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

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  • 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
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    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.
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