Conformity to continuous and discrete ordered traits

Heinrich Mora, Elisa; Denton, Kaleda K.; Palmer, Michael E.; Feldman, Marcus W.

doi:10.1073/pnas.2417078122

Conformity to continuous and discrete ordered traits

Elisa Heinrich Mora, Kaleda K. Denton, Michael E. Palmer, and Marcus W. Feldman https://orcid.org/0000-0002-0664-3803 [email protected]Authors Info & Affiliations

Contributed by Marcus W. Feldman; received August 21, 2024; accepted December 7, 2024; reviewed by Alberto Acerbi and Charles Efferson

January 17, 2025

122 (3) e2417078122

https://doi.org/10.1073/pnas.2417078122

PDF/EPUB

Significance

Conformist and anticonformist biases in acquiring cultural variants have been documented in humans and several nonhuman species. We introduce a framework for quantifying these biases when cultural traits are ordered, with greater and lesser values, and either continuous (e.g., level of a behavior) or discrete (e.g., number of displays of a behavior). Unlike previous models, we do not measure a cultural variant’s popularity by its distance to the population mean, but rather by its distance to other variants. We find that conformity can produce a variety of population distributions that need not center around the initial population’s mean variant. Anticonformity may lead to highly polarized or uniformly distributed populations, depending on its strength and on individuals’ precision when copying others.

Abstract

Models of conformity and anticonformity have typically focused on cultural traits with unordered variants, such as baby names, strategies (cooperate/defect), or the presence/absence of an innovation. There have been fewer studies of conformity to cultural traits with ordered variants, such as level of cooperation (low, medium, high) or proportion of time spent on a task (0% to 100%). In these studies of ordered cultural traits, conformity is defined as a preference for the mean trait value in a population even if no members of the population have variants near this mean; e.g., 50% of the population has variant 0 and 50% has variant 1, producing a mean of 0.5. Here, we introduce models of conformity to ordered traits, which can be either discrete or continuous. In these models, conformists prefer to adopt more popular cultural variants even if these variants are far from the population mean. To measure a variant’s “popularity” in cases where no two individuals share precisely the same variant on a continuum, we introduce a metric called k-dispersal; this takes into account a variant’s distance to its k closest neighbors, with more “popular” variants having lower distances to their neighbors. We demonstrate through simulations that conformity to ordered traits need not produce a homogeneous population, as has previously been claimed. Under some combinations of parameter values, conformity sustains substantial trait variation over many generations. Furthermore, anticonformity may produce a high level of polarization.

Cultural traits include information, beliefs, behaviors, customs, and practices that are acquired and transmitted through social mechanisms (1, 2). They may be continuous or discrete. For example, a continuous cultural trait could be “proportion of time spent helping others,” ranging from 0% to 100% and including all values in between. A discrete trait may be “level of cooperation: low, medium, high” or more simply, “cooperate or defect.” The latter is an example of a dichotomous cultural trait, i.e., a trait with two variants.

Most previous studies of conformity and anticonformity have focused on discrete, dichotomous cultural traits. For example, in a study on conformity in nine-spined sticklebacks, the two variants were “swim toward a yellow feeder” and “swim toward a blue feeder” (3). In an experiment on conformity for mate choice, fruit flies were painted pink or green, and observer fruit flies chose which color to mate with depending on the choices of demonstrator flies (4). In other experiments on conformity, the variants were the direction, to the left or right, that birds opened a puzzle box slider (5), and the choice of “same shape” or “different shapes” for people observing two shapes rotated at different angles (6).

For dichotomous cultural traits, the definitions of conformity and anticonformity are straightforward: conformity entails that the more popular of the two variants is adopted at a rate greater than its frequency, whereas anticonformity entails that it is adopted at a rate less than its frequency (7). Boyd and Richerson (7) formalized this definition in a mathematical model, which has been widely used in subsequent theoretical research (8–20). The model gives rise to the following population dynamics: Under conformity, the more popular of the two variants increases in frequency until everyone has it. Under sufficiently low levels of anticonformity, as in ref. 7, the two cultural variants stabilize at a population frequency of 50% each. With high levels of anticonformity, however, stable cycles between population states, and chaos, may occur (16).

For a discrete cultural trait with more than two variants, known as a polychotomous trait, the definitions of conformity and anticonformity are not the same as those for a dichotomous trait. Muthukrishna et al. (21) pointed out that in a model with two variants,

A_{1}

and

A_{2}

, the variant

A_{1}

must be present at a frequency above 50% to be the “more popular” variant that is preferred by conformists. However, in a population with four variants,

A_{1}, A_{2}, A_{3},

and

A_{4}

, the variant

A_{1}

need only be present at a frequency above 25% to suggest that it is more popular than some of the other variants. They concluded that “all current models and experiments may have been underestimating the strength of the conformist bias, because there are often more than 2 [variants] in the real world” (21), p. 11].

To illustrate the complexity inherent in conformity and anticonformity when there are more than two variants, consider a sample of

n = 10

individuals, referred to as “role models,” and imagine there are 4

A_{1}

, 3

A_{2}

, 2

A_{3}

, and 1

A_{4}

. Would a conformist adopt, say,

A_{2}

at a rate greater than its frequency or less than its frequency? In other words, is

A_{2}

“popular” or “unpopular”? Mesoudi (22) modeled conformity as a preference for only the most common variant among the role models; thus,

A_{2}

would be considered unpopular. This also means that in a more extreme example—say, 100

B_{1}

, 99

B_{2}

, and 1

B_{3}

—the variant

B_{2}

would be considered unpopular and adopted at a rate less than its frequency. In contrast, Denton et al. (23) denoted by

r

the number of distinct cultural variants in the sample of

n

, and classified a variant as “popular” if it occurs in more than

\frac{n}{r}

role models (see also ref. 24). Thus, in the example with variants

A_{i}

,

\frac{n}{r}

is

\frac{10}{4} = 2.5

, so variants

A_{1}

and

A_{2}

would be considered popular while

A_{3}

and

A_{4}

are unpopular (and similarly, in the example with variants

B_{i}

, both

B_{1}

and

B_{2}

would be considered popular). Further, in ref. 23, a conformist is more likely to acquire

A_{1}

than

A_{2}

because

A_{1}

is more popular than

A_{2}

, while an anticonformist is more likely to acquire

A_{4}

than

A_{3}

because

A_{4}

is rarer than

A_{3}

. A general formula for conformity and anticonformity to

n

role models and

m

variants is given by equation 17 in ref. 23.

The population dynamics of the model of ref. 23 for a discrete trait with

m > 2

cultural variants are as follows (see their table 2 for a summary and a comparison to the two-variant case). Under conformity, the

ℓ \geq 1

cultural variant(s) that are initially most popular (i.e., have a higher frequency than all others) increase in frequency until they reach an equilibrium of

\frac{1}{ℓ}

and all other variants disappear. Under sufficiently low levels of anticonformity, similar to the two-variant model (7), the frequencies of all

m

variants present in the population reach frequencies of

\frac{1}{m}

, whereas with sufficiently high levels of anticonformity, cycles and chaos are possible.

In comparison with research on discrete cultural traits, fewer studies have considered continuous cultural traits. A continuous trait was modeled in refs. 25–27, where group-mean transmission, referred to as “conformity” by Smaldino and Epstein (26) and Morgan and Thompson (27), entailed adopting a cultural variant equal to or near the group’s average value (or one’s belief about the group’s average value; see ref. 27). In ref. 26, the population could converge to an equilibrium in which everyone shared the same variant, even if individuals were not entirely conformist but adopted variants that deviated from the population mean (i.e., had a “distinctiveness preference”). Taking conformity to be a preference for the mean cultural variant is intuitive in some cases, particularly cases in which most individuals in the population adopt a cultural variant near this mean (such as, possibly, time spent on social media, working hours, or food portion sizes).

However, defining conformity in relation to the mean variant in a population is not analogous to the definition of conformity for discrete traits proposed by Boyd and Richerson (7) and used extensively throughout the conformity literature (8–20). For example, consider the cultural trait “proportion of resources shared with others.” Assume that 60% of the population shares all of their resources, whereas the other 40% of the population shares none. Under Boyd and Richerson’s definition, conformists would either choose to share all or none of their resources (i.e., the behaviors of others they have seen), and would have a greater-than-60% chance of sharing all their resources. On the other hand, if conformity were conceptualized as population-mean transmission, then conformists would be expected to share a fraction of

(60 % \times 100 %) + (40 % \times 0 %) = 60 %

of their resources—a behavior that they have not seen.

Here, we develop a framework for studying conformity and anticonformity to continuous cultural traits that does not rely on population-mean transmission but instead captures the idea that conformists preferentially adopt more popular cultural variants and anticonformists preferentially adopt less popular cultural variants. In a continuous-trait model, it is highly unlikely that two or more role models will have precisely the same variant. Therefore, “popular” cultural variants cannot be defined as being present in many role models. Instead, we define popular cultural variants with reference to their distance from one another, and introduce a measure that we call

k

-dispersal to quantify the popularity of a variant in a continuous-trait model. In Methods, we describe this metric as well as other features of our model.

The distinction “discrete” vs. “continuous” is not the only difference between the discrete-trait conformity models in refs. 7–20, 23, and 24 and the continuous-trait conformity models in refs. 25–27. Another difference is that these discrete-trait models deal with unordered variants whereas continuous-trait models deal with ordered variants. Examples of discrete, unordered variants include “choice of baby name,” “choice of tool type,” or “presence vs. absence of a behavior,” whereas ordered variants can fall on a number line, e.g., “percent of time allocated to a task” (see Table 1 for additional examples). To our knowledge, no models of conformity have considered ordered and discrete cultural traits, such as “education level: low, medium, high” or “number of individuals helped.” Therefore, we extend our model of continuous, ordered cultural traits to discrete, ordered cultural traits and explore the population dynamics of each of these models using simulations.

Table 1.

Examples of unordered (discrete) and ordered (discrete or continuous) traits

	Unordered	Ordered
Discrete	• Cooperate or defect	• Number of individuals helped
	• Vote for party A, B, or C	• Education level (high, medium, low)
	• Folklore motifs	• Number of flowers a bee visits per day
Continuous	Not applicable.	• Proportion of resources shared
		• Amount of time spent on a task
		• Skirt length

1. Methods

1.1. Model Assumptions and Overview.

Our model includes the following commonly made assumptions (e.g., see refs. 7 and 23). First, there are discrete and nonoverlapping generations of parents and offspring. Second, each offspring takes a random sample of

n

“role models” from the parent population, known as oblique cultural transmission (1). They could also sample from other offspring within their generation, known as horizontal cultural transmission (1), but as in previous models (7, 23), oblique and horizontal transmission can be mathematically equivalent. Note that we assume no natural selection.

Some previous models of conformity have assumed an infinitely large population size (7, 16, 17, 23) whereas others have considered a finite population size (18, 26). Here, we assume a finite population size, because in our model, incorporating an infinitely large population with an infinite number of cultural variants on a continuum would not be possible. Before detailing the rules that individuals use to acquire a cultural trait, the following is a brief overview of the entire model:

1.

A population comprises

N

individuals, each possessing a cultural variant represented by a value ranging between 0 and 1 (without loss of generality).

2.

Members of this population “reproduce” to generate

N

offspring, which initially do not have cultural variants.

3.

Then, each of the

N

offspring independently acquires a cultural variant through the following steps:

(a)

An offspring randomly samples

n

role models from the previous generation.

(b)

From these

n

role models, an offspring generates a probability distribution, which is based on its level of conformity, anticonformity, or a third, unbiased type of transmission known as random copying (these are described in detail in Sections 1.2 and 1.3).

(c)

According to this customized probability distribution, an offspring adopts a cultural variant.

Steps 1 through 3 are repeated, with the offspring population becoming the new adult population in each generation. A diagram of this overview is in SI Appendix, Fig. S1.1.

1.2. Models of Continuous Ordered Traits.

Here, we introduce models of conformity, anticonformity, and random copying of continuous traits, providing details about Step 3 from Section 1.1 above. Before introducing models of conformity and anticonformity, we discuss the baseline model of unbiased transmission, namely random copying.

1.2.1. Random copying.

In previous studies on discrete cultural traits, random copying occurs when individuals “choose another member of the previous generation at random and copy their cultural trait” (28) (p. 42). For continuous traits such as dress length, however, it is unlikely that one individual would copy the exact length in millimeters of another individual’s dress. Instead, we define random copying as follows. Assume, without loss of generality, that the continuous cultural trait has a minimum of 0 and a maximum of 1.

Definition 1.

Random copying for a continuous trait: An individual (a) samples the cultural variant of a member of the previous generation at random, (b) sets this value as the mean of a normal distribution with SD

σ

, (c) truncates this normal distribution by setting the minimum trait value to 0 and the maximum trait value to 1, and renormalizes so that the area under the curve is 1, and (d) samples a value at random according to this distribution.

Note that in Definition 1, after the truncation step (c), it is possible that the new truncated normal distribution does not have a SD of exactly

σ

or a mean of exactly the value,

x_{i}

, sampled in step (a). However, the new SD and mean of the truncated normal distribution do not tend to be far from these “target”

σ

and

x_{i}

values, respectively, unless

x_{i}

is near one of the boundaries of the distribution (0 or 1) (SI Appendix, section 2). Throughout this paper,

σ

or

x_{i}

refer to the SD and mean prior to the truncation step. In addition, if the target mean

x_{i}

is near one of the boundaries (0 or 1), the truncated normal distribution may have a higher peak near that boundary than the nontruncated normal distribution, which we refer to as an “edge effect;” see SI Appendix, Figs. S2.2–S2.4 for examples. This edge effect occurs whenever an individual samples from a normal distribution restricted to a particular domain and discards values that fall outside of that domain (SI Appendix, Fig. S2.3).

Two examples of the random copying model in Definition 1 with different values of

σ

are given in Fig. 1. This figure shows the probability density at which a single focal individual, in a single generation, will adopt a cultural variant ranging between 0 and 1 (on the

x

-axis), given a particular sample

x

of role models’ cultural variants. Specifically, there are

n = 6

role models with variants

x = (0.1, 0.15, 0.25, 0.6, 0.7, 0.9)

shown on the

x

-axis of Fig. 1 as black triangles. This figure does not depict the evolutionary process; in the full evolutionary model (see Results), every individual in the population of size

N

independently draws its own sample of

n

role models from the previous generation and constructs its own customized probability distribution according to Definition 1.

Fig. 1.

A single individual’s probability density of adopting a variant given a role model sample, assuming random copying according to *Definition* 1. A sample of $n = 6$ role models with variants $x = (0.1, 0.15, 0.25, 0.6, 0.7, 0.9)$ is shown in black triangles and dashed lines. (In whole-population simulations of our model, shown in *Results*, each individual independently draws its own sample of $n$ role models, and constructs its own probability density function that may differ from Fig. 1). (A) $σ = 0.1$ and (B) $σ = 0.05$ in *Definition* 1.

1.2.2. Conformity and anticonformity.

Similarly to the model of random copying, models of conformity and anticonformity entail that an individual (a) samples the cultural variant of a member of the previous generation, and then follows steps (b)–(d) from Definition 1. The key difference is in how the “member of the previous generation” is chosen; unlike in part (a) of Definition 1, not all members of the previous generation have an equal probability of being selected. As in discrete models of conformity and anticonformity (7), each individual in the population is assumed to first take a random sample of

n

role models from the previous generation, and then select a cultural variant from this sample with a probability that depends on their level of (anti)conformity and the role models’ cultural variants. In the remainder of this section, we will discuss how a cultural variant

x_{i}

from a sample of role models

x = (x_{1}, \dots, x_{n})

is chosen; i.e., how part (a) of Definition 1 is modified to accommodate conformity and anticonformity.

Past studies on discrete traits have defined conformity as the tendency to adopt a more common variant in a sample of

n

role models with a probability greater than its frequency, and a less common variant with a probability less than its frequency (7, 16, 23). For continuous cultural traits, however, if there is stochasticity in copying, it is unlikely that any two individuals will have exactly the same variant. Therefore, the frequency of individuals with a particular variant is a poor indicator of a variant’s popularity. Instead, we consider the proximity of cultural variants to other variants. For instance, if

n = 5

people’s hair lengths were observed and took the values (in inches)

x = (5, 12, 12.5, 13, 20),

then, although each variant has a frequency of

\frac{1}{n} = \frac{1}{5}

, a conformist would be more likely to adopt a variant around 12 to 13 inches than around 5 or 20 inches, as the latter are less densely clustered (i.e., more dispersed) from the other variants. This notion of proximity or distance between cultural variants is a unique property of ordered traits.

We quantify dispersal as follows:

Definition 2.

k-dispersal for an ordered trait: In a sample of

n

role models

x = (x_{1}, x_{2}, \dots, x_{n})

, consider variant

x_{i}

. The

k

-dispersal of variant

x_{i}

is the sum of the

k

shortest absolute distances between variant

x_{i}

and the other variants in

x

. Note that

k < n

.

For example, consider again the hair lengths (in inches)

x = (5, 12, 12.5, 13, 20)

. The first variant, 5, has a

k

-dispersal of

12 - 5 = 7

if

k = 1

; 14.5 if

k = 2

; and so on. In contrast, the third variant, 12.5, has a

k

-dispersal of 0.5 if

k = 1

and 1 if

k = 2

. Thus, variants around 12-13 have lower

k

-dispersals than variants 5 or 20.

Without loss of generality, assume again that the continuous trait lies between 0 and 1. Consider a sample

x = (x_{1}, x_{2}, \dots, x_{n})

of cultural variants observed in

n

role models, where the subscript

i

in

x_{i}

refers to the role model observed, not to the type of variant. Our goal is to define the probability of adopting cultural variant

x_{i}

in

x

, for a given

k

used in Definition 2, denoted by

P_{k} (x_{i} ∣ x)

. This probability should have the property that conformists are more likely to adopt more densely clustered variants (i.e., those with lower

k

-dispersals) than random copiers, and random copiers are more likely to adopt more densely clustered variants than anticonformists.

Let

d_{k} (x)

—a constant for a given role model sample

x

and value of

k

—denote the strength of (anti)conformity. When

d_{k} (x) > 0

, individuals exhibit conformity to the role model sample

x

; when

d_{k} (x) = 0

, there is random copying; and when

d_{k} (x) < 0

, there is anticonformity. Larger positive values of

d_{k} (x)

indicate stronger conformity and more negative values of

d_{k} (x)

indicate stronger anticonformity.

Under random copying (

d_{k} (x) = 0

), the value of

P_{k} (x_{i} ∣ x)

should be

\frac{1}{n}

; i.e., each role model’s cultural variant is adopted with the same probability, in accordance with Definition 1. Under conformity (

d_{k} (x) > 0

), a role model’s cultural variant

x_{i}

should be adopted with a probability greater than

\frac{1}{n}

(i.e.,

P_{k} (x_{i} ∣ x) > \frac{1}{n}

) if that variant is more densely clustered than the average amount of clustering in the sample, and less than

\frac{1}{n}

if it is less densely clustered than average. Under anticonformity (

d_{k} (x) < 0

), the opposite should be true, with less densely clustered variants being adopted with a probability greater than

\frac{1}{n}

. Let

g_{i, k} (x)

be a function that is positive if

x_{i}

is more densely clustered than the average level of clustering in the sample and negative if

x_{i}

is less densely clustered than average. Then

\begin{matrix} P_{k} (x_{i} ∣ x) & = \frac{1}{n} + \frac{g_{i, k} (x) d_{k} (x)}{n} \end{matrix}

[1]

(similar to equation 17a in ref. 23) satisfies the above requirements.

Now, we define

g_{i, k} (x)

using our notion of “dispersal” given in Definition 2: If variant

x_{i}

is more densely clustered near (i.e., less dispersed from) other cultural variants,

g_{i, k} (x)

is positive, whereas if

x_{i}

is less densely clustered near (i.e., more dispersed from) the other variants,

g_{i, k} (x)

is negative (similar to the definition for discrete, unordered traits in ref. 23 but substituting “more densely clustered” for “more frequent”). Denote the

k

-dispersal of variant

x_{i}

by

f_{i, k} (x)

, which is nonnegative, and denote the mean of all

k

-dispersals in the sample of role models by

\bar{f_{k}} (x) = \sum_{i = 1}^{n} f_{i, k} (x) / n

. Our function

g_{i, k} (x)

is given below and then explained line by line.

\begin{matrix} g_{i, k} (x) & = \{\begin{matrix} - \frac{f_{i, k} (x)}{\sum_{z \in I} z} if f_{i, k} (x) \in I, I = {z : z > \bar{f_{k}} (x)} \\ \frac{{[f_{i, k} (x)]}^{- 1}}{\sum_{z \in II} z^{- 1}} \begin{matrix} if f_{j, k} (x) > 0 \forall j and f_{i, k} (x) \in II, \\ II = {z : 0 < z < \bar{f_{k}} (x)} \end{matrix} \\ \frac{1}{\sum_{j} 1_{f_{j, k} (x) = 0}} if f_{i, k} (x) = 0 and \bar{f_{k}} (x) > 0 \\ 0 if \exists j, f_{j, k} (x) = 0 and 0 < f_{i, k} (x) < \bar{f_{k}} (x) \\ 0 if f_{i, k} (x) = \bar{f_{k}} (x) . \end{matrix} \end{matrix}

[2]

Group

I = {z : z > \bar{f_{k}} (x)}

contains those variant dispersal values

f_{j, k} (x)

in

x

that are higher than average (i.e., less densely clustered). Group

II = {z : 0 < z < \bar{f_{k}} (x)}

contains those variant dispersal values

f_{j, k} (x)

in

x

that are lower than average, but not zero (i.e., more densely clustered).

If the dispersal of a variant

x_{i}

is higher than the average dispersal of variants in the sample, i.e.,

f_{i, k} (x) > \bar{f_{k}} (x)

, then it belongs to Group

I

(row 1 in Eq. 2) and the value of

g_{i, k} (x)

will be negative. In addition,

g_{i, k} (x)

will become more negative as the dispersal

f_{i, k} (x)

increases. Therefore, an anticonformist with

d_{k} (x) < 0

will have a higher probability of adopting a cultural variant as its dispersal increases (Eq. 1).

To understand row 2 in Eq. 2, consider a case in which

f_{j, k} (x) > 0

for all

j

, and variant

x_{i}

is less dispersed than the average dispersal in the sample, so

0 < f_{i, k} (x) < \bar{f_{k}} (x)

. In this case,

g_{i, k} (x)

will be positive, and will increase as the dispersal

f_{i, k} (x)

decreases. Therefore, a conformist with

d_{k} (x) > 0

should have a higher probability of adopting a cultural variant

x_{i}

as

x_{i}

’s dispersal from other variants decreases (Eq. 1).

However, if

x_{i}

is so close to other cultural variants that its dispersal

f_{i, k} (x) \to 0

, then the numerator in row 2 of Eq. 2 will be very large, and so will at least one value within the sum in the denominator (as this sum includes the value of the numerator). If

f_{i, k} (x) = 0

, meaning that some cultural variants in the sample

x

are identical, then the expression in row 2 cannot be applied because it would entail division by 0; instead, row 3 in Eq. 2 is applied. The bold

1

in row 3 is an indicator function, meaning that the expression is 1 if the statement in the subscript, namely

f_{j, k} (x) = 0

, is true and 0 otherwise. Therefore, row 3 of Eq. 2 captures the idea that if

m

cultural variants have

k

-dispersals of 0, they each will have the same

g_{i, k} (x)

value, namely

\frac{1}{m}

. Rows 4 and 5 of Eq. 2 ensure that

\sum_{i = 1}^{n} g_{i, k} (x) = 0

so that in Eq. 1,

\sum_{i = 1}^{n} P_{k} (x_{i} ∣ x) = 1

.

Conformity and anticonformity can now be defined in terms of

n

, the number of role models;

k

, the value used in Definition 2;

σ

, the SD (similar to “copying error”); and

d_{k} (x)

, the function in Eq. 1 that gives the strength and direction of (anti)conformity.

Definition 3.

Conformity and anticonformity to a continuous trait: An individual samples uniformly at random

n

role models from the previous generation, and observes their cultural variants

x = (x_{1}, x_{2}, \dots, x_{n})

. For a specified

k

, the individual selects value

x_{i}

in

x

with a probability given by Eq. 1. This value is set as the mean of a normal distribution with SD

σ

, which is then renormalized to have a minimum of 0 and a maximum of 1 (as in Definition 1); finally, the individual samples a value from this distribution. There is conformity if

d_{k} (x) > 0

and anticonformity if

d_{k} (x) < 0

in Eq. 1.

The upper and lower bounds of the conformity coefficient

d_{k} (x)

are calculated in SI Appendix, section 2. Fig. 2 shows the probabilities of choosing a cultural variant given the role model sample

x = (0.1, 0.15, 0.25, 0.6, 0.7, 0.9)

, with either conformity (top row) or anticonformity (bottom row). Compared to the case of random copying shown in Fig. 1, conformity makes the tall peaks taller and the short peaks shorter, whereas anticonformity has the opposite effect.

Fig. 2.

An individual’s probability density of adopting a continuous cultural variant given a role model sample, with $k = 2$ and either conformity ( $d_{k} (x) = 0.9$ , (A and B) or anticonformity ( $d_{k} (x) = - 0.9$ , (C and D). The sample of $n = 6$ role models with variants $x = (0.1, 0.15, 0.25, 0.6, 0.7, 0.9)$ is shown in black triangles and dashed lines. The curve shows the probability density for a single individual to adopt a variant given this sample, following *Definition* 3. (A and C) $σ = 0.1$ and (B and D) $σ = 0.05$ .

In SI Appendix, sections 3 and 4, we extend this model in two ways. In SI Appendix, section 3, we consider a cultural trait that is circular rather than linear, meaning that any value that is chosen as the “minimum” (e.g., 0) lies next to the value chosen as the “maximum” (e.g., 1) on a circle rather than lying apart on a line. A circular trait may be, for example, time on a clock chosen for an event, month of the year chosen for a vacation, or day of the week chosen for doing laundry.

In SI Appendix, section 4, we propose a model of anticonformity that differs from Fig. 2. Fig. 2 captures the idea that anticonformists have a higher probability of copying cultural variants seen in some role models (namely, individuals with “unpopular” cultural variants) relative to the random copying strategy (Fig. 1), as is the case for many studies of discrete and unordered traits (7–20). However, this is not the only possible representation of anticonformity. In Fig. 2 C and D, the probability of adopting a variant not seen in any role models (e.g., 0.4) is fairly low. Efferson et al. (29) introduced a model of “strong anticonformity” to a dichotomous, unordered trait in which the probability of adopting a cultural variant that is not present in any role models is very high, and we generalize this idea to an ordered trait in SI Appendix, section 4 and Fig. 3. Rather than adjusting the peaks of Fig. 1, we invert the distribution in Fig. 1 (details in SI Appendix, section 4) so that a new individual has a higher chance of adopting trait values far from clusters of role model values.

Fig. 3.

Probability density function for a single individual to adopt a cultural variant given a role model sample under strong anticonformity. Under strong anticonformity, an individual is qualitatively “repelled” by its role models. In this model, a probability density function following *Definition* 1 is obtained, “flipped” vertically, and normalized (*SI Appendix*, section 4). In the figure, an example set of $n = 6$ role models with variants $x = (0.1, 0.15, 0.25, 0.6, 0.7, 0.9)$ is marked with black triangles and vertical dashed lines. (A) $σ = 0.1$ and (B) $σ = 0.05$ .

1.3. Models of Ordered Traits with Discrete Variation.

Consider an ordered trait that can take discrete values

α_{1}, α_{2}, \dots, α_{ℓ}

. For example, a bee might visit

0, 1, 2, \dots, 5, 000

flowers in a given day (but it could not visit, say,

2.75

flowers). Hence,

α_{1} = 0, α_{2} = 1, \dots α_{5001} = 5, 000

.

For the discrete versions of each of our models presented in Section 1.2 (Figs. 1–3), an individual first selects any cultural variant, following the previously mentioned protocols. For example, a conformist would sample

n

role models, observe their cultural variants

x

, and, for a specified

k

, select a value

x_{i}

following Eq. 1. Then, the individual rounds

x_{i}

to the nearest value

α_{1}, α_{2}, \dots, α_{ℓ}

. Analogs of Figs. 1–3 with 21 cultural variants are given in SI Appendix, Figs. S5.1–S5.3, respectively.

2. Evolutionary Dynamics

The parameters of our model are

n

, the number of sampled role models per individual;

σ

, the SD of the normal distribution in Definitions 1 and 3, or “precision” in trait adoption;

k

values in the measure of

k

-dispersal (Definition 2); conformity coefficients

d_{k} (x)

; population size,

N

; and number of generations for which simulations are run. Here, and in SI Appendix, section 7, the effects of different model parameters on the population dynamics are described.

In Fig. 4, parameters

k

,

n

, and

N

are held constant at 2, 5, and 10,000, respectively, and the effects of

d_{k} (x)

(rows) and

σ

(columns) on the population are shown after 100 generations for 5 different replicates (i.e., different populations), each displayed in a different color. SI Appendix, Fig. S7.1 shows the same populations after 1,000 generations, and SI Appendix, Fig. S7.2 is similar to Fig. 4 except that

k = 3

rather than

2

. In Fig. 4 and SI Appendix, Figs. S7.1 and S7.2, the cultural variants in all initial populations were sampled from a uniform distribution on

[0, 1]

and are shown in SI Appendix, Fig. S1.1A (i.e., all initial populations are the same).

Fig. 4.

Trait distributions in the population after 100 generations for a continuous trait (rows 1 to 5 from the *Top*) or nearly continuous trait (row 6, with “Strong A.C.” standing for strong anticonformity; see *SI Appendix*, section 4), with $k = 2$ , $n = 5$ , and $N = 10, 000$ . 5 replicates were performed, each beginning from the same initial population (*Top Right* of *SI Appendix*, Fig. S1.1A, which was generated by sampling from a uniform distribution on $[0, 1]$ ). Each of the 5 final distributions is plotted in a different color. For each of these 5 final population distributions with cultural variants $x = (x_{1}, x_{2}, \dots, x_{N})$ , the polarization index $F = \frac{Var (x)}{Mean (x) \cdot [1 - Mean (x)]}$ (see ref. 30) is calculated separately. The average of these 5 $F$ values, rounded to 3 decimal places, is shown in a yellow box in the *Top*-*Right* corner of each panel. Values of $d_{k} (x)$ and $σ$ are in row and column labels, respectively. The label “ $d_{k} (x) min$ ” refers to the case in which, for each $x$ , $d_{k} (x)$ is set to its lower bound (*SI Appendix*, section 2) plus $10^{- 5}$ . Similarly, “ $d_{k} (x) max$ ” means that $d_{k} (x)$ is set to its upper bound minus $10^{- 5}$ (*SI Appendix*, section 2). Histogram bin widths are 0.01, and $y$ -axes differ in some of the panels (e.g., left-most column’s *Top* two panels).

Moving from left to right across any row in Fig. 4 (also SI Appendix, Figs. S7.1 and S7.2), increasing the SD,

σ

, has a smoothing effect that results in peaks and valleys in the population distributions becoming less pronounced. This is intuitive, as increasing

σ

decreases the precision with which a given role model’s cultural variant is acquired, resulting in less clustering of cultural variants in the population.

Moving from the top row to the fifth row in Fig. 4, i.e., decreasing the level of conformity

d_{k} (x)

in Eq. 1, we see that populations become more “polarized.” Following Leimar et al. (30), we quantify the level of polarization in a population of size

N

with cultural variant distribution

x = (x_{1}, x_{2}, \dots, x_{N})

at generation 100 as

F = \frac{Var (x)}{Mean (x) \cdot [1 - Mean (x)]}

(shown in yellow boxes in the Top-Right corners of the panels in Fig. 4). The intuition behind this formula is that if all members of the population share the same cultural variant, then

F = 0

(no polarization), whereas if each member of the population has one of the extreme values (

x_{i} = 0

or

x_{i} = 1

) and not all members have the same variant, then

F = 1

. That increasing anticonformity results in greater levels of population polarization makes sense, as anticonformity is a preference for being different from the norm. Similarly, in Boyd and Richerson’s model of a dichotomous trait with

n = 3

role models (7), conformity homogenized populations (no polarization) whereas anticonformity resulted in populations with 50% of individuals adopting one cultural variant and 50% adopting the other.

With conformity (Top two rows of Fig. 4) and relatively low

σ

values of

0.01

and

0.05

, population distributions appear to peak around the center of the trait distribution. Notably, the mean cultural variant value in the initial population that was sampled from a uniform distribution on

[0, 1]

was 0.497 (SI Appendix, Fig. S1.1A). Thus, in these cases, conformity resulted in individuals becoming more similar to the population mean with lower population-level variance than the initial distribution. These dynamics are similar, but not identical, to previous models of conformity to a continuous trait in which conformity was defined as a preference for the mean variant (26, 27).

Specifically, in ref. 27, conformist transmission resulted in all individuals converging on a single cultural variant, which was the mean of their prior belief about the true population mean. In ref. 27, individuals’ priors were normally distributed with nonzero variance. In ref. 26, regardless of whether individuals were completely conformist or had some distinctiveness preference (i.e., some level of anticonformity), populations could also converge on a single cultural variant. In contrast, in our Fig. 4 and the discrete-trait analog, Fig. 5, none of the populations converged to a point in which all individuals shared the same cultural variant. However, if

σ = 0

(i.e., if role models’ cultural variants are copied with zero variance), then in both the continuous and discrete trait case, populations did converge on a single cultural trait value if

d_{k} (x) > 0

(conformity), but not if

d_{k} (x) \leq 0

(random copying or anticonformity) (SI Appendix, Fig. S7.3).

Fig. 5.

Similar to Fig. 4, except that the cultural trait is discrete rather than continuous. The trait can take values $0, 0.05, 0.1, 0.15, \dots, 1.0$ .

In both continuous and discrete models, with conformity and sufficiently high

σ

, different replicates of the simulation may have peaks to the Left or Right of the center (e.g., far Right, Top two panels in Fig. 4 and Top Right panel in Fig. 5), departing further from the expectations of models with population-mean transmission. In the continuous model where simulations ran for 1,000 generations rather than 100, some of these distributions shifted, but not necessarily closer to the center of the plot (SI Appendix, Fig. S7.1). As all simulations began from the same initial population (which was sampled uniformly on

[0, 1]

), whether the peak of the population distribution shifted to the left or to the right of the center over time was due to stochastic effects; for example, see SI Appendix, Fig. S7.4.

What causes population distributions to build up at the left or right of the central point, 0.5, for many generations? One hypothesis is that “edge effects,” i.e., boundary accumulation effects, play a role: In Figs. 4 and 5 and SI Appendix, Figs. S7.1–S7.4, the cultural trait had a minimum of 0 and a maximum of 1 (as opposed to a circular trait, discussed in SI Appendix, section 3). Consider what happens when an individual—either conformist or anticonformist—samples a role model whose variant

x_{i}

is near one edge of the variant domain; say 0. First, the individual sets

x_{i}

as the mean of a normal distribution with SD

σ

. However, because the individual cannot sample a variant below 0, the normal distribution is truncated and resampled until the value falls within the range

[0, 1]

(Definition 3). Although this truncation does not create a new mode at 0, it does amplify the peak of the normal distribution near 0 compared to a nontruncated normal distribution (SI Appendix, Figs. S2.2 and S2.3), or compared to a truncated normal distribution that is not as close to a boundary (SI Appendix, Fig. S2.4). Thus, when aggregating multiple truncated normal distributions within

[0, 1]

, distributions centered closer to the boundaries experience stronger density amplification than those centered near the middle. It is therefore possible for density to accumulate near the boundaries in the case of a trait restricted to

[0, 1]

.

The same is not true for a circular trait without boundaries. For a circular trait, sampled values below 0 wrap around to the other end of the distribution, near 1 (SI Appendix, section 3), so boundaries do not exert a stronger effect on the resulting density than any other points. If density build-ups occur (e.g., under conformity), they are equally likely to be centered anywhere in the range

[0, 1]

. Specifically, in the circular trait model (SI Appendix, Fig. S7.5), no panels in the bottom four rows have greater density near the boundaries than near other points, whereas in the linear trait model (Fig. 4), sufficiently small

σ

combined with

d_{k} (x) < 0

produces a density build-up near the boundaries. In the circular trait model under conformity, distributions may or may not be centered close to an edge, and if they are very close to an edge they “spill over” to the other edge (e.g., blue distribution in SI Appendix, Fig. S7.5 row 2, column 2), which does not occur in the linear trait model (Fig. 4).

In Figs. 4 and 5 and SI Appendix, Figs. S7.1 and S7.2, strong anticonformity (“Strong A.C.,” bottom row) produces markedly different results from the model of anticonformity defined in Eq. 1 (fourth and fifth rows). The key difference is that in the latter cases (fourth and fifth rows), anticonformists prefer to adopt cultural variants that are near those of some other individuals (namely, individuals with variants that have high

k

-dispersals; Fig. 2 C and D), whereas strong anticonformists prefer to adopt cultural variants that are far from those of all other individuals (Fig. 3). Thus, in a population of strong anticonformists, “clusters” of individuals adopting similar cultural variants cannot persist, because each individual prefers to be as different as possible from the others. Therefore, any peaks in the population distribution will likely flatten over time, and the resulting population will be near uniform. Indeed, even if the entire population begins with the same cultural variant, strong anticonformity will produce a near uniform distribution over time (SI Appendix, Fig. S7.6).

In the results described so far, the population size

N

is set at 10,000. In Fig. 6, the population size

N

is varied along with the conformity coefficients,

d_{k} (x)

, holding constant

k = 2

,

n = 5

,

σ = 0.05

, and the distribution from which the initial variants in the population are drawn (uniform). The final population distributions after 100 generations are shown and appear relatively similar with

N = 1, 000

,

N = 10, 000

, and

N = 20, 000

. With

N = 100

, distributions appear more “spiked” or less smooth than those with higher

N

values. However, the overall shapes of these distributions and their average polarization measures in most cases resemble those of distributions with higher

N

.

Fig. 6.

Effect of population size $N$ on population distributions after 100 generations. Here, the cultural trait is continuous (or nearly continuous in the *Bottom* row, where “Strong A.C.” stands for strong anticonformity), $k = 2$ , $n = 5$ , and $σ = 0.05$ . In each panel, 5 replications were performed, each beginning from the same initial population that was generated from sampling $N$ individuals ( $N$ is given in column labels) from a uniform distribution on $[0, 1]$ . Each of the 5 final distributions is plotted in a different color. For each of these 5 final distributions with cultural variants $x = (x_{1}, x_{2}, \dots, x_{N})$ , the polarization index $F = \frac{Var (x)}{Mean (x) \cdot [1 - Mean (x)]}$ (see ref. 30) is calculated, and then the 5 $F$ values are averaged to produce the value shown in a yellow box in the *Top* of each panel. Values of $d_{k} (x)$ are in row labels. The label “ $d_{k} (x) min$ ” refers to the case in which, for each $x$ , $d_{k} (x)$ is set to its lower bound (*SI Appendix*, section 2) plus $10^{- 5}$ . Similarly, “ $d_{k} (x) max$ ” means that $d_{k} (x)$ is set to its upper bound minus $10^{- 5}$ (*SI Appendix*, section 2). Histogram bin widths are 0.01, and $y$ -axes differ in some of the panels (e.g., left-most column).

Finally, in Fig. 7, we explore the effects of varying the initial distribution of cultural variants in the population, holding constant

k = 2

,

n = 5

,

σ = 0.05

, and

N = 10, 000

. Each initial population distribution is shown in the Top row. In the Bottom three rows of Fig. 7, with anticonformity, population distributions after 100 generations are very similar across columns (i.e., regardless of different initial conditions). With random copying, population distributions after 100 generations differ slightly in cases where initial distributions are skewed to the left (first column) or to the right (last column). However, with conformity, population distributions after 100 generations often depend sensitively on the initial population composition. There is one exception; comparing cases in which the population is initially uniform vs. initially clustered in the center of the trait distribution (columns 3 and 4, respectively), there are similar trait distributions after 100 generations. This is not simply due to the fact that initial distributions in columns 3 and 4 have similar means (0.497 and 0.500, respectively) because the initial distribution in column 2 also has a similar mean of 0.501; rather, both the mean and density of the initial distribution affect evolutionary trajectories.

Fig. 7.

Effect of initial population distributions (*Top* row) on distributions after 100 generations (rows 2 to 7), with $k = 2$ , $n = 5$ , $N = 10, 000$ , and $σ = 0.05$ . In the *Top* row, 5 initial population distributions are shown; these were sampled from the probability distributions in the column labels, where $B (α, β)$ denotes a beta distribution with parameters $α$ and $β$ , and $U [0, 1]$ denotes a uniform distribution on $[0, 1]$ . In rows 2 through 7, a panel in column $j$ shows 5 replications of the simulation, each beginning from the initial population shown in row 1, column $j$ and each in a different color. For each of these 5 final distributions with cultural variants $x = (x_{1}, x_{2}, \dots, x_{N})$ , the polarization index $F = \frac{Var (x)}{Mean (x) \cdot [1 - Mean (x)]}$ (see ref. 30) is calculated, and these 5 $F$ values are averaged to produce the value shown in a yellow box in the *Top* of each panel. Values of $d_{k} (x)$ are in row labels. The label “ $d_{k} (x) min$ ” refers to the case in which, for each $x$ , $d_{k} (x)$ is set to its lower bound (*SI Appendix*, section 2) plus $10^{- 5}$ . Similarly, “ $d_{k} (x) max$ ” means that $d_{k} (x)$ is set to its upper bound minus $10^{- 5}$ (*SI Appendix*, section 2). Histogram bin widths are 0.01, and $y$ -axes differ in some of the panels (e.g., row 1).

3. Discussion

Research on frequency-dependent cultural transmission to date has been primarily focused on unordered traits, with much less attention paid to ordered traits. Here, we have explored three types of cultural transmission, namely conformity, anticonformity, and random copying, and ordered traits that are continuous or discrete (Table 1).

At least three previous studies have considered conformity to a continuous ordered trait (25–27) (note that (25) did not use the term “conformity,” although their formalization is similar to that of refs. 26 and 27, which did). Morgan and Thompson (27) define conformity as “a tendency to adopt the mean trait value in a population with an expected squared error less than the population variance (thereby causing the population to homogenize)” (p. 2). They assume that all individuals know the variance of trait values in the entire population, but not the mean trait value in the population; instead, individuals estimate this mean by sampling a number of cultural role models (corresponding to

n

in the present study). They show that under their definition of conformity, population-level variation decreases over time until all individuals in the population share the same cultural trait value.

In the model of Smaldino and Epstein (26), an individual’s ideal trait value is the mean trait value in the population, which they denote by

\bar{x} (t)

, plus the product of the SD

σ (t)

of trait values in the population, and the individual

i

’s “distinctiveness preference,”

δ_{i}

. A nonzero distinctiveness preference corresponds to some extent of anticonformity, i.e., preferring to be different from the mean trait in the population, whereas “conformists prefer to be at the population mean” with

δ_{i} = 0

(26), p. 6]. Unlike Morgan and Thompson (27), all individuals in the model of Smaldino and Epstein (26) know both the mean trait value in the population,

\bar{x} (t)

, and the SD

σ (t)

of trait values in the population. They present several models that differ in the distribution of distinctiveness preferences,

δ_{i}

, across individuals, summarized in their Table 1. Among other findings, they show that if all members of the population have the same distinctiveness preference (

δ_{i} = δ \forall i

), then at equilibrium all individuals share the same cultural trait value. Therefore, if all individuals are entirely conformist (

δ_{i} = 0

) or share the same level of anticonformity, then, as in ref. 27, there is no variance in the trait distribution at equilibrium.

However, the definition of “conformity” for continuous traits proposed by Smaldino and Epstein (26) and Morgan and Thompson (27), namely population-mean transmission, is not analogous to the definition of conformity for discrete traits: a disproportionate tendency to copy the most common variant (7–20). The reason is that the mean cultural variant in a population may be nowhere near the most “popular” cultural variant, especially if the mean is affected by outliers or if the population is polarized (e.g., if most people occupy either the far left or far right on a political spectrum and the mean is in the middle). Therefore, we developed a model of (anti)conformity to ordered variants that focused on their popularity rather than their mean. For a continuous trait with infinitely many variants, counting how many individuals share precisely the same variant may not be a useful measure of popularity; instead, we assess popularity by evaluating how clustered cultural variants are in phenotypic space. Conformity can then be defined as a disproportionate tendency to adopt less dispersed (more clustered) cultural variants and anticonformity as the opposite tendency (Fig. 2).

In most of our models of a continuous trait, unlike in refs. 26 and 27, conformity does not lead all members of the population to converge on the same cultural variant. Instead, populations continue for hundreds of generations to display a range of different cultural variants (Fig. 4 and SI Appendix, Fig. S7.1), which may be narrow (as in the TopLeft of Fig. 4 and SI Appendix, Fig. S7.1) or broad (as in the Top Right of Fig. 4 and SI Appendix, Fig. S7.1). These distributions may be centered around a value that is close to the mean trait value in the initial population (e.g., the initial mean was 0.497 in Fig. 4 and SI Appendix, Fig. S7.1, similar to the Top Left panel) or around a different value (e.g., Top Right of Fig. 4 and SI Appendix, Fig. S7.1).

In other cases, population-level variation greatly decreases or disappears over time, aligning more closely with the results of refs. 26 and 27. In the case of a discrete rather than a continuous ordered trait, when there is conformist transmission and low error in copying role models’ cultural variants (i.e., low

σ

in Definition 3), populations reach a point where nearly all individuals share the same cultural variant (e.g., Fig. 5, Top Left). In addition, with either a discrete or a continuous trait and zero variance in trait adoption (

σ = 0

), as well as conformist transmission (

d_{k} (x) > 0

), populations reach a state in which 100% of individuals share the same cultural variant (SI Appendix, Fig. S7.3). However, in different replicates of the model (different colors in the figures), the cultural variant on which the population converges may differ. In addition, although this result (zero population-level variance) is similar to the results of refs. 26 and 27, it obtains under different conditions; in ref. 26, it occurred regardless of whether there was conformity or anticonformity, and in ref. 27, individuals had prior beliefs about the population mean that were normally distributed with nonzero variance and affected their trait adoption decisions.

In addition to the definition of conformity and anticonformity, another difference between our models and those of Smaldino and Epstein (26) is that we do not assume that individuals know the mean and SD of trait values in the whole population. In the real world, these values may be difficult to estimate as the number

N

of individuals in the population becomes large. Instead, we assume that individuals sample only a small number

n < N

of the population (as in refs. 7 and 27); e.g., an animal in a sparsely populated area may come into contact with few others in its lifetime, or a human may have a small number of social contacts.

When an individual adopts a cultural variant from a sample of

n

role models, as shown in Figs. 1 and 2, the parameter

σ

determines how similar their cultural variant is to one of the role models’ variants (i.e., the “precision” in variant adoption; see Definitions 1 and 3). For example, when observing a role model’s trait such as “level of cooperation,” the observer will not track all of that role model’s actions across its entire lifetime; instead, it will observe some actions and infer from these what the role model’s level of cooperation is likely to be, with some error determined by

σ

. Even for more directly observable cultural traits such as “dress length,” an observer likely will not use a measuring tape to get an exact number down to the millimeter, but rather will estimate the approximate length with some error. Thus, a nonzero value of

σ

may be more realistic than

σ = 0

, and the larger that

σ

is, the broader the spectrum of potential trait selections is. Therefore, in the evolutionary simulations, higher

σ

values produce smoother trait distributions, with lower peaks and higher valleys, compared to lower

σ

values (Figs. 4 and 5 and SI Appendix, Figs. S7.1, S7.2, and S7.5).

In most of our models, we have assumed that the cultural trait has a minimum and maximum value; e.g., the continuous trait “proportion of time spent performing a behavior” may range from 0 to 1 or the discrete trait “number of displays of a behavior” may range from 0 to 1,000. Without loss of generality, we set the hypothetical trait’s minimum to 0 and maximum to 1. In this case, we find that anticonformity according to Definition 3 may produce a significant build-up of cultural variants near the boundaries, especially when

σ

is low (e.g.,

σ = 0.01

and

d_{k} (x) < 0

in the fourth and fifth rows of Fig. 4). Under anticonformity, this occurs for many different initial population distributions (e.g., Fig. 7, fifth and sixth rows from the Top). However, if the cultural trait is instead represented as a circle without a minimum or maximum value (SI Appendix, section 3), this build-up does not occur, and anticonformity produces a relatively uniform distribution across the trait space (SI Appendix, Fig. S7.5). Similarly, when anticonformity is represented not as in Definition 3 but instead as strong anticonformity (e.g., Fig. 3 and SI Appendix, section 4), build-up of density near 0 and 1 does not occur, irrespective of whether the trait is linear or circular (Bottom row of Fig. 4 and SI Appendix, Fig. S7.5). This is likely due to the fact that in Definition 3, anticonformists prefer to be similar to some individuals (namely, those whose cultural variants are less clustered in the trait space; see the Bottom row of Fig. 2) whereas strong anticonformists prefer to be dissimilar to all individuals, so build-ups in the frequencies of cultural variants are not sustained over time with strong anticonformity.

Our extension of previous conceptions of conformity and anticonformity with discrete and unordered traits to ordered traits, which can be discrete or continuous, has produced an interesting array of theoretical evolutionary outcomes. In addition, quantifying conformity to ordered traits may be useful for understanding individuals’ behavior in real-world settings. For example, in many experiments, individuals are asked to make an estimation of a quantity [e.g., scatter plot correlations (31)] and are allowed to view the answers of others before making their final choice. It is possible that individuals are more likely to choose answers that are “clustered” near many of the demonstrators’ answers rather than choosing the mean answer of the demonstrators—especially if one demonstrator’s answer is very different from the others. Thus, future work could test how well our model (Definition 3) matches empirical data on estimation tasks compared to the model of conformity as group-mean transmission.

Future theoretical research could also expand on these models in several ways. For example, we considered a single population of size

N

, whereas in reality there may be different subgroups, each with a different level of conformity, interacting as a metapopulation. At a finer scale, different individuals within subgroups could vary in their tendencies to conform or anticonform, in which case

d_{k} (x)

could be chosen from a probability distribution. Exploring alternative approaches for the role model sampling process is another natural extension of our model. Rather than sampling

n

role models at random, an individual could select role models based on prestige, success, or similarity to themselves. Finally, the dispersal function could employ a nonlinear scale in the trait values.

Data, Materials, and Software Availability

There are no data underlying this work. The code for simulations is publicly available at https://github.com/ElisaHeinrich/ordered_conformity (32).

Acknowledgments

We thank Yoav Ram and Harrison Hartle for helpful suggestions that greatly improved the paper. This research was supported in part by the Morrison Institute for Population and Resource Studies at Stanford University; the Stanford Center for Computational, Evolutionary and Human Genomics; and the Santa Fe Institute. We acknowledge participants at the Modeling and Theory in Population Biology meeting of the Banff International Research Station (24htp001) for helpful suggestions.

Author contributions

E.H.M, K.K.D., and M.W.F. designed research; E.H.M., K.K.D., M.E.P., and M.W.F. performed research; E.H.M., K.K.D., M.E.P., and M.W.F. analyzed data; and E.H.M., K.K.D., M.E.P., and M.W.F. wrote the paper.

Competing interests

The authors declare no competing interest.

Supporting Information

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Information & Authors

Information

Published in

Proceedings of the National Academy of Sciences

Vol. 122 | No. 3
January 21, 2025

PubMed: 39823304

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Copyright

Data, Materials, and Software Availability

There are no data underlying this work. The code for simulations is publicly available at https://github.com/ElisaHeinrich/ordered_conformity (32).

Submission history

Received: August 21, 2024

Accepted: December 7, 2024

Published online: January 17, 2025

Published in issue: January 21, 2025

Keywords

Acknowledgments

We thank Yoav Ram and Harrison Hartle for helpful suggestions that greatly improved the paper. This research was supported in part by the Morrison Institute for Population and Resource Studies at Stanford University; the Stanford Center for Computational, Evolutionary and Human Genomics; and the Santa Fe Institute. We acknowledge participants at the Modeling and Theory in Population Biology meeting of the Banff International Research Station (24htp001) for helpful suggestions.

Author contributions

E.H.M, K.K.D., and M.W.F. designed research; E.H.M., K.K.D., M.E.P., and M.W.F. performed research; E.H.M., K.K.D., M.E.P., and M.W.F. analyzed data; and E.H.M., K.K.D., M.E.P., and M.W.F. wrote the paper.

Competing interests

The authors declare no competing interest.

Notes

Reviewers: A.A., Universita degli Studi di Trento; and C.E., Universite de Lausanne.

Authors

Affiliations

Elisa Heinrich Mora¹

Department of Biology, Stanford University, Stanford, CA 94305

Santa Fe Institute, Santa Fe, NM 87501

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Kaleda K. Denton¹

Department of Biology, Stanford University, Stanford, CA 94305

Santa Fe Institute, Santa Fe, NM 87501

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Michael E. Palmer

Department of Biology, Stanford University, Stanford, CA 94305

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Marcus W. Feldman² https://orcid.org/0000-0002-0664-3803 [email protected]

Department of Biology, Stanford University, Stanford, CA 94305

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Notes

2

To whom correspondence may be addressed. Email: [email protected].

1

E.H.M. and K.K.D. contributed equally to this work.

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E. Heinrich Mora,
K.K. Denton,
M.E. Palmer,
& M.W. Feldman,

Conformity to continuous and discrete ordered traits, Proc. Natl. Acad. Sci. U.S.A. 122 (3) e2417078122, https://doi.org/10.1073/pnas.2417078122 (2025).

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Abstract

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1. Methods

1.1. Model Assumptions and Overview.

1.2. Models of Continuous Ordered Traits.

1.2.1. Random copying.

1.2.2. Conformity and anticonformity.

1.3. Models of Ordered Traits with Discrete Variation.

2. Evolutionary Dynamics

3. Discussion

Data, Materials, and Software Availability

Acknowledgments

Author contributions

Competing interests

Supporting Information

References

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Data, Materials, and Software Availability

Submission history

Keywords

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