Evolutionary Dynamics of Collective Action

Part of the Mathematics and Biosciences in Interaction book series (MBI)


In the natural world, performing a given task which is beneficial to an entire group requires the cooperation of several individuals of that group who often share the workload required to perform the task. The mathematical framework to study the dynamics of collective action is game theory. We study the evolutionary dynamics of cooperators and defectors in a population in which groups of individuals engage in N-person, non-excludable public goods games. We analyze the N-person Prisoner’s dilemma (NPD), where the collective benefit increases proportional to the cost invested, and the N-person Snowdrift game (NSG), where the benefit is fixed but the cost is shared among those who contribute. We impose the existence of a threshold which must be surpassed before collective action becomes successful, and discuss the evolutionary dynamics in infinite and finite populations. In infinite populations, the introduction of a threshold leads, in both dilemmas, to a unified behavior, characterized by two interior fixed points. The fingerprints of the interior fixed points are still traceable in finite populations, despite evolution remaining active until the population reaches a monomorphic end-state. As the group size and population size become comparable, we find that spite dominates, making cooperation unfeasible


Evolution of Cooperation Collective Coordination Threshold Games Population Dynamics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Axelrod and W.D. Hamilton, The evolution of cooperation. Science 211 (1981), 1390–1396.MathSciNetGoogle Scholar
  2. 2.
    R. Boyd and P.J. Richerson, Culture and the Evolutionary Process. University of Chicago Press, USA, 1985.Google Scholar
  3. 3.
    P.E. Hammerstein, Genetic and Cultural Evolution of Cooperation. MIT press, Cambridge, MA., USA, 2003.Google Scholar
  4. 4.
    J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics. Cambridge Univ. Press, Cambridge, UK, 1998.Google Scholar
  5. 5.
    M.W. Macy and A. Flache, Learning dynamics in social dilemmas. Proc. Natl. Acad. Sci. USA 99 (2002), 7229–7236.CrossRefGoogle Scholar
  6. 6.
    J. Maynard Smith, Evolution and the Theory of Games. Cambridge University Press, Cambridge, UK, 1982.Google Scholar
  7. 7.
    M.A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life. The Belknap Press of Harvard University Press, Cambridge, MA, 2006.Google Scholar
  8. 8.
    M.A. Nowak, Five rules for the evolution of cooperation. Science 314 (2006), 1560– 1563.CrossRefGoogle Scholar
  9. 9.
    M.A. Nowak and K. Sigmund, Evolutionary dynamics of biological games. Science 303 (2004), 793–799.Google Scholar
  10. 10.
    H. Ohtsuki, C. Hauert, E. Lieberman, and M.A. Nowak, A simple rule for the evolution of cooperation on graphs and social networks. Nature 441 (2006), 502–505.CrossRefGoogle Scholar
  11. 11.
    F.C. Santos and J.M. Pacheco, Scale-free networks provide a unifying framework for the emergence of cooperation. Phys. Rev. Lett. 95 (2005), 098104.CrossRefGoogle Scholar
  12. 12.
    F.C. Santos, J.M. Pacheco, and T. Lenaerts, Evolutionary dynamics of social dilemmas in structured heterogeneous populations. Proc. Natl. Acad. Sci. USA 103 (2006), 3490–3494.CrossRefGoogle Scholar
  13. 13.
    F.C. Santos, M.D. Santos, and J.M. Pacheco, Social diversity promotes the emergence of cooperation in public goods games. Nature 454 (2008), 213–216.CrossRefGoogle Scholar
  14. 14.
    J.M. Pacheco, F.C. Santos, and F.A.C.C. Chalub, Stern-judging: A simple, successful norm which promotes cooperation under indirect reciprocity. PLoS Computational Biology 2 (2006), e178.CrossRefGoogle Scholar
  15. 15.
    B. Skyrms, The stag hunt. Proceedings and Addresses of the American Philosophical Association 75 (2001), 31–41.CrossRefGoogle Scholar
  16. 16.
    B. Skyrms, The Stag Hunt and the Evolution of Social Structure. Cambridge University Press, UK, 2004.Google Scholar
  17. 17.
    S. Bowles, Microeconomics: Behavior, Institutions and Evolution. Princeton University Press, USA, 2003.Google Scholar
  18. 18.
    M.M. Flood, Some experimental games, research memorandum RM-789. RAND Corporation, Santa Monica, CA (1952).Google Scholar
  19. 19.
    M. Dresher, The Mathematics of Games of Strategy: Theory and Applications. Prentice-Hall, Englewood Cliffs, NJ, 1961.Google Scholar
  20. 20.
    R. Sugden, The Economics of Rights, Co-operation and Welfare. Basil Blackell, Oxford, UK, 1986.Google Scholar
  21. 21.
    C. Boehm, Hierarchy in the Forest: The Evolution of Egalitarian Behavior. Harvard University Press, USA, 1999.Google Scholar
  22. 22.
    G. Hardin, The tragedy of the commons. Science 162 (1968), 1243–1248.Google Scholar
  23. 23.
    T.C. Schelling, Hockey helmets, concealed weapons, and daylight saving: A study of binary choices with externalities. J. Conflict Resolution 17 (1973), 381.CrossRefGoogle Scholar
  24. 24.
    R.M. Dawes, Social dilemmas. social dilemmas. social dilemmas. Annu. Rev. of Psychol. 31 (1980), 169–193.Google Scholar
  25. 25.
    R. Boyd and P.J. Richerson, The evolution of reciprocity in sizable groups. J. Theor. Biol. 132 (1988), 337–356.CrossRefMathSciNetGoogle Scholar
  26. 26.
    P. Kollock, Social dilemmas: The anatomy of cooperation. Annu. Rev. Sociol. 24 (1998), 183–214.[27] C. Hauert, F. Michor, M.A. Nowak, and M. Doebeli, Synergy and discounting of cooperation in social dilemmas. J. Theo. Bio. (2006), 195–202.Google Scholar
  27. 27.
    C. Hauert, A. Traulsen, H. Brandt, M.A. Nowak, and K. Sigmund, Via freedom fo coercion: The emergence of costly punishment. Science 316 (2007), 1905–1907.CrossRefMathSciNetGoogle Scholar
  28. 28.
    W.D. Hamilton, Biosocial anthropology. In Biosocial anthropology, 133–155, Malaby Press, London, UK, 1975.Google Scholar
  29. 29.
    P.E. Stander, Cooperative hunting in lions – the role of the individual. Behavioral Ecology and Sociobiology 29 (1992), 445–454.CrossRefGoogle Scholar
  30. 30.
    C. Boesch, Cooperative hunting roles among tai chimpanzees. Human Nature – an Interdisciplinary Biosocial Perspective 13 (2002), 27–46.Google Scholar
  31. 31.
    S. Creel and N.M. Creel, Communal hunting and pack size in African wild dogs, Lycaon-pictus. Anim. Behav. 50 (1995), 1325–1339.CrossRefGoogle Scholar
  32. 32.
    J. Maynard Smith and E. Szathm´ary, The Major Transitions in Evolution. Freeman, Oxford, UK, 1995.Google Scholar
  33. 33.
    B. Beding, The stone-age whale hunters who kill with their bare hands. Daily Mail, 12th April (2008).Google Scholar
  34. 34.
    R. Jervis, Cooperation under the security dilemma. World Politics 30 (1978), 167– 214.CrossRefGoogle Scholar
  35. 35.
    J. Bryant, Coordination theory, the stag hunt and macroeconomics. In J. Friedman (ed.), Problems of Coordination in Economic Activity, 207–225, Kluwer, Dordrecht, The Netherlands, 1994.Google Scholar
  36. 36.
    W.D. Hamilton, Selfish and spiteful behaviour in an evolutionary model. Nature 228 (1970), 1218–1220.CrossRefGoogle Scholar
  37. 37.
    J.M. Pacheco, F.C. Santos, M.O. Souza, and B. Skyrms, Evolutionary dynamics of collective action in n-person stag-hunt dilemmas. P. Roy. Soc. B: Biological Sciences 276 (2009).Google Scholar
  38. 38.
    M.O. Souza, J.M. Pacheco, and F.C. Santos, Evolution of cooperation under Nperson snowdrift games. J. Theor. Biol. 260 (2009), 581–588.CrossRefGoogle Scholar
  39. 39.
    S. Karlin and H.M.A. Taylor, A First Course in Stochastic Processes. 2nd edition, Academic, London, UK, 1975.Google Scholar
  40. 40.
    A. Traulsen, M.A. Nowak, and J.M. Pacheco, Stochastic dynamics of invasion and fixation. Phys. Rev. E: Stat. Nonlin. Soft. Matter. Phys. 74 (2006), 011909.CrossRefGoogle Scholar
  41. 41.
    A. Traulsen, M.A. Nowak, and J.M. Pacheco, Stochastic payoff evaluation increases the temperature of selection. J. Theor. Biol. 244 (2007), 349–356.CrossRefMathSciNetGoogle Scholar
  42. 42.
    A. Traulsen, J.M. Pacheco, and M.A. Nowak, Pairwise comparison and selection temperature in evolutionary game dynamics. J. Theor. Biol. 246 (2007), 522–529.CrossRefMathSciNetGoogle Scholar
  43. 43.
    M. Milinski, D. Semmann, H.J. Krambeck, and J. Marotzke, Stabilizing the Earth’s climate is not a losing game: Supporting evidence from public goods experiments. Proc. Natl. Acad. Sci. USA 103 (2006), 3994–3998.CrossRefGoogle Scholar
  44. 44.
    M. Milinski, R.D. Sommerfeld, H.J. Krambeck, F.A. Reed, and J. Marotzke, The collective-risk social dilemma and the prevention of simulated dangerous climate change. Proc. Natl. Acad. Sci. USA 105 (2008), 2291–2294.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Departamento de Matemática e AplicaçõesUniversidade do MinhoBragaPortugal
  2. 2.ATP group, CMAF Complexo InterdisciplinarUniversidade de LisboaLisboaPortugal
  3. 3.CENTRIA & Departamento de InformáticaUniversidade Nova de LisboaLisboaPortugal
  4. 4.Departamento de Matemática AplicadaUniversidade Federal FluminenseNiteróiBrasil
  5. 5.Logic and Philosophy of Science School of Social SciencesUniversity of California at IrvineIrvineUSA

Personalised recommendations