The Price of Altruism (50 page)

Unbeknownst to George Price in 1968, the simple covariance relationship

(1)
w
z = Cov (w, z)

had already been worked out and published independently by two other men, Robertson and Li, in 1966 and 1967 respectively.
²
But the uniqueness of the full Price equation stems from the inclusion of the further, expectation term. Here is the derivation:

Let there be a population in which each element is labeled by an index
i
. The frequency of elements with index
i
is
q
_i
, and each element with index
i
has some character,
z
_i
. Elements with a common index form a subpopulation that comprises a fraction
q
_i
of the total population, and no restrictions are placed on how the elements are grouped.

Now imagine a second, descendant, population with frequencies
q´
_i
and characters
z´
_i
. The change in the average character value,
between the mother and descendant populations is

This equation applies to anything that evolves because z can be defined however one likes. For this reason, the equation applies not only to genetics, but to any selection process whatsoever.

What is special about the Price equation is the way in which it associates statistically between entities in groups, a “mother” and “daughter” population. Instead of the value of
q
_i
obtaining from the frequency of elements with index
i
in the daughter population, it obtains from the proportion of the daughter population derived from the elements in with index
i
in the mother population. If we define the fitness of the element
i
as
w
_i
, the contribution to the daughter population from type
i
in the mother population, then
q´
_i
=
q
_i
w
_i
/
where
is the mean fitness of the mother population.

The character values
z´
_i
also use indices of the mother population. The value of
z´
_i
is the average character value of the descendants of index
i
. The way it is obtained is by weighing the character value of each entity of the index
i
in the daughter population by the fraction of the total fitness of
i
that it represents. The change in character value for descendants of
i
is defined as
zi
=
z´
_i
–z
_i
.

Equation (2) holds with these definitions for
q´
_i
and
z´
_i
. With a few substitutions and rearrangements we derive:

which, using standard definitions from statistics for covariance (Cov) and expectation (E) gives the full Price equation:

The two terms on the right-hand side of the equation can be thought of as the selection and transmission terms, respectively. Covariance between fitness and character represents the change in the character due to differential reproductive success, whereas the expectation term is a fitness-weighted measure of the change in character valued between the mother and daughter populations. The full equation, therefore, describes both selective changes within a generation and the response to selection.

But the addition of the expectation terms also allows to expand the equation to show selection working at different levels. Here is how the equation can expand itself:

where E and Cov are taken over their subscripts where there is ambiguity, and
j·i
are subsets of the group
i
with members that have index
j
.

The recursive expansion of equation (3) shows that the transmission is itself an evolutionary event that can be partitioned into selection among subgroups and transmission of those subgroups. The expansion of the trailing expectation term can continue (say from gene, to individual, to group, to species) until no change occurs during transmission in the final level. Meanwhile, however, it will be possible to see how much of the change in trait is due to selection at each of the levels below.

Consider the reduced form of the Price equation

w
z = Cov (
w, z
) =
_wz
V
_z

where w is fitness and z is a quantitative character. The equation shows that the change in the average value of a character,
z, depends on the covariance between the character and its fitness or, equivalently, the regression coefficient of fitness on the character multiplied by the variance of the character.

Since fitness itself is a quantitative character, z can be equal to fitness,
w. Then the regression
_wz
equals 1 and the variance, V
_w
is the variance in fitness. So the equation shows that the change in mean fitness,
w, is proportional to the variance in fitness, V
_w
. This is what most people took Fisher’s fundamental theorem to mean: The change in mean fitness of a population depends on the variance in fitness.

Price, however, showed that Fisher didn’t mean this in a general but rather in a very specific way. Fisher had defined mean fitness in a way different than usually construed. For him it related only to that portion of fitness dependent on the additive genetic variance. All other components relevant to phenotypic variance—including epistasis and dominance—were categorized as “environment” and left out of the equation. Since it was known that epistatic and dominance effects can reduce fitness, along with the obvious fact that the environment can degrade, George interpreted Fisher’s fundamental theorem as exactly true in its own terms, but not as biologically significant as Fisher had made it out to be.