Atomic Splittable Flow Over Time Games
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.
Flows Over Time
Deterministic Queuing
Atomic Splittable Games
Equilibria
Traffic
Cooperation
Theory of computation~Network flows
Theory of computation~Network games
Mathematics of computing~Network flows
Theory of computation~Quality of equilibria
4:1-4:16
Regular Paper
https://arxiv.org/abs/2010.02148
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.
Antonia
Adamik
Antonia Adamik
Technische Universität Berlin, Germany
Leon
Sering
Leon Sering
ETH Zürich, Switzerland
https://orcid.org/0000-0003-2953-1115
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).
10.4230/LIPIcs.SAND.2022.4
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.
Mor Armony and Nicholas Bambos. Queueing dynamics and maximal throughput scheduling in switched processing systems. Queueing systems, 44(3):209-252, 2003.
U. Bhaskar, L. Fleischer, and E. Anshelevich. A stackelberg strategy for routing flow over time. Games and Economic Behavior, 92:232-247, 2015.
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.
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.
R. E. Burkard, K. Dlaska, and B. Klinz. The quickest flow problem. Zeitschrift für Operations Research, 37(1):31-58, 1993.
Stefano Catoni and Stefano Pallottino. Traffic equilibrium paradoxes. Transportation Science, 25(3):240-244, 1991.
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.
R. Cominetti, J. Correa, and N. Olver. Long-term behavior of dynamic equilibria in fluid queuing networks. Operations Research, 2021.
Roberto Cominetti, José R. Correa, and Omar Larré. Dynamic equilibria in fluid queueing networks. Operations Research, 63(1):21-34, 2015.
Roberto Cominetti, José R Correa, and Nicolás E Stier-Moses. The impact of oligopolistic competition in networks. Operations Research, 57(6):1421-1437, 2009.
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.
José Correa and Nicolás E Stier-Moses. Wardrop equilibria. Wiley encyclopedia of operations research and management science, 2010.
Lisa Fleischer and Éva Tardos. Efficient continuous-time dynamic network flow algorithms. Operations Research Letters, 23(3):71-80, 1998.
L. R. Ford and D. R. Fulkerson. Constructing maximal dynamic flows from static flows. Operations research, 6:419-433, 1958.
L. R. Ford and D. R. Fulkerson. Flows in Networks. Princeton University Press, 1962.
David Gale. Transient flows in networks. Michigan Mathematical Journal, 6(1):59-63, 1959.
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.
Lukas Graf and Tobias Harks. A finite time combinatorial algorithm for instantaneous dynamic equilibrium flows. Mathematical Programming, pages 1-32, 2022.
Lukas Graf, Tobias Harks, and Leon Sering. Dynamic flows with adaptive route choice. Mathematical Programming, 2020.
Tobias Harks. Stackelberg strategies and collusion in network games with splittable flow. Theory of Computing Systems, 48(4):781-802, 2011.
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.
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.
Tobias Harks and Veerle Timmermans. Uniqueness of equilibria in atomic splittable polymatroid congestion games. Journal of Combinatorial Optimization, 36(3):812-830, 2018.
Alain Haurie and Patrice Marcotte. On the relationship between nash-cournot and wardrop equilibria. Networks, 15(3):295-308, 1985.
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.
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.
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.
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.
Patrice Marcotte. Algorithms for the network oligopoly problem. Journal of the Operational Research Society, 38(11):1051-1065, 1987.
E. Minieka. Maximal, lexicographic, and dynamic network flows. Operations Research, 21(2):517-527, 1973.
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.
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.
H. Pham and L. Sering. Dynamic equilibria in time-varying networks. In International Symposium on Algorithmic Game Theory, pages 130-145. Springer, 2020.
J Ben Rosen. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica: Journal of the Econometric Society, pages 520-534, 1965.
Tim Roughgarden. Selfish routing with atomic players. Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '05), pages 1184-1185, 2005.
Tim Roughgarden and Florian Schoppmann. Local smoothness and the price of anarchy in splittable congestion games. Journal of Economic Theory, 156:317-342, 2015.
Tim Roughgarden and Éva Tardos. How bad is selfish routing? Journal of the ACM, 49(2):236-259, 2002.
L. Sering. Nash flows over time. Technische Universitaet Berlin (Germany), 2020.
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.
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.
Jiří Sgall. Open problems in throughput scheduling. In European Symposium on Algorithms, pages 2-11. Springer, 2012.
Martin Skutella. An introduction to network flows over time. In Research trends in combinatorial optimization, pages 451-482. Springer, 2009.
W. S. Vickrey. Congestion theory and transport investment. The American Economic Review, 59(2):251-260, 1969. URL: http://www.jstor.org/stable/1823678.
http://www.jstor.org/stable/1823678
John Glen Wardrop. Some theoretical aspects of road traffic research. Proceedings of the Institution of Civil Engineers, 1(3):325-362, 1952.
Antonia Adamik and Leon Sering
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