Skip to main content

John Maynard Smith and the importance of consistency in evolutionary game theory

Abstract

John Maynard Smith was the founder of evolutionary game theory. He has also been the major influence on the direction of this field, which now pervades behavioural ecology and evolutionary biology. In its original formulation the theory had three components: a set of strategies, a payoff structure, and a concept of evolutionary stability. These three key components are still the basis of the theory, but what is assumed about each component is often different to the original assumptions. We review modern approaches to these components. We emphasis that if a game is considered in isolation, and arbitrary payoffs are assumed, then the payoffs may not be consistent with other components of the system which are not modelled. Modelling the whole system, including not only the focal game, but also the future behaviour of the players and the behaviour of other population members, allows a consistent model to be constructed. We illustrate this in the case of two models of parental care, showing how linking a focal game to other aspects of the system alters what is predicted.

This is a preview of subscription content, access via your institution.

References

  1. BalshineEarn S. and Earn D.J.D. (1997). An evolutionary model of parental care in St. Peter’s fish. J. Theor. Biol. 184: 423-431.

    Article  Google Scholar 

  2. Charnov E.L. 1982. The Theory of Sex Allocation. Princeton, Princeton University Press.

    Google Scholar 

  3. Charnov E.L. 1993. Life History Invariants. Oxford University Press, Oxford.

    Google Scholar 

  4. Christiansen F.B. 1991. On conditions for evolutionary stability for a continuously varying character. Am. Nat. 138: 37–50.

    Article  Google Scholar 

  5. Clutton-Brock T. and Godfray C. 1991. Parental investment. In: Krebs J.R. and Davies N.B. (eds), Behavioural Ecology: An Evolutionary Approach. Blackwell Science, Oxford, pp. 234–262.

    Google Scholar 

  6. Clutton-Brock T.H. 1991. The Evolution of Parental Care. Princeton University Press, Princeton, NJ, USA.

    Google Scholar 

  7. Clutton-Brock T.H. and Parker G.A. 1992. Potential reproductive rates and the operation of sexual selection. Q. Rev. Biol. 67: 437–456.

    Article  Google Scholar 

  8. Davies N.B. 1978. Territorial defence in the Speckled Wood butterfly (Pararge aegeria) – the resident always wins. Anim. Behav. 26: 138–147.

    Article  Google Scholar 

  9. Dawkins R. 1976. The Selfish Gene. Oxford University Press, Oxford.

    Google Scholar 

  10. Diekmann O., Jabin P.-E., Miscler S. and Perthane B. 2005. The dynamics of adaptation: an illuminating example and a Hamilton–Jacobi approach. Theor. Popul. Biol. 67: 257–271.

    Article  PubMed  Google Scholar 

  11. Enquist M. and Leimar O. 1983. Evolution of fighting behavior – decision rules and assessment of relative strength. J. Theor. Biol. 102: 387–410.

    Article  Google Scholar 

  12. Enquist M. and Leimar O. 1987. Evolution of fighting behavior – the effect of variation in resource value. J. Theor. Biol. 127: 187–205.

    Article  Google Scholar 

  13. Enquist M. and Leimar O. 1990. The evolution of fatal fighting. Anim. Behav. 39: 1–9.

    Article  Google Scholar 

  14. Enquist M., Leimar O., Ljungberg T., Mallner Y. and Segerdahl N. 1990. A test of the sequential assessment game – fighting in the cichlid fish Nannacara anomala. Anim. Behav. 40: 1–14.

    Article  Google Scholar 

  15. Eshel I. 1983. Evolutionary and continuous stability. J. Theor. Biol. 103: 99–111.

    Article  Google Scholar 

  16. Eshel I. 1996. On the changing concept of evolutionary population stability as a reflection of a changing point of view in the quantitative theory of evolution. J. Math. Biol. 34: 485–510.

    Article  PubMed  Google Scholar 

  17. Fisher R.A. 1930. The Genetical Theory of Natural Selection. Oxford University Press, Oxford.

    Google Scholar 

  18. Geritz S.A.H., Kisdi E., Meszena G. and Metz J.A.J. 1998. Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol. 12: 35–57.

    Article  Google Scholar 

  19. Geritz S.A.H., Metz J.A.J., Kisdi E. and Meszena G. 1997. Dynamics of adaptation and evolutionary branching. Phys. Rev. Lett. 78: 2024–2027.

    Article  Google Scholar 

  20. Grafen A. 1987. The logic of divisively asymmetric contests – respect for ownership and the Desperado effect. Anim. Behav. 35: 462–467.

    Article  Google Scholar 

  21. Grafen A. and Sibly R. 1978. A model of mate desertion. Anim. Behav. 26: 645–652.

    Article  Google Scholar 

  22. Hamilton W.D. 1967. Extraordinary sex ratios. Science 156: 477–488.

    PubMed  Article  Google Scholar 

  23. Hammerstein P. 1996. Darwinian adaptation, population genetics and the streetcar theory of evolution. J. Math. Biol. 34: 511–532.

    Article  PubMed  Google Scholar 

  24. Hammerstein P. and Riechert S.E. 1988. Payoffs and strategies in territorial contests: ESS analyses of two ecotypes of the spider Agelenopsis aperta. Evol. Ecol. 2: 115–138.

    Article  Google Scholar 

  25. Härdling R., Kokko H. and Arnold K.E. 2003. Dynamics of the caring family. Am. Nat. 161: 395–412.

    Article  PubMed  Google Scholar 

  26. Houston A.I. and Davies N.B. 1985. The evolution of cooperation and life history in the dunnock Prunella modularis. In: Sibly R.M. and Smith R.H. (eds), Behav. Ecol. Blackwell Scientific Publications, Oxford, pp. 471–487.

    Google Scholar 

  27. Houston A.I. and McNamara J.M. 1988. Fighting for food: a dynamic version of the Hawk–Dove game. Evol. Ecol. 2: 51–64.

    Article  Google Scholar 

  28. Houston A.I. and McNamara J.M. 1991. Evolutionarily stable strategies in the repeated hawk dove game. Behav. Ecol. 2: 219–227.

    Article  Google Scholar 

  29. Houston A.I. and McNamara J.M. 1999. Models of Adaptive Behaviour. Cambridge University Press, Cambridge.

    Google Scholar 

  30. Houston A.I. and McNamara J.M. 2002. A self-consistent approach to paternity and parental effort. Philos. Trans. R. Soc. Lond. B-Biol. Sci. 357: 351–362.

    Article  PubMed  Google Scholar 

  31. Houston A.I., Székely T. and McNamara J.M. 2005. Conflict between parents over care. Trends Ecol. Evol. 20: 33–38.

    Article  Google Scholar 

  32. Hurd P.L. and Enquist M. 1998. Conventional signalling in aggressive interactions: the importance of temporal structure. J. Theor. Biol. 192: 197–211.

    Article  Google Scholar 

  33. Jennings D.J., Gammell M.P., Payne R.J.H. and Hayden T.J. 2005. An investigation of assessment games during fallow deer fights. Ethology 111: 511–525.

    Article  Google Scholar 

  34. Johnstone R.A. 2004. Begging and sibling competition: how should offspring respond to their rivals? Am. Nat. 163: E388–E406.

    Article  Google Scholar 

  35. Johnstone R.A. and Roulin A. 2003. Sibling negotiation. Behav. Ecol. 14: 780–786.

    Article  Google Scholar 

  36. Kemp D.J. and Wiklund C. 2001. Fighting without weaponry: a review of male-male contest competition in butterflies. Behav. Ecol. Sociobiol. 49: 429–442.

    Article  Google Scholar 

  37. Kokko H. and Johnstone R.A. 2002. Why is mutual mate choice not the norm? Operational sex ratios, sex roles and the evolution of sexually dimorphic and monomorphic signalling. Philos. Trans. R. Soc. Lond. Ser. B-Biol. Sci. 357: 319–330.

    Article  Google Scholar 

  38. Krebs J.R. and Davies N.B. 1991. Behavioural Ecology: An Evolutionary Approach. Blackwell Science, Oxford.

    Google Scholar 

  39. Leimar O. 1997. Repeated games: a state-space approach. J. Theor. Biol. 184: 471–498.

    Article  Google Scholar 

  40. Leimar, O. Multidimensional convergence stability and the canonical adaptive dynamics. In: Dieckmann U. and J.A.J. Metz (eds), Elements of Adaptive Dynamics, Cambridge University Press, Cambridge. In press

  41. Lucas J.R. and Howard R.D. 1995. On alternative reproductive tactics in anurans – dynamic games with density and frequency-dependence. Am. Nat. 146: 365–397.

    Article  Google Scholar 

  42. Lucas J.R., Howard R.D. and Palmer J.G. 1996. Callers and satellites: chorus behaviour in anurans as a stochastic dynamic game. Anim. Behav. 51: 501–518.

    Article  Google Scholar 

  43. Maynard Smith J. 1968. Mathematical Ideas in Biology. Cambridge University Press, Cambridge.

    Google Scholar 

  44. Maynard Smith J. 1974. The theory of games and the evolution of animal conflicts. J. Theor. Biol. 47: 209–221.

    Article  PubMed  Google Scholar 

  45. Maynard Smith J. 1977. Parental investment: a prospective analysis. Anim. Behav. 25: 1–9.

    Article  Google Scholar 

  46. Maynard Smith J. 1982. Evolution and the Theory of Games. Cambridge University Press, Cambridge.

    Google Scholar 

  47. Maynard Smith J. (1989). Evolutionary Genetics. Oxford University Press,Oxford.

    Google Scholar 

  48. Maynard Smith J. and Hofbauer J. 1987. The Battle of the Sexes – a genetic model with limit-cycle behavior. Theor. Popul. Biol. 32: 1–14.

    Article  PubMed  Google Scholar 

  49. Maynard Smith J. and Parker G.A. 1976. The logic of asymmetric contests. Anim. Behav. 24: 159–175.

    Article  Google Scholar 

  50. Maynard Smith J. and Price G.R. 1973. The logic of animal conflicts. Nature 246: 15–18.

    Article  Google Scholar 

  51. McNamara J.M., Gasson C.E. and Houston A.I. 1999. Incorporating rules for responding into evolutionary games. Nature 401: 368–371.

    Google Scholar 

  52. McNamara J.M. and Houston A.I. 1986. The common currency for behavioral decisions. Am. Nat. 127: 358–378.

    Article  Google Scholar 

  53. McNamara J.M. and Houston A.I. 1990. State-dependent ideal free distributions. Evol. Ecol. 4: 298–311.

    Article  Google Scholar 

  54. McNamara J.M., Houston A.I., Barta Z. and Osorno J.L. 2003. Should young ever be better off with one parent than with two?. Behav. Ecol. 14: 301–310.

    Article  Google Scholar 

  55. McNamara J.M., Székely T., Webb J.N. and Houston A.I. 2000. A dynamic game-theoretic model of parental care. J. Theor. Biol. 205: 605–623.

    Article  PubMed  Google Scholar 

  56. Mesterton-Gibbons M. 1992. Ecotypic variation in the asymmetric Hawk–Dove game - when is bourgeois an evolutionarily stable strategy?. Evolutionary Ecology 6:198–222

    Article  Google Scholar 

  57. Mock D.W., Parker G.A. (1997). The Evolution of Sibling Rivalry. Oxford, Oxford University Press

    Google Scholar 

  58. Mylius S.D. 1999. What pair formation can do to the battle of the sexes: towards more realistic game dynamics. J. Theor. Biol. 197: 469–485.

    Article  PubMed  Google Scholar 

  59. Nabokov V. (2004). Pnin. New York, Knopf

    Google Scholar 

  60. Parker, G.A. 1982. Phenoype-limited evolutionarily stable strategies. Current Problems in Sociobiology. Cambridge University Press, Cambridge.

  61. Parker, G.A. 2006. Behavioural ecology: the science of natural history. In: Lucas J.R. and Simmons L.W. (eds), Essays on Animal Behaviour: Celebrating 50 years of Animal Behaviour. Elsevier Academic Press, Burlington, MA

  62. Parker G.A., Maynard Smith J. (1990). Optimality theory in evolutionary biology. Nature 348: 27–33.

    Article  Google Scholar 

  63. Parker G.A., Simmons L.W. (1996). Parental investment and the control of sexual selection: Predicting the direction of sexual competition. Proc. R. Soc. Lond. Ser. B-Biol. Sci. 263: 315–321

    Article  Google Scholar 

  64. Pereira H.M., Bergman A., Roughgarden J.(2003). Socially stable territories: The negotiation of space by interacting foragers. Am. Nat. 161: 143–152

    Article  PubMed  Google Scholar 

  65. Queller D.C. (1997). Why do females care more than males?. Proc. R. Soc. Lond. Ser. B-Biol. Sci. 264: 1555–1557

    Article  Google Scholar 

  66. Schuster P., Sigmund K. (1981). Coyness, philandering and stable strategies. Anim. Behav. 29: 186–192

    Article  Google Scholar 

  67. Selten R. (1980). A note on Evolutionarily Stable Strategies in asymmetric animal conflicts. J. Theor. Biol. 84: 93–101

    PubMed  Article  Google Scholar 

  68. Selten R. (1983). Evolutionary stability in extensive 2-person games. Math. Soc. Sci. 5: 269–363

    Article  Google Scholar 

  69. Shuster S.M., Wade M.J. (2003). Mating Systems and Strategies. Princeton, Princeton University Press.

    Google Scholar 

  70. Székely T., Webb J.N., Houston A.I., McNamara J.M. (1996). An evolutionary approach to offspring desertion. Curr. Ornithol. 13: 271–330

    Google Scholar 

  71. Taylor P.D.(1989). Evolutionary stability in one-parameter models under weak selection. Theor. Popul. Biol. 36: 125–143

    Article  Google Scholar 

  72. Taylor P.D.(1996). Inclusive fitness arguments in genetic models of behaviour. J. Math. Biol. 34: 654–674

    Article  PubMed  Google Scholar 

  73. Taylor P.D., Jonker L.B. (1978). Evolutionarily Stable Strategies and Game Dynamics. Math. Biosci. 40: 145–156

    Article  Google Scholar 

  74. Vollestad L.A., Peterson J., Quinn T.P.(2004). Effects of freshwater and marine growth rates on early maturity in male coho and Chinook salmon. Trans. Am. Fish. Soc. 133: 495–503

    Article  Google Scholar 

  75. Wachtmeister C.A., Enquist M. (1999). The evolution of female coyness – Trading time for information. Ethology 105: 983–992

    Article  Google Scholar 

  76. Wade M.J., Shuster S.M. (2002). The evolution of parental care in the context of sexual selection: a critical reassessment of parental investment theory. Am. Nat. 160: 285–292

    Article  Google Scholar 

  77. Webb J.N., Houston A.I., McNamara J.M., Székely T. (1999). Multiple patterns of parental care. Anim. Behav. 58: 983–993

    Article  PubMed  Google Scholar 

  78. Weissing F.J. (1996). Genetic versus phenotypic models of selection: Can genetics be neglected in a long-term perspective? J. Math. Biol. 34: 533–555

    Article  PubMed  Google Scholar 

  79. Werren J.H., Gross M.R., Shine R. (1980). Paternity and the evolution of male parental care. J. Theor. Biol. 82: 619–631

    Article  PubMed  Google Scholar 

  80. Yamamura N., Tsuji N. (1993). Parental care as a game. J. Evol. Biol. 6: 103–127

    Article  Google Scholar 

Download references

Acknowledgements

Our concerns about the consistency of one of the models of Wade and Shuster were first articulated in an unpublished manuscript co-authored by Tamás Székely and James Webb. Our thanks to them for many stimulating discussions of parental care and to the Leverhulme Trust for support.

Author information

Affiliations

Authors

Corresponding author

Correspondence to John M. McNamara.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Houston, A.I., McNamara, J.M. John Maynard Smith and the importance of consistency in evolutionary game theory. Biol Philos 20, 933 (2005). https://doi.org/10.1007/s10539-005-9016-4

Download citation

Keywords

  • Arbitrary payoffs
  • Consistency
  • Evolutionary game theory
  • Parental care