Nick Apperson wrote:
He is saying this (I think):
to read m moves deep with a branching factor of b you need to look at p
positions, where p is given by the following formula:
p = b^m (actually slightly different, but this formula is close enough)
which is:
log(p) = m log(b)
m = log(p) / log(b)
We assume that a doubling in time should double the number of positions
we can look at, so:
m(with doubled time) = log(2p) / log(b)
m(with doubled time) = log(2) * log(p) / log(b)
Your math is wrong (I think).
The correct equivalency for the last line would be:
m(with doubled time) = (log(2) + log(p)) / log(b)
So, as we can see we get a linear relationship no matter what b, the
branching factor, is. However, the slop on the line changes
accordingly.
But anyway that is not the exact relationship of interest here. There
is another key variable, that being the number of positions the player
can effectively evaluate in one period of time which is a subset of m,
not all of m and is presumed to be roughly fixed at some maximum value
relative to the strength of the player. Doubling this number and
comparing it to the next ply number of possibilities is the correct
relationship and this curve is much different depending on the branching
factor and not a linear relationship.
We are also assuming that the number of moves deep that
one reads is proportional to playing strength. This is in fact simply a
way of defining playing strength. There are many scales that could be
constructed where this is not the case, however, this is a good one to
pick from a theory perspective because the formula holds with any game
that has a branching factor. What you are talking about in terms of
using "sense" is another factor is strength that isn't really addressed
by the above math (in my opinion). There are certainly ways in which
one could say that by having a better sense of the game one can
effectively reduce the branching factor, but one could also say that
sense could be capable of restructuring the way people read a situation
so that it is possible to read much deeper than is allowed by simple
brute force search strategy. That is an issue for psychologists to
address. I am looking forward to good data from experiments on this
subject to answer some of these questions. I think until we have good
data for go, we won't really know. Just my opinion (guess it is the
scientist in me...)
Interesting thoughts.
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