j.ejor.2006.05.028. [21] T. Van Vuren, D. Van Vliet and M. Smith, Combined equilibrium in a network with partial route guidance, in "Traffic Control Methods" (eds. S. Yagar and S. E. Rowen), Engineering Foundation Press, New York, (1990), 375-387. [22] T. Van Vuren and D. Watling, A multiple user class assignment model for route guidance, Transportation Research Record, 1306 (1991), 22-31. [23] Q.-Y. Yan, J.-B. Li and J.-L. Zhang, Licensing schemes in Stackelberg model under asymmetric information of product costs, Journal of Industrial and Management Optimization, 3 (2007), 763-774. doi: 10.3934/jimo.2007.3.763. [24] H. Yang, X. N. Zhang and Q. Meng, Modeling private highways in networks with entry-exit based toll charges, Transportation Research Part B, 38 (2004), 191-213. doi: 10.1016/S0191-2615(03)00050-X. [25] H. Yang, X. N. Zhang and Q. Meng, Stackelberg games and multiple equilibrium behaviors on networks, Transportation Research Part B, 41 (2007), 841-861. doi: 10.1016/j.trb.2007.03.002. [26] J. Zhou, W. H. K. Lam and B. Heydecker, The generalized Nash equilibrium model for oligopolistic transit market with elastic demand, Transportation Research Part B, 39 (2005), 519-544. doi: 10.1016/j.trb.2004.07.003. [27] D. Zhu and P. Marcotte, Modified descent methods for solving the monotone variational inequality problem, Operations Research Letters, 14 (1993), 111-120. doi: 10.1016/0167-6377(93)90103-N.
 [1] Takeshi Fukao, Nobuyuki Kenmochi. Abstract theory of variational inequalities and Lagrange multipliers. Conference Publications, 2013, 2013 (special) : 237-246. doi: 10.3934/proc.2013.2013.237 [2] Xiaona Fan, Li Jiang, Mengsi Li. Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1795-1807. doi: 10.3934/jimo.2018123 [3] Dean A. Carlson. Finding open-loop Nash equilibrium for variational games. Conference Publications, 2005, 2005 (Special) : 153-163. doi: 10.3934/proc.2005.2005.153 [4] Yannick Viossat. Game dynamics and Nash equilibria. Journal of Dynamics and Games, 2014, 1 (3) : 537-553. doi: 10.3934/jdg.2014.1.537 [5] Qinglan Xia. An application of optimal transport paths to urban transport networks. Conference Publications, 2005, 2005 (Special) : 904-910. doi: 10.3934/proc.2005.2005.904 [6] Bin Zhou, Hailin Sun. Two-stage stochastic variational inequalities for Cournot-Nash equilibrium with risk-averse players under uncertainty. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 521-535. doi: 10.3934/naco.2020049 [7] Ouafa Belguidoum, Hassina Grar. An improved projection algorithm for variational inequality problem with multivalued mapping. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022002 [8] Pedro L. García, Antonio Fernández, César Rodrigo. Variational integrators for discrete Lagrange problems. Journal of Geometric Mechanics, 2010, 2 (4) : 343-374. doi: 10.3934/jgm.2010.2.343 [9] Jen-Yen Lin, Hui-Ju Chen, Ruey-Lin Sheu. Augmented Lagrange primal-dual approach for generalized fractional programming problems. Journal of Industrial and Management Optimization, 2013, 9 (4) : 723-741. doi: 10.3934/jimo.2013.9.723 [10] Grace Nnennaya Ogwo, Chinedu Izuchukwu, Oluwatosin Temitope Mewomo. A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 373-393. doi: 10.3934/naco.2021011 [11] Abd-semii Oluwatosin-Enitan Owolabi, Timilehin Opeyemi Alakoya, Adeolu Taiwo, Oluwatosin Temitope Mewomo. A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 255-278. doi: 10.3934/naco.2021004 [12] Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics and Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006 [13] Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091 [14] Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A penalty method for generalized Nash equilibrium problems. Journal of Industrial and Management Optimization, 2012, 8 (1) : 51-65. doi: 10.3934/jimo.2012.8.51 [15] Annamaria Barbagallo, Rosalba Di Vincenzo, Stéphane Pia. On strong Lagrange duality for weighted traffic equilibrium problem. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1097-1113. doi: 10.3934/dcds.2011.31.1097 [16] Marta Faias, Emma Moreno-García, Myrna Wooders. A strategic market game approach for the private provision of public goods. Journal of Dynamics and Games, 2014, 1 (2) : 283-298. doi: 10.3934/jdg.2014.1.283 [17] Sheng Zhang, Xiu Yang, Samy Tindel, Guang Lin. Augmented Gaussian random field: Theory and computation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 931-957. doi: 10.3934/dcdss.2021098 [18] Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449 [19] Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437 [20] Nassif Ghoussoub. A variational principle for nonlinear transport equations. Communications on Pure and Applied Analysis, 2005, 4 (4) : 735-742. doi: 10.3934/cpaa.2005.4.735

2020 Impact Factor: 1.801