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Journal of Mathematical Biology

, Volume 76, Issue 7, pp 1951–1973 | Cite as

The ESS and replicator equation in matrix games under time constraints

  • József GarayEmail author
  • Ross Cressman
  • Tamás F. Móri
  • Tamás Varga
Article

Abstract

Recently, we introduced the class of matrix games under time constraints and characterized the concept of (monomorphic) evolutionarily stable strategy (ESS) in them. We are now interested in how the ESS is related to the existence and stability of equilibria for polymorphic populations. We point out that, although the ESS may no longer be a polymorphic equilibrium, there is a connection between them. Specifically, the polymorphic state at which the average strategy of the active individuals in the population is equal to the ESS is an equilibrium of the polymorphic model. Moreover, in the case when there are only two pure strategies, a polymorphic equilibrium is locally asymptotically stable under the replicator equation for the pure-strategy polymorphic model if and only if it corresponds to an ESS. Finally, we prove that a strict Nash equilibrium is a pure-strategy ESS that is a locally asymptotically stable equilibrium of the replicator equation in n-strategy time-constrained matrix games.

Keywords

Evolutionary stability Monomorphic Polymorphic Replicator equation 

Mathematics Subject Classification

91A22 92D15 91A80 91A40 91A05 91A10 92D40 

Notes

Acknowledgements

This work was partially supported by the Hungarian National Research, Development and Innovation Office NKFIH [Grant Numbers K 108615 (to T.F.M.) and K 108974 and GINOP 2.3.2-15-2016-00057 (to J.G.)]. This project has received funding from the European Union Horizon 2020: The EU Framework Programme for Research and Innovation under the Marie Sklodowska-Curie Grant Agreement No. 690817 (to J.G., R.C. and T.V.). R.C. acknowledges support provided by an Individual Discovery Grant from the Natural Sciences and Engineering Research Council (NSERC).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • József Garay
    • 1
    • 2
    Email author
  • Ross Cressman
    • 3
  • Tamás F. Móri
    • 4
  • Tamás Varga
    • 5
  1. 1.MTA-ELTE Theoretical Biology and Evolutionary Ecology Research Group and Department of Plant Systematics, Ecology and Theoretical BiologyEötvös Loránd UniversityBudapestHungary
  2. 2.MTA Centre for Ecological Research, Evolutionary Systems Research GroupTihanyHungary
  3. 3.Department of MathematicsWilfrid Laurier UniversityWaterlooCanada
  4. 4.Department of Probability Theory and StatisticsEötvös Loránd UniversityBudapestHungary
  5. 5.MTA-SZTE Analysis and Stochastics Research Group, Bolyai InstituteUniversity of SzegedSzegedHungary

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