Cultural group selection for group-enthusiast physicists

This post is part of a series on understanding the models of cultural group selection, leading to Boyd et al. 2016's BBS paper and Joe Henrich's latest book The Weirdest People in the World: How the West Became Psychologically Peculiar and Particularly Prosperous (2020). The presentation of the formalism draws heavily from McElreath & Boyd 2008, ch.6.

What explains large-scale collaboration among unrelated individuals? For anthropologists, It is a mystery because reciprocity at that scale does not follow from our most beloved models in biology, aka evolutionary game theory and population genetic models. To solve this cooperation problem, they recast inclusive fitness theory into cultural group selection (CGS). Building on cultural evolution theory, they assert that (some) cultural behaviors have evolved because they provide group-level benefits, and these are crucial in explaining the evolutionary success of humans. What do they mean by that?

First, what are group-level cultural traits. In his book on the The WEIRDest People in the World (that’s us Westerners), Joe Henrich tells the story of how Ilahita, a traditional society from New Guinea, got big for that time and place (≫300 people). How? A collection of stories, myths, and rites of terror that have facilitated interactions among distantly related households.

For instance, every household raises pigs, but they find that eating their own pigs is disgusting because it would be like eating one of their own’s kid. This means that groups must exchange pigs at communal ceremonies. Furthermore, those ceremonies involve rites of passage for boys to become men (meaning boys can marry, learn about secret ritual knowledge, and climb the political ladder). The catch is that those rites must be performed by an opposite ritual group, meaning your success isn’t solely dependent on your own group.

Once you start looking for organizational norms, there are more than you know. llahita’s best of includes infusing terrors in their rites of passages because the Tambaran gods demand it (including going out and killing men from enemy communities), undertaking large community works (with some music-making and synchronous dance involved), and finally the ability to punish people who are doing ritual performance wrong.

From a network perspective, the idea of exchanging pigs with other groups, arguably, make more sense at the level of groups than at node-level. This is about how groups should behave towards each other. Of course, this is ingrained in individual psychology; people choose to punish one another because they don’t do what the gods want or experience disgust at the thought of eating their own pig. But sometimes stories and behaviors can be best explained at group-level.

But where is the math showing that, physicists ask.

Cultural group selection, actually

CGS is rooted in population genetics. Population genetic models are all about keeping track of copies of different alleles that might increase or decrease fitness under the effect of natural selection. A key relationship (theorem? model?) in the field is the Price equation:

\bar{w} \triangle p = \text{cov} (w_{i}, p_{i})+ E \big(w_i \triangle p_i)

Price equation

Recall from statistics that covariance takes different forms. It can be written as the average over the product of individual traits and fitness minus the product of their averages,

\text{cov}(w_i,p_i) = E(p_i w_i) - E(p_i)E(w_i)

It can also be written with a cofficient regression,

\text{cov}(w_i,p_i) = \text{var}(p_i)\frac{\text{cov}(w_i,p_i)}{\text{var}(p_i)} = \text{var}(p_i)\beta(w_i,p_i)

, revealing how without variance in allele frequencies you cannot have selection.

where wᵢ is individual fitness and pᵢ is the trait value. When assuming diploid individuals (two sets of chromosome only), together with low mutation rate, Price's equation boils down to

\bar{w} \triangle p = \text{cov} (w_{i}, p_{i})

. The price equation states that the average fitness of the population times the change in p is given by how individual trait value and fitness deviate together from their mean across generations. See Primer's blobs for a great video on the topic.

Cultural evolutionists now make a bold move to natural selection could (in principle) favor group-level traits, even in the face of individual cost:

The mean number of copies by group is

w_g = \frac{1}{n_g} \sum_i w_{ig}

, where n_g and p_g is the number of individuals and the frequency of realized behavior, respectively, in group g.

On the left, we represent a diploid individual, with its two set of genes that could be realized into one of two behaviors. In this world, we track the changes in individual allele frequency over generations, regardless of the population structure. On the right, we have two individuals as part of a group, each with a realized behavior. Below, we denote p_ig as the frequency of individual ᵢ with realized behavior, labeled S, in group _g. Accordingly, w_ig refers the number of copies of that behavior in that group.

Cultural group selection is a multilevel selection framework, which means we can decompose the above into nested components.

\bar{w} \triangle p

\text{cov} (w_{g}, p_{g})

E(w_{g} \triangle p_{g})

In group-land, the change of frequency of S is now equal to the following two terms. First, the covariance between allele frequency in g and the mean fitness in group g. Yet again, we are saying that selection on groups depends on var(p_g), or variance in S across groups.

\bar{w} \triangle p

\text{cov} (w_{g}, p_{g})

E(w_{g} \triangle p_{g})

The second term says that there is some change in frequency of S that comes from the average change in allele frequency within the group. In words, this change within group is an expectation over the product of the change in allele frequency in group g and its mean fitness. This term is where the individual component of natural selection sneaks back in.

w_{g} \triangle p_{g}

\text{cov} (w_{ig}, p_{ig})

\cancel{E(w_{ig} \triangle p_{ig})}

The product of the change in allele frequency in group g and its mean fitness, in turn, can be thought of as the sum of a covariance and the expectation of the change in frequency of allele at individual level. This time around, the covariance is between the allele frequency of individual i in group g and individual fitness of i in g. As we did before, if we assume no meiotic drive and low mutation rate within group at individual level (remember, this term disappear with haploid individuals), we get rid of the second term and plug back this expression in the original term. We get

\bar{w} \triangle p

\text{cov} (w_{g}, p_{g})

E(\text{cov} (w_{ig}, p_{ig}))

(General form of Price equation;
see Mcelreath & Boyd 2008 p.229 for a more detailed derivation)

This is the core of the multilevel selection idea of CGS. Basically, you can decompose the selection effect on behavior S across groups and individuals, looking at which part explains variance the most, as with something like ANOVA. Using the fact that we can write covariance such as $\text{cov}(x,y) = \text{var}(x)\beta(y,x)$ , cultural evolutionists like to use the following, more practical form

\bar{w} \triangle p_g = \text{var} (p_g) \beta(w_{g}, p_{g})+E\big[\text{var}(p_{ig}) \beta (w_{ig}, p_{ig})\big]

(Showing the regression coefficient)

what really matters is the relative strenght of selection within and between groups. The two terms have inverse signs, meaning that when one goes up the other one needs to go down. In Boyd et al. 2016, their main sketch of quantitative evidence is actually in the form of

\frac{\text{Group benefit}}{\text{Individual cost}} > \frac{1-F_{st}}{F_{st}}

(Showing the regression coefficient)

where F_st is the fraction of total variance that is between groups, aka var(p_g). The punchline should be clearer now,

For selection to favor group-level traits in the face of individual costs, we need a system that promote variation between groups while maintaining low variation within groups.

Anthropologists argue that group-based cultural systems are like that.

Return to Ilahita

We briefly come back to Ilahita’s stories to clarify a few points about the scope of CGS, mechanisms, and levels of selection.

Beyond stories and rites of terrors, Henrich argues that Ilahita’s scaling up is due to intergroup competition among (39) clans.

Intergroup competition is hypothesized as a key driver of cultural group selection. For instance, violent conflict such as wars is a great way to preserve strong between-group competition (increasing p_g) while maintaining conformity among your rank (reducing p_ig). Other mechanisms from cultural evolution theory further promote conditions for CGS to work, such as exhibiting preferentially learning from your peers (biased social learning; further reducing p_ig). We discuss in more depth the mechanism of CGS in another entry.

For fun and glory: the many lives of the Price’s equation

WIP; i'll add more interpretations and references when I have time

\triangle E_{i\in I}(z_i) = \text{cov}_{i\in I} (w_{i}, z_{i})+ E_{i \in I}(w_{i} \triangle z_i)

(Price equation; Gardner 2021)

In a nutshell, we are thinking about the covariance between average group fitness and average group allele frequency.

I want more!

Cultural group selection is connected to the following topics in McElreath & Boyd’s book:

Reciprocity and collective action (ch. 4.5)
- Altruistic punishment; second-order dilemma (punishing those who failed to punish; ch. 4.5.2). Second-order dilemma could be a nice use of adaptive higher-order interactions.
- More generally, n-person games (ch.4.5.1) and repeated interactions (ch. 4.1.1)
Costly signal theory; what if people fake their intent (ch.5.1)

Some papers making use of CGS:

Reviews of CGS:

Smith et al. 2020

In the next part of the series, we look at how CGS can relate to group-based master equations.

Humans & Golems

Cultural group selection for group-enthusiast physicists

Cultural group selection, actually

Return to Ilahita

For fun and glory: the many lives of the Price’s equation

I want more!