[Bug-gnubg] Recube efficiency in short bearoffs

[Øystein O Johansen]

Hi,

I believe the recube efficiency for short bearoffs (and maybe longer
bearoffs as well) can be improved. Currently the recube efficiency for all
positions evaluated by BEAROFF2 is set to 0.6. I believe this is way to
much.

Take this position as an example:

 GNU Backgammon  Position ID: sgAAQBwAAAAAAA
                 Match ID   : UQkAAAAAAAAA
 +13-14-15-16-17-18------19-20-21-22-23-24-+     O: gnubg
 |                  |   |    O  O     O    | OOO 0 points
 |                  |   |       O          | OO
 |                  |   |                  | OO
 |                  |   |                  | OO
 |                  |   |                  | OO
v|                  |BAR|                  |
 |                  |   |                  | XX
 |                  |   |                  | XX
 |                  |   |    X             | XX
 |                  |   |    X             | XX  On roll
 |                  |   |    X        X    | XXX 0 points
 +12-11-10--9--8--7-------6--5--4--3--2--1-+     X: user (Cube: 2)

An evaluation of this position says X wins 61.9% of all games. With a
recube efficiency of 0.6 the cubeful equities will be, according to
Janowskis formulaes:

E (no redouble) = Cv * ( p * (W+L+0.5x) - 1 )       (eq. 5)

E (redouble) = Cv * ( p * (W+L+0.5x) - 1 - 0.5x )   (eq. 6)

W = 1 and L = 1.

E (no redouble) = Cv * (0.619 * (2+(0.5*0.6) - 1) = 0.4237 * Cv
E (redouble)    = Cv * ((0.619 * 2.3) - 1 - 0.3 ) = 0.1237 * Cv

This is just the same as the eval text says:
No redouble         : +0.424
Redouble, pass      : +1.000   (+0.576)
Redouble, take      : +0.249   (-0.176)

Now I make a rollout of this position. 12960 games, cubeful, no trunc.

Cube analysis
Rollout cubeless equity  +0.239

1. Double, take         +0.392
2. Double, pass         +1.000  ( +0.608)
3. No double            +0.389  ( -0.004)
Proper cube action: Redouble, take
Rollout details:
Player user owns 2-cube:
  0.619 0.000 0.000 - 0.381 0.000 0.000 CL  +0.239 CF  +0.389
 [0.000 0.000 0.000 - 0.000 0.000 0.000 CL   0.000 CF  +0.002]
Player gnubg owns 4-cube:
  0.619 0.000 0.000 - 0.381 0.000 0.000 CL  +0.477 CF  +0.392
 [0.000 0.000 0.000 - 0.000 0.000 0.000 CL   0.000 CF  +0.003]
Full cubeful rollout with var.redn., with rotate rolls
12960 games, seed 1150602368 using Mersenne Twister
Play and cube: 0-ply cubeful [expert]

Because of the the bad estimation of the recube efficiency, the evaluation
that says no-double/take by a wide margin, is "proven" wrong by a rollout.
(At least it can't be a discussion after the rollout, the confidence is
about 80%)

Assuming this position is exactly at the borderline of a redouble or a
nodouble, and the cubeless winning probability is exactly 0.619, I can
solve what the recube efficiency should have been by setting the cubeful
equities equal to each other. I get 0.3447 which is far from the value of
0.6 in the EvalEfficiency function.
But then again it will be wrong to use this value as well since the recube
efficiency will not be the same for the two players as the race becomes
shorter.

I therefore believe we need some kind of function estimate the recube
efficeincy based on something. I also belive this something can be the sum
of the average number of rolls left. Finding a function like this, which
works fine, will require some work and research.

Of course this function may be different. If may be two functions. One for
each player. It may also be a function of several variables, like mean
number of rolls left for each player, and variance for each player. If we
can't find such function either, we may be able keep cubefuls equities in a
two sided bearoff database.

Joseph -- did you make some work on estimating a recube efficiency based on
the length of a race?

-Øystein

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