Dynamic Games and Applications

, Volume 5, Issue 3, pp 318–333 | Cite as

Lyapunov Functions for Time-Scale Dynamics on Riemannian Geometries of the Simplex



We combine incentive, adaptive, and time-scale dynamics to study multipopulation dynamics on the simplex equipped with a large class of Riemannian metrics, simultaneously generalizing and extending many dynamics commonly studied in dynamic game theory and evolutionary dynamics. Each population has its own geometry, method of adaptation (incentive), and time-scale (discrete, continuous, and others). Using information-theoretic measures of distance we give a widely-applicable Lyapunov result for the dynamics.


Evolutionary dynamics Evolutionary stability Lyapunov functions Riemannian geometry Time-scale calculus 



All plots in this paper we created with matplotlib and python code available at https://github.com/marcharper/python-ternary and https://github.com/marcharper/metric-incentive-dynamics. Marc Harper acknowledges support from the Office of Science (BER), U.S. Department of Energy, Cooperative Agreement No. DE-FC02-02ER63421. The authors thank the anonymous reviewers for a variety of helpful comments.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Genomics and ProteomicsUCLALos AngelesUSA
  2. 2.Department of MathematicsPomona CollegeClaremontUSA

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