[EMAIL PROTECTED] wrote:
Yes. Don's scalability argument states that ELO gain is proportional
to time doubling.
For me scalable use of time implies that time translates into depth.
The extra depth is:
m - m0 = log(2)/log(b).
So if the ELO gain for time doubling in Chess equals 100 over a wide
time scale and if Go has a 10 times larger branching factor than
Chess, then the ELO gain for time doubling in Go would equal 100/log
(10) = 43 (everything else assumed equal).
I'm not sure i agree with Don, but i just want so say that if he is
right, than mathematically he is also right with a larger branching
factor.
Yes, this seems obvious and to me it appears you are begging the
question - presupposing the conclusion. You said it yourself in the last
sentence: _if_ he is right then mathematically it follows for the larger
branching factor. Can't argue with that.
I was trying to compare a different relationship related to the
branching factor and other characteristics of Go to capacity of human
logical reasoning and thinking. The idea being to suggest a possible
explanation for why Go may be qualitatively different than Chess in this
regard.
So I'll attempt to put the relationship I was trying to describe with
words into a mathematical model and then further describe my thought
process.
Let b = branching factor
Let f = Effective avg. pruning factor(0-1), thus b*f is an effective avg
branching factor
Let t = length of thinking time
Let p = maximum ply or depth under consideration
Let n = avg. number of positions a player can effectively evaluate in
one unit of time (either explicitly or otherwise using whatever
reading/learning/patterns/etc. to his avail)
Both f and n can be considered idealized measures of skill and ability
of the player.
Let r = rough approximation (as this is a simplification/idealization)
of the ratio of coverage of the game tree to depth p and defined as:
r(b,f,t,p,n) = n*t/(b*f)^p for all n*t<=(b*f)^p, otherwise r=1.0
Obviously if you double the time and keep the depth constant the ratio
of coverage goes up in a linear relationship for all b. But as time is
increased, p is increasing presumably. Now the graph of r is not linear
and higher b results in a faster rate of decline. Now I understand that
this doesn't necessarily have anything to do with strength ratings.
So that is some background for the concept. Bear with me if this
borders on the obvious for a while. So we all know that Go evaluation
is very hard (for computers, but also for humans). You can't prune if
you can't evaluate in some sense however (not with certainty anyway).
You can't evaluate without understanding shapes/life and/or reading.
In chess these things are arguably quite a bit simpler. So with chess
with a much smaller starting branching factor and simpler more
left-brain devices for pruning and evaluating the cost/benefit of
looking deeper tends to have reasonable payback at relatively large
depths.
Contrast with Go, starting with a much higher branching factor and
lacking left brain (logical/reasoning) methods for pruning and
evaluating, depth tends to create more confusion and quickly exceeds the
brain's ability to keep track of exploding variations. However, as you
learn from experience you can recognize patterns for the different
concepts and balance with analysis to effectively prune and evaluate
position potential and group interaction and then you can go deeper with
some confidence level in your understanding of the status of the game.
Learning these skills while thinking about a particular game's next move
is not generally practical and even if possible would presumably require
enormous extra time. Yet without this ability you are left with a
massively rapid expanding game tree to search. Finally this is why I
think it may be the case that doubling human thinking time for Go might
not produce linear improvements.
Back to the model, we could add another variable perhaps:
Let c = reliability/certainty factor for the pruning and evaluations
done during the search. r*c might have some meaning...
And again I am not saying this is black and white. Chess and Go share
these same characteristics to different degrees. I believe Chess is
more logical/analytical and Go is more balanced analytical -
intuitive/holistic (yin-yang thing), thus each yields to both approaches
in different situations and ways.
But just because a rule of thumb holds for Chess doesn't mean it does
for Go. Of course I could be wrong, but I was just trying to introduce
reasonable doubt, since Don always seems so sure of himself ;-)
Matt
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