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Atomic Splittable Flow Over Time Games

Authors Antonia Adamik, Leon Sering

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Author Details

Antonia Adamik
  • Technische Universität Berlin, Germany
Leon Sering
  • ETH Zürich, Switzerland

Funding

  • Sering, Leon: 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).

Acknowledgements

We have considered atomic splittable flow over time games in different settings and under various assumptions in collaboration with several people. Unfortunately, most of these research directions were more challenging than expected and not as successful as the work at hand. Nonetheless, we want to thank Laura Vargas Koch, Veerle Timmermans, Björn Tauer, Tim Oosterwijk and Dario Frascaria for the excellent collaboration and inspiring discussions.

Cite AsGet BibTex

Antonia Adamik and Leon Sering. Atomic Splittable Flow Over Time Games. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SAND.2022.4

Abstract

In an atomic splittable flow over time game, finitely many players route flow dynamically through a network, in which edges are equipped with transit times, specifying the traversing time, and with capacities, restricting flow rates. Infinitesimally small flow particles controlled by the same player arrive at a constant rate at the player’s origin and the player’s goal is to maximize the flow volume that arrives at the player’s destination within a given time horizon. Here, the flow dynamics are described by the deterministic queuing model, i.e., flow of different players merges perfectly, but excessive flow has to wait in a queue in front of the bottle-neck. In order to determine Nash equilibria in such games, the main challenge is to consider suitable definitions for the players' strategies, which depend on the level of information the players receive throughout the game. For the most restricted version, in which the players receive no information on the network state at all, we can show that there is no Nash equilibrium in general, not even for networks with only two edges. However, if the current edge congestions are provided over time, the players can adapt their route choices dynamically. We show that a profile of those strategies always lead to a unique feasible flow over time. Hence, those atomic splittable flow over time games are well-defined. For parallel-edge networks Nash equilibria exists and the total flow arriving in time equals the value of a maximum flow over time leading to a price of anarchy of 1.

Subject Classification

ACM Subject Classification
  • Theory of computation → Network flows
  • Theory of computation → Network games
  • Mathematics of computing → Network flows
  • Theory of computation → Quality of equilibria
Keywords
  • Flows Over Time
  • Deterministic Queuing
  • Atomic Splittable Games
  • Equilibria
  • Traffic
  • Cooperation

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References

  1. Eitan Altman, Tamer Bacsar, Tania Jimenez, and Nahum Shimkin. Competitive routing in networks with polynomial costs. IEEE Transactions on Automatic Control, 47(1):92-96, 2002. Google Scholar
  2. Mor Armony and Nicholas Bambos. Queueing dynamics and maximal throughput scheduling in switched processing systems. Queueing systems, 44(3):209-252, 2003. Google Scholar
  3. U. Bhaskar, L. Fleischer, and E. Anshelevich. A stackelberg strategy for routing flow over time. Games and Economic Behavior, 92:232-247, 2015. Google Scholar
  4. Umang Bhaskar, Lisa Fleischer, Darrell Hoy, and Chien-Chung Huang. On the uniqueness of equilibrium in atomic splittable routing games. Mathematics of Operations Research, 40(3):634-654, 2015. Google Scholar
  5. Umang Bhaskar and Phani Raj Lolakapuri. Equilibrium computation in atomic splittable routing games. In 26th Annual European Symposium on Algorithms (ESA 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  6. R. E. Burkard, K. Dlaska, and B. Klinz. The quickest flow problem. Zeitschrift für Operations Research, 37(1):31-58, 1993. Google Scholar
  7. Stefano Catoni and Stefano Pallottino. Traffic equilibrium paradoxes. Transportation Science, 25(3):240-244, 1991. Google Scholar
  8. R. Cominetti, J. Correa, and O. Larré. Existence and uniqueness of equilibria for flows over time. In International Colloquium on Automata, Languages, and Programming, pages 552-563. Springer, 2011. Google Scholar
  9. R. Cominetti, J. Correa, and N. Olver. Long-term behavior of dynamic equilibria in fluid queuing networks. Operations Research, 2021. Google Scholar
  10. Roberto Cominetti, José R. Correa, and Omar Larré. Dynamic equilibria in fluid queueing networks. Operations Research, 63(1):21-34, 2015. Google Scholar
  11. Roberto Cominetti, José R Correa, and Nicolás E Stier-Moses. The impact of oligopolistic competition in networks. Operations Research, 57(6):1421-1437, 2009. Google Scholar
  12. José Correa, Andrés Cristi, and Tim Oosterwijk. On the price of anarchy for flows over time. In Proceedings of the 2019 ACM Conference on Economics and Computation, EC ’19, pages 559-577, New York, 2019. Association for Computing Machinery. Google Scholar
  13. José Correa and Nicolás E Stier-Moses. Wardrop equilibria. Wiley encyclopedia of operations research and management science, 2010. Google Scholar
  14. Lisa Fleischer and Éva Tardos. Efficient continuous-time dynamic network flow algorithms. Operations Research Letters, 23(3):71-80, 1998. Google Scholar
  15. L. R. Ford and D. R. Fulkerson. Constructing maximal dynamic flows from static flows. Operations research, 6:419-433, 1958. Google Scholar
  16. L. R. Ford and D. R. Fulkerson. Flows in Networks. Princeton University Press, 1962. Google Scholar
  17. David Gale. Transient flows in networks. Michigan Mathematical Journal, 6(1):59-63, 1959. Google Scholar
  18. Lukas Graf and Tobias Harks. The price of anarchy for instantaneous dynamic equilibria. In International Conference on Web and Internet Economics, pages 237-251. Springer, 2020. Google Scholar
  19. Lukas Graf and Tobias Harks. A finite time combinatorial algorithm for instantaneous dynamic equilibrium flows. Mathematical Programming, pages 1-32, 2022. Google Scholar
  20. Lukas Graf, Tobias Harks, and Leon Sering. Dynamic flows with adaptive route choice. Mathematical Programming, 2020. Google Scholar
  21. Tobias Harks. Stackelberg strategies and collusion in network games with splittable flow. Theory of Computing Systems, 48(4):781-802, 2011. Google Scholar
  22. Tobias Harks, Britta Peis, Daniel Schmand, Bjoern Tauer, and Laura Vargas Koch. Competitive packet routing with priority lists. ACM Transactions on Economics and Computation (TEAC), 6(1):4, 2018. Google Scholar
  23. Tobias Harks and Veerle Timmermans. Equilibrium computation in atomic splittable singleton congestion games. In International Conference on Integer Programming and Combinatorial Optimization, pages 442-454. Springer, 2017. Google Scholar
  24. Tobias Harks and Veerle Timmermans. Uniqueness of equilibria in atomic splittable polymatroid congestion games. Journal of Combinatorial Optimization, 36(3):812-830, 2018. Google Scholar
  25. Alain Haurie and Patrice Marcotte. On the relationship between nash-cournot and wardrop equilibria. Networks, 15(3):295-308, 1985. Google Scholar
  26. Martin Hoefer, Vahab S Mirrokni, Heiko Röglin, and Shang-Hua Teng. Competitive routing over time. In International Workshop on Internet and Network Economics, pages 18-29. Springer, 2009. Google Scholar
  27. J. Israel and L. Sering. The impact of spillback on the price of anarchy for flows over time. In International Symposium on Algorithmic Game Theory, pages 114-129. Springer, 2020. Google Scholar
  28. Max Klimm and Philipp Warode. Complexity and parametric computation of equilibria in atomic splittable congestion games via weighted block laplacians. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2728-2747. SIAM, 2020. Google Scholar
  29. Ronald Koch and Martin Skutella. Nash equilibria and the price of anarchy for flows over time. Theory of Computing Systems, 49(1):71-97, 2011. Google Scholar
  30. Patrice Marcotte. Algorithms for the network oligopoly problem. Journal of the Operational Research Society, 38(11):1051-1065, 1987. Google Scholar
  31. E. Minieka. Maximal, lexicographic, and dynamic network flows. Operations Research, 21(2):517-527, 1973. Google Scholar
  32. N. Olver, L. Sering, and L. Vargas Koch. Continuity, uniqueness and long-term behavior of Nash flows over time. In 2021 IEEE 62st Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 2021. Google Scholar
  33. Britta Peis, Bjoern Tauer, Veerle Timmermans, and Laura Vargas Koch. Oligopolistic competitive packet routing. In 18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  34. H. Pham and L. Sering. Dynamic equilibria in time-varying networks. In International Symposium on Algorithmic Game Theory, pages 130-145. Springer, 2020. Google Scholar
  35. J Ben Rosen. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica: Journal of the Econometric Society, pages 520-534, 1965. Google Scholar
  36. Tim Roughgarden. Selfish routing with atomic players. Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '05), pages 1184-1185, 2005. Google Scholar
  37. Tim Roughgarden and Florian Schoppmann. Local smoothness and the price of anarchy in splittable congestion games. Journal of Economic Theory, 156:317-342, 2015. Google Scholar
  38. Tim Roughgarden and Éva Tardos. How bad is selfish routing? Journal of the ACM, 49(2):236-259, 2002. Google Scholar
  39. L. Sering. Nash flows over time. Technische Universitaet Berlin (Germany), 2020. Google Scholar
  40. L. Sering and M. Skutella. Multi-source multi-sink Nash flows over time. In 18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems, volume 65, pages 12:1-12:20, 2018. Google Scholar
  41. L. Sering and L. Vargas Koch. Nash flows over time with spillback. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 935-945. SIAM, 2019. Google Scholar
  42. Jiří Sgall. Open problems in throughput scheduling. In European Symposium on Algorithms, pages 2-11. Springer, 2012. Google Scholar
  43. Martin Skutella. An introduction to network flows over time. In Research trends in combinatorial optimization, pages 451-482. Springer, 2009. Google Scholar
  44. W. S. Vickrey. Congestion theory and transport investment. The American Economic Review, 59(2):251-260, 1969. URL: http://www.jstor.org/stable/1823678.
  45. John Glen Wardrop. Some theoretical aspects of road traffic research. Proceedings of the Institution of Civil Engineers, 1(3):325-362, 1952. Google Scholar
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