Social media summary: Norm reinforcement, not just payoff biases or conformity, sustains arbitrary cultural traditions.

Payoff-biased copying and environmental variation

Consider an infinitely large group of Bayesian conformist individuals who repeatedly choose a value along a continuous dimension. Let the distribution of the value in the population be normal (with mean u and variance v). While done for simplicity, this assumption of normality is reasonable because many behavioural processes are influenced by numerous small independent effects, which tend to approximate a normal distribution per the Central Limit Theorem (McElreath, Reference McElreath2018). Furthermore, this assumption is unlikely to meaningfully change results as the focus of the model is primarily on the mean behaviour rather than the full distribution of behaviours.

Being conformists, individuals would like to adopt the mean value in the population (u) with an expected squared error equal to the population variance (v) divided by c, where c is the strength of the conformist tendency (c  ≥  1). Following Morgan and Thompson (Reference Morgan and Thompson2020), we assume that individuals estimate u by making observations of the population, which they combine with their prior beliefs (an evolved bias) according to Bayes’ rule to reach a conclusion. Note that this implementation differs from that of Boyd and Richerson (Reference Boyd and Richerson1985) where conformist individuals take N samples from their population's behavioural distribution and discard those outside the x% central confidence interval, effectively excluding ‘outliers’ from their decision-making process. In particular, this implementation can affect the mean if the distribution of values in the population is not symmetrical around the mean whereas our implementation cannot. However, given that we assume that the population distribution is symmetrical, neither implementation would change the mean value, only the variation, which will decrease. As a result, in both implementations, the variation steadily decreases, causing the population to conform to the mean value over time. For a schematic representation of this model see Figure 1.

Figure 1. Schematic representation of the payoff-biased copying and environmental variation model. Note that group behavioural distributions are updated based on the mean of the posterior with an assumed variance of 1. See the main text for the exact formulation of each step.

To incorporate payoff biased transmission, each value along the dimension has a corresponding fitness that affects the likelihood an individual with that belief is observed by others. The fitness landscape is a normal distribution with mean u _f and variance v _f. The distribution of values apparent to an observer (the observed distribution) is the true distribution multiplied by the fitness distribution, and is thus another normal distribution with mean u _o and variance v _o, where:

(1)$$u_o = \displaystyle{{\displaystyle{1 \over {v_f}}u_f + \displaystyle{1 \over v}u} \over {\displaystyle{1 \over {v_f}} + \displaystyle{1 \over v}}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;v_o = \displaystyle{1 \over {\displaystyle{1 \over {v_f}} + \displaystyle{1 \over v}}}$$

Note that the mean of the observed distribution is simply a weighted average of the means of the true distribution and fitness landscape (for the derivation of these equations, see supplementary materials).

All individuals are assumed to share the same prior belief concerning u, which is described as a normal distribution with mean u _p and variance v _p. Conceptually, the prior is a pre-evolved bias that individuals have towards particular behaviours, created by prior natural selection. Critically, this bias is shared among all individuals, regardless of group affiliations. Generally, we expect the prior to be quite weak or uninformative on the grounds that human behaviour is highly flexible and so it is unlikely that humans have powerful biologically evolved biases for the variety of specific tasks they engage in. Rather, biases are more likely to correspond to certain abstract preferences or predispositions that have a modest impact on a wide range of behaviours. For instance, a study by Ilie et al. (Reference Ilie, Ioan, Zagrean and Moldovan2008) found that the red team in first-person shooter games was more likely to win than the blue team, despite teams being assigned through random draws of individuals. This suggests that individuals may have an evolved bias pertaining to colours, with red fostering competitive actions across contexts (Ilie et al., Reference Ilie, Ioan, Zagrean and Moldovan2008).

Individuals then observe N models, sampled from the observed distribution, to update their beliefs about u and produce a posterior distribution. Given that the expected mean and variance of the values observed are u _o and v _o, the expected posterior is a normal distribution with mean u′ and variance v′ where:

(2)$${u}^{\prime} = \displaystyle{{\displaystyle{N \over {v_o}}u_o + \displaystyle{1 \over {v_p}}u_p} \over {\displaystyle{N \over {v_o}} + \displaystyle{1 \over {v_p}}}} = \displaystyle{{\displaystyle{N \over {v_f}}u_f + \displaystyle{N \over v}u + \displaystyle{1 \over {v_p}}u_p} \over {\displaystyle{N \over {v_f}} + \displaystyle{N \over v} + \displaystyle{1 \over {v_p}}}}$$

(3)$${v}^{\prime} = \displaystyle{1 \over {\displaystyle{N \over {v_o}} + \displaystyle{1 \over {v_p}}}} = \displaystyle{1 \over {\displaystyle{N \over {v_f}} + \displaystyle{N \over v} + \displaystyle{1 \over {v_p}}}}$$

Akin to equation (1), the mean of the posterior is simply a weighted average of the means of the observed distribution and the prior, which is equivalent to a weighted average of the means of the true distribution, fitness landscape and prior (for the derivation of these equations, see the supplementary materials).

Once the posterior is calculated, individuals take the most likely value (i.e. u′) as their point estimate of u and adopt it. As per our definition of conformist transmission, individuals adopt u′ with an expected squared error equal to vʹ/c, where c is the strength of conformist transmission. As such, after all individuals have conformed, the new value of u is u′ while v becomes vʹ/c. As this process repeats, variation decreases indefinitely, while the mean value in the population will evolve to an equilibrium which we can find by identifying conditions where there is no expected change in the mean value of the population (i.e. u = u′):

(4)$$u = {u}^{\prime} = \displaystyle{{\displaystyle{N \over {v_f}}u_f + \displaystyle{N \over v}u + \displaystyle{1 \over {v_p}}u_p} \over {\displaystyle{N \over {v_f}} + \displaystyle{N \over v} + \displaystyle{1 \over {v_p}}}} = \displaystyle{{\displaystyle{N \over {v_f}}u_f + \displaystyle{1 \over {v_p}}u_p} \over {\displaystyle{N \over {v_f}} + \displaystyle{1 \over {v_p}}}}$$

The equilibrium mean value (4) can be interpreted as a struggle between the fitness landscape (which pulls towards u _f with strength N/v _f) and the prior (which pulls towards u _p with strength 1/v _p). Intuitively, the ability of fitness-biased transmission to drive cultural dynamics relies on both the sharpness of the fitness function and the amount of data collected. Notice that, as v _p approaches 0, the prior becomes infinitely powerful, resulting in the population converging on the mean of their priors. Similarly, as N/v_f approaches infinity, the fitness landscape becomes infinitely powerful, causing the population to converge on the mean of the fitness landscape. As a result, the precise outcome depends on v _f, v _p and N, with the equilibrium ranging anywhere between u _p and u _f; as visualised in equation (5) and Figure 2 (for justification of equations (4) and (5), see supplementary materials):

(5)$$u = u_p + ( {u_f-u_p} ) \left({\displaystyle{{\displaystyle{N \over {v_f}}} \over {\displaystyle{N \over {v_f}} + \displaystyle{1 \over {v_p}}}}} \right)$$

Figure 2. The figure illustrates the equilibrium mean value or behaviour (u) reached based on the strength of the fitness landscape (N/v _f) and the influence of the prior (1/v _p), where N is the number of observations, v _f is the variance in fitness and v _p represents the variance in the prior. The equilibrium value (u) consistently lies between u _p and u _f. The colour coding in the figure indicates the closeness of the equilibrium value (u) to u _p and u _f, with u _p shown in dark red and u _f in white. When the fitness landscape is weak, i.e. n/v _f is low, its ability to influence decision-making is limited, leading to equilibria that are largely determined by the prior, thus moving u closer to u _p. This situation is akin to a world with only conformist transmission and biased priors, and as outlined by Morgan and Thompson (Reference Morgan and Thompson2020), priors dominate in this situation. On the other hand, when the prior is very weak, i.e. v _p is high, the fitness landscape significantly affects the equilibrium value in the population, resulting in u approximating u _f.

In sum, the payoff-biased transmission of continuous traditions creates equilibria that lie between values favoured by the prior (which favours u _p and is weighted by 1/v_p) and fitness landscape (which favours u _f and is weighted by N/v_f). This is a marked difference from the effect of conformist transmission alone, in which case the equilibrium is fully determined by the prior (Morgan & Thompson, Reference Morgan and Thompson2020). However, since the equilibrium is determined by the prior and fitness landscape it can be fully explained by ecological and cognitive factors, with culture being relegated to a mediator. This mediation is evident in the updated behavioural mean value (u′, see equation 2) which although initially influenced by the current mean value (u), ultimately converges on an equilibrium dictated solely by the priors and fitness landscape (see equation 5). Therefore, cultural evolution is simply how a population evolves to reach equilibrium; it does not affect the equilibrium value itself.

Norm reinforcement

The specific form of norm reinforcement that we consider here is the reinforcement of social markers that indicate group membership. Such indicators, including styles of dress and linguistic markers, are widespread in human groups and are argued to have led to the evolution of an ‘ethnic psychology’ (Moya & Henrich, Reference Moya and Henrich2016; Moya & Boyd, Reference Moya and Boyd2016). In some cases, such markers may serve no purpose and be reinforced solely through deliberate punishment of those who lack them. However, in many cases such markers, even if ecologically irrelevant, can serve important social functions and so be highly fitness relevant even in the absence of direct punishment. For instance, where groups are hostile to one another, clearly signalling your membership can help avoid accidental hostility from your own group mates and so increase your fitness. Such contexts may be common as recent evidence supports the cultural evolutionary prediction that warfare is disproportionately directed towards culturally dissimilar groups (Handley & Mathew, Reference Handley and Mathew2020). Even where groups coexist peacefully, they may have different norms for regulating common problems (e.g. how food is shared after collaborative foraging). In such cases it is important to interact selectively with group mates to avoid costly disagreements and so those who reliably indicate their group membership will experience greater fitness (McElreath et al., Reference McElreath, Boyd and Richerson2003). Thus, there are multiple mechanisms, both passive and active, by which norms concerning markers of group membership can be reinforced. Refer to Figures 3 and 4 for a visual representations of the model.

Figure 3. Schematic representation of the norm reinforcement model. See the main text for the exact formulation of each step.

Figure 4. Visualisations of the model's updating process. All panels show the prior in black. (a) The behavioural distribution (solid line), posterior distribution (dashed line) and noisy posterior distribution (dot–dash–dotted line) for one of two groups (blue and green) across one timestep (the fitness function is excluded to reduce clutter). It is noteworthy that the blue population begins with a bimodal distribution, which, over time, will diminish, resulting in a convergence towards a normal distribution. However, in this particular timestep, both the posterior and the smoothed posterior retain this bimodal characteristic, which the group will adopt as their behavioural distribution in the next timestep. (b) The behaviour distributions (solid lines) and fitness functions (dotted lines) for three groups (blue, green and yellow). With more than two groups, some groups end up ‘sandwiched’ between others. (c) The behaviour distributions (solid lines) and noisy posterior distributions (dot–dash–dotted lines) for three groups. Here the prior is strong enough to pull all groups towards each other, for this timestep at least.

Consider the same case described above with the following modifications: (1) rather than a single population, we consider n separate (infinitely large) sub-populations, with the behavioural distribution of the ith sub-population be denoted as Q _i, with mean u _i and variance v (for simplicity, we assume that all sub-populations have the same variance, although we do not assume the distributions are normal); and (2) fitness results from the reinforcement of norms, with an individual's fitness reflecting the degree to which their value (a) correctly indicates their group affiliation

$$\displaystyle{{Q_i} \over {\mathop \sum \nolimits_{\,j = 1}^n Q_j}}$$

and (b) resembles behaviours typical of their group (Q _i), the latter producing conformist effects. The fitness landscape of the ith sub-population, f _i, is therefore:

(6)$$f_i = Q_i\cdot \displaystyle{{Q_i} \over {\mathop \sum \nolimits_{\,j = 1}^n Q_j}}$$

This creates different, dynamic, fitness landscapes for each group that coevolve with their compositions. Note that the most prevalent behaviours in any given sub-population do not necessarily have the highest fitness. Rather, favoured behaviours are those that are group typical, but also predictive of group membership.

Individuals use their own group behavioural distribution, weighted by fitness, in order to engage in norm-reinforcing, conformist transmission. Thus, the observed distribution, for the ith group, O _i, is proportional to the true distribution in that group, multiplied by that group's fitness function:

(7)$$O_i = Q_if_i = {{Q_i^3 } \over {\mathop \sum \nolimits_{\,j = 1}^n Q_j}}$$

The observed distribution is not necessarily normal and has no standard form, thus it is computationally modelled, with details provided in the supplementary material.

Individuals then use Bayes’ rule to integrate the observed distribution (O _i) and their prior. Unlike the payoff-bias model, this integration cannot be done analytically since O _i is a non-standard distribution. Therefore, we define P as the prior normal distribution with mean u _p and variance v _p, and computationally calculate the expected posterior for the ith group using Bayes rule as follows:

(8)$$Q_i^{\prime} = {{O_i\cdot P} \over C}$$

The parameter C in the equation represents the integration constant that normalises the product of O _i and P, ensuring that it forms a proper distribution.

Finally, noise is added to the posterior and it becomes the behavioural distribution for the next iteration of the simulation. While it might seem reasonable for groups to adopt $Q_i^{\prime}$ exactly as their new behavioural distribution, doing so would eventually lead to the unrealistic result of groups without any internal variation. Thus, to ensure groups have a constant variance v, we apply the 1D diffusion equation to smooth/flatten the posterior ($Q_i^{\prime}$), adding noise until the distribution achieves the same level of variation as the initial distribution (v). We denote this noisy posterior distribution as $\widetilde{{Q_i^{\prime} }}$. There are many ways to simulate noise; however, the 1D diffusion equation was selected because it can preserve key characteristics of the posterior, such as the mean, mode or median (see the supplementary material for more details on the heat/diffusion equation). For instance, see Figure 4a, where it adds variance to the posterior while still preserving its multi-modal features. It should be noted that this work does not explicitly investigate the mechanisms that generate within-group variation; instead, we recognise its presence and assume that our groups maintain a level of variation over time.

We can iterate the above equations to numerically find the equilibria of an arbitrary number of groups ($Q_i = \widetilde{{Q_i^{\prime} }}$). The outcome of these iterations obeys the following observations: (1) all groups are pulled towards their shared prior (Figure 5a–c); (2) the intensity of this is inversely proportional to the variance of the prior (Figure 5d–f); (3) all groups are pushed away from each other due to the fitness effects of reinforcement (Figure 5a, b); (4) the intensity of this repulsion is proportional to the number of groups and the reciprocal of within-group variance (Figure 5c–e); (5) at equilibrium the distribution of groups around the peak of the prior (u _p) is symmetrical, with the centre of the distribution more densely populated than the tails (Figure 5a, b); and (6) regardless of the nature/shape of groups’ initial behavioural distributions, over time, they will converge towards normality (see supplementary materials).

Figure 5. (a) Cultural evolution of two groups, assuming u _p = 0, v _p = 5 and v = 1. Note that at equilibrium the two groups are symmetrically distributed around a mean value of 0 which is the most likely prior value (u _p = 0). Note also that the group represented by the blue line crosses the peak of the prior in order to ‘get away’ from the other group. (b) Cultural evolution of five groups, assuming u _p = 0, v _p = 5 and v = 1. The equilibrium distribution remains symmetrical. Note that the yellow group initially moves away from more plausible prior values in order to distance itself from the green group. (c) Cultural evolution of 10 groups, assuming u _p = 0, v _p = 5 and v = 1. At equilibrium, groups near the peak are more tightly clustered than those on the extremities. This is for two reasons: (1) groups near the peak of the prior are surrounded by other groups that causes a cultural ‘compression’; and (2) for groups on the extremities the prior is increasingly flat and so it exerts less pressure on their equilibrium. (d–f) The equilibrium variation between the mean values of each group (i.e. between-group variation), is indicated by the colour gradient from red to yellow, with yellow signifying high variation and red signifying low or none. This variation is a function of the prior variation (v _p) and of the variation within each group (v), assuming n =  2, 5 and 10, respectively. For each heatmap, the term ‘variation’ denotes the standard deviation between groups rather than the variance. Additionally, in (d), the equilibrium between-group variation is multiplied by 1.5 to enhance the gradient's visibility, making it easier to identify. This visual representation of the system's equilibrium states demonstrates that higher prior variation (v _p) tends to support the preservation of between-group variation, whereas lower values facilitate the convergence of groups.

In addition to these observations, we also explore how within-group variation and the variation in the prior affect the between-group variation. As observed in Figure 5d–f, a bifurcation exists in which groups continue to exhibit between-group variation indefinitely or collapse such that there is no between-group variation. The latter occurs when the prior's pull on groups is stronger than the repulsion from other groups, thus causing all groups to converge on their shared prior. The bifurcation is influenced by the parameters of prior variation (v _p) and within-group variation (v), as well as the number of groups (n). As n increases from 2 to 10, the region sustaining between-group variation expands, suggesting that larger numbers of groups promote the maintenance of this variation. The relationship between v and v _p is non-linear, indicating that changes in within-group variation can have disproportionate effects on the equilibrium state of between-group variation.

In sum, norm reinforcement sustains cultural variation in continuous traits in the face of a biased prior, provided the prior is not overwhelmingly strong. Arbitrary starting points are typically not stable; nonetheless, at equilibrium between group differences are highly stable and average values are unique to each group.