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Games 2010, 1(3), 189-220; doi:10.3390/g1030189

Shapley Polygons in 4 x 4 Games

Department of Mathematics, University Vienna, c/o Josef Hofbauer, Nordbergstra├če 15, A-1090 Wien, Austria
Received: 9 June 2010 / Accepted: 14 July 2010 / Published: 15 July 2010
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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.
Keywords: evolutionary game theory; Shapley polygon; best response dynamics; periodic solutions evolutionary game theory; Shapley polygon; best response dynamics; periodic solutions
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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Hahn, M. Shapley Polygons in 4 x 4 Games. Games 2010, 1, 189-220.

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