The problem with the Price equation

Abstract

In this paper, I will argue that the generality of the Price equation comes at a cost, and that is that the terms in it become meaningless. There are simple linear models that can be written in a Price equation-like form, and for those the terms in them have a meaningful interpretation. There are also models for which that is not the case, and in general, when no assumptions on the shape of the fitness function are made, and all possible models are allowed for, the regression coefficients in the Price equation do not allow for a meaningful interpretation. The failure to recognize that the Price equation, although general, only has a meaningful interpretation under restrictive assumptions, has done real damage to the field of social evolution, as will be illustrated by looking at an application of the Price equation to group selection.

This article is part of the theme issue ‘Fifty years of the Price equation’.

1. Selection and covariance

In order to illustrate the problem with the Price equation, I will first present three simple examples of models of selection. For all three, I will derive formulae that tell us what one step of selection does to the frequency of a set of genes. Then, I will write the Price equation and compare it to the three formulae. The Price equation is general, and does not make assumptions about how genes determine fitnesses. We will see that this generality comes at a cost; in contrast with the formulae for the three specific models, the Price equation does not allow for a meaningful interpretation of the terms in it, and is prone to misinterpretation. The Price equation is usually applied to models, but some language and concepts are borrowed from statistics, so we will also look at what the Price equation would do when applied to data.

In §2, we will do the same for a model of group selection.

(a). Models

(i). Model A

Take a very simple model of selection, where reproduction is asexual, and where some trait is determined by alleles at n loci. At locus j = 1, …, n individual i can have an allele that increases the trait by one unit (x_ij = 1), or one that does not (x_ij = 0). Fitness is linear in that trait, so effectively fitness is linear in the average $p_i=(1/n)\sum _{\hspace{0.17em}j=1}^n{x}_{ij}$, or, in terms of Price [1], the dose of the gene, or, in terms of Grafen [2], the p-score. In other words, if individual i has p-score $p_i=(1/n)\sum _{\hspace{0.17em}j=1}^n{x}_{ij}$, its fitness is $w_i=a+b\cdot p_i=a+b\cdot (1/n)\sum _{\hspace{0.17em}j=1}^n{x}_{ij}$ for some constants a and b.

If this is how selection operates, then what one round of selection does to the frequency of these alleles depends on the current state of the population. Suppose there are individuals i = 1, …, m with p-scores $p_i=\sum _{\hspace{0.17em}j=1}^n{x}_{ij}$. The current frequency of trait-increasing genes is

$$\overline{\hspace{0.17em}p}=\frac{1}{m}\sum _{i=1}^m{p}_i=\frac{1}{m}\sum _{i=1}^m\frac{1}{n}\sum _{\hspace{0.17em}j=1}^n{x}_{ij}.$$

The frequency of the trait-increasing genes in the offspring population is

$$\frac{\sum _{i=1}^m{p}_i{w}_i}{\sum _{i=1}^m{w}_i}=\frac{\sum _{i=1}^m{p}_i\cdot (a+b{p}_i)}{\sum _{i=1}^m(a+b{p}_i)}.$$

The change in frequency therefore is equal to

$$\mathrm{\Delta }p=\frac{\sum _{i=1}^m{p}_i\cdot (a+b{p}_i)}{\sum _{i=1}^m(a+b{p}_i)}-\frac{1}{m}\sum _{i=1}^m{p}_i$$

and therefore

$$\frac{\sum _{i=1}^m(a+b{p}_i)}{m}\mathrm{\Delta }p=[\frac{\sum _{i=1}^m{p}_i\cdot (a+b{p}_i)}{m}-\frac{\sum _{i=1}^m(a+b{p}_i)}{m}\frac{1}{m}\sum _{i=1}^m{p}_i].$$

If we write average fitness as

$$\overline{w}=\frac{\sum _{i=1}^m(a+b{p}_i)}{m},$$

we can rewrite this as

$$\begin{array}{rl}\overline{w}\mathrm{\Delta }p& =\frac{\sum _{i=1}^m{p}_i\cdot (a+b{p}_i)}{m}-(a+b\cdot \frac{1}{m}\sum _{i=1}^m{p}_i)\frac{1}{m}\sum _{i=1}^m{p}_i\\ & =\frac{\sum _{i=1}^m{p}_i\cdot b{p}_i}{m}-b\cdot \frac{1}{m}\sum _{i=1}^m{p}_i\frac{1}{m}\sum _{i=1}^m{p}_i\\ & =b\cdot [\frac{1}{m}\sum _{i=1}^m{p}_i^2-\frac{1}{m}\sum _{i=1}^m{p}_i\frac{1}{m}\sum _{i=1}^m{p}_i].\end{array}$$

If we write

$$\text{Var}(p)=\frac{1}{m}\sum _{i=1}^m{p}_i^2-\frac{1}{m}\sum _{i=1}^m{p}_i\frac{1}{m}\sum _{i=1}^m{p}_i,$$

we can shorten this to

$$\overline{w}\mathrm{\Delta }p=b\cdot \text{Var}(p).$$ 1.1

In earlier papers, I have pointed out that using the term ‘variance’ here is not entirely appropriate [3,4]. The term that is abbreviated to Var(p) here, really is a number that characterizes an aspect of the population state, and it is not the variance of a well-defined random variable. We can make it one, by drawing a random individual and letting the random variable be the p-score of that individual. But either way, it is important to realize that this number sums up a relevant aspect of the parent population, and nothing more.

If the fitness of individuals is linear in p_i, as it is in this model, then $\overline{w}\mathrm{\Delta }p$ is linear in Var(p). In other words, if we have two parent populations, with ‘variances’ Var₁(p) and Var₂(p), then ${\overline{w}}_1{\mathrm{\Delta }}_{1}p/{\overline{w}}_2{\mathrm{\Delta }}_{2}p={\text{Var}}_1(p)/{\text{Var}}_2(p)$. If the fitness of individuals does not follow this model, but a different one, then we cannot use w_i = a + bp_i, we do not arrive at the final equation, and $\overline{w}\mathrm{\Delta }p$ is not guaranteed to be, and typically is not, proportional to Var(p).

(ii). Model B

Assume that reproduction is again asexual, and that now there are two relevant traits. One is, as before, determined by alleles at n loci, and the p-score of individual i is $p_i=\frac{1}{n}\sum _{\hspace{0.17em}j=1}^n{x}_{ij}$. The value of the second trait, or the q-score of individual i is denoted by q_i. Fitness is linear in both traits; w_i = a + b · p_i + c · q_i for some constants a, b and c.

For this model, the change in frequency is equal to

$$\mathrm{\Delta }p=\frac{\sum _{i=1}^m{p}_i\cdot (a+b{p}_i+c{q}_i)}{\sum _{i=1}^m(a+b{p}_i+c{q}_i)}-\frac{1}{m}\sum _{i=1}^m{p}_i$$

and therefore

$$\begin{array}{rl}& \frac{\sum _{i=1}^m(a+b{p}_i+c{q}_i)}{m}\mathrm{\Delta }p\\ & =[\frac{\sum _{i=1}^m{p}_i\cdot (a+b{p}_i+c{q}_i)}{m}-\frac{\sum _{i=1}^m(a+b{p}_i+c{q}_i)}{m}\frac{1}{m}\sum _{i=1}^m{p}_i].\end{array}$$

If we write average fitness as

$$\overline{w}=\frac{\sum _{i=1}^m(a+b{p}_i+c{q}_i)}{m},$$

we can rewrite this as

$$\begin{array}{rl}\overline{w}\mathrm{\Delta }p& =\frac{\sum _{i=1}^m{p}_i\cdot (a+b{p}_i+c{q}_i)}{m}\\ & -(a+b\cdot \frac{1}{m}\sum _{i=1}^m{p}_i+c\cdot \frac{1}{m}\sum _{i=1}^m{q}_i)\frac{1}{m}\sum _{i=1}^m{p}_i\\ & =b\cdot [\frac{1}{m}\sum _{i=1}^m{p}_i^2-\frac{1}{m}\sum _{i=1}^m{p}_i\frac{1}{m}\sum _{i=1}^m{p}_i]\\ & +c\cdot [\frac{1}{m}\sum _{i=1}^m{p}_i{q}_i-\frac{1}{m}\sum _{i=1}^m{p}_i\frac{1}{m}\sum _{i=1}^m{q}_i].\end{array}$$

If we write

$$\text{Var}(p)=\frac{1}{m}\sum _{i=1}^m{p}_i^2-\frac{1}{m}\sum _{i=1}^m{p}_i\frac{1}{m}\sum _{i=1}^m{p}_i$$

and

$$\text{Cov}(p,q)=\frac{1}{m}\sum _{i=1}^m{p}_i{q}_i-\frac{1}{m}\sum _{i=1}^m{p}_i\frac{1}{m}\sum _{i=1}^m{q}_i,$$

we can shorten this to

$$\overline{w}\mathrm{\Delta }p=b\cdot \text{Var}(p)+c\cdot \text{Cov}(p,q).$$ 1.2

With the same reservation as before concerning the use of the terms ‘variance’ and ‘covariance’, this equation sums up the model. There are model parameters b and c, they are constant, and they reflect how fitness depends on trait values p and q. Besides those, there are Var(p) and Cov(p, q) that represent properties of the population state, which may, and typically will, change as selection happens, and that differ between different (hypothetical) population states.

The simple model from §1ai is nested in this one; if c = 0, then equation (1.2) reverts to equation (1.1). It may be worth noting that c being 0 and Cov(p, q) being 0 are very different reasons why the last term in equation (1.2) would become 0. In the first case, the value of q has no effect on anything, and the last term is 0 for any population state. In the second case, the value of q does have an effect on fitness, which for some population states—those with Cov(p, q) = 0—ends up having no effect on the total Δp.

(iii). Model C

Assume that reproduction is again asexual, and that there are again two relevant traits. Fitness, however, now depends on them in a different way, as the traits only have an effect on fitness if they are combined; w_i = a + d · p_iq_i for some constants a and d.

The change in frequency for this model is equal to

$$\mathrm{\Delta }p=\frac{\sum _{i=1}^m{p}_i\cdot (a+d{p}_i{q}_i)}{\sum _{i=1}^m(a+d{p}_i{q}_i)}-\frac{1}{m}\sum _{i=1}^m{p}_i$$

and therefore

$$\begin{array}{r}\frac{\sum _{i=1}^m(a+d{p}_i{q}_i)}{m}\mathrm{\Delta }p\\ & =[\frac{\sum _{i=1}^m{p}_i\cdot (a+d{p}_i{q}_i)}{m}-\frac{\sum _{i=1}^m(a+d{p}_i{q}_i)}{m}\frac{1}{m}\sum _{i=1}^m{p}_i].\end{array}$$

If we write average fitness as

$$\overline{w}=\frac{\sum _{i=1}^m(a+d{p}_i{q}_i)}{m},$$

we can rewrite this as

$$\begin{array}{rl}\overline{w}\mathrm{\Delta }p& =\frac{\sum _{i=1}^m{p}_i\cdot (a+d{p}_i{q}_i)}{m}-(a+d\cdot \frac{1}{m}\sum _{i=1}^m{p}_i{q}_i)\frac{1}{m}\sum _{i=1}^m{p}_i\\ & =d\cdot [\frac{1}{m}\sum _{i=1}^m{p}_i^2{q}_i-\frac{1}{m}\sum _{i=1}^m{p}_i{q}_i\frac{1}{m}\sum _{i=1}^m{p}_i].\end{array}$$

If we write

$$\text{Cov}(p,pq)=\frac{1}{m}\sum _{i=1}^m{p}_i^2{q}_i-\frac{1}{m}\sum _{i=1}^m{p}_i\frac{1}{m}\sum _{i=1}^m{p}_i{q}_i,$$

we can shorten this to

$$\overline{w}\mathrm{\Delta }p=d\cdot \text{Cov}(p,pq).$$ 1.3

With the same reservation as before concerning the use of the term ‘covariance’, this equation sums up the model.

(iv). Model D

For completeness, one could also formulate a more general model, which encompasses all previous ones as special cases; we would then assume that w_i = a + b · p_i + c · q_i + d · p_i q_i. In this case,

$$\overline{w}\mathrm{\Delta }p=b\cdot \text{Var}(p)+c\cdot \text{Cov}(p,q)+d\cdot \text{Cov}(p,pq).$$ 1.4

(v). The Price equation

The two aspects of the Price equation that are usually appreciated the most are its generality, and the fact that it is a tautology. We will, therefore, write the change in frequency, but this time without assuming anything about how fitness w relates to p. This is also what is usually done, including in the original papers by Price [1,5].

$$\mathrm{\Delta }p=\frac{\sum _{i=1}^m{p}_i{w}_i}{\sum _{i=1}^m{w}_i}-\frac{1}{m}\sum _{i=1}^m{p}_i.$$

This can be written as

$$\overline{w}\mathrm{\Delta }p=\frac{\sum _{i=1}^m{p}_i{w}_i}{m}-\frac{1}{m}\sum _{i=1}^m{p}_i\frac{1}{m}\sum _{i=1}^m{w}_i.$$

If we define the right-hand side as Cov(w, p), this can also be shortened to

$$\overline{w}\mathrm{\Delta }p=\text{Cov}(w,p),$$ 1.5

with the usual disclaimer about this not really being a covariance. Sometimes the ‘regression’ version of the Price equation is used, which, according to Price, would make this equation more ‘intuitively understandable’.¹ The regression form of equation (1.5) would be

$$\overline{w}\mathrm{\Delta }p=\frac{\text{Cov}(w,p)}{\text{Var}(p)}\cdot \text{Var}(p).$$

One may recognize Cov(w, p)/Var(p) as an estimator for b, in case the parent and the offspring population were to be data generated by Model A with noise. More about this possibility will follow in §1b, but here we assume that we are in a modelling context, and not in a statistical one. In either case, this term is usually referred to as a regression coefficient, and we will denote it here by β.

$$\overline{w}\mathrm{\Delta }p=\beta \cdot \text{Var}(p)$$ 1.6

(vi). Comparison

If we compare the formulae we got for Model A, B, C and D with the Price equation, we see something that requires reflection. The simplest model is summarized by

$$(1.1)\overline{w}\mathrm{\Delta }p=b\cdot \text{Var}(p).$$

The second one, model B, is summarized by

$$(1.2)\overline{w}\mathrm{\Delta }p=b\cdot \text{Var}(p)+c\cdot \text{Cov}(p,q).$$

The third is summarized by

$$(1.3)\overline{w}\mathrm{\Delta }p=d\cdot \text{Cov}(p,pq).$$

Encompassing all three of them, as done in model D, would make

$$(1.4)\overline{w}\mathrm{\Delta }p=b\cdot \text{Var}(p)+c\cdot \text{Cov}(p,q)+d\cdot \text{Cov}(p,pq).$$

Ever more general models lead to ever longer equations, and yet, when we use the Price equation, and move to total generality (and therefore well beyond D), we return to the form that the simplest model has, with β for b.

The first thing to note here is that this implies that β must be different from b, unless, of course, we are really considering model A, without allowing for more general ways in which fitnesses can depend on trait values. Moreover, β on the one hand and b, c and d on the other are really different things. In models A to D, the way trait values determine fitness is separated from the population state; the coefficients b, c and d reflect how fitness depends on trait values, and nothing else, while the different ‘variances’ and ‘covariances’ only reflect relevant properties of population states. In the Price equation, on the other hand, β will have to depend on everything; if the real model at hand is model B, C or D, for instance, β cannot be a constant, and will have to vary with the population state. That means that there is not really a meaningful interpretation that one can give to β, while an easy-to-make mistake would be to think of equation (1.6) in the same way as one would of equation (1.1).²

When the Price equation is ‘applied’ to any given model, one may be able to figure out how exactly the β in the Price equation depends on the model parameters and the population state together. For model B, for instance, one would then find that β = b + c(Cov(p, q)/Var(p)). That would be correct, but it is important to realize that in this case, nothing is really gained by using the Price equation; all that it offers is a roundabout route to understanding equation (1.2).

The generality of the Price equation—and this is the broader point—is at odds with β having a meaningful interpretation. Such a meaningful interpretation would have to apply to Model A, for which β = b, to Model B, for which β = b + c · Cov(p, q)/Var(p), to Model C, for which β = d · Cov(p, pq)/Var(p), and to all other ways in which trait values can determine fitnesses. Equation (1.6) implies that $\beta =\overline{w}\mathrm{\Delta }p/\text{Var}(p)$, or, in other words, β needs to be whatever it takes to make equation (1.6) hold. That means that all possible model complexity just ends up being absorbed by the complicated ways in which the β gets to depend on everything. The more models we allow for, the less of a coefficient β is, until, at full generality, it is just a completely meaningless term.

(vii). Denmark

Now there is no doubt that the Price equation is a tautology. Both for equation (1.5), and for the ‘regression or correlation form’ (1.6), the left-hand side is equal to the right-hand side. Also, there is no doubt that the Price equation is general; no assumption is made about how fitness w_i depends on the individual’s p-score p_i. What I would like to point out here, is that being a tautology and being general is in itself not that special. If we divide Cov(w, p), not by Var(p), but by the Planck constant h times the number of times Denmark won the Eurovision Song Contest, then we can rewrite equation (1.5) as

$$\overline{w}\mathrm{\Delta }p=\frac{\text{Cov}(w,p)}{h\cdot DK}\cdot h\cdot DK.$$

This is also a tautology, and it is also general. In this case, it is immediately clear that this is a useless expression, because it is obvious that the term Cov(w, p)/(h · DK) has no meaningful interpretation. In fact, we can make a million such general, but meaningless tautologies. The usefulness of a tautology (or a way to rewrite the change in frequency) depends on whether or not the right-hand side has a meaningful interpretation. With equation (1.6), the right-hand side does, in fact, have a meaningful interpretation, if the model we consider is Model A. In §1b, we will see that the right-hand side can also have a meaningful interpretation if the Price equation is applied to data that are generated by Model A with noise on top. But other than that, there is no scope for the Price equation being general and at the same time allowing for the right-hand side to have a meaningful interpretation.

(b). Statistics

So far, we have assumed that the Price equation is meant to be used in a theoretical context. This is also consistent with most of the literature. In this section, we will consider what happens when the Price equation is applied to data. There are a few reasons to do that.

The first is that Price himself was not entirely clear about whether the equation was to be used for deriving hypotheses from theory assumptions, or for testing them using data (see [1, p. 520]³ and [3, p. 416]). The second reason is that the Price equation, when applied to data can, in fact, have a meaningful interpretation when the data are generated by Model A plus noise. The third and most important reason is that both Price and later users of his equation borrow concepts and terminology from statistics, seemingly suggesting, or at least allowing for the thought, that the role these concepts play in the Price equation is in some way akin to the role they have in statistics. By looking at what the Price equation does when applied to data, we can see that this is by no means true, and that the use of these concepts in statistics actually suggests they should not be used the way they are in the Price equation literature.

(i). Noise and regression coefficients

When doing statistics, we assume that the data are generated by some underlying model, for some values of the parameters, with noise on top. This noise can obscure the true value of the parameters, and it can obscure which model generated the data. The task of statistics is to try to uncover the true values of the parameters, and to figure out what the true model is—insofar as the data allow us to do so. This is a two-step procedure; under the assumption that we have the right model specification, we need to find unbiased estimates of the parameters (preferably also with minimal variance), and then we also have to use the data to pick the right model specification.

(ii). Model A plus noise

The β in the Price equation is referred to, by Price and by others in the Price equation literature, as a regression coefficient. Regression coefficients feature in the parameter estimation step of the two-step procedure described above. This particular regression coefficient is the one that we get if we minimize the squared differences between the data and a + b · p_i, which is what the fitnesses would be if they follow Model A without the noise.

$$\underset{a,b}{min}\sum _{i=1}^m{(w_i-(a+b{p}_i))}^2.$$

The standard calculations, repeated in appendix A, imply that this minimum is indeed attained at

$$\hat{b}=\frac{\text{Cov}(w,p)}{\text{Var}(p)},$$

which makes $\hat{b}$ equal to the β from the Price equation. The Gauss–Markov theorem implies that $\hat{b}$ is a Best Linear Unbiased Estimator of b, under the assumption that the data are indeed generated by Model A plus noise

$$w_i=a+b\cdot p_i+{ϵ}_i,i=1,\dots ,m.$$

This implies that if the data are indeed generated by this model, the β in equation (1.6), the Price equation, does have a meaningful interpretation; it is an unbiased estimator of the true b.

Finally, it is worth realizing that if the data are indeed generated by Model A plus noise, then the only reason why the minimized squared differences are still larger than 0 is the noise. If there would be no noise, then we could just choose the true a and b—both of which one could straightforwardly derive from the data—and that would make all squared differences 0.

(iii). Model B plus noise

There are, however, also other model specifications. One alternative model specification is Model B plus noise

$$w_i=a+b\cdot p_i+c\cdot q_i+{ϵ}_i,i=1,\dots ,m.$$

If we assume that the data are generated by this model, we would minimize the squared differences between the data and a + bp_i + cq_i:

$$\underset{a,b,c}{min}\sum _{i=1}^m{(w_i-(a+b{p}_i+c{q}_i))}^2.$$

Standard calculations, repeated in appendix B, imply that this minimum is attained at

$$\hat{b}=\frac{\mathrm{Cov}(w,q)\cdot \text{Cov}(p,q)-\text{Cov}(w,p)\cdot \text{Var}(q)}{{[\text{Cov}(p,q)]}^2-\text{Var}(p)\cdot \text{Var}(q)}$$

and

$$\hat{c}=\frac{\mathrm{Cov}(w,p)\cdot \text{Cov}(p,q)-\text{Cov}(w,q)\cdot \text{Var}(p)}{{[\text{Cov}(p,q)]}^2-\text{Var}(p)\cdot \text{Var}(q)}.$$

The Gauss–Markov theorem implies that this $\hat{b}$ and $\hat{c}$ are the Best Linear Unbiased Estimators of b and c, respectively, under the assumption that the data are indeed generated by Model B plus noise.

(iv). Model C plus noise

Yet another model specification is Model C plus noise

$$w_i=a+d\cdot p_i{q}_i+{ϵ}_i,i=1,\dots ,m.$$

If we assume that the data are generated by this model, we would minimize the squared differences between the data and a + d · p_iq_i:

$$\underset{a,d}{min}\sum _{i=1}^m{(w_i-(a+d{p}_i{q}_i))}^2.$$

Standard calculations, repeated in appendix C, imply that this minimum is attained at

$$\hat{d}=\frac{\text{Cov}(w,pq)}{\text{Var}(pq)}.$$

The Gauss–Markov theorem implies that $\hat{d}$ is a Best Linear Unbiased Estimator of d, under the assumption that the data are indeed generated by Model C plus noise.

(v). Model specification

These three ordinary least squares (OLS) procedures are estimating parameter values under different assumptions concerning the model that generated the data. We do, however, also need to choose the right model. I will not describe what statistical tests one would do in order to choose the right model, but what I will do here is point out that applying the OLS minimization for one model to data that really are generated by another model, is something that statisticians will try to avoid at all times—and rightly so, because estimating parameters for a wrongly specified model makes all estimates meaningless. That is why the other step in the two-step procedure described above is to get the specification right.

(vi). Example 1

Figure 1 shows what happens when OLS for Model A is applied to data that really are generated by Model B, with $b=\frac{1}{3}$ and c = −1, plus noise. The blue lines in figure 1a represent Model B without noise, and the black dots represent data that are generated by Model B plus noise. Here, we have more observations for (p, q) = (0, 0) and (p, q) = (1, 1) than we have for (p, q) = (0, 1) and (p, q) = (1, 0). Model A does not take the values for q into account, and therefore it only minimizes the squared differences between the data ignoring q and the red line, both seen in figure 1b. It thereby estimates $\hat{b}$ to be close to $-\frac{1}{3}$, which is not correct, because the true value of the coefficient for p is $+\frac{1}{3}$. The reason for the bias in this estimator is a combination of two things: we have not included q in the specification, while q has a negative effect on fitness, and in this sample, p and q are correlated. Of course, this bias disappears if we choose Model B instead, and apply OLS to this model.

Figure 1.

Misspecifying the statistical model implies that we get wrong parameter estimates. (a) The black dots represent the data generated by $w=1+\frac{1}{3}p-q+ϵ$. The blue lines are the fitnesses for the data generating process without noise. (b) Fitting w = a + b · p to the data generated by $w=1+\frac{1}{3}p-q$ gets us an estimate of b that is close to $-\frac{1}{3}$ (the slope of the red line), while the effect really is $+\frac{1}{3}$ (the slope of the blue lines). The reason is that there are more observations for (p, q) equal to (0, 0) and (1, 1) than there are for (p, q) equal to (0, 1) and (1, 0). (Online version in colour.)

(vii). Example 2

Figure 2a shows what happens when OLS for Model B is applied to the data that really are generated by Model C, with d = 1, plus noise. The red lines represent a + b · p + c · q for the estimated values, which are $\hat{b}=1/2$ and $\hat{c}=1/2$. This is not correct; if one wants to state it in terms of the effect of p, one could say that the effect of p is 0 if q = 0, and 1 if q = 1, and not 1/2 across the board, as $\hat{b}=1/2$ suggests.

Figure 2.

Two ways to get wrong parameter estimates for one data generating process, both by misspecifying the statistical model. (a) Fitting w = a + b · p + c · q to the data generated by w = 1 + pq + ε. (b) Fitting w = a + b · p to the data generated by w = 1 + pq + ε. (Online version in colour.)

Figure 2b shows what happens when OLS for Model A is applied to data that really are generated by Model C, with d = 1, plus noise. The red line represents a + b · p for the estimated value, which is $\hat{b}=1/2$. This is even less correct; the squared errors are even larger than in figure 2a, and this suggests that q plays no role at all, while both specifications do not allow us to discover that q matters for the effect of p, or, in other words, that they only have an effect together.

Of course, these biases also disappear if we choose Model C and apply OLS to this model.

(viii). Squared differences

There can be two reasons why the squared differences between data (w_i in all examples) and a model (a + b · p_i for Model A, a + b · p_i + c · q_i for Model B, and a + d · p_iq_i for Model C in our examples) would not be zero. The first reason is noise. This reason for discrepancies between model and data is unavoidable and perfectly fine to have in statistics—although, of course, it would have to be minimized to get unbiased parameter estimates. The second reason for a gap between data and model can be that the model may be misspecified. This reason is not unavoidable, and the whole of statistics is actually geared towards avoiding it; one of the two steps in the two-step procedure described at the beginning of this section is to not misspecify the model.

If we now think away the noise, then the squared difference between w and the model will be 0 if the specification is right (and it is easy to get it right in the absence of noise). In other words, in the absence of noise, the only remaining reason why there could be squared differences that are not 0 would be misspecification, which is what statistics tries to avoid at all times, and which, again, without noise, is actually really easy to avoid. If we now go back to a modelling context, we understand that what the regression coefficient does in the Price equation is in no way, shape, or form the same as what it does in statistics. If the model is not Model A, then the regression coefficient belongs to a misspecified model, which is exactly what it is not supposed to do in statistics, where minimizing squared differences is meant to deal with noise, and nothing but noise.

2. Extension of covariance selection mathematics

In this section, we will apply the Price equation to group selection (see also [5–7]). We will take two classic perspectives: a group selection perspective and an inclusive fitness perspective. For both, we will ask ourselves how helpful the Price equation is, and for both we will encounter the same complications as in §1; we start with a linear model, in which everything has a meaningful interpretation, but, as the models we allow for become more general, the terms in the Price equation start losing their meaning.

One collateral finding is that for the derivation of Hamilton’s rule in a setting where one can, it matters if one formulates the fitness effects in ‘whole group’ or ‘others-only’ terms [8]. The current empirical literature on group selection uses a formula that is partly accurate for one, partly accurate for the other, but not completely accurate for either version. These two possibilities are equivalent, in the sense that one can go back and forth between them, but results will, obviously, look a bit different, depending on which version is used. This includes a difference between an F_ST with and an F_ST without replacement. To make sure that the difference is clear, we will first describe the two versions. Both apply to a setting in which there are n groups, all of which consist of m individuals.

(i). Whole-group public goods game

Here, it is assumed that the fitness of a cooperator in a group with k cooperators (including the individual itself) is w_C,k = 1 + (k/m)b − c, and that the fitness of a defector in a group with k cooperators is w_D,k = 1 + (k/m)b. Being a cooperator instead of a defector would then give m − 1 others in the group a fitness benefit of (1/m)b, adding up to an aggregate fitness benefit of ((m − 1)/m)b, at a net fitness cost to the individual itself of c − (1/m)b.

(ii). Others-only public goods game

Here, it is assumed that the fitness of a cooperator in a group with k cooperators (including the individual itself) is w_C,k = 1 + ((k − 1)/(m − 1))b − c, and that the fitness of a defector in a group with k cooperators is w_D,k = 1 + (k/(m − 1))b. Being a cooperator instead of a defector would then give m − 1 others in the group a fitness benefit of (1/(m − 1))b, adding up to an aggregate fitness benefit of b, at a fitness cost to the individual itself of c.

(iii). What is different between them and what is not

A difference between these formulations is that the parameters b and c coincide with the aggregate fitness benefits and the fitness costs in the others-only case, while the aggregate benefits are ((m − 1)/m)b, and the costs to the individual are c − (1/m)b in the whole group case.

What is the same, is how the average fitness in a group reacts to one individual changing from being a defector to being a cooperator. In both cases, the average fitness within the group would go up by (1/m)(b − c) as a result of an increase in the frequency of cooperators within that group of 1/m. Also, in a group that consists of defectors only, everyone has a payoff of 1 in either case, and in a group that consists of cooperators only, all have a payoff of 1 + b − c.

(a). A simple model of group selection

Suppose that there are n groups, all of which contain m individuals. We will denote the total number of individuals by N = nm. For the fitness of cooperators and defectors, we will first choose the whole group version of the public goods game, and then the others-only version.

In both instances, what one round of selection does to the frequency of cooperators depends on the current state, or, in other words, it depends on how many cooperators there currently are, and how they are distributed over the groups. Note that knowing the fitnesses does not by itself imply that we also know what the next population state will be, because that would also imply that we know where the offspring goes. But in that one step, we can go over all groups, count parent cooperators and parent defectors, compute numbers of offspring cooperators and offspring defectors, and compute the difference in relative frequencies.

In order to summarize the current population state, we use an indicator-like function to represent the type of the individual,

$$p_{i,j}=\{\begin{array}{cc}1& \text{if individual}\hspace{0.17em}j\hspace{0.17em}\text{in group}\hspace{0.17em}i\hspace{0.17em}\text{is a cooperator },\\ 0& \text{otherwise.}\end{array}$$

Sometimes, we will write $k_i=\sum _{\hspace{0.17em}j=1}^m{p}_{i,j}$ for the number of cooperators in group i. The average type in group i is

$${\overline{\hspace{0.17em}p}}_i=\frac{\sum _{\hspace{0.17em}j=1}^m{p}_{i,j}}{m}=\frac{k_i}{m}$$

and the average type in the population is

$$\overline{\hspace{0.17em}p}=\frac{\sum _{i=1}^n{\overline{\hspace{0.17em}p}}_i}{n}=\frac{\sum _{i,j}{p}_{i,j}}{N}=\frac{\sum _i{k}_i}{N}.$$

The number of offspring that individual j in group i has is denoted by w_i,j, with group and population averages similarly defined. The first few lines of the derivation, which are also the first lines of the derivation of the Price equation, do not use a specific fitness function yet.

$$\begin{array}{rl}\mathrm{\Delta }p& =\frac{\sum _{i,j}{p}_{i,j}{w}_{i,j}}{\sum _{i,j}{w}_{i,j}}-\frac{\sum _{i,j}{p}_{i,j}}{N},\\ \frac{\sum _{i,j}{w}_{i,j}}{N}\mathrm{\Delta }p& =\frac{\sum _{i,j}{p}_{i,j}{w}_{i,j}}{N}-\frac{\sum _{i,j}{p}_{i,j}}{N}\frac{\sum _{i,j}{w}_{i,j}}{N}\\ \text{and}\overline{w}\mathrm{\Delta }p& =\frac{\sum _i{\overline{\hspace{0.17em}p}}_i{\overline{w}}_i}{n}-\frac{\sum _{i,j}{p}_{i,j}}{N}\frac{\sum _{i,j}{w}_{i,j}}{N}+\frac{\sum _{i,j}{p}_{i,j}{w}_{i,j}}{N}-\frac{\sum _i{\overline{\hspace{0.17em}p}}_i{\overline{w}}_i}{n}\\ & =\frac{\sum _i{\overline{\hspace{0.17em}p}}_i{\overline{w}}_i}{n}-\frac{\sum _i{\overline{\hspace{0.17em}p}}_i}{n}\frac{\sum _i{\overline{w}}_i}{n}\\ & +\frac{1}{n}\sum _i[\frac{\sum _{\hspace{0.17em}j}{p}_{i,j}{w}_{i,j}}{m}-\frac{\sum _{\hspace{0.17em}j}{p}_{i,j}}{m}\frac{\sum _{\hspace{0.17em}j}{w}_{i,j}}{m}].\end{array}$$

(i). Whole group

Then, we can fill in the whole group public goods game fitness function

$$\begin{array}{rl}\overline{w}\mathrm{\Delta }p& =\frac{1}{n}\sum _i\frac{k_i}{m}\left(1+\frac{k_i}{m}(b-c)\right)-\frac{1}{n}\sum _i\frac{k_i}{m}\cdot \frac{1}{n}\sum _i\left(1+\frac{k_i}{m}(b-c)\right)\\ & +\frac{1}{n}\sum _i\left[\frac{k_i}{m}\left(1+\frac{k_i}{m}b-c\right)-\frac{k_i}{m}\left(1+\frac{k_i}{m}(b-c)\right)\right]\\ & =\sum _i\frac{k_i}{nm}\frac{k_i}{m}(b-c)-\sum _i\frac{k_i}{nm}\sum _i\frac{k_i}{nm}(b-c)\\ & +\frac{1}{n}\sum _i\left[\frac{k_i}{m}\left(\frac{k_i}{m}b-c\right)-\frac{k_i}{m}\left(\frac{k_i}{m}(b-c)\right)\right]\\ & =(b-c)\left[\sum _i\frac{k_i}{nm}\frac{k_i}{m}-\sum _i\frac{k_i}{nm}\sum _i\frac{k_i}{nm}\right]-c\frac{1}{n}\sum _i\left[\frac{k_i}{m}-{\left(\frac{k_i}{m}\right)}^2\right]\\ & =(b-c)\left[\frac{1}{n}\sum _i{({\overline{p}}_i)}^2-{\overline{p}}^2\right]-c\frac{1}{n}\sum _i\left[{\overline{p}}_i-{({\overline{p}}_i)}^2\right]\end{array}$$

The first term in square brackets is usually referred to as the between-group variance, and the second term in square brackets is typically referred to as the average within-group variance. They are numbers that characterize the state of the parent population, which are not random variables here. We can construct random variables, pertaining to the population state, for which these are indeed the variances. For the first one, we can draw one group, with all groups being equally likely to be drawn, and take as a random variable the share of cooperators in that group. For that random variable

$$\text{Var}(\overline{\hspace{0.17em}p})=\frac{1}{n}\sum _i{({\overline{\hspace{0.17em}p}}_i)}^2-{\overline{\hspace{0.17em}p}}^2$$

is the variance. For the second, for any group, we can draw one individual, with all individuals equally likely to be drawn, and take as a random variable whether or not this person is a cooperator. That makes

$$\overline{\text{Var}}(p)=\frac{1}{n}\sum _i[{\overline{\hspace{0.17em}p}}_i-{({\overline{\hspace{0.17em}p}}_i)}^2]$$

the average of the variances of these n random variables. With those definitions, we can shorten this equation to

$$\overline{w}\mathrm{\Delta }p=(b-c)\text{Var}(\overline{\hspace{0.17em}p})-c\cdot \overline{\text{Var}}(p).$$ 2.1

This equation reflects the group selection way of looking at the model, where the first compound term on the right-hand side represents between-group selection, which favours cooperators, and the second compound term represents within-group selection, which favours defectors; see Wilson & Wilson [9].⁴

Both terms feature, on the one hand, model parameters that are constant, and on the other hand, they contain expressions that represent the population state. In the first term, b − c is a model parameter that reflects how much an extra cooperator contributes to the average fitness in the group, and $\text{Var}(\overline{\hspace{0.17em}p})$ is the between-group variance, which is a property of that population state. In the second term, c is a model parameter that represents the difference within any given group between payoffs of cooperators and defectors, and $\overline{\text{Var}}(p)$ is the average within-group variance, which is a property of the population state. While b and c are constant, $\text{Var}(\overline{\hspace{0.17em}p})$ and $\overline{\text{Var}}(p)$ may change as selection happens, or may, and typically will, differ between different (hypothetical) population states.

From equation (2.1), it follows that Δp > 0 if and only if

$$\frac{\text{Var}(\overline{\hspace{0.17em}p})}{\text{Var}(\overline{\hspace{0.17em}p})+\overline{\text{Var}}(p)}b>c.$$ 2.2

One could be tempted to recognize Hamilton’s rule here, with $\text{Var}(\overline{\hspace{0.17em}p})/(\text{Var}(\overline{\hspace{0.17em}p})+\overline{\text{Var}}(p))$ for relatedness. It is important to realize, however, that b and c are only the parameters in the whole-group formulation, and not the costs and benefits of cooperation; the costs of cooperation are c − (1/m)b, and the aggregate benefits to the others in the group are ((m − 1)/m)b. Also $\text{Var}(\overline{\hspace{0.17em}p})/(\text{Var}(\overline{\hspace{0.17em}p})+\overline{\text{Var}}(p))$, even though this is how the F_ST is usually defined, is not equal to relatedness here (see [9] and below).

(ii). Others only

Instead, we can also fill in the others-only public goods game fitness function.

$$\begin{array}{rl}\overline{w}\mathrm{\Delta }p& =\frac{1}{n}\sum _i\frac{k_i}{m}\left(1+\frac{k_i}{m}(b-c)\right)-\frac{1}{n}\sum _i\frac{k_i}{m}\cdot \frac{1}{n}\left(\sum _{i}1+\frac{k_i}{m}(b-c)\right)\\ & +\frac{1}{n}\sum _i\left[\frac{k_i}{m}\left(1+\frac{k_i-1}{m-1}b-c\right)-\frac{k_i}{m}\left(1+\frac{k_i}{m}(b-c)\right)\right]\\ & =\sum _i\frac{k_i}{nm}\frac{k_i}{m}(b-c)-\sum _i\frac{k_i}{nm}\sum _i\frac{k_i}{nm}(b-c)\\ & +\frac{1}{n}\sum _i\left[\frac{k_i}{m}\left(\frac{k_i-1}{m-1}b-c\right)-\frac{k_i}{m}\left(\frac{k_i}{m}(b-c)\right)\right]\\ & =(b-c)\left[\sum _i\frac{k_i}{nm}\frac{k_i}{m}-\sum _i\frac{k_i}{nm}\sum _i\frac{k_i}{nm}\right]\\ & +b\frac{1}{n}\sum _i\left[\frac{k_i}{m}(\frac{k_i-1}{m-1}-\frac{k_i}{m})\right]-c\frac{1}{n}\sum _i\left[\frac{k_i}{m}-{\left(\frac{k_i}{m}\right)}^2\right]\\ & =(b-c)\left[\sum _i\frac{k_i}{nm}\frac{k_i-1}{m-1}-\sum _i\frac{k_i}{nm}\sum _i\frac{k_i}{nm}\right]\\ & -c\frac{1}{n}\sum _i\left[\frac{k_i}{m}\left(\frac{k_i}{m}-\frac{k_i-1}{m-1}\right)\right]-c\frac{1}{n}\sum _i\left[\frac{k_i}{m}-{\left(\frac{k_i}{m}\right)}^2\right]\\ & =(b-c)\left[\sum _i\frac{k_i}{nm}\frac{k_i-1}{m-1}-\sum _i\frac{k_i}{nm}\sum _i\frac{k_i}{nm}\right]\\ & -c\frac{1}{n}\sum _i\left[\frac{k_i}{m}\left(1-\frac{k_i-1}{m-1}\right)\right]\end{array}$$

Both terms in square brackets look quite a bit like their counterpart for the whole group case, with only one difference, and that is that in both a k_i/m term is replaced by (k_i − 1)/(m − 1). That makes them the without-replacement counterparts of the variances for the whole group case. We denote these without-replacement counterparts with an asterisk. Remembering that ${\overline{\hspace{0.17em}p}}_i=k_i/m$, we define

$${\text{Var}}^{\ast }(\overline{\hspace{0.17em}p})=\sum _i\frac{k_i}{nm}\frac{k_i-1}{m-1}-\sum _i\frac{k_i}{nm}\sum _i\frac{k_i}{nm}$$

and

$$\overline{{\text{Var}}^{\ast }}(p)=\frac{1}{n}\sum _i\left[\frac{k_i}{m}(1-\frac{k_i-1}{m-1})\right].$$

With those definitions, we can shorten this equation to

$$\overline{w}\mathrm{\Delta }p=(b-c)\hspace{0.17em}{\text{Var}}^{\ast }(\overline{\hspace{0.17em}p})-c\cdot \hspace{0.17em}\overline{{\text{Var}}^{\ast }}(p).$$ 2.3

Also in this equation, on the one hand, there are model parameters b and c that are constant, and on the other hand, there are expressions that represent the population state. That makes it a good summary of the model, but the compound terms here no longer represent between- and within-group selection.

From equation (2.3), it follows that Δp > 0 if and only if

$$\frac{{\text{Var}}^{\ast }(\overline{\hspace{0.17em}p})}{{\text{Var}}^{\ast }(\overline{\hspace{0.17em}p})+\overline{{\text{Var}}^{\ast }}(p)}b>c.$$ 2.4

This is, in fact, Hamilton’s rule; b and c here are actual fitness effects, and ${\text{Var}}^{\ast }(\overline{\hspace{0.17em}p})/({\text{Var}}^{\ast }(\overline{\hspace{0.17em}p})+\overline{{\text{Var}}^{\ast }}(p))$ does equal relatedness. The difference between this F_ST and the one from §2ai is that this one could be interpreted as the version without replacement (see [10]). The difference between the two versions disappears when groups are large; for m → ∞, aggregate benefits and (net) costs become the same in both versions, and also the difference between the two F_ST’s, disappears. Group selection, however, may very well take place with groups of limited size, where these differences are not negligible.

(b). The Price equation

The Price equation would share the first few lines of the derivation in common with the derivation for the two versions of the given model, and arrive at

$$\overline{w}\mathrm{\Delta }p=\frac{\sum _i{\overline{\hspace{0.17em}p}}_i{\overline{w}}_i}{n}-\frac{\sum _i{\overline{\hspace{0.17em}p}}_i}{n}\frac{\sum _i{\overline{w}}_i}{n}+\frac{1}{n}\sum _i[\frac{\sum _{\hspace{0.17em}j}{p}_{i,j}{w}_{i,j}}{m}-\frac{\sum _{\hspace{0.17em}j}{p}_{i,j}}{m}\frac{\sum _{\hspace{0.17em}j}{w}_{i,j}}{m}].$$

Again, the two aspects of the Price equation that are usually appreciated the most are its generality, and the fact that it is a tautology, so rather than filling in a model for how fitnesses w_i,j are determined, we leave the equation as it is, and abbreviate it as follows:

$$\overline{w}\mathrm{\Delta }p=\text{Cov}(\overline{w},\overline{\hspace{0.17em}p})+\overline{\text{Cov}}(w,p)$$ 2.5

with obvious definitions of $\text{Cov}(\overline{\hspace{0.17em}p},\overline{w})$ and $\overline{\text{Cov}}(w,p)$. The regression version of the Price equation for group selection then will become

$$\overline{w}\mathrm{\Delta }p=\frac{\text{Cov}(\overline{w},\overline{\hspace{0.17em}p})}{\text{Var}(\overline{\hspace{0.17em}p})}\cdot \text{Var}(p)+\frac{\overline{\text{Cov}}(w,p)}{\overline{\text{Var}}(p)}\cdot \overline{\text{Var}}(p).$$

Appendix D shows that, if the numbers were to be data generated by either the whole group or the others-only version of the model, plus noise, then $\text{Cov}(\overline{w},\overline{\hspace{0.17em}p})/\text{Var}(\overline{\hspace{0.17em}p})$ would be an unbiased estimator of the true difference between b and c. Appendix D also shows that, if the numbers were to be data generated by the whole group version of the model, plus noise, then $\overline{\text{Cov}}(w,p)/\overline{\text{Var}}(p)$ would be an unbiased estimator of the true −c, and if the numbers were to be data generated by the others-only version of the model, plus noise, then $\overline{\text{Cov}}(w,p)/\overline{\text{Var}}(p)$ would be an unbiased estimator of the true −(c + (1/(m−1))b). Here, we assume that we are in a modelling context and not in a statistical one, but these terms are regularly referred to as regression coefficients nonetheless, and we will denote them here by β − γ and γ, respectively,

$$\overline{w}\mathrm{\Delta }p=(\beta -\gamma )\cdot \text{Var}(\overline{\hspace{0.17em}p})-\gamma \cdot \overline{\text{Var}}(p).$$ 2.6

From equation (2.6), it follows that Δp > 0 if and only if

$$\frac{\text{Var}(\overline{\hspace{0.17em}p})}{\text{Var}(\overline{\hspace{0.17em}p})+\overline{\text{Var}}(p)}\beta >\gamma .$$ 2.7

This is what then is sometimes recognized as Hamilton’s rule, with $\text{Var}(\overline{\hspace{0.17em}p})/(\text{Var}(\overline{\hspace{0.17em}p})+\overline{\text{Var}}(p))$ for relatedness [11,12], but with our observations for the whole-group and the others-only version of the linear model in mind, a better candidate—because it would match equation (2.4) for the others-only version of the linear model—would be

$$\frac{{\text{Var}}^{\ast }(\overline{\hspace{0.17em}p})}{{\text{Var}}^{\ast }(\overline{\hspace{0.17em}p})+{\overline{\text{Var}}}^{\ast }(p)}\left[\frac{m-1}{m}\beta \right]>[\gamma -\frac{1}{m}\beta ].$$ 2.8

To more generally reiterate the point that equation (2.7) is not Hamilton’s rule, we can write the term $\gamma =\overline{\text{Cov}}(w,p)/\overline{\text{Var}}(p)$ as a weighted average of within-group differences in fitness between cooperators and defectors, which is done in appendix D. The fitness cost of cooperation, however, is not about the differences in fitness between cooperators and defectors within groups; the fitness cost is about differences between fitnesses of cooperators and what those fitnesses would have been, had they defected (and differences between fitnesses of defectors and what those fitnesses would have been, had they cooperated). For the whole-group version of the linear model, the fitness costs are c − (1/m)b, while this weighted average returns c, and for the others-only version the fitness costs are c, while this weighted average returns the value c + (1/m)b. Equation (2.7), therefore, does not match Hamilton’s rule in either case, because γ is not the fitness costs—unless groups are infinitely large, and (1/m)b tends to 0. For the linear case, one can, however, make a repair, and that would be equation (2.8), which then is Hamilton’s rule, with ((m−1)/m)β for benefits and γ − (1/m)β for costs.⁵

(i). Nonlinear fitness effects

For the linear public goods game, and specifically the whole group version, β − γ and γ in equation (2.6) coincided with parameters b − c and c. Also Hamilton’s rule, equation (2.8), holds, and does so in a meaningful way, which is a bit easier to see for the others-only version than it is for the whole-group version. For more complex models, however, it becomes ever harder to interpret β − γ and γ in equation (2.6) in a meaningful way. If we take a nonlinear model, then these ‘coefficients’ come to depend on everything in the model, as they did in §1 for all models other than Model A. Examples of nonlinear models would include one where the public good only materializes if all group members cooperate or, more generally, if the number of cooperators reaches a certain threshold. A richer set of examples can be found in Archetti & Scheuring and Archetti [13–15]. For all of those, all complexity of the model would end up being absorbed by complex ways in which β − γ and γ come to depend, both on model parameters and on the population state, and it becomes impossible to interpret them as coefficients.

Also for Hamilton’s rule, nonlinear fitness effects prevent it from holding in a meaningful way. In van Veelen et al. and van Veelen [16,17], we show that, when costs and benefits are defined using the regression method, Hamilton’s rule always holds, at the cost of Hamilton’s rule being meaningless (see also [18,19]). If we switch to the counterfactual method for computing costs and benefits (this is the classic definition, which for every individual compares its fitness with what it would have been, had it behaved differently), then Hamilton’s rule holds under the assumption of linearity, but not when fitnesses are nonlinear.

(ii). Cancellation effects at the group level

A final example of a model of group selection, where equation (2.6), although a tautology, is prone to hide rather than highlight the properties of the model, and where equation (2.8) gets the fitness effects wrong, is one where between-group competition is local and not global. Standard group selection models all assume that between-group competition is global (see, for example, [20–24]). We can, however, also assume that groups compete with each other locally.

If groups compete more intensely with neighbouring groups than they do with far-away groups, then the number of cooperators in neighbouring groups will negatively affect the fitness in any given group. Because local reproduction will imply that cooperative groups are typically also surrounded by cooperative groups, being a cooperative group will positively correlate with having cooperative neighbouring groups, and while being a cooperative group, everything else being equal, is good for average fitness within the group, being surrounded by cooperative groups is not. A similar effect at the individual level is called the cancellation effect [25–27], and in Akdeniz & van Veelen [10], we show that it can also exist at the group level.

In this case, the relevant variables that determine an individual’s fitness are not just whether or not the individual itself is a cooperator and how many others in the group are cooperators, but also how many cooperators there are in nearby groups. The first two variables determine fitness in the linear public goods game model we started with, which implicitly assumes global between-group competition, while the third plays no role there. For a model with local between-group competition, the first term on the right-hand side of equation (2.5) now not only reflects the fact that cooperative groups do better than defecting ones, as it did without local between-group competition, but also absorbs the cancellation effect at the group level. This works in the same way as in Example 1 in the previous section, where q was not included in the statistical model, and positively correlated with p. Similarly, not including the frequency of cooperators in the neighbouring groups as a variable here now means that we will understate the effect that the share of cooperators in a group has on the fitnesses of the group members, even if fitnesses are linear.

In the case where fitness effects are linear and between-group competition is local, Hamilton’s rule does hold, and it holds in a meaningful way; it is just not equation (2.8) (see [10]). This equation would apply if fitness effects are linear and being a cooperator only affects the fitness of the individual itself and the other individuals in the group. But with local between-group competition, the individuals in neighbouring groups are also affected, and this should also be reflected in Hamilton’s rule.

3. Discussion

In van Veelen [3], I argued that the Price equation is not a useful tool for doing theory, and also not for doing statistics. Price [1] is not entirely clear on whether the numbers that go into the Price equation are supposed to be data or follow from a model, so I explored both options and came up empty-handed for both. I argued that it can create an illusion of understanding, because it is tempting to think that the right-hand side explains the left-hand side, even though one is just a rewritten version of the other.

In van Veelen et al. [4], we emphasized that the Price equation holds for any transition, whether it is likely, unlikely, possible or impossible, while any modelling exercise would come down to deriving properties concerning the relative likelihood of transitions from one population state to the other, given the assumptions of the model. We also argued that no assumptions go into the Price equation, so no predictions can come out.

Here, I have taken a slightly different angle and argued that the right-hand side of the Price equation only has a meaningful interpretation for a certain model specification, and not in general. Even with something as seemingly simple and unambiguous as just separating within-group selection from between-group selection in a group selection context, it turns out that that is not as general as it seems. This separation implicitly assumes that there are no cancellation effects at the group level, and groups compete with each other in a global way.

There are settings in which the Price equation has a meaningful interpretation. If fitness is linear in some variable p, and p is uncorrelated with any other variable that has an effect on fitness, then the right-hand side of the Price equation, as reproduced in §1, has a meaningful interpretation. If, in a simple model of group selection, fitness is linear in the number of cooperators in the group, has a fixed fitness cost, and there is no cancellation happening at the group level, then the right-hand side of the Price equation for group selection in §2 also has a meaningful interpretation. As general as the Price equation is, it therefore only has a meaningful interpretation under relatively strict assumptions. If one explicitly states them, the Price equation is a natural expression to arrive at (see, for instance, [28]). But, the Price equation is not both general and meaningful.

(a). Real consequences

The failure to recognize that, although the Price equation is general, it only has a meaningful interpretation under restrictive assumptions, has left a trace of damage in the field of social evolution. It has led to long-lasting and widespread acceptance of incorrect or meaningless claims relating to inclusive fitness. Hamilton’s rule was originally derived within the context of a model ([29,30]; see also [16,31]). Later versions have used the ‘regression method’, inspired by the Price equation [6,32], for defining costs and benefits. This version, like the Price equation, ignores the specification issue, which means that it computes costs and benefits as if they follow a specific linear model, regardless of whether or not they do. Given that there can also be multiple linear specifications, this version of Hamilton’s rule is moreover not uniquely defined. With costs and benefits defined according to this regression method, Hamilton’s rule is claimed to be generally valid, that is, Hamilton’s rule would always get the direction of selection right. This is correct, but at the expense of Hamilton’s rule being meaningless if fitnesses do not follow the specific linear model, and at the cost of Hamilton’s rule not even being uniquely defined as soon as multiple linear specifications are possible. If costs and benefits are defined in the classic way, with the counterfactual method (which is well defined, and respects the variety of forms the fitness function can have), Hamilton’s rule does not always agree with the direction of selection [16,17].

The Price equation was also used to derive the claim that group selection and inclusive fitness are equivalent, again in the sense that Hamilton’s rule would always get the direction of selection right [5–7]. This claim is not correct, in a variety of ways [4,17,24,33]. Without the Price equation, we would probably not have had these claims in the first place.

Appendix A. Model A plus noise

Here, we assume that fitness is linear in the p-score, and that we have m noisy observations

$$w_i=a+b\cdot p_i+{ϵ}_i,i=1,\dots ,m.$$

We can estimate a and b by applying OLS, which means that we minimize

$$\underset{a,b}{min}\sum _{i=1}^m{(w_i-(a+b{p}_i))}^2.$$

The first-order conditions for this minimization are

$$\sum _{i=1}^m-2(w_i-(a+b{p}_i))=0$$

and

$$\sum _{i=1}^m-2p_i(w_i-(a+b{p}_i))=0,$$

which can be rewritten as

$$\sum _{i=1}^m{w}_i=ma+b\sum _{i=1}^m{p}_i$$

and

$$\sum _{i=1}^m{w}_i{p}_i=a\sum _{i=1}^m{p}_i+b\sum _{i=1}^m{p}_i^2.$$

Rewriting the first gives

$$a=\frac{1}{m}\sum _{i=1}^m{w}_i-b\frac{1}{m}\sum _{i=1}^m{p}_i.$$

Substituting in the second gives

$$\sum _{i=1}^m{w}_i{p}_i=(\frac{1}{m}\sum _{i=1}^m{w}_i-b\frac{1}{m}\sum _{i=1}^m{p}_i)\sum _{i=1}^m{p}_i+b\sum _{i=1}^m{p}_i^2.$$

Or, in other words,

$$b=\frac{\sum _{i=1}^m{w}_i{p}_i-(1/m)\sum _{i=1}^m{w}_i\sum _{i=1}^m{p}_i}{\sum _{i=1}^m{p}_i^2-(1/m)\sum _{i=1}^m{p}_i\sum _{i=1}^m{p}_i}.$$

We can divide both the enumerator and the denominator either by m − 1 or by m. If we divide both by m − 1, then the enumerator becomes the (unbiased) sample covariance, which we will denote by S_w,p, and the denominator becomes the (unbiased) sample variance, which we here will denote by $s_p^2$. Sometimes the estimator of a true parameter b is denoted by $\hat{b}$, which then would make the last equation read

$$\hat{b}=\frac{S_{w,p}}{s_p^2}.$$

If we divide both by m, then we get terms that in the Price equation literature typically are denoted by Cov(w, p) and Var(p), and that really are numbers that, in this case, characterize the data, and that are more accurately described as the biased sample variance and the biased sample covariance.

$$\hat{b}=\frac{\text{Cov}(w,p)}{\text{Var}(p)}.$$

Substituting back in the first equation gives

$$\hat{a}=\overline{w}-\frac{S_{w,p}}{s_p^2}\overline{\hspace{0.17em}p}=\overline{w}-\frac{\text{Cov}(w,p)}{\text{Var}(p)}\overline{\hspace{0.17em}p}.$$

The Gauss–Markov theorem states that $\hat{b}$ is a Best Linear Unbiased Estimator of b, under the assumption that the data are indeed generated by Model A plus noise.

Appendix B. Model B plus noise

Here, we assume that fitness is linear in the p-score and in the value of q, and that we have m noisy observations

$$w_i=a+b\cdot p_i+c\cdot q_i+{ϵ}_i,i=1,\dots ,m.$$

We can estimate a, b and c by applying OLS, which means that we minimize

$$\underset{a,b,c}{min}\sum _{i=1}^m{(w_i-(a+b{p}_i+c{q}_i))}^2.$$

The first-order conditions for this minimization are

$$\begin{array}{rl}& \sum _{i=1}^m-2(w_i-(a+b{p}_i+c{q}_i))=0,\\ & \sum _{i=1}^m-2p_i(w_i-(a+b{p}_i+c{q}_i))=0\\ \text{and}& \sum _{i=1}^m-2q_i(w_i-(a+b{p}_i+c{q}_i))=0,\end{array}$$

which can be rewritten as

$$\begin{array}{rl}& \sum _{i=1}^m{w}_i=ma+b\sum _{i=1}^m{p}_i+c\sum _{i=1}^m{q}_i,\\ & \sum _{i=1}^m{w}_i{p}_i=a\sum _{i=1}^m{p}_i+b\sum _{i=1}^m{p}_i^2+c\sum _{i=1}^m{p}_i{q}_i\\ \text{and}& \sum _{i=1}^m{w}_i{q}_i=a\sum _{i=1}^m{q}_i+b\sum _{i=1}^m{p}_i{q}_i+c\sum _{i=1}^m{q}_i^2.\end{array}$$

Rewriting the first gives

$$a=\frac{1}{m}\sum _{i=1}^m{w}_i-b\frac{1}{m}\sum _{i=1}^m{p}_i-c\frac{1}{m}\sum _{i=1}^m{q}_i.$$

With self-evident definitions of averages, we will write this as

$$a=\overline{w}-b\overline{\hspace{0.17em}p}-c\overline{q}.$$

Substituting in the second and third, dividing both by m, and rearranging gives

$$\text{Cov}(w,p)=b\cdot \text{Var}(p)+c\cdot \text{Cov}(p,q)$$

and

$$\text{Cov}(w,q)=b\cdot \text{Cov}(p,q)+c\cdot \text{Var}(q).$$

Rewriting the first of those gives

$$c=\frac{\text{Cov}(w,p)-b\cdot \text{Var}(p)}{\text{Cov}(p,q)}.$$

Substituting in the second of those, and multiplying left and right by Cov(p, q) gives

$$\text{Cov}(w,q)\cdot \text{Cov}(p,q)=b\cdot {[\text{Cov}(p,q)]}^2+(\text{Cov}(w,p)-b\cdot \text{Var}(p))\hspace{0.17em}\text{Var}(q),$$

which makes

$$\hat{b}=\frac{\text{Cov}(w,q)\cdot \text{Cov}(p,q)-\text{Cov}(w,p)\cdot \text{Var}(q)}{{[\text{Cov}(p,q)]}^2-\text{Var}(p)\cdot \text{Var}(q)}.$$

Filling this in in either of the two, or using symmetry, gives

$$\hat{c}=\frac{\text{Cov}(w,p)\cdot \text{Cov}(p,q)-\text{Cov}(w,q)\cdot \text{Var}(p)}{{[\text{Cov}(p,q)]}^2-\text{Var}(p)\cdot \text{Var}(q)}.$$

These are the biased sample variances and covariances; had we divided by m − 1 instead of m, we would get the unbiased sample variances and covariances (see appendix A). We get the $\hat{a}$ by substituting $\hat{b}$ and $\hat{c}$ back in the first equation. As before, the Gauss–Markov theorem states that $\hat{b}$ and $\hat{c}$ are Best Linear Unbiased Estimators of b and c, under the assumption that the data are indeed generated by Model B plus noise.

Appendix C. Model C plus noise

Here, we assume that fitness is linear in p · q, and that we have m noisy observations

$$w_i=a+d\cdot p_i{q}_i+{ϵ}_i,i=1,\dots ,m.$$

We can estimate a and d by applying OLS, which means that we minimize

$$\underset{a,d}{min}\sum _{i=1}^m{(w_i-(a+d{p}_i{q}_i))}^2.$$

Here, we can just follow appendix A, with d for b, and pq for p. If we do, we find

$$\hat{d}=\frac{\text{Cov}(w,pq)}{\text{Var}(pq)}$$

and

$$\hat{a}=\overline{w}-\frac{\text{Cov}(w,pq)}{\text{Var}(pq)}\overline{\hspace{0.17em}pq}.$$

The Gauss–Markov theorem states that $\hat{d}$ is a Best Linear Unbiased Estimator of d, under the assumption that the data are indeed generated by Model C plus noise.

Appendix D. Group selection statistics I

One approach to the estimation of b and c would be to estimate a model like the model in appendix B, with the fitness w_i,j of individual j in group i being explained by p_i—which indicates whether or not that individual is a cooperator itself—and by $p_{i,-j}=(1/(m-1))\sum _{k\ne j}{p}_{ik}$—which would be the share of cooperators in group i, excluding individual j itself. The formulae at the end of appendix B, with the right replacements for the variables, would then apply here too. The model there, however, does not allow for group-level noise. If we want to estimate the difference b − c, and allow for noise at the group-level as well as individual noise, then we can use the fact that, for both versions of the model, b − c is also the effect that one additional cooperator has on the total group payoff. That makes the average group payoff a linear function of its share of cooperators, and with the noise in the group average equal to the average noise in individual fitnesses in the group plus the group-level noise, this gives us m observations of group averages

$${\overline{w}}_i=(b-c){\overline{\hspace{0.17em}p}}_i+{\overline{ϵ}}_i.$$

Therefore, similar to appendix A, but now with group averages for individual values, the estimator of the difference b − c would be

$$\hat{b}-\hat{c}=\frac{S_{\overline{w},\overline{\hspace{0.17em}p}}}{s_{\overline{\hspace{0.17em}p}}^2}.$$

This could also be written as

$$\hat{b}-\hat{c}=\frac{\text{Cov}(\overline{w},\overline{\hspace{0.17em}p})}{\text{Var}(\overline{\hspace{0.17em}p})}\hspace{0.17em}$$

with, as before, this covariance and variance defined on the random variable that draws one of the observations, with all observations being equally likely to be drawn.

For estimating the c in the whole-group version of the public goods game, or for estimating c + (1/m)b in the others-only version, one could also use the structure in the independent variables, because, noise aside, within any group, the difference between the average payoff of cooperators and the average payoff of defectors should be c for the whole group version and c + (1/m)b for the others-only version. We can rewrite $\overline{\text{Cov}}(w,p)/\overline{\text{Var}}(p)$ as a weighted average of within-group differences in average fitness, where the difference for group i has weight ${\text{Var}}_i(p)/\sum _k{\text{Var}}_k(p)$, and where ${\text{Var}}_i(p)={\overline{\hspace{0.17em}p}}_i(1-{\overline{\hspace{0.17em}p}}_i)$

$$\begin{array}{rl}& \frac{\overline{Cov}(w,p)}{\overline{\mathrm{Var}}(p)}\\ & =\frac{1}{n}\sum _i\frac{\left[{\displaystyle \frac{1}{m}{\sum }_j{p}_{i,j}{w}_{i,j}-\frac{1}{m}{\sum }_{j=1}^m{p}_{i,j}\frac{1}{m}{\sum }_{j=1}^m{w}_{i,j}}\right]}{{\overline{p}}_i(1-{\overline{p}}_i)}\frac{{\overline{p}}_i(1-{\overline{p}}_i)}{\overline{\mathrm{Var}}(p)}\\ & =\sum _i\left[\frac{{\sum }_{j=1}^m{p}_{i,j}{w}_{i,j}}{m{\overline{p}}_i(1-{\overline{p}}_i)}-\frac{{\sum }_{j=1}^m{\overline{p}}_i{w}_{i,j}}{m{\overline{p}}_i(1-{\overline{p}}_i)}\right]\frac{{\mathrm{Var}}_i(p)}{n\cdot \overline{\mathrm{Var}}(p)}\\ & =\sum _i\left[\frac{{\sum }_{j=1}^m{p}_{i,j}{w}_{i,j}(1-{\overline{p}}_i)}{m{\overline{p}}_i(1-{\overline{p}}_i)}-\frac{{\sum }_{j=1}^m{\overline{p}}_i(1-p_{i,j})w_{i,j}}{m{\overline{p}}_i(1-{\overline{p}}_i)}\right]\frac{{\mathrm{Var}}_i(p)}{{\sum }_k{\mathrm{Var}}_k(p)}\\ & =\sum _i\left[\frac{{\sum }_{j=1}^m{p}_{i,j}{w}_{i,j}}{m{\overline{p}}_i}-\frac{{\sum }_{j=1}^m(1-p_{i,j})w_{i,j}}{m(1-{\overline{p}}_i)}\right]\frac{{\mathrm{Var}}_i(p)}{{\sum }_k{\mathrm{Var}}_k(p)}\\ & =\sum _i\left[\frac{{\sum }_{j=1}^m{p}_{i,j}{w}_{i,j}}{{\sum }_{j=1}^m{p}_{i,j}}-\frac{{\sum }_{j=1}^m(1-p_{i,j})w_{i,j}}{{\sum }_{j=1}^m(1-p_{i,j})}\right]\frac{{\mathrm{Var}}_i(p)}{{\sum }_k{\mathrm{Var}}_k(p)}\end{array}$$

Every group’s within-group difference is already an unbiased estimator of c for the whole group version, or c + (1/m)b for the other only version, but taking this weighted average reduces the variance of the estimator by using all the data. Note that the average within-group differences are more noisy in groups with few cooperators or groups with few defectors compared to groups with more equal amounts of cooperators and defectors. The groups with more equal amounts therefore get more weight in order to minimize the variance of the estimator.

Appendix E. Group selection statistics II

In Group selection statistics I, we have assumed that the statistics is meant to recover parameters a, b and c in the fitness function. Some statistical exercises, however, aim at establishing how conducive the group-structured population is to the evolution of cooperation [11,34]. They, therefore, want to estimate the F_ST, which is then informative about the critical b/c ratio above which cooperation would evolve. There are three remarks to be made here. The first is that this assumes that the effects of cooperation are well described by a linear fitness function. The second is that it assumes that there is no cancellation effect or, in other words, between-group competition is global [10]. Finally, if b and c are the actual fitness costs and benefits, then the F_ST to be estimated is the second one, ‘without replacement’, and not, as is the norm, the first. For groups that are large (as they can be assumed to be in a variety of other situations where F_STs are computed), the difference is negligible, but with groups that are not large, the difference is not that small.

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