1. Introduction
Hamilton's Rule was used to
support organism fitness altruism 
(OFA) in nature after classical
group selection failed to be able 
to do so over 50 years ago.
It was argued the rule can measure
when an altruistic gene can spread 
within one population. 
Gene centric Neo Darwinistic
viewpoints such as Dawkins'
selfish geneism and Wilson's
sociobiology remain dependent
on Hamilton's logic.   


2. Discussion
Using Hamilton's Rule
when:

	rb>c ... (i)

where:

r = a defined measure of relatedness
b = a defined measure of resources 
c = the cost of b in fitness units

it remains argued that an altruistic gene 
can spread within one population.
 
The values r,b and c only constitute
biological variables. Because no constant 
term exists within Hamilton's rule
no frame of reference exists within 
it. This being the case, although 
the rule can validly measure when 
one gene relatively spreads, 
(one gene increases relative to
another gene in the same population) it 
cannot measure if that gene has absolutely 
increased or decreased. Thus a gene may 
relatively increase at a cost so great that 
its representation within one population 
absolutely decreases. If repeated, such
a gene is only selecting for its own extinction.
In such a case many other genes besides the
inclusive fitness gene are also being carried 
into extinction. Conversely, a gene that just 
relatively decreases could absolutely increase 
because the cost c was smaller than the actor's 
absolute fitness gain producing a net increase 
i.e. not a decrease for the actor. 

It appears that Hamilton's rule cannot
answer a most pressing biological question: 
what is the logic for an altruistic 
in fitness gene to spread? The answer appears 
to be that it can only be measured to spread 
using Hamilton's logic when the paid altruistic 
cost c constitutes a mutualistic investment and 
not a donation. If this is the case then the rule 
has been misused to support OFA within nature 
ever since it replaced classical group selection.

Definitions.
A donation: When the cost c 
returns less than c to the donor.

An Investment: When the cost c  
returns more than c to the donor.

In order to be able to determine a true
altruistic phenotype the proposed altruist 
should not make an absolute fitness gain
constituting an investment.
The total fitness of the proposed 
altruist should be able to be measured to 
_decrease_ and not _increase_ via the rule.
Clearly this is not possible within the
rule as it stands. If the total fitness of 
the proposed altruist increases and does
not decrease even though a cost c was 
paid for the transfer of b resources such
an event can only constitute an investment 
by the supposed altruist and not a donation. 
Without the total organism fitness of the 
proposed altruist, the rule cannot 
distinguish between c as a valid donation
or c an investment when such a difference
is required for any rule that is used to
differentiate between OFA
and OFM (organism fitness mutualism).

Only one case exists within Hamilton's
rule where the total fitness of the
actor has been included within it:

	rb > c(max)

This is the case whereby all the resources
that are normally employed to reproduce
and raise to fertile adulthood all
of the actors own infertile offspring 
becomes transferred as b resources. 

Allowing the total fitness of the actor
to be K, then c(max) = K. In this single
case the actor must appear sterile-like, 
i.e. be capable of reproduction but 
fail to normally reproduce. 

Only when K is paid as the maximal 
possible cost can altruism be 
absolutely proven within the rule:

	rb > K

A relative fitness is just a comparison
of minimally, two fitness totals. In
the rule these two totals are rb and
c which are compared by simple
subtraction. However, in Hamilton's
rule they are not compared directly
they are only compared indirectly via 
a missing  baseline fitness m which 
has been added to both sides of 
the rule so that it becomes mathematically 
deleted:

	rb + m > c + m   ..(i)

reducing to:

	    rb > c       ..(ii)

Although such a baseline fitness
m remains mathematically 
deleted from the rule, it is always
critical for any valid _biological_ 
application of that rule. 
This is because the cost c cannot 
be larger than the total fitness 
of the actor K, setting a limit to 
the rule where OFA can only be 
absolutely proven when K = the 
maximum cost c. Only in this
instance does no basline fitness
exist within Hamilton's rule, i.e.
m = 0. Thus only in this one instance
has all the fitness of the actor
become entirely visible within the 
rule. 

The deleted baseline fitness m 
must always be equal or smaller 
than K otherwise fitness is being 
added to the rule from nowhere, i.e. 
such extra fitness is just an illegal 
addition. Also, unless c + m is the 
same as K (the maximum fitness of the actor), 
fitness has been unaccountably deleted.

	K = c + m       .. (iii)

Since the actor only pays 
the maximum possible cost 
when  c = K and only this 
one case proves OFA via 
the rule then:

       rb > K

alone, proves OFA using the
rule. Substituting for K:

	 rb > c + m


	 rb-c > m	    .. (iv)

The value rb-c is
inclusive fitness 
where Hamilton's rule
suggests:

	rb-c > 0

Thus the rule remains in 
error by the amount m when
all the actor's fitness is 
explicitly included within the 
rule allowing it to measure
when an altruistic gene can
absolutey spread within one 
population.

For any altruistic gene to be 
_proven_ to spread, inclusive fitness 
must be larger than the missing 
baseline fitness m. Since this 
baseline fitness remains deleted 
it is not possible to know when 
this is, or is not the case within
the rule. Because inclusive fitness
(rb-c) is only a relative fitness
it cannot be maximised, i.e.
inclusive fitness cannot constitute
a valid fitness maximand that can
be compared to K the Darwinian
fitness maximand to determine what 
a organism may be selected to
do as measured via Hamilton's rule.

3. Conclusion:
Hamilton's rule remains incomplete.
Until m is given a real value
and is explicitly included on 
just one side of the rule it 
is not possible for the rule to
measure when an altruistic in
fitness gene can be proven
to spread, i.e. spread absolutely.


Regards,

John Edser
Independent Researcher

PO Box 288
Church Pt
NSW 2105
Australia

edser{at}tpg.com.au
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