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Shapley Polygons in 4 x 4 Games

Author

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  • Martin Hahn

    () (Department of Mathematics, University Vienna, c/o Josef Hofbauer, Nordbergstra├če 15, A-1090 Wien, Austria)

Abstract

We study 4 x 4 games for which the best response dynamics contain a cycle. We give examples in which multiple Shapley polygons occur for these kinds of games. We derive conditions under which Shapley polygons exist and conditions for the stability of these polygons. It turns out that there is a very strong connection between the stability of heteroclinic cycles for the replicator equation and Shapley polygons for the best response dynamics. It is also shown that chaotic behaviour can not occur in this kind of game.

Suggested Citation

  • Martin Hahn, 2010. "Shapley Polygons in 4 x 4 Games," Games, MDPI, Open Access Journal, vol. 1(3), pages 1-32, July.
  • Handle: RePEc:gam:jgames:v:1:y:2010:i:3:p:189-220:d:8994
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    More about this item

    Keywords

    evolutionary game theory; Shapley polygon; best response dynamics; periodic solutions;

    JEL classification:

    • C - Mathematical and Quantitative Methods
    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • 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

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