Journal of Theoretical Biology 352 (2014) 6–15

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Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi

The effect of cultural interaction on cumulative cultural evolution Wataru Nakahashi n School of Advanced Sciences, The Graduate University for Advanced Studies, Shonan Village, Hayama, Kanagawa 240-0193, Japan

H I G H L I G H T S








Developed a new analytical model of cultural evolution. Studied what factors would affect cultural evolutionary speed in one population. Also investigated the effect of cultural interaction on cultural evolution. Cultural interaction sometimes accelerates cultural evolution. The analytical method is confirmed by individual-based simulations.

art ic l e i nf o

a b s t r a c t

Article history: Received 28 October 2013 Received in revised form 23 January 2014 Accepted 24 February 2014 Available online 5 March 2014

Cultural transmission and cultural evolution are important for animals, especially for humans. I developed a new analytical model of cultural evolution, in which each newborn learns cultural traits from multiple individuals (exemplars) in parental generation, individually explores around learned cultural traits, judges the utility of known cultural traits, and adopts a mature cultural trait. Cultural evolutionary speed increases when individuals explore a wider range of cultural traits, accurately judge the skill level of cultural traits (strong direct bias), do not strongly conform to the population mean, increase the exploration range according to the variety of socially learned cultural traits (condition dependent exploration), and make smaller errors in social learning. Number of exemplars, population size, similarity of cultural traits between exemplars, and one-to-many transmission have little effect on cultural evolutionary speed. I also investigated how cultural interaction between two populations with different mean skill levels affects their cultural evolution. A population sometimes increases in skill level more if it encounters a less skilled population than if it does not encounter anyone. A less skilled population sometimes exceeds a more skilled population in skill level by cultural interaction between both populations. The appropriateness of this analytical method is confirmed by individual-based simulations. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Social learning Exploratory individual learning Cumulative culture Gene–culture coevolution Explosion of culture

1. Introduction The great success of our species is supported by highly developed cultures, which have evolved over time through repeated bouts of innovation, improvement, and social transmission. Recently, cultural evolution of nonhuman animals has also been reported using time series data (Allen et al., 2013; Van de Waal et al., 2013). As population genetics is necessary for investigating the evolution of genetic traits, theoretical research on cultural transmission is important for cultural evolution. While genetic traits are most often transmitted from parents to offspring, cultural traits are transmitted through social learning, which is a more complex process than heredity. Therefore, theoretical

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http://dx.doi.org/10.1016/j.jtbi.2014.02.032 0022-5193 & 2014 Elsevier Ltd. All rights reserved.

models different from population genetics are necessary for investigating cultural evolution. Since the 1970s, many researchers have studied cultural evolution from a theoretical standpoint (Cavalli-Sforza and Feldman, 1973, 1981; Boyd and Richerson, 1985). Because cultural evolution encompasses many factors, previous theoretical research on cultural evolution covers various topics. The most well researched topic is the evolution of learning strategies that could promote and support cultural evolution. Many studies have focused on the effect of environmental stability (Boyd and Richerson, 1988; Rogers, 1988; Feldman et al., 1996; Henrich and Boyd, 1998; Enquist et al., 2007; Nakahashi, 2007, 2010, 2013a, 2013b; McElreath and Strimling, 2008) and spatial structure (Aoki and Nakahashi, 2008; Aoki, 2010; Rendell et al., 2010; Kobayashi and Wakano, 2012; Nakahashi et al., 2012) on the evolution of individual learning and various types of social learning. These studies discuss how culture coevolves with learning strategies by obtaining the frequency of adaptive cultural trait (or skill level) at

W. Nakahashi / Journal of Theoretical Biology 352 (2014) 6–15

equilibrium. Many researchers have also studied how culture (skill level) would evolve without considering the coevolution of learning strategies (Henrich, 2004; Powell et al., 2009; Kobayashi and Aoki, 2012). These studies discuss what factors accelerate cultural evolutionary speed under various conditions. In this paper, I focus on the latter issue. Henrich (2004) has presented a simple analytical model to show that larger population size facilitates cumulative cultural evolution toward higher skill levels. Kobayashi and Aoki (2012) extended Henrich's model to show that a larger number of acquaintances and overlapping-generations also facilitate cultural evolution. In these studies, each newborn is assumed to learn a cultural trait from the most highly skilled individual in the population (Henrich, 2004), or among its acquaintances (Kobayashi and Aoki, 2012), but its mature skill level may disagree with that of the cultural parent because of inaccurate social learning and exploratory individual learning. The directly biased social learning rule of choosing the most skilled individual among candidates as the cultural parent is called the best-of-k (Aoki et al., 2011). Although humans tend to learn from a successful individual (Henrich and Broesch, 2011; Mesoudi, 2011), the most successful individual does not always have the highest skill level in the real world. The success of individuals is determined by many factors, including a chance factor. In many cases the best behavior is unclear. In fact, meaningless cultural traits are sometimes transmitted by hitchhiking successful individuals (Mesoudi and O'Brien, 2008; Henrich and Broesch, 2011). Moreover, humans have other social learning biases (Laland, 2004). For example, we tend to avoid adopting extreme cultural traits (Boyd and Richerson, 1985). That is, the cultural parent is not always the most skilled individual among candidates. We have to investigate whether the results of Henrich (2004) and Kobayashi and Aoki (2012) hold even when more realistic social learning rules are assumed. Another shortcoming of the previous studies on cultural evolution is that they assumed only one well-mixed population. In human history, cultural changes are often brought about by cultural interactions with other regional populations. For example, European art culture was strongly affected by ukiyo-e introduced from Japan. The direction of cultural evolution often varies in each region because of different environments, different initial conditions, different learning strategies, and random drift. Therefore, cultural difference between regions is sometimes very large and it is interesting to investigate how cultural interaction affects cultural evolution. Although the effect of structured population on cultural evolution has been theoretically studied (Lehmann et al., 2010; Kobayashi and Wakano, 2012), these studies have focused on skill level at equilibrium (or ESS). Human culture is evolving even now; therefore it may be better to investigate cultural evolutionary speed instead of the skill level of the equilibrium population. Nakahashi (2013c) investigated how cultural interaction affects cultural evolutionary speed. However, since the paper was written for anthropologists, it mainly discussed how the model results could apply to the Paleolithic human cultural evolution, and did not show detailed mathematical derivations. Moreover, although many analytical results were obtained by using mathematical approximations, whether the approximations were appropriate was not confirmed by individual-based simulations. In this paper, I have shown the details of the mathematical derivations of Nakahashi (2013c) by including some additional cases, and confirm the analytical results by individual-based simulations. Instead of discussing the application of the model to human cultural evolution, I focused on mathematically interesting points of the model results. The models may apply to cultural evolution in other animals.

7

2. Model 2.1. Basic model As in Henrich (2004) and Kobayashi and Aoki (2012), we assume that every mature individual has one cultural trait (skill) and that skill level is expressed as a real number. A newborn learns cultural traits from multiple individuals (exemplars) in the parental generation. The number of exemplars from which each newborn learns is k (k Z 2). All newborns have the same number of exemplars. Next, he/she explores around each socially learned cultural trait symmetrically with dispersion ϕ2 . That is, when he/she learns from an exemplar with skill level ze , the skill level of “explored” cultural traits distributes with mean ze and variance ϕ2 . Exploratory individual learning is the essential process for human cumulative culture (Henrich, 2004; Kobayashi and Aoki, 2012; Nakahashi, 2013a, 2013b). For example, when we learn how to broil meat from a cultural parent, we may basically follow his/her recipe but may add minor changes on broiling time and the amount of sauce to find the most delicious recipe for ourselves. Here, the mean skill level is assumed to be unchanged in this process, but the effect of its change is discussed in Section 2.6. During exploratory individual learning, he/she compares and judges the utilities of explored cultural traits according to the following criterion to adopt his/her mature cultural trait. An experimental study showed that humans add minor changes in cultural traits and check their utilities to find the best cultural trait through an individual learning process (Mesoudi and O'Brien, 2008). We consider weak directional selection (preference) so that cultural traits of higher skill level have slightly larger utility. The relative probability that he/she adopts a cultural trait of level z is assumed to be a linear function of the utility of cultural trait, wðzÞ ¼ 1 þ az

ð1Þ

where a is small. That is, the exemplar with the highest skill level is not always chosen as a cultural parent, but the exemplar with higher skill level is more likely to be chosen. Larger a implies that each newborn can accurately judge (or strongly rely on) the skill level of cultural traits (strong direct bias). Note that every function can be approximated as this linear function provided selection is weak. All individuals are assumed to have the same criterion (preference). Then, as shown in Appendix A, the expected skill level of the mature cultural trait of individual i is Eðz0i Þ ¼ zi þ

s2i þ ϕ2 ; 1=a þ zi

ð2Þ

where zi and s2i are the mean and variance of skill level in his/her exemplars, respectively, which are assumed to be uncorrelated. Assuming that exemplars are randomly chosen from the parental generation, we have Eðzi Þ ¼ z and Eðs2i Þ ¼ s2 ðk  1Þ=k, where z and s2 are the population mean and variance of skill level in the parental generation respectively. As shown in Appendix B, the expected population mean in the offspring generation becomes Eðz0 Þ ¼ z þ

s2 ðk  1Þ=k þ ϕ2

ð3Þ

1=a þ z

provided the selection is weak (as⪡1). Therefore the expected generational change of mean skill level in the population is Δz ¼ Eðz0 Þ  z ¼

s2 ðk  1Þ=k þ ϕ2 1=a þ z

;

ð4Þ

which is defined as cultural evolutionary speed. This equation implies that the population mean and variance of skill level in the parental generation affect cultural evolutionary speed.

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Let us next obtain the variance of skill level in the population at steady state. The variance increases due to exploration by each individual around the socially learned cultural traits. It may also change due to the selection of cultural traits as in Eq. (1), but we can neglect this effect because we assume weak selection. To obtain the steady state variance, we have to consider effects that decrease the variance. First, we consider the effect of a finite population size. That is, the variance decreases to ðN  1Þ=N times its previous value due to the sampling effect where N is the population size. Note that since we assume weak selection, each individual selects his/her cultural parent almost at random. Then, assuming that the variance of skill level in the parental generation is s2 , the variance in the offspring generation becomes 0

s2 ¼

N 1 2 s þ ϕ2 : N 20

^ ¼s ¼s Since s variance: 2

ð5Þ 2

at steady state, we have the steady state

s^ 2 ¼ Nϕ2 :

Nϕ2 ðk  1Þ=k þ ϕ2 : 1=a þ z

0

s2 ¼ q

s2 k

ð6Þ

ð7Þ

Therefore, cultural evolutionary speed increases when population size and the number of exemplars are large, which is analogous to Henrich (2004) and Kobayashi and Aoki (2012), although the effect of the number of exemplars was not analytically obtained in previous models. However, the sampling effect may not be the main factor that reduces the variety of cultural traits in humans when population size is large. That is, humans tend to avoid adopting extreme cultural traits, which may mainly reduce the variety of cultural traits in the human population. For example, Sherif (1936) showed that when participants estimated a movement of pinpoint of light

þ ð1  qÞðs2 þϕ2 Þ:

ð8Þ

0

^ ¼ s2 ¼ s2 at steady state, we have the steady state Since s variance 2

Substituting this into (4), we have the steady state cultural evolutionary speed: Δz ¼

in a dark room, their estimates converged to a common value through interaction of participants and the leveling-off of extreme estimates. Although this tendency is often modeled as conformist transmission (preference for common cultural traits: Henrich and Boyd (1998), Nakahashi (2007), Nakahashi et al. (2012)), in this paper I model this by introducing the blending effect (preference for the mean cultural trait) proposed by Boyd and Richerson (1985). For simplicity, hereafter we neglect the sampling effect because it is sufficiently weak provided that the population size is large and the blending effect is not weak. Each individual sometimes (with probability q) blends socially learned cultural traits to adopt the weighted sample mean. Assuming that population size is sufficiently large for neglecting the sampling effect, as shown in Appendix C, the recursion of the variance of skill level in the population is

s^ 2 ¼

k 1q 2 ϕ : k1 q

ð9Þ

Substituting this into (4), we have the steady state cultural evolutionary speed Δz ¼

1 ϕ2 : 1=a þ z q

ð10Þ

Therefore, cultural evolutionary speed increases when the exploration range (ϕ2 ) is large, the strength of selection of cultural traits (a) is large, and the probability of blending (q) is small, which is illustrated in Figs. 1 and 2. In other words, when individuals have higher creativity to explore a wider range of cultural traits, accurately judge the skill level of cultural traits (strong direct bias), and do not strongly conform to the population mean, culture evolves faster. Hence, high creativity supported by individual learning ability can accelerate cultural evolution, which

Fig. 1. The effect of parameters on cultural evolutionary speed, Δz. Each figure shows the effect of (A) exploration range, ϕ2 , (B) strength of selection of cultural traits, a, (C) probability of blending, q, and (D) strength of condition dependent exploration, r. Parameters are z ¼ 100 and ϕ2 ¼ 0:1, a ¼ 0:01, q ¼ 0:5, r ¼ 0:0 when each parameter is not a variable.

W. Nakahashi / Journal of Theoretical Biology 352 (2014) 6–15

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and the recursion of the variance of skill level in the population is 0

s2 ¼ q

1 þ Rðk  1Þ 2 s þ ð1 qÞðs2 þϕ2 Þ: k

ð12Þ

Thus, the steady state variance is

s^ 2 ¼

1 k 1 q 2 ϕ ; 1  R k 1 q

ð13Þ

and the steady state cultural evolutionary speed is Δz ¼

1 ϕ2 : 1=a þ z q

ð14Þ

That is, relatedness does not affect the speed. This can be intuitively explained as follows. When each individual tends to sample exemplars with similar cultural traits, the “effective” number of exemplars decreases. Since the number of exemplars does not affect cultural evolutionary speed, relatedness does not affect it either. Fig. 2. Generational changes of skill level under various conditions. Parameters are ϕ2 ¼ 0:1, a ¼ 0:01, q ¼ 0:5, r ¼ 0:0, and zð0Þ ¼ 100 under the baseline condition, and each curve shows the effect of an increase of each parameter.

is analogous to Aoki et al. (2011) and Kobayashi and Aoki (2012). However, the number of exemplars (k) does not affect cultural evolution, which is in contrast to Kobayashi and Aoki (2012). Of course, the result that the number of exemplars does not affect cultural evolution is dependent on the assumption of blending. However, even if we assume that the decrease of variance in every generation is a constant fraction (i.e., independent of k), the steady state cultural evolutionary speed becomes analogous to (7). Then, the influence of k is proportional to ðk  1Þ=k, so that the difference of k has only a small effect on cultural evolutionary speed provided k is not small.

2.2. Vertical transmission This model is useful for evaluating other factors that may affect human cultural evolution. First, let us consider vertical transmission. In this case, if cultural traits are transmitted from both parents (k ¼ 2), the results do not change (but note that when cultural traits with higher utility tend to increase the fitness (number of offspring) of their carriers, vertical transmission may accelerate cultural evolutionary speed). However, if cultural traits are transmitted from one parent (k ¼ 1), Eq. (4) provides that 2 Δz ¼ 1=aϕ þ z, so that the population variance has no effect on cultural evolutionary speed, and the speed is slower than for oblique transmission with k Z 2. That is, when a cultural trait is transmitted from father to son and from mother to daughter, culture does not evolve quickly. Learning from multiple exemplars is important for fast cultural evolution.

2.3. Relatedness of cultural traits Second, let us consider the effect of “relatedness”. Although the above basic model assumes that each newborn randomly samples exemplars from the parental generation, he/she may sample exemplars with similar cultural traits. For example, if the population is subdivided, each newborn tends to sample exemplars with similar cultural traits. Let the relatedness (correlation) of skill level between two exemplars, j and h, be R; i.e., R ¼ Corrðzj ; zh Þ (Boyd and Richerson, 1985). Then, as shown in Appendix D, the expected variance of skill level in exemplars is Eðs2i Þ ¼

k1 ð1  RÞs2 ; k

ð11Þ

2.4. One-to-many (teacher) transmission Third, let us consider the effect of one-to-many (teacher) transmission. Although the basic model assumes that every individual in the population can be chosen as an exemplar by the next generation, only certain individuals may in fact be an exemplar (teacher) in human society. Note that in the basic model also, a fraction ð1  k=NÞN  e  k of the parental population does not function as exemplars because of the random sampling of exemplars, but here we consider a more limited teacher population. Let the number of teachers be T (rigorously speaking they are candidates to be teachers and a fraction ð1  k=TÞN  e  kN=T of them are not sampled). Provided the mean skill level of teachers is the same as the population mean, this situation is the same as sampling from a teacher population with variance ðT  1=TÞs2 . Then, the expected variance of skill level in exemplars is Eðs2i Þ ¼

k1 T 1 2 s: k T

ð15Þ

Therefore the recursion of the variance of skill level in the population is
 0 T  1 s2 T 1 2 þ ð1  qÞ ð16Þ s2 ¼ q s þ ϕ2 : T k T Thus, the steady state variance is

s^ 2 ¼

Tkð1  qÞ ϕ2 : k þ ðT  1Þðk  1Þq

ð17Þ

It follows that the steady state cultural evolutionary speed is
 2 1 ð1 qÞk ϕ 1 : ð18Þ Δz ¼ 1=a þ z k þ ðT  1Þðk  1Þq q That is, speed slightly decreases as the number of teachers decreases, but this effect is almost negligible when the number of teachers is not small. So, one-to-many transmission does not essentially affect the speed of cultural evolution, a result that is in contrast to Lycett and Gowlett (2008) but consistent with Aoki et al. (2011). However, this does not mean that the selection of teachers does not affect speed. If “specialists” tend to be selected as teachers (Pigeot, 1990), they may have higher skill level, so cultural evolution may speed up. In the basic model, direct bias of choosing a cultural parent with higher skill level (larger a) accelerates cultural evolution. In other words, the selection of teachers accelerates cultural evolution not by the effect of one-tomany transmission (limited number of teachers) but by that of direct bias.

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2.5. Condition dependent exploration

3. Interaction of two populations

Fourth, let us consider the effect of condition dependent exploration. Although the basic model assumes that the exploration range is constant under every condition, humans tend to explore more widely when the skill levels of exemplars are distributed more widely. That is, we may not try many different behaviors (strongly rely on social learning) if exemplars have similar behaviors (e.g., the effect of unanimity, Asch, 1955), but may try many if they have various behaviors. For example, a fisherman might only fish at one spot if other anglers are concentrated there, whereas he/she might explore many spots to find the best one if other anglers are widespread. Let the exploration variance be rs2i þ ϕ2 (instead of the constant ϕ2 in the basic model). Then, the recursion of the variance of skill level in the population is
 0 s2 k1 2 ð19Þ s2 ¼ q þ ð1  qÞ s2 þ r s þ ϕ2 : k k

The above models address the cultural evolution of one population. Here, we are interested in cultural interactions of two (or more) populations that may have different skill levels and learning strategies. Since the model contains no assumptions on the distribution of skill level (except the weak selection assumption), we can consider a situation where in a population encounters another population to interact culturally. Let us consider a situation where population M with mean zM and variance s2M comes into contact with population A with mean zA and variance s2A . If a newborn of population M samples an exemplar from population A with probability pM (strength of interaction), as shown in Appendix E, population M can be considered to have an “exemplar population” with mean ð1  pM ÞzM þ pM zA and variance ð1  pM Þs2M þ pM s2A þ pM ð1  pM ÞðzM  zA Þ2 . Similar considerations apply to population A. By setting parameters as in the basic model with subscripts M and A for populations M and A, respectively, we can trace the generational changes of mean and variance of skill levels in populations M and A as follows:

When r 4 q=ð1  qÞ, the variance and cultural evolutionary speed increase to infinity. When r o q=ð1  qÞ, the steady state variance is

s^ 2 ¼

k 1q ϕ2 : k  1 q  rð1 qÞ

þ

ð20Þ

1 ϕ2 : 1=a þ z q  rð1  qÞ

½ð1  pM Þs2M þ pM s2A þ pM ð1  pM ÞðzM  zA Þ2 ðkM  1Þ=kM þ ϕ2M 1=aM þ ð1  pM ÞzM þ pM zA ð25aÞ

2 2 s20 M ¼ ðqM =kM þ 1  qM Þ½ð1  pM ÞsM þ pM sA

The steady state cultural evolutionary speed is Δz ¼

z0M ¼ ð1  pM ÞzM þ pM zA

ð21Þ

That is, when the exploration range is strongly affected by the variance of skill level in exemplars, culture evolves fast, which is illustrated in Figs. 1D and 2. This is because condition dependent exploration increases the variance of skill level in the population, which accelerates cultural evolutionary speed.

þpM ð1  pM ÞðzM  zA Þ2  þ ð1  qM Þϕ2M z0A ¼ ð1  pA ÞzA þ pA zM þ

½ð1  pA Þs2A þ pA s2M þ pA ð1  pA ÞðzM  zA Þ2 ðkA  1Þ=kA þ ϕ2A 1=aA þ ð1  pA ÞzA þpA zM ð25cÞ

2 2 s20 A ¼ ðqA =kA þ 1 qA Þ½ð1  pA ÞsA þ pA sM

þpA ð1  pA ÞðzM  zA Þ2  þ ð1  qA Þϕ2A 2.6. Error of social learning Fifth, let us consider the error of social learning. Henrich (2004) and Kobayashi and Aoki (2012) assumed that the mode of mature skill level is smaller than that of the cultural parent because of the error of social learning. Including the error of social learning, ε, into the basic model, we have Eðz0 Þ ¼ z  ε þ

s2 ðk  1Þ=k þ ϕ2 1=a þ z

:

ð22Þ

Since the recursion of the variance is the same as the basic model, cultural evolutionary speed at steady state is Δz ¼

1 ϕ2  ε: 1=a þ z q

ð23Þ

Skill level then converges to an equilibrium, ϕ2 1 z^ ¼  : qε a

ð24Þ

That is, skill level evolves to a higher level when social learning ability is higher (ε is smaller). Moreover, high improvement ability (large ϕ2 and a) brings about high skill level at equilibrium, which is analogous to Nakahashi (2013a, 2013b). It may be more realistic that error increases when individuals try to explore more widely. If we assume ε ¼ cϕ2 , skill level converges to an equilibrium, z^ ¼ ð1=qcÞ  ð1=aÞ. As in above case, higher social learning ability (small c) brings about higher skill level, although the exploration range (ϕ2 ) has no effect in this case.

ð25bÞ

ð25dÞ

When does cultural interaction with another population accelerate the cultural evolution of a focal population? When does a less skilled population exceed a more skilled population in skill level by cultural interaction between both populations? 3.1. Explosion of culture First, let us consider the condition that cultural interaction with another population (population A) accelerates the cultural evolution of a focal population (population M). In order to obtain the effect of cultural interaction analytically, we assume that cultural interaction occurs only once (one generation) and selection is sufficiently weak. We compare the cultural evolutions of two populations, one that interacts with an outside population and one that does not, and obtain the condition that the mean skill level of the interacting population ultimately exceeds that of the non-interacting population. Hereafter, we delete subscript M from all parameters of population M. As shown in Appendix F, mean skill level of the interacting population ultimately exceeds that of the non-interacting population when a½s2A  s2 þ ð1  pÞg 2  4 qg:

ð26Þ

where g is the initial difference of skill levels between the two ð0Þ populations (g ¼ zð0Þ M  zA ). That is, even when the outside population has lower mean skill level than that of the focal population (g 4 0), the mean skill level of the interacting population can ultimately exceed that of the non-interacting population when the variance of skill level in the outside population (s2A ) is large. Similarly, even when the outside population has higher mean skill

W. Nakahashi / Journal of Theoretical Biology 352 (2014) 6–15

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level (g o 0), the mean skill level of the interacting population may ultimately be exceeded by that of the non-interacting population when s2A is small. We call the phenomenon that the mean skill level of a population that interacts with a less skilled population ultimately exceeds that of a non-interacting population as “explosion of culture.” The explosion of culture is more likely to occur when the variance of skill level in the outside population is large. Let us assume that the variance of skill level in the outside population is the same as that in the focal population (s2A ¼ s2 ). Then, when the outside population has higher mean skill level (g o 0), condition (26) is always satisfied. That is, cultural interaction with a more skilled population always benefits the focal population. On the other hand, cultural interaction with a less skilled population does not always adversely affect the focal population. The explosion of culture can occur. When s2A ¼ s2 , the condition for the explosion of culture to occur becomes agð1  pÞ 4 q:

ð27Þ

Since the condition ag⪡1 is necessary for the weak selection assumption, condition (27) is never satisfied under the weak selection assumption. However, as shown in Appendix G, when we include condition dependent exploration, the condition for the explosion of culture to occur becomes agð1  pÞð1 þrÞ 4 q  rð1  qÞ

ð28Þ

provided r o q=ð1  qÞ. This condition can be satisfied under the weak selection assumption when r is large. The explosion of culture is more likely to occur when the effect of the variance of skill level in exemplars on the exploration range (r) is large, the difference in skill levels between two populations (g) is large, the strength of interaction (p) is small, the tendency of blending (q) is small, and the selection (a) is strong.

simulation, we assume that when an exemplar has skill level ze , the skill level of explored cultural traits distributes as ( pffiffiffi pffiffiffi pffiffiffi 1=2 3ϕ for ze  3ϕ rz r ze þ 3ϕ tðz; ze Þ ¼ : ð30Þ 0 otherwise That is, we consider a uniform distribution with mean ze and variance ϕ2 . Then, since wðzÞ ¼ 1 þaz, we have E½tðz; ze ÞwðzÞ ¼ wðze Þ:

3.2. Surpassing the master Next, let us consider the condition that a less skilled population (population A) exceeds a more skilled population (population M) in skill level by cultural interaction between both populations. That is, when does the scholar surpass the master? In order to obtain the condition analytically, we assume that cultural interaction occurs only once (one generation), selection is sufficiently weak, and learning strategies (abilities) are the same between both populations. The strengths of interaction, pM and pA , and the ð0Þ initial skill levels of both populations, zð0Þ M and zA , only differ. We delete subscripts M and A from all parameters that are not different between both populations. Then, as shown in Appendix H, provided r o q=ð1 qÞ and pM þ pA o1, a less skilled population ultimately exceeds a more skilled population in skill level when agð1 þ rÞðpA  pM Þ 4q rð1  qÞ

Fig. 3. Comparison of the mean skill level values obtained by analytical recursions (broken lines) and by individual-based simulations (solid lines). Cultural interaction with the more and less skilled populations occurs between generations 101 and 110. Parameters are aM ¼ aA ¼ 0:1, ϕ2M ¼ ϕ2A ¼ 0:1, qM ¼ qA ¼ 0:6, kM ¼ kA ¼ 10, ð0Þ r M ¼ r A ¼ 1:2, N M ¼ N A ¼ 1000, pM ¼ 0:01, pA ¼ 0:1, zð0Þ M ¼ 100, and zA ¼ 70.

ð29Þ

where g is the initial difference of skill levels between the two ð0Þ  zð0Þ populations (g ¼ zM A ). That is, when the less skilled population is more eager to learn the outside population's culture than the more skilled population (pA 4 pM ), a “surpassing the master” may occur. Then, the surpassing the master is more likely to occur when the effect of the variance of skill level in exemplars on the exploration range (r) is large, the difference in skill levels between two populations (g) is large, the tendency of blending (q) is small, and the selection (a) is strong.

4. Individual-based simulation Since above analytical results are obtained by using mathematical approximations, it is better to confirm whether the approximations are appropriate by individual-based simulations. To run the simulation, we have to set the function of exploration. In this

ð31Þ

That is, the probability that an exemplar with ze is chosen as a cultural parent is proportional to wðze Þ. We consider the cultural interaction of two populations that have different initial skill levels. We set the initial distribution of skill level to be a uniform distribution with steady state variance. After many generations of independent cultural evolution, two populations interact with each other in some generations, and again both evolve independently. In this study, the mean skill level values obtained by recursions (25) and individual-based simulations were compared (Fig. 3). Before the interaction, the evolutionary speed was slow and both values showed good correspondence, but after the interaction, disagreement of both values became large. That is, the recursions were an appropriate approximation when selection was weak, but sometimes inappropriate when selection was strong. Provided strong condition dependent exploration was included, the disagreement was not large and the overall tendency almost held. Fig. 3 also shows that the explosion of culture and surpassing the master actually occurred in individual-based simulations. The mean skill level of the more skilled population was accelerated by cultural interaction with the less skilled population, and that of the less skilled population ultimately exceeded than that of the more skilled population.

5. Discussion In this paper, I have studied the factors that affect cultural evolution. I have shown that cultural evolutionary speed increases when individuals have higher creativity to explore a wider range of cultural traits, accurately judge the skill level of cultural traits (strong direct bias), do not strongly conform to the population mean, increase the exploration range when the variance of socially

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W. Nakahashi / Journal of Theoretical Biology 352 (2014) 6–15

learned cultural traits is large (condition dependent exploration), and make smaller errors in social learning, provided individuals learn cultural traits from multiple exemplars. The number of exemplars, population size, relatedness (similarity) of cultural traits between exemplars, and one-to-many (teacher) transmission have little effect on cultural evolutionary speed. In other words, cultural evolutionary speed is mainly accelerated by high learning abilities provided the difference among the utilities of cultural traits is small (weak selection) and population size is not small. The importance of learning abilities on cultural evolution can be explained intuitively as follows. Exploratory individual learning is similar to mutation in population genetics, so that wider exploration range accelerates cultural evolution, as higher mutation rate accelerates genetic evolution (Fisher, 1958). Condition dependent exploration also widens the range and accelerates cultural evolution. The process of judgment and choice of cultural traits are similar to the process of natural selection in population genetics, so that stronger direct bias for cultural traits accelerates cultural evolution, as stronger selection accelerates genetic evolution (Fisher, 1958). Obviously, cultural evolution speeds down when skill level of cultural traits deteriorates in social learning process. Some results are different from claims made in previous studies that population size and the number of acquaintances strongly affect cultural evolutionary speed (Henrich, 2004; Kobayashi and Aoki, 2012). This is because, in those models, individuals learned exclusively from an exemplar with the highest skill level available to them. Therefore, selection is extremely strong, which is different from the assumption used in my model. In a situation where the skill level of behaviors (cultural traits) is clear and everyone can recognize best behavior, their model is appropriate. However, in the real world, the skill level of behaviors is often unclear and it is difficult to identify the best behavior. In such a situation, we may avoid learning extreme cultural traits and tend to depend on the sample mean. My model is realistic in such a situation. Therefore, we can consider that population size and the number of acquaintances affect cultural evolutionary speed only when the skill level of cultural traits is clear or population size is very small. This result suggests that evolutionary speed of practical technologies and that of impractical items have different properties. Practical technologies may evolve fast when population size and number of exemplars are large because their utility is clear, while the evolutionary speed of impractical items may be constant because their utility is unclear. In fact, if we compare modern cultures with the Middle Ages, the evolutionary speed of practical technologies such as vehicles and calculators is accelerating due to population increase, but that of impractical items such as arts and fashions are relatively constant. However, population size may also have other consequences that are not studied in this paper. Lehmann et al. (2011) considered the number of independent cultural traits carried by individuals and that by populations and showed that both numbers increase as population size increases. Nakahashi and Feldman (2014) showed that division of labor likely evolves when group size is large, which may also increase the variety of cultural traits in the population. Therefore, if the variety of cultural traits has positive effect on the evolution of each cultural trait, larger population may cause faster cultural evolution. For example, we can suppose that the exploration range of a cultural trait is stimulated by other kinds of cultural traits, which is similar to condition dependent exploration in my model, or a combination of different kinds of cultural traits produces a new cultural trait. Since my model does not consider these situations, new models of cultural evolution may be necessary to study how the variety of cultural traits affects cultural evolution.

My model also shows that conformity (blending) has negative effect on cultural evolution. This result is analogous to results in previous studies. Henrich (2001) considered the diffusion of innovations and showed that conformist transmission produces a “long tail” S-shaped adoption curve. That is, conformity inhibits the diffusion of innovations when the frequency of adopters is low. Lehmann et al. (2011) showed that conformist transmission decreases the number of independent cultural traits carried by individuals and that by populations. Aoki et al. (2011) showed that conformist (many-to-one) transmission decelerates cultural evolution in the cultural Moran model. Although these previous studies did not consider cumulative cultural evolution, the negative effect of conformity on cultural evolution seems robust. This is because conformity homogenizes the culture of population, which prevents innovations. I have confirmed the appropriateness of approximation in the mathematical analysis by using individual-based simulations. I have shown that generational changes of mean skill level obtained by the mathematical approximation show good correspondence with that obtained by numerical simulations when selection is weak. That is, our approximation in the analytical method is sufficiently appropriate under the weak selection condition. Next, I have considered the situation where two populations encounter to interact culturally. Previous research on cultural evolution did not include cultural interactions (Henrich, 2004; Kobayashi and Aoki, 2012). The effect of cultural interaction on cultural evolution was first investigated in this model. I have shown that an “explosion of culture” may occur under certain conditions. That is, when a more and a less skilled population interact culturally, it sometimes accelerates the cultural evolution of the more skilled population. I have analytically shown that the explosion of culture sometimes occurs when the variance of skill level in the outside population is large or the exploration range is affected by the variance of skill level in exemplars. The explosion of culture may be counterintuitive because it implies that cultural interaction with a less skilled population can accelerate cultural evolution of a more skilled population. However, if you consider the situation that the less skilled population has a larger variance of skill level (so that the highest skill of the less skilled population is higher than that of the more skilled population), you may easily imagine that the interaction can trigger the explosion of culture. In the present model, the skill level of population is defined as the mean skill level, which may disturb our intuition. Humans may reexamine their own behaviors when observing unfamiliar behaviors, which is modeled as condition dependent exploration in this paper. Cultural interaction with a less skilled population entails the increase of variance and the decrease of mean, and the former effect sometimes exceeds the latter effect. In other words, when a newborn observes individuals with lower skill levels, they sometimes function as a “rotten apple” to decrease skill level, but sometimes function as a “negative exemplar” of how not to behave, so that cultural interaction with a less skilled population can both decelerate and accelerate cultural evolution. I have also shown that a “surpassing the master” may occur under certain conditions. That is, a less skilled population sometimes exceeds a more skilled population in skill level by cultural interaction between both populations. I have analytically shown that the surpassing the master can occur when the less skilled population is more eager to learn the outside population's culture than the more skilled population. Individual-based simulations showed that the explosion of culture and surpassing the master could actually occur (Fig. 3). That cultural interaction could provoke the explosion of culture and surpassing the master may provide many new perspectives about human cultural evolution. The surpassing the master may

W. Nakahashi / Journal of Theoretical Biology 352 (2014) 6–15

13

s2 ðk  1Þ=k þ ϕ2

have played important roles in human cultural evolution. For example, the position of the most advanced regional population was often replaced by another regional population in human history. As shown in the model, if a less advanced population eagerly imports cultures of a more advanced population but the more advanced population does not, the less advanced population may eventually surpass the more advanced population. Advanced populations might tend to disregard less advanced populations, which would cause the surpassing. It is important for every population to interact with different populations and to experience culture shock, which may broaden the perspective of individuals and induce new cultural innovations. Although the explosion of culture may also have played important roles in human cultural evolution, it may be difficult to detect this effect because non-interacting populations were generally nonexistent provided cultural interaction occurred. One possible example of the explosion of culture has been pointed out by Nakahashi (2013c). In the Upper Paleolithic, certain artistic behaviors of modern humans emerged first in Europe rather than Africa, which is called the artistic explosion (reviewed in Balter (2012)). Only European modern humans encountered a less artistic population, Neanderthals.

Since each individual samples k exemplars, the skill level of blenders distributes with mean z0 and variance s2 =k. That is, the weighted sample mean is z0 and the variance is approximately the error variance because selection is weak. Note that we assume a symmetrical distribution of exploration so that exploration does not affect the variance of skill level in blenders. On the other hand, the skill level of non-blenders distributes with mean z0 and variance s2 þ ϕ2 . Since a fraction q of population is blenders, the recursion of the variance of skill level in the population is

Acknowledgments

Eðs2i Þ ¼ E

I thank Dr. Shiro Horiuchi and Dr. Yutaka Kobayashi for their fruitful suggestion. This research was supported in part by the MEXT Grant no. 22101004 “Replacement of Neanderthals by Modern Humans: Testing Evolutionary Models of Learning”.

¼ zþ

Appendix C

0

s2 ¼ q

s2 k

þ ð1  qÞðs2 þ ϕ2 Þ:

ðC:1Þ

Appendix D

¼E

!

k

∑ ðzi U j  zi Þ2 =k

j¼1 k

! 2

∑ ðzi U j zÞ =k  E½ðzi zÞ2 

j¼1

¼ s2  ¼

Appendix A

ðB:1Þ

1=a þz

1 þ Rðk  1Þ 2 s k

k1 ð1  RÞs2 k

ðD:1Þ

because The distribution of skill level of explored cultural traits of individual i is written as xi ðzÞ. Then, the expected skill level of mature cultural trait of individual i is R zxi ðzÞwðzÞdz Eðz0i Þ ¼ R : ðA:1Þ xi ðzÞwðzÞdz When the mean and variance of skill level in individual i's exemplars are zi and s2i , respectively, xi ðzÞ has mean zi and variance s2i þ ϕ2 . Since wðzÞ ¼ 1 þ az, we have R zxi ðzÞwðzÞdz Eðz0i Þ ¼ R xi ðzÞwðzÞdz R ½zxi ðzÞ þ az2 xi ðzÞdz ¼ R ½xi ðzÞ þ azxi ðzÞdz R R zxi ðzÞdz þ a z2 xi ðzÞdz R ¼ R xi ðzÞdz þa zxi ðzÞdz ¼

zi þ aðs2i þ ϕ2 þ z2i Þ 1 þazi

¼ zi þ

s2i þ ϕ2

1=a þ zi

s2 k 1

Rs2 þ k k 1 þRðk  1Þ 2 ¼ s k

E½ðzi  zÞ2  ¼

ðD:2Þ

Then, the variance of skill level in blenders is approximately the error variance, ð1 þ Rðk  1ÞÞs2 =k, so that the recursion 0 of the variance of skill level in the population is s2 ¼ qð1 þ Rðk 1ÞÞs2 =k þ ð1  qÞðs2 þ ϕ2 Þ. Appendix E The exemplar population consists of, on average, a proportion 1  pM of population M individuals and pM of population A individuals. Therefore, the mean skill level of exemplar population is Eexemplar ¼ ð1  pM ÞzM þpM zA :

ðE:1Þ

Then, the variance of skill level in exemplar population is ðA:2Þ

Appendix B When selection is weak (as⪡1), we can assume 1=a⪢zi  z so that ! s2i þ ϕ2 0 0 Eðz Þ ¼ EðEðzi ÞÞ ¼ Eðzi Þ þ E 1=a þ zi ! 2 2 si þ ϕ 1 Eðs2 þ ϕ2 Þ  zþ ¼ z þE 1=a þ z i 1=a þz þ ðzi  zÞ

V exemplar ¼ ð1  pM Þ½ðzM  Eexemplar Þ2 þ s2M  þ pM ½ðzA  Eexemplar Þ2 þ s2A  ¼ ð1  pM Þs2M þ pM s2A þ pM ð1  pM ÞðzM  zA Þ2 :

ðE:2Þ

Appendix F We assume that cultural interaction occurs only once (one generation) and selection is sufficiently weak. Also, we delete subscript M from all parameters of population M (focal population). Then, writing the initial mean skill level of population A as z  g, population M has exemplar population with mean z  pg and variance ð1  pÞs2 þ ps2A þ pð1 pÞg 2 . Under what conditions can skill level of this population ultimately exceed that of a non-

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W. Nakahashi / Journal of Theoretical Biology 352 (2014) 6–15

interacting population that has exemplar population mean z and variance s2 ? Assume that 1=a1þ z  a always holds (weak selection). Then, since the difference in variance values between the interacting and non-interacting populations is p½s2A  s2 þ ð1  pÞg 2 , from (4) and (25a), the difference in generational increases of skill level by selection between the two populations is

variance s2 þpM ð1  pM Þg 2 , and population A has that with mean ð0Þ zM  ð1  pA Þg and variance s2 þ pA ð1  pA Þg 2 . That is, the initial difference in mean is gð1  pM  pA Þ and that in variance is ½pA ð1 pA Þ  pM ð1 pM Þg 2 . Then, solving as in Appendix F, provided r o q=ð1  qÞ, the mean skill level of a less skilled population (population A) ultimately exceeds that of a more skilled population (population M) when

p½s2A  s2 þ ð1  pÞg 2 ðk 1Þ=k k1 :  ap½s2A  s2 þ ð1  pÞg 2  U k 1=a þ z

agð1 þ rÞ½pA ð1  pA Þ  pM ð1  pM Þ 4 ½q  rð1  qÞð1  pM  pA Þ:

ðF:1Þ

From (8) and (25b), in the next generation, the difference in variances between the two populations decreases to  q 1  q þ p½s2A  s2 þ ð1  pÞg 2 : ðF:2Þ k Finally, the difference in total increases of skill level by selection between the two populations is k1 1  qi ap½s2A  s2 þ ð1  pÞg 2  U ∑ 1  qþ : ap½s2A  s2 þ ð1  pÞg 2  U ¼ k k q i¼0 ðF:3Þ This exceeds the difference in skill level at the first generation, pg, when a½s2A  s2 þ ð1  pÞg 2  4 qg:

ðF:4Þ

Appendix G When we include condition dependent exploration, (25) becomes z0M ¼ ð1  pM ÞzM þpM zA þ

½ð1  pM Þs2M þ pM s2A þ pM ð1  pM ÞðzM  zA Þ2 ð1 þ r M ÞðkM  1Þ=kM þ ϕ2M 1=aM þ ð1  pM ÞzM þ pM zA

ðG:1aÞ 2 2 s20 M ¼ ½qM =kM þ ð1  qM Þð1 þ r M ðkM  1Þ=kM Þ½ð1  pM ÞsM þ pM sA

þ pM ð1  pM ÞðzM zA Þ2  þð1 qM Þϕ2M

ðG:1bÞ

z0A ¼ ð1  pA ÞzA þ pA zM þ

½ð1  pA Þs2A þ pA s2M þ pA ð1  pA ÞðzM  zA Þ2 ð1 þ r A ÞðkA  1Þ=kA þ ϕ2A 1=aA þð1 pA ÞzA þ pA zM ðG:1cÞ

2 2 s20 A ¼ ½qA =kA þ ð1  qA Þð1 þ r A ðkA  1Þ=kA Þ½ð1  pA ÞsA þ pA sM

þ pA ð1  pA ÞðzM  zA Þ2  þ ð1  qA Þϕ2A

ðG:1dÞ

Then, solving as in Appendix F, provided r oq=ð1  qÞ, the mean skill level of an interacting population ultimately exceeds that of a non-interacting population when agð1  pÞð1 þ rÞ 4 q  rð1  qÞ:

ðG:2Þ

That is, explosion of culture is more likely to occur when the exploration range is strongly affected by the variance of skill level in exemplars.

Appendix H We assume that cultural interaction occurs only once (one generation), selection is sufficiently weak, and learning strategies (abilities) are the same between both populations. The strengths of interaction, pM and pA , and the initial skill levels of both populað0Þ tions, zð0Þ M and zA only differ. We delete subscripts M and A from all parameters that are not different between both populations. Then, writing the initial mean skill level of population A as zð0Þ M g, population M has exemplar population with mean zð0Þ M pM g and

Since pA ð1  pA Þ pM ð1  pM Þ ¼ ðpA  pM Þð1 pM  pA Þ, pM þ pA o 1, condition (H.1) becomes agð1 þ rÞðpA  pM Þ 4 q  rð1  qÞ:

ðH:1Þ

provided ðH:2Þ

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