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Properties of equilibria in transport problems with complex interactions between users

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  • Iryo, Takamasa
  • Watling, David

Abstract

It is well known that uniqueness and stability are guaranteed properties of traffic equilibria in static user-equilibrium traffic assignment problems, if the link travel utilities are assumed to be strictly monotonically decreasing with respect to the link traffic volumes. However, these preferable properties may not necessarily hold in a wide range of transport problems with complex interactions, e.g. asymmetric interactions (including dynamic traffic assignment), social interactions, or with economies of scale. This study aims to investigate such solution properties of transport models with complex interactions between users. Generic formulations of models are considered in this study, both for utility functions and for the evolutionary dynamics relevant to the stability analysis. Such an analysis for a generic formulation is mathematically challenging due to the potential non-differentiability of the dynamical system, precluding the application of standard analyses for smooth systems. To address this issue, this study proposes a transport system with two alternatives and two user groups. While it is a simple model whose dynamics can be depicted on a plane, it also includes the core components of transport models, i.e. multiple choices and user-classes. This study classifies all possible formulations into nine cases with respect to the signs (i.e. positive or negative) of interactions between users. Then, the evolutionary dynamics of each case is mathematically analysed to examine stability of equilibria. Finally, the solution properties of each case is revealed. Multiple equilibria exist in many cases. In addition, cases with no stable equilibrium are also found, yet even in such cases we are able to characterise the circumstances in which the different kinds of unstable behaviour may arise.

Suggested Citation

  • Iryo, Takamasa & Watling, David, 2019. "Properties of equilibria in transport problems with complex interactions between users," Transportation Research Part B: Methodological, Elsevier, vol. 126(C), pages 87-114.
    • Handle: RePEc:eee:transb:v:126:y:2019:i:c:p:87-114
    DOI: 10.1016/j.trb.2019.05.006
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    References listed on IDEAS

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    1. Mohring, Herbert, 1972. "Optimization and Scale Economies in Urban Bus Transportation," American Economic Review, American Economic Association, vol. 62(4), pages 591-604, September.
    2. Iryo, Takamasa, 2011. "Multiple equilibria in a dynamic traffic network," Transportation Research Part B: Methodological, Elsevier, vol. 45(6), pages 867-879, July.
    3. Watling, David, 1996. "Asymmetric problems and stochastic process models of traffic assignment," Transportation Research Part B: Methodological, Elsevier, vol. 30(5), pages 339-357, October.
    4. Giulio Cantarella & Pietro Velonà & David Watling, 2015. "Day-to-day Dynamics & Equilibrium Stability in A Two-Mode Transport System with Responsive bus Operator Strategies," Networks and Spatial Economics, Springer, vol. 15(3), pages 485-506, September.
    5. Yang, Fan & Zhang, Ding, 2009. "Day-to-day stationary link flow pattern," Transportation Research Part B: Methodological, Elsevier, vol. 43(1), pages 119-126, January.
    6. Mounce, Richard, 2006. "Convergence in a continuous dynamic queueing model for traffic networks," Transportation Research Part B: Methodological, Elsevier, vol. 40(9), pages 779-791, November.
    7. William A. Brock & Steven N. Durlauf, 2001. "Discrete Choice with Social Interactions," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 68(2), pages 235-260.
    8. Smith, Mike & Mounce, Richard, 2011. "A splitting rate model of traffic re-routeing and traffic control," Transportation Research Part B: Methodological, Elsevier, vol. 45(9), pages 1389-1409.
    9. Hofbauer, Josef & Sandholm, William H., 2007. "Evolution in games with randomly disturbed payoffs," Journal of Economic Theory, Elsevier, vol. 132(1), pages 47-69, January.
    10. Watling, David P., 2016. "A route-swapping dynamical system and Lyapunov function for stochastic user equilibriumAuthor-Name: Smith, Michael J," Transportation Research Part B: Methodological, Elsevier, vol. 85(C), pages 132-141.
    11. Michael J. Smith, 1984. "The Stability of a Dynamic Model of Traffic Assignment---An Application of a Method of Lyapunov," Transportation Science, INFORMS, vol. 18(3), pages 245-252, August.
    12. Iryo, Takamasa, 2016. "Day-to-day dynamical model incorporating an explicit description of individuals’ information collection behaviour," Transportation Research Part B: Methodological, Elsevier, vol. 92(PA), pages 88-103.
    13. Ren-Yong Guo & Hai Yang & Hai-Jun Huang, 2018. "Are We Really Solving the Dynamic Traffic Equilibrium Problem with a Departure Time Choice?," Transportation Science, INFORMS, vol. 52(3), pages 603-620, June.
    14. Maness, Michael & Cirillo, Cinzia, 2016. "An indirect latent informational conformity social influence choice model: Formulation and case study," Transportation Research Part B: Methodological, Elsevier, vol. 93(PA), pages 75-101.
    15. Terry L. Friesz & David Bernstein & Nihal J. Mehta & Roger L. Tobin & Saiid Ganjalizadeh, 1994. "Day-To-Day Dynamic Network Disequilibria and Idealized Traveler Information Systems," Operations Research, INFORMS, vol. 42(6), pages 1120-1136, December.
    16. Smith, M. J., 1984. "Two alternative definitions of traffic equilibrium," Transportation Research Part B: Methodological, Elsevier, vol. 18(1), pages 63-65, February.
    17. Patrick McCarthy, 2001. "Effect of speed limits on speed distributions and highway safety: A survey of recent literature," Transport Reviews, Taylor & Francis Journals, vol. 21(1), pages 31-50, January.
    18. Benjamin G. Heydecker, 1983. "Some Consequences of Detailed Junction Modeling in Road Traffic Assignment," Transportation Science, INFORMS, vol. 17(3), pages 263-281, August.
    19. Iryo, Takamasa, 2019. "Instability of departure time choice problem: A case with replicator dynamics," Transportation Research Part B: Methodological, Elsevier, vol. 126(C), pages 353-364.
    20. Fukuda, Daisuke & Morichi, Shigeru, 2007. "Incorporating aggregate behavior in an individual's discrete choice: An application to analyzing illegal bicycle parking behavior," Transportation Research Part A: Policy and Practice, Elsevier, vol. 41(4), pages 313-325, May.
    21. Sandholm, William H., 2001. "Potential Games with Continuous Player Sets," Journal of Economic Theory, Elsevier, vol. 97(1), pages 81-108, March.
    22. Ying, Jiang Qian & Yang, Hai, 2005. "Sensitivity analysis of stochastic user equilibrium flows in a bi-modal network with application to optimal pricing," Transportation Research Part B: Methodological, Elsevier, vol. 39(9), pages 769-795, November.
    23. Juan Antonio Carrasco & Bernie Hogan & Barry Wellman & Eric J Miller, 2008. "Collecting Social Network Data to Study Social Activity-Travel Behavior: An Egocentric Approach," Environment and Planning B, , vol. 35(6), pages 961-980, December.
    24. Sandholm, William H., 2003. "Evolution and equilibrium under inexact information," Games and Economic Behavior, Elsevier, vol. 44(2), pages 343-378, August.
    25. Zhang, Ding & Nagurney, Anna, 1996. "On the local and global stability of a travel route choice adjustment process," Transportation Research Part B: Methodological, Elsevier, vol. 30(4), pages 245-262, August.
    26. Maness, Michael & Cirillo, Cinzia & Dugundji, Elenna R., 2015. "Generalized behavioral framework for choice models of social influence: Behavioral and data concerns in travel behavior," Journal of Transport Geography, Elsevier, vol. 46(C), pages 137-150.
    27. He, Xiaozheng & Guo, Xiaolei & Liu, Henry X., 2010. "A link-based day-to-day traffic assignment model," Transportation Research Part B: Methodological, Elsevier, vol. 44(4), pages 597-608, May.
    28. Frank Goetzke & Tilmann Rave, 2011. "Bicycle Use in Germany: Explaining Differences between Municipalities with Social Network Effects," Urban Studies, Urban Studies Journal Limited, vol. 48(2), pages 427-437, February.
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