Jim's Hamilton's rule prize

From: Name And Address Supplied (name_and_address_supplied_at_hotmail.com)
Date: 12/20/04 

Date: Mon, 20 Dec 2004 21:38:59 +0000 (UTC)

> It just has to he reasonable. It has to make sense.
>
> One approach you might take is that my objections to
> it are themselves unreasonable.
>
> Jim

Okay, then here is my attempt.

The change in the mean trait value due to selection is given by
Cov[w/wbar,z] where z is trait value, w is fitness, and wbar is mean
fitness (Price 1970). The trait is favoured when Cov[w/wbar,z]>0, i.e.
when Regression[w,z]>0, assuming variance in fitness and variance in
trait value are nonzero. Let Z be the trait value of one's social
environment, say, the average over all social partners. Then the
condition is:

 PartialRegression[w,z|Z]+PartialRegression[w,Z|z]*Regression[Z,z]>0

which is Hamilton's rule

 -c + b * r > 0

or

 r b > c.

This is exact, and completely general. Do I win?