jimmcginn{at}yahoo.com (Jim McGinn) wrote in message
news:...

> I made an offer of 10 thousand dollars to anybody 
> that can demonstrate the validity of Hamilton's 
> nonsense.  Joe started to approach the problem but 
> then bowed out when I insisted that he provide 
> justification for the assumptions that he tried to 
> slip in the back door of his argument.  
>


Change in mean trait value of the population due to selection equals
Cov[w/wbar,z] (Price 1970). Cov[w/wbar,z] =
Regression[w,z]*Variance[z]/wbar, where z is trait value, w is
fitness, wbar is mean fitness. Provided there is variation in z, then
Sign[Change in mean trait value due to selection] =
Sign[Regression[w,z]]. Therefore, mean trait value is increased by
selection when:

 Regression[w,z]>0
 
i.e. 

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

where z' is partner trait value. Since the three components above are
the proper definitions of -c, b and r in Hamilton's rule, then we have

 rb>c

Jim, email me, and I'll let you know where to send the money, okay?
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