Abstract
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
Mathematics Subject Classification (2000). Primary 92D50; Secondary 91A22.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Axelrod and W.D. Hamilton, The evolution of cooperation. Science 211 (1981), 1390–1396.
R. Boyd and P.J. Richerson, Culture and the Evolutionary Process. University of Chicago Press, USA, 1985.
P.E. Hammerstein, Genetic and Cultural Evolution of Cooperation. MIT press, Cambridge, MA., USA, 2003.
J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics. Cambridge Univ. Press, Cambridge, UK, 1998.
M.W. Macy and A. Flache, Learning dynamics in social dilemmas. Proc. Natl. Acad. Sci. USA 99 (2002), 7229–7236.
J. Maynard Smith, Evolution and the Theory of Games. Cambridge University Press, Cambridge, UK, 1982.
M.A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life. The Belknap Press of Harvard University Press, Cambridge, MA, 2006.
M.A. Nowak, Five rules for the evolution of cooperation. Science 314 (2006), 1560– 1563.
M.A. Nowak and K. Sigmund, Evolutionary dynamics of biological games. Science 303 (2004), 793–799.
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.
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.
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.
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.
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.
B. Skyrms, The stag hunt. Proceedings and Addresses of the American Philosophical Association 75 (2001), 31–41.
B. Skyrms, The Stag Hunt and the Evolution of Social Structure. Cambridge University Press, UK, 2004.
S. Bowles, Microeconomics: Behavior, Institutions and Evolution. Princeton University Press, USA, 2003.
M.M. Flood, Some experimental games, research memorandum RM-789. RAND Corporation, Santa Monica, CA (1952).
M. Dresher, The Mathematics of Games of Strategy: Theory and Applications. Prentice-Hall, Englewood Cliffs, NJ, 1961.
R. Sugden, The Economics of Rights, Co-operation and Welfare. Basil Blackell, Oxford, UK, 1986.
C. Boehm, Hierarchy in the Forest: The Evolution of Egalitarian Behavior. Harvard University Press, USA, 1999.
G. Hardin, The tragedy of the commons. Science 162 (1968), 1243–1248.
T.C. Schelling, Hockey helmets, concealed weapons, and daylight saving: A study of binary choices with externalities. J. Conflict Resolution 17 (1973), 381.
R.M. Dawes, Social dilemmas. social dilemmas. social dilemmas. Annu. Rev. of Psychol. 31 (1980), 169–193.
R. Boyd and P.J. Richerson, The evolution of reciprocity in sizable groups. J. Theor. Biol. 132 (1988), 337–356.
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.
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.
W.D. Hamilton, Biosocial anthropology. In Biosocial anthropology, 133–155, Malaby Press, London, UK, 1975.
P.E. Stander, Cooperative hunting in lions – the role of the individual. Behavioral Ecology and Sociobiology 29 (1992), 445–454.
C. Boesch, Cooperative hunting roles among tai chimpanzees. Human Nature – an Interdisciplinary Biosocial Perspective 13 (2002), 27–46.
S. Creel and N.M. Creel, Communal hunting and pack size in African wild dogs, Lycaon-pictus. Anim. Behav. 50 (1995), 1325–1339.
J. Maynard Smith and E. Szathm´ary, The Major Transitions in Evolution. Freeman, Oxford, UK, 1995.
B. Beding, The stone-age whale hunters who kill with their bare hands. Daily Mail, 12th April (2008).
R. Jervis, Cooperation under the security dilemma. World Politics 30 (1978), 167– 214.
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.
W.D. Hamilton, Selfish and spiteful behaviour in an evolutionary model. Nature 228 (1970), 1218–1220.
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).
M.O. Souza, J.M. Pacheco, and F.C. Santos, Evolution of cooperation under Nperson snowdrift games. J. Theor. Biol. 260 (2009), 581–588.
S. Karlin and H.M.A. Taylor, A First Course in Stochastic Processes. 2nd edition, Academic, London, UK, 1975.
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.
A. Traulsen, M.A. Nowak, and J.M. Pacheco, Stochastic payoff evaluation increases the temperature of selection. J. Theor. Biol. 244 (2007), 349–356.
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.
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.
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Basel AG
About this chapter
Cite this chapter
Pacheco, J.M., Santos, F.C., Souza, M.O., Skyrms, B. (2011). Evolutionary Dynamics of Collective Action. In: Chalub, F., Rodrigues, J. (eds) The Mathematics of Darwin’s Legacy. Mathematics and Biosciences in Interaction. Springer, Basel. https://doi.org/10.1007/978-3-0348-0122-5_7
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0122-5_7
Published:
Publisher Name: Springer, Basel
Print ISBN: 978-3-0348-0121-8
Online ISBN: 978-3-0348-0122-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)