IDEAS home Printed from https://ideas.repec.org/a/gam/jgames/v15y2024i6p41-d1534570.html
   My bibliography  Save this article

End Behavior of the Threshold Protocol Game on Complete and Bipartite Graphs

Author

Listed:
  • Alexandra Fedrigo

    (Department of Mathematical Sciences, University of Alabama in Huntsville, 301 Sparkman Drive, Huntsville, AL 35899, USA)

Abstract

The threshold protocol game is a graphical game that models the adoption of an idea or product through a population. There are two states players may take in the game, and the goal of the game is to motivate the state that begins in the minority to spread to every player. Here, the threshold protocol game is defined, and existence results are studied on infinite graphs. Many generalizations are proposed and applied. This work explores the impact of graph topology on the outcome of the threshold protocol game and consequently considers finite graphs. By exploiting the well-known topologies of complete and complete bipartite graphs, the outcome of the threshold protocol game can be fully characterized on these graphs. These characterizations are ideal, as they are given in terms of the game parameters. More generally, initial conditions in terms of game parameters that cause the preferred game outcome to occur are identified. It is shown that the necessary conditions differ between non-bipartite and bipartite graphs because non-bipartite graphs contain odd cycles while bipartite graphs do not. These results motivate the primary result of this work, which is an exhaustive list of achievable game outcomes on bipartite graphs. While possible outcomes are identified, it is noted that a complete characterization of when game outcomes occur is not possible on general bipartite graphs.

Suggested Citation

  • Alexandra Fedrigo, 2024. "End Behavior of the Threshold Protocol Game on Complete and Bipartite Graphs," Games, MDPI, vol. 15(6), pages 1-16, December.
    • Handle: RePEc:gam:jgames:v:15:y:2024:i:6:p:41-:d:1534570
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2073-4336/15/6/41/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2073-4336/15/6/41/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Kandori Michihiro & Rob Rafael, 1995. "Evolution of Equilibria in the Long Run: A General Theory and Applications," Journal of Economic Theory, Elsevier, vol. 65(2), pages 383-414, April.
    2. Hisashi Ohtsuki & Christoph Hauert & Erez Lieberman & Martin A. Nowak, 2006. "A simple rule for the evolution of cooperation on graphs and social networks," Nature, Nature, vol. 441(7092), pages 502-505, May.
    Full references (including those not matched with items on IDEAS)

    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. Sanjeev Goyal & Fernando Vega-Redondo, 2000. "Learning, Network Formation and Coordination," Econometric Society World Congress 2000 Contributed Papers 0113, Econometric Society.
    2. Konno, Tomohiko, 2013. "An imperfect competition on scale-free networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(21), pages 5453-5460.
    3. Hofbauer, Josef & Sorger, Gerhard, 1999. "Perfect Foresight and Equilibrium Selection in Symmetric Potential Games," Journal of Economic Theory, Elsevier, vol. 85(1), pages 1-23, March.
    4. Trenchard, Hugh, 2015. "The peloton superorganism and protocooperative behavior," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 179-192.
    5. Carlos Alós-Ferrer & Georg Kirchsteiger & Markus Walzl, 2010. "On the Evolution of Market Institutions: The Platform Design Paradox," Economic Journal, Royal Economic Society, vol. 120(543), pages 215-243, March.
    6. , & , & ,, 2008. "Monotone methods for equilibrium selection under perfect foresight dynamics," Theoretical Economics, Econometric Society, vol. 3(2), June.
    7. R. Bentley & Michael O’Brien & Paul Ormerod, 2011. "Quality versus mere popularity: a conceptual map for understanding human behavior," Mind & Society: Cognitive Studies in Economics and Social Sciences, Springer;Fondazione Rosselli, vol. 10(2), pages 181-191, December.
    8. Jorge Peña & Yannick Rochat, 2012. "Bipartite Graphs as Models of Population Structures in Evolutionary Multiplayer Games," PLOS ONE, Public Library of Science, vol. 7(9), pages 1-13, September.
    9. Jehiel, Philippe, 1998. "Learning to Play Limited Forecast Equilibria," Games and Economic Behavior, Elsevier, vol. 22(2), pages 274-298, February.
    10. Hehenkamp, Burkhard & Kaarbøe, Oddvar M., 2004. "Equilibrium selection in supermodular games with mean payoff technologies," Working Papers in Economics 08/04, University of Bergen, Department of Economics.
    11. Peng Liu & Haoxiang Xia, 2015. "Structure and evolution of co-authorship network in an interdisciplinary research field," Scientometrics, Springer;Akadémiai Kiadó, vol. 103(1), pages 101-134, April.
    12. Aslihan Akdeniz & Matthijs van Veelen, 2019. "The cancellation effect at the group level," Tinbergen Institute Discussion Papers 19-073/I, Tinbergen Institute.
    13. Oechssler, Jorg, 1997. "An Evolutionary Interpretation of Mixed-Strategy Equilibria," Games and Economic Behavior, Elsevier, vol. 21(1-2), pages 203-237, October.
    14. Ennio Bilancini & Leonardo Boncinelli, 2020. "The evolution of conventions under condition-dependent mistakes," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 69(2), pages 497-521, March.
    15. Michael Foley & Rory Smead & Patrick Forber & Christoph Riedl, 2021. "Avoiding the bullies: The resilience of cooperation among unequals," PLOS Computational Biology, Public Library of Science, vol. 17(4), pages 1-18, April.
    16. Zhao, Zhengwu & Zhang, Chunyan, 2023. "The mechanisms of labor division from the perspective of task urgency and game theory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 630(C).
    17. Lessard, Sabin & Lahaie, Philippe, 2009. "Fixation probability with multiple alleles and projected average allelic effect on selection," Theoretical Population Biology, Elsevier, vol. 75(4), pages 266-277.
    18. Quan, Ji & Cui, Shihui & Chen, Wenman & Wang, Xianjia, 2023. "Reputation-based probabilistic punishment on the evolution of cooperation in the spatial public goods game," Applied Mathematics and Computation, Elsevier, vol. 441(C).
    19. Yasuhiro Shirata, 2020. "Evolution of a Collusive Price in a Networked Market," Dynamic Games and Applications, Springer, vol. 10(2), pages 528-554, June.
    20. Maruta, Toshimasa, 1997. "On the Relationship between Risk-Dominance and Stochastic Stability," Games and Economic Behavior, Elsevier, vol. 19(2), pages 221-234, May.

    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:gam:jgames:v:15:y:2024:i:6:p:41-:d:1534570. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.