Price equation explained

In the theory of evolution and natural selection, the Price equation (also known as Price's equation or Price's theorem) describes how a trait or allele changes in frequency over time. The equation uses a covariance between a trait and fitness, to give a mathematical description of evolution and natural selection. It provides a way to understand the effects that gene transmission and natural selection have on the frequency of alleles within each new generation of a population. The Price equation was derived by George R. Price, working in London to re-derive W.D. Hamilton's work on kin selection. Examples of the Price equation have been constructed for various evolutionary cases. The Price equation also has applications in economics.^[1]

The Price equation is a mathematical relationship between various statistical descriptors of population dynamics, rather than a physical or biological law, and as such is not subject to experimental verification. In simple terms, it is a mathematical statement of the expression "survival of the fittest".

Statement

The Price equation shows that a change in the average amount

of a trait in a population from one generation to the next (

\Deltaz

) is determined by the covariance between the amounts

z_i

of the trait for subpopulation

and the fitnesses

w_i

of the subpopulations, together with the expected change in the amount of the trait value due to fitness, namely

E(w_i\Deltaz_i)

\Delta{z}=

	1
	w

\operatorname{cov}(w_i,z_i)+

	1
	w

\operatorname{E}(w_i\Deltaz_i).

Here

is the average fitness over the population, and

\operatorname{E}

and

\operatorname{cov}

represent the population mean and covariance respectively. 'Fitness'

is the ratio of the average number of offspring for the whole population per the number of adult individuals in the population, and

w_i

is that same ratio only for subpopulation

If the covariance between fitness (

w_i

) and trait value (

z_i

) is positive, the trait value is expected to rise on average across population

. If the covariance is negative, the characteristic is harmful, and its frequency is expected to drop.

The second term,

E(w_i\Deltaz_i)

, represents the portion of

\Deltaz

due to all factors other than direct selection which can affect trait evolution. This term can encompass genetic drift, mutation bias, or meiotic drive. Additionally, this term can encompass the effects of multi-level selection or group selection. Price (1972) referred to this as the "environment change" term, and denoted both terms using partial derivative notation (∂_NS and ∂_EC). This concept of environment includes interspecies and ecological effects. Price describes this as follows:

Proof

Suppose we are given four equal-length lists of real numbers^[2]

n_i

z_i

n_i'

z_i'

from which we may define

w_i=n_i'/n_i

n_i

and

z_i

will be called the parent population numbers and characteristics associated with each index i. Likewise

n_i'

and

z_i'

will be called the child population numbers and characteristics, and

w_i'

will be called the fitness associated with index i. (Equivalently, we could have been given

n_i

z_i

w_i

z_i'

with

n_i'=w_in_i

.) Define the parent and child population totals:

n \stackrel{def

}\;\sum_i n_i

n' \stackrel{def

}\;\sum_i n_i'

and the probabilities (or frequencies):^[3]

q_{i \stackrel{def}

}\;n_i/n

q_{i' \stackrel{def}

}\;n_i'/n'

Note that these are of the form of probability mass functions in that

\sum_iq_i=\sum_iq_i'=1

and are in fact the probabilities that a random individual drawn from the parent or child population has a characteristic

z_i

. Define the fitnesses:

w_{i \stackrel{def}

}\;n_i'/n_iThe average of any list

x_i

is given by:

E(x_i)=\sum_iq_ix_i

so the average characteristics are defined as:

z \stackrel{def

}\;\sum_i q_i z_i

z' \stackrel{def

}\;\sum_i q_i' z_i'

and the average fitness is:

w \stackrel{def

}\;\sum_i q_i w_iA simple theorem can be proved:

q_iw_i=\left(

	n_i	\right)\left(
	n

	n_i'
	n_i

\right)=\left(

	n_i'
	n'

\right)\left(

	n'
	n

\right)=q

i'\left(	n'
	n

\right)

so that:

w=	n'
	n

\sum_iq_i'=

	n'
	n

and

q_iw_i=wq_i'

The covariance of

w_i

and

z_i

is defined by:

\operatorname{cov}(w_i,z_{i) \stackrel{def}

}\;E(w_i z_i)-E(w_i)E(z_i) = \sum_i q_i w_i z_i - w zDefining

\Deltaz_i \stackrel{def

}\; z_i'-z_i, the expectation value of

w_i\Deltaz_i

E(w_i\Deltaz_i)=\sumq_iw_i(z_i'-z_i)=\sum_iq_iw_iz_i'-\sum_iq_iw_iz_i

The sum of the two terms is:

\operatorname{cov}(w_i,z_i)+E(w_i\Deltaz_i)=\sum_iq_iw_iz_i-wz+\sum_iq_iw_iz_i'-\sum_iq_iw_iz_i=\sum_iq_iw_iz_i'-wz

Using the above mentioned simple theorem, the sum becomes

\operatorname{cov}(w_i,z_i)+E(w_i\Deltaz_i)=w\sum_iq_i'z_i'-wz=wz'-wz=w\Deltaz

where

\Deltaz \stackrel{def

}\;z'-z.

Derivation of the continuous-time Price equation

Consider a set of groups with

i=1,...,n

that are characterized by a particular trait, denoted by

x_i

. The number

n_i

of individuals belonging to group

experiences exponential growth:

= f_n_

where

f_i

corresponds to the fitness of the group. We want to derive an equation describing the time-evolution of the expected value of the trait:

\mathbb(x) = \sum_p_x_ \equiv \mu, \quad p_ =

Based on the chain rule, we may derive an ordinary differential equation:

\begin &= \sum_ + \sum_ \\&= \sum_ x_ + \sum_ p_ \\&= \sum_ x_ + \mathbb\left(\right)\end

A further application of the chain rule for

dp_i/dt

gives us:

= \sum_, \quad = \begin -p_/N, \quad &i\neq j \\ (1-p_)/N, \quad &i=j \end

Summing up the components gives us that:

\begin &= p_\left(f_ - \sum_p_f_\right) \\&= p_\left[f_{i} - \mathbb{E}(f)\right]\end

which is also known as the replicator equation. Now, note that: $\begin\sum_ x_ &= \sum_ p_x_\left[f_{i} - \mathbb{E}(f)\right] \\&= \mathbb\left\ \\&= \text(x,f)\end$ Therefore, putting all of these components together, we arrive at the continuous-time Price equation: $\mathbb(x) = \underbrace_ + \underbrace_$

Simple Price equation

When the characteristic values

z_i

do not change from the parent to the child generation, the second term in the Price equation becomes zero resulting in a simplified version of the Price equation:

w\Deltaz=\operatorname{cov}\left(w_i,z_i\right)

which can be restated as:

\Deltaz=\operatorname{cov}\left(v_i,z_i\right)

where

v_i

is the fractional fitness:

v_i=w_i/w

This simple Price equation can be proven using the definition in Equation (2) above. It makes this fundamental statement about evolution: "If a certain inheritable characteristic is correlated with an increase in fractional fitness, the average value of that characteristic in the child population will be increased over that in the parent population."

Applications

The Price equation can describe any system that changes over time, but is most often applied in evolutionary biology. The evolution of sight provides an example of simple directional selection. The evolution of sickle cell anemia shows how a heterozygote advantage can affect trait evolution. The Price equation can also be applied to population context dependent traits such as the evolution of sex ratios. Additionally, the Price equation is flexible enough to model second order traits such as the evolution of mutability. The Price equation also provides an extension to Founder effect which shows change in population traits in different settlements

Dynamical sufficiency and the simple Price equation

Sometimes the genetic model being used encodes enough information into the parameters used by the Price equation to allow the calculation of the parameters for all subsequent generations. This property is referred to as dynamical sufficiency. For simplicity, the following looks at dynamical sufficiency for the simple Price equation, but is also valid for the full Price equation.

Referring to the definition in Equation (2), the simple Price equation for the character

can be written:

w(z'-z)=\langlew_iz_i\rangle-wz

For the second generation:

w'(z''-z')=\langlew'_iz'_i\rangle-w'z'

The simple Price equation for

only gives us the value of

for the first generation, but does not give us the value of

and

\langlew_iz_i\rangle

, which are needed to calculate

z''

for the second generation. The variables

w_i

and

\langlew_iz_i\rangle

can both be thought of as characteristics of the first generation, so the Price equation can be used to calculate them as well:

\begin{align} w(w'-w)&=\langle

	2\rangle
w
	i

-w²\\ w\left(\langlew'_iz'_i\rangle-\langlew_iz_{i\rangle\right)}&=\langle

	2
w
	i

z_i\rangle-w\langlew_iz_{i\rangle
\end{align}}

The five 0-generation variables

\langlew_iz_i\rangle

\langle

	2\rangle
w
	i

, and

\langle

	2z
w
	i

must be known before proceeding to calculate the three first generation variables

, and

\langlew'_iz'_i\rangle

, which are needed to calculate

z''

for the second generation. It can be seen that in general the Price equation cannot be used to propagate forward in time unless there is a way of calculating the higher moments

\langle

	n\rangle
w
	i

and

\langle

	nz
w
	i\rangle

from the lower moments in a way that is independent of the generation. Dynamical sufficiency means that such equations can be found in the genetic model, allowing the Price equation to be used alone as a propagator of the dynamics of the model forward in time.

Full Price equation

The simple Price equation was based on the assumption that the characters

z_i

do not change over one generation. If it is assumed that they do change, with

z_i

being the value of the character in the child population, then the full Price equation must be used. A change in character can come about in a number of ways. The following two examples illustrate two such possibilities, each of which introduces new insight into the Price equation.

Genotype fitness

We focus on the idea of the fitness of the genotype. The index

indicates the genotype and the number of type

genotypes in the child population is:

n'_i=\sum_jw_jin_j

which gives fitness:

w_i=

	n'_i
	n_i

Since the individual mutability

z_i

does not change, the average mutabilities will be:

\begin{align} z&=

	1
	n

\sum_iz_in_i\\ z'&=

	1
	n'

\sum_iz_in'_{i
\end{align}}

with these definitions, the simple Price equation now applies.

Lineage fitness

In this case we want to look at the idea that fitness is measured by the number of children an organism has, regardless of their genotype. Note that we now have two methods of grouping, by lineage, and by genotype. It is this complication that will introduce the need for the full Price equation. The number of children an

-type organism has is:

n'_i=n_i\sum_jw_ij

which gives fitness:

w_i=

	n'_i
	n_i

=\sum_jw_ij

We now have characters in the child population which are the average character of the

-th parent.

z'_j=

	\sum_in_iz_iw_ij
	\sum_in_iw_ij

with global characters:

\begin{align} z&=

	1
	n

\sum_iz_in_i\\ z'&=

	1
	n'

\sum_iz_in'_{i
\end{align}}

with these definitions, the full Price equation now applies.

Criticism

The use of the change in average characteristic (

z'-z

) per generation as a measure of evolutionary progress is not always appropriate. There may be cases where the average remains unchanged (and the covariance between fitness and characteristic is zero) while evolution is nevertheless in progress. For example, if we have

z_i=(1,2,3)

n_i=(1,1,1)

, and

w_i=(1,4,1)

, then for the child population,

n_i'=(1,4,1)

showing that the peak fitness at

w₂₌₄

is in fact fractionally increasing the population of individuals with

z_i=2

. However, the average characteristics are z=2 and z'=2 so that

\Deltaz=0

. The covariance

cov(z_i,w_i)

is also zero. The simple Price equation is required here, and it yields 0=0. In other words, it yields no information regarding the progress of evolution in this system.

A critical discussion of the use of the Price equation can be found in van Veelen (2005),^[4] van Veelen et al. (2012),^[5] and van Veelen (2020).^[6] Frank (2012) discusses the criticism in van Veelen et al. (2012).^[7]

Cultural references

Price's equation features in the plot and title of the 2008 thriller film WΔZ.

The Price equation also features in posters in the computer game BioShock 2, in which a consumer of a "Brain Boost" tonic is seen deriving the Price equation while simultaneously reading a book. The game is set in the 1950s, substantially before Price's work.

Notes and References

Knudsen . Thorbjørn . 2004 . General selection theory and economic evolution: The Price equation and the replicator/interactor distinction . Journal of Economic Methodology . 11 . 2 . 147–173 . 10.1080/13501780410001694109 . 154197796 . 2011-10-22 .
The lists may in fact be members of any field (i.e. a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do
Frank . Steven A. . 1995 . George Price's Contributions to Evolutionary Genetics . J. Theor. Biol. . 175 . 3. 373–388 . 10.1006/jtbi.1995.0148. 7475081 . 1995JThBi.175..373F . Mar 19, 2023.
van Veelen . December 2005 . M. . On the use of the Price equation . Journal of Theoretical Biology . 237 . 4 . 412–426 . 10.1016/j.jtbi.2005.04.026 . 15953618. 2005JThBi.237..412V .
van Veelen . M. . García . J. . Sabelis . M.W. . Egas . M. . April 2012 . Group selection and inclusive fitness are not equivalent; the Price equation vs. models and statistics . Journal of Theoretical Biology . 299 . 64–80 . 10.1016/j.jtbi.2011.07.025 . 21839750. 2012JThBi.299...64V .
van Veelen . March 2020 . M. . The problem with the Price equation . Philosophical Transactions of the Royal Society B . 375 . 1797 . 1–13 . 10.1098/rstb.2019.0355 . 32146887. 7133513 . free .
Frank . S.A. . 2012 . Natural Selection IV: The Price equation . Journal of Evolutionary Biology . 25 . 6 . 1002–1019 . 1204.1515 . 10.1111/j.1420-9101.2012.02498.x . 22487312 . 3354028.

Price equation explained

Statement

Proof

Derivation of the continuous-time Price equation

Simple Price equation

Applications

Dynamical sufficiency and the simple Price equation

Full Price equation

Genotype fitness

Lineage fitness

Criticism

Cultural references

See also

Further reading

Notes and References