Journal of Mathematical Biology

, Volume 30, Issue 1, pp 73–87

Cross entropy minimization in uninvadable states of complex populations

  • Immanuel M. Bomze
Article

Abstract

Selection is often. viewed as a process that maximizes the average fitness of a population. However, there are often constraints even on the phenotypic level which may prevent fitness optimization. Consequently, in evolutionary game theory, models of frequency dependent selection are investigated, which focus on equilibrium states that are characterized by stability (or uninvadability) rather than by optimality. The aim of this article is to show that nevertheless there is a biologically meaningful quantity, namely cross (fitness) entropy, which is optimized during the course of evolution: a dynamical model adapted to evolutionary games is presented which has the property that relative entropy decreases monotonically, if the state of a (complex) population is close to an uninvadable state. This result may be interpreted as if evolution has an “order stabilizing” effect.

Key words

Frequency dependent selection Evolutionary game theory Replicator dynamics 

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References

  1. Akin, E.: Exponential families and game dynamics. Can J. Math. 34, 374–405 (1982)Google Scholar
  2. Bishop, D. T., Cannings, C.: A generalized war of attrition. J. Theor. Biol. 70, 85–124 (1978)Google Scholar
  3. Bomze, I. M.: A functional analytic approach to statistical experiments. London: Longman 1990aGoogle Scholar
  4. Bomze, I. M.: Dynamical aspects of evolutionary stability. Monatsh. Math. 110, 189–206 (1990b)Google Scholar
  5. Bomze, I. M., Pötscher, B. M.: Game theoretical foundations of evolutionary stability. Berlin Heidelberg New York: Springer 1989Google Scholar
  6. Csiszár, R. I.: On topological properties of F-divergences. Stud. Sci. Math. Hung. 2, 329–339 (1967)Google Scholar
  7. Csiszár, R. I.: I-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3, 146–158 (1974)Google Scholar
  8. Desharnais, R. A.: Natural selection, fitness entropy, and the dynamics of coevolution. Theor. Popul. Biol. 30, 309–340 (1986)Google Scholar
  9. Desharnais, R. A., Costantino, R. F.: Natural selection and fitness entropy in a density-regulated population. Genetics 101, 317–329 (1982)Google Scholar
  10. Fano, R. M.: Transmission of information. New York: Wiley 1961Google Scholar
  11. Ginzburg, L. R.: A macro-equation of natural selection. J. Theor. Biol. 67, 677–686 (1977)Google Scholar
  12. Good, I. J.: The population frequencies of species and the estimation of population parameters. Biometrika 40, 237–264 (1953)Google Scholar
  13. Hines, W. G. S.: Three characterizations of population strategy stability. J. Appl. Probab. 17, 333–340 (1980a)Google Scholar
  14. Hines, W. G. S.: Strategy stability in complex populations. J. Appl. Probab. 17, 600–610 (1980b)Google Scholar
  15. Hines, W. G. S.: Strategy stability in complex randomly mating diploid populations. J. Appl. Probab. 19, 653–659 (1982)Google Scholar
  16. Hofbauer, J., Schuster, P., Sigmund, K.: A note on evolutionary stable strategies and game dynamics. J. Theor. Biol. 81, 609–612 (1979)Google Scholar
  17. Hofbauer, J., Sigmund, K.: The theory of evolution and dynamical systems. Cambridge: Cambridge University Press 1988Google Scholar
  18. Hutson, V., Law, R.: Evolution of recombination in populations experiencing frequency-dependent selection with time delay. Proc. R. Sec. Lond., Ser. B 213, 345–359 (1981)Google Scholar
  19. Iwasa, Y.: Free fitness that always increases in evolution. J. Theor. Biol. 135, 265–281 (1988)Google Scholar
  20. Kullback, S.: Information theory and statistics. New York0: Wiley 1959Google Scholar
  21. Kullback, S., Leibler, R. A.: On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951)Google Scholar
  22. Lang, S.: Differential manifolds. Reading, Mass.: Addison-Wesley 1972Google Scholar
  23. LeCam, L.: Asymptotic methods in statistical decision theory. Berlin Heidelberg New York: Springer 1986Google Scholar
  24. Maynard Smith, J.: Evolution and the theory of games. Cambridge: Cambridge University Press 1982Google Scholar
  25. Maynard Smith, J.: Can a mixed strategy be stable in a finite population? J. Theor. Biol. 130, 247–260 (1988)Google Scholar
  26. Maynard Smith, J., Price, G. R.: The logic of animal conflict. Nature (London) 246, 15–18 (1973)Google Scholar
  27. Price, G. R.: Selection and covariance. Nature (London) 277, 520–521 (1970)Google Scholar
  28. Reiss, R.-D.: Approximate distributions of order statistics. Berlin Heidelberg New York: Springer 1989Google Scholar
  29. Rényi, A.: Foundations of probability theory. San Francisco: Holden-Day 1970Google Scholar
  30. Shaffer, M. E.: Evolutionarily stable strategies for a finite population and a variable contest size. J. Theor. Biol. 132, 469–478 (1988)Google Scholar
  31. Shore, J. E., Johnson, R. W.: Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy. IEEE Trans. Inf. Theory IT-26, 26–37 (1980); corrections, IT-27, 942 (1981)Google Scholar
  32. Sigmund, K.: Game dynamics, mixed strategies and gradient systems. Theor. Popul. Biol. 32, 114–126 (1987)Google Scholar
  33. Strasser, H.: Mathematical theory of statistics. Berlin: de Gruyter 1985Google Scholar
  34. Taylor, P., Jonker, L.: Evolutionarily stable strategies and game dynamics. Math. Biosci. 40, 145–156 (1978)Google Scholar
  35. Thomas, B.: Evolutionarily stable sets in mixed strategist models. Theor. Popul. Biol. 28, 332–341 (1985)Google Scholar
  36. Vickers, G. T.: Cannings, C.: On the definition of an evolutionarily stable strategy. (Preprint 1987)Google Scholar
  37. Wehrl, A.: General properties of entropy. Rev. Mod. Phys. 50, 221–260 (1978)Google Scholar
  38. Zeeman, E. C.: Dynamics of the evolution of animal conflicts. J. Theor. Biol. 89, 249–270 (1981)Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Immanuel M. Bomze
    • 1
  1. 1.Institut für Statistik und InformatikUniversität WienWienAustria

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