Cooperation in Social Dilemmas

  • Dirk Helbing
Part of the Understanding Complex Systems book series (UCS)


Game theory formalizes certain interactions between physical particles or between living beings in biology, sociology, and economics, and quantifies the outcomes by payoffs. The prisoner’s dilemma (PD) describes situations in which it is profitable if everybody cooperates rather than defects (free-rides or cheats), but as cooperation is risky and defection is tempting, the expected outcome is defection. Nevertheless, some biological and social mechanisms can support cooperation by effectively transforming the payoffs. Here, we study the related phase transitions, which can be of first order (discontinous) or of second order (continuous), implying a variety of different routes to cooperation. After classifying the transitions into cases of equilibrium displacement, equilibrium selection, and equilibrium creation, we show that a transition to cooperation may take place even if the stationary states and the eigenvalues of the replicator equation for the PD stay unchanged. Our example is based on adaptive group pressure, which makes the payoffs dependent on the endogeneous dynamics in the population. The resulting bistability can invert the expected outcome in favor of cooperation.


Stationary Solution Mutual Cooperation Equilibrium Selection Replicator Equation Indirect Reciprocity 
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.



This work was partially supported by the Future and Emerging Technologies programme FP7-COSI-ICT of the European Commission through the project QLectives (grant no.: 231200).


  1. 1.
    J. von Neumann, O. Morgenstern, Theory of Games and Economic Behavior (Princeton University, Princeton, 1944)Google Scholar
  2. 2.
    R. Axelrod, The Evolution of Cooperation (Basic, New York, 1984)Google Scholar
  3. 3.
    J. Hofbauer, K. Sigmund, Evolutionary Games and Population Dynamics (Cambridge University Press, Cambridge, 1998)Google Scholar
  4. 4.
    J.W. Weibull, Evolutionary Game Theory (MIT Press, Cambridge, MA, 1996)Google Scholar
  5. 5.
    N.F. Johnson, P.M. Hui, R. Jonson, T.S. Lo, Phys. Rev. Lett. 82, 3360 (1999)Google Scholar
  6. 6.
    D. Challet, M. Marsili, R. Zecchina, Phys. Rev. Lett. 84, 1824 (2000)Google Scholar
  7. 7.
    G. Szabó, C. Hauert, Phys. Rev. Lett. 89, 118101 (2002)Google Scholar
  8. 8.
    C. Hauert, M. Doebeli, Nature 428, 643 (2004)Google Scholar
  9. 9.
    J.C. Claussen, A. Traulsen, Phys. Rev. Lett. 100, 058104 (2008)Google Scholar
  10. 10.
    C.P. Roca, J.A. Cuesta, A. Sánchez, Phys. Rev. Lett. 97, 158701 (2006)Google Scholar
  11. 11.
    D. Helbing, W. Yu, PNAS 106, 3680 (2009)Google Scholar
  12. 12.
    D. Helbing, T. Vicsek, New J. Phys. 1, 13 (1999)Google Scholar
  13. 13.
    D. Helbing, T. Platkowski, Europhys. Lett. 60, 227 (2002)Google Scholar
  14. 14.
    G. Szabó, G. Fath, Phys. Rep. 446, 97 (2007)Google Scholar
  15. 15.
    J.M. Pacheco, A. Traulsen, M.A. Nowak, Phys. Rev. Lett. 97, 258103 (2006)Google Scholar
  16. 16.
    F.C. Santos, J.M. Pacheco, Phys. Rev. Lett. 95, 098104 (2005)Google Scholar
  17. 17.
    J. Gómez-Gardeñes, M. Campillo, L.M. Floría, Y. Moreno, Phys. Rev. Lett. 98, 108103 (2007)Google Scholar
  18. 18.
    S. VanSegbroeck, F.C. Santos, T. Lenaerts, J.M. Pacheco, Phys. Rev. Lett. 102, 058105 (2009)Google Scholar
  19. 19.
    J. Berg, A. Engel, Phys. Rev. Lett. 81, 4999 (1998)Google Scholar
  20. 20.
    A. Traulsen, J.C. Claussen, C. Hauert, Phys. Rev. Lett. 95, 238701 (2005)Google Scholar
  21. 21.
    H. Ohtsuki, M.A. Nowak, J.M. Pacheco, Phys. Rev. Lett. 98, 108106 (2007)Google Scholar
  22. 22.
    J. Eisert, M. Wilkens, M. Lewenstein, Phys. Rev. Lett. 83, 3077 (1999)Google Scholar
  23. 23.
    M. Eigen, P. Schuster, The Hypercycle (Springer, Berlin, 1979)Google Scholar
  24. 24.
    R.A. Fisher, The Genetical Theory of Natural Selection (Oxford University Press, Oxford, 1930)Google Scholar
  25. 25.
    M. Opper, S. Diederich, Phys. Rev. Lett. 69, 1616 (1992)Google Scholar
  26. 26.
    V.M. de Oliveira, J.F. Fontanari, Phys. Rev. Lett. 89, 148101 (2002)Google Scholar
  27. 27.
    J.Y. Wakano, M.A. Nowak, C. Hauert, PNAS 106, 19 (2009)Google Scholar
  28. 28.
    M.A. Nowak, Science 314, 1560 (2006)Google Scholar
  29. 29.
    A. Traulsen, C. Hauert, H. De Silva, M.A. Nowak, K. Sigmund, PNAS 106(3), 709 (2009)Google Scholar
  30. 30.
    H. Ohtsuki, M.A. Nowak, J. Theor. Biol. 243, 86–97 (2006)Google Scholar
  31. 31.
    O. Gurerk, B. Irlenbusch, B. Rockenbach, Science 312, 108–111 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Dirk Helbing
    • 1
  1. 1.CLU E1ETH ZurichZurichSwitzerland

Personalised recommendations