Conditions that favour cumulative cultural evolution

Abstract

The emergence of human societies with complex language and cumulative culture is considered a major evolutionary transition. Why such a high degree of cumulative culture is unique to humans is perplexing given the potential fitness advantages of cultural accumulation. Here, Boyd & Richerson’s (1996 Why culture is common, but cultural evolution is rare. Proc. Br. Acad. 88, 77–93) discrete-cultural-trait model is extended to incorporate arbitrarily strong selection; conformist, anti-conformist and unbiased frequency-dependent transmission; random and periodic environmental variation; finite population size; and multiple ‘skill levels.’ From their infinite-population-size model with success bias and a single skill level, Boyd and Richerson concluded that social learning is favoured over individual learning under a wider range of conditions when social learning is initially common than initially rare. We find that this holds only if the number n of individuals observed by a social learner is sufficiently small, but with a finite population and/or a combination of success-biased and conformist or unbiased transmission, this result holds with larger n. Assuming social learning has reached fixation, the increase in a population’s mean skill level is lower if cumulative culture is initially absent than initially present, if population size is finite, or if cultural transmission has a frequency-dependent component. Hence, multiple barriers to cultural accumulation may explain its rarity.

This article is part of the theme issue ‘Human socio-cultural evolution in light of evolutionary transitions’.

1. Introduction

Culture encompasses norms, attitudes and patterns of behaviour that are socially learned and transmitted among individuals [1,2]. A growing body of empirical evidence suggests that culture is not unique to humans, but is also present in such non-human animals as chimpanzees [3], humpback whales [4,5], orcas [5], French grunts [6], bluehead wrasse [7], cowbirds [8] and many others. Furthermore, non-human animals may exhibit similar cultural transmission biases to humans. Conformist transmission has been documented in humans [9–11], great tits [12,13] (also see [14,15]), sticklebacks [16], and fruit flies [17], and success- or pay-off-biased transmission has been observed in humans [18,19], sticklebacks [20,21], capuchin monkeys [22] and vervet monkeys [23]. In most species capable of cultural transmission, culture appears to be non-cumulative, meaning that cultural traits do not appear to become increasingly sophisticated, complex, or effective through incremental improvement over time [2,24]. Non-cumulative cultural traits, which are ‘within the inventive capacity of a single individual’ [24, p. 379], might be learned individually in addition to socially [25]. For example, great tits exhibited social learning when solving a puzzle box in the presence of skilled demonstrators, but some were also capable of solving the puzzle box when demonstrators were not present [12].

Accumulation of culture may be unique to humans [24,26,27], although some have suggested that it is present in a small number of non-human animals such as birds and chimpanzees [2,25]. Nevertheless, humans exhibit by far the greatest degree of cumulative culture, and human societies have been considered a major evolutionary transition (MET) [28–30]. This cumulative culture is supported by physiological, cognitive and behavioural adaptations, such as the capacity for spoken and written language, prolonged (and often bi-parental) childcare, strong social structures and more, which allow the preservation and accurate long-term transmission of larger quantities of information compared to what is possible in non-human animals. This body of knowledge has allowed humans to create technologies and modify environments to meet their needs in a variety of ways seen nowhere else in nature, constituting a major system transition [31] referred to as the Anthropocene. As no single individual possesses the suite of knowledge that has enabled humans’ vast ecological dominance, cumulative culture can be considered a distributed adaptation [32,33]. Given the benefits of this distributed adaptation—e.g. defence against predators and diseases; protection against temperature extremes and other environmental stressors; increase in food quantity, security and nutritional value; and many others [31]—it is puzzling that cumulative culture is not more common in nature.

Boyd & Richerson [25] explored the question of why culture is common in animals but cumulative culture is rare using mathematical models. They concluded that a barrier to the evolution of cumulative culture was the difficulty for social learning to increase in frequency when initially rare. When social learning is rare, most animals learn individually and cumulative culture is initially absent. Therefore, the benefit of social learning may be too small to offset the costly development of neural circuitry required for social learning. Boyd and Richerson suggested that many instances of ‘culture’ in nature may be a product of ‘local enhancement’: juveniles are exposed to environmental conditions similar to those experienced by adults and therefore learn similar behaviours in a non-social, individual manner (see also [34]). Note that in [25,34], local enhancement is considered to be a form of social learning although animals learn individually; here, however, we do not consider local enhancement to be social learning or to produce culture.

Many theoretical studies have explored conditions under which social learning can increase in frequency when initially rare. Feldman et al. [35] showed that social learning can be favoured over individual learning when environmental changes occur at sufficiently long intervals. In [36,37], a third strategy—innate behaviour—was included, and social learning was favoured over individual learning or innate behaviour when the intervals between environmental changes were of intermediate length. In these models, social learners copied behaviours from members of the adult generation at random (i.e. there was unbiased oblique transmission). Others have allowed social learning to be success-biased, pay-off-biased, conformist or anti-conformist [25,34,38–44]. Success and pay-off bias refer to the preferential adoption of more successful individuals’ cultural variants, or variants with higher pay-offs, respectively [45,46] (these are the same if individuals’ pay-offs dictate their success), whereas conformity or anti-conformity occur when a more common cultural variant is copied at a rate greater or lower than its frequency, respectively [47]. Henrich & Boyd [34] suggested that stronger conformity favours the evolution of social learning, but Wakano and Aoki’s analysis produced a different result [41] (also see [39,40]). Kendal et al. [42] found that either conformist or anti-conformist social learning could be favoured over individual learning provided environmental change was sufficiently infrequent, whereas in some cases, pay-off-biased social learning could evolve in less stable environments.

In the first part of this paper, we explore the conditions under which social learning can evolve—a prerequisite to the evolution of cumulative culture—by extending the discrete-trait model in [25]. First, we generalize this model: rather than restricting our analysis to weak selection, as in [25], selection of any strength is allowed; a general number n of ‘role models’ observed by social learners is incorporated in all derivations rather than assuming n = 1 for the case where social learning is near fixation, as in [25]; and in addition to success bias, we incorporate conformity, anti-conformity or unbiased frequency-dependent transmission (hereafter, ‘unbiased transmission’) into social learning. Second, in addition to the kind of deterministic environmental change in [25], random and periodic environmental variation are investigated, as in [35–37]. Finally, both finite and infinite population sizes are explored. We determine whether social learning is favoured over individual learning under a more stringent range of conditions when social learning is initially rare than initially common, as in [25]. We show that if cultural transmission is entirely success-biased and the population size is infinite, as in [25], this holds only if the number of role models is sufficiently small. However, if social learning includes both success-biased and conformist or unbiased transmission, and/or if the population size is finite, this can be true with a greater number of role models. Our results hold with any of the three forms of environmental variation that we study.

In the second part of the paper, we investigate the transition from simple social learning to cumulative cultural evolution. In [25], cumulative culture is incorporated in a continuous-trait model, assuming that there is constant variance in the distribution of phenotypes (although mechanisms to maintain such variance, such as random mutation, are not incorporated), and that social learners imitate a random individual from the previous generation. Others have shown that ‘adaptive filtering’ (i.e. the removal of maladaptive cultural variants) may be crucial for the accumulation of culture, and have suggested that success bias or conformity may enable such adaptive filtering [48]. However, in another study [49], the rate at which individuals adopted adaptive variants was higher with unbiased transmission than conformity. Finally, in [24], it was much more difficult for culture to accumulate in a finite than infinite population.

In our model of cumulative culture, cultural transmission can be entirely success-biased, or a combination of success-biased and conformist, anti-conformist, or unbiased transmission. Both finite and infinite population sizes are explored. Our results show that cultural accumulation can be hindered by finite population sizes, as in [24], as well as frequency-dependent transmission and, in some cases, small numbers of sampled role models.

2. Models of non-cumulative culture: framing and extensions

We first present a deterministic, discrete-time model of a population of individual and social learners with a dichotomous trait whose variants are ‘skilled’ and ‘unskilled,’ assuming a very large population size and non-overlapping generations. In the parental generation, let

$$\begin{array}{rl}& x_0=\text{frequency of unskilled social learners,}\\ & x_1=\text{frequency of skilled social learners; }\end{array}$$

$$\begin{array}{rl}& y_0=\text{frequency of unskilled individual learners,}\\ & y_1=\text{frequency of skilled individual learners}.\end{array}$$

The model parameters are given in table 1 and the model is illustrated in figure 1. As in [25], we assume an infinite number of environmental states, so no state change occurs more than once, and that there is one distinct skill, which confers a fitness benefit, to be learned in each environmental state. We also assume that the phenotype ‘individual learner’ or ‘social learner’ is passed from a parent to its offspring vertically and uni-parentally (e.g. genetic, asexual transmission) as in [25,35–37]. Although these phenotypes may in reality be determined by multiple genes or alleles, the assumption that a phenotype is genetically determined by one allele is common in biological models (known as the ‘phenotypic gambit’) [50]. Hence, the frequencies x₀′, x₁′, y₀′, y₁′ in the offspring generation are:

$$\begin{array}{rl}T{x}_0^{\prime }& =(x_0+x_1)\{\gamma +(1-\gamma )[1-\rho \pi (x_1+y_1)\\ & -(1-\rho )\varphi (x_1+y_1)]\}(1-\delta )(1-K-C_I)\end{array}$$ 2.1a

$$\begin{array}{rl}T{x}_1^{\prime }& =(x_0+x_1)\{\{\gamma +(1-\gamma )[1-\rho \pi (x_1+y_1)\\ & -(1-\rho )\varphi (x_1+y_1)]\}\delta (1-K-C_I+D)\\ & +(1-\gamma )[\rho \pi (x_1+y_1)\\ & +(1-\rho )\varphi (x_1+y_1)](1-K-C_S+D)\}\end{array}$$ 2.1b

$$T{y}_0^{\prime }=(y_0+y_1)(1-\delta )(1-C_I)$$ 2.1c

$$T{y}_1^{\prime }=(y_0+y_1)\delta (1-C_I+D),$$ 2.1d

where T is a normalizing factor, namely the sum of the right-hand sides, and π(x₁ + y₁) and ϕ(x₁ + y₁) are given by equations (2.2) and (2.3) below, respectively.

Table 1.

Model parameters^a.

parameter	range	definition
γ	(0, 1)	probability of environmental change
CI	$(0,\frac{1}{2})$	fitness cost of individual learning
K	$(0,\frac{1}{2})$	fitness cost of capacity for social learning (i.e. neural circuitry)
CS	$(0,\frac{1}{2})$	fitness cost of social learning (i.e. copying other individuals)
D	(0, ∞)	fitness benefit of learning the skill
δ	(0, 1)	probability of learning a skill individually
n	{3, 4, …}	number of adult role models sampled by a social learner
π(x1 + y1)	(0, 1)	success-biased learning efficiency
ρ	(0, 1)	probability of success-biased rather than frequency-dependent social learning
ϕ(x1 + y1)	(0, 1)	frequency-dependent learning efficiency
A(j)	( − j, n − j)	conformity coefficient, where j is the number of role models that carry the
		variant in the majority of the sample of n role models and $\frac{n}{2}<j<n$

^aParameters in rows 1-8 are similar to the parameters in [25] (although in [25] it is assumed that π(x₁ + y₁) is constant, denoted by π); those in rows below 8 are used in the present study but not in [25]. The upper bound of $\frac{1}{2}$ for C_I, K and C_S ensures that fitnesses are always positive.

Figure 1.

Illustration of the model. A naive individual learner (orange) suffers an individual learning cost C_I and receives a fitness benefit D if learning is successful, which occurs with probability δ. A naive social learner (green) suffers a fitness cost K owing to the development of neural machinery required for social learning. It samples n role models at random from the adult generation, and if the environment has not changed, some adults may be skilled. (If the environment has just changed, all adults will be unskilled, because there is one distinct skill to be learned in each environmental state.) If the social learner acquires the skill from its sample of role models, it receives a fitness benefit D and pays a social learning cost C_S; otherwise, it can switch to individual learning with cost C_I and probability δ of receiving fitness benefit D. (Online version in colour.)

As in [25], each social learner randomly samples n adult ‘role models,’ and the success-biased learning efficiency, π(x₁ + y₁), is the probability that at least one of these role models is skilled. In extending the model of [25] by removing the assumption of weak selection, we let the probability that one or more of n role models is skilled be

$$\pi (x_1+y_1)=1-{(1-x_1-y_1)}^n,$$ 2.2

rather than their appendix, equation A1.3 (see our electronic supplementary material, appendix A for the derivation and explanation of the differences).

Frequency-dependent transmission can be conformist, anti-conformist or unbiased, depending on the coefficient A(j) in equation (2.3) below. A(j) represents the strength and direction of the departure from unbiased transmission, and j = k, k + 1, …, n − 1 is the number of role models that carry the variant in the majority of the sample, so k = (n + 1)/2 if n is odd and k = (n/2) + 1 if n is even (see [51] and Box 7.4 in [47]). Following [47], if the frequency of skilled individuals in the parental generation is x₁ + y₁ and the number of role models sampled is n ≥ 3, the probability of acquiring the skill through frequency-dependent social learning is

$$\begin{array}{rl}\varphi (x_1+y_1)& =x_1+y_1+\sum _{\hspace{0.17em}j=k}^{n-1}\frac{A(j)}{n}\left(\begin{array}{c}n\\ j\end{array}\right)[{(x_1+y_1)}^j{(1-x_1-y_1)}^{n-j}\\ & -{(x_1+y_1)}^{n-j}{(1-x_1-y_1)}^j],\end{array}$$ 2.3

where −j < A(j) < n − j. In equation (2.3), A(j) > 0 entails conformity, A(j) < 0 entails anti-conformity, and A(j) = 0 entails unbiased transmission.

In addition to the kind of environmental variation assumed in [25], where γ is a fixed parameter (see equations (2.1)), we explore the ‘random environmental change’ and ‘periodic environmental change’ models of Aoki et al. [37] (also see [35,36]). In the random environmental change model, the parameter formerly known as γ in equations (2.1) becomes the indicator function 1_γ, which takes the value 1 with probability γ and 0 otherwise. Therefore, this model is not deterministic, and the model in equations (2.1) can be regarded as an approximation to the random environmental change model where 1_γ is set to be the expected value E(1_γ) = γ. In the periodic environmental change model, the parameter γ in equations (2.1) also becomes 1_γ, but takes the value 1 once every fixed period of $\frac{1}{\gamma }$ generations and 0 otherwise. The recursions with random and periodic environmental change are in the electronic supplementary material, equations (B1) and (B2) in appendix B.

Finally, in addition to the above models, which assume a very large population, we model a small population of size N. In this case, multinomial sampling (as in the Wright–Fisher model) is incorporated into equations (2.1), or the electronic supplementary material (B1) and (B2) for the random and periodic environmental change models, to produce the offspring frequencies x₀′, x₁′, y₀′, y₁′ in each generation.

3. Models of non-cumulative culture: analysis and simulations

(a) . Increase of social learning when it is rare

Using equations (2.1), with (2.2) and (2.3), we determine the conditions under which social learning increases in frequency when initially rare. Assume that the population is initially composed entirely of individual learners, i.e. x₀ = x₁ = 0 so that y₀ + y₁ = 1. Then x₀′ = x₁′ = 0 and

$$T{y}_0^{\prime }=(1-\delta )(1-C_I)$$ 3.1a

and

$$T{y}_1^{\prime }=\delta (1-C_I+D),$$ 3.1b

where T = 1 − C_I + δD. At equilibrium, y₀′ = y₀ and y₁′ = y₁, and these frequencies are denoted by $y_0^{\ast }$ and $y_1^{\ast }$, respectively. From equations (3.1), an equilibrium $y_1^{\ast }$ is attained after a single generation (because y₀ and y₁ do not appear on the right-hand side of the recursions), where

$$y_1^{\ast }=\frac{\delta (1-C_I+D)}{1-C_I+\delta D}.$$ 3.2

As selection weakens, i.e. C_I and D approach zero, $y_1^{\ast }$ approaches δ, as in [25]. In the electronic supplementary material, appendix C, we derive the conditions for invasion by social learning of the equilibrium $(x_0,x_1,y_0,y_1)=(0,0,1-y_1^{\ast },y_1^{\ast })$, namely:

Result 3.1. —

Social learning increases in frequency when initially rare if K, the cost of the capacity for social learning, is sufficiently small such that

$$[\rho \pi (y_1^{\ast })+(1-\rho )\varphi (y_1^{\ast })](1-\gamma )[D(1-\delta )+C_I-C_S]>K$$ 3.3

where π(x₁ + y₁) $\simeq$ $1-{(1-y_1^{\ast })}^n=\pi (y_1^{\ast })$ with $y_1^{\ast }$ given in equation (3.2). Similarly, from (2.3), ϕ(x₁ + y₁) $\simeq$ $\varphi (y_1^{\ast })$.

If social learning is entirely success-biased, as in [25], then ρ = 1. In this case, inequality (3.3) becomes similar to inequality (1) in [25], except that their first term is 1 − (1 − δ)ⁿ rather than $1-{(1-y_1^{\ast })}^n$ here. Their derivation is an approximation of ours under weak selection, when C_I and D approach zero (see equation (3.2)). However, here we see that if C_I and D approach zero then, as ρπ(x₁ + y₁) + (1 − ρ)ϕ(x₁ + y₁) > 0, 1 − γ > 0 and −C_S < 0, inequality (3.3) does not hold and therefore social learning cannot increase.

The relationship between inequality (3.3) and the fitness parameters C_I and D, as well as the individual learning probability δ, is complicated because, by equation (3.2), these parameters appear in $y_1^{\ast }$. Decreasing the environmental change rate γ and decreasing the fitness costs associated with social learning, K and C_S, facilitate invasion by social learning. From inequality (3.3), since [ρπ(x₁ + y₁) + (1 − ρ)ϕ(x₁ + y₁)](1 − γ) > 0, a necessary condition for invasion is D(1 − δ) + C_I − C_S > 0, so lower values of γ, K and C_S make invasion by social learning easier.

In addition, all else being equal, increasing the success-biased learning efficiency, $\pi (y_1^{\ast })$, and/or the frequency-dependent learning efficiency, $\varphi (y_1^{\ast })$, facilitates invasion by social learning. In equation (2.3) with $y_1=y_1^{\ast }$ and x₁ = 0, if $y_1^{\ast }>\frac{1}{2}$ (i.e. the skill is common among individual learners), then stronger conformity (greater positive A(j)) or weaker anti-conformity (less negative A(j)) increases the probability $\varphi (y_1^{\ast })$ that a frequency-dependent social learner will become skilled, and hence facilitates invasion by social learning. If $y_1^{\ast }<\frac{1}{2}$, then weaker conformity or stronger anti-conformity facilitates invasion (see [51]). From equation (3.2), $y_1^{\ast }>\frac{1}{2}$ if D > (1/δ)(1 − 2δ)(1 − C_I).

(b) . Increase of individual learning when it is rare

Here, we obtain conditions for a population composed only of social learners to be stable to invasion by individual learning. In exploring the case where social learning is common, n = 1 was assumed in [25]. Here, we first derive the condition under which the equilibrium $(x_0,x_1,y_0,y_1)=(1-x_1^{\ast },x_1^{\ast },0,0)$ is stable or unstable assuming a general number n of role models. The analysis is similar to that in §3a and is given in the electronic supplementary material, appendix D; we find the following result:

Result 3.2. —

Fixation of social learning is stable to invasion by individual learning if K is sufficiently small; namely

$$\begin{array}{r}[\rho \pi (x_1^{\ast })+(1-\rho )\varphi (x_1^{\ast })](1-\gamma )[D(1-\delta )+C_I-C_S]>K.\end{array}$$ 3.4

Overall, inequalities (3.3) and (3.4) relate to the increase of social learning when it is initially rare or when it is initially near fixation, respectively, but do not tell us whether social learning will increase from an intermediate frequency. To address this, we conducted simulations.

(c) . Numerical simulations

The aim of these simulations is to determine how often social learning increases when it is initially rare compared to when it is initially common, i.e. present in more than 50% of the population, among many simulations with different combinations of parameter values (described in the electronic supplementary material, appendix E). In figure 2, each plotted value represents the fraction of 2328 simulations for which the frequency of social learning, x₀ + x₁, increases. For finite-population models (columns 3 and 4 in figure 2), the population size is N = 100. In the electronic supplementary material, appendix F, some examples with N = 10 000 are shown to produce a similar result. The code for all simulations is available at https://github.com/kaleda/MET, and part of this code relies on [52,53].

Figure 2.

Conditions for increase in frequency of social learning (SL). In columns 1 and 2, the population size is infinite and social learning is either initially rare (frequency 10⁻⁷) or common (frequency 0.5 + 10⁻⁷), respectively. In columns 3 and 4, the population size N is 100 and social learning is initially present in 1 or 51 individuals, respectively. Individuals can have success bias only (ρ = 1) (red), or a combination of success bias and frequency-dependent transmission with ρ = 0.4. From equation (2.3), where (n/2) < j < n, there is conformity with A(j) = n − j − 0.1 shown in yellow; unbiased transmission with A(j) = 0 in orange; and anti-conformity with A(j) = −j + 0.1 in green. Each point shows the fraction of the 2328 simulations in which the ‘final’ frequency of social learning is greater than its initial frequency. With a fixed rate of environmental change as in [25] (a–d), the ‘final’ frequency is the equilibrium frequency of social learning; in the infinite periodic model (i,j), it is the average frequency of social learning over one period at equilibrium; and in the finite periodic model (k,l) and the random model (e–h), simulations are run for 100 000 generations, and the frequency of social learning is averaged over the last 50 000 generations. (Online version in colour.)

One of the main conclusions of [25], where it was assumed that social learners were entirely success biased (ρ = 1) and the population size was infinite, is that social learning is favoured over individual learning under a wider range of conditions when social learning is initially common than when it is initially rare. The red points in figures 2a,b show that this holds only if the number of sampled role models, n, is relatively small. However, figure 2 shows that with a finite population (c,d), and/or if ρ = 0.4 and frequency-dependent transmission is conformist (yellow points) or unbiased (orange points), social learning increases when common under a wider range of conditions than when it is rare for n = 3, …, 10. Thus, the main conclusion of [25] can hold, but with different assumptions from those in their study.

4. Models of cumulative culture: framing and extensions

In the previous sections, there was one skill with fitness benefit D in each environmental state. Here, we incorporate r ‘levels’ of the skill, with fitness benefits D₁ < D₂ < · · · < D_r for levels 1, 2, …, r, respectively. Previous models of cumulative culture have similarly incorporated discrete levels of a single trait; e.g. in [26], there are an infinite number of possible levels, whereas in [54] there are 10 levels. Castro & Toro [55] point out that although a broad category of behaviour such as ‘obtaining food’ might have many levels of complexity, specific skills, for example gathering fruit or digging up tubers, might have a small number of possible complexity levels, which they take to be four.

In each generation, an individual that has acquired a skill may improve its skill by increasing it one level at ‘innovation rate’ u. (Because the probability of acquiring a basic, level 1, skill need not be the same as the probability of improving a skill, u may not equal δ from table 1.) Therefore, through individual learning, fitness benefit D₁ can be acquired with probability δ(1 − u) and D₂ can be acquired with probability δu, but fitness benefits greater than D₂ cannot be acquired. Through social learning, an individual may, for example, copy a level ℓ skill and then improve it to receive fitness benefit D_ℓ+1. In the next generation, another social learner may copy this level ℓ + 1 skill and improve it so that its fitness benefit is D_ℓ+2, and so on. If the environment changes, however, no individuals from the previous generation are skilled so any fitness benefits owing to prior accumulation of culture disappear. Finally, the time and energy spent attempting to improve one’s skill if it is below the maximum skill level, r, incur an ‘innovation cost’ I that applies whether or not innovation was successful.

To determine factors that hinder cultural evolution besides the difficulty for social learning to increase in frequency, assume that the population is composed entirely of social learners and that this population state is stable. Let x₀ denote the frequency of unskilled social learners and x₁, x₂, …, x_r denote the frequency of social learners with skill level 1, 2, …, r, respectively. The probability that a social learner acquires a skill at level i under success-biased transmission is π(x, i) and under frequency-dependent transmission it is ϕ(x, i), where x = (x₀, x₁, …, x_r). Hence, if the environment changes with probability γ as in equations (2.1), the recursions are given by equations (4.1) below. Note that if a social learner does not acquire the skill socially, it attempts individual learning in which case it can acquire skill levels 0, 1 or 2 only; thus, a term involving C_I appears in the recursions for x₀, x₁ and x₂:

$$\begin{array}{rl}& T{x}_0^{{\hspace{0.17em}}^{\prime }}=[\gamma +(1-\gamma )(1-\sum _{i=1}^r[\rho \pi (\mathit{x},i)+(1-\rho )\varphi (\mathit{x},i)])]\\ & \times (1-\delta )(1-K-C_I)\end{array}$$ 4.1a

$$\begin{array}{c}T{x^{\prime }}_1=\{\left[\gamma +(1-\gamma )\left(1-\sum _{i=1}^r[\rho \pi (x,i)+(1-\rho )\varphi (x,i)]\right)\right].\hfill \\ \delta (1-K-C_I-I+D_1)+(1-\gamma )[\rho \pi (x,1)+(1-\rho )\hfill \\ \varphi (x,1)](1-K-C_s-I+D_1)\}(1-u)\hfill \end{array}$$ 4.1b

$$\begin{array}{c}T{x^{\prime }}_2=\left[\gamma +(1-\gamma )\left(1-\sum _{i=1}^r[\rho \pi (x,i)+(1-\rho )\varphi (x,i)]\right)\right].\hfill \\ \delta u(1-K-C_I-I+D_2)\hfill \\ +(1-\gamma )\{[\rho \pi (x,1)+(1-\rho )\varphi (x,1)]u+[\rho \pi (x,2)\hfill \\ +(1-\rho )\varphi (x,2)](1-u)\}(1-K-C_s-I+D_2)\hfill \end{array}$$ 4.1c

$$\begin{array}{rl}T{x}_{\ell }^{\prime }& =(1-\gamma )\{[\rho \pi (\mathit{x},\ell -1)+(1-\rho )\varphi (\mathit{x},\ell -1)]u\\ & +[\rho \pi (\mathit{x},\ell )+(1-\rho )\varphi (\mathit{x},\ell )](1-u)\}(1-K-C_S-I+D_{\ell })\end{array}$$ 4.1d

$$\begin{array}{rl}T{x}_r^{\prime }& =(1-\gamma )\{[\rho \pi (\mathit{x},r-1)+(1-\rho )\varphi (\mathit{x},r-1)]u(1-K-C_S-I+D_r)\\ & +[\rho \pi (\mathit{x},r)+(1-\rho )\varphi (\mathit{x},r)](1-K-C_S+D_r)\}.\end{array}$$ 4.1e

Equation (4.1d) is repeated for ℓ = 3, …, r − 1, and T is the sum of all right-hand sides. The electronic supplementary material, appendix G describes an example calculation and the expression π(x, i) (where success-biased social learners acquire the highest skill level in their sample of n role models). Frequency-dependent social learners can acquire a skill from a sample of n role models following the formulation in eqn. (17) of [52]. Random and periodic environmental changes are incorporated as in previous, non-cumulative models.

5. Models of cumulative culture: simulations

Here, our objective is to understand why a population of social learners might not accumulate culture. First, social learning is set to be initially common at a frequency of 0.5 + 10⁻⁷ and an equilibrium in the frequencies of unskilled social, skilled social, unskilled individual and skilled individual learners is reached in the absence of cumulative culture, using the same conditions as in the previous infinite-population simulations shown in figure 2. This ‘burn-in’ period allows realistic initial phenogenotype frequencies to be reached for each set of parameter values. Note that for this step, we only use the infinite-population model (which appears to be a good approximation to the finite-population model when social learning is common; see figure 2) so that the outcome of the burn-in period is deterministic and hence stochastic differences in results do not affect the subsequent steps. For the infinite-population model with random environmental change, the periodic environmental change model (which is deterministic) is used for the burn-in period. This was shown to be a good approximation to the random environmental change model (figure 2).

Combinations of parameter values for which social learning is not fixed at equilibrium are omitted from subsequent simulations. Then, we assume that there has been a recent environmental change so that all individuals are unskilled, and the new environment has five available skill levels. Thus, all initial frequencies are set to zero apart from x₀, which is 1. In another case, we assume that the lowest level of cumulative culture, which cannot be learned individually (level 3) is initially present in 10% of the population owing to stochastic events, i.e. x₃ = 0.1, so initially x₀ = 0.9. The mean level of culture present in the population [26] prior to an environmental change (or prior to the expected time of an environmental change, depending on the model) is calculated for each simulation, and subsequently averaged over all simulations with the relevant conditions (specified in figure 3) to produce the ‘grand mean,’ plotted in figure 3. Skill level i + 1, where i ≥ 0, is assumed to confer a fitness benefit of D_i+1 = D + βi, where D values are in the electronic supplementary material, appendix E. We take β = 0.1, innovation cost I = 0.08, and rate of successful innovation u = 0.001. Figure 3 shows the results for the periodic environmental change model, where an environmental change occurs at exactly 1/γ generations. The results for the other two models of environmental change are very similar to those in figure 3 and are shown in the electronic supplementary material, figure S2 of appendix H.

Figure 3.

Grand mean of the level of culture present in all simulated replicates prior to an environmental change in the periodic environmental change model. First, 2328 simulations of the non-cumulative model are run with the same parameter values as in figure 2, an infinite population, an initial social learning frequency of 0.5 + 10⁻⁷, and either success bias (red), or a combination of success bias and unbiased transmission (orange), success bias and conformity (yellow) or success bias and anti-conformity (green). Of these, the combinations of parameter values that allow social learning to reach fixation are then used in the corresponding cumulative culture model, where all learners are social. Initially, either all learners are unskilled, so cumulative culture (CC) is absent ((a) for infinite and (c) for finite populations), or 90% are unskilled and 10% have a level 3 skill ((b) for infinite and (d) for finite populations), making the initial average level of culture either 0 or 3(0.1) = 0.3 (dashed black line), respectively. As with the non-cumulative culture models (figure 2), under unbiased transmission conformity coefficients are zero; under conformity they take their maximum values minus 0.1; under anti-conformity they take their minimum values plus 0.1 (for maxima and minima, see [52]), and ρ = 0.4 in these models. In finite-population models, the population size N is 100 and random drift is modelled using multinomial sampling as in the Wright–Fisher model. (Online version in colour.)

Figure 3 and the electronic supplementary material, figure S2 show that it can be easier for culture to accumulate if some individuals with cumulative culture are initially present in the population rather than absent, and if population sizes are infinite rather than finite. In all treatments, purely success bias (shown in red) favours cultural accumulation to a greater degree than the other types of transmission biases. With purely success bias (red), a combination of success bias and unbiased transmission (orange), or a combination of success bias and conformity (yellow), increasing the number of role models, n, tends to facilitate the accumulation of culture, but with success and anti-conformist bias (green), the opposite can occur (e.g. see figure 3d).

6. Discussion

Our goal has been to elucidate conditions that favour the evolution of social learning as well as cumulative culture—critical features of human evolution that are likely to have facilitated the formation of complex societies, institutions, technology, and language, rendering humans an MET [28–30]. METs encompass fusions of formerly independent entities into novel types of ‘individuals,’ as well as novel types of information storage and transmission [29]. There is debate about whether human societies may eventually qualify as the former [56–60], but little dispute that human language exemplifies the latter [30,61,62]. Considering the advantages that complex language and other forms of cumulative culture (e.g. technology) confer on humans, why this MET happened only once, and why cumulative culture is rare in other species, remain mysterious.

One reason that has been suggested to account for this discrepancy is that the conditions under which social learning can invade a population of individual learners are stringent, but if social learning does invade and increase in frequency, then the conditions that favour social learning are much more relaxed [25]. However, others have argued that the conditions which favour social learning are not necessarily the same as those that favour cumulative culture [48]. Here, two types of models were explored. The first built on the discrete-trait model of [25], where there was a dichotomous cultural trait (skilled/unskilled) and both social and individual learners in the population, and revealed conditions that allowed social learning to evolve. In the second model, we postulated a polychotomous trait with several ‘skill levels’ and an unskilled state. To find barriers to the accumulation of culture other than the difficulty for social learning to evolve, in this model, we assumed that the population was comprised entirely of social learners, and that this fixation state was stable to invasion by individual learning. These two models revealed several hindrances to the evolution of social learning and cultural accumulation, discussed below.

In the discrete-trait model of [25], social learners are success biased, the population size is infinite, the rate of environmental change is deterministic, and selection is weak. In deriving the condition for social learning to be favoured when it is common (but not when it is rare), it is assumed in [25] that offspring sample n = 1 adult role model. This ‘common’ frequency is near fixation, and other frequencies at which social learning is more common than individual learning are not studied. It is concluded that ‘selection cannot favour a capacity for observational [social] learning when rare’ [25, p. 88] if it is more costly than individual learning. In our infinite-population model with success-biased social learning and deterministic environmental change (as in [25]), arbitrarily strong selection, and n ≥ 3 role models, we obtain a different result from that of [25]. As n increases, the number of conditions under which social learning increases in frequency when initially rare increases, and with large enough n, it is comparable to the number of conditions under which social learning increases when it is initially more common than individual learning, near 50% frequency (red points in figure 2a,b). In these simulations, social learning is more costly than individual learning.

A likely reason for this result (figure 2a) is that in the present model of success bias, following [25], a social learner acquires the skill if at least one of the n role models is skilled. Thus, as n becomes large, the probability that one or more role models is skilled becomes high and it becomes more likely that a social learner will receive a benefit. If success bias were imperfect (i.e. did not lead to the adoption of the more advantageous phenotype in the sample of n with probability 1), as in [42,63], results would probably differ. In our model, a probability ρ < 1 of success bias and a probability 1 − ρ of unbiased transmission together can be conceptualized as ‘imperfect success bias’ (orange points in figure 2). In this case, social learning increased when it was common in more cases than it increased when it was rare for n = 3, …, 10, differing from the red points in figure 2a,b with perfect success bias. Note that in figure 2, all three forms of environmental change appear to produce similar results, suggesting that the deterministic rate, γ, of environmental change in [25] provides a good approximation to the random and periodic models.

In figure 2, success bias was also reduced by allowing individuals to exhibit conformity or anti-conformity with probability 1 − ρ = 0.6 (yellow and green points, respectively). Previous research has shown that the combination of conformity and success bias can be more adaptive than either kind of bias alone [63]. However, in the present model with completely accurate success bias, reduction of this bias can only decrease the probability of acquiring the skill from the sample of role models and therefore make it more difficult for social learning to increase. Therefore, within each plot in figure 2, the red points are above the yellow, green and orange points. Figure 2 shows that the inclusion of conformity can produce more stringent conditions for social learning to increase when rare than when common for n = 3, …, 10 role models, unlike in some cases of entirely success-biased transmission (figure 2a,b,e,f,i,j). With anti-conformity, an infinite population, and sufficiently many role models, the reverse can be true; social learning can increase when initially rare but not when initially common (figure 2a,b,e,f,i,j), as anti-conformity favours rare phenotypes. Finally, in all finite-population models, it is much more difficult for social learning to increase when rare (figure 2c,g,k) than when common (figure 2d,h,l; note the discontinuous y-axis).

The difficulty for social learning to increase in frequency may pose a barrier to the evolution of cumulative culture. However, there is increasingly strong evidence that some species can learn socially: social learning has been widely observed in species of primates, birds, fishes, cetaceans and insects [64], and conformity [12,16,65] or a combination of conformity and anti-conformity [17] (discussed in [66], p. 10) has also been documented in non-human animals. We now turn to the question of why a population of social learners might not accumulate culture.

Comparing figure 3b,d and 3a,c, we see that drift hinders cultural accumulation, as in [24]. A rare, advantageous cultural innovation that is present in a parental generation may not be passed on to the offspring generation in a finite population, but will in an infinite population. Points in figure 3b (or 3d), where cumulative culture is initially present in 10% of the population, are greater than the corresponding points in figure 3a (or figure 3c), where cumulative culture is initially absent. Thus, even a small accumulation of culture that initially occurred by chance could facilitate further accumulation. In addition, as in [26], success bias combined with a sufficiently high number of role models can facilitate cultural accumulation (e.g. figure 3b); increasing n increases the probability that an advantageous cultural variant will be represented in the sample, and success bias causes the adoption of such a variant to occur at a higher rate than frequency-dependent transmission (as in the model of non-cumulative culture discussed above).

In addition to success bias, some have suggested that conformity facilitates cultural accumulation [67], although others have shown that conformity hinders cultural accumulation relative to unbiased frequency-dependent transmission [49]. In figure 3, the level of cultural accumulation with a combination of success bias and conformity (yellow points) can be similar to that with a combination of success bias and unbiased transmission (orange points) (e.g. these points frequently overlap in figures 3c,d), or can be lower in the former case (e.g. figure 3a). Whether the results in figure 3 are robust to different choices of parameter values, however, remains to be investigated.

Dean et al. [67] suggest that conformity may facilitate cultural accumulation to a greater degree than unbiased frequency-dependent transmission if the population is subdivided into groups, as in the model of [34]. Henrich & Boyd assert that conformity ‘creates and maintains group boundaries and cultural differences through time’ [34, p. 231], which can increase the strength of group selection and therefore favour cultural variants that increase group fitness [68]. However, in [51], it was shown that introducing or increasing conformity can also decrease between-group differences, so conformity may not always increase the strength of group selection. Future studies that incorporate population subdivision into the present model could determine whether or not conformity could promote cultural accumulation through group selection, despite disfavouring rare variants, and whether the effects of anti-conformity would differ from those in the present study.

Another possible extension of the present model would be the incorporation of ‘selfish’ social learners that imitate others but do not pay the fitness costs of attempting innovation themselves, as in [24]. Kobayashi et al. [24] found that even low amounts of unbiased oblique transmission greatly hindered cultural accumulation relative to vertical transmission, as only the latter enabled the ‘privatization’ of beneficial cultural innovations among individuals that were likely to share the non-selfish phenotype. Another mechanism that may promote the privatization of beneficial variants is population subdivision, and it would be interesting to see whether conformity, anti-conformity, unbiased transmission and/or success bias hinder the spread of selfish individuals in such a model. Finally, additional extensions to our cumulative culture model include the incorporation of an infinite number of cultural variants, as in [26,49], or a continuous cultural trait, as in [25], as well as allowing individuals to not only improve cultural variants through innovation but also combine them to generate new variants [69].

In summary, cumulative culture may be rare in nature in part owing to the difficulty for social learning to evolve, and in part owing to factors that make cultural accumulation difficult among social learners. Factors that hinder both the invasion by social learning into a population of individual learners and the accumulation of culture among social learners include small population sizes and imperfect success-biased transmission owing to the inclusion of frequency-dependent transmission. It is possible that humans crossed the threshold from non-cumulative to cumulative culture by chance, or that this was facilitated by features of human evolution, such as large brains capable of accurately distinguishing more successful cultural variants, or tendencies to aggregate in large groups. Once culture began to accumulate, even to a small extent, its further accumulation may have occurred more readily than if it had been initially absent. Eventually, this process probably accelerated and gave rise to the vast array of languages, technologies and institutions that we see today.

Data accessibility

The data are provided in the electronic supplementary material [70].

Authors' contributions

K.K.D.: conceptualization, formal analysis, investigation, methodology, visualization, writing—original draft, writing—review and editing; Y.R.: conceptualization, formal analysis, methodology, writing—original draft, writing—review and editing; M.W.F.: conceptualization, formal analysis, funding acquisition, investigation, methodology, project administration, supervision, writing—original draft, writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration

We declare we have no competing interests.

Funding

This work was supported in part by the Morrison Institute for Population and Resource Studies at Stanford University; the Stanford Center for Computational, Evolutionary and Human Genomics; the John Templeton Foundation; and the Minerva Stiftung Center for Lab Evolution.