in article [email hidden], Perplexed in Peoria at
[email hidden] wrote on 6/8/04 8:20 AM:
Quoted message said:"Guy Hoelzer" <[email hidden]> wrote in message
"]news:[email hidden]...
Quoted message said:in article [email hidden], Name And
Address Supplied at [email hidden]
wrote on 6/5/04 10:43 PM:
Quoted message said:"Malcolm" <[email hidden]> wrote in
message news:<[email hidden]>...
<snip>
> 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".
This statement makes so many mean field approximations, I am
not sure that it retains any meaningful relationship to
reality. For example, selection biases delta-p (the
frequency of the altruism allele) to be negative when the
allele is present in only a single copy in the population,
even if rb>c. This example seems very straight forward to
me, but if you don't believe me then you can convince
yourself with any form of a numerical simulation. This
example provides proof that Hamilton's Rule does not hold
for all frequencies of the altruism allele. A more extensive
analysis will show you that Hamilton's Rule fails as a
quantitative predictor of allele frequency change, except
when the allele is in a low frequency range (but not too
low), simply because "r" becomes an increasingly inaccurate
indicator of the probability that another individual shares
the allele.
Quoted message said:The frequency of the allele in the population makes
absolutely no difference in whether it is advantageous.
It is never advantageous to have an altruism allele,
by definition. It is only advantageous to interact
with an altruist.
Quoted message said: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).
The "dilution factor" idea is an interesting one, but it too
makes restrictive assumptions about the social structure. If
I remember Hamilton (1964) correctly, I think he was just
doing a little thinking outside of the constraining box of
his toy model of kin selection. I don't think he was
attempting to systematically generalize his results.
Quoted message said: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.
I never claimed that a selfish allele could invade a
population of altruists under the structure of Hamilton's
model. You see the frequency dependence in the quantitative
errors the model makes in estimating the rate of allele
frequency change when the altruism allele becomes too
common. It is at the very low end of the allele frequency
range where selection acts in a qualitatively different
manner from that predicted by the rb>c rule.
Note, again, that it is fundamentally wrong to imagine that
there is ever an advantage to an individual's fitness
conferred directly by the altruism allele. This possibility
is assumed away by Hamilton's model.
Quoted message said: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.
OK. Now you have jumped to characterizing the allele as
specifically causing altruism towards siblings. That is a
new twist, but I will go with it for now.
Quoted message said: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.
You raise a number of factors here that can facilitate the
evolution of altruism, but none of them are part and parcel
of Hamilton's model, so you are constructing an "iron man"
in an attempt to save the "man made of straw." I would also
argue that your "iron man" is pretty rusty. For example,
your claim that the offspring of the first altruist will
enjoy the benefits of sibling altruism is hollow if the
first altruist has fewer than two fertile offspring as a
result of their altruistic behavior. Like so many others
before you, I think you are trying to defend Hamilton's
seductive kin selection model by imbuing it with poorly
defined attributes that were not actually part of the
original. Hamilton's model is a beautiful and heuristically
important toy that is not worthy of worship. Some of its
shortcomings are easy to see, so let's not pretend they
aren't there. We will serve Hamilton's legacy better by
standing on his shoulders and looking beyond the reach of
his toy than we would by letting the ghost of Hamilton sit
on us and limit our perceptions to the insides of the toy.
Cheers,
Guy
