"Guy Hoelzer"  wrote in message
news:ca2shs$v4a$1{at}darwin.ediacara.org...
> in article c9uauo$2mc4$1{at}darwin.ediacara.org, Name And Address Supplied at
> name_and_address_supplied{at}hotmail.com wrote on 6/5/04 10:43 PM:
>
> > "Malcolm"  wrote in message
> > news:...
> >
> > 
> >
> >> The technical term used is "identical by descent",
which maybe doesn't
help
> >> much.
> >>
> >> Imagine we have a new mutation which has been going for only five or
six
> >> generations and is thus still very rare. The chance of this new
mutation
> >> being in a relative is obviously given by the 1/2, 1/8 metric and not
the
> >> 99% one.
> >>
> >> The point is that every new allele starts off as just such a rare
mutation,
> >> so we use the more restricted definition of
"related" when calculating
> >> whether altruism is adaptive.
> >
> > I don't see your point. The implication seems to be that the
> > relatedness appropriate to Hamilton's rule will increase as the allele
> > becomes more frequent. That's clearly not the case. Hamilton's rule
> > makes no assumption about allele frequencies, except for pq>0.
>
> This is not correct.  Hamilton's Rule makes plenty of cryptic assumptions,
> which is a primary source of confusion.  Let me give examples from both
ends
> of the frequency spectrum showing why Hamilton's Rule assumes a limited
> window of frequency for the altruism mutation.  The Rule assumes that both
> the cost on benefit of the altruistic act affect the frequency of the
> altruism allele in a deterministic and invariant way.  When the mutation
is
> present in only one copy, only the cost of the altruistic behavior affects
> the fate of the allele; so Hamilton's Rule is an invalid and
> overly-optimistic model in this case.  When the altruism allele is very
> common, then the coefficient of relatedness (r) is a very poor predictor
of
> the presence of the altruism allele in a behavioral partner; thus the
> benefit of altruism does not affect the frequency of the altruism allele
> with probability "r" under these conditions either, as assumed by
Hamilton's
> Rule.  For any given social structure and phenotypic expression of an
> "altruism mutation", there would be an optimum frequency of
the altruism
> allele corresponding to the maximum effect of kin selection; but I have
> never seen this calculated for a given situation.  It would also be
helpful
> to see an analysis of the rate at which the effectiveness of kin selection
> diminishes as you move away from this optimum.

You are wrong about Hamilton's rule being frequency sensitive in its
applicability.  The rule applies equally well at all frequencies.
If rb > c, then it is advantageous to have the "altruistic allele".
The frequency of the allele in the population makes absolutely no
difference in whether it is advantageous.  Where the frequency DOES
make a difference is in just how big of an advantage it is.  Hamilton [1964]
talks about a "dilution factor" in discussing this.  That is,
"r" is
still the correct factor in the rule, but there is a positive frequency
dependent dilution factor "d" such that the selection coefficient will
be proportional to d(rb-c).

At the high end of the frequency range, you can see your error by
asking whether the selfish allele can invade a population of altruists.
Notice that this simply multiplies "b" and "c" by -1. 
The rule still
applies and predicts that the selfish allele will be disadvantageous.

At the low end, your error is more subtle.  There is still an advantage
to carrying the only (dominant) gene for sibling altruism in the
population.  This advantage is realized in your own fitness, assuming
that you compute fitness in the way that Edser and Hamilton [1964]
recommend - that is by counting the number of offspring that survive
to maturity.  Your offspring are more likely to survive, because they
will (often) have altruistic siblings.  And even if you define fitness
as birth-to-birth, altruism genes are no more problematic than genes
for parental care.  For cross-generation effects, you really need to
have a way of defining overall fitness that includes the expected
personal fitnesses of your offspring.
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