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The Impact of Spillback on the Price of Anarchy for Flows over Time

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Algorithmic Game Theory (SAGT 2020)

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

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Abstract

Flows over time enable a mathematical modeling of traffic that changes as time progresses. In order to evaluate these dynamic flows from a game theoretical perspective we consider the price of anarchy (PoA). In this paper we study the impact of spillback effects on the PoA, which turn out to be substantial. It is known that, in general, the PoA is unbounded in the spillback setting. We extend this by showing that it is still unbounded even when considering networks with unit edge capacities and that the Braess ratio can be arbitrarily large.

In contrast to that, we show that on a fixed network the PoA as a function of the flow amount is bounded by a constant and also upper bound the PoA for the set of networks where the outflow capacities satisfy certain constraints depending on the quickest flow. This upper bound only depends on the worst spillback factor of the Nash flows over time of the given network. It therefore provides a way to quantify the impact of spillback to the quality of the dynamic equilibria.

In addition, we show the surprising fact that the introduction of spillback behavior can actually speed up dynamic equilibria in some networks.

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Grant BR 4744/2-1 and Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689).

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Notes

  1. 1.

    All results from Sect. 4 can also be translated to the total delay price of anarchy measuring the arrival times of all particles combined, similarly as is done in  [1].

  2. 2.

    We imagine i as a natural number. But since it is an open question whether the event point converges to a finite limit, it is possible to expand the index set to the ordinal numbers up to \(\omega ^\omega \). In this case the i-th phase should be defined as \((\theta _i, \theta _{i + 1})\) as it is not possible to determine a predecessor of an ordinal number. For the sake of simplicity however, we stick to the definition where \((0, \theta _1)\) is the first phase.

  3. 3.

    Note, that in  [7] an even more general result is shown for the Koch-Skutella model.

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Authors and Affiliations

  1. Technische Universität Berlin, Berlin, Germany

    Jonas Israel & Leon Sering

Authors
  1. Jonas Israel
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  2. Leon Sering
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Correspondence to Leon Sering .

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Editors and Affiliations

  1. Institute of Mathematics, Augsburg University, Augsburg, Germany

    Tobias Harks

  2. Institute of Mathematics, Technical University Berlin, Berlin, Germany

    Max Klimm

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Israel, J., Sering, L. (2020). The Impact of Spillback on the Price of Anarchy for Flows over Time. 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_8

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  • DOI: https://doi.org/10.1007/978-3-030-57980-7_8

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