Journal of Mathematical Biology

, Volume 32, Issue 6, pp 597–615 | Cite as

Sequential methods for generating patterns of ESS's

  • M. Broom
  • C. Cannings
  • G. T. Vickers


A finite conflict with given payoff matrix may have many ESS's (evolutionarily stable strategies). For a given set of pure strategies { 1, 2, ...,n} a set of subsets of these is called a pattern, and if there exists ann ×n matrix which has ESS's whose supports (i.e. the playable strategies) precisely match the elements of the pattern, then the pattern is said to be attainable. In [5] and [10] some methods were developed to specify when a pattern was, or was not, attainable. The object here is to present a somewhat different method which is essentially recursive. We derive certain results which allow one to deduce from the attainability of a pattern for givenn the attainability of other patterns forn+1, and by induction for anyn+r.

Key words

PRIMARY 90D05 SECONDARY 92A15 ESS Patterns Polymorphism 


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  1. 1.
    Abakuks, A.: Conditions for evolutionarily stable strategies. J. Appl. Probab.17, 559–562 (1980)Google Scholar
  2. 2.
    Bishop, D. T., Cannings, C.: Models of animal conflict. Adv. Appl. Probab.8, 616–621 (1976)Google Scholar
  3. 3.
    Cannings, C., Broom, M., Vickers, G. T.: Adding pairs in conflicts. (in preparation)Google Scholar
  4. 4.
    Cannings, C., Tyrer, J. P., Vickers, G. T.: Routes to polymorphism. J. Theoret. Biol. (1994)Google Scholar
  5. 5.
    Cannings, C., Vickers, G. T.: Patterns of ESS's 2.J. Theor. Biol.132, 409–420 (1988)Google Scholar
  6. 6.
    Cannings, C., Vickers, G. T.: Patterns and invasions of Evolutionarily Stable Strategies. J. Appl. Math. Comput.32, 227–253 (1990)Google Scholar
  7. 7.
    Cannings, C., Vickers, G. T.: The genealogy of patterns. IMS18, 193–204 (1991)Google Scholar
  8. 8.
    Haigh, J.: Game theory and evolution. Adv. Appl. Probab.7, 8–10 (1975)Google Scholar
  9. 9.
    Maynard Smith, J., Price, G. R.: The logic of animal conflict. Nature246, 15–18 (1973)Google Scholar
  10. 10.
    Vickers, G. T., Cannings, C.: Patterns of ESS's 1. J. Theor. Biol.132, 387–408 (1988)Google Scholar
  11. 11.
    Zeeman, E. C.: Population dynamics from game theory. In: Nitecki, A. Robinson, C. (eds.) Global theory of Dynamical Systems, pp. 471–497. Berlin Heidelberg New York: Springer 1980Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • M. Broom
    • 1
    • 2
  • C. Cannings
    • 1
  • G. T. Vickers
    • 2
  1. 1.School of Mathematics and Statistics, Division of Probability and StatisticsUniversity of SheffieldSheffieldUK
  2. 2.School of Mathematics and Statistics, Division of Applied and Computational MathematicsUniversity of SheffieldSheffieldUK

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