Price equation

The Price equation ( original: Price Price's equation or equation ) is a covariance equation which is a mathematical description of evolution and natural selection. Situated she has the American George R. Price, as in London, he worked for kin selection to an alternative derivation of William D. Hamilton's work in 1967.

Meanwhile, place the equation in the economic theory application.

Details

Consider now a population with the elements i (i is an index that numbers the elements. ) Element i has the fitness.

Finally, it is a measure of a property of the element I, whose evolution is to be viewed. As concrete examples, one could imagine that the distance in which a hunter prey can still distinguish visually from an equal-sized non- prey, or simply the maximum travel speed or physical strength.

The Price equation now states that

Applies. Here, the average fitness and the change in the average capacity. The term cov ( wi, zi) is the covariance of the with regard to the fitness of the population, and E (w, z i ) characteristic is the expected value (average value) of the health of a single individual multiplied by the change in the plane of that individual property.

In the special case - so if the fitness is even looked at the property - is the Price equation, Fisher's fundamental theorem of a new formulation of natural selection.

: It is customary to use instead the notation to indicate that the covariance is a property of the entire data set (w, z), so all data pairs in this calculation respond. With the change in notation can be seen immediately but that the calculation at the same level takes place as the calculation of the expected value.
The last term ( the expected value ) is only non-zero when the value of the transition from one generation to the next property may change. However, remains constant, we obtain the simplified Price equation: .
Conversely, if z i is nonzero, you can use the Price equation in itself - just put in the expected value at the end - by dividing each group i in further subgroups ij. ( The introduction of the second index j mathematically represents this further subdivision dar. ):

This operation is possible because the values represent the same with only one index averages for the two index values , as do the values for those without an index only index. (See also the example of the evolution of altruism below.) To each of the index were to avoid misunderstandings attached to the operators " E" and " cov ", act on these operators.
This recursive insertion can be repeated as often in principle. Practically a level of selection will eventually be reached in which does not change.
Group ( tribe, herd, swarm, people ), individual genes, proteins are an example of a four -stage selection process. In this way, it becomes clear that the Price equation is the mathematical foundation of multilevel selection theory of ( among others ) David Sloan Wilson and Elliott Sober. (See story.)

History

Although the importance of the equation - as the whole work Prices - was quickly captured by Hamilton and some other experts in the theory of evolution, she moved in 1995 by an article by Steven A. Frank back into the center of scientific interest. It plays a central role in multilevel selection theory of Wilson and Sober, outlined in her book of 1999. Swartz James finally caught the equation with his biography Prices (2000) finally out of the mists of history. ( Cf. references)

Proof of Price 's equation

For the proof, the following definitions are needed. Be the number of occurrences of the real number pair.

The average or expected value of any variable x is then:

And the covariance between and is:

It now lies in front of a population of organisms, all of which have a genetic trait that is described by a real number z .. Then groups of individuals within the population can be defined that have the same value z. The index i describe the group of property, and ' is the number of individuals which make up the group. The total population size is n - the sum of all:

The average value of the z property is then:

Assume now that the population evolved fully a generation. All individuals of the parent generation had disappeared and in a selection process, the least adapted individuals of generation child had been removed from the self-reproducing population. After reproduction and selection, the size of the population to the value n'i have changed. General designate dashed values now values the child generation, ungestrichelte values , the values of the parent generation. The fitness of group i is now defined as the ratio of the sizes of child to parent generation:

Wherein the average fitness (second "=" with the equation (4 ) )

Is. The total size of the child generation is n ', where:

Whereby the equation ( 5)

Will. The average value of z ' of the subject property in the child generation is:

Here are z'i the (possibly modified ) values of the considered property in the child generation. From the equations ( 1) and (2) follows:

And

To write Equation (7 ) using equation (8) in order to obtain:

Equation ( 1) to write the first term on the right side of equation (9):

And with equation (4) the authors engage in equation (10 ), a further transformation:

And in a third step one applies to the right side of equation (11), the equations (5a) and (6 ) to give:

If, then equation (12 ) into equation ( 9), one obtains the Price equation:

Example: Evolution of altruism

The Price equation is able to describe in an elegant way the evolution of a predisposition toward altruism. For this purpose, altruism is defined as a behavior that on the one hand, the fitness ( reproductive success ) of the altruistic individual reduced, on the other hand, the average fitness of the group to which it belongs, the altruistic individual increases. An individual behaves altruistically towards another individual, it is assumed that both belong to the same group.

General derivation

Consider a hierarchy of groups:

The total population is divided into groups and is designated by the index i ( numbered ).
Each of these groups a number of subgroups I have to be designated by the index j.

Individuals are therefore assigned with two indices i and j that indicate to which group and which sub- group an individual belongs.

Be the number of individuals of type ij.
Is the degree of altruism, the one each individual subgroup ij shows towards all members of the group i. This value is constant from generation to generation. So suppose. It should be noted that a basic premise of this model is that altruism is shown only within their own group. Altruism evolves only under the external pressure of a competing group. At the same time Altruismusbeziehungen define the extent and limits of the group.

In this model, the fitness is defined as follows:

The term is the fitness that the individual loses by his own altruism. She was proportional to the degree of Altruismuses which the individual against members of their own group i shows.

The term is the fitness that the individual gains through the altruism of the other members of his group i. This gain is proportional to the average altruism of the group to their members.

For the studies on the evolution of altruism, it is necessary that a and b are positive numbers. As part of the group i described the behavior of an individual is only altruistic when.

The size of a group is derived from the sum of the sizes of its sub- groups:

The respective mean values for the groups resulting from the sum of all subgroups of a group and normalized to the size of each group as:

It follows by the definition of the fitness wi immediately:

Z'i calculated completely analogously to zi, but may ( in contrast to the constancy of z'ij ) take a different value:

The total size of the population is:

For the global average values, ie the averages across all groups, you have to add all the groups and all subgroups and then divide by the total size of the population:

In the child generation is the total size of the population:

Calculated exactly as described in:

So you can bring the Price equation to use again. In this case, you need the version in which the Price equation was used once in yourself:

In the first step results in a trivial simplification of the adoption:

Now you can express from the section proof of Price 's equation by expectation values of the covariances as defined by equation (2):

Where products are not in the expected values , ( A.2 ), ( A.3 ), ( A.7 ) and ( A.8 ) can be the spelling with the equations change to match the left hand side of the equation. In addition, the expectation value of a sum equals the sum of the expected values of the two terms:

It can be seen that the first and the fourth term of the right side of the equation cancel each other.

Up to this point the only Price equation was transformed. No elements of the model have so far been used. Now, however, the equations (A.1 ) and ( A.8 ) are used:

Multiplying out and separate letter of expectation values are obtained:

Some of the expectations are very easy to calculate using the above equations:

The first and third terms cancel each other. The parameter K thus disappears completely from the right side of the equation. Moreover, one can the fourth term with the help of variance ( Short Definition: cov ( x, x) = var ( x ), ie the covariance of a variable with itself) rewrite:

One newly added expectation value can be again easily calculated:

It summarizes and divided by the whole equation, which gives the change in the considered property " altruism " in the child generation compared to the parental generation, and exclusively in function of parameters and variables of the parents' generation:

The advantage of this notation with the variance is that the variance is always greater than or equal to zero.

The first term of this equation represents the advantage that each group has its altruistic members. He is greater than zero, if b > a.

The second term is the loss of altruistic members again, to each group. It is in any case to a loss because both variables A and the variance is greater than zero and thus makes the minus the loss or at best, the non- winning is inevitable.

Recall now back to Fishers fundamental theorem of natural selection: " The increase in the average fitness of each organism that follows at any time from the natural selection and is mediated by a change in gene frequency is exactly equal to the genetic variance in fitness at that time. "They have to do it in the result equation with two variances. The variance seen in the average Altruismusneigung about the groups can ( b> a) lead to an increase in altruism. So you have to do it with an effect of group selection. However, the average variance in the Altruismusneigung within a group through a reduction in the number of altruists to a reduction in the overall average Altruismusneigung. Here one has therefore to do it with an effect of individual selection. So there are two levels of selection present. Which dominates the net determine the parameters a and b, and the two variances.

Finally, it should be noted that the increase in altruism is the smaller, the greater the tendency to altruism is already. Consider this, the denominator of the earnings equation.

The parameter k has the meaning of a general inertia: the larger it is, the slower it will run the change (in time). However, it can not reverse the general trend. The better adapted to the individuals of the population investigated so - for reasons other than their Altruismusneigung - already are, the slower will increase their Altruismusneigung, even if the tendency due to the remaining relevant parameters of the system goes in this direction.

Concrete elaboration with numerical values and the importance of the definition of what is a group

To make it clear how important it is to identify the groups correctly, numbers are used for the parameters now and the groups defined explicitly. ( For the choice of parameters is decisive only b> a and that otherwise cause the bills to " beautiful " results. ) Let

In this case, equation ( A.21 ) is very symmetrical written as:

There were two groups, which are composed of two members each. Now, two cases should be considered.

Case 1:

And Case 2:

In the first case, the members of a group are identical in terms of their Altruismusneigung. In the second case they are not, for the groups are identical in terms of their members. In each there is an egoist and an altruist. If we now calculate for Case 1, we obtain, in the second case, however, completely opposite. This comes about as follows for Case 1: The selfish group remains the same in every generation with 2 members. The altruistic group, however, doubled in each generation their size. The effect is so so great ( " double " ), since (ba ) / k = 1, is the property of Altruismusneigung with this parameter choice and exactly 50 % of the total fitness and together account for all other effects also only a half. In case 2, the population development constitutes such a way that the one altruist remains in both groups in each generation, but the number of egoists doubled from generation to generation. The deeper reason for this fundamental difference between Case 1 and Case 2 is that in Case 1, the individuals of the considered property are identical with respect to ( variance = 0) and the groups differ in the averages of their individuals (large variance), in case 2 however, the different individuals in a group, but the groups are identical in their average values. The crucial factor is the question of where the larger variance is (see " Fishers fundamental theorem of natural selection " ): If the individuals differ from each other in a group more than the groups ( average of its members ), they are in direct competition with each other within their group (individual selection). Are the individuals within the groups, however, very similar and the groups differ greatly from one another, then reduce the selection pressure within the groups and the individuals are indirectly through their group membership with the members of the other groups in competition ( group selection ). This is the case, the - in this model - the emergence of altruism within groups promotes or altruistic groups can grow faster than non- altruistic groups.

It can be seen that one is " What will become? " Already can make a crucial mistake in identifying the relevant groups to the question. If one believes, for example, that for the question of the evolution of altruism nationality is (consider, for example, two Austrians and two Swiss ) decisive, one can arrive at false conclusions, when in reality the affiliation to the quantities "Star Trek fan "and" starship Enterprise fan " and each one of the considered Swiss and Austrians Star Trek fan and the other starship Enterprise fan. As noted above, the question is crucial for the summary to groups to apply the Price equation, where altruism is exercised. And the groups of such arts must be found internally homogeneous than the groups with each other, otherwise altruism will disappear in the course of generations. On the one hand, therefore, must altruists (and possibly also the non- altruists, ie parasites) come together to form groups, on the other hand - a basic prerequisite of evolutionary biology - must be at least partially genetically determined altruistic inclinations, so that a kind of altruistic is. Does not apply to the first condition, the parasites assert themselves in each group.

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