I've been following the discussion of Hamilton's Rule for several weeks now and have been struck by
the argument engendered by what seems to me to be a very simple model, albeit one based upon one of
those small but brilliant insights that occur in science. Its being a damp, drizzly November in my
soul (and a cold and rainy December in the real world), I could, like Cato, produce a philosophical
flourish and fall upon my sword, or, like Ishmael, take to the sea; for me, I'll plunge into this
sbe morass.
It seems best for me to outline what I understand to be Hamilton's Rule, in the hopes that this may
clarify points of the discussion. I add the caveat that it has been years since I read the original
papers by Hamilton, and that my knowledge of the population genetics and animal behavior literature
is current only as of ten (or more) years ago.
The initial problem arose because of observations of animal behavior which suggested that some
animals behaved altruistically in nature. By this it was meant that individuals behaved in ways that
appeared to decrease their fitness, either by putting their lives at risk, or by reducing the number
of offspring they could produce; at the same time, the result of the behavior appeared to benefit
other group members in the sense of increasing their fitness. Examples that were cited included
alarm calling, as seen in bird flocks or mammalian groups such as prairie dog colonies, and the
forgoing of reproduction in order to aid other group members' reproductive efforts, as in the social
hymenoptera. Whether these, and other, behaviors are indeed altruistic is a valid empirical
question, to which I'll return later, but whether or not individual examples turn out to fill the
altruistic bill has no bearing on the internal logic of Hamilton's model.
These examples presented a problem for population genetics. Imagine a population evolving along
traditional lines. Traits, including heritable behavioral patterns, increase or decrease in
frequency according to their effects on individuals' survival and reproduction. Now a behavior
emerges that we would recognize as altruistic: the individual exhibiting the behavior suffers a loss
in fitness; it could die in the act, or perhaps lose resources and produce fewer fertile offspring.
At the same time, the act increases the fitness of those in the group. How could this trait increase
and become established in the population? According to the traditional population genetic models, it
could not. If individuals with the trait are producing fewer offspring than those without the trait,
the trait would die out along with the individuals exhibiting it.
Hamilton's insight was that some of the individuals aided by the altruistic behavior may, in fact,
carry the trait themselves. Thus, the sacrifice by the donor (i.e., the one exhibiting the behavior)
may increase the fitness of recipients who also carry the trait. Might there be conditions under
which such a trait could actually increase in frequency?
As it turned out, the conditions are simple. If rb > c, the trait can increase. There are three
variables in this inequality. Let's start with b and c, which are measured in the same units, namely
fitness. b is the benefit, in fitness, to the recipient of the altruistic act, and c is the cost, in
fitness, to the altruistic actor. Now saying that b and c are measured in fitness and actually
measuring them are two different things, and a large thread could be started on the measurement of
fitness. The actual measurement does not affect the validity of the rule. We can say, without loss
of generality, that we could measure fitness by the number of fertile offspring produced. Thus, if I
dive into the water to save you, and do it but die in the attempt, we have clear costs and benefits,
but what are the fitness costs and benefits? Well, if I am older perhaps I could expect to produce
one more offspring (let's assume my previous, surviving offspring are off and on their own and no
longer need my help). You, on the other hand, are younger and can expect to produce two more
offspring before you die, if only you hadn't fallen into the river. Here the cost to me is one
fitness unit, and the benefit to you is two fitness units.
For the example above, in which we were dealing with two individuals, it is clear that b and c must
be measured in whole units (given our measure of fitness), but for a population as a whole, this
need not be the case. b and c can be measured in average gains and losses. Obviously if the cost is
0.25, an individual cannot produce the remaining 0.75 offspring, but if we look at the population
(which is, in fact, the unit in which we measure evolution and natural selection) we can say that
individuals who perform this behavior produce 75% of the offspring that they would produce if they
didn't perform it.
Now we come to r. For simplicity, Hamilton worked with, I believe, a single locus model; i.e., one
allele was responsible for the difference between altruistic and non-altruistic types. This is not
as far-fetched as it might appear. When an eagle flies overhead, some prairie dogs bark an alarm
call. We might think of this as a threshold in behavior. If the stimulus is of the right type and
intensity, the dog is likely to call. Could we not imagine that a small change in neurotransmitter
efficacy, or lowered thresholds of neuroreceptors might cause a dog to call when others would not?
In any event, this just-so story is irrelevant to the validity of the model. To continue, Hamilton
defined r as the genetic relatedness of the recipient to the donor; i.e.,
share half their genes, uncles/aunts with nieces/nephews share 0.25, cousins 0.125, etc. What r is
is the probability of the recipient of the altruism also carrying the allele in question. In
reality, altruistic behaviors are likely much more complicated than the simple single locus model.
This does not really affect Hamilton's inequality. If we think of r as the probability of the
recipient carrying the altruistic trait, the inequality holds whatever method of inheritance is
involved (though measuring it might be a practical problem).
In essence then, Hamilton's rule says that if the performance of the altruistic act leads to more
offspring that carry the altruistic trait than would be the case if the act was never performed, the
incidence of the trait will increase in the population. Simple. The rule uncovers the fact that
fitness, in the sense of an individual's ability to pass its traits on to the next generation, can
involve more than just one's direct offspring. This additional part of one's fitness needs to be
added to the traditional measure to give inclusive fitness; it was Hamilton's great insight that in
looking at the spread of traits in populations, the influence of a trait might spread beyond direct
descendants.
I would like to point out that Hamilton's rule is a (mathematical) model. It is not an hypothesis,
and is not something to be tested or refuted. Given certain assumptions and conditions (the main
condition being rb > c), the results are foreordained; it is a mathematical certainty. This is no
different from the Hardy-Weinberg equilibrium equations, or, in fact, Darwin's model of natural
selection. These models are mathematically true. What is open to testing is whether or not the
assumptions and conditions are ever met in nature, and how strong the effects will be when faced
with opposing forces. These are empirical questions, and must be tested on a case by case basis. The
hypothesis that would be tested is that an observed instance of altruistic behavior had evolved
because of its effects on the inclusive fitness of those with the trait. This could be refuted by
field observations and/or experiments. It might turn out that Hamilton's rule is unimportant in the
history of evolution because the conditions required are rarely found, but it would not, and can
not, refute the mathematical logic of the model.
In fact, many instances of altruistic behavior are arguable. Much work done on alarm calling has
shown that there is a very small, or no, cost involved in the behavior. In fact, in some species
it has been suggested that an alarm call, by causing panic in a group, can actually aid the
caller by confusing a predator and/or offering up many potential targets while the caller is the
only one that knows where the predator is. It is my recollection that prairie dogs do not alarm
call until they have run to their own borrow entrance, and are thus in relative safety. Of
course, because the cost (c) is so low in this case, Hamilton's inequality would require very
little benefit to the recipients to be valid. As I argued above, these questions must be
addressed on a case by case basis.
This has gone on much longer than planned, although I did mention its being a cold and rainy
December, and I have free time. I would like to make one more point. Much has been made in this
newsgroup about concerns that Hamilton's rule addresses relative fitness effects whereas the
population might be declining due to absolute fitness losses. I'm not quite clear on this argument;
it seems that the concern is that although the altruistic trait may be increasing relative to the
non-altruistic one, the whole population may be declining due to the costs to altruistically
behaving individuals. I would point out that when rb > c holds, it is a necessary consequence that b
Quoted message said:= c (remember that r must be <= 1); this means that more offspring are produced if the altruistic
behavior is performed than if it is not. Thus the population is in no danger from the altruistic
behavior, although it may indeed be declining due to other factors.
It is important to note that Hamilton's model addresses the initial spread of a rare, new trait. The
idea is that the altruistic behavior, if exceedingly costly, is rarely performed. How often am I
faced with the problem of jumping in a river to save someone? Not yet. Anyway, what this means is
that the propensity for altruism, once arisen, can be passively spread through a group; each time
conditions allow its expression, there is a disproportionate fitness gain to others likely to have
the same propensity. Thus over time the trait can increase its relative frequency.
There is much more that can be said about the ramifications of Hamilton's rule (especially regarding
the special case of haplodiploidy in the social hymenoptera), and other models of altruistic
behavior, but I fear I've long overwritten my welcome. I hope this post has helped some to clarify a
few of the points made in the unending arguments on this newsgroup concerning this model. I look
forward to being taken to task for all the errors I've most likely committed above. Have fun.
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