Re: Perpetually Perplexed

From: Perplexed in Peoria (jimmenegay_at_sbcglobal.net)
Date: 01/22/05 

Date: Sat, 22 Jan 2005 13:01:41 -0500 (EST)

"Guy Hoelzer" <hoelzer@unr.edu> wrote in message news:csormc$fo8$1@darwin.ediacara.org...
> in article csieil$1fek$1@darwin.ediacara.org, Perplexed in Peoria at
> jimmenegay@sbcglobal.net wrote on 1/17/05 11:41 PM:
>
> > "William Morse" <wdmorse@twcny.rr.com> wrote in message
> > news:csfjsq$hu2$1@darwin.ediacara.org...
> > [Responding to Hoelzer]
> >> The argument is fine if your initial statement was that Hamilton's rule
> >> "works better with" perfectly graded altruism. However your statement was
> >> that Hamilton's rule "relies on" perfectly graded altruism. That clearly
> >> isn't true. Yes the threshold point goes down - which may well help
> >> explain why in practice there is little "pure" kin directed altruism in
> >> nature - but there will still be a threshold point, and I think that is
> >> what Jim is saying.
> >
> > Yes, it is basically what I was saying. If "perfect grading" is taken
> > to be that grading that maximizes summation(rb-c) over all social
> > interactions, then perfect grading is not an *assumption* of
> > Hamilton's rule. It could conceivably be a *consequence* of the
> > rule, if the rule is cojoined with a host of additional "adaptationist"
> > assumptions that tend to promote optimization in nature.
>
> There seems to be confusion over the point of Hamilton's rule. I suppose
> that this is not surprising give all the obfuscation of this issue on sbe
> over the last few months (and years).

No sh*t!

> Hamilton's Rule, as opposed to his
> model, is IMHO specifically about identifying a tipping point below which
> (inclusive) selection tends to decrease the frequency of the altruism allele
> and above which it tends to increase the frequency. From this point of
> view, I still argue that my point was correct. If you are assuming a
> different meaning of "Hamilton's Rule", or if you still think that my point
> was incorrect, I would like to see a justification for your view.

I agree on the meaning of the rule.

I still think that your point is incorrect. Justification below.

> To reiterate my contention, the tipping point at which kin selection would
> favor an increase in the frequency of an altruism allele, as a function of
> the ratio b/c, increases as the association between r and the extent of
> altruism expresses becomes less precise. Hamilton's Rule (tipping point:
> rb-c = 0) assumes that the extent of altruism is perfectly correlated with r
> on a per act basis.

You will have to explain what you mean by "tipping point ... as a function
of the ratio b/c". My understanding of a "tipping point" is this: As r
increases (or decreases), you eventually come to a tipping point. This
point is reached when r = c/b. Or, alternatively, you can increase (or
decrease) b/c to arrive at the tipping point. If you travel along this
axis, the tipping point is reached at b/c = 1/r. In talking about a tipping
point being given by some function of b/c, you must imagine that we are
traveling to the tipping point along some other axis. What axis?

I think that I understand that you want to talk about "grading" as a
smooth dependency of some aspect of the behavior on r. That is, we are
taking r to be the independent variable, and some aspect of the behavior
as dependent. OK. There are several metrics of the behavior that might
be "graded".

a. The frequency of the behavior might depend on r. That is, in the case
most favorable for altruism, the donor might exhibit the behavior more
frequently toward close relatives. Whether this is due to kin recognition
or viscosity makes little difference.
b. The ratio b/c might depend on r. That is, in the case most favorable
for altruism, the ratio is lower for more distant relatives. This might
happen in the kin recognition case if the donor refuses to help distant
relatives if the situation is one in which the costs are high and the
benefits are low, though the donor would help a close relative in the
same situation. (This is not necessarily the same thing as case a above.
The donor might compensate by increasing the frequency of less costly
forms of altruism toward distant relatives, thus leaving the overall
frequency independent of r.)
c. Both the frequency and the ratio b/c are independent of r, but the
donor grades the magnitude of b and c so that close relatives get large
b's at high cost, but distant relatives receive small b's at low cost.

Any of these forms of grading can result in a situation favorable to
altruism, even if the grading is imperfect. And Hamilton's rule,
properly understood, can be used to summarize the grading and answer
the question as to whether the grading does, in fact, favor altruism.

To see this, you must realize that the proper expression of Hamilton for
this kind of model involves indexing and summing over all instances of
the behavior. Let the index i range over all instances of altruistic
behavior caused by the donor's genome. That is, we are implicitly
saying that if the donor did not carry the "gene for" altruism, then
none of those behaviors would have happened. We don't include behaviors
that would have happened whether the donor carried the gene or not.

Then Hamilton's rule is:
   Summ (r[i] * b[i]) > Summ (c[i])
where the summations are over i.

The result you get from this calculation will certainly depend upon
any grading that exists. However, the calculation (i.e. the formula)
can handle any grading that happens to be present, perfect or imperfect.

The result of the calculation will give the correct answer as to
whether the behavior is favored by (inclusive) selection, again
regardless of whether the grading is perfect.

Now, you may be unhappy with my assumption that the index set i
is determined by a hypothetical - the issue of whether the behavior
would have happened if the genome had happened to be different.

OK. Lets try again with another set of assumptions. This time, the
index set ranges over ALL social interactions. Furthermore, we
will not assume that we are comparing an altruistic allele to a
selfish one. Instead we will compare two alleles that may both
be altruistic (or may not) but they result in behavior that is
"graded" differently. We are still comparing actual behavior to
hypothetical behavior (no way around THAT problem), but now the
difference is in the c's and b's. Or, rather, the difference IS
the c's and b's. Each c[i] and each b[i] is the result of a
subtraction of a hypothetical fitness (of the donor for c and of
the recipient for b) from the corresponding actual fitness.

Now, with these assumptions, Hamilton's rule takes the same form
as before:
   Summ (r[i] * b[i]) > Summ (c[i])

Again, the calculation can handle any grading, perfect or imperfect.
The important point is that each application of the rule involves
a comparison between two alleles, and hence between two situations -
only one of which could be the actual situation.

Now, if we make the standard adaptationist assumptions:
a. Plenty of time for selection to work.
b. Alleles will eventually be generated by mutation that will
   cause any particular grading that you might imagine, if
   compatible with the biophysical constraints.
c. The environment is sufficiently stable that past adaptation
   is indicative of current adaptation.
d. As a consequence of a, b, and c, evolution is complete and
   the current situation is finally optimal.
well, if you accept those assumptions, then "perfect" grading
might well be a consequence of Hamilton.

It occurs to me that one source of our disagreement might be
that you are approaching the rule from an ecologist's perspective
whereas I am approaching it from a pop gen perspective. I am
seeking to justify the rule, you are puzzling over how to apply
it.

If so, then you might be led to claim that perfect grading is
diagnostic - situations in which near-perfect grading exist
are ones in which Hamilton's rule applies, whereas situations
in which the grading is manifestly imperfect are probably
situations in which the rule is inapplicable.

How could the rule be inapplicable? Well, the rule assumes that
the altruistic behavior is unilateral - it doesn't apply to
Triver's reciprocal altruism. Furthermore, the rule cannot
usefully be applied if there are hidden direct benefits or
costs to the donor which we fail to include in our calculation
of c.

Is the test for perfect grading a good diagnostic for the
applicability of the rule. Well, it IS one possible diagnostic
which might be used, but I doubt that it is a particularly
good one.

> There were other, smaller, logical lapses in Dr. Hoelzer's remarks,
> > but they pale in significance compared to his confusion of assumptions
> > and consequences.
>
> This is quite a statement when I have yet to see a single instance of my
> "confusion of assumptions and consequences," let alone other "lapses of
> logic."

I hope that the above explains why I claimed a "confusion of assumptions
and consequences". The rule does not assume perfect grading, but
perfect grading may be a consequence of the rule, if further adaptationist
assumptions are made.

Regarding the "smaller lapses", I will document them, if you continue to
insist.

> Note that I tried to be explicit about assumptions, which are not
> lapses of logic if false. My logic depended explicitly on the validity of
> the assumptions.

Agreed. But that is only to say again that you were discussing
consequences, rather that discussing what assumptions are embedded
in Hamilton's logic.

Relevant Pages

  • Re: Perpetually Perplexed
    ... >> selection tends to decrease the frequency of the altruism allele ... I agree with your last sentence, but this is not the kind of grading that I ... > happen in the kin recognition case if the donor refuses to help distant ... then you might be led to claim that perfect grading is ...
    (sci.bio.evolution)
  • Re: Perpetually Perplexed
    ... disagree that Hamilton's rule relies on perfect grading. ... > likely that selection would favor the allele under Hamilton's model. ... the degree to which altruism can be adaptive. ...
    (sci.bio.evolution)
  • Re: Perpetually Perplexed
    ... > then I am arguing that the dispensing of altruism is assumed to be perfectly ... To the extent that the grading of altruism with kinship is not ... then too much personal fitness is lost through altruism toward more ... > be optimally altruistic toward closer relatives. ...
    (sci.bio.evolution)
  • Re: Perpetually Perplexed
    ... creating an altruism allele will tend to selectively increase in frequency, ... To the extent that the grading of altruism with kinship is not ... then too much personal fitness is lost through altruism toward more ... be optimally altruistic toward closer relatives. ...
    (sci.bio.evolution)