Social norm complexity and past reputations in the evolution of cooperation


Indirect reciprocity is the most elaborate and cognitively demanding1 of all known cooperation mechanisms2, and is the most specifically human1,3 because it involves reputation and status. By helping someone, individuals may increase their reputation, which may change the predisposition of others to help them in future. The revision of an individual’s reputation depends on the social norms that establish what characterizes a good or bad action and thus provide a basis for morality3. Norms based on indirect reciprocity are often sufficiently complex that an individual’s ability to follow subjective rules becomes important4,5,6, even in models that disregard the past reputations of individuals, and reduce reputations to either ‘good’ or ‘bad’ and actions to binary decisions7,8. Here we include past reputations in such a model and identify the key pattern in the associated norms that promotes cooperation. Of the norms that comply with this pattern, the one that leads to maximal cooperation (greater than 90 per cent) with minimum complexity does not discriminate on the basis of past reputation; the relative performance of this norm is particularly evident when we consider a ‘complexity cost’ in the decision process. This combination of high cooperation and low complexity suggests that simple moral principles can elicit cooperation even in complex environments.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Norm complexity.
Figure 2: Cooperation index of leading norms.
Figure 3: Cooperation index versus norm complexity.
Figure 4: Average behavioural complexity.


  1. 1

    Nowak, M. A. & Sigmund, K. Evolution of indirect reciprocity. Nature 437, 1291–1298 (2005)

    CAS  ADS  Article  Google Scholar 

  2. 2

    Rand, D. G. & Nowak, M. A. Human cooperation. Trends Cogn. Sci. 17, 413–425 (2013)

    Article  Google Scholar 

  3. 3

    Alexander, R. D. The Biology of Moral Systems (Transaction Publishers, 1987)

  4. 4

    Feldman, J. Minimization of Boolean complexity in human concept learning. Nature 407, 630–633 (2000)

    CAS  ADS  Article  Google Scholar 

  5. 5

    Chater, N. & Vitányi, P. Simplicity: a unifying principle in cognitive science? Trends Cogn. Sci. 7, 19–22 (2003)

    Article  Google Scholar 

  6. 6

    Feldman, J. The simplicity principle in perception and cognition. Wiley Interdiscip. Rev. Cogn. Sci. 7, 330–340 (2016)

    Article  Google Scholar 

  7. 7

    Ohtsuki, H. & Iwasa, Y. How should we define goodness?—reputation dynamics in indirect reciprocity. J. Theor. Biol. 231, 107–120 (2004)

    MathSciNet  Article  Google Scholar 

  8. 8

    Ohtsuki, H. & Iwasa, Y. The leading eight: social norms that can maintain cooperation by indirect reciprocity. J. Theor. Biol. 239, 435–444 (2006)

    MathSciNet  Article  Google Scholar 

  9. 9

    Brandt, H. & Sigmund, K. The logic of reprobation: assessment and action rules for indirect reciprocation. J. Theor. Biol. 231, 475–486 (2004)

    MathSciNet  Article  Google Scholar 

  10. 10

    Nowak, M. A. & Sigmund, K. Evolution of indirect reciprocity by image scoring. Nature 393, 573–577 (1998)

    CAS  ADS  Article  Google Scholar 

  11. 11

    Ohtsuki, H., Iwasa, Y. & Nowak, M. A. Indirect reciprocity provides only a narrow margin of efficiency for costly punishment. Nature 457, 79–82 (2009)

    CAS  ADS  Article  Google Scholar 

  12. 12

    Dunbar, R. Grooming, Gossip, and the Evolution of Language (Harvard Univ. Press, 1998)

  13. 13

    Sommerfeld, R. D., Krambeck, H.-J., Semmann, D. & Milinski, M. Gossip as an alternative for direct observation in games of indirect reciprocity. Proc. Natl Acad. Sci. USA 104, 17435–17440 (2007)

    CAS  ADS  Article  Google Scholar 

  14. 14

    Skyrms, B. Signals: Evolution, Learning and Information (Oxford Univ. Press, 2010)

  15. 15

    Kandori, M. Social norms and community enforcement. Rev. Econ. Stud. 59, 63–80 (1992)

    MathSciNet  Article  Google Scholar 

  16. 16

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

    CAS  Article  Google Scholar 

  17. 17

    Wegener, I . & Teubner, B. The Complexity of Boolean Functions Vol. 1 (B. G. Teubner, 1987)

  18. 18

    McCluskey, E. J. Minimization of Boolean functions. Bell Labs Tech. J. 35, 1417–1444 (1956)

    MathSciNet  Article  Google Scholar 

  19. 19

    Rendell, L. et al. Why copy others? Insights from the social learning strategies tournament. Science 328, 208–213 (2010)

    CAS  ADS  MathSciNet  Article  Google Scholar 

  20. 20

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

  21. 21

    Ohtsuki, H. & Iwasa, Y. Global analyses of evolutionary dynamics and exhaustive search for social norms that maintain cooperation by reputation. J. Theor. Biol. 244, 518–531 (2007)

    MathSciNet  Article  Google Scholar 

  22. 22

    Santos, F. P., Santos, F. C. & Pacheco, J. M. Social norms of cooperation in small-scale societies. PLOS Comput. Biol. 12, e1004709 (2016)

    ADS  Article  Google Scholar 

  23. 23

    Pacheco, J. M., Santos, F. C. & Chalub, F. A. C. Stern-judging: a simple, successful norm which promotes cooperation under indirect reciprocity. PLOS Comput. Biol. 2, e178 (2006)

    ADS  Article  Google Scholar 

  24. 24

    Hamlin, J. K. Moral judgment and action in preverbal infants and toddlers evidence for an innate moral core. Curr. Dir. Psychol. Sci. 22, 186–193 (2013)

    Article  Google Scholar 

  25. 25

    Hamlin, J. K., Wynn, K., Bloom, P. & Mahajan, N. How infants and toddlers react to antisocial others. Proc. Natl Acad. Sci. USA 108, 19931–19936 (2011)

    CAS  ADS  Article  Google Scholar 

  26. 26

    Resnick, P., Kuwabara, K., Zeckhauser, R. & Friedman, E. Reputation systems. Commun. ACM 43, 45–48 (2000)

    Article  Google Scholar 

  27. 27

    Dellarocas, C. Reputation mechanism design in online trading environments with pure moral hazard. Inform. Syst. Res. 16, 209–230 (2005)

    Article  Google Scholar 

  28. 28

    Ho, C.-J ., Zhang, Y ., Vaughan, J. & Van Der Schaar, M. Towards Social Norm Design for Crowdsourcing Markets. Report No. WS-12-08 (AAAI, 2012)

  29. 29

    Zhang, Y. & van der Schaar, M. Peer-to-peer multimedia sharing based on social norms. Signal. Process. Image Commun. 27, 383–400 (2012)

    Google Scholar 

  30. 30

    Karnaugh, M. The map method for synthesis of combinational logic circuits. Trans. AIEE Part I 72, 593–599 (1953)

    MathSciNet  Google Scholar 

  31. 31

    Fishman, M. A. Indirect reciprocity among imperfect individuals. J. Theor. Biol. 225, 285–292 (2003)

    Article  Google Scholar 

  32. 32

    Roberts, G. Evolution of direct and indirect reciprocity. Proc. R. Soc. Lond. B 275, 173–179 (2008)

    Article  Google Scholar 

  33. 33

    Sherratt, T. N. & Roberts, G. The importance of phenotypic defectors in stabilizing reciprocal altruism. Behav. Ecol. 12, 313–317 (2001)

    Article  Google Scholar 

  34. 34

    Umans, C. The minimum equivalent DNF problem and shortest implicants. J. Comput. Syst. Sci. 63, 597–611 (2001)

    MathSciNet  Article  Google Scholar 

  35. 35

    Vigo, R. A note on the complexity of Boolean concepts. J. Math. Psychol. 50, 501–510 (2006)

    MathSciNet  Article  Google Scholar 

  36. 36

    Feldman, J. The simplicity principle in human concept learning. Curr. Dir. Psychol. Sci. 12, 227–232 (2003)

    Article  Google Scholar 

  37. 37

    Null, L. & Lobur, J. The Essentials of Computer Organization and Architecture Ch. 3 (Jones & Bartlett Publishers, 2014)

  38. 38

    Santos, F. P., Pacheco, J. M. & Santos, F. C. Evolution of cooperation under indirect reciprocity and arbitrary exploration rates. Sci. Rep. 6, 37517 (2016)

    CAS  ADS  Article  Google Scholar 

  39. 39

    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)

    CAS  ADS  MathSciNet  Article  Google Scholar 

  40. 40

    Stewart, A. J. & Plotkin, J. B. Collapse of cooperation in evolving games. Proc. Natl Acad. Sci. USA 111, 17558–17563 (2014)

    CAS  ADS  Article  Google Scholar 

  41. 41

    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)

    ADS  Article  Google Scholar 

  42. 42

    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)

    CAS  Article  Google Scholar 

  43. 43

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

    ADS  Article  Google Scholar 

Download references


This work was supported by Fundação para a Ciência e Tecnologia (FCT) through grants SFRH/BD/94736/2013, PTDC/EEI-SII/5081/2014, PTDC/MAT/STA/3358/2014, UID/BIA/04050/2013 and UID/CEC/50021/2013. We are grateful to A. P. Francisco and M. Janota for comments.

Author information




F.P.S., F.C.S. and J.M.P. conceived the project. F.P.S. performed the mathematical and numerical analysis. F.P.S., F.C.S. and J.M.P. analysed the results and wrote the paper. All authors contributed to all other aspects of the project.

Corresponding author

Correspondence to Jorge M. Pacheco.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Additional information

Reviewer Information Nature thanks C. Efferson, E. Fehr, G. Szabó and A. Tavoni for their contribution to the peer review of this work.

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

Extended data figures and tables

Extended Data Figure 1 Cooperation index of third- and fourth-order norms.

In the space of fourth-order norms (red bars), only a small fraction of norms (about 0.2% of 216) foster high levels of cooperation (η > 0.9), as conveyed by the complementary cumulative distribution function (CCDF; see inset for a close-up of the tail). In the space of third-order norms (blue bars), about 2% of norms (of a total of 28) promote high levels of cooperation (η > 0.9). Other parameters: Z = 50, ε = α = χ = 0.01, μ = 1/Z, b = 5, c = 1, γ = 0.

Extended Data Figure 2 The most cooperative norms.

a, c, e, g, Data from simulations in which individuals pay a complexity cost cc proportional to the complexity κs of the strategy that they employ (cc = s/10 = γκs). b, d, f, h, Data when no complexity cost is involved. Irrespective of whether the previous reputation of the recipient (ad) or the donor (eh) is used as the fourth consideration (as the fourth-order bit; see Extended Data Fig. 3), or whether there is a complexity cost involved, the highest levels of cooperation are already achieved for κ = 4. Moreover, when we plot norm performance (in terms of the cooperation index), separating norms according to their complexity κ (for κ ≤ 5; c, d, g and h) it becomes apparent that fourth-order norms are generally outperformed by lower order norms. Furthermore, paying a complexity cost is most detrimental to the more sophisticated fourth-order norms, which no longer promote cooperation under indirect reciprocity. Other parameters: Z = 50, ε = α = χ = 0.01, μ = 1/Z, b = 5, c = 1.

Extended Data Figure 3 Alternative ways of defining a social norm.

a, b, We consider norms that attribute a new reputation (outer ring) on the basis of (i) the action of the donor (first-order bit; blue ring); (ii) the current (actual) reputation of the receiver (second-order bit; yellow ring); (iii) the current (actual) reputation of the donor (third-order bit; pink ring); and (iv) the previous reputation of either the recipient (a) or the donor (b) (fourth-order bit; green ring). In a and b, there are 216 norms in total. c, Number of bits (layers, l) used for each norm order, and the corresponding number of possible strategies (s) and norms. Because actions are taken using the same information used by a norm to attribute a new reputation, we consider 28 different strategies for norms up to fourth-order.

Extended Data Figure 4 Robustness of results to parameter variations.

A full list and detailed description of all model parameters is provided in Supplementary Information. ad, Norm performance (in terms of the complexity index) as a function of population size Z (a), the benefit-to-cost ratio b/c in the donation game in which individuals interact (b), the private assessment error probability χ (c) and the reputation assignment probability τ (d). Here we use the previous reputation of the recipient as the fourth-order bit (as in the main text) and investigate, within the space of fourth-order norms, the performance of the (second- and third-order) leading eight norms together with the (first-order) image score and (zeroth-order) all good norms. The norms ‘ss’ and ‘sj’ denote simple standing and stern judging; ‘ss + sj’ has the first 8 bits equal to the third-order representation of simple standing and the last 8 equal to the third-order representation of stern judging; and ‘sj + ss’ is defined similarly; see Supplementary Table 4 for details of these norms. Other parameters: Z = 50, ε = α = χ = 0.01, μ = 1/Z, b = 5, c = 1, γ = 0.

Extended Data Figure 5 Global versus local mutation schemes.

b, We consider a mutation scheme in which a new strategy is adopted with probability μ (drawn from a UPD) when a mutation occurs39,40,41,42. c, Alternatively, we consider a local mutation (in each strategy), whereby with probability μL (drawn from a UPD) one bit changes. a, For the leading eight norms8, we find that the same conclusions are attained regardless of the mutation scheme considered. Other parameters: Z = 50, ε = α = χ = 0.01, b = 5, c = 1, γ = 0.

Extended Data Figure 6 Pseudo code for the Monte Carlo simulations used to calculate the cooperation index under each norm.

Given the large number of norms considered, we used Runs = 15 and Gens = 1.5 × 104 in Figs 2, 3, 4 and Extended Data Figs 1 and 2, and Runs = 50 and Gens = 105 in Extended Data Figs 4 and 5.

Supplementary information

Supplementary Information

This file contains a Supplementary Discussion, Supplementary Tables 1-4 and Supplementary References. (PDF 345 kb)

Life Sciences Reporting Summary (PDF 71 kb)

PowerPoint slides

Source data

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Santos, F., Santos, F. & Pacheco, J. Social norm complexity and past reputations in the evolution of cooperation. Nature 555, 242–245 (2018).

Download citation

Further reading


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.