Price Equation - Proof of The Price Equation

Proof of The Price Equation

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:
\operatorname{E}(x_i)\ \stackrel{\mathrm{def}}{=}\ \frac{\sum_i x_i n_i}{\sum_i n_i}
  • The covariance between the and values is:
\operatorname{cov}(x_i,y_i) \ \stackrel{\mathrm{def}}{=}\ \frac{\sum_i n_i~~}{\sum_i n_i} = \operatorname{E}(x_iy_i)-\operatorname{E}(x_i)\operatorname{E}(y_i)

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:

n = \sum_i n_i\,

The average value of the characteristic is defined as:

z \ \stackrel{\mathrm{def}}{=}\ \operatorname{E}(z_i) = \frac{\sum_i z_i n_i}{n}

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:

n' = \sum_i n'_i\,

The fitness of group i will be defined to be the ratio of children to parents:

w_i = \frac{n_i'}{n_i}

with average fitness of the population being

w \ \stackrel{\mathrm{def}}{=}\ \operatorname{E}(w_i) = \frac{\sum_i w_i n_i}{n} = \frac{\sum_i \frac{n_i'}{n_i} n_i}{n} = \frac{\sum_i n_i'}{n} = \frac{n'}{n}

The average value of the child characteristic will be z' where:

z' = \frac{\sum_i z'_i n_i'}{n'}

where z′i are the (possibly new) values of the characteristic in the child population. Equation (2) shows that:

\operatorname{cov}(w_i,z_i)=\operatorname{E}(w_iz_i)-wz

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

\operatorname{E}(w_i\,\Delta z_i)=\operatorname{E}(w_iz'_i)-\operatorname{E}(w_iz_i)

Combining Equations (7) and (8) leads to

\operatorname{cov}(w_i,z_i)+\operatorname{E}(w_i\,\Delta z_i) = \bigl(\operatorname{E}(w_iz_i)-wz\bigr) + \bigl(\operatorname{E}(w_iz'_i)-\operatorname{E}(w_iz_i)\bigr) = \operatorname{E}(w_iz'_i)-wz

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:

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