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.
Highlights
Identifying cycles in games is easy, but it is not easy to analyse the qualitative behaviour of these cycling structures
It is shown in this paper that the limit set of the time averages for the replicator equation is a subset of the maximal invariant set for the best response dynamics
Almost all orbits for the best response dynamics cross a transition face and so this allows us for the best response dynamics to give results for almost all orbits
Summary
Identifying cycles in games is easy, but it is not easy to analyse the qualitative behaviour of these cycling structures. For example even the structure of the simple RSP game can either lead to convergence to a Nash equilibrium or to convergence to a periodic orbit for the best response dynamics. A strong connection was found between the limit set of time averages of the orbits for the replicator equation and the ω− limits of the best response dynamics. Cycles for low dimensional games for the replicator equation are for example thoroughly analysed in [3,4] In these papers conditions are given under which so called heteroclinic cycles are attracting or repelling. In this paper we are especially interested in games with a full better reply cycle, which means a cycle exists using all pure strategies This means that no pure strategy can become a Nash equilibrium. All other orbits (especially those x, with V (x) < V (N1234 )) converge via s to N1234 in finite time
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
