Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Review Article
  • Published:

Partners and rivals in direct reciprocity

Nature Human Behaviour volume 2, pages 469–477 (2018)Cite this article

Subjects

A Publisher Correction to this article was published on 27 March 2018

This article has been updated

Abstract

Reciprocity is a major factor in human social life and accounts for a large part of cooperation in our communities. Direct reciprocity arises when repeated interactions occur between the same individuals. The framework of iterated games formalizes this phenomenon. Despite being introduced more than five decades ago, the concept keeps offering beautiful surprises. Recent theoretical research driven by new mathematical tools has proposed a remarkable dichotomy among the crucial strategies: successful individuals either act as partners or as rivals. Rivals strive for unilateral advantages by applying selfish or extortionate strategies. Partners aim to share the payoff for mutual cooperation, but are ready to fight back when being exploited. Which of these behaviours evolves depends on the environment. Whereas small population sizes and a limited number of rounds favour rivalry, partner strategies are selected when populations are large and relationships stable. Only partners allow for evolution of cooperation, while the rivals’ attempt to put themselves first leads to defection.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on SpringerLink
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Repeated interactions allow evolution of cooperation.
Fig. 2: Eight strategies for the repeated PD.
Fig. 3: Adaptive players versus ZD strategies.
Fig. 4: Partners and rivals.
Fig. 5: Evolution favours partners or rivals.

Similar content being viewed by others

Change history

  • 27 March 2018

    In the version of this Review Article originally published, in Fig. 4 an arrow pointing from ALLC to ALLD was mistakenly omitted. This has now been corrected in all versions of the Review Article.

References

  1. von Neumann, J. & Morgenstern, O. Theory of Games and Economic Behavior (Princeton Univ. Press, Princeton, NJ, 1944).

    Google Scholar 

  2. Nash, J. F. Equilibrium points in n-person games. Proc. Natl Acad. Sci. USA 36, 48–49 (1950).

    Article  PubMed  CAS  Google Scholar 

  3. Flood, M. M. Some experimental games. Manage. Sci. 5, 5–26 (1958).

    Article  Google Scholar 

  4. Rapoport, A. & Chammah, A. M. Prisoner’s Dilemma (Univ. Michigan Press, Ann Arbor, MI, 1965).

  5. Trivers, R. L. The evolution of reciprocal altruism. Q. Rev. Biol. 46, 35–57 (1971).

    Article  Google Scholar 

  6. Nowak, M. A. Evolutionary Dynamics (Harvard Univ. Press, Cambridge, MA, 2006).

    Google Scholar 

  7. Sigmund, K. The Calculus of Selfishness (Princeton Univ. Press, Princeton, NJ, 2010).

    Book  Google Scholar 

  8. Nowak, M. A. Five rules for the evolution of cooperation. Science 314, 1560–1563 (2006).

    Article  PubMed  PubMed Central  Google Scholar 

  9. Smale, S. The prisoner’s dilemma and dynamical systems associated to non-cooperative games. Econometrica 48, 1617–1634 (1980).

    Article  Google Scholar 

  10. Mailath, G. J. & Samuelson, L. Repeated Games and Reputations (Oxford Univ. Press, Oxford, 2006). Extensive compendium on repeated games from an economics point of view, which gives an excellent overview on the folk theorem literature.

    Book  Google Scholar 

  11. Abreu, D. Extremal equilibria of oligopolistic supergames. J. Econ. Theory 39, 191–225 (1986).

    Article  Google Scholar 

  12. Bernheim, D. & Whinston, M. D. Multimarket contact and collusive behavior. Rand J. Econ. 21, 1–26 (1990).

    Article  Google Scholar 

  13. Cable, D. M. & Shane, S. A prisoner’s dilemma approach to entrepreneur–venture capitalist relationships. Acad. Manage. Rev. 22, 142–176 (1997).

    Article  Google Scholar 

  14. Majeski, S. J. Arms races as iterated prisoner’s dilemma games. Math. Soc. Sci. 7, 253–266 (1984).

    Article  Google Scholar 

  15. Aumann, R. J. War and peace. Proc. Natl Acad. Sci. USA 103, 17075–17078 (2006).

    Article  PubMed  CAS  Google Scholar 

  16. Wilkinson, G. S. Reciprocal food-sharing in the vampire bat. Nature 308, 181–184 (1984).

    Article  Google Scholar 

  17. Stephens, D. W., McLinn, C. M. & Stevens, J. R. Discounting and reciprocity in an iterated prisoner’s dilemma. Science 298, 2216–2218 (2002).

    Article  PubMed  CAS  Google Scholar 

  18. Milinski, M. Tit for tat in sticklebacks and the evolution of cooperation. Nature 325, 433–435 (1987).

    Article  PubMed  CAS  Google Scholar 

  19. Ben-Porath, E. The complexity of computing a best response automaton in repeated games with mixed strategies. Games Econ. Behav. 2, 1–12 (1990).

    Article  Google Scholar 

  20. Papadimitriou, C. H. On players with a bounded number of states. Games Econ. Behav. 4, 122–131 (1992).

    Article  Google Scholar 

  21. Friedman, J. A non-cooperative equilibrium for supergames. Rev. Econ. Stud. 38, 1–12 (1971).

    Article  Google Scholar 

  22. Aumann, R. J. in Essays in Game Theory and Mathematical Economics in Honor of Oskar Morgenstern (eds Henn, R. & Moeschlin, O.) 11–42 (Wissenschaftsverlag, Mannheim, 1981).

  23. Maynard Smith, J. Evolution and the Theory of Games (Cambridge Univ. Press, Cambridge, 1982).

    Book  Google Scholar 

  24. Selten, R. & Hammerstein, P. Gaps in Harley’s argument on evolutionarily stable learning rules and in the logic of “Tit for Tat”. Behav. Brain Sci. 7, 115–116 (1984).

    Article  Google Scholar 

  25. Boyd, R. & Lorberbaum, J. No pure strategy is evolutionary stable in the iterated prisoner’s dilemma game. Nature 327, 58–59 (1987).

    Article  Google Scholar 

  26. Bendor, J. & Swistak, P. Types of evolutionary stability and the problem of cooperation. Proc. Natl Acad. Sci. USA 92, 3596–3600 (1995).

    Article  PubMed  CAS  Google Scholar 

  27. van Veelen, M., Garcia, J., Rand, D. G. & Nowak, M. A. Direct reciprocity in structured populations. Proc. Natl Acad. Sci. USA 109, 9929–9934 (2012).

    Article  PubMed  Google Scholar 

  28. Garcia, J. & van Veelen, M. In and out of equilibrium I: Evolution of strategies in repeated games with discounting. J. Econ. Theory 161, 161–189 (2016).

    Article  Google Scholar 

  29. Boyd, R. Mistakes allow evolutionary stability in the repeated prisoner’s dilemma game. J. Theor. Biol. 136, 47–56 (1989).

    Article  PubMed  CAS  Google Scholar 

  30. Fudenberg, D. & Maskin, E. Evolution and cooperation in noisy repeated games. Am. Econ. Rev. 80, 274–279 (1990).

    Google Scholar 

  31. Nowak, M. A. & Sigmund, K. Chaos and the evolution of cooperation. Proc. Natl Acad. Sci. USA 90, 5091–5094 (1993).

    Article  PubMed  CAS  Google Scholar 

  32. Imhof, L. A., Fudenberg, D. & Nowak, M. A. Evolutionary cycles of cooperation and defection. Proc. Natl Acad. Sci. USA 102, 10797–10800 (2005).

    Article  PubMed  CAS  Google Scholar 

  33. Imhof, L. A. & Nowak, M. A. Stochastic evolutionary dynamics of direct reciprocity. Proc. R. Soc. London Ser. B 277, 463–468 (2010).

    Article  Google Scholar 

  34. Nowak, M. A., Sasaki, A., Taylor, C. & Fudenberg, D. Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 646–650 (2004). Introduces finite population size to evolutionary game dynamics.

    Article  PubMed  CAS  Google Scholar 

  35. Imhof, L. A., Fudenberg, D. & Nowak, M. A. Tit-for-tat or win-stay, lose-shift? J. Theor. Biol. 247, 574–580 (2007).

    Article  PubMed  PubMed Central  Google Scholar 

  36. Garcia, J. & Traulsen, A. The structure of mutations and the evolution of cooperation. PLoS ONE 7, e35287 (2012).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  37. Kurokawa, S. & Ihara, Y. Emergence of cooperation in public goods games. Proc. R. Soc. London Ser. B 276, 1379–1384 (2009).

    Article  Google Scholar 

  38. Martinez-Vaquero, L. A., Cuesta, J. A. & Sanchez, A. Generosity pays in the presence of direct reciprocity: a comprehensive study of 2×2 repeated games. PLoS ONE 7, e35135 (2012).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  39. Baek, S. K., Jeong, H. C., Hilbe, C. & Nowak, M. A. Comparing reactive and memory-one strategies of direct reciprocity. Sci. Rep. 6, 25676 (2016).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  40. Axelrod, R. The Evolution of Cooperation (Basic Books, New York, NY, 1984). Axelrod’s tournament and the success of tit-for-tat have been transformative for the field; this book contains a detailed analysis of the tournament’s results .

    Google Scholar 

  41. Molander, P. The optimal level of generosity in a selfish, uncertain environment. J. Conflict Resolut. 29, 611–618 (1985).

    Article  Google Scholar 

  42. Nowak, M. A. & Sigmund, K. Tit for tat in heterogeneous populations. Nature 355, 250–253 (1992).

    Article  Google Scholar 

  43. Nowak, M. A. & Sigmund, K. A strategy of win-stay, lose-shift that outperforms tit-for-tat in the prisoner’s dilemma game. Nature 364, 56–58 (1993). Describes a simple yet surprisingly powerful strategy to maintain cooperation in noisy repeated games, win-stay, lose-shift.

    Article  PubMed  CAS  Google Scholar 

  44. Kraines, D. P. & Kraines, V. Y. Pavlov and the prisoner’s dilemma. Theory Decis. 26, 47–79 (1989).

    Article  Google Scholar 

  45. Lindgren, K. in The Economy as an Evolving Complex System II (eds Arthur, W. B., Durlauf, S. N. & Lane, D. A.) 337–368 (Addison-Wesley, Reading, MA, 1997).

  46. Hauert, C. & Schuster, H. G. Effects of increasing the number of players and memory size in the iterated prisoner’s dilemma: a numerical approach. Proc. R. Soc. London Ser. B 264, 513–519 (1997).

    Article  Google Scholar 

  47. Pinheiro, F. L., Vasconcelos, V. V., Santos, F. C. & Pacheco, J. M. Evolution of all-or-none strategies in repeated public goods dilemmas. PLoS Comput. Biol. 10, e1003945 (2014).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  48. Hilbe, C., Martinez-Vaquero, L. A., Chatterjee, K. & Nowak, M. A. Memory-n strategies of direct reciprocity. Proc. Natl Acad. Sci. USA 114, 4715–4720 (2017).

    Article  PubMed  CAS  Google Scholar 

  49. Fischer, I. et al. Fusing enacted and expected mimicry generates a winning strategy that promotes the evolution of cooperation. Proc. Natl Acad. Sci. USA 110, 10229–10233 (2013).

    Article  PubMed  Google Scholar 

  50. Yi, S. D., Baek, S. K. & Choi, J.-K. Combination with anti-tit-for-tat remedies problems of tit-for-tat. J. Theor. Biol. 412, 1–7 (2017).

    Article  PubMed  Google Scholar 

  51. Duersch, P., Oechssler, J. & Schipper, B. When is tit-for-tat unbeatable? Int. J. Game Theory 43, 25–36 (2013).

    Article  Google Scholar 

  52. Rapoport, A., Seale, D. A. & Colman, A. M. Is tit-for-tat the answer? On the conclusions drawn from axelrod’s tournaments. PLoS ONE 10, e0134128 (2015).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  53. Bendor, J. In good times and bad: reciprocity in an uncertain world. Am. J. Polit. Sci. 31, 531–558 (1987).

    Article  Google Scholar 

  54. Nowak, M. A., Sigmund, K. & El-Sedy, E. Automata, repeated games and noise. J. Math. Biol. 33, 703–722 (1995).

    Article  Google Scholar 

  55. Brandt, H. & Sigmund, K. The good, the bad and the discriminator — errors in direct and indirect reciprocity. J. Theor. Biol. 239, 183–194 (2006).

    Article  PubMed  Google Scholar 

  56. Hofbauer, J. & Sigmund, K. Evolutionary Games and Population Dynamics (Cambridge Univ. Press, Cambridge, 1998).

    Book  Google Scholar 

  57. Cressman, R. Evolutionary Dynamics and Extensive Form Games (MIT Press, Cambridge, MA, 2003).

    Google Scholar 

  58. Nowak, M. A. & Sigmund, K. The evolution of stochastic strategies in the prisoner’s dilemma. Acta Appl. Math. 20, 247–265 (1990).

    Article  Google Scholar 

  59. Press, W. H. & Dyson, F. Iterated prisoner’s dilemma contains strategies that dominate any evolutionary opponent. Proc. Natl Acad. Sci. USA 109, 10409–10413 (2012). Using innovative mathematical concepts, Press and Dyson show there are strategies for the repeated prisoner’s dilemma that allow players to extort their opponents.

    Article  PubMed  Google Scholar 

  60. Chen, J. & Zinger, A. The robustness of zero-determinant strategies in iterated prisoner’s dilemma games. J. Theor. Biol. 357, 46–54 (2014).

    Article  PubMed  Google Scholar 

  61. Pan, L., Hao, D., Rong, Z. & Zhou, T. Zero-determinant strategies in iterated public goods game. Sci. Rep. 5, 13096 (2015).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  62. McAvoy, A. & Hauert, C. Autocratic strategies for iterated games with arbitrary action spaces. Proc. Natl Acad. Sci. USA 113, 3573–3578 (2016).

    Article  PubMed  CAS  Google Scholar 

  63. McAvoy, A. & Hauert, C. Autocratic strategies for alternating games. Theor. Popul. Biol. 113, 13–22 (2016).

    Article  PubMed  Google Scholar 

  64. Ichinose, G. & Masuda, N. Zero-determinant strategies in finitely repeated games. J. Theor. Biol. 438, 61–77 (2018).

    Article  PubMed  Google Scholar 

  65. Hilbe, C., Nowak, M. A. & Sigmund, K. The evolution of extortion in iterated prisoner’s dilemma games. Proc. Natl Acad. Sci. USA 110, 6913–6918 (2013). Based on computer simulations, this article suggests that extortionate strategies can only succeed in small populations, or when two populations evolve at different rates.

    Article  PubMed  Google Scholar 

  66. Hilbe, C., Nowak, M. A. & Traulsen, A. Adaptive dynamics of exortion and compliance. PLoS ONE 8, e77886 (2013).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  67. Szolnoki, A. & Perc, M. Evolution of extortion in structured populations. Phys. Rev. E 89, 022804 (2014).

    Article  CAS  Google Scholar 

  68. Szolnoki, A. & Perc, M. Defection and extortion as unexpected catalysts of unconditional cooperation in structured populations. Sci. Rep. 4, 5496 (2014).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  69. Adami, C. & Hintze, A. Evolutionary instability of zero-determinant strategies demonstrates that winning is not everything. Nat. Commun. 4, 2193 (2013).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  70. Wu, Z.-X. & Rong, Z. Boosting cooperation by involving extortion in spatial prisoner’s dilemma games. Phys. Rev. E 90, 062102 (2014).

    Article  CAS  Google Scholar 

  71. Hilbe, C., Röhl, T. & Milinski, M. Extortion subdues human players but is finally punished in the prisoner’s dilemma. Nat. Commun. 5, 3976 (2014).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  72. Xu, B., Zhou, Y., Lien, J. W., Zheng, J. & Wang, Z. Extortion can outperform generosity in iterated prisoner’s dilemma. Nat. Commun. 7, 11125 (2016).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  73. Milinski, M., Hilbe, C., Semmann, D., Sommerfeld, R. D. & Marotzke, J. Humans choose representatives who enforce cooperation in social dilemmas through extortion. Nat. Commun. 7, 10915 (2016).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  74. Hilbe, C., Hagel, K. & Milinski, M. Asymmetric power boosts extortion in an economic experiment. PLoS ONE 11, e0163867 (2016).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  75. Hilbe, C., Wu, B., Traulsen, A. & Nowak, M. A. Cooperation and control in multiplayer social dilemmas. Proc. Natl Acad. Sci. USA 111, 16425–16430 (2014).

    Article  PubMed  CAS  Google Scholar 

  76. Stewart, A. J. & Plotkin, J. B. From extortion to generosity, evolution in the iterated prisoner’s dilemma. Proc. Natl Acad. Sci. USA 110, 15348–15353 (2013). This study shows that large evolving populations favour the emergence of generous strategies, and introduces the important concept of evolutionary robustness.

    Article  PubMed  Google Scholar 

  77. Stewart, A. J. & Plotkin, J. B. Collapse of cooperation in evolving games. Proc. Natl Acad. Sci. USA 111, 17558–17563 (2014). Describes all evolutionary robust strategies for iterated 2 × 2 games.

    Article  PubMed  CAS  Google Scholar 

  78. Stewart, A. J. & Plotkin, J. B. Small groups and long memories promote cooperation. Sci. Rep. 6, 26889 (2016).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  79. Stewart, A. J., Parsons, T. L. & Plotkin, J. B. Evolutionary consequences of behavioral diversity. Proc. Natl Acad. Sci. USA 113, E7003–E7009 (2016).

    Article  CAS  PubMed  Google Scholar 

  80. Akin, E. in Ergodic Theory, Advances in Dynamics (ed. Assani, I.) 77–107 (de Gruyter, Berlin, 2016). Based on the mathematical formalism of zero-determinant strategies, Akin characterized all memory-1 partner strategies (called ‘good strategies’ in this article).

  81. Akin, E. What you gotta know to play good in the iterated prisoner’s dilemma. Games 6, 175–190 (2015).

    Article  Google Scholar 

  82. Akin, E. Good strategies for the iterated prisoner’s dilemma: Smale vs. Markov. J. Dynam. Games 4, 217–253 (2017).

    Article  Google Scholar 

  83. Hilbe, C., Traulsen, A. & Sigmund, K. Partners or rivals? Strategies for the iterated prisoner’s dilemma. Games Econ. Behav. 92, 41–52 (2015). Introduces the notion of competitive rival strategies, and describes partner and rival strategies for the repeated prisoner’s dilemma with discounted payoffs.

    Article  PubMed  PubMed Central  Google Scholar 

  84. van den Berg, P., Molleman, L. & Weissing, F. J. Focus on the success of others leads to selfish behavior. Proc. Natl Acad. Sci. USA 112, 2912–2917 (2015).

    Article  PubMed  CAS  Google Scholar 

  85. Dal Bó, P. Cooperation under the shadow of the future: experimental evidence from infinitely repeated games. Am. Econ. Rev. 95, 1594–1604 (2005).

    Google Scholar 

  86. Blonski, M., Ockenfels, P. & Spagnolo, G. Equilibrium selection in the repeated prisoner’s dilemma: axiomatic approach and experimental evidence. Am. Econ. J. Microecon 3, 164–192 (2011).

    Article  Google Scholar 

  87. Fudenberg, D., Dreber, A. & Rand, D. G. Slow to anger and fast to forgive: cooperation in an uncertain world. Am. Econ. Rev. 102, 720–749 (2012).

    Article  Google Scholar 

  88. Doebeli, M. & Hauert, C. Models of cooperation based on the prisoner’s dilemma and the snowdrift game. Ecol. Lett. 8, 748–766 (2005).

    Article  Google Scholar 

  89. Nowak, M. A. Evolving cooperation. J. Theor. Biol. 299, 1–8 (2012).

    Article  PubMed  Google Scholar 

  90. Frean, M. R. The prisoner’s dilemma without synchrony. Proc. R. Soc. London Ser. B 257, 75–79 (1994).

    Article  CAS  Google Scholar 

  91. Nowak, M. A. & Sigmund, K. The alternating prisoner’s dilemma. J. Theor. Biol. 168, 219–226 (1994).

    Article  Google Scholar 

  92. Zagorsky, B. M., Reiter, J. G., Chatterjee, K. & Nowak, M. A. Forgiver triumphs in alternating prisoner’s dilemma. PLoS ONE 8, e80814 (2013).

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  93. Roberts, G. & Sherratt, T. N. Development of cooperative relationships through increasing investment. Nature 394, 175–179 (1998).

    Article  PubMed  CAS  Google Scholar 

  94. Wahl, L. M. & Nowak, M. A. The continuous prisoner’s dilemma: I. Linear reactive strategies. J. Theor. Biol. 200, 307–321 (1999).

    Article  PubMed  CAS  Google Scholar 

  95. Killingback, T. & Doebeli, M. The continuous prisoner’s dilemma and the evolution of cooperation through reciprocal altruism with variable investment. Am. Nat. 160, 421–438 (2002).

    PubMed  Google Scholar 

  96. Gokhale, C. S. & Traulsen, A. Evolutionary games in the multiverse. Proc. Natl Acad. Sci. USA 107, 5500–5504 (2010).

    Article  PubMed  Google Scholar 

  97. Traulsen, A., Nowak, M. A. & Pacheco, J. M. Stochastic dynamics of invasion and fixation. Phys. Rev. E 74, 011909 (2006).

    Article  CAS  Google Scholar 

  98. Fudenberg, D. & Imhof, L. A. Imitation processes with small mutations. J. Econ. Theory 131, 251–262 (2006).

    Article  Google Scholar 

  99. Wu, B., Gokhale, C. S., Wang, L. & Traulsen, A. How small are small mutation rates? J. Math. Biol. 64, 803–827 (2012).

    Article  PubMed  Google Scholar 

  100. Boerlijst, M. C., Nowak, M. A. & Sigmund, K. Equal pay for all prisoners. Am. Math. Mon. 104, 303–307 (1997).

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the European Research Council Start Grant 279307: Graph Games (to K.C.), Austrian Science Fund (FWF) Grant P23499-N23 (to K.C.), FWF NFN Grant S11407-N23 Rigorous Systems Engineering/Systematic Methods in Systems Engineering (to K.C.), Office of Naval Research Grant N00014-16-1- 2914 (to M.A.N.) and the John Templeton Foundation (M.A.N.). C.H. acknowledges generous support from the ISTFELLOW programme.

Author information

Authors and Affiliations

  1. Program for Evolutionary Dynamics, Harvard University, Cambridge, MA, USA

    Christian Hilbe & Martin A. Nowak

  2. IST Austria, Klosterneuburg, Austria

    Christian Hilbe & Krishnendu Chatterjee

  3. Department of Organismic and Evolutionary Biology, Department of Mathematics, Harvard University, Cambridge, MA, USA

    Martin A. Nowak

Authors
  1. Christian Hilbe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Krishnendu Chatterjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Martin A. Nowak
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors conceived the study, performed the analysis and wrote the manuscript.

Corresponding author

Correspondence to Christian Hilbe.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figures 1 and 2

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hilbe, C., Chatterjee, K. & Nowak, M.A. Partners and rivals in direct reciprocity. Nat Hum Behav 2, 469–477 (2018). https://doi.org/10.1038/s41562-018-0320-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41562-018-0320-9

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing