Still, I happened to hear a couple of times comments from top
worldclass players, saying that this table slighly overestimated
leader's chances, which made me switch to using Snowie ME table.
I would find it interesting to lead an informal 'survey' if newsgroup
participants could mention which table they favor (Snowie, Woolsey,
Zadeh) and on what grounds?
Also, has any kind of 'scientific' comparison between ME tables ever
been published in BG litterature?
French_Kiss
I use the WoolseyHeinrich MET, because I can remember it! Well, almost. I
learnt the 1away & 2away values, and use Neil's Numbers for the rest.
http://www.gammonline.com/demo/equity.htm
The most scientific comparison I know of is by Kit Woolsey. GammOnLine
subscribers can check out this discussion by Kit
Woolsey.http://www.gammonline.com/members/Jan01/articles/sneq.htm. Kit
writes,
"Snowie assumes a gammon rate of 26%, while for my table I assumed a gammon
rate of about 21%. I agree that Snowie's gammon rate is more accurate in
theory. In practice most backgammon players (even experts) do not play
aggressively enough for a gammon, since they are more concerned about
winning the game. Thus the empirical gammon rate is considerably lower than
Snowie's theoretical rate."

Ian
Btw, my favorite method of coming up with the Woolsey numbers is the Turner
method:
50 + (24/T + 3) * D where T is the number of points the trailer has to go
and D is the differences in scores. This gets to within 1% of the Woolsey
number for all scores up to 11a, 11a, except you should at 2% to the
results for 2a 5a to 2a 8a. Also, you just need to memorize the
Crawford score sequence: 30,25,17,15,10,9,6,5,3,3,2,2,1,1 as the formula
doesn't work for those scores.
Gregg
"Olivier" <ocroi...@hotmail.com> wrote in message
news:b65eadae.01120...@posting.google.com...
50% + (85*D)/(T+6) very accurate for 3 away and more
JeanPierre
"Olivier" <ocroi...@hotmail.com> a Ã©crit dans le message news:
b65eadae.01120...@posting.google.com...
> I believe the vast majority of players is using Woolsey's ME table, as
> I did for a long while.
>
> Still, I happened to hear a couple of times comments from top
> worldclass players, saying that this table slighly overestimated
> leader's chances, which made me switch to using Snowie ME table.
>
> I would find it interesting to lead an informal 'survey' if newsgroup
> participants could mention which table they favor (Snowie, Woolsey,
> Zadeh) and on what grounds?
>
These METs are calculated assuming efficient cube handling by both
players. It's true that if cube is turned exactly at player's take
point it is equally correct to take or drop. But of course this
does not mean that we would get the same MET assuming 1) efficient
and 2) non efficient cube handling in our calculations.
I have calculated MET using gammon rate g=.26 and Point Per Game
PPG=2.2 as explained on ref 1, see Table 1 below.
Table 1: Match equity table. Non efficient doubles (x=.58)
Gammon rate [g=.2] 0.2600
Winning % [w=.5] 0.5000 dELO 0.0000
Cube efficiency [x=.5] 0.5750 PPG 2.2000
Cube deterrent [a=.2] 0.2000 ppg 0.0000
Skill function const 1.0679
1 2 3 4 5 6 7
1 50.0 68.5 75.0 81.8 84.3 89.2 90.9
2 31.5 50.0 60.7 68.6 75.2 81.1 85.0
3 25.0 39.3 50.0 58.2 65.8 72.4 77.5
4 18.2 31.4 41.8 50.0 57.8 64.7 70.5
5 15.8 24.8 34.2 42.2 50.0 57.1 63.2
6 10.8 18.9 27.6 35.3 42.9 50.0 56.4
7 9.1 15.0 22.5 29.5 36.8 43.6 50.0
Differences between Snowie's MET (efficient dbls) and the MET above
are rather small and certainly it is true that your performance
will not be much different either you use the MET above or the one
of Snowie. Note that you should do also adjustments in your cube
handling and MET depending on your opponent's skill level.
Things might be slightly different in a case of backgammon bots
because of their capability to estimate well game equities and also
their opponents playing skills (ref 2). The small errors on Snowie's
MET will produce rather big errors on Gammon Price (GP) tables
(ref 3), see Table 2 below, where is presented difference of GP
tables calculated using 1) efficient and 2) non efficient cube
handling.
Table 2: Difference of gammon price tables GP[x=1]GP[x=.58]
2 3 4 5 6 7
2 .0 .0 .05 .13 .07 .13
3 .0 .16 .23 .11 .12 .12
4 .09 .0 .01 .02 .03 .07
5 .11 .05 .03 .03 .03 .01
6 .0 .0 .01 .0 .02 .02
7 .09 .01 .02 .0 .01 .01
I do not know the details of JellyFish's or Snowie's cube handling
algorithm but I know that their double/take decisions are based on
MET that is calculated assuming efficient doubling. Data on Table 2
will suggest that scores 3away naway and 2away naway could be
the bots' weak point. How well JellyFish and Snowie clear these and
other match scores can be now rolled out with Dueller (ref 4).
References:
1) "Match Equity Calculator V2.0" by Matti RintaNikkola
http://groups.google.com/groups?selm=
233281d4.01073...@posting.google.com&output=gplain
2) "Rating in Snowie2 and Snowie3" by Douglas Zare
http://groups.google.com/groups?selm=
3B6EBACD...@math.columbia.edu
3) "Gammon price" by Ron Karr and David Montgomery
http://www.bkgm.com/rgb/rgb.cgi?view+124
http://www.bkgm.com/rgb/rgb.cgi?view+125
4) "JellyFish And Snowie Battle It Out!" by Tony Lezard
http://www.gammonvillage.com/news/article_display.cfm?resourceid=1155

Posted from ganimede.alcanet.it [194.243.74.13]
via Mailgate.ORG Server  http://www.Mailgate.ORG
http://www.msoworld.com/mindzine/news/classic/bg/cubeformulae.html
and
http://www.msoworld.com/mindzine/news/classic/bg/match_equities.html
Michael
"JeanPierre SEIMAN" <sei...@iprolink.ch> wrote in message
news:9v0tqc$stt$1...@unlisys.unlisys.net...
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Matti RintaNikkola wrote:
> [...]
> Table 1: Match equity table. Non efficient doubles (x=.58)
>
> Gammon rate [g=.2] 0.2600
> Winning % [w=.5] 0.5000 dELO 0.0000
> Cube efficiency [x=.5] 0.5750 PPG 2.2000
> Cube deterrent [a=.2] 0.2000 ppg 0.0000
> Skill function const 1.0679
>
> 1 2 3 4 5 6 7
> 1 50.0 68.5 75.0 81.8 84.3 89.2 90.9
> 2 31.5 50.0 60.7 68.6 75.2 81.1 85.0
> 3 25.0 39.3 50.0 58.2 65.8 72.4 77.5
> 4 18.2 31.4 41.8 50.0 57.8 64.7 70.5
> 5 15.8 24.8 34.2 42.2 50.0 57.1 63.2
> 6 10.8 18.9 27.6 35.3 42.9 50.0 56.4
> 7 9.1 15.0 22.5 29.5 36.8 43.6 50.0
By the way, I doubt the trailer wins less at 2away 4away than at Crawford
2away. I believe that either the 31.4% for 2away 4away is too low (very
likely) or the 31.5% for Crawford 2away is too high. If that's the output
of a model of optimal cube usage, then that model should be revised.
Your table has quite a few noticeable differences with the table used by
Snowie and with the WoolseyHeinrich MET. For example, your value for
trailing 2away 3away is 39.3%, compared to 40.58% for Snowie. That results
in a significantly different take point on an initial double at 3away
3away.
In your model, how frequently does the 2away 3away game
a) end the match with a win for the trailer?
b) end the match with a win for the leader?
c) go to Crawford 2away?
d) go to Crawford 3away?
e) go to 2away 2away?
For each of these questions, once one has the answer, there is a subdivision
into the possible ways that one can get there. For example, if the trailer
wins the match, did this happen on an undoubled backgammon, a doubled
gammon, or a 4cube?
I'm suspicious of recursive calculations of match equity values which are
based on assumptions that match play resembles money play too much, ignoring
the checker play choices available, and how valuable it is to be too good to
double. A very simple example is the value at Crawford 2away, which is
often calculated by assuming that the trailer wins 50% of the games, of
which the same percentage are gammons as in cubeless money play. I don't
believe those assumptions; maybe the errors cancel, and maybe not. At 2away
3away, in a gammonish position the leader almost can't use the doubling
cube, but in a race the cube is almost as powerful for the leader as for the
trailer. When I lead 2away 3away, I often win undoubled gammons; I never
do that for money. Does your model notice that introducing the Jacoby Rule
at this match score would affect the play tremendously?
Incidentally, I'd say that 2away 3away, 3away 4away, and 4away 5away
are all far from money play.
Douglas Zare
Why do you believe that trailer should win more at 2away 4away than at
Crawford 2away?
I would rather call the MET above as the output of the MET calculation
procedure explained on references 1 and 2 than the output of a model of
optimal cube usage. Surely the procedure can be improved, I might do it
later, or even different approach can be used to derive more accurate
MET, but this is not what I would like to start to speak about now.
> Your table has quite a few noticeable differences with the table used by
> Snowie and with the WoolseyHeinrich MET.
This is true in a case of MET for equally skilled players, see Table 2
below. Table 2 should be almost equal with Snowie's MET. The small
differences between the tables are due the different 'free drop
vigorish' (see ref 2). I believe Snowie's table is calculated assuming
(correctly) non zero free drop vigorish while the tables calculated
using MEC assume (incorrectly) zero free drop vigorish. Non zero free
drop vigorish could be easily added also in MEC.
Table 2: Match equity table. Efficient doubles (x=1) & equally skilled
players.
Gammon rate [g=.2] 0.2600
Winning % [w=.5] 0.5000 dELO 0.0000
Cube efficiency [x=.5] 1.0000 PPG 4.4352
Cube deterrent [a=.2] 0.0000 ppg 0.0000
Skill function const 0.5292
1 2 3 4 5 6 7
1 50.0 68.5 75.0 81.8 84.3 89.2 90.9
2 31.5 50.0 59.5 66.4 73.7 79.5 83.5
3 25.0 40.5 50.0 57.1 64.6 71.0 75.8
4 18.2 33.6 42.9 50.0 57.5 64.0 69.4
5 15.8 26.3 35.4 42.5 50.0 56.7 62.3
6 10.8 20.5 29.0 36.0 43.3 50.0 55.9
7 9.1 16.5 24.2 30.6 37.7 44.1 50.0
In a case of MET for unequally skilled players the differences become
more noticeable. Were you having difficulties to choose MET between the
ones shown in Tables 1 and 2? In a case of unequally skilled players the
choice should be clear..., see Tables 3 and 4 below.
Table 3: Match equity table. Inefficient doubles (x=.58) & unequally
skilled players.
Gammon rate [g=.2] 0.2600
Winning % [w=.5] 0.5500 dELO 139.4403
Cube efficiency [x=.5] 0.5750 PPG 2.2000
Cube deterrent [a=.2] 0.2000 ppg 0.2596
Skill function const 1.0406
1 2 3 4 5 6 7
1 55.0 73.3 79.8 85.8 88.0 92.2 93.6
2 36.7 55.0 66.9 74.5 80.8 86.0 89.4
3 30.3 45.7 57.2 65.3 72.8 79.0 83.6
4 22.8 37.8 49.3 57.7 65.8 72.5 77.9
5 20.2 31.1 41.9 50.4 58.7 65.8 71.8
6 14.5 24.7 35.1 43.6 52.0 59.4 65.8
7 12.5 20.3 29.7 37.7 46.0 53.4 60.0
Table 4: Match equity table. Efficient doubles (x=1) & unequally skilled
players.
Gammon rate [g=.2] 0.2600
Winning % [w=.5] 0.5500 dELO 139.4403
Cube efficiency [x=.5] 1.0000 PPG 4.4352
Cube deterrent [a=.2] 0.0000 ppg 0.4435
Skill function const 0.5330
1 2 3 4 5 6 7
1 55.0 73.3 79.8 85.8 88.0 92.2 93.6
2 36.7 55.0 65.0 71.5 78.7 83.9 87.6
3 30.3 46.3 56.1 63.1 70.7 76.7 81.2
4 22.8 39.0 49.0 56.1 64.0 70.3 75.5
5 20.2 31.9 41.9 49.2 57.2 63.8 69.4
6 14.5 25.6 35.3 42.6 50.7 57.5 63.5
7 12.5 21.3 30.3 37.3 45.2 51.8 58.0
> For example, your value for
> trailing 2away 3away is 39.3%, compared to 40.58% for Snowie. That
> results
> in a significantly different take point on an initial double at 3away
> 3away.
Yes, that is true. Tony's match series are already starting to relieve
bots' difficulties in this match score, see ref 3.
>
> In your model, how frequently does the 2away 3away game
(ala Snowie)
> x=.58 x=1
> a) end the match with a win for the trailer? .07 .12 (estimation)
> b) end the match with a win for the leader? .29 .50
> c) go to Crawford 2away? .22 .38 (estimation)
> d) go to Crawford 3away? .21 0
> e) go to 2away 2away? .21 0
This information is not saved during the calculations but in this
specific score it can be calculated easily even with a pen and piece of
paper. I will show how you can do this calculations. Point distribution
that MEC uses is equal
q(n) = (1x)q (n) + xq (n),
 +
where x is cube efficiency parameter, q is the point distribution of

cubeless and gammonless game and q is the point distribution that is
+
obtained assuming continues, zero volatility game with efficient doubles
as explained in ref 2. q(n) is the probability that player will win n
points (if n<0 means that he has lost n points). From the efficient
doubles approximation follows directly that the minimum win would be
equal 2 so using the numbers of Table 1 we will obtain
q(1) = q(1) = (1x)p = .21 (case d & e above)
We can easily calculate also leader's probability that he will win the
match (case b above) because the used approximations, see above, do not
change the players game winning probabilities. So leader's probability
to win the match is
q(>=2) = xp = .29 (case b)
The 'remaining probability' .29 will be divided between cases a and c.
I will not calculate it, you can do it yourself if you like, but I just
wrote my (probably poor) estimation: case a) .29*.25 and case c)
.29*.75. See your question Table above, where I have added also the
point distribution that Snowie uses in his MET calculations.
> I'm suspicious of recursive calculations of match equity values which are
> based on assumptions that match play resembles money play too much,
ignoring
> the checker play choices available, and how valuable it is to be too good
to
> double.
1) I'm not aware any other way than recursive calculations to obtain MET
2) assumptions that resembles money play too much Me too I'm suspicious
of this kind of assumptions... anyway using simple money game point
distribution in calculations will lead more accurate MET than MET
that is calculated assuming effective doublings.
3) ignoring the checker play choices available. I do not know how you
would like to implement this in MET calculation. Surely at some
scores there are more checker play choices available than at others
(i.e. naway 1away vs naway maway). I think that taking this into
account somehow in calculations just complicate the procedure without
bringing more accuracy.
4) how valuable it is to be too good to double. See 3) above.
> Does your model notice that introducing the Jacoby Rule
> at this match score would affect the play tremendously?
See reference 2
> Incidentally, I'd say that 2away 3away, 3away 4away, and 4away 5away
> are all far from money play.
I agree... I would like to add in this list also 3away 3away and
4away 4away.
References:

1) "Match Equity Calculator V2.0" by Matti RintaNikkola
http://groups.google.com/groups?selm=
233281d4.01073...@posting.google.com&output=gplain
2) "How to Compute a Match Equity Table" by Tom Keith
http://www.bkgm.com/articles/met.html
3) "A thought on bot cube usage" by M.RintaNikkola
http://groups.google.com/groups?selm=
808087c90ceddfda1fc...@mygate.mailgate.org&output=gplain
Lucky New Year to all ;)
Matti
PS. I will not read r.g.b next two and half weeks...

Posted from [212.177.60.14]
Matti RintaNikkola wrote:
The trailer could simply double immediately, after which a single win is worth
50% mwc, a gammon is worth 100% mwc, and a loss is worth 0% mwc, just as for
Crawford 2away.
So, if your table says that 2away 4away is less advantageous for the trailer,
this must mean that it is correct to drop sometimes if the trailer doubles
immediately, and that the equity saved by dropping is greater than the equity the
trailer gains from not having do double immediately... or else your table is
wrong.
Hmm, 31, 62, I'd take. After my opponent's 31, I wouldn't double. It looks
like the equity of trailing 2away 4away is greater than the equity at Crawford
2away.
I think your model is missing some basic understanding of match play.
> I would rather call the MET above as the output of the MET calculation
> procedure explained on references 1 and 2 than the output of a model of
> optimal cube usage. Surely the procedure can be improved, I might do it
> later, or even different approach can be used to derive more accurate
> MET, but this is not what I would like to start to speak about now.
>
> > Your table has quite a few noticeable differences with the table used by
> > Snowie and with the WoolseyHeinrich MET.
>
> This is true in a case of MET for equally skilled players, see Table 2
> below. Table 2 should be almost equal with Snowie's MET. The small
> differences between the tables are due the different 'free drop
> vigorish' (see ref 2). I believe Snowie's table is calculated assuming
> (correctly) non zero free drop vigorish while the tables calculated
> using MEC assume (incorrectly) zero free drop vigorish. Non zero free
> drop vigorish could be easily added also in MEC.
Interesting. According to Chris Yep, one way that Snowie's table errs is in
assuming 0 value for the free drop in certain contexts. For example, the entry at
Crawford 3away in Snowie's table is 25.22%, almost exactly the value that one
would get by assuming 50% wins and 0.9% backgammons if the free drop had no value
at PostCrawford 2away. However, it does have significant valueI think most
theorists say more value than the backgammons, and that the entry should be less
than 25%. I'm not sure if I agree that the backgammon rate should be so low at
this match score.
> Table 2: Match equity table. Efficient doubles (x=1) & equally skilled
> players.
>
> Gammon rate [g=.2] 0.2600
> Winning % [w=.5] 0.5000 dELO 0.0000
> Cube efficiency [x=.5] 1.0000 PPG 4.4352
> Cube deterrent [a=.2] 0.0000 ppg 0.0000
> Skill function const 0.5292
>
> 1 2 3 4 5 6 7
> 1 50.0 68.5 75.0 81.8 84.3 89.2 90.9
> 2 31.5 50.0 59.5 66.4 73.7 79.5 83.5
> 3 25.0 40.5 50.0 57.1 64.6 71.0 75.8
> 4 18.2 33.6 42.9 50.0 57.5 64.0 69.4
> 5 15.8 26.3 35.4 42.5 50.0 56.7 62.3
> 6 10.8 20.5 29.0 36.0 43.3 50.0 55.9
> 7 9.1 16.5 24.2 30.6 37.7 44.1 50.0
This matches Snowie's table pretty closely, though I don't necessarily believe
either.
> In a case of MET for unequally skilled players the differences become
> more noticeable. Were you having difficulties to choose MET between the
> ones shown in Tables 1 and 2? In a case of unequally skilled players the
> choice should be clear..., see Tables 3 and 4 below.[...]
I wasn't attempting to choose. I was saying that I don't believe Table 1, and
what might be erroneous entries in Table 1 might lead one to misplay.
For the moment, let's focus on modelling correct play rather than, in addition,
constructing a model of errors.
> > For example, your value for
> > trailing 2away 3away is 39.3%, compared to 40.58% for Snowie. That
> > results
> > in a significantly different take point on an initial double at 3away
> > 3away.
>
> Yes, that is true. Tony's match series are already starting to relieve
> bots' difficulties in this match score, see ref 3.
The bots' difficulties? Are you asserting that 39.3% is correct, and that
Snowie's table is wrong, or just pointing out that Snowie and Jellyfish play so
differently that at least one must be wrong?
Incidentally, that one might lose more than a point per game in a short session
does not mean that one's game would be improved by passing every double.
Similarly, that in that session Jellyfish might have done better to pass every
double in one category does not mean it was erring by takingand of course,
there are many takes in match play that are money passes, as the points have
different meanings.
> >
> > In your model, how frequently does the 2away 3away game
> (ala Snowie)
> > x=.58 x=1
> > a) end the match with a win for the trailer? .07 .12 (estimation)
> > b) end the match with a win for the leader? .29 .50
> > c) go to Crawford 2away? .22 .38 (estimation)
> > d) go to Crawford 3away? .21 0
> > e) go to 2away 2away? .21 0
This is very interesting. Have you tested either, or an arbitrary interpolation,
against data from actual play? I strongly doubt that the leader should
never/rarely win one point. Often the leader plays on for the gammon, but doesn't
get it. The trailer can double early, but never losing his/her market? I don't
think that's plausible either. I suppose that these suggest that the values in
the first column should be more plausible, but these are from the model which
said that 2away 4away is worse than Crawford 2away. In my experience, the
trailer wins the match with a doubled gammon much more than 7% of the time, never
mind winning the match on a 4cube or backgammon. I'd guess more than 10% of the
time.
So, when you say x=1, q=q+, and the leader never gets an undoubled gammon. On the
other hand, I don't see why the trailer wins the match this game only 12% of the
time. Surely in this model, every time the trailer wins a gammon, the trailer
wins the match, and that happens 13% of the time if your gammon rate is 0.26.
Further, the trailer also wins the match with a single win with the cube on 4,
which can happen if the leader doubles but doesn't win, which happens a
significant fraction of the time. If 12% is right, then the right value in that
match equity table is more like 38%.
It looks to me like you are assuming that the leader must win 50% of the time,
and then afterwards you divide the wins among the different possible point
values. In fact, the trailer may win most games in match play. Of course, the
trailer might trade wins for extra gammons, but the trailer often can make more
use of the doubling cube, and the leader's take point is much higher in gammonish
situations than the trailer's, and on 4cubes and higher. I think this assumption
is behind some of the inconsistencies.
As a test, suppose the gammon rate in this game only were 100%. Then the trailer
could double immediately, and turn the match into DMP, 50% mwc. By your method,
the mwc with perfectly inefficient doubles would be 34.25 (1/2 chance to get lead
Crawford 2away), worse than the 37.5% of 0% gammons. With perfectly efficient
racing doubles, I believe with your assumptions you would obtain a 50% winning
chance for the trailer. So, in combining the two you would get something less
than 50% mwc (58% of the way from 34.25 to 50%), when the correct value is
clearly 50%. The cube efficiency of the trailer is tremendously underestimated in
your model: the trailer's doubles kill the leader's gammons. On the other hand,
the leader gets remarkably little value out of the doubling cube.
Incidentally, another flaw with this type of calculations is the assumption that
the gammon rates at the time that one doubles are close to the gammon rates
overall. A high gammon rate can provoke a double from the trailer in a nearly
symmetric position.
> > I'm suspicious of recursive calculations of match equity values which are
> > based on assumptions that match play resembles money play too much,
> ignoring
> > the checker play choices available, and how valuable it is to be too good
> to
> > double.
>
> 1) I'm not aware any other way than recursive calculations to obtain MET
Kit Woolsey's method (with Hal Heinrich) was not recursive, at least within the 5
point match. He looked at actual data from a thousand matches. He also adjusted
some of the numbers slightly to make them more plausible and easier to remember.
However, I expect that a recursive method can work. However, it should be based
on the results of actual backgammon games. It should not be based on
generalizations from money play which produce a lower value trailing 2away
4away than at Crawford 2away.
> 2) assumptions that resembles money play too much Me too I'm suspicious
> of this kind of assumptions... anyway using simple money game point
> distribution in calculations will lead more accurate MET than MET
> that is calculated assuming effective doublings.
How effective can the leader's double be at 2away 3away in a gammonish
situation? It's a huge flaw in the model if you assume that the leader always
cashes or doubles in rather than playing for an undoubled gammon.
> 3) ignoring the checker play choices available. I do not know how you
> would like to implement this in MET calculation. Surely at some
> scores there are more checker play choices available than at others
> (i.e. naway 1away vs naway maway). I think that taking this into
> account somehow in calculations just complicate the procedure without
> bringing more accuracy.
I don't know what you mean by the differences at naway 1away vs. naway maway.
What would you say the differences are at those match scores?
At 2away 4away, not only is there the threat that the trailer will double,
converting the game to Crawford 2away, but before the initial double, the
leader's gammon price is very high. So to me, 2away 4away is sometimes farther
from money play than Crawford 2away. On the other hand, 3away 5away usually
resembles money play, more so than 4away 5away.
> 4) how valuable it is to be too good to double. See 3) above.
>
> > Does your model notice that introducing the Jacoby Rule
> > at this match score would affect the play tremendously?
>
> See reference 2
I've seen it. I don't see how it answers my question, unless you mean that your
model also would not be affected by the introduction of the Jacoby Rule, despite
the fact that this would greatly change match play. (I mean introducing the
Jacoby Rule except in the Crawford game. I played in one minor online hypergammon
tournament in which the Jacoby Rule was enforced even in the Crawford game, which
was quite a change.)
If you have an oversimplified model, then the output can't be trusted. The
oversimplifications might not be fixable by increasing the complexity elsewhere.
Your errors might cancel, but you can't count on that.
> > Incidentally, I'd say that 2away 3away, 3away 4away, and 4away 5away
> > are all far from money play.
>
> I agree... I would like to add in this list also 3away 3away and
> 4away 4away.
Those are significantly more like money in the sense that the gammon prices are
equal for the players, unlike for 3away 4away. So, you wouldn't call these
scores "money game like" anymore? I don't mean to imply that you said that about
2away 3away, but you did for 3away 4away and 4away 5away in a post here in
October, "JellyfishSnowie: Game statistics". It was an interesting post, but I
don't think one can group together the results from doubles at 3away 4away and
3away 3away.
I don't mind the application of mathematical models to backgammon, far from it.
However, I think one should make the underlying assumptions clear so that more
people can have an informed level of confidence or skepticism.
Douglas Zare
I don't understand in what sense you mean that Kit's calculation
wasn't recursive. For each score he look at the distribution
of subsequent scores, and combined the mwcs from those scores.
Each entry recurses one level, and each of those entries is also
calculated based on recursion, down to some simple base cases.
David
David Montgomery wrote:
You are right, that can be called recursive as well. I wanted to emphasize that there
was extra input at each match score that was derived from real matches rather than a
set model.
There can be a model of backgammon that allows correct recursive calculations. I don't
know how easily one can adjust something based on a onedimensional continuous limit.
Continuity often does not agree with becoming too good to double, or with lastroll
doubles.
Douglas Zare
> Interesting. According to Chris Yep, one way that Snowie's table errs is in
> assuming 0 value for the free drop in certain contexts. For example, the
> entry at Crawford 3away in Snowie's table is 25.22%, almost exactly the
> value that one would get by assuming 50% wins and 0.9% backgammons if the
> free drop had no value at PostCrawford 2away.
>
ok. Then the small differences between the tables are due the different
backgammon rates. The table below is calculated assuming zero
backgammon rate.
>
> > Table 2: Match equity table. Efficient doubles (x=1) & equally skilled
> > players.
> >
> > Gammon rate [g=.2] 0.2600
> > Winning % [w=.5] 0.5000 dELO 0.0000
> > Cube efficiency [x=.5] 1.0000 PPG 4.4352
> > Cube deterrent [a=.2] 0.0000 ppg 0.0000
> > Skill function const 0.5292
> >
> > 1 2 3 4 5 6 7
> > 1 50.0 68.5 75.0 81.8 84.3 89.2 90.9
> > 2 31.5 50.0 59.5 66.4 73.7 79.5 83.5
> > 3 25.0 40.5 50.0 57.1 64.6 71.0 75.8
> > 4 18.2 33.6 42.9 50.0 57.5 64.0 69.4
> > 5 15.8 26.3 35.4 42.5 50.0 56.7 62.3
> > 6 10.8 20.5 29.0 36.0 43.3 50.0 55.9
> > 7 9.1 16.5 24.2 30.6 37.7 44.1 50.0
>
> This matches Snowie's table pretty closely, though I don't necessarily
> believe either.
Good... lets try to look "behind these numbers".
As you have shown in previous message the model that has been used in
MEC (ref 1) to implement inefficient cube handling in MET calculation is
too rude to give reliable equities in certain match scores. In order to
come out with more accurate model the procedure that calculates match
equities need to be rewritten. I will explain here using also examples
the algorithm of the new equity calculation procedure. The procedure
will be based on continues and zero volatility approximation of
Backgammon game. Match equities will be calculated using players' point
distributions, which have to be calculated in every match score. In
order to solve point distribution of a particular score we need to know
players' take points (TP) of all the possible doubles. Take points can
be solved as explained on reference 2. Using the assumption of
continues and zero volatility game and the solved take points we can
calculate the doubling and redoubling probabilities (see details on ref
4). For example the probability that player 1 (P1) will double first is
p  TP1
P("P1 dbls 1st") =  , (1)
1  TP1  TP2
where p is the cubeless game winning probability of player 1, TP1 and
TP2 are take points of the first double for players 1 and 2. Knowing
all doubling and redoubling probabilities we can calculate the
probabilities of the possible final stakes of the score. See reference
5 where probabilities of possible final stakes are calculated in a case
of money game. In match play things are slightly more complicate
because of players' take points are not constant and equal as in a case
of money game. When the probabilities of final stakes, gammon rate (g)
and cube efficiency (x) are known we can write the point distribution
and solve the match equity.
o Example 1: Score 2away 3away, equally skilled players.
Game winning probability p=.5 and gammon rate g=.25.
Player P1 needs 2 points in order to win a match and player P2
needs 3 points.
Using continues and zero volatility game approximation:
 Take points values TP1=.34 and TP2=.25 have been read from reference
6: table "Market Windows".
 Using equation (1) we can solve the doubling probabilities:
P("P1 dbls 1st") = .39 and
P("P2 dbls 1st") = .61.
 When P1 doubles and double is taken the cube is turned immediately
to 4. In this case P1 will win with probability 1  .25
 When P2 doubles and double is taken he will win with probability
1  .34 No redoubles.
We will obtain the following point distributions:
1) efficient doubles (x=1)
P1 dbls 1st P2 dbls 1st tot
1   
2  30.2 30.2
> 4 9.7 10.1 19.8
1   
2  15.6 15.6
<4 29.2 5.2 34.4 => M[2,3] = 40.5
2) inefficient doubles (x=.5)
P1 dbls 1st P2 dbls 1st tot
1  30.5 30.5
2  15.1 15.1
> 4 4.9 5.0 9.9
1 19.5  19.5
2  7.8 7.8
<4 14.6 2.6 17.2 => M[2,3] = 40.4
3) all doubles are dropped (x=0)
P1 dbls 1st P2 dbls 1st tot
1  61.0 61.0
1 39.0  39.0 => M[2,3] = 40.3
Match winning probabilities M[2,3] above have been calculated using
calculated point distribution and MET shown in Table 2.
o Note 1. Match equity does not depend on cube efficiency x if doubling
is done exactly at players' take points. Small variation in match
winning probabilities M[2,3] above is due the rounding errors and
inaccuracies of used parameter values. Thus it is enough to solve
probability P("P1 dbl 1st") and match winning probability can be
calculated assuming that all the doubles are dropped (x=0) from
M[m,n] = P("P1 dbls 1st")M[m,n1] + [1  P("P1 dbls 1st")]M[m1,n] (2)
Note also that if we know match equities doubling probability can be
calculated from
M[m,n]  M[m1,n]
P("P1 dbls 1st") =  (3)
M[m,n1]  M[m1,n]
 o o o 
How good is the continuos and zero volatility game approximation?
There are not many ways to get answer to this question: either you
develop more accurate model or test the approximation against match
statistics. I have chosen the latter method and used Big Brother match
data analyzed by Peter Frankhauser (ref 6). First I compared P("P1 dbls
1st") probabilities obtained from equation (1) to the experimental data
(ref 7: Table 4).
In reference 7 the doubling probability is calculated using only games
where cube is turned ignoring completely games where either of players'
game has became too good to double and the cube is never turned.
Fortunately we can calculate using the data of Peter's analysis the
fraction of games where cube is never turned, see Table 6 below. As
experimental doubling probability we have used the probability
P("P1 doubles 1st or his game has became too good to double"). IMHO it
is rather difficult to estimate whose (leader's/trailer's) game has
became too good to double at scores maway naway (m>2, n>2) so in these
scores have been simply used the doubling probability obtained from
reference 7: Table 4. At scores 2away naway have been assumed that
trailer's game has never became too good to double and the experimental
doubling probability has been calculated using the data of reference 7:
Table 4 and Table 6 below.
Theoretical doubling probabilities have been calculated using
p=.5, the values for TP1 and TP2 are again read from reference 6, Table
"Market Windows". The differences between these two doubling
probabilities are shown on Table 3 below.
Table 3: Measured P("P1 dbls 1st")  Calculated P("P1 dbls 1st") (eq 1)
[%]
3 4 5 6 7 # of average
2 8.3 6.5 13.1 14.7 7.7 records diff.
3 3.4 3.2 2.4 4.9 2away naway 482 9.6
4 .9 4.8 5.1 maway naway 1105 .8
5 2.0 3.9
6 8.3 (m>2, n>2)
Clearly the differences are biggest on scores 2away naway. In
practice at these scores trailer doubles much before (or equivalently
leader doubles much later) than what is predicted from continuos and
zero volatility game approximation. Next I calculated difference
between experimental (ref 6: Table 2) and theoretical METs (Table 2),
see Table 4 below. I have calculated differences only on match scores
that interest us. Here differences are much smaller than on Table 3.
Table 4: "Measured MET"  "Calculated MET" [%]
3 4 5 6 7 # of average
2 2.4 5.3 2.6 8.4 2.4 records diff.
3 1.3 1.2 7.3 .9 2away naway 482 1.3
4 1.0 .1 3.9 maway naway 1105 .1
5 8.1 1.0
6 2.8 (m>2, n>2)
o Note 2: Big Brother match statistics is not statistics of equally
skilled players. For example at score 2away 3away we have more
records where underdog is also less skilled player than leader.
 o o o 
What's going on these 2away naway scores?
Let's take closer look to one of them i.e. on score 2away 3away where
we have statistics of 203 games. In this score the measured probability
that leader will double first is .307. This probability can be
calculated also using the data of the measured MET (ref 7: Table 2) and
equation (3):
42.9  50.0
P("leader dbls 1st at score 2aw 3aw")=  = .267
23.7  50.0
On the other hand continuos and zero volatility game model predicts (eq
1) for the probability .39! Also in trailer's match winning chances
there is a clear difference between the measured and theoretical values
at score 2aw 3aw: 42.9% vs 40.5. In order to explain these
differences let's make two hypothesis: Hypothesis 1) players play
correctly, game model fails and Hypothesis 2) players make cube handling
errors, model works correctly.
o Hypothesis 1: Players play correctly but model fails (i.e. doubling
probability function  see eq 1).
Let's use the following doubling probabilities and take points:
P("leader doubles 1st") = .31
P("trailer doubles 1st") = .69
TP1 = .25 (trailer's take point)
TP2 = .34 (leader's take point)
Note that above take points are equal to those predicted by the game
model. In order to assure that these are correct take points lets
calculate match equities using different cube efficiency values: x=1,
x=.5 and x=0 (g=.25)
x=1: x=.5: x=0:
leader trailer tot leader trailer tot tot
dbls dbls dbls dbls
1    1  34.5 34.5 1 69.0
2  34.2 34.2 2  17.1 17.1 1 31.0
> 4 7.8 11.4 19.2 > 4 3.8 5.7 9.5
1    1 15.5  15.5
2  17.6 17.6 2  8.8 8.8
<4 23.2 5.8 29.0 <4 11.7 2.9 14.6
=>M[2,3]=42.6 =>M[2,3]=42.3 =>M[2,3]=42.3
Ok.
o Hypothesis 2: Players make cube handling errors and the game model
works correctly. In order to get out from equation (1) the cubing
probabilities:
P("leader doubles 1st") = .31
P("trailer doubles 1st") = .69
we need to modify players' calculated doubling points. Lets assume
that in reality trailer doubles much before leader's take point
(.34). I have chosen following doubling points:
leader doubles when trailer's game winning chances (gwc) are TP1 = .25
trailer doubles when leader's gwc are TP2 = .39
Using these doubling points we will get from equation (1) the above
cubing probabilities.
Now if both players incorrectly drops all the doubles we will obtain
for trailer's match winning probability M[2,3] = 42.3% (just like in
hypothesis 1, above)
If trailer doubles when leader's game winning chances are as high as
39% leader should take every double, see below:
leader dbls trailer dbls tot
x=.5 x=1
1    (g=.25)
2  31.6 31.6
> 4 3.9 10.5 14.4
1 15.5  15.5
2  20.2 20.2
<4 11.6 6.7 18.3 => M[2,3] = 39.9%
 o o o 
Why would continuos and zero volatility game model fail at scores 2away
naway?
There has been already discussion of the weaknesses of the game model in
this thread. However the main reason why the model fails at scores
2away naway is not yet mentioned. IMHO the reason is that the cube
value is not taken account in the model currently used. In the game
model the cube rises only the stakes, it does not give any additional
winning advantage to the cube owner. Generally when player takes the
double his winning chances will be higher than just before the double
because after the double only he will have access to the cube. Score
2away naway is exception from this general rule because at this score
trailer when doubled will give always dead cube to the leader and on the
other hand when leader doubles trailer can redouble immediately to 4.
Lets take a closer look to the middle game doubles at the scores maway
naway and 2away naway. In order to estimate the cube value (CV) and
error of the game model we will divide doubling action in two phases:
Phase 1) cube is turned but not yet shipped (i.e. no one has access to
the cube) and Phase 2) cube is turned and shipped (i.e. only taker has
access to the cube). Now lets write players game winning chances (gwc)
at these two doubling phases in a case of optimal double:
maway naway: (m>2 and n>2)
doubler's gwc taker's gwc
Phase 1: 1  TP + CV TP  CV
Phase 2: 1  TP TP (TP taker's take point)
2away naway: (n>2)
1) leader dbls leader's gwc trailer's gwc
Phase 1: 1  TP2 + CV2 TP2  CV2
Phase 2: 1  TP2 TP2
2) trailer dbls trailer's gwc leader's gwc
Phase 1: 1  TP1 TP1
Phase 2: 1  TP1 TP1
In the game model cube is turned when doubler's gwc are 1  TP (Phase 1)
and because in the model cube does not have any value doubler's gwc do
not change when double is completed. In reality doubler's gwc in the
middle game doubles should be higher than 1  TP (Phase 1) i.e. doubles
should be made later than what the game model predicts.
Taking account the cube value we can write doubling probability formula
(1) as
p  TP1 + CV1
P("P1 dbls 1st") =  , (4)
1  TP1  TP2 + CV1 + CV2
where CV>0 is the cube value correction. Player's take points TP1 and
TP2 are obtained from the continuos and zero volatility game model. The
values of CV should be estimated for every match score for example using
experimental data.
o Note 3: Take point values TP1 and TP2 are depending from redoubles
and so similar correction discussed above should be done to all the
possible doubles of the score.
 o o o 
How much does the correction change the doubling probabilities?
Rough idea about the magnitude of the cube value parameter CV can be
obtained using the information of reference 8:
"Given two even players, I think starting with the cube on your side
is worth about .175ppg. If your opponent can't reship to 4 after your
double, I would estimate the spot is worth .25ppg" by David Montgomery
Assuming that PPG (Point Per Game) is about 2.1 we can convert above
values in terms of gwc: .175ppg ~ 8% gwc and .25ppg ~ 12% gwc.
I calculated again the differences between theoretical and experimental
doubling probabilities now using equation (4). In a case of 2away
naway I used the cube values CV1=.0, CV2=.12 and in other scores
CV1=CV2=.08, see Table 5.
Table 5: Measured P("P1 dbls 1st")  Calculated P("P1 dbls 1st")(eq4)
[%]
3 4 5 6 7 # of average
2 .5 .1 5.6 7.8 1.2 records diff.
3 2.1 4.5 .7 2.2 2away naway 482 1.8
4 1.1 5.3 3.8 maway naway 1105 .2
5 1.5 5.1
6 7.8 (m>2, n>2)
 o o o 
> > > For example, your value for
> > > trailing 2away 3away is 39.3%, compared to 40.58% for Snowie. That
> > > results in a significantly different take point on an initial
> > > double at 3away 3away.
> >
> > Yes, that is true. Tony's match series are already starting to relieve
> > bots' difficulties in this match score, see ref 3.
>
> The bots' difficulties? Are you asserting that 39.3% is correct, and that
> Snowie's table is wrong, or just pointing out that Snowie and Jellyfish play so
> differently that at least one must be wrong?
IMHO 3away 3away is so far the most interesting score in JellyFish
vs Snowie match series (ref 9). It is true that they play differently
but from a few data that we have shows also that they both probably fail
in their doubling/taking decisions in this score. However much more data
is needed to be sure. More than asserting that 39.9% is correct (which
you have shown correctly to be wrong :) I have been suspecting that
40.5% is wrong. That is simply because I do not know details on bots'
doubling algorithm but only the fact that they both use MET which is
calculated as explained on ref 2 and that their cube handling at score
3away 3away seems a bit weak to me when comparing to the other scores.
> Incidentally, that one might lose more than a point per game in a short session
> does not mean that one's game would be improved by passing every double.
> Similarly, that in that session Jellyfish might have done better to pass every
> double in one category does not mean it was erring by takingand of course,
> there are many takes in match play that are money passes, as the points have
> different meanings.
This I agree completely.
> > > I'm suspicious of recursive calculations of match equity values which are
> > > based on assumptions that match play resembles money play too much,
> > > ignoring the checker play choices available, and how valuable it is
> > > to be too good to double.
> >
> > 1) I'm not aware any other way than recursive calculations to obtain MET
>
> Kit Woolsey's method (with Hal Heinrich) was not recursive, at least within the 5
> point match. He looked at actual data from a thousand matches. He also adjusted
> some of the numbers slightly to make them more plausible and easier to remember.
> However, I expect that a recursive method can work. However, it should be based
> on the results of actual backgammon games.
Unfortunately I haven't seen Kit Woolsey's analyses of Hal Heinrich's
database nor derivation of his MET. However a thousand matches to
derive MET seems to me quite fewsee Peter Frankhauser's analyses of
1035 matches (ref 7) and his MET (2512 matches ref 10). One problem of
measuring MET (i.e. deriving from match database) is that you need huge
databases to find METs for unequally skilled players.
> > 2) assumptions that resembles money play too much Me too I'm suspicious
> > of this kind of assumptions... anyway using simple money game point
> > distribution in calculations will lead more accurate MET than MET
> > that is calculated assuming effective doublings.
After I went through the details of the model (see above) I'd like to
correct also my comments: MET that is calculated assuming effective
doublings when correctly done will lead more accurate table than MET
that is calculated using money game point distribution. However using
money game point distribution in MET calculation will lead more accurate
table than the one obtained from original MEC (ref 11) when calculating
MET for players of unequal playing skills.
> How effective can the leader's double be at 2away 3away in a gammonish
> situation? It's a huge flaw in the model if you assume that the leader always
> cashes or doubles in rather than playing for an undoubled gammon.
This is true if you use the model to solve point distributions but I
think that it will have few influence to the MET, see example 1 above.
> > 3) ignoring the checker play choices available. I do not know how you
> > would like to implement this in MET calculation. Surely at some
> > scores there are more checker play choices available than at others
> > (i.e. naway 1away vs naway maway). I think that taking this into
> > account somehow in calculations just complicate the procedure without
> > bringing more accuracy.
>
> I don't know what you mean by the differences at naway 1away vs. naway maway.
> What would you say the differences are at those match scores?
At score naway 1away games have more checker play choices available
than games at scores maway naway simply because these games are longer
(no cube drops).
> At 2away 4away, not only is there the threat that the trailer will double,
> converting the game to Crawford 2away, but before the initial double, the
> leader's gammon price is very high. So to me, 2away 4away is sometimes farther
> from money play than Crawford 2away. On the other hand, 3away 5away usually
> resembles money play, more so than 4away 5away.
ok. How would you implement these in MET calculation?
> > > Does your model notice that introducing the Jacoby Rule
> > > at this match score would affect the play tremendously?
> >
> > See reference 2
>
> I've seen it. I don't see how it answers my question, unless you mean that your
> model also would not be affected by the introduction of the Jacoby Rule, despite
> the fact that this would greatly change match play.
Right, continuos and zero volatility game approximation implies directly
that if both players play correctly game will never became too good to
double and so introduction of Jacoby rule does not change anything.
I agree that Jacoby rule would change a lot leader's doubling decisions
at scores 2away naway. What is not so clear to me is how much it would
change MET. IMHO change might be even so small that it can ignored.
Everything is depending on the "average" volatility of the game.
Because we have match data available lets take a look how much
Jacoby rule could change the game. Without Jacoby rule the fraction of
games that are finished without cube never turned (according to Big
Brother see ref 7: Tables 3 & 5), see Table 6 below.
Table 6: Game has became too good to double cube never turned [%]
3 4 5 6 7
2 6.4 6.7 5.3 5.3 5.4
3 2.9 3.6 .9 4.3 3.5
4 3.4 .0 5.1 1.8
5 2.8 .0 .8
6 .0 1.5
7 1.4
Introducing Jacoby rule would change Table 6 completely because then
games would never became too good to double i.e. Table 6 would be full
of zeros. Change might seem rather small. However at scores 2away
naway change is very clear from leader's point of view. We know
that at these scores the probability that leader doubles or become too
good to double is about .30. On the other hand we know that for trailer
it's very rear to become too good to double at these scores so when
leader becomes good enough to turn the cube he is about every fifth time
too good to double.
> If you have an oversimplified model, then the output can't be trusted. The
> oversimplifications might not be fixable by increasing the complexity elsewhere.
> Your errors might cancel, but you can't count on that.
AFAIK continuos and zero volatility game model is the most accurate
model currently available. If you can't trust it and want more accurate
model you have to develop it yourself or wait that someone else would do
it. IMHO the MET that can be obtained from the model is accurate enough
for practical use because of variation of gammon rates and players'
skill differences. Gammon rate and skill differences you have to
estimate anyway yourself and adjust your MET accordingly.
> > > Incidentally, I'd say that 2away 3away, 3away 4away, and 4away 5away
> > > are all far from money play.
> >
> > I agree... I would like to add in this list also 3away 3away and
> > 4away 4away.
>
> Those are significantly more like money in the sense that the gammon prices are
> equal for the players, unlike for 3away 4away. So, you wouldn't call these
> scores "money game like" anymore? I don't mean to imply that you said that about
> 2away 3away, but you did for 3away 4away and 4away 5away in a post here in
> October, "JellyfishSnowie: Game statistics". It was an interesting post, but I
> don't think one can group together the results from doubles at 3away 4away and
> 3away 3away.
I still would group together these scores as I have done in my post
"JellyfishSnowie: Game statistics" (ref 12). I think also that calling
this game group as "money game like scores" is better than for example
"score group A". I also mentioned that games in this group differs from
money games. IMHO one can group together results from doubles at 3away
4away and 3away 3away if there is few data available. It depends also
how one is going to use the results and what conclusions one wants to
make.
> I don't mind the application of mathematical models to backgammon, far from it.
> However, I think one should make the underlying assumptions clear so that more
> people can have an informed level of confidence or skepticism.
Also I like the application of the mathematical models to backgammon.
I do not mind if someone posts an application, model or an idea that is
not so well formulated or proved. If the idea is good surely the missing
parts will be put in their places this is an interactive media.
Best Regards,
Matti RintaNikkola
 o o o 
References:

1) "Match Equity Calculator V2.0" by Matti RintaNikkola
http://groups.google.com/groups?selm=
233281d4.01073...@posting.google.com&output=gplain
2) "How to Compute a Match Equity Table" by Tom Keith
http://www.bkgm.com/articles/met.html
3) "A thought on bot cube usage" by M.RintaNikkola
http://groups.google.com/groups?selm=
808087c90ceddfda1fc...@mygate.mailgate.org&output=gplain
4) "Underdoubling dice" by Bill Taylor
http://www.bkgm.com/rgb/rgb.cgi?view+429
5) "Doubling probabilities etc." by M.RintaNikkola
http://groups.google.com/groups?as_umsgid=
233281d4.01071...@posting.google.com
http://groups.google.com/groups?selm=
233281d4.0110182245.59e874ba%40posting.google.com&output=gplain
6) "Match Play Doubling Strategy" by Tom Keith
http://www.bkgm.com/articles/mpd.html
7) "Some statistics on the BigBrother Matches" by Peter Frankhauser
http://groups.google.com/groups?selm=
4dofu0%24slm%40omega.gmd.de&output=gplain
8) "Value of the cube" by Gary Wong & David Montgomery
http://groups.google.com/groups?as_umsgid=
wt4stuz...@brigantine.CS.Arizona.EDU
http://groups.google.com/groups?as_umsgid=
6umpl5%24...@krackle.cs.umd.edu
9) "JellyFish And Snowie Battle It Out!" by Tony Lezard
http://www.gammonvillage.com/news/article_display.cfm?resourceid=1155
http://www.mantis.co.uk/~tony/backgammon/
10) "MatchTable from BBmatches" by Peter Frankhauser
http://groups.google.com/groups?selm=
31EE2718.2781E494%40darmstadt.gmd.de&output=gplain
11) "Match equity calculator" by Claes Thornberg
http://www.bkgm.com/rgb/rgb.cgi?view+114
12) "JellyFish  Snowie: Game statistics" by M. RintaNikkola
http://groups.google.com/groups?selm=
233281d4.01102...@posting.google.com&output=gplain

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