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 be 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 there will be at least 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 Δz_{i} defined by
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 Δz. That is,
Note that this is not the average value of Δz_{i}. Also take w to be the average fitness of this population. The Price equation states:
where the functions and 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
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.
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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 notation 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:
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′_{i}. Primes will be used to denote child parameters, unprimed variables denote parent parameters.
The total number of children is n' where:
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′_{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 Δz_{i} so that Δz_{i} = z'_{i} − z_{i}. As seen in Equation (1), the expected value operator is linear, so

Combining Equations (7) and (8) leads to

but from Equation (1) gives:
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:
When the characters 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:
which can be restated as:
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 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."
As an example of the simple Price equation, consider a model for the evolution of sight. Suppose z_{i} is a real number measuring the visual acuity of an organism. An organism with a higher z_{i} will have better sight than one with a lower value of z_{i}. Let us say that the fitness of such an organism is w_{i}=z_{i} 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 z_{i} = 3,2,1:
i  0  1  2 
n_{i}  10  20  30 
z_{i}  3  2  1 
Using Equation (4), the child population (assuming the character z_{i} doesn't change)
i  0  1  2 
n_{i}  30  40  30 
z_{i}  3  2  1 
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:
which indicates that the trait of sightedness is increasing in the population. Applying the Price equation we have (since z′_{i}= z_{i}):
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:
For the second generation:
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 〈w '_{i} z '_{i} 〉 which are needed to calculate z″ for the second generation. The variables w ' and 〈w '_{i} z '_{i} 〉 can both be thought of characteristics of the first generation, so the Price equation can be used to calculate them as well:
The five 0generation variables w, z, 〈w_{i} z_{i} 〉, 〈w^{2}_{i} 〉, and 〈w^{2}_{i} z_{i} 〉 which must be known before proceeding to calculate the three first generation variables w ', z ', and 〈w '_{i} z '_{i} 〉, 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 (〈w^{n}_{i} 〉 and 〈w^{n}_{i} z_{i} 〉) 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.
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 'sicklecell trait'. The reason that the sicklecell 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 sicklecell trait, that individual (a "carrier") is highly resistant to malaria, while a person who has neither gene carrying the sicklecell trait will be susceptible to malaria. Let's see what the Price equation has to say about this.
Let z_{i}=i be the number of sicklecell genes that organisms of type i have so that z_{i} = 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 n_{0}n_{1}/(n_{0}+n_{1}) and the number of i–i matings is n^{2}_{i}/[2(n_{0}+n_{1})] 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 n_{i}=[n_{0},n_{1},n_{2}]. If b is the birth rate, then after reproduction there will be
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 sicklecell anemia. The fitness coefficients are then:
To find the concentration n_{1} of carriers in the population at equilibrium, with the equilibrium condition of Δ z=0, the simple Price equation is used:
where f=n_{1}/n_{0}. At equilibrium then, assuming f is not zero:
In other words the ratio of carriers to noncarriers will be equal to the above constant nonzero value. In the absence of malaria, a=1 and f=0 so that the sicklecell gene is eliminated from the gene pool. For any presence of malaria, a will be smaller than unity and the sicklecell 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:
In a 2sex species or deme with sexes 1 and 2 where z_{1} = 1, z_{2} = 0, z is the relative frequency of sex 1. Since all individuals have one parent of each sex, the fitness of each sex is proportional to the other sex's size. 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, Δz = a / (a + b) − z so that z' = a / (a + b).
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.
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 w_{i} by a model equation:
where z_{i} is a measure of altruism, the az_{i} 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:
where var(z_{i}) is the variance of z_{i} which is just the covariance of z_{i} with itself:
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. n_{ij} will specify the number of individuals of type ij. Let z_{ij} be the degree of altruism expressed by individual j of group i towards the members of group i. Let's specify the fitness w_{ij} by a model equation:
The a z_{ij} term is the fitness the organism loses by being altruistic and is proportional to the degree of altruism z_{ij} that it expresses towards members of its own group. The b z_{i} term is the fitness that the organism gains from the altruism of the members of its group, and is proportional to the average altruism z_{i} expressed by the group towards its members. Again, in studying altruistic (rather than spiteful) behavior, it is expected that a and b are positive numbers. Note that the above behavior is altruistic only when az_{ij} >bz_{i}. Defining the group averages:
and global averages:
It can be seen that since the z_{i} and z_{i} are now averages over a particular group, and since these groups are subject to selection, the value of Δz_{i} = z′_{i}−z_{i} will not necessarily be zero, and the full Price equation will be needed.
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 ingroup loss of altruists, assuming that the altruism is not uniform across the group. The first term is:
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 ingroup losses, especially if the variance of the ingroup altruism is low. In order for this effect to be significant, there must be a spread in the average altruism of the groups.
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: A_{0}, A_{m}, B_{0}, and B_{m}. If an organism's single food gene is of the A type, then the organism can utilize Afood 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 Bfood only, and its survival is proportional to β. A_{0} and A_{m} are both Aalleles, but organisms with the A_{0} gene produce offspring with A_{0}genes only, while organisms with the A_{m} gene produce (1−3m) offspring with the A_{m} gene, and m organisms of the remaining three gene types. Likewise, B_{0} and B_{m} are both Balleles, but organisms with the B_{0} gene produce offspring with B_{0}genes only, while organisms with the B_{m} gene produce (1−3m) offspring with the B_{m} gene, and m organisms of the remaining three gene types.
Let i=0,1,2,3 be the indices associated with the A_{0}, A_{m}, B_{0}, and B_{m} genes respectively. Let w_{ij} be the number of viable typej organisms produced per typei organism. The w_{ij} matrix is: (with i denoting rows and j denoting columns)
α  0  0  0 
mα  (1−3m)α  mβ  mβ 
0  0  β  0 
mα  mα  mβ  (1−3m)β 
Mutators are at a disadvantage when the food supplies α and β are constant. They lose every generation compared to the nonmutating genes. But when the food supply varies, even though the mutators lose relative to an A or B nonmutator, 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 zscore as:
in other words, 0 for nonmutator genes, 1 for mutator genes. There are two cases:
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:
which gives fitness:
Since the individual mutability z_{i} does not change, the average mutabilities will be:
with these definitions, the simple Price equation now applies.
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 itype organism has is:
which gives fitness:
We now have characters in the child population which are the average character of the ith parent.
with global characters:
with these definitions, the full Price equation now applies.
Price's equation features in the plot and title of the 2008 thriller film WΔZ (http://www.imdb.com/title/tt0804552).

