RE: [Bug-gnubg] New formula for estimating bearoff GWC

[Joachim Matussek]

>> Hi Joachim,

>> I'm finding your article very stimulating. Thanks for putting in the hard 
>> work. I'm still thinking  about it, but there  are a few things I'd like to 
>> discuss already.

Hello Ian,

thank you for your answer. A nice catalogue of questions. This is the style of 
constructive discussion I was looking for.

A short preamble to the history of my work. I decided to improve my own play 
and wanted to start at the end of a backgammon game. Bearoff cube seem to be a 
technical aspect of backgammon and I didn´t find any easy to use formula for 
calculating winning chances. There are several approaches but most of them only 
tell if it is a double/take/pass in moneygame. Walter Trice recommends the Ward 
method over Thorp. Additionally he gives information about effective pipcount 
and some examples how to calculate it exactly. But he doesn´t provide a formula 
for general positions. 

I just had found a tool to calculate bilinear fits which I had looked for 
because I am doing a FIBS rating study similar to Kees van den Doel´s. Then I 
realized it can also handle n-fold linear fits and I had the idea of applying 
it to bearoff positions. I wasn´t sure at all if it would give any useful 
results because bearoffs are only semi-linear. All that happened last thursday. 
I wrote some scripts and got some fast results that looked promising. I thought 
it was worth publishing and better have discussions about it than working on my 
own for a long time.

>> 1 EPC Count

>> 1.1 Why does the formula stop at 8 chequers? Is it because that's how far 
>> you've tested, or does it break down later?

My starting point were 8 checkers. I thought I would calculate positions with 
more checkers by using the Ward method. Of course i tried it with more checkers 
(up to 13). This results in different coefficients. I was pretty sure that the 
accuracy for middle range bearoff positions would suffer. So i stopped thinking 
about it. Another reason is that I am working with MS Excel and it is a hard 
job to handle 3002 positions within one table. 

>> 1.2 Your wastage per pip is linear. Douglas Zare uses a non-linear 
>> adjustment for chequers on the ace-point 0, 1, 2,  2, 2 ..., and something 
>> similar for the 2 point. Have you tried any  such method?

My wastage per pip is linear except of my correction terms. I know that Trice´s 
and Zare´s formulas for (near-to-)n-roll positions are pretty accurate so i 
just excluded these positions of my work. It was hard to find a correction term 
that would be easy to remember. It seems that these kind of position are those 
with the highest deviation from my estimate.

>> 1.3 You make no allowances for gaps. This seems a likely source of 
>> inaccuracy. Perhaps Chuck Bower's of a  "useless gap" might be useful.

I found that e.g. 000033 has an error of 1.02 pips. This is clearly the result 
of the gap on the 4 point.

>> 1.4 You penalise stacks equally, irrespective of their position. This seems 
>> unnatural. With only 8 chequers and a stack, you are likely to have gaps 
>> elsewhere, so perhaps this ties in with the previous point.

Indeed I penalise stacks equally. But it seems that stacks on lower points 
(1,2,3) are penalised enough by their coefficient. Another reason is that my 
correction terms were found by trial and error. I am sure that there are 
possible improvements and I would feel happy if someone finds some that are 
easy to use.

>> 1.5 Have you any idea which type of positions give the largest error? . Do 
>> they err in a particular direction?  Nearly 10% are off by a pip or more, 
>> and I worry that they may be the less trivial positions.

The highest deviations occur for near-to-n-roll position which I excluded. The 
average error seems to have no direction because of the nature of my method. 
Linear fits give an average solution.

>> 2 EPC to GWC

>> 2.1 Is this part of your article only concerned with 8 chequer positions? 
>> I'm assuming not.

No, this part is not limited to 8 checkers. I used n-roll positions up to 15 
checkers as well as pip positions.

>> 2.2 Your table of %/pip is a great idea. I was planning on constructing such 
>> a table myself (I don't have the tools to extract from the database so I was 
>> planning on just using some hand-selected examples).

>> 2.3 How many samples were taken for each epc?

I have taken only one example for each epc. The distance between these was 
chosen at about 3 pips. For roll positions it is obvious that is almost 
impossible to construct many more of them. Even the positions between the 
integer roll positions have a little quality of pip positions.

>> 2.4 As I understand it, you have taken examples with a 1 epc difference 
>> between the players. I believe that the  %/pip is not linear as the pip 
>> difference changes. I believe the %/pip differs between an even race and a 
>> close take/pass. (I don't know the direction; this is something I'm planning 
>> to look at.) I also believe the %/pip tails off >> as you move further into 
>> pass territory.

No, perhaps my explanation wasn´t clear enough. I evaluated all positions 
against all positions subdivided into 4 classes (pip-pip, pip-roll, roll-pip, 
roll-roll). This gave a broad range of results. You are true... the %/pip ratio 
gets smaller when entering the pass territory deeper. Thus I only used the 
values between 20% and 80% and averaged them. They seem to be pretty constant. 
Most of the times I excluded the %/pip near 50% GWC because they tend to 
fluctuate because of the division by near to 0.

>> I would be extremely interested in a table with samples at a 4 pip deficit 
>> (i.e. even race), even pips (some   advantage to roller), borderline 
>> take/pass, and larger pass (say 5 pips).

>> 2.5 The 0.5% and 1% adjustments for pip-roll and roll-roll positions are 
>> very exciting because they are so simple to remember and use. How confident 
>> are you with these.

Pip positions lead to broad distribution of rolls needed to bearoff. Roll 
positions are peeked distributions. Because of the appearance of my chart I am 
pretty confident about its accuracy. Of course you have to do some 
interpolation between the adjustments depending on the position type.

>> Regards
>> Ian Shaw

I would be interested in any improvements to my method. If anybody is 
interested in my scripts and my Excel sheets I will share them via mail. Any 
co-work is appreciated.

Regards,

Joachim Matussek


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