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Existence and Complexity of Approximate Equilibria in Weighted Congestion Games

Author

Listed:
  • George Christodoulou

    (School of Informatics, Aristotle University of Thessaloniki, 541 24 Thessaloniki, Greece)

  • Martin Gairing

    (Department of Computer Science, University of Liverpool, Liverpool L69 3BX, United Kingdom)

  • Yiannis Giannakopoulos

    (Department of Data Science, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany)

  • Diogo Poças

    (Laboratório de Sistemas de Grande Escala, Fundação para a Ciência e Tecnologia, 1749-016 Lisboa, Portugal)

  • Clara Waldmann

    (Operations Research Group, Technical University of Munich, 80333 Munich, Germany)

Abstract

We study the existence of approximate pure Nash equilibria ( α -PNE) in weighted atomic congestion games with polynomial cost functions of maximum degree d . Previously, it was known that d -PNE always exist, whereas nonexistence was established only for small constants, namely, for 1.153-PNE. We improve significantly upon this gap, proving that such games in general do not have Θ ˜ ( d ) -PNE, which provides the first superconstant lower bound. Furthermore, we provide a black-box gap-introducing method of combining such nonexistence results with a specific circuit gadget, in order to derive NP-completeness of the decision version of the problem. In particular, by deploying this technique, we are able to show that deciding whether a weighted congestion game has an O ˜ ( d ) -PNE is NP-complete. Previous hardness results were known only for the special case of exact equilibria and arbitrary cost functions. The circuit gadget is of independent interest, and it allows us to also prove hardness for a variety of problems related to the complexity of PNE in congestion games. For example, we demonstrate that the question of existence of α -PNE, in which a certain set of players plays a specific strategy profile, is NP-hard for any α < 3 d / 2 , even for unweighted congestion games. Finally, we study the existence of approximate equilibria in weighted congestion games with general (nondecreasing) costs, as a function of the number of players n . We show that n -PNE always exists, matched by an almost tight nonexistence bound of Θ ˜ ( n ) , which we can again transform into an NP-completeness proof for the decision problem.

Suggested Citation

  • George Christodoulou & Martin Gairing & Yiannis Giannakopoulos & Diogo Poças & Clara Waldmann, 2023. "Existence and Complexity of Approximate Equilibria in Weighted Congestion Games," Mathematics of Operations Research, INFORMS, vol. 48(1), pages 583-602, February.
    • Handle: RePEc:inm:ormoor:v:48:y:2023:i:1:p:583-602
    DOI: 10.1287/moor.2022.1272
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    References listed on IDEAS

    as
    1. Juliane Dunkel & Andreas S. Schulz, 2008. "On the Complexity of Pure-Strategy Nash Equilibria in Congestion and Local-Effect Games," Mathematics of Operations Research, INFORMS, vol. 33(4), pages 851-868, November.
    2. Tobias Harks & Max Klimm, 2012. "On the Existence of Pure Nash Equilibria in Weighted Congestion Games," Mathematics of Operations Research, INFORMS, vol. 37(3), pages 419-436, August.
    3. Conitzer, Vincent & Sandholm, Tuomas, 2008. "New complexity results about Nash equilibria," Games and Economic Behavior, Elsevier, vol. 63(2), pages 621-641, July.
    4. Gilboa, Itzhak & Zemel, Eitan, 1989. "Nash and correlated equilibria: Some complexity considerations," Games and Economic Behavior, Elsevier, vol. 1(1), pages 80-93, March.
    5. José R. Correa & Andreas S. Schulz & Nicolás E. Stier-Moses, 2004. "Selfish Routing in Capacitated Networks," Mathematics of Operations Research, INFORMS, vol. 29(4), pages 961-976, November.
    6. repec:cup:cbooks:9781316779309 is not listed on IDEAS
    7. Roughgarden,Tim, 2016. "Twenty Lectures on Algorithmic Game Theory," Cambridge Books, Cambridge University Press, number 9781316624791, August.
    8. Roughgarden,Tim, 2016. "Twenty Lectures on Algorithmic Game Theory," Cambridge Books, Cambridge University Press, number 9781107172661, August.
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