IDEAS home Printed from https://ideas.repec.org/p/amz/wpaper/2021-23.html
   My bibliography  Save this paper

Best-Response Dynamics, Playing Sequences, And Convergence To Equilibrium In Random Games

Author

Listed:
  • Pangallo, Marco
  • Heinrich, Torsten
  • Jang, Yoojin
  • Scott, Alex
  • Tarbush, Bassel
  • Wiese, Samuel
  • Mungo, Luca

Abstract

We analyze the performance of the best-response dynamic across all normal-form games using a random games approach. The playing sequence—the order in which players update their actions—is essentially irrelevant in determining whether the dynamic converges to a Nash equilibrium in certain classes of games (e.g. in potential games) but, when evaluated across all possible games, convergence to equilibrium depends on the playing sequence in an extreme way. Our main asymptotic result shows that the best-response dynamic converges to a pure Nash equilibrium in a vanishingly small fraction of all (large) games when players take turns according to a fixed cyclic order. By contrast, when the playing sequence is random, the dynamic converges to a pure Nash equilibrium if one exists in almost all (large) games.

Suggested Citation

  • Pangallo, Marco & Heinrich, Torsten & Jang, Yoojin & Scott, Alex & Tarbush, Bassel & Wiese, Samuel & Mungo, Luca, 2021. "Best-Response Dynamics, Playing Sequences, And Convergence To Equilibrium In Random Games," INET Oxford Working Papers 2021-23, Institute for New Economic Thinking at the Oxford Martin School, University of Oxford.
    • Handle: RePEc:amz:wpaper:2021-23
    as

    Download full text from publisher

    File URL: https://www.inet.ox.ac.uk/files/Best-Response-Dynamics-Playing-Sequences-And-Convergence-To-Equilibrium-In-Random-Games.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Pangallo, Marco & Farmer, J. Doyne & Heinrich, Torsten, "undated". "Best reply structure and equilibrium convergence in generic games," INET Oxford Working Papers 2017-07, Institute for New Economic Thinking at the Oxford Martin School, University of Oxford, revised Mar 2018.
    2. Rinott, Yosef & Scarsini, Marco, 2000. "On the Number of Pure Strategy Nash Equilibria in Random Games," Games and Economic Behavior, Elsevier, vol. 33(2), pages 274-293, November.
    3. Babichenko, Yakov, 2013. "Best-reply dynamics in large binary-choice anonymous games," Games and Economic Behavior, Elsevier, vol. 81(C), pages 130-144.
    4. McLennan, Andrew & Berg, Johannes, 2005. "Asymptotic expected number of Nash equilibria of two-player normal form games," Games and Economic Behavior, Elsevier, vol. 51(2), pages 264-295, May.
    5. Blume Lawrence E., 1993. "The Statistical Mechanics of Strategic Interaction," Games and Economic Behavior, Elsevier, vol. 5(3), pages 387-424, July.
    6. Ben Amiet & Andrea Collevecchio & Marco Scarsini & Ziwen Zhong, 2021. "Pure Nash Equilibria and Best-Response Dynamics in Random Games," Mathematics of Operations Research, INFORMS, vol. 46(4), pages 1552-1572, November.
    7. Stanford, William, 1997. "On the distribution of pure strategy equilibria in finite games with vector payoffs," Mathematical Social Sciences, Elsevier, vol. 33(2), pages 115-127, April.
    8. Andrew McLennan, 2005. "The Expected Number of Nash Equilibria of a Normal Form Game," Econometrica, Econometric Society, vol. 73(1), pages 141-174, January.
    9. Klaus Kultti & Hannu Salonen & Hannu Vartiainen, 2011. "Distribution of pure Nash equilibria in n-person games with random best replies," Discussion Papers 71, Aboa Centre for Economics.
    10. Vincent Boucher, 2017. "Selecting Equilibria using Best-Response Dynamics," Economics Bulletin, AccessEcon, vol. 37(4), pages 2728-2734.
    11. Pei, Ting & Takahashi, Satoru, 2019. "Rationalizable strategies in random games," Games and Economic Behavior, Elsevier, vol. 118(C), pages 110-125.
    12. Arieli, Itai & Babichenko, Yakov, 2016. "Random extensive form games," Journal of Economic Theory, Elsevier, vol. 166(C), pages 517-535.
    13. Stanford, William, 1999. "On the number of pure strategy Nash equilibria in finite common payoffs games," Economics Letters, Elsevier, vol. 62(1), pages 29-34, January.
    14. William Stanford, 1996. "The Limit Distribution of Pure Strategy Nash Equilibria in Symmetric Bimatrix Games," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 726-733, August.
    15. repec:ebl:ecbull:v:3:y:2002:i:22:p:1-6 is not listed on IDEAS
    16. Friedman, James W. & Mezzetti, Claudio, 2001. "Learning in Games by Random Sampling," Journal of Economic Theory, Elsevier, vol. 98(1), pages 55-84, May.
    17. Takahashi, Satoru, 2008. "The number of pure Nash equilibria in a random game with nondecreasing best responses," Games and Economic Behavior, Elsevier, vol. 63(1), pages 328-340, May.
    18. Powers, Imelda Yeung, 1990. "Limiting Distributions of the Number of Pure Strategy Nash Equilibria in N-Person Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 19(3), pages 277-286.
    19. Dindos, Martin & Mezzetti, Claudio, 2006. "Better-reply dynamics and global convergence to Nash equilibrium in aggregative games," Games and Economic Behavior, Elsevier, vol. 54(2), pages 261-292, February.
    20. Candogan, Ozan & Ozdaglar, Asuman & Parrilo, Pablo A., 2013. "Dynamics in near-potential games," Games and Economic Behavior, Elsevier, vol. 82(C), pages 66-90.
    21. Tetsuo Yamamori & Satoru Takahashi, 2002. "The pure Nash equilibrium property and the quasi-acyclic condition," Economics Bulletin, AccessEcon, vol. 3(22), pages 1-6.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Mimun, Hlafo Alfie & Quattropani, Matteo & Scarsini, Marco, 2024. "Best-response dynamics in two-person random games with correlated payoffs," Games and Economic Behavior, Elsevier, vol. 145(C), pages 239-262.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Torsten Heinrich & Yoojin Jang & Luca Mungo & Marco Pangallo & Alex Scott & Bassel Tarbush & Samuel Wiese, 2021. "Best-response dynamics, playing sequences, and convergence to equilibrium in random games," Papers 2101.04222, arXiv.org, revised Nov 2022.
    2. Torsten Heinrich & Yoojin Jang & Luca Mungo & Marco Pangallo & Alex Scott & Bassel Tarbush & Samuel Wiese, 2023. "Best-response dynamics, playing sequences, and convergence to equilibrium in random games," International Journal of Game Theory, Springer;Game Theory Society, vol. 52(3), pages 703-735, September.
    3. Ben Amiet & Andrea Collevecchio & Marco Scarsini & Ziwen Zhong, 2021. "Pure Nash Equilibria and Best-Response Dynamics in Random Games," Mathematics of Operations Research, INFORMS, vol. 46(4), pages 1552-1572, November.
    4. Takahashi, Satoru, 2008. "The number of pure Nash equilibria in a random game with nondecreasing best responses," Games and Economic Behavior, Elsevier, vol. 63(1), pages 328-340, May.
    5. Tom Johnston & Michael Savery & Alex Scott & Bassel Tarbush, 2023. "Game Connectivity and Adaptive Dynamics," Papers 2309.10609, arXiv.org, revised Nov 2023.
    6. Pei, Ting & Takahashi, Satoru, 2019. "Rationalizable strategies in random games," Games and Economic Behavior, Elsevier, vol. 118(C), pages 110-125.
    7. Ben Amiet & Andrea Collevecchio & Kais Hamza, 2020. "When "Better" is better than "Best"," Papers 2011.00239, arXiv.org.
    8. Samuel C. Wiese & Torsten Heinrich, 2022. "The Frequency of Convergent Games under Best-Response Dynamics," Dynamic Games and Applications, Springer, vol. 12(2), pages 689-700, June.
    9. Klaus Kultti & Hannu Salonen & Hannu Vartiainen, 2011. "Distribution of pure Nash equilibria in n-person games with random best replies," Discussion Papers 71, Aboa Centre for Economics.
    10. Arieli, Itai & Babichenko, Yakov, 2016. "Random extensive form games," Journal of Economic Theory, Elsevier, vol. 166(C), pages 517-535.
    11. Szabó, György & Borsos, István & Szombati, Edit, 2019. "Games, graphs and Kirchhoff laws," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 521(C), pages 416-423.
    12. Rinott, Yosef & Scarsini, Marco, 2000. "On the Number of Pure Strategy Nash Equilibria in Random Games," Games and Economic Behavior, Elsevier, vol. 33(2), pages 274-293, November.
    13. Jonathan Newton, 2018. "Evolutionary Game Theory: A Renaissance," Games, MDPI, vol. 9(2), pages 1-67, May.
    14. Mimun, Hlafo Alfie & Quattropani, Matteo & Scarsini, Marco, 2024. "Best-response dynamics in two-person random games with correlated payoffs," Games and Economic Behavior, Elsevier, vol. 145(C), pages 239-262.
    15. Heinrich, Torsten & Wiese, Samuel, 2020. "The Frequency of Convergent Games under Best-Response Dynamics," INET Oxford Working Papers 2020-24, Institute for New Economic Thinking at the Oxford Martin School, University of Oxford.
    16. Sung-Ha Hwang & Jonathan Newton, 2017. "Payoff-dependent dynamics and coordination games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 64(3), pages 589-604, October.
    17. Carlos Alós-Ferrer & Nick Netzer, 2015. "Robust stochastic stability," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 58(1), pages 31-57, January.
    18. Andrea Collevecchio & Tuan-Minh Nguyen & Ziwen Zhong, 2024. "Finding pure Nash equilibria in large random games," Papers 2406.09732, arXiv.org, revised Aug 2024.
    19. Stanford, William, 2010. "The number of pure strategy Nash equilibria in random multi-team games," Economics Letters, Elsevier, vol. 108(3), pages 352-354, September.
    20. Noga Alon & Kirill Rudov & Leeat Yariv, 2021. "Dominance Solvability in Random Games," Working Papers 2021-84, Princeton University. Economics Department..

    More about this item

    Keywords

    Best-response dynamics; equilibrium convergence; random games;
    All these keywords.

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:amz:wpaper:2021-23. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: INET Oxford admin team (email available below). General contact details of provider: https://edirc.repec.org/data/inoxfuk.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.