Snowie match equity chart
Harald Retter
rate of 26%. I couldn't find those equities summarized in
a diagram, so I spent some minutes to create one. As I am
not willed to memorize more than that, I did it up to -9.
Here is the result:
Your opponent needs
You
need| 1* 1 2 3 4 5 6 7 8 9
----+--------------------------------------------------
1* | 50.0 -- 51.5 68.5 69.5 81.3 81.9 88.8 89.1 93.3
1 | -- 50.0 68.8 75.1 82.0 84.3 89.1 90.8 93.6 95.0
2 | 48.5 31.5 50.0 60.4 68.5 75.3 81.0 85.1 88.5 90.9
3 | 31.5 24.9 39.6 50.0 60.4 65.3 71.7 76.8 81.2 84.6
4 | 30.5 18.0 31.5 39.6 50.0 57.9 64.6 70.3 75.2 79.4
5 | 18.7 15.7 24.7 34.7 42.1 50.0 56.8 62.8 68.1 72.9
6 | 18.1 10.9 19.0 28.3 35.4 43.2 50.0 56.3 61.9 67.1
7 | 11.2 09.2 14.9 23.2 29.7 37.2 43.7 50.0 55.6 61.0
8 | 10.9 06.4 11.5 18.8 24.8 31.9 38.1 44.4 50.0 55.5
9 | 06.7 05.0 09.1 15.4 20.6 27.1 32.9 39.0 44.5 50.0
* = Post Crawford
-- = Retter-Paradox ;-)
While typing all those numbers, I was quite surprised how
similar they seemed to be to the Woolsey-table, I used to
work with (and actually still do). If my memory is right,
Kits tables where computed assuming a gammon rate of 20%.
Quite a difference, but the tables do not seem to reflect
on it much. So I created a second diagramm to display the
differences between the two versions. I took the numbers,
which are printed in "Cubes and Gammons..." by Ortega and
Kleinman and subtracted them from Snowie-values:
YON
YN | 1* 1 2 3 4 5 6 7 8 9
----+--------------------------------------------------
1*| # -- # # # # # # # #
1 | -- 0 -1.5 +0.1 -1.1 -0.9 -1.1 -0.5 -0.6 -0.1
2 | # +1.5 0 +0.6 +0.7 +0.5 +0.2 +0.3 +0.5 +0.1
3 | # -0.1 -0.6 0 +1.5 -0.5 +0.2 +0.4 +0.7 +0.2
4 | # +1.1 -0.7 -1.5 0 -0.1 +0.2 0 -0.2 0
5 | # +0.9 -0.5 +0.5 +0.1 0 -0.5 -0.5 -0.3 -0.4
6 | # +1.1 -0.2 -0.2 -0.2 +0.5 0 +0.1 -0.4 -0.2
7 | # +0.5 -0.3 -0.4 0 +0.5 -0.1 0 -0.6 -0.3
8 | # +0.6 -0.5 -0.7 +0.2 +0.3 +0.4 +0.6 0 +0.2
9 | # +0.1 -0.1 -0.2 0 +0.4 +0.2 +0.3 -0.2 0
# = only Snowie-data available
Oh, how nice, only four matchscores (-1-2/-1-4/-1-6/-3-4)
differ more than 1%point, the largest is 1.5%points. Once
these figures should get "accepted", there is not as much
to relearn as I feared. Most discrepancies appear at -1-T
(T = points to go for Trailer). As expected the increased
gammon rate favours the Trailer at these scores, with him
only having usage of those extra-points.
I am confused by the result for -4-3. A doubled gammon at
this score should be a big shot for the trailer, winning
the match exactly outright. His matchequity should raise
assuming a larger gammon rate. Food for thought, who has
an idea?
Regards, Harald Retter
stig...@yahoo.com
"Harald Retter" <harald...@okay.net> wrote:
> I am confused by the result for -4-3. A doubled gammon at
> this score should be a big shot for the trailer, winning
> the match exactly outright. His matchequity should raise
> assuming a larger gammon rate. Food for thought, who has
> an idea?
>
> Regards, Harald Retter
>
>
Hi Harald (and the world)
Funny, I'd just made a table with Snowie before I read your post.
I only bothered to do it up to 5 though. Thanks for the table.
But at -4,-3 I got 42.2%. And at -1,-3 I got 68.5%. It looks
like you've put the wrong number at -4,-3 because its the same as for
-3,-2.
And I don't think the equitytable used by Snowie is calculated
with a gammonrate of 26%. I think they've actually rolled out
the initial position _with_ the cube.
Woolseys table says 41% at the score.
Stig Eide
-----== Posted via Deja News, The Leader in Internet Discussion ==-----
http://www.dejanews.com/rg_mkgrp.xp Create Your Own Free Member Forum
Harald Retter
I hope it is accurate now. BTW the score -4-3 doesn't con-
fuse me anymore. ;-)
Your opponent needs
You
need| 1* 1 2 3 4 5 6 7 8 9
----+--------------------------------------------------
1* | 50.0 -- 51.5 68.5 69.5 81.3 81.9 88.8 89.1 93.3
1 | -- 50.0 68.5 75.1 82.0 84.3 89.1 90.8 93.6 95.0
2 | 48.5 31.5 50.0 60.4 68.5 75.3 81.0 85.1 88.5 90.9
3 | 31.5 24.9 39.6 50.0 57.8 65.3 71.7 76.8 81.2 84.6
4 | 30.5 18.0 31.5 42.2 50.0 57.9 64.6 70.3 75.2 79.4
5 | 18.7 15.7 24.7 34.7 42.1 50.0 56.8 62.8 68.1 72.9
6 | 18.1 10.9 19.0 28.3 35.4 43.2 50.0 56.3 61.9 67.1
7 | 11.2 09.2 14.9 23.2 29.7 37.2 43.7 50.0 55.6 61.0
8 | 10.9 06.4 11.5 18.8 24.8 31.9 38.1 44.4 50.0 55.5
9 | 06.7 05.0 09.1 15.4 20.6 27.1 32.9 39.0 44.5 50.0
stig...@yahoo.com schrieb in Nachricht <6ri0us$3ns$1...@nnrp1.dejanews.com>...
Chuck Bower
(snip)
>While typing all those numbers, I was quite surprised how
>similar they seemed to be to the Woolsey-table, I used to
>work with (and actually still do). If my memory is right,
>Kits tables where computed assuming a gammon rate of 20%.
(snip)
Well, one (or BOTH!) of our memories is in error. Here
is what mine tells me: Kit analyzed matches from Hal Heinrich's
extensive database and built a table. (Note: some may have
noticed that I always refer to this table as "the Woolsey-Heinrich
Match Equity Table". Now you know why!)
After constructing the table empirically Kit 'smoothed' the numbers.
I once read (or was told, but sorry, I don't remember by whom)
that the Woolsey-Heinrich MET is consistent with a theoretical
table with 23% gammons. Of course, maybe what Harald is concluding
is that there is little difference between MET's when the gammon
fraction differs by a few percent. So tables generated theoretically
from 20%, 23%, or 26% gammon fractions look about the same. Is
that true??
Chuck
bo...@bigbang.astro.indiana.edu
c_ray on FIBS
Harald Retter
Chuck Bower schrieb in Nachricht <6rk0g3$kqf$1...@flotsam.uits.indiana.edu>...
>In article <6rhjv8$lfs$1...@trader.ipf.de>,
>Harald Retter <harald...@okay.net> wrote:
>
> (snip)
>>While typing all those numbers, I was quite surprised how
>>similar they seemed to be to the Woolsey-table, I used to
>>work with (and actually still do). If my memory is right,
>>Kits tables where computed assuming a gammon rate of 20%.
> (snip)
>
> Well, one (or BOTH!) of our memories is in error. Here
>is what mine tells me: Kit analyzed matches from Hal Heinrich's
>extensive database and built a table. (Note: some may have
>noticed that I always refer to this table as "the Woolsey-Heinrich
>Match Equity Table". Now you know why!)
>
>After constructing the table empirically Kit 'smoothed' the numbers.
>I once read (or was told, but sorry, I don't remember by whom)
>that the Woolsey-Heinrich MET is consistent with a theoretical
>table with 23% gammons.
Just assume a rate of 20% and examine the score of -2-1:
T wins a gammon and the match outright 10% of the time.
He wins a plain game 40% of the time coming to -1-1, where
his chances are 50%. 50% * 40% = 20%.
His entire equity is 10% + 20% = 30% (The Woolsey-Heinrich-figure)
The same works for -3-1:
T wins a gammon 10% of the time, comming to -1-1 and leaving 50%.
10% * 50% = 5%
T wins a plain game 40%, -2-1 Postcrawford, leaving 50%.
40% * 50% = 20%
(I assume the possibility for T to win a BG outright and the worth
of the Freedrop for L after 40% single wins of T about equalize.)
5% + 20% = 25% (Kits figure again)
This brought me to the assumption, it was computed with 20% gammon
rate. While I might be wrong, where is my fault and what is the
calculation with 23%, that leads you to this figures?
Best wishes, Harald Retter
Chuck Bower
Harald Retter <harald...@okay.net> wrote:
>
>Chuck Bower schrieb in Nachricht <6rk0g3$kqf$1...@flotsam.uits.indiana.edu>...
>>In article <6rhjv8$lfs$1...@trader.ipf.de>,
>>Harald Retter <harald...@okay.net> wrote:
>>
>> (snip)
>>>While typing all those numbers, I was quite surprised how
>>>similar they seemed to be to the Woolsey-table, I used to
>>>work with (and actually still do). If my memory is right,
>>>Kits tables where computed assuming a gammon rate of 20%.
>> (snip)
>>
>> Well, one (or BOTH!) of our memories is in error. Here
>>is what mine tells me: Kit analyzed matches from Hal Heinrich's
>>extensive database and built a table. (Note: some may have
>>noticed that I always refer to this table as "the Woolsey-Heinrich
>>Match Equity Table". Now you know why!)
>>
>>After constructing the table empirically Kit 'smoothed' the numbers.
>>I once read (or was told, but sorry, I don't remember by whom)
>>that the Woolsey-Heinrich MET is consistent with a theoretical
>>table with 23% gammons.
>
>Just assume a rate of 20% and examine the score of -2-1:
>
>T wins a gammon and the match outright 10% of the time.
>He wins a plain game 40% of the time coming to -1-1, where
>his chances are 50%. 50% * 40% = 20%.
>His entire equity is 10% + 20% = 30% (The Woolsey-Heinrich-figure)
Assuming leader will win this game half of the time ("equal players")
then a gammon fraction (ratio of trailer's gammon wins to trailer's
total wins) of f leads one to the following formula for trailer's
match winning chances (MWC) at 2-away, 1-away (-2,-1) Crawford:
trailer's MWC = (1+f)/4
As Harald points out, if you set MWC to exaclty 30%, you get f = 0.2.
(Harald continues:)
>The same works for -3-1:
>
>T wins a gammon 10% of the time, comming to -1-1 and leaving 50%.
>10% * 50% = 5%
>T wins a plain game 40%, -2-1 Postcrawford, leaving 50%.
>40% * 50% = 20%
>(I assume the possibility for T to win a BG outright and the worth
>of the Freedrop for L after 40% single wins of T about equalize.)
>
>5% + 20% = 25% (Kits figure again)
Here, working forwards doesn't really tell you much. Again, if we
assume that leader will win the game half the time, then after this game is
over, trailer will be at the -1,-1 score when s/he scores a gammon:
0.5*f*0.5
and will be at the -2,-1 score for the remainder of his/her wins. IF
we assume that the free drop has no value, then this adds the following
to trailers MWC:
0.5*(1-f)*0.5
So, trailer's MWC at the -3,-1 match score, assuming equal players and
no value to the free drop (and no likelihood of backgammons, same
assumptions that Harald made) you just add the two numbers above and get:
trailers's MWC at -3,-1 Crawford = 0.25.
In words, this match score doesn't depend on gammon fraction (if
you make the above assumptions) so you can't work backwards and get the
gammon fraction from the MWC. Another way to think of this is that EVERY
match equity table derived assuming equal players, no backgammons, and no
value to a free drop has 25% as trailers MWC at the -3,-1 score.
(Harald finishes:)
>
>This brought me to the assumption, it was computed with 20% gammon
>rate. While I might be wrong, where is my fault and what is the
>calculation with 23%, that leads you to this figures?
Well, there you go, Harald. Instead of just trusting my memory, you make
me do some homework. I've never calculated a match equity table (MET).
(Maybe I should learn how, but I like the Woolsey-Heinrich, so I've never
seen much to gain by doing so.) But, Harald, you did a nice job calculating
a couple, right? Maybe you could do my work for me? ;)
Being at home now (not at work, where I am doomed to trust my less
than perfect memory) I looked up the groundbreaking article: "Inside the
Data Base: Everything You Always Wanted to Know About Match Equities" by
Kit Woolsey, INSIDE BACKGAMMON vol. 2, #2, March-April 1992. There is a
lot of interesting things in that article (so why don't you all go buy
this volume and read it!) but I will try and summarize the relevant points:
1) Using data provided by Hal Heinrich from his database of "over 1000
matches" (Note, I think this database is much bigger today than it was in
1992), Kit tallied all matches which met one of the following two criteria:
a) Crawford game with trailer needing an even number of points to win, or
b) Post-Crawford with trailer needing three or more points to win.
Here he found that under these conditions, trailer won 41 gammons and 153
simple games. 41/(41+153) = 21%. He concluded that this was a good number
to use for gammon fraction among top level players (the protagonists in
Heinrich's data).
2) Kit compared a MET he created in in 1980 with statistics from Hal's
database and concluded that some of his old MET values were OK, but he
realized that he had underestimated the trailer's MWC.
3) Kit later says:
So... back to the drawing board. Using these results to give me a
likely distribution at various scores, I rewrote the computer program
to generate new match equity figures.
The output of that computer program, Chart 5 in the article (but also found
on page 12 in his book "How to Play Tournament Backgammon"), is the now
almost universally used Woolsey(-Heinrich) Match Equity Table.
One might guess that Kit used 21% for gammon fraction (which, BTW, is
consistent with 30% MWC for trailer at -2,-1, when rounded), but he never
explicity says in the article (that I can see, anyway) exactly how the
program works, nor exactly what input assumptions were used. So, we still
don't have the complete answer, but...
Maybe Kit will chime in now. Hopefully his memory of 1992 is better
than mine!