Bulletin of Mathematical Biology

, Volume 69, Issue 3, pp 1093–1118 | Cite as

Game Dynamics with Learning and Evolution of Universal Grammar

  • W. Garrett Mitchener
Orginal Article


We investigate a model of language evolution, based on population game dynamics with learning. First, we examine the case of two genetic variants of universal grammar (UG), the heart of the human language faculty, assuming each admits two possible grammars. The dynamics are driven by a communication game. We prove using dynamical systems techniques that if the payoff matrix obeys certain constraints, then the two UGs are stable against invasion by each other, that is, they are evolutionarily stable. Then, we prove a similar theorem for an arbitrary number of disjoint UGs. In both theorems, the constraints are independent of the learning process. Intuitively, if a mutation in UG results in grammars that are incompatible with the established languages, then the mutation will die out because mutants will be unable to communicate and therefore unable to realize any potential benefit of the mutation. An example for which these theorems do not apply shows that compatible mutations may or may not be able to invade, depending on the population's history and the learning process. These results suggest that the genetic history of language is constrained by the need for compatibility and that mutations in the language faculty may have died out or taken over due more to historical accident than to any straightforward notion of relative fitness.


Population game dynamics Replicator equation Language dynamical equation Learning Evolution Evolutionary stability Universal grammar Metastrategy 


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  1. Adger, D., 2003. Core Syntax: A Minimalist Approach. Oxford University Press, Oxford.Google Scholar
  2. Briscoe, E.J., 2000. Grammatical acquisition: Inductive bias and coevolution of language and the language acquisition device. Language 76(2), 245–296.CrossRefGoogle Scholar
  3. Cangelosi, A., Parisi, D., editors, 2002. Simulating the evolution of language. Springer-Verlag.Google Scholar
  4. Chomsky, N., 1988. Language and problems of knowledge. MIT Press.Google Scholar
  5. Guckenheimer, J., Holmes, P., 1990. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag.Google Scholar
  6. Hauser, M.D., 1996. The evolution of communication. Harvard University Press, Cambridge, MA.Google Scholar
  7. Hofbauer, J., Sigmund, K., 1998. Evolutionary games and population dynamics. Cambridge University Press.Google Scholar
  8. Hurford, J.R., Studdert-Kennedy, M., Knight, C., editors, 1998. Approaches to the evolution of language. Cambridge University Press.Google Scholar
  9. Jackendoff, R., 1999. Possible stages in the evolution of the language capacity. Trends Cogn. Sci. 3, 272–279.CrossRefGoogle Scholar
  10. Jackendoff, R., 2002. Foundations of language. Oxford University Press, Oxford.Google Scholar
  11. Kirby, S., 2001. Spontaneous evolution of linguistic structure: an iterated learning model of the emergence of regularity and irregularity. IEEE Trans. Evol. Comput. 5(2), 102–110.CrossRefGoogle Scholar
  12. Komarova, N.L., Nowak, M.A., 2001a. The evolutionary dynamics of the lexical matrix. Bull. Math. Biol. 63(3), 451–485.CrossRefGoogle Scholar
  13. Komarova, N.L., Nowak, M.A., 2001b. Natural selection of the critical period for language acquisition. Proc. Royal Soc. London, Ser. B 268, 1189–1196.Google Scholar
  14. Komarova, N.L., Niyogi, P., Nowak, M.A., 2001. The evolutionary dynamics of grammar acquisition. J. Theor. Biol. 209(1), 43–59.CrossRefGoogle Scholar
  15. Lai, C.S.L., Fisher, S.E., Hurst, J.A., Vargha-Khadem, F., Monaco, A.P., 2001. A forkhead-domain gene is mutated in a severe speech and language disorder. Nature 413(6855), 519–523.CrossRefGoogle Scholar
  16. Lightfoot, D., 1999. The development of language: Acquisition, Changes and Evolution. Blackwell Publishers.Google Scholar
  17. Mitchener, W.G., 2003a. Bifurcation analysis of the fully symmetric language dynamical equation. J. Math. Biol. 46, 265–285.zbMATHCrossRefGoogle Scholar
  18. Mitchener, W.G., 2003b. A mathematical model of human languages: The interaction of game dynamics and learning processes. PhD thesis, Princeton University.Google Scholar
  19. Mitchener, W.G., Nowak, M.A., 2003. Competitive exclusion and coexistence of universal grammars. Bull. Math. Biol. 65(1), 67–93.CrossRefGoogle Scholar
  20. Mitchener, W.G., Nowak, M.A., 2004. Chaos and language. Proc. Royal Soc. London, Biol. Sci. 271(1540), 701–704, DOI 10.1098/rspb.2003.2643.Google Scholar
  21. Niyogi, P., Berwick, R.C., 1997a. Evolutionary consequences of language learning. Linguist. Philos. 20, 697–719.CrossRefGoogle Scholar
  22. Niyogi, P., Berwick, R.C., 1997b. A dynamical systems model for language change. Complex Syst. 11, 161–204. URL Scholar
  23. Nowak, M.A., Krakauer, D.C., 1999. The evolution of language. Proc. Natl. Acad. Sci. U. S. A. 96, 8028–8033.Google Scholar
  24. Nowak, M.A., Krakauer, D.C., Dress, A., 1999a. An error limit for the evolution of language. Proc. Royal Soc. London, Ser. B. 266, 2131–2136.Google Scholar
  25. Nowak, M.A., Plotkin, J., Krakauer, D.C., 1999b. The evolutionary language game. J. Theor. Biol. 200, 147–162.CrossRefGoogle Scholar
  26. Nowak, M.A., Plotkin, J., Jansen, V.A.A., 2000. Evolution of syntactic communication. Nature 404(6777), 495–498.CrossRefGoogle Scholar
  27. Nowak, M.A., Komarova, N.L., Niyogi, P., 2001. Evolution of universal grammar. Science 291(5501), 114–118.CrossRefGoogle Scholar
  28. Nowak, M.A., Komarova, N.L., Niyogi, P., 2002. Computational and evolutionary aspects of language. Nature 417(6889), 611–617.CrossRefGoogle Scholar
  29. Pinker, S., 1990. The Language Instinct. W. Morrow and Company, New York.Google Scholar
  30. Pinker, S., Bloom, P., 1990. Natural language and natural selection. Behav. Brain Sci. 13, 707–784.Google Scholar
  31. Plotkin, J., Nowak, M.A., 2000. Language evolution and information theory. J. Theor. Biol. 205, 147–159.CrossRefGoogle Scholar
  32. Radford, A., 2004. Minimalist Syntax: Exploring the Structure of English. Cambridge University Press, Cambridge.Google Scholar
  33. Tesar, B., Smolensky, P., 2000. Learnability in Optimality Theory. MIT Press.Google Scholar
  34. Trapa, P.E., Nowak, M.A., 2000. Nash equilibria for an evolutionary language game. J. Math. Biol. 41, 172–188.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Department of MathematicsCollege of CharlestonCharlestonUSA

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