Let me guess: you teach philosophy!
Yeah, I couldn't recall the theorem, I had to look it up. I'm rusty but it was still pretty easy to follow. Price's theorem, and it proves exactly what you said analytically. My major was math, analysis in particular, with a penchant for statistics and probability. In the end, I just love the calculus. I'd love to teach it, but I can't afford the Phd. and it's a little late for me anyway.
There's also a neat thing on Science Channel, Dark Matters or some such. Talks about dark science --weapons engineers, mad scientists --and George Price, who wrote the theorem.
Nah, you need a Phd to teach, and then there are no jobs. Think about the last time you saw an add for "philosopher wanted'. A BA was enough academia for me to know I had no interest in going further. I do appreciate math though. Unfortunately Im mildly numerically dyslexic so it makes it tough to pursue physics etc. Your never too old ....I didn't go back to school until I was 40 and finished at 45.
Ive seen Price's theorem some time past. Interesting, but again, mathematically, a bit over my head.
Example: Evolution of altruism[edit]
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 wi by a model equation:
w_i = \frac{n'_i}{n_i} = k - a z_i + b z
where zi is a measure of altruism, the azi 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:
w\Delta z = -a\operatorname{var}\left(z_i\right)
where var(zi) is the variance of zi which is just the covariance of zi with itself:
\operatorname{var}(z_i) \;\stackrel{\mathrm{def}}{=}\; \operatorname{E}(z_i^2) - \operatorname{E}(z_i)^2
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. nij will specify the number of individuals of type ij. Let zij be the degree of altruism expressed by individual j of group i towards the members of group i. Let's specify the fitness wij by a model equation:
w_{ij} = \frac{n'_{ij}}{n_{ij}} = k - a z_{ij} + b z_i
The a zij term is the fitness the organism loses by being altruistic and is proportional to the degree of altruism zij that it expresses towards members of its own group. The b zi term is the fitness that the organism gains from the altruism of the members of its group, and is proportional to the average altruism zi 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 azij >bzi. Defining the group averages:
\begin{align} n_i &= \sum_j n_{ij} \\ z_i &= \frac{1}{n_i}\sum_j z_{ij}n_{ij} \\ w_i &= \frac{1}{n_i}\sum_j w_{ij}n_{ij} = k + (b - a)z_i \\ n_i'&= \sum_j n_{ij}' = n_i[k + (b - a)z_i] \\ z_i'&= \frac{1}{n_i'}\sum_j z_{ij}n_{ij}' \end{align}
and global averages:
\begin{align} n &= \sum_{ij} n_{ij} = \sum_i n_i \\ z &= \frac{1}{n}\sum_{ij} z_{ij}n_{ij} = \frac{1}{n}\sum_i z_in_i \\ w &= \frac{1}{n}\sum_{ij} w_{ij}n_{ij} = \frac{1}{n}\sum_i w_in_i \\ n'&= \sum_{ij} n_{ij}' = \sum_i n_i' \\ z'&= \frac{1}{n'}\sum_{ij} z_{ij}n_{ij}' = \frac{1}{n'}\sum_i z_i'n_i' \end{align}
It can be seen that since the zi and zi are now averages over a particular group, and since these groups are subject to selection, the value of Δzi = z′i−zi will not necessarily be zero, and the full Price equation will be needed.
\Delta z = \operatorname{cov}\left(\frac{w_i}{w}, z_i\right) + \operatorname{E}\left(w_i\,\Delta \frac{z_i}{w}\right)
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 in-group loss of altruists, assuming that the altruism is not uniform across the group. The first term is:
\operatorname{cov}\left(\frac{w_i}{w}, z_i\right) = \left(b - a\right)\operatorname{var}(z_i)
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 in-group losses, especially if the variance of the in-group altruism is low. In order for this effect to be significant, there must be a spread in the average altruism of the groups.
