"William Morse"  wrote in message
news:cp0pte$1909$1{at}darwin.ediacara.org...
> an588{at}freenet.carleton.ca (Catherine Woodgold) wrote in
> news:coebn4$1m2i$1{at}darwin.ediacara.org:
>
> > phillip smith (deletethis-phills{at}ihug.co.nz) writes:
> >> I should have said I am working on a replacement for fitness( see
> >> reply to JE). I think fitness has fatal errors. We keep using it
> >> because we have had no choice. Not that I am sure I an do better.
> >
> > Here is a definition of the fitness of a given gene in context.
> > For this definition to make sense, I assume a multiple-universe
> > model:  that due to quantum-mechanical fluctuations, a given universe
> > at a given point in time develops into multiple future versions
> > (possbilities), over which a meaningful probability metric
> > can be defined.
> >
> > Consider a gene A in a particular individual X in a particular
> > environment.  Consider also all the copies of gene A, for
> > example in siblings, cousins, and other members of the population.
> >
> > Consider the universe U containing individual X.  Consider also
> > the fictional universe U-prime which is exactly like U except that
> > individual X does not exist.
> >
> > Go forward in time 5 generations, and count up
> > the average number of copies of gene A in the population, averaged
> > over the alternate universes with a probabilistic weighting.
> >
> > The fitness of gene A in individual X is the average number
> > of copies of A in the futures of universe U, minus the average
> > number of copies of A in the futures of universe U-prime.
> > Descendents of cousins etc. are also included in the count.
>
> That's a great definition - up to 5 generations. What happens if all the
> descendants at 6 generations die without issue? Is the fitness 0, or do
> we have an arbitrary cutooff at 4 generations?
>
> So what we would like to have is a definition that, like yours, takes
> into account alternate future environments, but that can be extended
> farther into the future. I have a rather fuzzy mathematical intuition
> that if you somehow combine Markov chains with a discount factor for
> expected future gains you might come up with a reasonable approximation
> of a fitness function - recognising that fitness is always going to be
> ultimately only definable in retrospect.

I think that you are on the wrong track with your talk of a "discount
factor".  It seems to me that what we have are a sequence of fitness
definitions.  There is first the traditional simple one-generation
definition - count direct offspring (with a factor of 1/2 if the
organism is sexual).  Then there is the two-generation definition -
count grand children, then take the square root to get average growth
per generation.  Take the cube root for the three-generation definition. 
And so on.  Now, once you have this infinite sequence of
fitness
numbers, you may wish to take some kind of weighted average.  Perhaps
that is what you mean by a discount factor - you intend to give the
greatest weight to the short term fitness numbers.  But, depending on
your application, there may be some reason to use some other "envelope"
rather than an exponentially declining one.

In economics, a "discount factor" is used to come up with a "current
value" of some infinite stream.  But if you recall that any fitness
must be some kind of growth rate per unit time, you will realize that
the growth *rate* is automatically finite, even if the growth is
unbounded.
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