A road pricing game is a game where various stakeholders and/or regions with different (and usually conflicting) objectives compete for toll setting in a given transportation network to satisfy their individual objectives. We investigate some classical game theoretical solution concepts for the road pricing game. We establish results for the road pricing game so that stakeholders and/or regions playing such a game will beforehand know what is obtainable. This will save time and argument, and above all, get rid of the feelings of unfairness among the competing actors and road users. Among the classical solution concepts we investigate is Nash equilibrium. In particular, we show that no pure Nash equilibrium exists among the actors, and further illustrate that even “mixed Nash equilibrium" may not be achievable in the road pricing game. The paper also demonstrates the type of coalitions that are not only reachable, but also stable and profitable for the actors involved.
[1] O. Johansson, "Optimal road-pricing: Simultaneous treatment of time
losses, increased fuel consumption, and emissions," Transportation
Research Part D: Transport and Environment, vol. 2, pp. 77-87, June 1997.
[2] Y. Yin and S. Lawphongpanich, "Internalizing emission externality
on road networks," Transportation Research Part D: Transport and
Environment, vol. 11, pp. 292-301, July 2006.
[3] Q. Heaney, M. O-Mahony, and E. Gibbons, "External Costs Associated
with Interregional Transport," Transportation Research Record,
vol. 1659, pp. 79-86, Jan. 1999.
[4] X. Guo and H. Yang, "User heterogeneity and bi-criteria system
optimum," Transportation Research Part B: Methodological, vol. 43,
pp. 379-390, May 2009.
[5] Y. Yin and H. Yang, "Optimal Tolls with a Multiclass, Bicriterion Traffic
Network Equilibrium," Transportation Research Record, vol. 1882,
pp. 45-52, Jan. 2004.
[6] C.-C. Lu, X. Zhou, and H. Mahmassani, "Variable Toll Pricing and
Heterogeneous Users: Model and Solution Algorithm for Bicriterion Dynamic
Traffic Assignment Problem," Transportation Research Record,
vol. 1964, pp. 19-26, Jan. 2006.
[7] D. M. Newbery, "Road Damage Externalities and Road User Charges,"
Econometrica, vol. 56, no. 2, pp. 295-316, 1988.
[8] A. Sumalee, S. Shepherd, and A. May, "Road user charging design:
dealing with multi-objectives and constraints," Transportation, vol. 36,
pp. 167-186, Feb. 2009.
[9] H. Yang, F. Xiao, and H. Huang, "Competition and Equilibria of Private
Toll Roads with Elastic Demand," Transportation Research Board 85th
Annual Meeting., 2006.
[10] L. Zhang and D. Levinson, "Road pricing with autonomous links,"
Transportation Research Record: Journal of the Transportation Research
Board, vol. 1932, no. Transportation Research Board of the National
Academies, Washington, D.C., pp. 147-155, 2005.
[11] F. Xiao, H. Yang, and D. Han, "Competition and efficiency of private
toll roads," Transportation Research Part B: Methodological, vol. 41,
pp. 292-308, Mar. 2007.
[12] A. de Palma and R. Lindsey, "Private toll roads: Competition under
various ownership regimes," The Annals of Regional Science, vol. 34,
pp. 13-35, Mar. 2000.
[13] B. De Borger, S. Proost, and K. Van Dender, "Congestion and Tax
Competition in a Parallel Network," 2004.
[14] T.-L. Liu, J. Chen, and H.-J. Huang, "Existence and efficiency of
oligopoly equilibrium under toll and capacity competition," Transportation
Research Part E: Logistics and Transportation Review, vol. 47,
pp. 908-919, Nov. 2011.
[15] S.-i. Mun and K.-j. Ahn, "Road pricing in a serial network," Journal of
Transport Economics and Policy (JTEP), vol. 42, no. 3, pp. 367-395,
2008.
[16] K. Small and E. Verhoef, The Economics of Urban Transportation. Harwood
Fundamentals of Pure and Applied Economics Series, Routledge,
second ed., 2007.
[17] E. Verhoef, "Second-best Road Pricing through Highway Franchising,"
Journal of Urban Economics, vol. 62, pp. 337-61, 2007.
[18] D. Wu, Y. Yin, and H. Yang, "The independence of volume-capacity
ratio of private toll roads in general networks," Transportation Research
Part B: Methodological, vol. 45, pp. 96-101, Jan. 2011.
[19] S.-i. Mun and S. Nakagawa, "Pricing and investment of cross-border
transport infrastructure," Regional Science and Urban Economics,
vol. 40, pp. 228-240, July 2010.
[20] J. Y. T. Wang, H. A. I. Yang, and E. T. Verhoef, "Strategic Interactions
of Bilateral Monopoly on a Private Highway," Networks and Spatial
Economics, vol. 4, pp. 203-235, 2004.
[21] X. Zhang, H. M. Zhang, H.-j. Huang, L. Sun, and T.-Q. Tang, "Competitive
, cooperative and Stackelberg congestion pricing for multiple regions
in transportation networks," Transportmetrica, vol. 7, no. 4, pp. 297-320,
2011.
[22] A. E. Ohazulike, M. C. J. Bliemer, G. J. Still, and E. C. van Berkum,
"Multi-objective road pricing: A cooperative and competitive bilevel
optimization approach," in T.P. Alkim & T. Arentze e.a. (Eds.), 11th
Trail Congress Connecting People - Integrating Expertise, (Delft), T.P.
Alkim & T. Arentze e.a. (Eds.). TRAIL (ISBN 978-90-5584-139-4).,
2010.
[23] A. E. Ohazulike, M. C. J. Bliemer, G. Still, and E. C. V. Berkum, "Multi-
Objective Road Pricing : A Game Theoretic and Multi-Stakeholder
Approach," in 91st annual meeting of the Transportation Research
Board, Washington D.C., 2012.
[24] M. Beckmann, C. McGuire, and C. Winsten, Studies in the Economics
of Transportation. New Haven. CT: Yale University Press, New Haven.
CT, 1955.
[25] S. Mardle and K. M. Miettinen, Nonlinear Multiobjective Optimization,
vol. 51. Massachusetts: Kluwer Academic Publishers, Feb. 2000.
[26] G. P. Liu, J.-B. Yang, and J. F. Whidborne, Multiobjective Optimization
and Control. Hertfordshire, England: Research studies press Ltd, 2003.
[27] M. B. Yildirim and D. W. Hearn, “A first best toll pricing framework for
variable demand traffic assignment problems,” Transportation Research
Part B: Methodological, vol. 39, pp. 659–678, Sept. 2005.
[28] A. E. Ohazulike, “Multi-Objective Road Pricing Problem: A Cooperative
and Competitive Bilevel Optimization Approach,” Master Thesis,
University of Twente, 2009.
[29] S. Leyffer and T. Munson, “Solving multi-leader-common-follower
games,” Optimization Methods and Software, vol. 25, pp. 601–623, Aug.
2010.
[30] J. Nash, “Non-Cooperative Games,” Annals of Mathematics, vol. 54,
no. 2, pp. 286–295, 1951.
[31] N. Nisan, T. Roughgarden, E. Tardos, and V. V. Vazirani, Algorithmic
game theory. New York, USA: Cambridge University Press, 2007.
[32] K.-K. Tan, J. Yu, and X.-Z. Yuan, “Existence theorems of nash equilibria
for non-cooperative n-person games,” International Journal of Game
Theory, vol. 24, pp. 217–222, Sept. 1995.
[33] X. Hu and D. Ralph, “Using EPECs to Model Bilevel Games in
Restructured Electricity Markets with Locational Prices,” Operations
Research, vol. 55, pp. 809–827, Sept. 2007.
[1] O. Johansson, "Optimal road-pricing: Simultaneous treatment of time
losses, increased fuel consumption, and emissions," Transportation
Research Part D: Transport and Environment, vol. 2, pp. 77-87, June 1997.
[2] Y. Yin and S. Lawphongpanich, "Internalizing emission externality
on road networks," Transportation Research Part D: Transport and
Environment, vol. 11, pp. 292-301, July 2006.
[3] Q. Heaney, M. O-Mahony, and E. Gibbons, "External Costs Associated
with Interregional Transport," Transportation Research Record,
vol. 1659, pp. 79-86, Jan. 1999.
[4] X. Guo and H. Yang, "User heterogeneity and bi-criteria system
optimum," Transportation Research Part B: Methodological, vol. 43,
pp. 379-390, May 2009.
[5] Y. Yin and H. Yang, "Optimal Tolls with a Multiclass, Bicriterion Traffic
Network Equilibrium," Transportation Research Record, vol. 1882,
pp. 45-52, Jan. 2004.
[6] C.-C. Lu, X. Zhou, and H. Mahmassani, "Variable Toll Pricing and
Heterogeneous Users: Model and Solution Algorithm for Bicriterion Dynamic
Traffic Assignment Problem," Transportation Research Record,
vol. 1964, pp. 19-26, Jan. 2006.
[7] D. M. Newbery, "Road Damage Externalities and Road User Charges,"
Econometrica, vol. 56, no. 2, pp. 295-316, 1988.
[8] A. Sumalee, S. Shepherd, and A. May, "Road user charging design:
dealing with multi-objectives and constraints," Transportation, vol. 36,
pp. 167-186, Feb. 2009.
[9] H. Yang, F. Xiao, and H. Huang, "Competition and Equilibria of Private
Toll Roads with Elastic Demand," Transportation Research Board 85th
Annual Meeting., 2006.
[10] L. Zhang and D. Levinson, "Road pricing with autonomous links,"
Transportation Research Record: Journal of the Transportation Research
Board, vol. 1932, no. Transportation Research Board of the National
Academies, Washington, D.C., pp. 147-155, 2005.
[11] F. Xiao, H. Yang, and D. Han, "Competition and efficiency of private
toll roads," Transportation Research Part B: Methodological, vol. 41,
pp. 292-308, Mar. 2007.
[12] A. de Palma and R. Lindsey, "Private toll roads: Competition under
various ownership regimes," The Annals of Regional Science, vol. 34,
pp. 13-35, Mar. 2000.
[13] B. De Borger, S. Proost, and K. Van Dender, "Congestion and Tax
Competition in a Parallel Network," 2004.
[14] T.-L. Liu, J. Chen, and H.-J. Huang, "Existence and efficiency of
oligopoly equilibrium under toll and capacity competition," Transportation
Research Part E: Logistics and Transportation Review, vol. 47,
pp. 908-919, Nov. 2011.
[15] S.-i. Mun and K.-j. Ahn, "Road pricing in a serial network," Journal of
Transport Economics and Policy (JTEP), vol. 42, no. 3, pp. 367-395,
2008.
[16] K. Small and E. Verhoef, The Economics of Urban Transportation. Harwood
Fundamentals of Pure and Applied Economics Series, Routledge,
second ed., 2007.
[17] E. Verhoef, "Second-best Road Pricing through Highway Franchising,"
Journal of Urban Economics, vol. 62, pp. 337-61, 2007.
[18] D. Wu, Y. Yin, and H. Yang, "The independence of volume-capacity
ratio of private toll roads in general networks," Transportation Research
Part B: Methodological, vol. 45, pp. 96-101, Jan. 2011.
[19] S.-i. Mun and S. Nakagawa, "Pricing and investment of cross-border
transport infrastructure," Regional Science and Urban Economics,
vol. 40, pp. 228-240, July 2010.
[20] J. Y. T. Wang, H. A. I. Yang, and E. T. Verhoef, "Strategic Interactions
of Bilateral Monopoly on a Private Highway," Networks and Spatial
Economics, vol. 4, pp. 203-235, 2004.
[21] X. Zhang, H. M. Zhang, H.-j. Huang, L. Sun, and T.-Q. Tang, "Competitive
, cooperative and Stackelberg congestion pricing for multiple regions
in transportation networks," Transportmetrica, vol. 7, no. 4, pp. 297-320,
2011.
[22] A. E. Ohazulike, M. C. J. Bliemer, G. J. Still, and E. C. van Berkum,
"Multi-objective road pricing: A cooperative and competitive bilevel
optimization approach," in T.P. Alkim & T. Arentze e.a. (Eds.), 11th
Trail Congress Connecting People - Integrating Expertise, (Delft), T.P.
Alkim & T. Arentze e.a. (Eds.). TRAIL (ISBN 978-90-5584-139-4).,
2010.
[23] A. E. Ohazulike, M. C. J. Bliemer, G. Still, and E. C. V. Berkum, "Multi-
Objective Road Pricing : A Game Theoretic and Multi-Stakeholder
Approach," in 91st annual meeting of the Transportation Research
Board, Washington D.C., 2012.
[24] M. Beckmann, C. McGuire, and C. Winsten, Studies in the Economics
of Transportation. New Haven. CT: Yale University Press, New Haven.
CT, 1955.
[25] S. Mardle and K. M. Miettinen, Nonlinear Multiobjective Optimization,
vol. 51. Massachusetts: Kluwer Academic Publishers, Feb. 2000.
[26] G. P. Liu, J.-B. Yang, and J. F. Whidborne, Multiobjective Optimization
and Control. Hertfordshire, England: Research studies press Ltd, 2003.
[27] M. B. Yildirim and D. W. Hearn, “A first best toll pricing framework for
variable demand traffic assignment problems,” Transportation Research
Part B: Methodological, vol. 39, pp. 659–678, Sept. 2005.
[28] A. E. Ohazulike, “Multi-Objective Road Pricing Problem: A Cooperative
and Competitive Bilevel Optimization Approach,” Master Thesis,
University of Twente, 2009.
[29] S. Leyffer and T. Munson, “Solving multi-leader-common-follower
games,” Optimization Methods and Software, vol. 25, pp. 601–623, Aug.
2010.
[30] J. Nash, “Non-Cooperative Games,” Annals of Mathematics, vol. 54,
no. 2, pp. 286–295, 1951.
[31] N. Nisan, T. Roughgarden, E. Tardos, and V. V. Vazirani, Algorithmic
game theory. New York, USA: Cambridge University Press, 2007.
[32] K.-K. Tan, J. Yu, and X.-Z. Yuan, “Existence theorems of nash equilibria
for non-cooperative n-person games,” International Journal of Game
Theory, vol. 24, pp. 217–222, Sept. 1995.
[33] X. Hu and D. Ralph, “Using EPECs to Model Bilevel Games in
Restructured Electricity Markets with Locational Prices,” Operations
Research, vol. 55, pp. 809–827, Sept. 2007.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:55481", author = "Anthony E. Ohazulike and Georg Still and Walter Kern and Eric C. van Berkum", title = "Multi-Stakeholder Road Pricing Game: Solution Concepts", abstract = "A road pricing game is a game where various stakeholders and/or regions with different (and usually conflicting) objectives compete for toll setting in a given transportation network to satisfy their individual objectives. We investigate some classical game theoretical solution concepts for the road pricing game. We establish results for the road pricing game so that stakeholders and/or regions playing such a game will beforehand know what is obtainable. This will save time and argument, and above all, get rid of the feelings of unfairness among the competing actors and road users. Among the classical solution concepts we investigate is Nash equilibrium. In particular, we show that no pure Nash equilibrium exists among the actors, and further illustrate that even “mixed Nash equilibrium" may not be achievable in the road pricing game. The paper also demonstrates the type of coalitions that are not only reachable, but also stable and profitable for the actors involved.
", keywords = "Road pricing game, Equilibrium problem with equilibrium constraint (EPEC), Nash equilibrium, Game stability.", volume = "6", number = "3", pages = "266-12", }
