Price equation

The Price equation (original: Price equation or Price's equation ) is a covariance equation that is a mathematical description of evolution and natural selection . It was established by the American George R. Price when he was working in London in 1967 on an alternative derivation of William D. Hamilton's work on the selection of kin .

The equation is now also used in economic theory .

Details

Now consider a population with the elements . ( is an index that numbers the elements.) Have the element fitness . ${\displaystyle i}$ ${\displaystyle i}$ ${\displaystyle i}$ ${\displaystyle w_{i}}$

${\displaystyle z_{i}}$after all, be a measure of a property of the element whose evolution is to be considered. As concrete examples, one could imagine that the distance in which a hunter can visually distinguish a prey from a non-prey of the same size, or simply the maximum speed of movement or physical strength. ${\displaystyle i}$ ${\displaystyle z_{i}}$

The Price equation now says that

${\displaystyle {\overline {w}}\Delta {\overline {z}}=\operatorname {cov} (w_{i},z_{i})+\operatorname {E} (w_{i}\Delta z_{i}),\,\!}$

applies. Where is the average fitness and is the change in average property. The term is the covariance of the property with respect to fitness in the population and is the expected value ( mean ) of the fitness of a single individual multiplied by the change in the property of that individual. ${\displaystyle {\overline {w}}}$ ${\displaystyle \Delta {\overline {z}}={\overline {z'}}-{\overline {z}}}$ ${\displaystyle \operatorname {cov} (w_{i},z_{i})}$ ${\displaystyle \operatorname {E} (w_{i}\Delta z_{i})}$

In the special case - when fitness itself is the property under consideration - the Price equation is a reformulation of Fisher's fundamental theorem of natural selection . ${\displaystyle z_{i}=w_{i}}$

Explanations and Notes

• ${\displaystyle \operatorname {cov} (w_{i},z_{i})}$: it is common to use the notation instead to indicate that the covariance is a property of the entire data set , i.e. that all data pairs are included in this calculation. With the changed notation, you can immediately see that the calculation takes place on the same level as the calculation of the expected value. ${\displaystyle \operatorname {cov} (w,z)}$ ${\displaystyle (w_{i},z_{i})}$ ${\displaystyle (w_{i},z_{i})}$
• The last term (the expected value) is only non-zero if the value of the property can change during the transition from one generation to the next. Remains , however, constant, we obtain the simplified price equation: . ${\displaystyle z_{i}}$ ${\displaystyle {\overline {w}}\Delta {\overline {z}}=\operatorname {cov} (w_{i},z_{i})}$
• If, on the other hand, Δ z _{i is} different from zero, the Price equation can be used in itself - more precisely in the expected value at the end - by dividing each group into further subgroups . (The introduction of the second index mathematically represents this further subdivision.): ${\displaystyle \Delta \ z_{i}}$ ${\displaystyle i}$ ${\displaystyle ij}$ ${\displaystyle j}$

${\displaystyle {\overline {w}}\Delta {\overline {z}}=\operatorname {cov_{i}} (w_{i},z_{i})+\operatorname {E_{i}} (\operatorname {cov_{j}} (w_{ij},z_{ij})+\operatorname {E_{j}} (w_{ij}\Delta z_{ij})),\,\!}$

• This process becomes possible because the values ​​with only one index represent averages for the values ​​with two indexes just as the values ​​without an index do for those with only one index. (See also the example on the evolution of altruism below.) In order to avoid misunderstandings , the index on which these operators act was added to the operators and . ${\displaystyle E}$ ${\displaystyle \operatorname {cov} }$
• This recursive insertion can in principle be repeated any number of times. In practice, at some point a level of selection will be reached in which there is no change. ${\displaystyle z_{i}}$
• Group (tribe, pack, swarm, colony), 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 the multilevel selection theory of ( inter alia) David Sloan Wilson and Elliott Sober . (See the History section .)

history

Although the meaning of the equation - like the whole of Prices - was quickly grasped by Hamilton and several other experts in the theory of evolution, it was not until 1995 that an article by Steven A. Frank brought it back to the center of scientific interest. It plays a central role in the multilevel selection theory of Wilson and Sober, presented in their book from 1999. James Swartz finally got the equation out of the darkness of history with his biography Prices (2000). (See references)

Proof of the Price equation

The following definitions are needed for the proof. Let be the number of occurrences of the real number pair . ${\displaystyle n_{i}}$ ${\displaystyle (x_{i},y_{i})}$

• The mean or expected value of any quantity x is then:

${\displaystyle {\overline {x}}=\operatorname {E} (x_{i})\equiv {\frac {\sum _{i}x_{i}n_{i}}{\sum _{i}n_{i}}}~~~~~~~~~~~~~~~(1)}$

• And the covariance between and is: ${\displaystyle x_{i}}$ ${\displaystyle y_{i}}$

${\displaystyle \operatorname {cov} (x_{i},y_{i})\equiv {\frac {\sum _{i}n_{i}~[x_{i}-\operatorname {E} (x_{i})]~[y_{i}-\operatorname {E} (y_{i})]}{\sum _{i}n_{i}}}=\operatorname {E} (x_{i}y_{i})-\operatorname {E} (x_{i})\operatorname {E} (y_{i})~~~~~~~~~~~~~~~(2)}$

We now have a population of organisms, all of which have a genetic property that is described by a real number . Then groups of individuals can be defined within the population that have the same value . Let the index describe the group with the property and ' be the number of individuals that make up the group. Let the total size of the population be - the sum of all : ${\displaystyle z}$ ${\displaystyle z}$ ${\displaystyle i}$ ${\displaystyle z_{i}}$ ${\displaystyle n_{i}}$ ${\displaystyle n}$ ${\displaystyle n_{i}}$

${\displaystyle n=\sum _{i}n_{i}\,}$

The average value of the property is then: ${\displaystyle z}$

${\displaystyle {\overline {z}}={\frac {\sum _{i}z_{i}n_{i}}{n}}~~~~~~~~~~~~~~~~~~(3)}$

Assume now that the population has completely evolved by one generation. All individuals of the parent generation had disappeared and in a selection process the least adapted individuals of the child generation were removed from the reproducing population. After reproduction and selection, the size of the population changed to the value . In general, dashed values ​​now denote values ​​of the child generation, unbroken values ​​the values ​​of the parent generation. The fitness of a group is now defined as the ratio of the sizes of the child to the parent generation: ${\displaystyle n_{i}'}$ ${\displaystyle i}$

${\displaystyle w_{i}={\frac {n_{i}'}{n_{i}}}~~~~~~~~~~~~~~~~~(4)}$

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

${\displaystyle {\overline {w}}={\frac {\sum _{i}w_{i}n_{i}}{n}}={\frac {\sum _{i}n_{i}'}{n}}~~~~~~~~~~~~~(5)}$

is. The total size of the child generation is where: ${\displaystyle n'}$

${\displaystyle n'=\sum _{i}n'_{i}\,}$

giving equation (5) to

${\displaystyle {\overline {w}}={\frac {n'}{n}}~~~~~~~~~~~~~(5a)}$

becomes. The average value of the property under consideration in the child generation is: ${\displaystyle z'}$

${\displaystyle {\overline {z'}}={\frac {\sum _{i}z'_{i}n_{i}'}{n'}}~~~~~~~~~~~~~(6)}$

Here are the (possibly changed) values ​​of the property under consideration in the child generation. From equations (1) and (2) it follows: ${\displaystyle z'_{i}}$

${\displaystyle \operatorname {cov} (w_{i},z_{i})=\operatorname {E} (w_{i}z_{i})-{\overline {w}}~{\overline {z}}\,~~~~~~~~~~~~~~~(7)}$

and

${\displaystyle \operatorname {E} (w_{i}\Delta z_{i})=\operatorname {E} (w_{i}z'_{i})-\operatorname {E} (w_{i}z_{i})\,~~~~~~~~~~~~~~~(8)}$

If you rewrite equation (7) with the help of equation (8), you get:

${\displaystyle \operatorname {cov} (w_{i},z_{i})+\operatorname {E} (w_{i}\Delta z_{i})=\operatorname {E} (w_{i}z'_{i})-{\overline {w}}~{\overline {z}}\,~~~~~~~~~~~~~~~(9)}$

With equation (1) one writes the first term on the right side of equation (9):

${\displaystyle \operatorname {E} (w_{i}z'_{i})={\frac {\sum _{i}w_{i}z'_{i}n_{i}}{n}}~~~~~~~~~~~~~~~(10)}$

And with equation (4) one carries out another transformation in equation (10):

${\displaystyle \operatorname {E} (w_{i}z'_{i})={\frac {\sum _{i}w_{i}z'_{i}n_{i}}{n}}={\frac {\sum _{i}z'_{i}n'_{i}}{n'}}~{\frac {n'}{n}}~~~~~~~~~~~~~~~(11)}$

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

${\displaystyle \operatorname {E} (w_{i}z'_{i})={\overline {z'}}~{\overline {w}}~~~~~~~~~~~~~~~(12)}$

If you now insert equation (12) into equation (9), you get the Price equation:

${\displaystyle \operatorname {cov} (w_{i},z_{i})+\operatorname {E} (w_{i}\Delta z_{i})={\overline {w}}~{\overline {z'}}-{\overline {w}}~{\overline {z}}={\overline {w}}~\Delta {\overline {z}}\,}$

Example: evolution of altruism

The Price equation can elegantly describe the evolution of a predisposition towards altruism . For this purpose, altruism is defined as behavior which, on the one hand, reduces the fitness (reproductive success) of the altruistic individual and, on the other hand, increases the average fitness of the group to which the altruistic individual belongs. If one 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 identified with the index (numbered consecutively). ${\displaystyle i}$
• Each of these groups has a number of subgroups called the index . ${\displaystyle i}$ ${\displaystyle j}$

Individuals are therefore assigned with two indices and , which indicate which group and which subgroup an individual belongs to. ${\displaystyle i}$ ${\displaystyle j}$

• ${\displaystyle n_{ij}}$be the number of individuals of the type . ${\displaystyle ij}$
• ${\displaystyle z_{ij}}$be the degree of altruism exhibited by each individual in the subgroup towards all members of the group . This value is constant from generation to generation. So suppose that: . It should be noted that a basic requirement of this model is that altruism is only shown within one's own group. Altruism only develops under external pressure from a competing group. At the same time, altruistic relationships define the scope and boundaries of the group. ${\displaystyle ij}$ ${\displaystyle i}$ ${\displaystyle z'_{ij}=z_{ij}}$

In this model, fitness is defined as follows: ${\displaystyle w_{ij}}$

${\displaystyle w_{ij}={\frac {n'_{ij}}{n_{ij}}}=k-az_{ij}+bz_{i}~~~~~~~~~~~~~~~~~~~~~~(A.1)}$

The term is the fitness that the individual loses through his or her own altruism. It is proportional to the degree of altruism the individual shows towards members of his own group . ${\displaystyle az_{ij}}$ ${\displaystyle z_{ij}}$ ${\displaystyle i}$

The term is the fitness that the individual gains through the altruism of the other members of his group . This gain is proportional to the group's average altruism towards its members. ${\displaystyle bz_{i}}$ ${\displaystyle i}$ ${\displaystyle z_{i}}$

To study the evolution of altruism, it is necessary that and are positive numbers. In the context of the group , the described behavior of an individual is only altruistic if is. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle i}$ ${\displaystyle az_{ij}>bz_{i}}$

The size of a group results from the sum of the sizes of its subgroups:

${\displaystyle n_{i}=\sum _{j}n_{ij}\,}$

The respective average values ​​for the groups result from the sum of all subgroups of a group and normalized to the size of the respective group as:

${\displaystyle z_{i}={\frac {\sum _{j}z_{ij}n_{ij}}{n_{i}}}=\operatorname {E_{j}} (z_{ij})~~~~~~~~~~~~~~~~~~~~~~(A.2)}$

${\displaystyle w_{i}={\frac {\sum _{j}w_{ij}n_{ij}}{n_{i}}}={\frac {n'_{i}}{n_{i}}}=k+(b-a)z_{i}=\operatorname {E_{j}} (w_{ij})~~~~~~~~~~~~~~~~~~~~~~(A.3)}$

From this the definition of fitness w _i immediately results :

${\displaystyle n_{i}'=\sum _{j}n_{ij}'=n_{i}(k+(b-a)z_{i})~~~~~~~~~~~~~~~~~~~~~~(A.4)}$

z ' _{i is} calculated completely analogously to , but can (in contrast to the constancy of ) take on a different value: ${\displaystyle z_{i}}$ ${\displaystyle z'_{ij}}$

${\displaystyle z_{i}'={\frac {\sum _{j}z_{ij}n_{ij}'}{n_{i}'}}=\operatorname {E_{j}} (z'_{ij})~~~~~~~~~~~~~~~~~~~~~~(A.5)}$

The total size of the population is:

${\displaystyle n=\sum _{ij}n_{ij}=\sum _{i}n_{i}~~~~~~~~~~~~~~~~~~~~~~(A.6)}$

For the global averages, i.e. the averages over all groups, you have to add over all groups or all subgroups and then divide by the total size of the population:

${\displaystyle {\overline {z}}={\frac {\sum _{ij}z_{ij}n_{ij}}{n}}={\frac {\sum _{i}z_{i}n_{i}}{n}}=\operatorname {E_{i}} (z_{i})~~~~~~~~~~~~~~~~~~~~~~(A.7)}$

${\displaystyle {\overline {w}}={\frac {\sum _{ij}w_{ij}n_{ij}}{n}}={\frac {\sum _{i}w_{i}n_{i}}{n}}={\frac {n'}{n}}=k+(b-a){\overline {z}}=\operatorname {E_{i}} (w_{i})~~~~~~~~~~~~~~~~~~~~~~(A.8)}$

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

${\displaystyle n'=\sum _{ij}n_{ij}'=\sum _{i}n_{i}'\ =n(k+(b-a){\overline {z}})~~~~~~~~~~~~~~~~~~~~~~(A.9)}$

${\displaystyle {\overline {z'}}}$is calculated completely analogously to : ${\displaystyle {\overline {z}}}$

${\displaystyle {\overline {z'}}={\frac {\sum _{ij}z_{ij}n_{ij}'}{n'}}={\frac {\sum _{i}z_{i}'n_{i}'}{n'}}=\operatorname {E_{i}} (z'_{i})~~~~~~~~~~~~~~~~~~~~~~(A.10)}$

With this one can apply the Price equation again. In this case you need the version in which the Price equation has been inserted into itself:

${\displaystyle {\overline {w}}\Delta {\overline {z}}=\operatorname {cov_{i}} (w_{i},z_{i})+\operatorname {E_{i}} (\operatorname {cov_{j}} (w_{ij},z_{ij})+\operatorname {E_{j}} (w_{ij}\Delta z_{ij}))~~~~~~~~~~~~~~~~~~~~~~(A.11)}$

In the first step, a trivial simplification results from the assumption : ${\displaystyle \Delta z_{ij}=0}$

${\displaystyle {\overline {w}}\Delta {\overline {z}}=\operatorname {cov_{i}} (w_{i},z_{i})+\operatorname {E_{i}} (\operatorname {cov_{j}} (w_{ij},z_{ij}))~~~~~~~~~~~~~~~~~~~~~~(A.12)}$

Now one can express the covariances according to the definition according to equation (2) from the section Proof of the Price equation by expectation values:

${\displaystyle {\overline {w}}\Delta {\overline {z}}=\operatorname {E_{i}} (w_{i}z_{i})-\operatorname {E_{i}} (w_{i})\operatorname {E_{i}} (z_{i})+\operatorname {E_{i}} (\operatorname {E_{j}} (w_{ij}z_{ij})-\operatorname {E_{j}} (w_{ij})\operatorname {E_{j}} (z_{ij}))~~~~~~~~~~~~~~~~~~~~~~(A.13)}$

Where there are no products in the expected values, the notation can be changed using equations (A.2), (A.3), (A.7) and (A.8) so that it corresponds to the left side of the equation. In addition, the expected value of a sum is equal to the sum of the expected values ​​of the two terms:

${\displaystyle {\overline {w}}\Delta {\overline {z}}=\operatorname {E_{i}} (w_{i}z_{i})-{\overline {w}}\,{\overline {z}}+\operatorname {E_{i}} (\operatorname {E_{j}} (w_{ij}z_{ij}))-\operatorname {E_{i}} (w_{i}z_{i})~~~~~~~~~~~~~~~~~~~~~~(A.14)}$

It can be seen that the first and fourth terms on the right-hand side of the equation cancel each other out.

${\displaystyle {\overline {w}}\Delta {\overline {z}}=-{\overline {w}}\,{\overline {z}}+\operatorname {E_{i}} (\operatorname {E_{j}} (w_{ij}z_{ij}))~~~~~~~~~~~~~~~~~~~~~~(A.15)}$

Up to this point the Price equation has just been transformed. No elements of the model have been used so far. But now equations (A.1) and (A.8) are used:

${\displaystyle {\overline {w}}\Delta {\overline {z}}=-(k+(b-a){\overline {z}}){\overline {z}}+\operatorname {E_{i}} (\operatorname {E_{j}} ((k-az_{ij}+bz_{i})z_{ij}))~~~~~~~~~~~~~~~~~~~~~~(A.16)}$

By multiplying the expected values ​​and writing them separately, we get:

${\displaystyle {\overline {w}}\Delta {\overline {z}}=-k{\overline {z}}-(b-a){\overline {z}}^{2}+k\operatorname {E_{i}} (\operatorname {E_{j}} (z_{ij}))-a\operatorname {E_{i}} (\operatorname {E_{j}} (z_{ij}^{2}))+b\operatorname {E_{i}} (\operatorname {E_{j}} (z_{i}z_{ij}))~~~~~~~~~~~~~~~~~~~~~~(A.17)}$

Some of the expected values ​​can be calculated very easily using the equations given above:

${\displaystyle {\overline {w}}\Delta {\overline {z}}=-k{\overline {z}}-(b-a){\overline {z}}^{2}+k{\overline {z}}-a\operatorname {E_{i}} (\operatorname {E_{j}} (z_{ij}^{2}))+b\operatorname {E_{i}} (z_{i}^{2})~~~~~~~~~~~~~~~~~~~~~~(A.18)}$

The first and third terms cancel each other out. The parameter thus disappears completely from the right-hand side of the equation. In addition, the fourth term can be rewritten using the variance (short definition: cov (x, x) = var (x), i.e. the covariance of a variable with itself): ${\displaystyle k}$

${\displaystyle {\overline {w}}\Delta {\overline {z}}=-(b-a){\overline {z}}^{2}-a\operatorname {E_{i}} (\operatorname {var_{j}} (z_{ij})+(\operatorname {E_{j}} (z_{ij}))^{2})+b\operatorname {E_{i}} (z_{i}^{2})~~~~~~~~~~~~~~~~~~~~~~(A.19)}$

The one newly added expected value can easily be calculated again:

${\displaystyle {\overline {w}}\Delta {\overline {z}}=-(b-a){\overline {z}}^{2}-a\operatorname {E_{i}} (\operatorname {var_{j}} (z_{ij}))-a\operatorname {E_{i}} (z_{i}^{2})+b\operatorname {E_{i}} (z_{i}^{2})~~~~~~~~~~~~~~~~~~~~~~(A.20)}$

One summarizes and divides the whole equation by , whereby one obtains the change in the considered property "altruism" in the child generation compared to the parent generation, exclusively depending on parameters and variables of the parent generation: ${\displaystyle {\overline {w}}}$

${\displaystyle \Delta {\overline {z}}={\frac {(b-a)\operatorname {var_{i}} (z_{i})-a\operatorname {E_{i}} (\operatorname {var_{j}} (z_{ij}))}{k+(b-a){\overline {z}}}}~~~~~~~~~~~~~~~~~~~~~~(A.21)}$

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 each group has from its altruistic members. It is greater than zero if is. ${\displaystyle b>a}$

The second term reflects the loss of altruistic members each group has. In any case, it is a loss, since both variables and the variance are greater than zero and thus the minus makes the loss or at best the non-profit inevitable. ${\displaystyle a}$

Recall Fisher's fundamental theorem of natural selection: “The increase in the mean fitness of every organism that follows from natural selection at any point in time and is mediated by a change in the frequency of genes is exactly the same as the genetic increase Variance in fitness at this point in time. ”You are dealing with two variances in the result equation. The variance in the average tendency towards altruism seen across the groups can ( ) lead to an increase in altruism. So you are dealing with an effect of group selection . The average variance in the tendency towards altruism within a group leads to a reduction in the number of altruists, however, in a reduction in the overall average tendency towards altruism. Here one is consequently dealing with an effect of individual selection. So both levels of selection are present. Which net dominates are determined by the parameters and , as well as the two variances. ${\displaystyle b>a}$ ${\displaystyle a}$ ${\displaystyle b}$

Finally, it should be noted that the increase in altruism is smaller, the greater the tendency towards altruism. Consider the denominator of the result equation.

The parameter has the meaning of a general inertia: the larger it is, the slower the change will take place (over time). However, he cannot reverse the general trend. The better adapted the individuals are to the population under consideration - also for reasons other than their tendency to altruism - the slower their tendency to altruism will increase, even if the tendency is in this direction based on the remaining relevant parameters of the system. ${\displaystyle k}$

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

In order to make it clear how important it is to correctly identify the groups, numbers are now used for the parameters and the groups are explicitly defined. (The only decisive factor for the choice of the parameters is and that otherwise the calculations lead to “nice” results.) Let it be ${\displaystyle b>a}$

${\displaystyle a=1~}$

${\displaystyle b=2~}$

${\displaystyle k=1~}$

In this case equation (A.21) can be written very symmetrically as:

${\displaystyle \Delta {\overline {z}}={\frac {\operatorname {var_{i}} (\operatorname {E_{j}} (z_{ij}))-\operatorname {E_{i}} (\operatorname {var_{j}} (z_{ij}))}{1+{\overline {z}}}}~~~~~~~~~~~~~~~~~~~~~~(a.1)}$

There are two groups, each of which should consist of two members. Two cases will now be considered.

Case 1:

${\displaystyle z_{00}=z_{01}=0~}$

${\displaystyle z_{10}=z_{11}=1~}$

and case 2:

${\displaystyle z_{00}=z_{10}=0~}$

${\displaystyle z_{01}=z_{11}=1~}$

In the first case, the members of a group are identical in terms of their tendency towards altruism. In the second case, they are not, but the groups are identical in terms of their members. In each there is an egoist and an altruist. If one now calculates for case 1, one obtains, but in the second case it is completely opposite . This comes about for case 1 as follows: The selfish group remains the same in every generation with 2 members. The altruistic group, on the other hand, doubles in size with each generation. The effect is so great ("doubling") because it represents the property of the tendency to altruism with this parameter selection and at exactly 50% of the overall fitness and all other effects together make up only half. In case 2, the population development is presented in such a way that the one altruist in both groups is retained in each generation, but the number of egoists doubles from generation to generation. The deeper reason for this fundamental difference between case 1 and case 2 is that in case 1 the individuals are identical in terms of the property under consideration (variance = 0) and the groups differ in the averages of their individuals (large variance), in case 2 however, the individuals in a group differ, but the groups are identical in their average values. The decisive question is where the greater variance is (cf. “Fisher's fundamental theorem of natural selection”): If the individuals in a group differ from each other more than the groups (on average of their members), then they are more direct Competition with one another within their group (individual selection). However, if the individuals within the groups are very similar and the groups differ greatly from one another, then the selection pressure within the groups is reduced and the individuals are indirectly in competition with the members of the other groups through their group membership (group selection). This is the case, which - in this model - promotes the emergence of altruism within the groups or allows altruistic groups to grow faster than non-altruistic groups. ${\displaystyle \Delta z}$ ${\displaystyle \Delta z=+1/6}$ ${\displaystyle \Delta z=-1/6}$ ${\displaystyle (b-a)/k=1}$ ${\displaystyle 0\leq z_{ij}\leq 1}$

You can see that when you ask “What will develop?” You can make a decisive mistake when identifying the relevant groups. If one believes, for example, that nationality is decisive for the question of the evolution of altruism (for example, consider two Austrians and two Swiss), one can come to wrong conclusions if in reality it is belonging to the crowd “Star Trek fan "And" Spaceship Enterprise Fan "and one of the Swiss and Austrian considered is a Star Trek fan and the other is a Spaceship Enterprise fan. As noted above, where altruism is being practiced is critical to grouping for the purpose of applying the Price equation. And the groups found in this way must be internally more homogeneous than the groups among themselves, otherwise altruism will disappear in the course of the generations. On the one hand, altruists (and if possible also non-altruists, i.e. parasites) have to come together in groups, on the other hand - an elementary prerequisite of evolutionary biology - altruistic tendencies have to be at least partially genetically determined so that one kind becomes more altruistic. If the first condition does not apply, the parasites prevail in each group.

However, every individual, especially every human being, is part of a myriad of sets. To verify this statement, just look at the number of categories that exist for the biography articles on Wikipedia and a large number of which can be assigned to a single person at the same time. Each of these categories can have a certain relevance as a group in the sense of the evolution of altruism; H. the members of the category show some kind of altruistic behavior among themselves. The model presented here therefore represents a strong simplification of reality.

literature

• GR Price : Selection and Covariance . In: Nature . 227, 1970, pp. 520-521. doi : 10.1038 / 227520a0 .

• GR Price : Fisher's "fundamental theorem" made clear . In: Annals of Human Genetics . 36, 1972, pp. 129-140.

• SA Frank: George Price's contributions to Evolutionary Genetics . In: Journal of Theoretical Biology . 175, 1995, pp. 373-388.

• SA Frank: The Price Equation, Fisher's Fundamental Theorem, Kin Selection, and Causal Analysis . In: evolution . 51, No. 6, 1997, pp. 1712-1729.

• M. Grafen: Developments of the price equation and natural selection under uncertainty . In: Proc. R. Soc. London B . 267, 2000, pp. 1223-1227.

• M. Van Veelen: On the use of the price equation . In: Journal of Theoretical Biology . 237, 2005, pp. 412-426. doi : 10.1016 / j.jtbi.2005.04.026 .
• Elliott Sober and David Sloan Wilson (1999) Unto Others: The Evolution and Psychology of Unselfish Behavior. ISBN 0-6749-30479 ;
• James Swartz (2000) Death of an Altruist: Was the man who found the selfless gene too good for this world? . Lingua Franca 10.5: 51-61