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The ESS and replicator equation in matrix games under time constraints.

József Garay1,2, Ross Cressman3, Tamás F Móri4

  • 1MTA-ELTE Theoretical Biology and Evolutionary Ecology Research Group and Department of Plant Systematics, Ecology and Theoretical Biology, Eötvös Loránd University, Pázmány Péter sétány 1/c, Budapest, 1117, Hungary. garayj@caesar.elte.hu.

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Summary

Evolutionarily stable strategies (ESS) in time-constrained matrix games connect to polymorphic equilibria. A polymorphic state matching the ESS is an equilibrium, and strict Nash equilibria are stable ESS in these games.

Keywords:
Evolutionary stabilityMonomorphicPolymorphicReplicator equation

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Area of Science:

  • Evolutionary Game Theory
  • Mathematical Biology
  • Game Theory

Background:

  • Introduced matrix games under time constraints.
  • Characterized monomorphic evolutionarily stable strategies (ESS) in these games.

Purpose of the Study:

  • Investigate the relationship between ESS and the existence/stability of polymorphic equilibria.
  • Analyze the conditions for stable polymorphic states in time-constrained matrix games.

Main Methods:

  • Theoretical analysis of evolutionary game dynamics.
  • Application of the replicator equation to polymorphic models.
  • Characterization of Nash equilibria and ESS.

Main Results:

  • Identified a connection between ESS and polymorphic equilibria, where a polymorphic state matching the ESS is an equilibrium.
  • Demonstrated that for two-strategy games, a polymorphic equilibrium is locally asymptotically stable if and only if it corresponds to an ESS.
  • Proved that strict Nash equilibria are pure-strategy ESS and locally asymptotically stable equilibria in n-strategy time-constrained matrix games.

Conclusions:

  • Established a link between monomorphic ESS and stable polymorphic equilibria in time-constrained matrix games.
  • Highlighted the significance of ESS as a predictor of stable evolutionary outcomes.
  • Extended the understanding of evolutionary stability to complex game scenarios.