To appear in the Journal of Cognition
and Culture (2007)
Nicolas Claidière & and Dan Sperber
Institut
Jean Nicod, Paris
The role of attraction in cultural
evolution
(Reply to J. Henrich and R. Boyd, “On
modeling cognition and culture”, Journal of Cognition
and Culture, 2 (2), 2002).[1]
Abstract: Henrich and Boyd (2002) were the first to propose a formal model of
the role of attraction in cultural evolution. They came to the surprising
conclusion that, when both attaction and selection are at work, final outcomes
are determined by selection alone. This result is based on a determistic view
of cultural attraction, different from the probabilistic view introduced in
Sperber (1996). We defend this probabilistic view, show how to model it, and argue
that, when both attraction and selection are at work, both affect final
outcomes.
Two naturalistic research programmes relevant to the
explanation of cultural phenomena that started in the 70s — the evolutionary
approach of Boyd and Richerson (1985, Richerson and Boyd 2005), and their
collaborators, and the cognitive approach of Atran (1990, 2002), Boyer (1994,
2001), Hirschfeld (1996), Sperber (1996), and their collaborators — have to a certain extent converged over the
years, the first, more evolutionary programme going into greater detail into
the cognitive bases of cultural evolution, and the second, more cognitive
programme paying an ever increasing attention to the evolution of mind and
culture. Part of the reason why this relative convergence went almost unnoticed
is the fact that these programmes were generally pursued in mutual ignorance
with no discussion of the work in the other tradition, or, worse with
misrepresentation, as when Sperber extended his criticisms addressed to Dawkins
and memeticists to the work of Boyd and Richerson without attending to the
relevant differences between these two approaches.
In their article “On modeling cognition and culture,”
Henrich and Boyd (2002) open a serious discussion of the cognitive approach.
They overestimate, however, the points of divergence: we happen to agree with
much of what they present as objections. Let us, to illustrate this point, add
comment in square brackets and in italics to their concluding paragraph:[2]
The
crux of Sperber, Atran and Boyer’s position is that the transmission of culture requires
domain specific cognitive mechanisms [yes, with qualifications], and that therefore
population dynamic models of culture proceed from untenable assumptions [some population dynamic
models,
memetic ones in particular, proceed from untenable assumptions, but they need not; what we want is to contribute to improving these models, not reject them]. We accept that social
learning, like all other forms of learning, requires innate expectations about
objects in the environment and the nature of relationships among them. How these
innate structures shape the human mind is obviously of great importance for
understanding human culture. The mistake is to see these ideas as incompatible
with making population dynamic models of cultural change [this is a mistake we have never been tempted
to make]. It will never be enough
to focus on the mind and ignore the interactions between different minds [of course]. To keep track of such interactions
some kind of population dynamic models will be necessary. What is needed is
both more effort by coevolutionary theorists to incorporate rich cognition into
formal models of social learning, and more effort by cognitive scientists to
consider how innate cognitive structure interacts with social processes and the
cognition of social learning to influence the epidemiology of representations
and its associated behavioral products [total
agreement].
Henrich and Boyd
article presents and discuss three models. The second and the third models
illustrate the claims that population-scale conformity-biased and prestige-biased
transmission can play a role in compensating for high error rates in
inter-individual transmission and in securing adaptive cultural evolution, and
that discrete units of transmission are not necessary for this to happen.
Contrary to what Henrich
and Boyd seem
to expect, we[3] are in general agreement with these
claims.
Still, there is an important point of disagreement between
Henrich and Boyd and us regarding the respective roles of attraction and
selection in cultural evolution. They argue, with the use of the first model
presented in their article, that, to put it succinctly, in cultural evolution,
selection trumps attraction. We reply that what looks like a demonstration is
in fact based on quite inadequate modeling of attraction. Our response is in
two parts, a first part where the arguments are presented informally, and a
second, more formal part presenting and discussing models and simulations.
1 – The arguments
The idea of cultural attraction was introduced in
Sperber 1996, ch. 5. It is intended to help reconcile two observations:
1)
at the micro-level,
transmission of information among humans is generally not a copying process and
typically results in modifications of the information transmitted;
2)
at the macro-level, cultural
information is relatively stable within whole populations and often across
generations.
The micro-processes of transmission are not faithful enough to come
near explaining this macro-stability (unlike the faithfulness of gene
replication that does provide the core of the explanation of the relative inertia
of gene pools).
As we just mentioned, the approach defended by Henrich
and Boyd identifies mechanisms — conformity-biased and prestige-biased
transmission — that can contribute to the explanation of this macro-stability.
These mechanisms tend to favor some cultural contents not because of properties
of these contents, but because of their distribution in the population either
as contents adopted by the majority, or as contents adopted by the most
prestigious individuals. The idea of attraction, on the other hand, aims at
explaining the relative prevalence and stability of cultural contents as a
function of properties of the contents themselves. We believe that both kinds
of phenomena — distribution-based transmission biases and content-based
attraction — play a role in explaining cultural stability and evolution, and we
leave for another occasion the discussion of what their respective roles might
be.
Here is an account of the idea of cultural attraction simplified
as much as possible for the purpose of this discussion. When an individual
acquires a new cultural item (e.g. a skill, a belief, or a norm), she never
just copies the variant or variants she observes; rather, drawing on the
information transmitted and her own background knowledge, inferential
abilities, and interests, she construct a variant of her own. This variant is
likely to depart from the variants on which it is based both because some
information may be lost in the process, and because the goal of acquisition is
generally to acquire not a replica of other people’s variants, but, rather, a
piece of knowledge or a skill that suits the individual own dispositions and
preferences. It would be misleading therefore to talk of these departures from
model or models in cultural transmission as “failures to replicate”,
“mutations”, or “noise”. Even if these departures from the model often do
involve poor cognitive or behavioral performance, they occur not as accidents
or malfunctions but as normal outcomes of the constructive processes involved
in cultural transmission.
If each individual variant of a cultural item departed
at random from the variants that had inspired it (and in the absence or
insufficiency of compensating factors such as the biases described by Henrich
and Boyd), it is hard to see how cultural items would ever reach the minimal
level of stability within a population over time without which the very notion
of culture does not make sense at all. If, on the other hand, individual variants
do not depart at random from their model, but tend to gravitate around the same
positions in the space of possibilities, then, even without any strict replication
ever, one would end up with clusters of cultural items around these attractors
and therefore at least the modicum of stability that culture presupposes.
Attractors as points or areas in the space of
possibilities are abstract objects similar in this respect to proportions or
centers of gravity. They exist because there are concrete factors of attraction
that affect the probability that individual variants of a cultural item will
depart from their models in one direction rather than in another and that cause
all the variants of a given item to gravitate around the same point. Factors of
attractions can be of different kinds. At the most general level, they may have
to do with psychological dispositions or with environmental constraints and
affordances (contrary to what Henrich and Boyd suggest, it has never been part
of the theory that factors of attraction should be exclusively cognitive). Attractors
themselves can and do change over time as an effect of the factors that explain
them, but they change in historical time, that is, slowly enough to uphold the
relative stability of culture.
To illustrate in the simplest possible way (and in a manner
that will help us discuss Henrich and Boyd’s model) the idea of attraction and
its relationship to replication and selection, consider a schematic version of
the evolution of cigarette consumption in a population (see figure 1a — this is
not meant to be realistic, but just to make the idea more concrete, and the
presentation in the text of the article will be informal, with formal details presented
in Appendix 1). Members of some population smoke each between zero and 30 cigarettes
per day, so there are 31 variants of their smoking pattern. Every year, a new
age cohort of youngsters joins this population and select, from among the
members of the cohort just above them, a person whose smoking pattern they would
like to adopt. Depending on their smoking pattern, some people have a greater
probability than others of being selected as models to imitate. More
specifically, let us assume that relatively light smokers who smoke 10
cigarettes a day are the people most likely to be selected as models. This
probability of an individual being selected as model given his or her smoking
pattern is represented in figure 1a as a black curve.[4] New smokers, however, end
up, in less than a year, with a variant that may differ from that of the model
they selected. This is so for a variety of reasons, in particular because of
the lack of correct estimation of the smoking pattern of the people they chose
to imitate, because of carelessness in imitative behavior, and, above all, because
of the fact that smoking is an addictive acquired taste so that people tend
either not to smoke at all or to smoke more cigarettes than they intended to.
Figure 1 a: The cigarettes model, with two
peaks of attraction and one peak of selection (details in Appendix 1)
People’s smoking pattern is likely to depart from the variant
they selected not at random, but, we assume, in the direction of one of two
attractors. One attractor is abstinence, or zero cigarette, and the other,
based on the addictive properties of tobacco, is at 25 cigarettes per day. The
0-cigarette attractor has a strong effect on people who choose to imitate non-smokers
and who tend to remain non-smokers themselves, and also on people who select as
models smokers of one to five cigarettes per day, and who are likely to end up as
non-smokers. So, the 0-cigarette variant is a very strong but very local
attractor. Even so, some people decide to imitate a non-smoker but end-up,
through weakness of will, becoming smokers themselves. Attraction is a probabilistic.
The 25-cigarettes attractor is also quite strong and has much wider effect. The
people who select as models smokers smoking from 7 to 30 cigarettes per day
tend to end up smoking a number of cigarettes between the variant they selected
and 25. Even so, some people who decide to imitate a light or even a heavy
smoker end up non-smokers. Again, this is an improbable but not an impossible
outcome. The attractive force of different smoking patterns is represented in
figure 1a as a grey curve.
This toy model illustrates several interesting
properties and cases:
1)
The curve of attraction
indicates probabilities of transformation in one direction rather than another.
2)
A curve flat on both side of a
given variant (as around the 7-cigarettes variant) indicates that
transformations in either direction are equally probable.
3)
A curve slanted in the same
direction on both sides of the variant indicates that the variant is more
attractive than variants on the descending side and less attractive than variants
on the ascending side (as for, say, 15).
4)
An attractor is a peak in the
curve of attraction, such that the neighboring variants on both sides (or just
on one side, if it is at one end of the range of possibilities) are less
attractive than it is (as for 0 and 25).
5)
An attractor with very steep
curve on both sides (or just on one side, if it is at one end of the range of
possibilities) indicates that when this variant is selected as a model, it is
very likely to be replicated. In other terms a very steep attractor is
equivalent to a replicator (as for 0)
Imagine that each age cohort has 310 members and that,
in the initial cohort at time t0, each of the 31 variants is followed
by exactly 10 people. We can ask how the relative success of each variant will
evolve with successive cohorts. If there was only attraction and no selection,
we would expect after some time the distribution of smoking patterns to
correspond to the attraction curve. A simulation with 200 time steps and 10
runs confirms this prediction (see figure 1b). If there was only selection, no
attraction, and accurate copying of the model, we would expect to find that, after
a few time steps, the population is concentrated at the selection peak of 10 cigarettes/day,
and this is indeed what we found (this result being trivial, the data is not
shown). On the other hand, if there was selection but inaccurate copying of the
model, we would expect to find most of the population concentrated around the
selection peak and this is what we found (see figure 1c).
The more interesting situation is that where both attraction and selection are at work. Imagine that, in such a situation, we track the “descendants” — descent being through selection as a model — of an individual A smoking 8 cigarettes a day. We might observe that, because selection at this point is quite strong, 2 individuals in the second age cohort, B and C, select A as a model. Because, at that point,attraction is nearly symmetrical B might end up smoking 5 cigarettes, and C 10 cigarettes. Now, a third age cohort arrives and, because selection is lower for 5 cigarettes than for 10, only one individual, D, might select B (who smokes 5 cigarettes) as model, and 3 other individuals, E, F, and G, might select C (who smokes the 10 cigarettes). D, imitating the 5 cigarettes pattern, might end up smoking 0 cigarette since attraction toward 0 is high at that point. E, F, And G, imitating the 10 cigarettes pattern, might end up smoking 13,8, and 12 cigarettes respectively because attraction is relatively flat at that point. With such lines of descent, we should not be surprised if both selection and attraction had an effect of the distribution of the population among the various smoking patterns, with the 10-cigarettes pattern being better represented than if there were no selection, and the 0 and 25 patterns, and those in their neighborhood being better represented than if there was no attraction. This is indeed what we found (see figure 1d). Of course, with different parameters, we might render the effect of selection or those of attraction negligible, but the point we have illustrated so far is that, in principle, when both attraction and selection are at work, they may both have noticeable effects on the distribution of variants in the population.
Even without this example, it seems intuitively
implausible that, when both attraction and selection are involved in a cultural
evolution process, only attraction or only selection should systematically determine
the final outcome. Henrich and Boyd claim however to have demonstrated that, in
particular when attaction is strong, the final outcome is determined by
selection alone.
Figure 1b: The cigarette model with
attraction and without selection: distribution of the population after 200
steps (details in Appendix 1)
Figure 1c: The cigarette model with selection and inaccurate copying, and without
attraction: distribution of the population after 200 steps (details in Appendix
1)
Figure 1d: The cigarette model with both
attraction and selection: distribution of the population after 200 steps
(details in Appendix 1)
Henrich and Boyd, while granting the reality of
attraction, suggest that the dynamics of cultural evolution reduce to that of
replication and selection where selective forces determine the ultimate outcome.
If this were correct, the notion of attraction might still be relevant to a
detailed description of the processes involved—and in particular, as we will
see, of its initial stages—, but not to modeling the dynamics of cultural
evolution. The argument is based on the use of a formal model that scholars
interested in culture and cognition but with no competence in modeling may not
have fully understood, let alone felt confident enough to evaluate. They may have
been left with the idea that a demonstration had been given of a surprising and
even paradoxical conclusion that would severely limit the claim of relevance to
cultural evolution of the cognitive approach. This is not so. It is not so, to
begin with, because such models cannot yield such decisive conclusions. They are
great tools for asking novel questions about cultural evolution, imagining
possible answers, and sharpening our conceptual tools. They allow
demonstrations of what happens in the model. On the other hand, in the
absence of a clear methodology for judging the fit between the model and the
reality it purports to represent and to test non-trivial predictions of the
model on the basis of (preferably quantifiable) empirical evidence, these
models don’t demonstrate or even provide compelling argument about what is
actually the case in the real world. This should not be understood as a
criticism, but as a reminder. So, even if the model used by Henrich and Boyd
were adequate, what it would show — and this would be interesting enough — is
that attraction might work in a manner such that, quite generally, its
effects on cultural dynamics would collapse into those of replication plus
selection. As it happens, their model is, we believe, based on
misunderstandings and is not a good tool to explore the issue.
Henrich and Boyd’s model assumes a population whose
members hold mental representations the content of which is a value x represented
by real numbers between 0 and 1. During each time period, people in the
population choose each an individual as their model and try to acquire his or
her representation. However people construal of this representation is biased
towards one of two attractors, which are situated at the two ends of the
continuum, i.e. at 0 and at 1. There is an arbitrary cut-off point m between 0
and 1 such that, when the variant selected has a value between 0 and m, people
invariably end up with a representation that is closer to 0 than the variant
selected, and when the variant selected has a value between m and 1, people
invariably end up with a representation that is closer to 1 than the variant
selected (see figure 2 reproduced from Henrich and Boyd’s figure 1)
Figure 2: Henrich and Boyd’s model. Detailed description in section 2
To make all this a bit more concrete, let us translate
this into a version of our cigarette model (we take it that the fact that one
model involves a continuous variable between 0 and 1 and the other 31 discrete
variants between 0 and 30 is irrelevant to the issue at hand). We have the same
general situation regarding the transmission of smoking patterns as in our
initial model, but there are only two attractors at 0 and at 30 cigarettes, and
there is a cut-off point at, say, 17 cigarettes. People who decide to imitate
someone who smokes less than 17 cigarettes end up smoking even less than their
model, whereas people who choose to imitate someone who smokes 17 or more
cigarettes ends up smoking more than their model. There is no probabilistic
element left regarding the direction of attraction. Attraction is wholly in one
direction or wholly in the other. The population is therefore partitioned into
two groups, those under the 17-cigarettes threshold who are attracted towards
0, and those at or above this threshold who are attracted towards 30.
Whereas in our initial model, anyone at any variant
could be attracted in either direction and just the probability of transformation
in one direction rather than the other changed from one variant to another,
here the direction of transformation is a sure thing. This is not strong probabilistic
attraction, but deterministic attraction. Departing from Sperber’s notion of
“attraction” defined in terms of greater probabilities of transformations
towards, rather than away from a given point or “attractor” (Sperber 1996:112),
Henrich and Boyd’s understanding of “attraction” is not probabilistic but
determistic (an understanding possibly “attracted” towards the standard deterministic
notion of “attraction” in systems dynamics). They do talk of stronger or weaker
force of attraction, but actually, what they mean by “force” of attraction is
not the relative probability of departing from the model in one direction
rather than another, but the variable size of the departure from the model
always in one and the same direction, that of the attractor. With a “stronger
attractor” so understood descendents of a given variant will reach the
attractor in fewer steps than with a “weaker attractor”, but, in any case,
after a shorter or longer time interval, all items will be at an attractor, and
there will be no role left for attraction.
Deterministic cultural attraction is to regular,
probabilistic cultural attraction what black holes are to regular physical
attraction. Nothing ever gets out of a black hole. No line of cultural descent
ever moves in any direction other than that of its attractor. The descendants
of variants below 17 cigarettes will, after a few time periods, end up non-smokers
and stay so forever. The descendents of variants at or above 17cigarettes will,
after a few time periods, end up at 30 cigarettes per day and stay there
forever. As we noted, very steep
attraction — i.e. a much higher probability of change in one direction rather
than the other — culminates in attractors that are equivalent to replicators.
In Henrich and Boyd’s model not only are the two end points, 0 and 1 (or, in
our cigarette version of their model, 0 and 30) perfect replicators, but so are
also two other, less obvious traits, that of being attracted towards 0 and that
of being attracted towards 1 (0 and 30 in our version). No wonder that
replicator dynamics seems uniquely relevant to the evolution of the model!
What about selection in Henrich and Boyd’s model? They
assume that, in selecting whom to emulate, individuals are likely to prefer someone
whose representation has a higher value. The selective force increases continuously
from 0 to 1. As a result, people whose representation has a value above m
are all more likely to be selected as models to be imitated than any people
whose representation has a value below m, and people altogether most
likely to be selected as models are those with the representation 1, which also
happens to be an attractor. Translating into the cigarette model, this would
mean that the greater the number of cigarette an individual smokes, the greater
his or her likelihood to be imitated, with selective force, i.e. the
probability of being imitated, peaking at the maximum number of 30 cigarettes
per day. All variants at or above 17-cigarettes would be more likely to be
selected than any variant under that threshold.
Henrich and Boyd’s model has three relevant peculiarities:
1)
The variants in the model fall
into two groups, above and below a threshold, and the trait of belonging to one
or the other of these two groups strictly replicates.
2)
Attraction is determistically
towards 0 in the group below the threshold, and towards 1 in the group above
the threshold, which the effect that 0 and 1 are strict replicators.
3)
Selective force is wholly in
favor of the upper group and peaks at his attractor.
Given these three peculiarities, it should be intuitively clear that:
1)
With each time period, there
will tend to be more people with variants in the upper group selected as models,
until all the people have variants in this upper group.
2)
The variants in the upper group
will evolve toward the upper attractor until this perfect replicator is the
only variant represented in the population: deterministic attraction
self-eliminates.
3)
Moreover, if attraction is
strong enough, it self-eliminates in a few steps and, from early on, the
process is simply one of selection between two replicators.
So, in Henrich and Boyd model the only variant remaining in the end
is 1, and in the cigarette version, it is 30 cigarettes a day. The fact that,
in both versions, 0 was also an attractor does not make any difference to this
ultimate outcome, since selection favors the higher group and attractor .
Henrich and Boyd used formal considerations and equations, but, in fact, their conclusions regarding what happens in their model follow quite commonsensically from plain properties of this model that can be informally understood. However, nothing of interest follows regarding the relationship between attraction and selection in cultural evolution, because what obtains in this model is an artifact linked to the peculiarities of the model. To give just one intuitive illustration of this, there is no a priori reason why selective force should peak at an attractor (it does not in our initial cigarette model). Imagine, then, the following variation of Henrich and Boyd’s model: everything is as they describe it except that maximum selective force is at the threshold m, the selective force of the variants above and below the threshold have on average the same probability of being selected, and, in particular, the selective force of 1 and of 0 are equal. It should be intuitively obvious that, in this case, however strong the selective forces, they would not matter at all to the ultimate outcome, which would be exclusively determined by initial conditions, attraction, and drift (with all the descendents of variants below the threshold ending up at attractor 0, and all descendents of variants above the threshold ending up at attractor 1). If Henrich and Boyd had used this modified model (which is of course quite arbitrary, but so is their own model), and had generalized from it, they would have come to the surprising and equally unwarranted conclusion that, when you have both selective force and attraction at work, in the end, only attraction matters.
Even informally, it seems clear that the model used by Henrich
and Boyd has such peculiar properties (in particular the non-probabilistic
character of attraction and the coincidence of the selective peak with an
attractor) that it does not help, unlike many of other models developed by
Boyd, Richerson, and their collaborators (including the two other models in the
article under discussion), get a better grasp of questions and possible answers
in the study of cultural evolution. Henrich and Boyd’s model is even less
capable of giving any support to the implausible theoretical claim that, even
in the presence of strong attraction, only selection determines the final
outcome.
In the next section, we present a formal treatment of
our arguments and show that by manipulating the parameters of Henrich and Boyd
own model, one may reach very different conclusions. We first show that the
results of Henrich and Boyd do not depend on what they call the force of
attraction or of selection but just on the peculiarities of their model. We
then extend their model and show that, when the representation most selected does
not coincide with an attractor, the outcome is not anymore that predicted by
selection alone. And finally, by making attraction probabilistic, we show that,
in general, the outcome depends on the relative strength of both attraction and
selection.
2 – Models and simulations
2.1 – Confirming Henrich and Boyd’s own
results
First we replicated Henrich and Boyd’s
own simulation, using the same parameters (see figure 3a). This served both to
confirm their results and to establish that we were following the same procedure.
What is represented here (and in
figure 2 above borrowed from Henrich and Boyd), is the evolution of a pool of
mental representations in a population. The content of these representations is
a real number x between 0 and 1. During each time period, people in the population observe the
behavior of another individual, infer from this behavior the mental
representation of the model, and adopt the mental representation they have inferred their model must have. Not all individuals have
the same probability of being selected as model. Rather, the probability that an individual
be selected as a model increases with the value of his or her representation
and equals 1+sx. People’s inferences are moreover biased towards
attractors, which
happen to be x
= 0 and x = 1. As a result, instead of inferring the
actual value
of a representation x, people interpret it as
having the value x + Δx. Which of the two attractors biases the
interpretation of a given representation x is determined by a point m between 0
and 1 that marks the limit between the two basins of attraction of the two
attractors. If x is greater than m, it is attracted toward attractor 1. If x is
smaller than m, it is attracted toward attractor 0. The “force” of attraction —
we have questioned this use of the notion of force in the first section and
won’t raise the issue again here — is expressed by a number, β0
for attractor 0 and β1 for attractor 1. If x < m, then Δx
= - β0*x, and if x > m, then Δx = β1*(1-x).
Using the same parameters as Henrich and Boyd (i.e., m =
0.6, s = 0.05, β0 = β1 = 0.5, n = 200), we
indeed replicate their results. The evolution of the pool of representations
fits the prediction of replicator dynamics, and attraction plays a negligible
role. Before reading too much into this result, one should pay attention to the
two curves indicating the average value of x in group 0 (containing all and
only variants below m) and in group 1 (containing all and only variants above
m). They indicate that after about 10 time periods
(see the shaded area), all the representations have either the value 1 or the
value 0 and are not anymore subject to attraction. From the 10-steps point in
the time scale, the process involves only replicators and there is no way
attraction could play any role at all. Given this, the fact that the dynamics
at work is plain replicator dynamics is quite trivial. As selection favor
representations with value 1 over representations with value 0, in the end, all
representations have a value of 1 (as can be seen from the distribution at time
t = 250).
Figure 3a: Replication of the simulation used by Henrich and Boyd in
support of the claim that weak selection override even strong attraction The
left frame represents the evolution through time of values of x as observed and
as predicted by replicator dynamics with the following parameters: m = 0.6, s =
0.05, β0 = β1 = 0.5, n = 200. With these
parameters, attraction self-eliminates in about 10 time steps (shaded area). Thereafter
(unshaded area), only selection is at work. The right frame represents the
distribution of representations after 250 time steps for the 10 simulations.
2.2 – When attraction is weaker or when selection is stronger: Same
outcome
What would happen if attraction was much “weaker” in Henrich
and Boyd sense, while still being non-probabilistic? Intuitively, it would take
many more steps to eliminate the impact of attraction, but, selection would still
be the sole determinant of the final outcome. We performed a simulation with
the same value as before except for β0 and β1
which were divided by 20. As the shaded area in figure 3b shows, it does take
more steps to get rid of the values between 0 and 1, and during all these
steps, the dynamic of the population does not follow replicator dynamic. However,
once practically all representations have values 0 or 1 and are therefore not
subject to attraction anymore, the dynamics converges with replicator dynamics
and the end result is solely determined by selection (see the distribution
graph).
Figure 3b: If attraction is weak,
it takes more steps (shaded area) for it to self-eliminate. Still, once all
representations have converged to 0 or 1, selection determines the same outcome
as previously. The left frame represents the evolution through time of values
of x as observed and as predicted by replicator dynamics with parameters as in
Fig. 3a except β0 = β1 = 0.025. The right frame
represents the distribution of representations after 250 times step for the 10
simulations.
Raising the selection by increasing s does, on the other
hand, make the population dynamics even closer to that of replicators, and the
equilibrium is reached much faster (since this result is quite trivial, the data
is not shown).
So far, our simulations show that the end result of the
model of Henrich and Boyd does not depend on the force of either attraction or
selection. The claim that the final outcome is determined only by selection is
in fact related to two artifacts of the model: first attraction is non
probabilistic and second selection happens to favor an attractor. What would
happen if we altered these two special features of Henrich and Boyd’s model?
2.3 – When selection does not peak at an attractor: Different
outcome
We believe that Henrich and Boyd’s would-be
demonstration that selection determines the final outcome irrespective of
attraction is an artifact of their choice of selective function and, even more
importantly, of the non-probabilistic character of attraction in their model.
We first present simulations where we leave their attraction parameters
untouched but where we modify their selection function and in particular their
selection peak.
There is no principled reason to assume that attractors, that is, points towards which transformations tend to be biased, should coincide with variants most likely to be selected as models. After all, in real life, people typically choose as models the most skilled performers (craftsmen, warriors, artists, and so on) even though their own performance tends to be biased towards easier and less admirable outcomes. Henrich and Boyd used a linear function of x as the selective function (viz. w(x) = 1 + s*x) which makes the value 1, which happens to be an attractor in their model, the one most likely to be selected. To keep attraction and selection properly apart, we used a Gaussian function of x as the selective function: w(x) = exp(-(x-µ).^2/(2*σ^2)). In Fig 4, it is holders of the representation x = 0.7 who are the most likely to be chosen as models. However, far from converging towards 0.7, in fine all representations have a value of 1, that is, the value of one of the two attractors. Why it should be so is not mysterious. The selection peak (0.7) is above m (0.6), and therefore variant 1 is favored by selection over variant 0. In group 1 however, the force of selection is dominated by that of deterministic attraction, and variants favored by selection are eliminated in favor of variants favored by attraction, i.e. variants with the value of 1. In this case therefore the final outcome is the combined effect of attraction, which eliminated all variants other than 0 and 1 (including 0.7, the variant most favored by selection), and of selection, which favored 1 over 0 (see figure 4).
Figure 4: Selection peaks at x =
0.7, while the attractors are at 0 and 1. Because selection favors values
closer to 1 over values closer to 0 the mean representation value in the
population converges toward 1. The left frame represents the evolution through
time of values of x as observed and as predicted by replicator dynamics with
the following parameters: µ = 0.70, σ = 2, β0 = β1
= 0.5, m = 0.6, n = 200. The shaded area corresponds to the time span where
attraction has some effect. The right frame represents the distribution of
representations after 250 times step for the 10 simulations.
2.4 – When attraction is probabilistic: Different
outcome
The very idea of attraction is intended to capture the observation that, in cultural transmission, departures from the model are not purely random and tend to be biased in certain direction. To reintroduce stochasticity in the idea of attraction while staying as close as possible to Henrich and Boyd model, we allow for the representation value acquired by an individual to vary between an interval of [x – r + Δx ; x + r + Δx].[5] To help visualize the effect of this probabilistic reinterpretation of attraction, we show in figure 5a the lines of descent of three individual representations: two obeying a non-probabilistic force of attraction à la Henrich and Boyd and beginning, one, just above the cut-off point m, and the other just below it, and a third representation with a random initial value and subject to probabilistic attraction. Without some positive degree of randomness, attraction is a deterministic mechanism that drives representations values toward 0 or 1 at a speed depending on the ‘force’ of attraction (in figure 5a attraction toward 0 is 3 times ‘stronger’ than attraction toward 1). With randomness, attraction is the probability for a representation to have a certain value given the value of the model from which it is inferred. As the figure well illustrates, with probabilistic attraction all values have a certain probability of being reached. But since, in this model, the attraction bias towards 0 is three times greater than the one towards 1, overall, values closer to 0 are more often reached.
Figure 5a: Attraction with and
without a degree of randomness. The two thick lines represent the lines of
descent, in the absence of randomness, of two representations, one with an
initial value above m (here 0.6) converging toward 1, and the other with an
initial value below m converging toward 0. The thin line represents the line of
descent, with a degree of randomness added to attraction, of a representation
with an arbitrary initial value. All values between 0 and 1 can be reached by
this line of descent. Parameters are as follows: r = (0 for thick lines and 0.2
for thin one), β0 = 0.1, β1 = 0.03, m = 0.6.
If we represent now the whole
population (n=200) with probabilistic attraction and otherwise the same
parameters as in figure 5a, we observe that all values are reached but that
they are more or less represented depending on the force of attraction.
Figure 5b: Evolution of the
population with no selection and probabilistic attraction three times stronger
toward 0 than toward 1. The left frame represents the evolution through time of
values of x as observed with the same parameters as in 5a except r=0.2. The
right frame represents the distribution of representations after 250 times step
for the 10 simulations.
What if we add to the parameters of figure 5b a weak selection force peaking at 0.7? Both selection and attraction are important factors, with selection favoring values close to 0.7 and attraction favoring values close to 0 or to 1. Because attraction remains dominant, the most often selected variants (close to 0.7) are immediately attracted toward 1 or 0 (see figure 5c). If we increase selection, we expect values around 0.7 (and therefore also around 1) to be better represented. Strong selection may indeed force the dynamics to look like replicator dynamics for mean values, but attraction remains crucial to account for the distribution we observe at equilibrium (fig 5d). Only with selection quite strong and probabilistic attraction quite weak could attraction be ignored. In general however, when you have both attraction and selection at work, both contribute to the evolution of the population. If Henrich and Boyd had shown otherwise, it would indeed have been surprising, but they have not.
Figure 5c: Adding weak selection
to attraction changes the distribution of representations in the population
(see fig 5b for comparison) but it does not bring the population dynamic close
to replicator dynamics. Both selection and attraction are important to explain
the equilibrium distribution we observe (see the right frame). Selection favors
values close to 0.7 and attraction values close to 0 or 1. Parameters are as
follows: µ = 0.7, σ = 1.5, r = 0.2, β0 = 0.1, β1
= 0.03, m = 0.6, n = 200.
Figure 5d: Stronger selection may
drive the dynamic closer to the replicator dynamic (see fig 5c and 5b for
comparison) but it still does not account for the distribution we observe in
the right frame. Parameters as in Fig 5c, except σ = 0.4
APPENDIX 1: The cigarette model
The ‘cigarette model’
informally presented in the text was meant to illustrate as simply as possible
ordinary relationships between attraction and selection. Here we explain the
model in more technical details.
Principles
Members of a population
may each smoke between 0 and 30 cigarettes a day, so there are 31 different
cigarettes patterns. Initially each smoking pattern is equally represented by
10 individuals, thus the size of the population is 310. Every year, a new age
cohort of 310 youngsters joins this population and each select, from among the
preceding age cohort, the individual whose smoking pattern he or she want to imitate.
Imitation is imperfect and individuals typically end up, in less than a year,
with a smoking pattern different from that of the individual they chose to
imitate. Departure from the model are not purely random and tend to be in the
direction of attractors. Thus, the first uniform distribution progressively
changes with time due to both selection and attraction.
Selection
Depending on their
smoking pattern, some people have a greater probability than others of being
selected as models to imitate. More precisely, we suppose that the likelihood
of an individual smoking x cigarette a day to be selected as a model is given
by the following function:
W(x) is greatest for x = 10 and decreases before and after that peak value (see the selection curve in figure 1a). This simply means that people smoking 10 cigarettes a day have a higher chance of being selected as models than others. In particular, if selection alone were at work and imitation were accurate, other smoking patterns, because of their lower probability of being selected, would progressively disappear, and all individuals would end up smoking 10 cigarettes per day.
Randomness
Imitation, however is
not perfect. Consider first the case where the probability of a departure from
the model is equal in both directions (towards smoking a greater or a lesser
number of cigarettes than the model) and decreases with the distance from the
model. For instance, an individual trying to imitate a person who smokes 10
cigarettes a day, has the same probability to end up smoking 8 or 12 cigarettes
and a lesser probability of ending up smoking 6 or 14 cigarettes than 8 or 12. To
model this case, we define a probability function r(y,x):
Here x is the value selected and r(y,x) is the
probability that an individual having selected a model smoking x cigarettes a
day ends up smoking y cigarettes a day (y varying between 0 and 30). Notice,
that whatever the smoking pattern of the individual imitated, the imitator may
end up with any of the 31 patterns, but the probabilities are quite different
for each pattern. For instance, if individual A selects as model an individual
smoking 5 cigarettes a day, the probability that A will smoke 6 cigarettes by
the end of the year is r(6,5) = 0.17, while the probability that A will end up
smoking 10 cigarettes a day is r(10,5) = 0.09. Given that the first age cohort
is uniformly distributed and the probability of going either to the left or to
the right is the same, we would of course expect, in the absence of selection,
to find a uniform distribution of patterns. With randomness combined with
selection ( and W(x) as characterized above), we find the pattern illustrated
in figure 1c: most of the population is
concentrated around the selection peak, as one would expect.
Probabilistic attraction
We are interested in the
case where people’s smoking pattern is likely to depart from the variant they
selected not at random, but, we assume, in the direction of two attractors (0
and 25). We stipulate that people smoking less than 5 cigarettes are strongly
attracted toward 0 and people smoking more than 5 cigarettes are progressively
attracted toward 25. To represent this case, we redefine the probability
function r(y,x) as follows:
In this equation, the first term 
represent the attractor
0. Thus, r(y,x) is high when y is close to 0 and decreases rapidly when y
increases. The second term 
represents the attractor
25. Thus, r(y,x) is high when y is close to 25 and decreases progressively as y
depart from 25 (see figure 1a, the first term is mainly responsible of the part
below 5 of the attraction function, the second of the part above 5). Finally,
the third term, 
, correspond to the previous randomness function. Now for
instance, the probability that an individual selecting a person smoking 5
cigarettes a day as model should end up smoking 6 cigarettes r(6,5) = 0.15 is
lower than the probability of that individual ending up smoking 4 cigarettes
r(4,5) = 0.17 because attraction is lower towards 1 than towards 0 at this
point. As before, r(y,x) is never 0 which means that there is always a certain
probability to end up smoking any given pattern. What we expect, if attraction
is acting alone (that is, without selection,) is that the most frequent
patterns will be 25 and 0 cigarettes and those close to them (see figure 1b).
Considering both attraction and selection
If we have both selection
and probabilistic attraction (each with the parameters specified above) in
play, we would expect both to affect the distribution of variants in the long
run and indeed this is what we observe (see figure 1d).
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