Cooperation, social norm internalization, and hierarchical societies

Many animal and human societies exhibit hierarchical structures with different degrees of steepness. Some of these societies also show cooperative behavior, where cooperation means working together for a common benefit. However, there is an increasing evidence that rigidly enforced hierarchies lead to a decrease of cooperation in both human and non-human primates. In this work, we address this issue by means of an evolutionary agent-based model that incorporates fights as social interactions governing a dynamic ranking, communal work to produce a public good, and norm internalization, i.e. a process where acting according to a norm becomes a goal in itself. Our model also includes the perception of how much the individual is going to retain from her cooperative behavior in future interactions. The predictions of the model resemble the principal characteristics of human societies. When ranking is unconstrained, we observe a high concentration of agents in low scores, while a few ones climb up the social hierarchy and exploit the rest, with no norm internalization. If ranking is constrained, thus leading to bounded score differences between agents, individual positions in the ranking change more, and the typical structure shows a division of the society in upper and lower classes. In this case, we observe that there is a significant degree of norm internalization, supporting large fractions of the population cooperating in spite of the rank differences. Our main results are robust with respect to the model parameters and to the type of rank constraint. We thus provide a mechanism that can explain how hierarchy arises in initially egalitarian societies while keeping a large degree of cooperation.


S1 Unconstrained model: Individual groups
The results reported in the main article offer an overall picture of the whole population, by aggregating the results of different groups. However, it is essential to take the study one level down and look at processes and mechanisms at the individual group level, to assess to what extent the aggregate results describe them well. Therefore, we now look at this individual group level, beginning with Figure S1 (a), where we can see representative examples showing that, as previously mentioned, most of the individuals in the upper half do not follow the norm of cooperating in the collective action. In contrast, the rest of the individuals follow the norm. This is not always true, inasmuch of in Figure S1 (b) there is an individual with rank 0 who is following the norm, and an individual with rank 0 who is not following the norm. Therefore, the behavior is subject to noise, but the proportion of individuals misbehaving for their rank is very low if we compare these figures with Figure 3 (c) from the main article, where the aggregated behavior is shown. The same applies to the case of bigger groups but, due to the increased number of individuals, the behavior is more consistent, and those who have a high (respectively, low) rank and cooperate (respectively, do not cooperate) are less noticeable.
Also, within this perspective of what goes on in each group, we observe that the payoffs' distribution correlates with the expected behavior: within the groups, the individuals with higher rank have more payoffs   than those in lower ranks (according to Figures S3 and S4). This is consistent with the fact that the higher the rank, the more probable it is for an individual to win a fight and, therefore, to accumulate more payoffs. There is still noise in the distribution, as in Figure S4 (b), an individual with low rank has more payoffs than those above it.
Finally, attending to the individual distribution of fitness values concerning the position in the hierarchy, Figures S5 and S6 show that the individuals with more probability of having offspring the next generation are those with the higher ranks. This happens in both small and big groups. With these figures and the ones in S1-S2, the individual behavior in the reproduction stage can be predicted: those who reproduce with higher probability are those who do not follow the norm, in agreement with the observed behavior of the internalization parameter in Figure 3 (b) from the main article. In any event, looking at these individual examples, and at the rest of those in the simulations, we can conclude that the aggregate behavior represents well the general features of individual groups.   The difference between the constrained model and the unconstrained model can be seen in the payoff distributions from Figures S9 and S10. For the constrained model, all the individuals have, approximately, the same amount of payoffs when the groups are small ( Figures S9), with an increment in the difference when the size of the group increases (see figures S10).
There exists a critical difference between this model and the previous, unconstrained one, that can be seen in Figures S11 and S12. The individuals in small groups (S11) have a fitness distribution that is not correlated with their position in the hierarchy. Some individuals have the highest fitness even when its rank is the lowest (see Figure S11 (a) or (b)). This translates into a higher probability of selecting an individual   with a low rank for reproducing (and thus, more probable that cooperates in the CA). The mechanism making this happen is the same that increases the internalization parameter in Figure 5 from the main article: the mobility of the individual within the hierarchy, from high ranks-where it can increase its accumulated payoffs, making it more probable to be selected at the individual level when reproducing-to low ranks-where it cooperates more due to its rank, and that makes its group increase its success in the CA problem, thus being more probable to be selected at the group level-. This does not happen when the group size is larger, as in Figures S12, so the increase in the size mitigates this effect.
We have looked at different trajectories from randomly selected individuals ( Figure S13 and S14) to provide grounds for our hypothesis: moving individuals in the hierarchy increases internalization by forcing them to cooperate in low ranks. In these plots, two individuals from the same group are selected at random, and their trajectories are plotted. In Figure S13 (a) we observe different behavior from the ones in figures (b) to (b). In figure (a), the individuals exchange their positions: one of them increased its rank up to the maximum and then suddenly decreased it to the minimum possible. However, the second individual maintained a low rank while the first was on the highest possible rank; and slowly increased it up to the maximum in a few rounds. Individuals transitioning from low ranks to high ones and vice versa are indeed observed in these models.
On the contrary, in Figure S14 (a), we observe individuals who had a constant behavior of increasing or decreasing its rank, and in figure (b), we observe an initial perturbation before increasing their ranks. Thus, rank trajectories in unconstrained systems are steady. In Figure S13 (b) we observe initial movement in the trajectories before one of them quickly rose to the top of the hierarchy, but the difference with small models is that we do not observe as many individuals transitioning from the top to the bottom of the hierarchy as in models with smaller group size.

S3 Endogenously constrained model: Individual groups
Figures below show, for individual groups, the ranks distributions, payoff distributions per rank, and the relationship between fitness and ranks. Figure S15 shows the plots for small groups (size n = 8), and figure S16 shows the ones for big groups (size n = 24). Groups have been randomly selected, and the behavior is consistent between all of them. Figure S15: Simulation results for small groups, n = 8. Columns from left to right correspond to λ = 0.1, 0.25, 0.5 and 0.75. Top to bottom rows correspond to: Two rows of score histograms for two different groups; where blue indicates free riders-do not follow the norm of cooperating-and red indicates norm followers-cooperators. Two rows of payoff with respect the score plots. Two rows of fitness v. ranks plots. Figure S16: Simulation results for large groups, n = 24. Columns from left to right correspond to λ = 0.1, 0.25, 0.5 and 0.75. Top to bottom rows correspond to: Two rows of score histograms for two different groups; where blue indicates free riders-do not follow the norm of cooperating-and red indicates norm followers-cooperators. Two rows of payoff with respect the score plots. Two rows of fitness v. ranks plots.