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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.

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


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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.

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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.

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Correspondence to Jorge M. Pacheco.

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The authors declare no competing financial interests.

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

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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.

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

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