Abstract
We consider a dimorphic population state, P, which is a convex combination of two Dirac measures δ x and δ y , in evolutionary games with a continuous strategy space. We first establish necessary and sufficient conditions for this dimorphic population state, P, to be a rest point of the associated replicator dynamics. We provide sufficient conditions for the replicator dynamics trajectory to converge to P when it originates from the line \(L =\{\eta \delta _{x} + (1-\eta )\delta _{y}: 0 <\eta < 1\}\). If the trajectory emanates from a point outside L, then we derive sufficient conditions for the trajectory to converge to L in the special case where each point in L is a rest point. We have, also, obtained condition for the trajectory to stay away from the line L in the limit. Furthermore, main results are illustrated using examples.
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Acknowledgements
The authors thank an anonymous referee for pointing out an error in an example in the preliminary version of this paper. We also acknowledge the support of NBHM through the project “Evolutionary Stability in Games with Continuous Action Spaces”.
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Hingu, D., Mallikarjuna Rao, K.S., Shaiju, A.J. (2016). Evolutionary Stability of Dimorphic Population States. In: Thuijsman, F., Wagener, F. (eds) Advances in Dynamic and Evolutionary Games. Annals of the International Society of Dynamic Games, vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-28014-1_12
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DOI: https://doi.org/10.1007/978-3-319-28014-1_12
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