Re: [Bug-gnubg] MET and extrapolation bug ?

[Joern Thyssen]

On Sun, Dec 15, 2002 at 02:08:05PM +0100, Øystein Johansen wrote
> GNU Backgammon  Position ID: Zg74CRCzmcGAJA
>                     Match ID   : cAlgAnAAGAAA
>     +24-23-22-21-20-19------18-17-16-15-14-13-+  O: GNU Backgammon
>     | X  O  X  O  X  O |   |                X |  7 points
>     |    O     O     O |   |                X |
>     |                O |   |                  |
>     |                  |   |                  |
>     |                  |   |                  |
>     |                  |BAR|                  |v 19 point match (Cube: 1)
>     |                  |   |                6 |
>     |                  |   |                O |
>     |                  |   |                O |
>     | X     X  X     X |   |    X           O |  On roll
>     | X  O  X  X     X |   |    X     O     O |  3 points
>     +-1--2--3--4--5--6-------7--8--9-10-11-12-+  X: Oystein
> 
> 
> Hi,
> 
> The above position was posted to GammOnLine some days ago. As Albert replies 
> in one of his posts:
> 
> If a MET up to 15 is used, and GNU will use extrapolation from further 
> scores, 
> the 2-ply evaluation says Double/Take. If I use Jacobs table, which goes to 
> 25-away, GNU evaluates the position as a clear pass.
> 
> Please look at the numbers as the difference is so high that I suspect there 
> is a bug somewhere. ( Maybe a bug in the extrapolation algorithm? )

I think the "bug" is in the match equity table! 

For example, Woolsey's table has 2 digits of accuracy, the typical
relative accuray is round 0.5%. Calculating the various points (double
point, cash point, too good point etc) requires the subtraction of two
values. The error on the subtraction is 1%.  For the above match score
the differences are in the order of 4% (risk for taking cube) and 17%
(risk + gain) (gammonless!). The two values are divided with eachother.
The result

take point = (4% +/- 1%) / (17% +/- 1%) = 24% +/- 7%

The relative error on the take point is 30%...

Having just one more digit gives 24% +/- 1%, and another digit 24% +/-
0.1%.

In order to avoid propagation of rounding errors it's vital to have one
or more digits. The book "Gammons and cubes near the end of th ematch"
by Ortega and Kleinmann start by adding one arbitrary digit to Woolsey's
MET for that particular reason. 

THe position still evaluates to a take with the Ortega/Kleinman table.
However, it's closer to a pass, than using Woolsey's. The equity for
double, take is around 0.98. The relative error in the calculation of
the various points is roughly 4%, so the resulting equity is (roughly)
0.98 +/- 0.04.

In conclusion, I don't think there is a bug in the MET extrapolation,
but the example demonstrates that the calculation of take points etc
(due to the "expensive" subtraction) requires good values in the MET.

Jørn