Skip to main content

Dynamic Equilibria in Time-Varying Networks

Part of the Lecture Notes in Computer Science book series (LNISA,volume 12283)

Abstract

Predicting selfish behavior in public environments by considering Nash equilibria is a central concept of game theory. For the dynamic traffic assignment problem modeled by a flow over time game, in which every particle tries to reach its destination as fast as possible, the dynamic equilibria are called Nash flows over time. So far, this model has only been considered for networks in which each arc is equipped with a constant capacity, limiting the outflow rate, and with a transit time, determining the time it takes for a particle to traverse the arc. However, real-world traffic networks can be affected by temporal changes, for example, caused by construction works or special speed zones during some time period. To model these traffic scenarios appropriately, we extend the flow over time model by time-dependent capacities and time-dependent transit times. Our first main result is the characterization of the structure of Nash flows over time. Similar to the static-network model, the strategies of the particles in dynamic equilibria can be characterized by specific static flows, called thin flows with resetting. The second main result is the existence of Nash flows over time, which we show in a constructive manner by extending a flow over time step by step by these thin flows.

Keywords

  • Nash flows over time
  • Dynamic equilibria
  • Deterministic queuing
  • Time-varying networks
  • Dynamic traffic assignment

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689).

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-57980-7_9
  • Chapter length: 16 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   59.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-57980-7
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   79.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.

References

  1. Bhaskar, U., Fleischer, L., Anshelevich, E.: A stackelberg strategy for routing flow over time. Games Econ. Behav. 92, 232–247 (2015)

    MathSciNet  CrossRef  Google Scholar 

  2. Cao, Z., Chen, B., Chen, X., Wang, C.: A network game of dynamic traffic. In Proceedings of the 2017 ACM Conference on Economics and Computation, pp. 695–696 (2017)

    Google Scholar 

  3. Cominetti, R., Correa, J.R., Larré, O.: Existence and uniqueness of equilibria for flows over time. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6756, pp. 552–563. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22012-8_44

    CrossRef  MATH  Google Scholar 

  4. Cominetti, R., Correa, J., Larré, O.: Dynamic equilibria in fluid queueing networks. Oper. Res. 63(1), 21–34 (2015)

    MathSciNet  CrossRef  Google Scholar 

  5. Cominetti, R., Correa, J., Olver, N.: Long term behavior of dynamic equilibria in fluid queuing networks. In: Eisenbrand, F., Koenemann, J. (eds.) IPCO 2017. LNCS, vol. 10328, pp. 161–172. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-59250-3_14

    CrossRef  Google Scholar 

  6. Correa, J., Cristi, A., Oosterwijk, T.: On the price of anarchy for flows over time. In Proceedings of the 2019 ACM Conference on Economics and Computation, pp. 559–577 (2019)

    Google Scholar 

  7. Fleischer, L., Tardos, É.: Efficient continuous-time dynamic network flow algorithms. Oper. Res. Lett. 23(3–5), 71–80 (1998)

    MathSciNet  CrossRef  Google Scholar 

  8. Ford, L.R., Fulkerson, D.R.: Constructing maximal dynamic flows from static flows. Oper. Res. 6, 419–433 (1958)

    MathSciNet  CrossRef  Google Scholar 

  9. Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (1962)

    MATH  Google Scholar 

  10. Gale, D.: Transient flows in networks. Michigan Math. J. 6(1), 59–63 (1959)

    MathSciNet  CrossRef  Google Scholar 

  11. Graf, L., Harks, T., Sering, L.: Dynamic flows with adaptive route choice. Math. Program. (2020)

    Google Scholar 

  12. Harks, T., Peis, B., Schmand, D., Tauer, B., Vargas Koch, L.: Competitive packet routing with priority lists. ACM Trans. Econo. Comp. 6(1), 4 (2018)

    MathSciNet  MATH  Google Scholar 

  13. Koch, R.: Routing Games over Time. Ph.D. thesis, Technische Universität Berlin (2012). https://doi.org/10.14279/depositonce-3347

  14. Koch, R., Skutella, M.: Nash equilibria and the price of anarchy for flows over time. In: Mavronicolas, M., Papadopoulou, V.G. (eds.) SAGT 2009. LNCS, vol. 5814, pp. 323–334. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04645-2_29

    CrossRef  Google Scholar 

  15. Koch, R., Skutella, M.: Nash equilibria and the price of anarchy for flows over time. Theor. Comput. Syst. 49(1), 71–97 (2011)

    MathSciNet  CrossRef  Google Scholar 

  16. Macko, M., Larson, K., Steskal, L.: Braess’s paradox for flows over time. Theor. Comput. Syst. 53(1), 86–106 (2013)

    MathSciNet  CrossRef  Google Scholar 

  17. Minieka, E.: Maximal, lexicographic, and dynamic network flows. Oper. Res. 21(2), 517–527 (1973)

    MathSciNet  CrossRef  Google Scholar 

  18. Peis, B., Tauer, B., Timmermans, V., Vargas Koch, L.: Oligopolistic competitive packet routing. In: 18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (2018)

    Google Scholar 

  19. Scarsini, M., Schröder, M., Tomala, T.: Dynamic atomic congestion games with seasonal flows. Oper. Res. 66(2), 327–339 (2018)

    MathSciNet  CrossRef  Google Scholar 

  20. Sering, L., Skutella, M.: Multi-source multi-sink Nash flows over time. In: 18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems, vol. 65, pp. 12:1–12:20 (2018)

    Google Scholar 

  21. Sering, L., Vargas Koch, L.: Nash flows over time with spillback. In: Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 935–945. SIAM (2019)

    Google Scholar 

  22. Skutella, M.: An introduction to network flows over time. In: Cook, W., Lovász, L., Vygen, J. (eds.) Research trends in combinatorial optimization, pp. 451–482. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-540-76796-1_21

  23. Vickrey, W.S.: Congestion theory and transport investment. Am. Econ. Rev. 59(2), 251–260 (1969)

    Google Scholar 

  24. Wardrop, J.G.: Some theoretical aspects of road traffic research. Proc. Inst. Civil Engineers 1(5), 767–768 (1952)

    CrossRef  Google Scholar 

  25. Wilkinson, W.L.: An algorithm for universal maximal dynamic flows in a network. Oper. Res. 19(7), 1602–1612 (1971)

    MathSciNet  CrossRef  Google Scholar 

  26. Yagar, S.: Dynamic traffic assignment by individual path minimization and queuing. Transp. Res. 5(3), 179–196 (1971)

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Leon Sering .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Pham, H.M., Sering, L. (2020). Dynamic Equilibria in Time-Varying Networks. In: Harks, T., Klimm, M. (eds) Algorithmic Game Theory. SAGT 2020. Lecture Notes in Computer Science(), vol 12283. Springer, Cham. https://doi.org/10.1007/978-3-030-57980-7_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-57980-7_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-57979-1

  • Online ISBN: 978-3-030-57980-7

  • eBook Packages: Computer ScienceComputer Science (R0)