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Replicator equations induced by microscopic processes in nonoverlapping population playing bimatrix games.

Archan Mukhopadhyay1, Sagar Chakraborty1

  • 1Department of Physics, Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India.

Chaos (Woodbury, N.Y.)
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Summary

This study explores the microscopic origins of discrete replicator equations using game theory and a stochastic birth-death process. It reveals how these models emerge from finite populations playing games in the limit of infinite size.

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Area of Science:

  • Evolutionary Game Theory
  • Mathematical Biology
  • Population Genetics

Background:

  • Replicator equations model strategy evolution in populations.
  • Discrete versions are crucial for computational and theoretical analysis.
  • Understanding their microscopic basis is key to validating these models.

Purpose of the Study:

  • To investigate the microscopic foundations of discrete replicator equations.
  • To connect stochastic processes in finite populations to deterministic game dynamics.
  • To derive discrete replicator maps from fundamental principles.

Main Methods:

  • Introducing frequency-dependent selection into the Wright-Fisher process.
  • Modeling a finite, non-overlapping generation population playing a bimatrix game.
  • Utilizing connections between master, Fokker-Planck, and Langevin equations.

Main Results:

  • Established a stochastic framework for discrete replicator dynamics.
  • Demonstrated the emergence of deterministic replicator maps from microscopic interactions.
  • Provided a theoretical link between population-level game dynamics and individual-level stochasticity.

Conclusions:

  • The Wright-Fisher process with game-theoretic selection provides a microscopic basis for discrete replicator equations.
  • The study bridges the gap between stochastic population dynamics and deterministic evolutionary game theory.
  • This work offers a robust foundation for analyzing evolutionary dynamics in finite populations.