"John Edser"  wrote in message
news:...



Okay, so we got this far last time . . . 

> NAS:-
> Hamilton's rule is only meant to be applied to behaviours
> which we assume are possible.
> 
> JE:-
> Biology is suddenly included when it
> suits NAS argument but excluded when it
> doesn't.
> 
> NAS:-
> We can assume a strategy
> set, and work out which of the possible strategies is the
> best for the gene to take.  You can't argue that the 
> winning strategy might not be possible if you have already
> opted to include it in the strategy set!  
> 
> JE:-
> Hamilton rule does not set limits
> to the number of recipients helped.

Not explicitly, no; but then Hamilton's rule is only valid insofar as
we have correctly defined r, b and c.  As I pointed out previously, we
can consider any amount x of altruism, but introduce accelerating
costs or decelerating benefits.  Say that providing total help x gives
an accelerating cost C[x] such that C > 0, C' = dC/dx > 0, but gives a
linear benefit B[x] such that B > 0, B' = 0.  Then the optimum level
of altruism satisfies

 r B[x] = C[x]

Hamilton's rule here will often predict an intermediate value of x is
the ESS; it will not suggest that organisms should be altruistic to an
infinite number of relatives.
   
> Of course it _is_ limited, in _reality_,
> The whole idea is to introduce _absolute_
> limits into the algebra of the _rule_
> and not just outside of it, as you are doing.

I think you'll find that c[x] and b[x], and r[x], are all *inside*
Hamilton's rule.

> Setting limits outside the rule keeps
> the rule arbitrary. The limits must
> be present within the rule as a term.
> 
> You just snipped as: "lots of incomprehensible 
> nonsense" the problem of how help is apportioned 
> when resources are absolutely limited and recipients 
> are only chosen at random. Either you are just 
> assuming resources are infinite so almost anything
> now becomes possible within Hamilton's rule, or you
> simply can't/won't (or both), see that apportioning 
> largess _randomly_ presents a real problem for the 
> rule. If resources are limited so that an infinite
> number of recipients cannot be helped and 
> only a random number of recipients are provided 
> with only random amounts of help, either too many
> or too few recipients will receive too much
> or too little help, since help cannot be apportioned
> strictly relative to relatedness. This means a net 
> _loss_ in fitness must result compared to using  exactly
> the same limited resources to just reproduce yourself
> (kin select yourself). A random system of _resource_
> allocation simply does not work within Hamilton's rule
> except when you kin select yourself. This problem
> has never been addressed.
> 

Yes it has; but it is an extreme case. It is not generally true that b
= c, although it can be.

> > JE:-
> > 2) The Search For A Missing Constant Within
> > Hamilton's Rule.
>  
> > the terms have no consistent relationship 
> > to each other so their relationships can vary 
> > every time the inequality is evaluated. This is 
> > the case for Hamilton's rule since r,b and c are 
> > all just variables. Thus Hamilton's rule, as it
> > stands, is just arbitrary. 
> 
> NAS:-
> r, b and c are not variables. 
> 
> JE:-
> Then they have to be constants, which
> very clearly, they are NOT. A term 
> either must remain fixed or it can 
> vary, it can't do both.

As i said before, they are functions, containing parameters and the
variable or variables under optimization.  They are in themselves
variable, but in a limited, well-defined way, strictly in accordance
with variation in the maximand; Hamilton's rule is not arbitrary.

> 
> NAS:-
> They are functions, containing
> parameters and the variable or variables 
> under optimization.  
> 
> JE:-
> This makes no difference. Either each 
> term AS IT IS PRESENTED WITHIN THE
> RULE remains the same number (a constant) 
> or it does not (a variable), over every
> use of the rule.

Well, of course b c and r are not necessarily constants over all
biological scenarios, but they can be considered to be over some
well-defined sets of situations, so that the rule can be used
comparatively.

> 
> Not a single constant exists within
> Hamilton's rule so the rule is
> arbitrary.
> 
> >snip irrelevant mathematics to the point above<
>  
> > JE:-
> > This was because all these numbers = 1, i.e. in simple,
> > additive gene terms 2 brothers or 8 cousins have 
> > a probability of 1 for having a genomic set of genes 
> > replicated from yourself. 
> 
> NAS:-
> No! there is some non zero probability that 
> they will not carry any of your genes.
> 
> JE:-
> There is "some non zero probability that 
> they will not carry any of your genes". 
> Haldane just decided to delete
> it for the use of his comment, which
> he is entirely free to do since all probabilities 
> are just possibilities and not certainties. 
> To what extent they need to approximate 
> was up to Haldane. For Haldane, his
> use of probability was sufficient to say
> what he said, for the reason's I stated.

No, Haldane was talking about the expectation, not the probabilities. 
On average, 2 full sibs or 4 cousins shared the same number of copies
of his genes as he did himself.  The probability that they shared the
same number is less than one though, and not trivially so.  There is a
significant probability that they will have a lower number, or even
no, copies of his genes.  And there is a significant probability that
they will, together, carry more copies of his genes than he does.  The
expectation, he says, is that the number will the same.  This was my
point: it is the expectation, not the probability, that matters;
although expectation is a sum of outcomes weighted by their
probabilities.

> 
> NAS:-
> I suppose that 4 brothers and 16 cousins have probability 2 of
> having a genomic set of genes replicated from yourself?!?!
> 
> JE:-
> It is utterly pointless allowing a 
> probability > 1. 

My point exactly, but it followed from your logic.

> However a probability 
> of 1 is not a certainty but can usefully 
> be suggested to approximate one, which
> all Haldane did re: his comment.
> 
> > JE:-
> > A probability of 1 means
> > this genomic set of genes is equivalent to your
> > own set but only as far as just the simple _addition_ 
> > of genes are concerned. Given this over simplification,
> > Haldane's comment neglected the biological fact that if 
> > he remained alive long enough to reproduce himself say, 
> > 3 times he would be fitter than any of his other options 
> > since rb=0.5*3 = 1.5 representing the gene fitness of 
> > the 3 organisms normally reproduced.
> 

But this three offspring he might have produced is taken into account
by the c component of Hamilton's rule.  Haldane was making the
assumption that he and his brothers/cousins all had the same
reproductive value.  This is why he mentioned brothers and cousins,
rather than relatives in older or younger generations; in the latter
case there is clearly going to be a difference in reproductive values.

> NAS:-
> This nonsense you are spouting stems from your gross misunderstanding
> of
> what b is actually representative of in Hamilton's rule, and perhaps
> also
> an unfamiliarity with the concept of 'reproductive value'.
> 
> JE:-
> Hamilton's rule is not
> about genes being helped 
> "in vitro"...
>

You are a master of the non sequitur, John.
 
> One fine day you may realise that
> you have to put more biology back
> into Hamilton's rule and not just
> take more and more biology out of it. 

Perhaps I've already realised this.  Which is why i acknowledge that r
is more than ibd (something which you have failed to understand).  And
it is why i acknowledge that b and c are not necessarily constants,
that the biological details influence all components of Hamilton's
rule.  This does not make the rule intractible though; it is still a
highly powerful tool when you know how to use it.

> The recipients and donors are both 
> organisms and all rb help must be measured 
> in extra organisms replicated by each
> organism recipient. This causes all gene 
> freq. changes. Gene freq. changes are 
> just a result from, and not the cause of, 
> these actions. You have testable cause 
> and affect within Hamilton's rule
> _entirely_ reversed. It was reversed
> to make the mathematics easier. 

In a sense there is a reversal, but this is true in much of Darwinian
evolutionary biology.  We want to understand why a trait we observe is
 as it is.  We could assume that it is adaptive, and hence increases
in frequency when it is perturbed from a frequency of 1.  We use
Hamilton's rule to assess when kin selection will cause a trait to do
this.

> Since
> you have been so badly taught, nobody
> said to you that you must reverse the 
> modelling causation you were assuming, 
> in order to do any valid _biology_.
> I suggest you demand an explanation
> from your teachers as to why they
> omitted such a fundamental.
>

Sadly, Hamilton isn't alive today to answer my questions.  But a close
reading of his work - something which you obviously have not taken the
time to do - is often a very good substitute, as he had a knack of
seeing potential stumbling points his readers might struggle over or
misunderstand.
 
> > JE:-
> > I am sure Haldane would not choose to die for 2 
> > brothers (a), or 8 cousins (b), compared to reproducing 
> > himself 3 times in the normal way before he dies.
> 
> 
> NAS:-
> Assume all reproductive values are equal (they need not be, but this
> is an illustration, so i don't want to make things too complicated for
> you). If Haldane doesn't do the reproductive act, he will have three extra
> offspring.  If he does do it, he won't have these offspring.  Then c =
> 3. Because we assume equal reproductive values, then each of the
> recipientsis expected to have three extra offspring if they are saved by an
> altruistic Haldane.
>   
> JE:-
> If you assume reproductive values are equal
> then you are assuming a free lunch exists
> within Hamilton's rule. If Haldane only
> supplies 100% of his resources to helping
> himself reproduce and nobody else then
> these resources only allow a total of 3 
> organisms to be reproduced (yes, all the 
> genes helped are actually in organisms and
> organisms are very *EXPENSIVE*). The total 
> fitness of Haldane is 3 organisms (not 3 very
> cheap genes "in vitro"), no more and 
> no less. However, as if by magic, when _exactly_ 
> the same resources that allowed Haldane to only
> reproduce himself 3 times are now apportioned 
> between 2 brothers or 8 cousins, these resources 
> suddenly allow EACH of to increase their organism 
> reproductive efforts by 3 _each_. In the case of 
> brothers the resources donated have suddenly doubled 
> and in the case of the cousins, increased by magic
> eight times! This logic is termed "creative 
> accounting". Enron accountants tried to get 
> away with it but failed but Stalin perfected it
> and got away with it for his entire lifetime.
> It is as old as the pyramid selling con trick 
> which is against the law in Australia. I suggest 
> you refer back to your teachers for clarification 
> re: classic model _misuse_. 
>

Calm down John.  By assuming values of b and c you are assuming that a
trait exists which can take c from the bearer and turn it into b units
of fitness (in total) for the recipient(s). We've been through that
already.  Sometimes, yes, there will be a limit on how much bigger b
can be than c, but it is not true that b always be equal to c. 
Consider the costs and benefits of sentry duty in, say, meerkats.  Non
zero sum games do exist in nature, and there is a huge huge literature
on them.
 
> >snip creative accounting mathematics<
>  
> >JE:-
> >Biology 
> > does not limit Haldane to just helping other's
> > to reproduce his genes unless he stipulates he
> > always was, and will always remain, sterile.
> 
> NAS:-
> Hence the 'c' component.
> 
> JE:-
> The donor must reproduce
> a minimum of one organism 
> when c does not = 0.
>

If the donor only has a very slim chance of reproduction in the
remainder of its lifetime, then c can be close to zero, but not zero,
and certainly not greater than or equal to one.  If it can give up
this chance by ploughing what resources it has into helping a relative
produce more offspring, then it is adaptive to do so.
   
> >  JE:-
> > Hamilton argued that in _every_ case the kin fitness 
> > (rb) is greater than its cost (c) in organism 
> > fitness, then the freq. of the altruistic gene 
> > must increase. However, this is not the case. If 
> > the missing total fitness of the donor 
> > is now included within Hamilton's rule as
> > the maximum K, for his altruistic gene 
> > to spread, the kin fitness of the donor must 
> > be greater than the donor's total fitness minus 
> > the organism cost of kin fitness:
> > 
> > 		rb > K-c	
> > and:
> > 
> > 		rb+c > K
> 
> NAS:-
> This is ridiculous.  
> 
> JE:-
> Good to see healthy scepticism
> and an unbiased appreciation of
> different views.
>

Don't cut me off mid sentence to make it look like I hadn't
substantiated the comment with reason.
 
> NAS:-
> Baseline fitness is a constant and hence
> vanishes from Hamilton's rule, which is a 
> measure of marginal fitness.
> 
> JE:-
> I refer you to your inflationary
> accounting. Unless you invoke creative 
> accounting no resources exist to pay 
> for any sustainable altruistic gene 
> increases unless you assume resources
> are infinite. 

You assume a false dichotomy.  It does not follow that in order to
assume b can be greater than c that you must assume infinite
resources.

And if by creating accounting you mean a nonzero sum game, then i
think we are pretty much in agreement, excepting of course that i
acknowledge that such scenarios are abundant in nature, whereas you
seem to think that they cannot exist.

>Total fitness must be 
> included as a term within Hamilton's 
> rule and not just outside of it or a free 
> lunch has to be provided to increase 
> the freq. of the altruistic gene.
>

This free lunch being b - c?  If this is really all that your argument
boils down to, John, then i suggest that you look into the extensive
evolutionary and economic literature on nonzero sum games. I've given
you two examples already in this thread, and there are many many more
to be found out there.
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