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Multi-Stakeholder Road Pricing Game: Solution Concepts

Authors: Anthony E. Ohazulike, Georg Still, Walter Kern, Eric C. van Berkum


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: Nash Equilibrium, Road pricing game, Equilibrium problem with equilibrium constraint (EPEC), Game stability

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[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.