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

Article

Abstract

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.

Keywords

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

References

  1. 1.
    Amari S, Cichocki A (2010) Information geometry of divergence functions. Bull Pol Acad Sci 58(1):183–195Google Scholar
  2. 2.
    Amari S, Nagaoka H (2007) Methods of information geometry, vol 191. American Mathematical Society, ProvidenceGoogle Scholar
  3. 3.
    Bartosiewicz Z, Piotrowska E (2011) Lyapunov functions in stability of nonlinear systems on time scales. J Differ Equ Appl 17(03):309–325MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bohner M, Peterson A (2001) Dynamic equations on time scales, vol 160. Springer, BostonCrossRefMATHGoogle Scholar
  5. 5.
    Bohner M, Peterson A (2002) Advances in dynamic equations on time scales. Springer, BostonGoogle Scholar
  6. 6.
    Bomze I (1991) Cross entropy minimization in uninvadable states of complex populations. J Math Biol 30(1):73–87MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Brown G (1951) Iterative solution of games by fictitious play. Act Anal Prod Alloc 13(1):374–376Google Scholar
  8. 8.
    Cressman R (1997) Local stability of smooth selection dynamics for normal form games. Math Soc Sci 34(1):1–19MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cressman R (2003) Evolutionary dynamics and extensive form games, vol 5. MIT Press, CambridgeMATHGoogle Scholar
  10. 10.
    DaCunha J (2005) Stability for time varying linear dynamic systems on time scales. J Comput Appl Math 176(2):381–410MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Davis J, Gravagne I, Marks R, Ramos A (2010) Algebraic and dynamic lyapunov equations on time scales. In: 42nd Southeastern Symposium on System Theory (SSST), pp 329–334. IEEEGoogle Scholar
  12. 12.
    Fryer D (2012) The Kullback-Leibler divergence as a Lyapunov function for incentive based game dynamics. arXiv preprint arXiv:1207.0036
  13. 13.
    Fryer D (2012) On the existence of general equilibrium in finite games and general game dynamics. arXiv preprint arXiv:1201.2384
  14. 14.
    Fudenberg D (1998) The theory of learning in games, vol 2. MIT press, CambridgeMATHGoogle Scholar
  15. 15.
    Harper M (2011) Escort evolutionary game theory. Physica D 240(18):1411–1415MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hernando A, Hernando R, Plastino A, Plastino AR (2013) The workings of the maximum entropy principle in collective human behaviour. J R Soc Interface 10(78):20120758CrossRefGoogle Scholar
  17. 17.
    Hofbauer J (2011) Deterministic evolutionary game dynamics, vol 69. American Mathematical Society, New Orleans, pp 61–69CrossRefGoogle Scholar
  18. 18.
    Hofbauer J, Sigmund K (1990) Adaptive dynamics and evolutionary stability. Appl Math Lett 3(4):75–79MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  20. 20.
    Hofbauer J, Sigmund K (2003) Evolutionary game dynamics. Bull Am Math Soc 40(4):479–519MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Joosten R (2013) Paul samuelson’s critique and equilibrium concepts in evolutionary game theory. In: Petrosjan L, Mazalov V (eds) Game theory and applications, vol 16. NOVA Publishers, New York, p 8Google Scholar
  22. 22.
    Joosten R, Roorda B (2008) Generalized projection dynamics in evolutionary game theory. Technical report, Papers on economics and evolutionGoogle Scholar
  23. 23.
    Joosten R, Roorda B (2011) On evolutionary ray-projection dynamics. Math Methods Oper Res 74(2):147–161MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Lahkar R, Sandholm W (2008) The projection dynamic and the geometry of population games. Games Econ Behav 64(2):565–590MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Matsui A (1992) Best response dynamics and socially stable strategies. J Ecoomic Theory 57(2):343–362MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Nagurney A, Zhang D (1995) Projected dynamical systems and variational inequalities with applications, vol 2. Springer, BerlinMATHGoogle Scholar
  27. 27.
    Nagurney A, Zhang D (1997) Projected dynamical systems in the formulation, stability analysis, and computation of fixed-demand traffic network equilibria. Transport Sci 31(2):147–158MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Naudts J (2004) Estimators, escort probabilities, and phi-exponential families in statistical physics. J Ineq Pure Appl Math 5(4):102MathSciNetGoogle Scholar
  29. 29.
    Sandholm W (2011) Population games and evolutionary dynamics. MIT press, CambridgeGoogle Scholar
  30. 30.
    Sandholm W, Dokumacı E, Lahkar R (2008) The projection dynamic and the replicator dynamic. Games Econ Behav 64(2):666–683CrossRefMATHGoogle Scholar
  31. 31.
    Sorin S (2012) On some global and unilateral adaptive dynamics. AMS Short Course Evol Game Dyn 69:81–110MathSciNetCrossRefGoogle Scholar
  32. 32.
    Taylor P, Jonker L (1978) Evolutionary stable strategies and game dynamics. Math Biosci 40(1):145–156MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Tsallis C (1988) Possible generalization of boltzmann-gibbs statistics. J Stat Phys 52(1):479–487MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Weibull J (1997) Evolutionary game theory. MIT press, CambridgeGoogle Scholar

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

Personalised recommendations