Price equation

The Price equation (also known as Price's equation) is a covariance equation which is a mathematical description of evolution and natural selection. The Price equation was derived by George R. Price, working in London to rederive W.D. Hamilton's work on kin selection.

Suppose there is a population of $$n$$ individuals over which the amount of a particular characteristic varies. Those $$n$$ individuals can grouped by the amount of the characteristic that each displays. In this case, at most there will be $$n$$ groups of $$n$$ distinct values of the characteristic, and at least there will be 1 group of a single shared value of the characteristic. Index each group with $$i$$ so that the number of members in the group is $$n_i$$ and the value of the characteristic shared among all members of the group is $$z_i$$. Now assume that having $$z_i$$ of the characteristic is associated with having a fitness $$w_i$$ where the product $$w_i n_i$$ represents the number of offspring in the next generation. Denote this number of offspring from group $$i$$ by $$n_i'$$ so that $$w_i=n_i'/n_i$$. Let $$z_i'$$ be the amount of the characteristic displayed by the offspring from group $$i$$. Denote the amount of change in characteristic in group $$i$$ by $$\Delta z_i$$ defined by



\Delta{z_i} \ \stackrel{\mathrm{def}}{=}\ z_i' - z_i $$

Now take $$z$$ to be the average characteristic value in this population and $$z'$$ to be the average characteristic value in the next generation. Define the change in average characteristic by $$\Delta{z}$$. That is,



\Delta{z} \ \stackrel{\mathrm{def}}{=}\ z' - z $$

Note that this is not the average value of $$\Delta{z_i}$$. Also take $$w$$ to be the average fitness of this population. The Price equation states:



w\Delta{z}=\operatorname{cov}(w_i,z_i)+\operatorname{E}(w_i\Delta z_i), \,\! $$

where the functions $$\operatorname{E}$$ and $$\operatorname{cov}$$ are respectively defined in Equations (1) and (2) below but are like the sample versions of the expected value and covariance operators from probability (see Sample mean and covariance). Note that this is really a difference equation relating the average value of a characteristic in one generation to the average value of the characteristic in the very next generation. In fact, assuming that $$w$$ is not zero, it is often useful to write it as



\Delta{z}=\frac{\operatorname{cov}(w_i,z_i)}{w}+\frac{\operatorname{E}(w_i\Delta z_i)}{w}\, $$

In the specific case that characteristic $$z_i = w_i$$ (i.e., the fitness is the characteristic of interest), then Price's equation reformulates Fisher's fundamental theorem of natural selection.

Price's equation is, importantly, a tautology. It is a statement of mathematical fact between certain variables, and its value lies in the insight gained by assigning certain values encountered in evolutionary genetics to the variables. For example, the statement "if every pair of birds has two offspring, then among ten pairs of birds there will be twenty offspring" is a tautology. It doesn't really impart any new information about birds so much as it organizes our concepts about birds and their offspring. The Price equation is much more sophisticated than the above statement, but at its core, it too is a mathematically provable tautology.

The Price equation also has applications in economics.

Proof of the Price Equation
To prove the Price equation, the following definitions are needed. If $$n_i$$ is the number of occurrences of a pair of real numbers $$x_i$$ and $$y_i$$, then:


 * The average or expected value of the $$x_i$$ values is:


 * The covariance between the $$x_i$$ and $$y_i$$ values is:

The notation $$\langle x_i \rangle = \operatorname{E}( x_i )$$ will also be used when convenient.

Suppose there is a population of organisms all of which have a genetic characteristic described by some real number. For example, high values of the number represent an increased visual acuity over some other organism with a lower value of the characteristic. Groups can be defined in the population which are characterized by having the same value of the characteristic. Let subscript $$i$$ identify the group with characteristic $$z_i$$ and let $$n_i$$ be the number of organisms in that group. The total number of organisms is then $$n$$ where:



n = \sum_i n_i\, $$

The average value of the characteristic is $$z$$ defined as:

Now suppose that the population reproduces, all parents are eliminated, and then there is a selection process on the children, by which less fit children are removed from the reproducing population. After reproduction and selection, the population numbers for the child groups will change to n&prime;i. Primes will be used to denote child parameters, unprimed variables denote parent parameters.

The total number of children is n'  where:



n' = \sum_i n'_i\, $$

The fitness of group i will be defined to be the ratio of children to parents:

with average fitness of the population being

The average value of the child characteristic will be z'  where:

where z&prime;i are the (possibly new) values of the characteristic in the child population. Equation (2) shows that:

Call the change in characteristic value from parent to child populations $$\Delta z_i$$ so that $$\Delta z_i = z'_i - z_i$$. As seen in Equation (1), the expected value operator $$\operatorname{E}$$ is linear, so

Combining Equations (7) and (8) leads to

but from Equation (1) gives:


 * $$\operatorname{E}(w_iz'_i)=\frac{\sum_i w_iz'_in_i}{n}$$

and from Equation (4) gives:

Applying Equations (5) and (6) to Equation (10) and then applying the result to Equation (9) gives the Price Equation:


 * $$\operatorname{cov}(w_iz_i)+\operatorname{E}(w_i\Delta z_i)=wz'-wz=w\Delta z\,$$

The Simple Price Equation
When the characters zi 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\Delta z = \operatorname{cov}\left(w_i,z_i\right) $$

which can be restated as:



\Delta z = \operatorname{cov}\left(v_i,z_i\right) $$

where vi is the fractional fitness: vi= wi/w. This simple Price equation can be proven using the definition in Equation (2) above. It makes this fundamental and tautological 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."

Example: The Evolution of Sight
As an example of the simple Price equation, consider a model for the evolution of sight. Suppose zi is a real number measuring the visual acuity of an organism. An organism with a higher zi will have better sight than one with a lower value of zi. Lets say that the fitness of such an organism is wi=zi which means the more sighted it is, the fitter it is, that is, the more children it will produce. Beginning with the following description of a parent population composed of 3 types: (i = 0,1,2) with sightedness values zi = 3,2,1:
 * {| class="wikitable"

Using Equation (4), the child population (assuming the character zi doesn't change)
 * align="center"|i
 * align="center"|0
 * align="center"|1
 * align="center"|2
 * align="center"|ni
 * align="center"|10
 * align="center"|20
 * align="center"|30
 * align="center"|zi
 * align="center"|3
 * align="center"|2
 * align="center"|1
 * }
 * align="center"|2
 * align="center"|1
 * }
 * {| class="wikitable"

We would like to know how much average visual acuity has increased or decreased in the population. From Equation (3), the average sightedness of the parent population is z = 5/3. The average sightedness of the child population is z' = 2, so that the change in average sightedness is:
 * align="center"|i
 * align="center"|0
 * align="center"|1
 * align="center"|2
 * align="center"|ni
 * align="center"|30
 * align="center"|40
 * align="center"|30
 * align="center"|zi
 * align="center"|3
 * align="center"|2
 * align="center"|1
 * }
 * align="center"|2
 * align="center"|1
 * }



\Delta z \ \stackrel{\mathrm{def}}{=}\ z'-z = 1/3 \,\! $$

which indicates that the trait of sightedness is increasing in the population. Applying the Price equation we have (since z&prime;i= zi):



\Delta z = \operatorname{cov}\left(w_i,z_i\right)/w = 1/3 $$

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 z can be written:



w(z'-z)=\langle w_i z_i\rangle - wz $$

For the second generation:



w'(z''-z')=\langle w'_i z'_i\rangle - w'z' $$

The simple Price equation for z only gives us the value of z ' for the first generation, but does not give us the value of w '  and &lang;w 'i z 'i &rang; which needs to be calculated z ' '  for the second generation. w ' and &lang;w 'i z 'i &rang; can both be thought of characteristics of the first generation, so the Price equation can be used to calculate them as well:


 * $$w(w'-w)=\langle w_i^2\rangle - w^2$$

w(\langle w'_i z'_i\rangle-\langle w_i z_i\rangle)=\langle w_i ^2 z_i\rangle - w\langle w_i z_i\rangle $$

The five 0-generation variables w, z, &lang;wi zi &rang;, &lang;w2i &rang;, and &lang;w2i zi &rang;, which must be known before proceeding to calculate the three first generation variables w ', z ', and &lang;w 'i z 'i &rang;, 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 (&lang;wni &rang; and &lang;wni zi &rang;) 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.

Example: The Evolution of Sickle Cell Anemia


As an example of dynamical sufficiency, consider the case of sickle cell anemia. Each person has two sets of genes, one set inherited from the father, one from the mother. Sickle cell anemia is a blood disorder which occurs when a particular pair of genes both carry the 'sickle-cell trait'. The reason that the sickle-cell gene has not been eliminated from the human population by selection is because when there is only one of the pair of genes carrying the sickle-cell trait, that  individual (a "carrier") is highly resistant to malaria, while a person who has neither gene carrying the sickle-cell trait will be susceptible to malaria. Let's see what the Price equation has to say about this.

Let zi=i be the number of sickle-cell genes that organisms of type i have so that zi = 0 or 1 or 2. Assume the population sexually reproduces and matings are random between type 0 and 1, so that the number of 0–1 matings is n0n1/(n0+n1) and the number of i–i matings is n2i/[2(n0+n1)] where i = 0 or 1. Suppose also that each gene has a 1/2 chance of being passed to any given child and that the initial population is ni=[n0,n1,n2]. If b is the birth rate, then after reproduction there will be


 * $$b\left(\frac{n_0^2/2+n_0n_1/2+n_1^2/8}{n_0+n_1}\right)$$ type 0 children (unaffected)
 * $$b\left(\frac{n_0n_1/2+n_1^2/4}{n_0+n_1} \right)$$ type 1 children (carriers)


 * $$b\left(\frac{n_1^2/8}{n_0+n_1} \right)$$ type 2 children (affected)

Suppose a fraction a of type 0 reproduce, the loss being due to malaria. Suppose all of type 1 reproduce, since they are resistant to malaria, while none of the type 2 reproduce, since they all have sickle-cell anemia. The fitness coefficients are then:


 * $$w_0=ab\left(\frac{n_0^2/2+n_0n_1/2+n_1^2/8}{n_0(n_0+n_1)}\right)$$
 * $$w_1=b\left(\frac{n_0n_1/2+n_1^2/4}{n_1(n_0+n_1)} \right)$$
 * $$w_2=0\,$$

To find the concentration n1 of carriers in the population at equilibrium, with the equilibrium condition of Δ z=0, the simple Price equation is used:



0=\operatorname{cov}(w_i/w,z_i) = \frac{f(2-2a-af)}{(1+f)(2a+2f+af)} $$

where f=n1/n0. At equilibrium then, assuming f is not zero:



f=\frac{n_1}{n_0}=\frac{2(1-a)}{a} $$

In other words the ratio of carriers to non-carriers will be equal to the above constant non-zero value. In the absence of malaria, a=1 and f=0 so that the sickle-cell gene is eliminated from the gene pool. For any presence of malaria, a will be smaller than unity and the sickle-cell gene will persist.

The situation has been effectively determined for the infinite (equilibrium) generation. This means that there is dynamical sufficiency with respect to the Price equation, and that there is an equation relating higher moments to lower moments. For example, for the second moments:


 * $$ \langle w_i^2z_i \rangle = \frac{\langle w_i z_i \rangle}{z}$$
 * $$ \langle w_i^2 \rangle = \frac{-b z^2 w^2 + 2 b z^2 w \langle w_i z_i

\rangle + b z \langle w_i z_i \rangle^2 - b z^2 \langle w_i z_i \rangle^2 - 4 \langle w_i z_i \rangle^3} { b z^2 - 4 z \langle w_i z_i \rangle} $$

Example: sex ratios
In a 2-sex species or deme with sexes 1 and 2 where $$ z_1= 1$$, $$ z_2 = 0$$, $$z$$ is the relative frequency of sex 1. The simplest model of the sex ratio is achieved by assuming that the fitness of an individual is proportional to the relative frequency of the sex with which it mates. Consider proportionality constants $$ a$$ and $$ b$$ such that $$ w_1 = a(1 - z)$$ and $$ w_2 = bz$$. This gives $$ w = (a + b)z(1 - z)$$ and $$ cov(w_i, z_i) + wz = E(w_i,z_i) = az(1 - z)$$, so $$ cov(w_i, z_i) = az(1 - z) - (a + b)z^2(1 - z) = z(1 - z)(a - (a + b)z) $$. Hence, $$ \Delta z = a/(a + b) - z $$ so that $$ z' = a/(a + b) $$.

The Full Price Equation
The simple Price equation was based on the assumption that the characters zi do not change over one generation. If it is assumed that they do change, with zi 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.

Example: The Evolution of Altruism
To study the evolution of a genetic predisposition to altruism, altruism will be defined as the genetic predisposition to behavior which decreases individual fitness while increasing the average fitness of the group to which the individual belongs. First specifying a simple model, which will only require the simple Price equation. Specify a fitness wi by a model equation:



w_i = \frac{n'_i}{n_i} = k - a z_i + b z $$

where zi is a measure of altruism, the azi term is the decrease in fitness of an individual due to altruism towards the group and bz is the increase in fitness of an individual due to the altruism of the group towards an individual. Assume that a and b are both greater than zero. From the Price equation:



w\Delta z = -a~\operatorname{var}\left(z_i\right) $$

where var(zi) is the variance of zi which is just the covariance of zi with itself:



\operatorname{var}(z_i) \ \stackrel{\mathrm{def}}{=}\ \operatorname{E}(z_i^2)-\operatorname{E}(z_i)^2 $$

It can be seen that, by this model, in order for altruism to persist it must be uniform throughout the group. If there are two altruist types the average altruism of the group will decrease, the more altruistic will lose out to the less altruistic.

Now assuming a hierarchy of groups which will require the full Price equation. The population will be divided into groups, labelled with index i and then each group will have a set of subgroups labelled by index j. Individuals will thus be identified by two indices, i and j, specifying which group and subgroup they belong to. nij will specify the number of individuals of type ij. Let zij be the degree of altruism expressed by individual j of group i towards the members of group i. Let's specify the fitness wij by a model equation:



w_{ij} = \frac{n'_{ij}}{n_{ij}} = k - a z_{ij} + b z_i $$

The a zij term is the fitness the organism loses by being altruistic and is proportional to the degree of altruism zij that it expresses towards members of its own group. The b zi term is the fitness that the organism gains from the altruism of the members of its group, and is proportional to the average altruism zi expressed by the group towards its members. Again, in studying study altruistic (rather than spiteful) behavior, it is expected that a and b are positive numbers. Note that the above behavior is altruistic only when azij >bzi. Defining the group averages:


 * $$n_i = \sum_j n_{ij}\,$$


 * $$z_i = \frac{\sum_j z_{ij}n_{ij}}{n_i}$$


 * $$w_i = \frac{\sum_j w_{ij}n_{ij}}{n_i}=k+(b-a)z_i$$


 * $$n_i'= \sum_j n_{ij}'=n_i(k+(b-a)z_i)\,$$


 * $$z_i'= \frac{\sum_j z_{ij}n_{ij}'}{n_i'}$$

and global averages:


 * $$n = \sum_{ij} n_{ij} = \sum_i n_i\,$$


 * $$z = \frac{\sum_{ij} z_{ij}n_{ij}}{n} = \frac{\sum_i z_in_i}{n}$$


 * $$w = \frac{\sum_{ij} w_{ij}n_{ij}}{n} = \frac{\sum_i w_in_i}{n}$$


 * $$n'= \sum_{ij} n_{ij}' = \sum_i n_i'\,$$


 * $$z'= \frac{\sum_{ij} z_{ij}n_{ij}'}{n'} = \frac{\sum_i z_i'n_i'}{n'}$$

It can be seen that since the zi and zi are now averages over a particular group, and since these groups are subject to selection, the value of Δzi = z&prime;i−zi will not necessarily be zero, and the full Price equation will be needed.



\Delta z = \operatorname{cov}(w_i/w,z_i)+\operatorname{E}(w_i\Delta z_i/w)\, $$

In this case, the first term isolates the advantage to each group conferred by having altruistic members. The second term isolates the loss of altruistic members from their group due to their altruistic behavior. The second term will be negative. In other words there will be an average loss of altruism due to the in-group loss of altruists, assuming that the altruism is not uniform across the group. The first term is:



\operatorname{cov}(w_i/w,z_i)=\left(b-a\right)\operatorname{var}(z_i) $$

In other words, for b>a there may be a positive contribution to the average altruism as a result of a group growing due to its high number of altruists and this growth can offset in-group losses, especially if the variance of the in-group altruism is low. In order for this effect to be significant, there must be a spread in the average altruism of the groups.

Example: The Evolution of Mutability
Suppose there is an environment containing two kinds of food. Let α be the amount of the first kind of food and β be the amount of the second kind. Suppose an organism has a single allele which allows it to utilize a particular food. The allele has four gene forms: A0, Am, B0, and Bm. If an organism's single food gene is of the A type, then the organism can utilize A-food only, and its survival is proportional to α. Likewise, if an organism's single food gene is of the B type, then the organism can utilize B-food only, and its survival is proportional to β. A0 and Am are both A-alleles, but organisms with the A0 gene produce offspring with A0-genes only, while organisms with the Am gene produce (1−3m) offspring with the Am gene, and m organisms of the remaining three gene types. Likewise, B0 and Bm are both B-alleles, but organisms with the B0 gene produce offspring with B0-genes only, while organisms with the Bm gene produce (1−3m) offspring with the Bm gene, and m organisms of the remaining three gene types.

Let i=0,1,2,3 be the indices associated with the A0, Am, B0, and Bm genes respectively. Let wij be the number of viable type-j organisms produced per type-i organism. The wij matrix is: (with i denoting rows and j denoting columns)


 * {| class="wikitable"


 * align="center"|α
 * align="center"|0
 * align="center"|0
 * align="center"|0
 * align="center"|mα
 * align="center"|(1−3m)α
 * align="center"|mβ
 * align="center"|mβ
 * align="center"|0
 * align="center"|0
 * align="center"|β
 * align="center"|0
 * align="center"|mα
 * align="center"|mα
 * align="center"|mβ
 * align="center"|(1−3m)β
 * }
 * align="center"|mα
 * align="center"|mβ
 * align="center"|(1−3m)β
 * }


 * $$\mathbf{w}=

\begin{bmatrix} \alpha & 0            &0       & 0      \\ m\alpha & (1-3m)\alpha & m\beta & m\beta \\ 0      & 0            & \beta  & 0      \\ m\alpha & m\alpha     & m\beta & (1-3m)\beta \end{bmatrix} $$

Mutators are at a disadvantage when the food supplies α and β are constant. They lose every generation compared to the non-mutating genes. But when the food supply varies, even though the mutators lose relative to an A or B non-mutator, they may lose less than them over the long run because, for example, an A type loses a lot when α is low. In this way, "purposeful" mutation may be selected for. This may explain the redundancy in the genetic code, in which some amino acids are encoded by more than one codon in the DNA. Although the codons produce the same amino acids, they have an effect on the mutability of the DNA, which may be selected for or against under certain conditions.

With the introduction of mutability, the question of identity versus lineage arises. Is fitness measured by the number of children an individual has, regardless of the children's genetic makeup, or is fitness the child/parent ratio of a particular genotype?. Fitness is itself a characteristic, and as a result, the Price equation will handle both.

Suppose we want to examine the evolution of mutator genes. Define the z-score as:



z_i = \left[0,1,0,1\right] $$

in other words, 0 for non-mutator genes, 1 for mutator genes. There are two cases:

Example: Genotype Fitness
Lets focus on the idea of the fitness of the genotype. The index i indicates the genotype and the number of type i genotypes in the child population is:

n'_i = \sum_j w_{ji}n_j\, $$ which gives fitness:

w_i=\frac{n'_i}{n_i} $$ Since the individual mutability zi does not change, the average mutabilities will be:


 * $$z = \frac{\sum_i z_i n_i}{n}$$
 * $$z' = \frac{\sum_i z_i n'_i}{n'}$$

with these definitions, the simple Price equation now applies.

Example: 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 i-type organism has is:

n'_i = n_i\sum_j w_{ij}\, $$ which gives fitness:



w_i=\frac{n'_i}{n_i} = \sum_j w_{ij} $$

We now have characters in the child population which are the average character of the i-th parent.

z'_j = \frac{\sum_i n_i z_i w_{ij} }{\sum_i n_i w_{ij}} $$ with global characters:


 * $$z = \frac{\sum_i z_i n_i}{n}$$
 * $$z' = \frac{\sum_i z_i n'_i}{n'}$$

with these definitions, the full Price equation now applies.