Re: Hamilton's Rule In The Mirror Evaded


<name_and_address_supplied@xxxxxxxxxxx> wrote
>
> The general aim of John Edser's contributions to this forum is not to
> kindly help the interested lay-person to understand evolutionary
> theory, but rather to try to convince the non-expert that neodarwinism
> is fundamentally flawed and that it should be scrapped. The criticisms
> that he levels display an astounding ignorance of the development of
> the field over the last 40 years.
>
One of the things I have complained about is John Edser's tendency to
mention experts who contribute to the group by name, and demand attention
from them. In my opinion this is contributing to an unpleasant atmosphere.
However I can't have it both ways, and then accuse him of targetting
innocent non-experts.
Secondly, there is often a case for taking things from first principles.
Ultimately it doesn't matter to anyone except the historian whether Hamilton
put his rule in the mirror or not. What is important is whether we can
reverse the sign of the variables, and what that means.
>
> So when someone else weighs in, in support of Edser, suggesting that he
> has hit on some fundamental problem that requires more attention, it is
> relevant for me and others to point out that the 'problem' has infact
> been resolved in Hamilton's original work.
>
It's a valid criticism of me that maybe I should have looked up Hamilton's
papers instead of relying on the textbooks I have lying around the house,
which are not very many in number.

In philosophy one must read a text "sympathetically".Lawyers and politicians
look for the weak points in an opposing argument and say why they are weak.
Serious thinkers look for the strong points and say why they are strong.
>
> Hamilton's rule summarises the effects of the action on all individuals
> affected. Thus, "rb-c>0" is not Hamilton's rule, for the competitive
> scenario you have outlined. In reality we would have something like:
>
> r1 b - r2 d - c > 0
>
> where r1 is the relatedness to the direct recipient of the mutualistic
> behaviour, r2 is the relatedness to whoever has to pay for the
> increased competition, and d is the effect of increased competition, in
> terms of loss of fitness to competitors. If r2 d is sufficiently large,
> then Hamilton's rule does not necessarily predict the mutualism is
> favoured.
>
> So, again, it is not Hamilton's rule that is flawed here, but John's
> understanding of it.
>
Washburn strikes.

Hamilton's rule is rb - c > 0, as long as r is defined in such a way that
average relatedness is 0. In practise the population size N is usually large
enough that this using IBD is close enough to the truth. The beauty of it is
that we don't need to measure the r2 d term. The whole reason b is a benefit
is because it confers a competitive advantage in a world of limited
resources. (d is in fact a vector which sums to minus b, r2 is another
vector giving the relatedness of everyone disadvantaged, c is the special
case of self where r2[i] is unity).

Why did I say Washburn strikes? Because your rule works if we use the
definition of r proposed by Washburn.

Now let's reverse the sign of c and set r to zero.

In mathematical terms, we still have the same situation. Average relatedness
is zero, so the r2 d term can be ignored. However we've ignored the biology.
There are a few individuals for whom r is high, and a huge number for whom r
is tiny. Opportunities for altruistic acts where b exceeds c by a factor of,
say, sixteen are likely to be very rare. Mutualistic acts where the only
constraint is that b be positive and c be negative are likely to be much
more common.

This seems to be why Hamilton fixes up the rule

"The mean relatedness to the entire species population other than self
is -1/(N-1) where N is the population."

This has the unfortunate effect that I am more related to an octopus (r = 0)
than to an Eskimo ( r = -1/6 billion). But it again eliminates the r2 d
term, assuming that the population competes uniformly for the same
resources.

When I posted the original reply to John I wasn't aware that Hamilton had
solved the problem by allowing r to have a small negative value.

So it's been a worthwhile discussion for me. I hope that other people have
found something of value. Discussions are much more productive when everyone
makes an effort to be nice.




.

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