Fundamental Theory for Evolutionary Games

  • Jun Tanimoto
Part of the Evolutionary Economics and Social Complexity Science book series (EESCS, volume 6)


In this chapter, we take a look at the appropriate treatment of linear dynamical systems, which you may be familiar with if you have taken some standard engineering undergraduate classes. The discussion is then extended to non-linear systems and their general dynamic properties. In this discussion, we introduce the 2-player and 2-strategy (2 × 2) game, which is the most important archetype among evolutionary games. Multi-player and 2-strategy games are also introduced. In the latter parts of this chapter, we define the dilemma strength, which is useful for the universal comparison of the various reciprocity mechanisms supported by different models.


Nash Equilibrium Equilibrium Point Payoff Matrix Replicator Dynamic Mutual Cooperation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Akiyama, E., and Y. Aruka. 2004. The effect of agents memory on evolutionary phenomena—The Avatamsaka game and four types 2 × 2 dilemma games. Proceedings of 9th Workshop on Economics and Heterogeneous Interacting Agents, CD-ROM.Google Scholar
  2. Alexander, R. 1987. The biology of moral systems. New York: Aldine De Gruyter.Google Scholar
  3. Barabasi, A.L., and R. Albert. 1999. Emergence of scaling in random networks. Science 286: 509–512.MathSciNetCrossRefPubMedADSGoogle Scholar
  4. Browning, L., and A. Colman. 2004. Evolution of coordinated alternating reciprocity in repeated dynamic games. Journal of Theoretical Biology 229: 549–557.MathSciNetCrossRefPubMedGoogle Scholar
  5. Crowley, P. 2001. Dangerous games and the emergence of social structure: evolving memory-based strategies for the generalized hawk-dove game. Behavioral Ecology 12(6): 753–760.CrossRefGoogle Scholar
  6. Ebel, H., and S. Bornholdt. 2002. Coevolutionary games on networks. Physical Review E 66: 056118.CrossRefADSGoogle Scholar
  7. Hamilton, W.D. 1963. The evolution of altruistic behavior. American Naturalist 97: 354–356.CrossRefGoogle Scholar
  8. Hamilton, W.D. 1964. The genetical evolution of social behavior. Journal of Theoretical Biology 7: 1–16.CrossRefPubMedGoogle Scholar
  9. Hardin, G. 1968. Tragedy of the commons. Science 162(3859): 1243–1248.CrossRefPubMedADSGoogle Scholar
  10. Hassell, M.P., H.N. Comins, and R.M. May. 1994. Species coexistence and self-organizing spatial dynamics. Nature 313: 10–11.Google Scholar
  11. Luce, R.D., and H. Raiffa. 1957. Game and decisions: Introduction and critical survey. New York: Wiley.Google Scholar
  12. Masuda, N. 2007. Participation costs dismiss the advantage of heterogeneous networks in evolution of cooperation. Proceedings of the Royal Society B 274: 1815–1821.PubMedCentralMathSciNetCrossRefPubMedGoogle Scholar
  13. Maynard Smith, J. 1976. Group selection. Quarterly Review of Biology 51: 277–283.CrossRefGoogle Scholar
  14. Maynard Smith, J. 1982. Evolution and the theory of games. Cambridge: Cambridge University Press.MATHCrossRefGoogle Scholar
  15. Németh, A., and K. Takács. 2010. The paradox of cooperation benefits. Journal of Theoretical Biology 264: 301–311.MathSciNetCrossRefPubMedGoogle Scholar
  16. Nowak, M.A. 2006a. Evolutionary dynamics. Cambridge, MA: Belknap Press of Harvard University Press.MATHGoogle Scholar
  17. Nowak, M.A. 2006b. Five rules for the evolution of cooperation. Science 314: 1560–1563.PubMedCentralCrossRefPubMedADSGoogle Scholar
  18. Nowak, M.A., and R.M. May. 1992. Evolutionary games and spatial chaos. Nature 359: 826–829.CrossRefADSGoogle Scholar
  19. Nowak, M.A., A. Sasaki, C. Taylor, and D. Fundenberg. 2004. Emergence of cooperation and evolutionary stability in finite populations. Nature 428: 646–650.CrossRefPubMedADSGoogle Scholar
  20. Nowak, M.A., and K. Sigmund. 1998. Evolution of indirect reciprocity by image scoring. Nature 393: 573–577.CrossRefPubMedADSGoogle Scholar
  21. Ohtsuki, H., C. Hauert, E. Lieberman, and M.A. Nowak. 2006. A simple rule for the evolution of cooperation on graphs and social networks. Nature 441: 502–505.PubMedCentralCrossRefPubMedADSGoogle Scholar
  22. Ohtsuki, H., and M.A. Nowak. 2006. The replicator equation on graphs. Journal of Theoretical Biology 243: 86–97.PubMedCentralMathSciNetCrossRefPubMedGoogle Scholar
  23. Price, G.R. 1970. Selection and covariance. Nature 227: 520–521.CrossRefPubMedADSGoogle Scholar
  24. Ren, J., W.X. Wang, and F. Qi. 2007. Randomness enhances cooperation: a resonance-type phenomenon in evolutionary games. Physical Review E 75: 045101(R).CrossRefADSGoogle Scholar
  25. Santos, F.C., and J.M. Pacheco. 2005. Scale-free networks provide a unifying framework for the emergence of cooperation. Physical Review Letter 95: 098104.CrossRefADSGoogle Scholar
  26. Santos, F.C., J.M. Pacheco, and T. Lenaerts. 2006. Cooperation prevails when individuals adjust their social ties. PLoS Computational Biology 2: 1284–1291.CrossRefGoogle Scholar
  27. Slatkin, M., and M.J. Wade. 1978. Group selection on a quantitative character. Proceedings of the National Academy of Science of the United States of America 75: 3531–3534.MATHMathSciNetCrossRefADSGoogle Scholar
  28. Tanimoto, J. 2005. Environmental dilemma game to establish a sustainable society dealing with an emergent value system. Physica D 200: 1–24.CrossRefADSGoogle Scholar
  29. Tanimoto, J. 2009. A simple scaling of the effectiveness of supporting mutual cooperation in donor-recipient games by various reciprocity mechanisms. Biosystems 96: 29–34.CrossRefPubMedGoogle Scholar
  30. Tanimoto, J. 2014. Mathematical analysis of environmental system. Tokyo: Springer.MATHCrossRefGoogle Scholar
  31. Tanimoto, J., and H. Sagara. 2007a. Relationship between dilemma occurrence and the existence of a weakly dominant strategy in a two-player symmetric game. Biosystems 90(1): 105–114.CrossRefPubMedGoogle Scholar
  32. Tanimoto, J., and H. Sagara. 2007b. A study on emergence of coordinated alternating reciprocity in a 2x2 game with 2-memory length strategy. Biosystems 90(3): 728–737.CrossRefPubMedGoogle Scholar
  33. Tanimoto, J., and A. Yamauchi. 2010. Does “game participation cost” affect the advantage of heterogeneous networks for evolving cooperation? Physica A 389: 2284–2289.CrossRefADSGoogle Scholar
  34. Taylor, M., and M.A. Nowak. 2007. Transforming the dilemma. Evolution 61(10): 2281–2292.PubMedCentralCrossRefPubMedGoogle Scholar
  35. Traulsen, A., and M.A. Nowak. 2006. Evolution of cooperation by multilevel selection. Proceedings of the National Academy of Science of the United States of America 103: 10952–10955.CrossRefADSGoogle Scholar
  36. Trivers, R. 1971. The evolution of reciprocal altruism. Quarterly Review of Biology 46: 35–37.CrossRefGoogle Scholar
  37. Trivers, R. 1985. Social evolution. Menlo Park: Benjamin/Cummings.Google Scholar
  38. Watts, D.J., and S.H. Strogatz. 1998. Collective dynamics of “small-world” networks. Nature 393: 440–442.CrossRefPubMedADSGoogle Scholar
  39. Weibull, J.W. 1997. Evolutionary game theory. Cambridge, MA: MIT Press.Google Scholar
  40. Williams, G.C. 1996. Adaption and natural selection: A critique of some current evolutionary thought. Princeton: Princeton University Press.Google Scholar
  41. Wilson, D.S. 1975. A theory of group selection. Proceedings of the National Academy of Science of the United States of America 72: 143–146.MATHCrossRefADSGoogle Scholar
  42. Wynne-Edwards, V.C. 1962. Animal dispersion in relation to social behavior. Edinburg: Oliver and Boyd.Google Scholar
  43. Yamauchi, A., J. Tanimoto, and A. Hagishima. 2010. What controls network reciprocity in the prisoner’s dilemma game? Biosystems 102(2–3): 82–87.CrossRefPubMedGoogle Scholar
  44. Yamauchi, A., J. Tanimoto, and A. Hagishima. 2011. An analysis of network reciprocity in Prisoner’s Dilemma games using full factorial designs of experiment. Biosystems 103: 85–92.CrossRefPubMedGoogle Scholar

Copyright information

© Springer Japan 2015

Authors and Affiliations

  • Jun Tanimoto
    • 1
  1. 1.Graduate School of Engineering SciencesKyushu University InterdisciplinaryFukuokaJapan

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