"Perplexed in Peoria"  wrote in
news:cp5748$2jln$1{at}darwin.ediacara.org: 

> 
> "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.

(Josh - sorry for not abbreviating the post, but I wanted to leave in the 
original spark for my suggestion) 

[moderator's note: Cool dat. - JAH]

What I was trying to get at was how to make a reasonable approximation of 
current fitness given that the future is uncertain - which was in 
response to Catherine Woodgold's interesting idea to try to use multiple-
universe models to calculate a fitness value. 

Future uncertainty, in my limited understanding of economics, is what the 
discount factor is for. Even with zero inflation, a dollar tomorrow is 
not worth a dollar today. If it were, banks would charge you a fixed fee 
rather than compound interest for a loan. Markov chains alone do not 
provide for this - they only provide a prediction for expected future 
variation - and it is the unexpected future variation (an asteroid 
impact, the evolution of flight) that cheapens the current value of 
future fitness. 

Catherine Woodgold chose an arbitrary cut-off point of 5 generations, and 
this may in fact be a valid way of "discounting" future fitness - the 
additional fitness after 5 generations may in general be minimal. I think 
this ignores the very high fitness associated with certain phenotypes, 
e.g. flying ability or eusocialism, which is why I suggested a discount 
rate. Now I was not thinking of fitness as a growth rate, and perhaps I 
should be, but I still don't see how that resolves the problem of 
incorporating future uncertainty into a current measure. As Guy noted in 
another follow, fitness is not really a measurable quantity so much as a 
concept useful in modeling evolution. I believe it is grounded in reality 
but, as I noted above, can only really be measured in retrospect. I also 
believe that defining fitness only over the short term is not 
particularly helpful in understanding evolution, so it is worthwhile to 
try to develop a definition that takes into account expected future 
variation.

Yours,

Bill Morse
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