Impacts of drift and population bottlenecks on the cultural transmission of a neutral continuous trait: an agent based model

Accepted Manuscript Impacts of Drift and Population Bottlenecks on the Cultural Transmission of a Neutral Continuous Trait: An Agent Based Model Adam N. Rorabaugh PII:

S0305-4403(14)00194-0

DOI:

10.1016/j.jas.2014.05.016

Reference:

YJASC 4074

To appear in:

Journal of Archaeological Science

Received Date: 21 January 2014 Revised Date:

17 April 2014

Accepted Date: 19 May 2014

Please cite this article as: Rorabaugh, A.N., Impacts of Drift and Population Bottlenecks on the Cultural Transmission of a Neutral Continuous Trait: An Agent Based Model, Journal of Archaeological Science (2014), doi: 10.1016/j.jas.2014.05.016. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Impacts of Drift and Population Bottlenecks on the Cultural Transmission of a Neutral Continuous Trait: An Agent Based Model

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Author: Adam N. Rorabaugh Department of Anthropology, Washington State University PO Box 644910, Washington State University, Pullman WA 99164-4910 [email protected]

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Abstract: Although there is increasing interest in the connections between copying error and the generation of variation of continuous cultural traits, the complex interplay between forces of evolutionary drift and copying error in continuous traits remains under-examined. Here, an agent based model is provided that examines the effects of drift and population bottlenecks on the production of variation in a selectively neutral metric attribute under vertical, unbiased, and prestige biased modes of transmission. The provided model has implications for inferring demographic change or restricted forms of knowledge in the production of technologies.

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Key Words: Cultural transmission, cultural drift, prestige bias, continuous cultural attributes, population bottlenecks, agent-based modeling

ACCEPTED MANUSCRIPT 1. Introduction Understanding factors that influence the reproduction of technological traditions is a fundamental issue in archaeology. Renewed interest in evolutionary frameworks has provided key insights into the the processes that guide and regulate how culture is transmitted (e.g.

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Bentley et al 2004, Bettinger and Eerkens 1999, Boyd and Richerson 1985, Eerkens and Lipo 2007, Gandon et al 2014, Hamilton and Buchanan 2009, Henrich and Henrich 2007, Henrich and Gil-White 2001, Kandler and Shennan 2013, Kempe et al 2012, Kohler et al 2004, Lipo

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2001, Lipo and Madsen 2001, Neiman 1995, Shennan 2002, Shennan and Wilkinson 2001). Modeling cultural transmission requires archaeologists to encompass all of the processes

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which can occur during social learning and inheritance. This involves examining the diversity of copying rules and the processes by which the information being copied is transformed. Such processes include processes and the perceptual and mechanical errors that arise in making copies. Here, I focus on furthering our understanding of copying errors in material

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culture, which is necessary in order to examine how other evolutionary processes such as selection from environmental or social constraints that sort or winnow cultural variation (Eerkens and Lipo 2005: 317).

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This study expands on previous research examining the generation of variation due to copying errors during social learning (e.g. Bettinger and Eerkens 1997, Eerkens and Bettinger

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2001, Eerkens and Lipo 2005, Eerkens 2000, Hamilton and Buchanan 2009, Kempe et al 2012). Previous models examining visual copying error have ignored the effects of drift, also known as sampling error in finite populations. By ignoring drift, the impact of fluctuating population size and in particular the effect of population bottlenecks, sharp reductions in population size, on the generation of variation due to copying error has not been investigated by archaeologists. Population bottlenecks occur in situations where population sizes decrease, and can drastically reduce the cultural variation of a population (e.g. Premo 2012, Premo and 1

ACCEPTED MANUSCRIPT Kuhn 2010). Population bottlenecks can be the result of many factors ranging including, but not limited to, group fissioning, migration of a subset of a population to a new region, epidemic disease, and warfare. Population bottlenecks and expansion can be empirically observed with archaeo-demographic estimates, and there is a growing body of studies on

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population reconstruction (e.g. Bocquet-Appel 2011, Bosma et al 2013, Hoppa and Vaupel 2002, Kohler and Glaude 2008).

An agent based model is employed to examine the effect of finite populations that

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change through time on the production and maintenance of variance in artificial societies. The model is parametrized with several idealized population curves to provide expected

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values for metric variation, measured using the coefficient of variation, under different mechanisms of cultural transmission including vertical, unbiased, and prestige biased transmission (Boyd and Richerson 1985). Like all models this study is a deliberate simplification of reality, but illuminates issues of equifinality that can arise in the

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archaeological record from how variation is patterned by behavioral processes such as biased learning and changes in population size.

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1.1 Cultural Transmission and the Material Record

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According to models of gene-culture coevolution (dual inheritance), socially learned behaviors can affect selective pressures and drive genetic evolution in populations (Boyd and Richerson 1985). Culture and genes are subject to selective pressures (although they can be quite different) but are inherited through unique pathways. Unlike genetic inheritance, culture may be acquired from individuals other than biological parents. A key contribution of the gene-culture coevolutionary approach is the attempt to characterize the different pathways in which culture is transmitted.

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ACCEPTED MANUSCRIPT Each pathway of cultural transmission has empirical repercussions for the spatial and temporal patterning of socially learned human behaviors (Cavalli-Sforza and Feldman 1981, Boyd and Richerson 1985). Our efforts to understand specific learning contexts in prehistory can be confounded by equifinality. The principle of equifinality is that any given final state,

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result, or outcome may be reached through more than one process, set of initial conditions, or starting points (Premo 2010: 31). In other words, the types of data we use to infer forms of cultural transmission may be patterned in similar ways by greatly different social and non-

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social processes.

To elaborate, strong functional constraints, learning from a key individual (such as in

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an apprenticeship context), conforming to the majority, and parent to child learning can result in less diversity through time (Eerkens and Lipo 2007: 251-252). In contrast, individual experimentation (guided variation), unbiased learning from peers, choosing the least common cultural variant (anti-conformist transmission), or the reintroduction of old, no longer used,

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styles (reworking a discovered artifact or intentionally reusing past styles) may all result in increased diversity (e.g. Boyd and Richerson 1985, Eerkens et al 2006, Henrich and Henrich

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2007, Shennan 2000).

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1.2 Learning and Population Structure Another key factor that patterns cultural variation is population structure.

Archaeologists have increasingly acknowledged the effects of population structure on learning (e.g. Crema et al 2014, Holman et al 2007, Kandler et al 2012, Lipo and Madsen 2001, Lipo et al 1997, Neiman 1995, Perrault and Brantingham 2011, Premo 2012, Premo and Kuhn 2010, Shennan 2000). Population size has been shown to affect rates of innovation and the loss of cultural traits due to drift (Henrich 2004, Lycett and Norton 2010, Powell et al

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ACCEPTED MANUSCRIPT 2010). The implications of these studies emphasizes why understanding factors such as population bottlenecks affect learning is crucial for studying how variation is produced and reproduced in the archaeological record. Most recently, the role of spatial distance between populations has been examined (Premo and Scholnick 2011). Changes in population, the

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spatial scale of social learning, and mechanism of cultural transmission all influence the

number of individuals who pass on cultural information, teachers. As the number of teachers is analogous to the number of breeding individuals in genetic populations, a concept from

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population genetics, the effective size of a population (Ne) (Wright 1931, 1938, Crow and Kimura 1970) has been fruitful in past archaeological discussions (Bentley et al 2004, Lipo

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and Madsen 2001, Neiman 1995). As a result of sampling error effective population sizes, the number of teachers in a population, may be much smaller than the actual population (Neiman 1990: 206-207). Recently, some researchers have referred to population in terms of the number of cultural artifacts present (e.g. Kandler and Shennan 2013: 19). I refer to population

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as the total number of potential learners in the population. Effective population is the number of teachers out of this overall population of learners, as originally formulated by Neiman (1995).

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In many respects the formulation of effective population in cultural transmission

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studies, if defined as the number of teachers in a population, is not fundamentally different from that in population genetics. The congruence of these forms of effective population is a result of the underlying emergent process of descent with modification shared by genes and culture. Researchers in archaeology, anthropology, psychology, and biology have argued that cultural variation and change can be characterized as a product of descent with modification (e.g. Eerkens and Lipo 2007, Mesoudi et al 2004; O'Brien and Lyman 2000; Shennan 2000). There is variation in socially transmitted information in populations, that variation is

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ACCEPTED MANUSCRIPT inherited through social learning, and that variation is winnowed in subsequent generations as a result of natural selection, cultural selection, and drift (Lycett 2011: 145). When all of these factors are present descent with modification must occur (Mesoudi et al. 2004). The fundamental concept of effective population size being the subset of individuals transmitting

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information, whether biological or cultural, is the same between the biological and social formulations of descent with modification.

Borrowing from neutral theory (Kimura and Crow 1964, Crow and Kimura 1970,

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Neiman 1990) Neiman (1995) introduces a model that considers the impact of effective

population size and innovation rates on stylistic diversity. This model has been successfully

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applied towards examining mechanisms of cultural transmission in ceramic technologies (e.g. Kohler et al. 2004, Shennan and Wilkinson 2001). Although Neiman's (1995) model, based on the well-mixed Wright-Fisher infinite alleles model with neutral variants, may be suitable for examining diversity in discrete technological styles, it is not appropriate for examining

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variation in continuous artifact traits such as the width or length of formed lithic tools. Eerkens and Lipo (2005) present an alternative model, drawing from quantitative genetics literature that describes the effects of selection and drift on continuous traits.

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Eerkens and Lipo (2005) examine the cumulative effects of visual copying errors on metric

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attributes. However, their approach ignores drift because there is no chance in their model for individuals to be over or underrepresented by sampling error. The model presented here is intended to expand on their work by providing a way to integrate population dynamics, specifically by examining the impact of finite populations and population bottlenecks on the copying of metric attributes of the material record. The agent-based models presented here are fully analogous to Neiman's (1995) neutral model with drift but are for continuous traits.

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ACCEPTED MANUSCRIPT 2. Constructing a Model Examining Impact of Effective Population (Ne) on Drift in a Metric Attribute 2.1 Model Parameters Here, Eerkens and Lipo's (2005) model of copying errors in a metric attribute, which

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Hamilton and Buchanan (2009) call the “accumulated copying error” (ACE) model, has been included in an agent-based simulation (programmed in PERL) that also allows for changes in the population size of an artificial society of social learners. This addition I term this the

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accumulated copy error with drift (ACED) model. In this model, during each time step an offspring generation of agents learns (i.e. copies) a single metric attribute (a floating point

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value greater than 0) that represents a selectively neutral (Dunnell 1978), or pay-off independent (Brantingham 2007), character or trait found on a piece of material culture. A conservative estimate for a copying error range of ±3% is based on the Weber fraction for visual measurement of line length and consideration of error resulting from the

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recollection of mental templates (Coren et al 1994, Eerkens 2000, Eerkens and Bettinger 2001, Eerkens and Lipo 2005). Individuals attempting to produce an artifact of a particular size without an independent scale or ruler (direct comparison) are unable to perceive

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differences less than 1.7% (Eerkens 2000). Error in size estimation for remembered objects

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(mental templates) increases through time from initial observation (Kerst and Howard 1978, 1984, Moyer et al 1978). Experimental approaches suggest that an error size of ±3% (Eerkens 2000) reflects situations where one or few individuals responsible for the production of specific artifacts recalled designs from memory. A copying error range of ±3% was chosen for the model. This error range is based on a uniform probability distribution that is iteratively bounded between -3% to +3%. In other words, there is an equal probability of adding or subtracting any value between -3% to +3%

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ACCEPTED MANUSCRIPT to each simulated metric attribute each time step per agent. The stochastic processes of copying error through time for each simulated tool forms a Markov chain of cumulative metric variation through time. Although this model leads to a slight positive skew in mean attribute size and slight

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negative skew in median attribute size, this is justifiable as it accurately reflects emergent biases in visual copying confirmed in an experimental study by Kempe et al. (2012: 3).

Copying error is relative to the size of the object being transmitted, large objects will have

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higher error while small objects have lower error. Although most Markov chains have a small attribute size, leading to a slight negative skew in median attribute size, those chains that

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increase in size have an increased magnitude in their copying error. Although this does increase the chance for large artifacts to become small, reducing the mean, this is countered by the minority of Markov chains that become exponentially larger through time. The result is a slight positive skew in mean attribute size. This is an underlying bias in visual copying

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and not just a result of the types of models used to examine visual copying. However, in actual data sets involving artifacts used in daily life, extreme high and low values are likely to be winnowed by selective pressures. An examination of this process is however beyond the

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scope of this study. The presence of these biases in this agent based model are consistent with

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the deterministic models presented by Hamilton and Buchanan (2009) and Kempe et al (2012) where median attribute size decreases through time. In this model, the coefficient of variation (CV) of one simulated metric attribute is

used to examine the amount of metric variation present in each time step. CV is an expression of the standard deviation as a percentage of the mean of the parent distribution (Sokal and Rohlf 1995: 58-60, Thomas 1986: 82-85). CV was chosen over other summary statistics for continuity with Eerkens and Lipo's (2005) study. For each time step, the mean and 95%

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ACCEPTED MANUSCRIPT confidence interval for the CV of 30 runs is reported. This number of runs provides a sufficient sample size to be confident in the results while not being overly taxing in processing time. Several forms of cultural transmission are examined in the model: 1) vertical

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transmission (learning from direct biological parents) without drift, 2) vertical transmission with drift, 3) unbiased transmission (learning from a randomly selected individual) with drift, 4) conformist transmission (copying the most common variant in the population), 5) and

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prestige biased transmission (learning from a subset of the population viewed as either highly skilled or prestigious). Table 1 outlines how each form of cultural transmission is

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parameterized in the model. Each of these social learning rules can be considered as affecting the effective population size (Ne) of unique teachers out of the total population.

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INSERT TABLE 1 HERE

Previous approaches (Eerkens and Lipo 2005, Hamilton and Buchanan 2009, Kempe et al 2012) examining visual copying error have excluded drift as a factor. To illustrate the

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importance of drift, a version of the vertical transmission model without drift is provided. For

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the vertical transmission without drift model, all individuals have guaranteed reproductive success by having at least one offspring barring the stochastic effects of a bottleneck that decreases population size.

The lack of sampling error from the absence of reproductive variability means that

for each time step a runaway process of increasing trait variance through time should occur barring strong bottleneck effects. When reproductive variability is added to the vertical transmission model by drawing the number of offspring for each agent for each time step

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ACCEPTED MANUSCRIPT from a Poisson distribution with a mean of 1, the result is mathematically equivalent to the unbiased model with drift where sampling error is introduced by agents randomly selecting teachers. Due to this equivalency, and the unrealistic assumptions of the vertical transmission without drift model, only unbiased and biased transmission are considered in this study.

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However, one simulation of vertical transmission without drift is provided to illustrate the runaway process.

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2.2 Assessing Effect of Population Bottlenecks

In order to examine the impact of population bottlenecks on different forms of

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transmission several sets of simulations were constructed. To account for changes in population sizes, several rules were established. As total populations expand new agents randomly select a tool to copy from a teacher in the pool of teachers in the previous time step. When populations decrease an agent is randomly chosen to die, as agents are considered as

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having equal fitness values. Innovation still occurs when populations decrease, but when population sizes are smaller it is at a reduced rate compared to periods of population expansion. This is a result of the number of innovators being lower in small populations.

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Four simulations were run: 1) constant sized population 2) exponentially increasing

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population 3) short bottleneck 4) long bottleneck. The initial parameters for each model are outlined in Table 2. A small set of scenarios were chosen that highlight major dynamics. For the purposes of better understanding how Ne patterns the coefficient of variation,

the iterated results of simulations with different effective population sizes are presented. To record Ne for each time step, each agent reports a “taught” flag if their cultural variant is transmitted. The unbiased transmission model with drift is used for this set of simulations, for a total population range from 10 to 10,000 individuals iterated by 250. Computational

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ACCEPTED MANUSCRIPT constraints from the exponential memory costs of larger populations was a deciding factor for the upper limit of 10,000 individuals. Mean Ne and mean coefficient of variation over 10,000 time steps for 30 runs are reported (Fig. 1). Mean Ne is 63.1% (σ=0.27%) of the total population under unbiased transmission.

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This means that a total population of 1500 individuals has an Ne approaching 1000

individuals. There is a monotonic but non-linear relationship between Ne and mean

coefficient of variation significant at a 0.01 level (Spearman's R=0.689, p=0.001). CV rapidly

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increases as the number of teachers in a population increases from 1-2000 individuals. After which, the monotonic increase in CV is less rapid but continues although there is

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considerable dispersion. The amount of dispersion makes it difficult to estimate the effective population size in an arbitrarily large populations. As a result, I limit the majority of my simulations to examining the effects of bottlenecks in smaller population sizes where the dynamics can be clearly observed. The same pattern of a monotonically increasing CV with a

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presented later.

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high degree of dispersion is also apparent in the exponentially increasing population model

There is also a strong relationship between the strength of biased transmission

(conformist or prestige) and reduced effective population size (Fig. 2). Strength of transmission was iterated by 5% in a constant sized population of 500 individuals over 10,000 time steps for 30 runs. Effective population sizes for both types of biased transmission were relatively consistent between runs (σ=1.5 for conformist, σ=1.3 for prestige). Differences in mean effective population size by type of transmission bias (conformist

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ACCEPTED MANUSCRIPT and prestige) were found to not be significant at a 0.01 level according to a KolmogorovSmirnov test (K-S statistic=0.167 p=0.289). This is only the case if the prestige indicator trait is transmitted to a small portion of the population, in this case 5% of the total population. If an indicator trait is more frequent in the population the effective population size will be larger

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and there will be a clear departure from conformist transmission. Due to the equifinality

between strong conformist transmission and strong prestige bias, I only report the results of conformist transmission for the first set of simulations with biased transmission (constant

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sized population) for comparison.

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INSERT FIG. 2 HERE

Although cumulative variation through time should be a dominant pattern in each

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model, population bottlenecks should result in the accumulation of variation being constrained, or even reduced, for all forms of transmission. Unbiased transmission is expected to be more severely impacted by changes in population size, as strong prestige bias

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already results in a low Ne and would not be impacted as greatly by demographic shifts unless a teacher was impacted by population loss.

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In most circumstances, there should not be an observable difference in the mean coefficient of variation between a group of 10 individuals under unbiased transmission, and a group of 1,000 individuals under strong (99%) prestige biased transmission as they would have an equivalent Ne. In contrast, 1,000 individuals under unbiased transmission would have a considerably higher Ne and higher mean coefficient of variation than a population of 10 individuals under strong prestige bias. Due to this relationship, effective population is used as a heuristic to compare differences between the number of teachers in a population to the 11

ACCEPTED MANUSCRIPT overall population and not used in the strict biological sense. In this model, agents copy from teachers in one-to-one sessions. However, in reality teachers may produce varying quantities of objects following a drift-like processes, which may influence non-social learning (copying these observed objects) and social learning

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(teacher selecting some variants as examples for learners over others). Since I assume one-toone learning, the potential effects from the number of generated objects by teachers is not

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considered in this model.

3. Results 3.1 Simulations without Bottlenecks

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INSERT TABLE 2 HERE

Constant Sized Population: Vertical Transmission Without Drift, and Unbiased Transmission

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Although effective population sizes are not directly measured in the simulations, a contrast in effective population sizes between transmission models can be inferred in Fig. 3, which compares a version of the vertical transmission model without drift from reproductive

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variation. Of note is in the vertical transmission without drift model, where there is vertical

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transmission of fully independent lineages, the effective population size is equivalent to the total population, a clear violation of the effective population size concept. In the vertical transmission without drift model, the mean of the coefficient of variation appears to increase monotonically through time with minor fluctuations resulting from the inherent stochasticity of the agent-based model. However, there is a non-linear relationship with time where the slope of the curve decreases through time (e.g. Eerkens and Lipo 2005). The dispersion in the model obscures this pattern, but the results of the vertical transmission without drift model

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ACCEPTED MANUSCRIPT are consistent with Eerkens and Lipo's findings (2005). In contrast, unbiased learning, which is, once again, mathematically equivalent to vertical transmission with drift, reaches equilibrium after 400 time steps where the mean

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remains relatively stable through time.

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Constant Sized Population: Unbiased and Biased (Conformist and Prestige) Transmission For each mode of social learning the mean coefficient of variation accumulates

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through time in a different manner, patterned by differences in effective population size. In the model without bottlenecks (Fig. 4), unbiased learning reaches equilibrium after 400 time steps where the mean remains relatively stable through time. Both prestige biased and conformist forms of transmission reach equilibria over fewer time steps (20). This model

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suggests that equilibria are quickly reached with smaller effective population sizes. Just as vertical transmission with drift and unbiased transmission are mathematically equivalent, conformist and prestige bias have equivalent impacts on effective population size provided

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that the prestige indicator trait is present in only a small subset (<5%) of the population. For

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the remaining simulations I examine the relationship between strong prestige bias and unbiased learning, but the results from prestige bias are consistent with strong conformist learning.

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Exponentially Increasing Population

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ACCEPTED MANUSCRIPT The results of the exponentially increasing population model (Fig. 5) show a monotonic increase in both unbiased and prestige biased transmission. Both forms of learning have increased dispersion through time in the exponential model, as both forms of transmission are adjusting to new equilibria as Ne increases with each time step. This model

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indicates that rapid population expansion may result in a relatively high degree of dispersion and variation in both biased and unbiased transmission compared to constant sized

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populations.

3.2 Simulations with Bottlenecks Short Bottleneck

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Mean coefficient of variation is greatly reduced for unbiased transmission during a

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population bottleneck (Fig. 6). Although the bottleneck is relatively brief (500 time steps), its impacts on mean CV last five times longer than the bottleneck, with mean coefficient of variation recovering to pre-bottleneck levels at time step 6250 (the effects of the bottleneck

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lasting for 1250 time steps). Unbiased transmission exhibits reduced mean coefficient of

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variation during the length of the bottleneck, with mean coefficient of variation only increasing once the bottleneck has ended and with mean variation recovering to its prebottleneck state 500 time steps after the bottleneck ends. Prestige biased transmission, with an already small Ne, appears unaffected by the bottleneck. Of note is the equifinality between unbiased transmission with a small Ne and prestige biased transmission in the model, indicated by overlapping 95% CIs during the bottleneck. Each bottleneck also results in founder effects as the traits not winnowed by drift dominate the population. Although the

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ACCEPTED MANUSCRIPT affect of this on CV, a is reduced mean and 95% confidence interval, this can result in post bottleneck mean attribute values being strongly deviating from those from before the bottleneck. The impact of population bottlenecks and founder effects on continuous attributes

novel area of research.

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INSERT FIG. 6 HERE

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is not an area explored in population genetics, as genes are discrete traits, and so this is a

Long Bottleneck

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The results of the long bottleneck model (with a bottleneck four times the length, lasting 2000 time steps) are generally comparable to the short bottleneck (Fig. 7). As with the short bottleneck model, unbiased transmission exhibits reduced mean coefficient of variation through the duration of the bottleneck (2750 time steps), and recovers 500 time steps after the

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end of the bottleneck (time step 7750). In other words, for both the short and long models, mean coefficient of variation fully recovers 500 time steps after the bottleneck ends. This suggests that the recovery time from a bottleneck appears dependent on the magnitude of a

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bottleneck and not its length. As with the short bottleneck model, prestige biased transmission

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appears unaffected, while unbiased transmission has an overlapping 95% CI with prestige bias during the bottleneck.

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Based on the results of both bottleneck models, recovery from bottlenecks in social learning appears to be relatively rapid, compared to biological populations, at least for the

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ACCEPTED MANUSCRIPT specific case of visual copying of artifacts. This is in direct contrast to genetic transmission, where bottlenecks can have long term effects on variation. Future modeling efforts comparing genetic and cultural bottleneck recovery times are warranted.

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4. Discussion

Developing models to examine the production of variation in the archaeological

record is crucial to understanding the production and reproduction of material and immaterial

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knowledge. The one provided here, the accumulated copying error with drift (ACED) model is a synthesis of approaches examining the relationships between demography and small scale

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cumulative copying error through time in stylistic attributes. Although the issues modeled here have been discussed implicitly (e.g. Boyd and Richerson 1985, Cavalli-Sforza and Feldman 1981), these simulations highlight how bottlenecks in populations can have clear ramifications for behavioral interpretations of metric variation in artifacts. Namely, changes

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in population structure and learning context can have equifinal impacts on material culture which can hinder our ability to interpret forms of social learning in the past. As unbiased transmission is heavily structured by the effective population size, a basic

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understanding of population structure and drift is critical in interpreting patterns of cultural

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transmission. Although studies have attempted to characterize the presence of conformity or prestige bias in past societies (e.g. Bettinger and Eerkens 1997, Eerkens and Lipo 2005, Hamilton and Buchanan 2009, Kempe et al 2012), these models have omitted the effects of drift. All human societies have finite population sizes, which has clear ramifications for how the production of cultural variation is patterned. In addition, without an estimated demographic baseline to ensure the absence of strong bottlenecks interpretive caution may be warranted as this study suggests that a

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ACCEPTED MANUSCRIPT decrease in population may result in equifinality between unbiased and prestige biased transmission. This is clearly a tall order archaeologically when considering populations that remain in the same location for long periods of time and regions where determining population structure is difficult due to the nature of the record. Similarly, contextual data

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regarding the magnitude of transmission bias is necessary. This may be impossible for some regions, but in the case of ethnographic or text-aided archaeology such estimates may be able to be derived. These challenges only highlight the need for interpretive caution.

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I suggest that framing our interpretations based on how behaviors pattern effective population sizes may be a more fruitful endeavor that better fits the archaeological data. For

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instance, the variance detected by Hamilton and Buchanan (2009) in Clovis points is consistent with a cultural bottleneck that may be the result of biased transmission or population structure resulting in a small local Ne for tool producers (e.g. Premo and Scholnick 2011). Based on the models with drift provided here, the actual mode of

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transmission in their study could be vertical, unbiased, or biased but they are detecting a clear bottleneck of cultural information. Likewise, although drift is not considered by Eerkens and Lipo (2005) it can provide an additional source of variation increasing and reducing

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processes. Drift may be an alternative to the interpretations they provide for the role of visual

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copying in Great Basin projectile technologies and Woodland ceramics. Relative differences in effective population should, however, be considered on an ordinal scale and take into account how factors such as subsistence, the complexity of technological manufacturing, and material quality may also increase or decrease variation. Although a decrease in the effective population of learners may result in an equifinal pattern for unbiased and biased learning, this does not necessarily have to be equated with demographic collapse. Ideological factors such as more restricted forms of teaching may

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ACCEPTED MANUSCRIPT result in a lowered effective population for specific technologies. Punctuated reduction of metric variation may be attributable restricted teaching if other lines of evidence suggest no shifts in population structure.

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5. Conclusions

The coefficient of variation of simulated metric attributes was found to be strongly patterned by effective population size, or number of teachers. Several cultural transmission

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processes were found to have equifinal effects on effective population size, and the

coefficient of variation. Vertical transmission with drift and unbiased transmission were

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found to be indistinguishable. Similarly, strong conformist and prestige biased transmission were found to be equifinal. In addition, population bottlenecks since they reduce effective population size also result in equifinality between unbiased transmission in a small population and biased forms of learning.

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This paper is intended to highlight the importance of examining the interplay between the cognitive processes of humans and how they affect cultural variation and the impacts of drift whether from demographic changes or ideological changes. The models provided yield

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hypothetical distributions to test against the material record, and also caution us to consider

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that quite varied behaviors may have equifinal effects on variation in material culture. However, this model highlights the shared underlying processes of how population dynamics affect the number of teachers, or effective population size. The overall implications of this model are that we need to exercise additional caution

when interpreting variation reducing processes in cultural transmission. This model can not be directly applied to create predictions from observed archaeological data. It does underscore the need for contextual data such as demographic, ethnographic, or textual

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ACCEPTED MANUSCRIPT information to interpret whether the observed metric variation in a piece of material culture is the result of behavioral processes or can be more parsimoniously explained by population structure. By viewing transmission in terms of effective population size, more nuanced

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interpretations that consider how social organization and settlement patterns structure

variation in artifacts can be considered. If we consider how social factors can impact cultural effective population sizes, we can truly begin to apply evolutionary perspectives to the

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production and reproduction of knowledge in societies where elite prerogatives had a key role

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in structuring learning practices such as complex hunter gatherers and states.

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ACCEPTED MANUSCRIPT Acknowledgments This paper was originally presented in a symposium titled “Recent Developments in Cultural Transmission Theory and its Applications” at the 2012 Annual Meeting of the Society for American Archaeology, Memphis, TN. The author wishes to thank for their feedback at

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various stages of this project Dr. Colin Grier and Dr. Luke Premo, Kristin Safi for the

opportunity to participate in the SAA session, and Dr. Fraser Neiman and Mark Madsen for their insightful comments on an early draft and all of the internal and anonymous reviewers

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for their comments and critiques.

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Table 1. Cultural Transmission Mode Parameters for Model Mode of Transmission Agent Decision All members of population transmit a 'parent' flag each Vertical- No Drift time step. Cultural variant selected from parental individual from previous time step. Vertical- With Drift Members of population transmit a 'parent' flag each time step based on a Poisson distribution with a mean of 1. Cultural variant selected from parental individual from previous time step. Unbiased Randomly select a cultural variant from any individual from previous time step. A set % of agents (strength of conformist bias) will Conformist select the most common cultural variant (the values for all variants are rounded to 3% bins) from the previous time step to copy. Otherwise agents randomly select a cultural variant from any individual from previous time step. Prestige Biased 1% of the population transmits a 'prestigious' flag each time step to agent marked with 'parent' flag (prestige is vertically transmitted with drift). A % of agents (strength of prestige bias) then randomly select a cultural variant from 'prestigious' individuals from previous time step.

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Constant Sized Population Exponentially Increasing Population

Short Bottleneck Long Bottleneck

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Table 2. Initial Model Parameters Initial N Total Time Steps Bottleneck N Bottleneck Notes Length (in Time Steps) 500 10000 n/a n/a 50 10000 n/a n/a Fixed growth rate of 0.1% per time step. 500 10000 50 500 500 10000 50 2000

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Bottlenecks occur at time step 5000.

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Fig. 1. Relationship Between Effective Population Size and Coefficient of Variation (30 Runs)

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Fig. 2. Iterated Strength of Transmission Bias Impact on Mean Effective Population Size for Conformist and Prestige Biased Transmission in a Constant Sized Population (N=500, 30 Runs) Yellow: Conformist Transmission Blue: Prestige Biased Transmission

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Fig. 3. 95% CI and Mean Coefficients of Variation for Model With Constant Sized Population (N=500, 30 Runs) Green: Vertical transmission without drift Red: Unbiased transmission Gray line: Population Trend

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Fig. 4. 95% CI and Mean Coefficients of Variation for Model With Constant Sized Population (N=500, 30 Runs) Red: Unbiased transmission Blue: 90% Prestige Biased transmission Yellow: 90% Conformist Transmission Gray line: Population Trend

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Fig. 5. 95% CI and Mean Coefficients of Variation for Model Using Exponentially Increasing Population Curve (30 Runs) Red: Unbiased transmission Blue: 90 % Prestige Biased transmission Gray line: Population Trend

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Fig. 6. 95% CI and Mean Coefficients of Variation for Model With Short Bottleneck (30 Runs) Red: Unbiased transmission Blue: 90% Prestige Biased transmission Gray line: Population Trend

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Fig. 7. 95% CI and Mean Coefficients of Variation for Model With Long Bottleneck (30 Runs) Red: Unbiased transmission Blue: 90% Prestige Biased transmission Gray line: Population Trend

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Highlights:

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Models the effects of drift on the learning of a continuous metric attribute.
Strong conformist and prestige biased transmission are indistinguishable.
Vertical cultural transmission without drift results in a runaway effect.
Vertical cultural transmission with drift is same as unbiased with drift.
Unbiased and prestige biased learning are same during a population bottleneck.