idea to estimate the percentage of drawn positions in chess

[Dann Corbit]

Another way to approach the problem is with tablebase files
Examine the number of draws with two men as a percentage
Examine the number of draws with three men as a percentage
Examine the number of draws with four men as a percentage
Examine the number of draws with five men as a percentage
Examine the number of draws with six men as a percentage
Examine the number of draws with seven men as a percentage
Look to see if there is a recognizable sequence or pattern
If discovered, see if the pattern holds for 8 men, whenever that is completed.
If the pattern holds, compute the sequence for 9 to 32 men.
For each of the 31 sets, compute the totals.
Sum the results.

[Uri Blass]

Another way to approach the problem is with tablebase files
Examine the number of draws with two men as a percentage
Examine the number of draws with three men as a percentage
Examine the number of draws with four men as a percentage
Examine the number of draws with five men as a percentage
Examine the number of draws with six men as a percentage
Examine the number of draws with seven men as a percentage
Look to see if there is a recognizable sequence or pattern
If discovered, see if the pattern holds for 8 men, whenever that is completed.
If the pattern holds, compute the sequence for 9 to 32 men.
For each of the 31 sets, compute the totals.
Sum the results.

I do not expect the 9 men tablebase files in the near future so I do not think we can get results by this way.
Theb best way to get an estimate is to find the best engine for these random positions and to give it to play against itself and find the percentage of draws.

I believe in 99% of the cases results at 1 minute per move are going to be the same as the theoretical result and it may be interesting also to compare results at different time controls to see if the estimate is changed for some direction when we increase the time control.

[towforce]

Another way to approach the problem is with tablebase files
Examine the number of draws with two men as a percentage
Examine the number of draws with three men as a percentage
Examine the number of draws with four men as a percentage
Examine the number of draws with five men as a percentage
Examine the number of draws with six men as a percentage
Examine the number of draws with seven men as a percentage
Look to see if there is a recognizable sequence or pattern...

In this post, link, I used the method of differences (link) to see whether there might be a reasonable fit of a polynomial of degree n-1 (where n is the number of known values), but unfortunately, it doesn't look as though there is. You might be able to make a fit using other mathematical functions, but it's likely to be coincidence ("this combination of functions just happens to make a good fit to this data, but won't deliver new values accurately").

It might be worth trying a logarithmic fit or a reciprocal regression, but given that there isn't a good polynomial fit, my intuition tells me that other functions won't fit well either.

[chesskobra]

The problem (in crude approximation) is this: Place m pieces (some white and some black) on the board in all possible ways. In what fraction of the arrangements the material is equal. The draw rate will likely behave somewhat like that asymptotically (n x n board, m not too small), and the behaviour will not be polynomial. If you ask a similar question for coin toss: in n tosses of a fair coin, what is the probability of (heads = tails) (assume n even), then this will proportional to 1/sqrt(n). We may modify the question: how many sequences of black, white and blue, with black + white = m and black + white + blue = n, have the same number of blacks and whites. This can be answered easily, and the asymptotics in m will not tbe anything like a polynomial. One may also try to calculate 'expected imbalance of material' in a random position with m pieces on n x n board. This is a diffusion problem.