Price equation
This article may be too technical for most readers to understand. (February 2014) 
The Price equation (also known as Price's equation or Price's theorem) describes how a trait or gene 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 proportion of genes within each new generation of a population. The Price equation was derived by George R. Price, working in London to rederive W.D. Hamilton's work on kin selection. The Price equation also has applications in economics.^{1}
Contents
Statement
The Price equation shows that a change in trait z (Δz) is determined by the covariance between each value of z (z_{i}) and fitness (w_{i}) and the expected change in the trait value due to fitness (w_{i}), (E(w_{i} Δz_{i})):
where w is the mean fitness, and E and cov represent the population mean and covariance respectively.
The covariance term captures the effects of natural selection; if the covariance between fitness (w_{i}) and the trait value (z_{i}) is positive then the trait value is predicted to increase on average over population i. If the covariance is negative then the trait is deleterious and is predicted to decrease in frequency.
The second term (E(w_{i} Δz_{i})) is more nuanced and represents factors other than direct selection that can affect trait evolution. This term can encompass genetic drift, mutation bias, or meiotic drive. Additionally, this term can encompass the effects of multilevel selection or group selection.
Price equation and group selection
The Price equation given above examines how the trait value of individuals (z_{i}), denoted by the subscript i, relates to the fitness of that individual (w_{i}).The first step in adapting the Price equation to group selection is to change the subscript on each term to reflect the average trait value (z_{g}), and corresponding mean fitness (w_{g}) of each groups (g), as opposed to each individual:
Giving the estimated change in a trait between groups. To incorporate changes within groups, set Δz_{g} to individual form of the Price equation giving:
Using this formulation, the Price equation can yield insights into the evolution of altruistic traits. By definition, altruistic traits increase group fitness while decreasing individual fitness. Therefore the covariance between the group average of an altruistic trait (z_{g}) and group average fitness (w_{g}) is positive (var(w_{g},z_{g})>0), and the covariance between the individual investment in an altruistic trait and that individuals fitness is negative (cov(w_{i},z_{i})<0). Therefore, the Price equation shows that high between group variance (var(z_{g})) coupled with low within group variance (var(w_{gi},z_{gi})) is required for altruistic traits to spread due to natural selection.
Proof of the Price equation
Suppose there is a population of individuals over which the amount of a particular characteristic varies. Those individuals can be grouped by the amount of the characteristic that each displays. There can be as few as just one group of all individuals (consisting of a single shared value of the characteristic) and as many as groups of one individual each (consisting of distinct values of the characteristic). Index each group with so that the number of members in the group is and the value of the characteristic shared among all members of the group is . Now assume that having of the characteristic is associated with having a fitness where the product represents the number of offspring in the next generation. Denote this number of offspring from group by so that . Let be the average amount of the characteristic displayed by the offspring from group . Denote the amount of change in characteristic in group by defined by
Now take to be the average characteristic value in this population and to be the average characteristic value in the next generation. Define the change in average characteristic by . That is,
Note that this is not the average value of (as it is possible that ). Also take 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 and are equivalent to the traditional definitions of sample mean and covariance; however, they are not meant to be statistical estimates of characteristics of a population. In particular, the Price equation is a deterministic difference equation that models the trajectory of the actual mean value of a characteristic along the flow of an actual population of individuals. Assuming that the mean fitness is not zero, it is often useful to write it as
In the specific case that characteristic (i.e., fitness itself is the characteristic of interest), then Price's equation reformulates Fisher's fundamental theorem of natural selection.
To prove the Price equation, the following definitions are needed. If is the number of occurrences of a pair of real numbers and , then:
 The mean of the values is:
 The covariance between the and values is:
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 identify the group with characteristic and let be the number of organisms in that group. The total number of organisms is then where:
The average value of the characteristic is 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 so that . As seen in Equation (1), the expected value operator is linear, so
Combining Equations (7) and (8) leads to
Now, let's compute the first term in the equality above. From Equation (1), we know that:
Substituting the definition of fitness, (Equation (4)), we get:
Next, substituting the definitions of average fitness () from Equation (9), and average child characteristics () from Equation (10) gives the Price Equation:
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:
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 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 of the Price equation
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.
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:
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 as 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.
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.
Proof: 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 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.
Genotype fitness
We 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.
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 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.
Criticism of the use of the Price equation
A critical discussion of the use of the Price equation can be found in van Veelen (2005) "On the use of the Price equation" ^{2} and van Veelen et al. (2012) "Group selection and inclusive fitness are not equivalent; the Price equation vs. models and statistics".^{3} A discussion of this criticism can be found in Frank (2012) ^{4}
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.
See also
References
 Inline references
 ^ Knudsen, Thorbjørn (2004). "General selection theory and economic evolution: The Price equation and the replicator/interactor distinction". Journal of Economic Methodology (Taylor and Francis Journals) 11 (2): 147–173. doi:10.1080/13501780410001694109. Retrieved 20111022.
 ^ "On the use of the Price equation". J. Theor. Biol. 237: 412–26. December 2005. doi:10.1016/j.jtbi.2005.04.026. PMID 15953618.
 ^ "Group selection and inclusive fitness are not equivalent; the Price equation vs. models and statistics". J. Theor. Biol. 299: 64–80. April 2012. doi:10.1016/j.jtbi.2011.07.025. PMID 21839750.
 ^ Frank, S A (2012). "Natural Selection. IV. The Price Equation". Journal of Evolutionary biology: 1002–1019. doi:10.1111/j.14209101.2012.02498.
 General references
 Frank, S.A. (1995). "George Price's contributions to Evolutionary Genetics" (PDF). Journal of Theoretical Biology 175 (3): 373–388. doi:10.1006/jtbi.1995.0148. PMID 7475081.
 Frank, S.A. (1997). "The Price Equation, Fisher's Fundamental Theorem, Kin Selection, and Causal Analysis" (PDF). Evolution 51 (6): 1712–1729. doi:10.2307/2410995. JSTOR 2410995.
 Gardner, A. (2008). "The Price equation" (PDF). Curr. Biol. 18 (5): R198–R202. doi:10.1016/j.cub.2008.01.005. PMID 18334191.
 Grafen, A. (2000). "Developments of the Price equation and natural selection under uncertainty" (PDF). Proceedings of the Royal Society B 267 (1449): 1223–1227. doi:10.1098/rspb.2000.1131. PMC 1690660. PMID 10902688.
 Langdon, W. B. (1998). "Evolution of GP Populations: Price's Selection and Covariance Theorem". Genetic Programming and Data Structures: 167–208.
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 Price, G.R. (1970). "Selection and covariance". Nature 227 (5257): 520–521. doi:10.1038/227520a0. PMID 5428476.
 Price, G.R. (1972). "Extension of covariance selection mathematics". Annals of Human Genetics 35 (4): 485–490. doi:10.1111/j.14691809.1957.tb01874.x. PMID 5073694.
 Price, G.R. (1972). "Fisher's "fundamental theorem" made clear". Annals of Human Genetics 36 (2): 129–140. doi:10.1111/j.14691809.1972.tb00764.x. PMID 4656569.
 Van Veelen, Matthijs (2005). "On the use of the Price equation". Journal of Theoretical Biology 237 (4): 412–426. doi:10.1016/j.jtbi.2005.04.026. PMID 15953618.
 van Veelen, Matthijs; Julián García, Maurice W. Sabelis, and Martijn Egas (2010). "Call for a return to rigour in models (correspondence)". Nature 467 (7316): 661. doi:10.1038/467661d. PMID 20930826.
 van Veelen, Matthijs; Julián García, Maurice W. Sabelis, and Martijn Egas (April 2012). "Group selection and inclusive fitness are not equivalent; the Price equation vs. models and statistics". Journal of Theoretical Biology 299: 64–80. doi:10.1016/j.jtbi.2011.07.025. PMID 21839750.
 Day, T. (2006). "Insights from Price's equation into evolutionnary epidemiology". DIMACS Series in Discrete Mathematics and Theoretical Computer Science 71: 23–43.
 "Radiolab: "The Good Show"". Season 9. Episode 1. December 14, 2011. WNYC. Missing or empty
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