idea to estimate the percentage of drawn positions in chess

[towforce]

Another approach:

* calculate proportion of draws in the 4, 5, 6, and 7 man tablebases

* from this result find an expression that allows you to calculate the proportion of draws for n pieces on the board

* use a database of chess positions to work out the proportion of positions with each number of pieces

* hence calculate the number of drawn positions

This is probably the easiest way to do the job - but, with a sample size of only 4, the second step is a bit dodgy.

[jefk]

yet another approach: random movers.
'engine' play with simply random moves

But if both sides play random then you also don't get wins quickly, of course.
As subset you can do games with one decent engine (2800 or so) vs random
play and then the decent engine win 99.999x pct of the time.

Nevertheless the nr of possible positions of the random vs random games
is much higher, and that's which the theoretical draw rate will be lower.

Which confirms our knowledge that draw rate is increasing with rating. And
at sufficient rating (and sufficient time usage and a decent opening book,
not containing any mistakes) all games will be drawn. All ? yep all.

[Ajedrecista]

Hello:

yet another approach: random movers.
'engine' play with simply random moves

[...]

The approach of random games was tested before:

One billion random games (2011).
One billion random games (2013).
Random game mating probabilities (2015).

Regards from Spain.

Ajedrecista.

[chesskobra]

Uri asked for an estimate for the percentage of drawn positions in chess. We have to distinguish between random games and random positions. WDL statistics for random games may not be even remotely similar to the theoretical WDL for random positions. For example, take a tree in which root (a) has two children b and c, and c has two children d and e. Now the probability of a random game ending on b is 1/2. Probability of each position (sampled directly from {a,b,c,d,e}) is 1/5. Now assign values W to b and D to each of d and e. Now the probability that a random game is a win for one side is 1/2. Probability that a random position is winning is 2/5.

In fact we have to also clearly define what is a random game. If we have a tree with n end vertices, then there are precisely n possible games, and an engine playing random moves does not sample uniformly from the n games. In fact an engine choosing a move uniformly randomly from the available moves on each turn will produce scholar's mate with very high probability compared to any other game.

[jefk]

WDL statistics for random games may not be even remotely similar to the theoretical WDL for random positions

Correct, while you easily can get a lost position if you play random moves, the
other side will not utilize this if it's also playing random moves; thus the high
draw rate (insufficient material) is much higher than when taking all possible positions
(which should be a possible result of a game, of course)
So while i mentioned random games, i should have mentioned also one engine 2800
and the other other one random, then of course the random mover always is losing.

Thus i maintain that in the widest tree, the nr of drawn positions is relatively small;
not that this matters much, because in the narrow tree of the current top engines i maintain
that the draw percentage is 100 pct. Some may tend to think about whatever Alfasuper
zero with a rating of 5000, or a Superbetazero or Superfish 23.6 or so rated 6000, the
reality is that ratings are converging, and somewhere (for slow games) the max is around
3900 or so (or lower) with all draws. Too bad for Ajedrista, while for me this is certain;
in the academic community they probably will also acknowledge within ten yrs
or so that chess is ultraweakly solved and within 15-20 yrs possibly with a weak
solution (will take some computer time, i admit and i would first do draughts).

[fkarger]

It is always difficult to predict the future but lets make some estimation.

The number of legal chess positions is around 10^40 to 10^50.
The number of legal games is around 10^120.

Lets attack the smaller number: 10^40.
To confidently solve this you basically need a 32 men table base (otherwise: how to prove?).

Currently there are around 10^22 bytes of permanent storage available on the whole planet.
This will increase but for example HDD storage did not grow exponentially for some time.

So I guess it could be difficult to solve chess.

[chesskobra]

So I guess it could be difficult to solve chess.

Yes, but how is this relevant to estimating the percentage of drawn positions? I am saying this only because some people here would jump on to this discussion and rehash their usual BS arguments.

[fkarger]

You are right. It is not very relevant for the original question.

Only in the sense that you would also need that 32 men table base in order to definitely
know the percentage of drawn positions in chess.

[chesskobra]

But estimating the percentage should be possible.

[fkarger]

Yes, maybe the following procedure could help:

1) Generate a lot of legal random positions (e.g. by generating random games)
2) Estimate for any of the generated positions if it is a draw by using Stockfish at depth d
3) Increase d
4) Redo 2) and 3) until no resources left

After that you could probably observe that the estimations of Stockfish change
less and less and you could extrapolate the truth (for d = Infinity).