Is this a double? Revisited. Jury is out?
James H. Cochrane
+13-14-15-16-17-18-+---+19-20-21-22-23-24-+
| X O O | | O O O X |
| X O O | | O O O |
| O | | |
| | | |64
| X | | X |
| X X | | X X O X |
| O X X | | X X O X O |
+-12-11-10-9--8--7-+---+-6--5--4--3--2--1-+
X 5-away, O 3-away. X on roll. Cube action?
The original post generated a lively discussion which I believe is
instructive to players of intermediate strength and beyond. Much
appreciated. It was interesting that many contributors never ventured
an opinion on the doubling decision, but made several good points on
the factors going into it or mitigating against it.
My own thinking was based chiefly on the fact that there were a few
market losers, a -5,-3 score, and some gammon advantage in the
neighborhood of two or three times that of opponent’s and bearing in
mind that both the gammon and the recube vig were biased in my favor
(I would not double for money or at an even score). But I wanted to
quantify the advantage and the gammon advantage, and get some
bottom-line opinions.
As for quantification, Donald Kahn has kindly taken the time to
perform some JF rollouts (see below). As for opinions, I’ll cite below
excerpts from those who ventured one, leaving out the very good
follow-up discussions pertaining to issues of gammon factor and recube
vig by them and others.
Stu Katz:
Two features would deter me from doubling. O has an advanced anchor
that inhibits blitzing and priming. X still needs to escape through
the broken prime and O will certainly hit loose on 20 if possible to
prevent the escape and go for a full prime. The anchor on 4 allows X
to play aggressively since it limits gammon threat.
As O I would definitely take although it would be with some
discomfort. As X I would favor holding the cube because I think O will
win too often from this position, but it's tempting.
Brian Sheppard:
It seems quite a bit premature to me.
First, I think that trailing in the match is not yet a factor.
I say this because the double does not involve "wasted" points
for either side, and the redouble is largely free of wasted points
as well. (A point is "wasted" if you do not need it to win the
match.) So I am using money-game considerations here.
Kit Woolsey:
So, what does it all mean in the actual position. It looks like O has
a clear take now, even at the match score. However, let's look at a
favorable sequence for X (say X hits and O flunks). For money O would
perhaps have a very close pass. Thus for money it appears that a
double would be premature, since in the best variations X will only
lose his market by a very small amount. Not true at the match score!
Since as we have seen O's drop point is considerably higher than it
would be for money, if X gets that favorable sequence he will lose his
market by a considerable amount. Thus, the cost of not doubling and
being wrong is quite great at the match score, while it is very small
for money.
Putting it all together, it looks like X's double is very reasonable.
Is it theoretically correct? I have no idea -- my guess would be yes,
but it wouldn't surprise me at all if that proved to be wrong. In
practice I would definitely double, since there is always the chance
that my opponent might pass (in addition to maybe being the
theoretically correct action anyway).
Donald Kahn:
I did some rollouts of the position. At Level 5, the rollouts were
1296 times, all the way to the end of the game (not truncated). This
is the only kind of rollout that gives information on different cube
positions. I also made one Level 6, 144 games (which equates to
1900+) in order to compare with the Level 5.
The normal Level 5 rollout shows that cubeless, the favorite wins
60.3%, including 17.4 gammons. Other wins 39.7% with 7.9% gammons.
This is rated "No double" by JF's evaluation function. If doubled and
taken, the underdog now has 49.2% chances of winning, that is, when
his recube power is normal (as for money, when he is a winner if he
can get to 77.5% in the game.) At the score however, his power is
less than normal because the other player will take with as little as
15% chances. When I rolled it out with this cranked in, the taker won
46.1% rather than 49.2% as above. But this is still plenty, and is
indicative that maybe the original double is a little premature.
The Level 6 rollout, which is cubeless, gave 59.6 to 40.4 for the
favorite, which is very close to Level 5, and confirms the conclusions
above, right or wrong.
Me( Jim Cochrane):
Using these Level 5 numbers:
If X doubles:
X wins a gammon 17.4% to attain -1, -3 which is 75.8%.
X wins 2 (100 - 46.1 - 17.4) 36.5% to attain -3, -3 which is 50% (but
some of these win 4 because of redouble/take. Refinement "A" discussed
below.)
X loses a gammon 7.9% to end the match.
X loses 2 (46.1 - 7.9) 38.2% to attain -5, -1 which is 15.4% (but
sometimes this must include losing 4 and the match by accepting a
redouble. Refinement "B" discussed below.)
Ignoring the refinements, X wins the match 17.4 x .758 + 36.5 x .5 + 0
+ 38.2 x .154 = 37.3%
Taking into account the games where O redoubles and X takes means
estimating what fraction of the time O reaches a position on roll
where O’s chances are somewhere over 80% but less than 85% and gammon
is ruled out.
A. Suppose the redouble/take takes place in fraction f of the
continuations above where X wins a non-gammon. Then X wins 2 points
(1-f)*36.5% of the time and wins 4 points f*36.5% of the time. My
guess is that f is quite low (these are reversals of fortune for X
where gammon is ruled out and the tiny redoubling window cited above
is reached on O’s roll, and X wins). How about f=0.1? Then, instead of
36.5 x 0.5 (18.25%), this category yields for O: 32.85 x .5 + 3.65 x
.758 = 19.2%, gaining about 1%.
B. Suppose that the redouble/take takes place in fraction g of the
continuations above where X loses a non-gammon. Then X loses 2 points
(1-g)*38.2% of the time and loses 4 points (and match) g*38.2% of the
time. These are also reversals of fortune for X where the tiny
redoubling window is reached and gammon has been ruled out, but are in
the cases where JF says O wins. My guess is that the window is reached
a fairly good part of the time. Let’s be generous(?) and say g=0.4.
Then, instead of 38.2% x .154 (5.9%), X’s chance decline to 22.9% x
154 = 3.5%, a decline of 2.4%
These refinements cause a net decrease of 1.4% to yield X winning the
match 35.9%.
If X holds back:
X wins a gammon 17.4% to attain -3, -3 which is 50%
X wins 1 (60.3 - 17.4) 42.9% to attain -4, -3 which is 41.6%.
X loses a gammon 7.9% to attain -5, -1 which is 15.3%.
X loses 1 (100 - 60.3 - 7.9) 31.8% to attain -5, -2 which is 23.9%.
In this "no double" case , X wins the match 17.4 x 0.5 + 42.9 x 0.305
+ 7.9 x .153 + 31.8 x 0.239 = 35.4%
WOW!
These estimates say that it still a double (35.9% compared to 35.4%).
Yet the JF evaluation function said no double! It seems to me that the
jury is still out. Do my allowances for the fractions f and g unduly
favor the double decision? Is the JF evaluation function unreliable in
assessing the recube vig for O? Have I erred above?
I’d love to see an even more extensive JF analysis. But at this point
I am tempted to conclude that, theoretically, the double is a coin
flip, except for Kit Woolsey’s point that the opponent might pass, so
in practice, double. (In the actual game however, opponent severely
criticized the double and snatched the cube).
Thanks again to all who responded, and in advance for any new inputs.
Brian Sheppard
James H. Cochrane <code...@erols.com> wrote in article
<341ca5ce...@news.erols.com>...
> +13-14-15-16-17-18-+---+19-20-21-22-23-24-+
> | X O O | | O O O X |
> | X O O | | O O O |
> | O | | |
> | | | |64
> | X | | X |
> | X X | | X X O X |
> | O X X | | X X O X O |
> +-12-11-10-9--8--7-+---+-6--5--4--3--2--1-+
> X 5-away, O 3-away. X on roll. Cube action?
>
> + 38.2 x .154 = 37.3%
>
>
> These refinements cause a net decrease of 1.4% to yield X winning the
> match 35.9%.
>
> + 7.9 x .153 + 31.8 x 0.239 = 35.4%
The calculation you just did makes doubling look good
because the "hold back" case doesn't include the equity gained by
keeping access to the cube. In effect, you are comparing "double
now" versus "double never."
Your calculation shows that these two options are almost equal,
and I propose that this confirms that doubling is quite premature.
The basic problem with doubling is that X has few "market losers."
Sequences like hit-fan are less than 10%. Escaping the back man
followed by a missed shot is about 10%. Total: 20%. And note that
these are not resounding market-losers. In a money-game O could very
reasonably take those doubles. If O has to drop them in this match
situation, then it is only because O has little redoubling equity.
Please note that X has a few bad sequences, too. 5-2, 6-2, 4-5 leave
shots next turn. And O's 1-1, 2-2, 3-3, 6-5, 5-5, 6-6 start an attack
against X's back man. X's market-losing chances are not much greater
than the chance of these poor sequences.
From a technical perspective this double is bad. Kit says that X
should double as a practical matter because O might drop. But I
note that not one commentator in this newsgroup would actually
drop as O, so I have to wonder about that.
Brian
David Montgomery
In article <01bcc1ca$77b7a590$3ac032cf@polaris> "Brian Sheppard" <!bri...@mstone.com> writes:
>James H. Cochrane <code...@erols.com> wrote in article
><341ca5ce...@news.erols.com>...
>> | X O O | | O O O X |
>> | X O O | | O O O |
>> | O | | |
>> | | | |64
>> | X | | X |
>> | X X | | X X O X |
>> | O X X | | X X O X O |
>> +-12-11-10-9--8--7-+---+-6--5--4--3--2--1-+
>> X 5-away, O 3-away. X on roll. Cube action?
>
>should double as a practical matter because O might drop. But I
>note that not one commentator in this newsgroup would actually
>drop as O, so I have to wonder about that.
>
>Brian
I guess it's a small matter, but although I lean toward not
doubling, I think the jury is still out on this cube (based on
the evidence I've seen in the newsgroup) -- I wouldn't call
this double bad.
There are a lot of small factors that add up to a lot of
justification for the cube.
First, the rollout numbers were X wins 60.3%, wins G 17.4%,
loses G 7.9%. By my rough estimates, X is about 8-10% below
the winning chances he would need to cash the game at the score.
In a money game, usually you need to be within 7% (very roughly)
to offer an initial double, so from this it looks like you
are a little short, but not much.
Second, per Tom Keith's early/late ratio, its more important to
double-in here than for money, so perhaps being 8-10% away from
the cash point is sufficient.
Third, by doubling you take away the cube from O to some degree.
With a centered cube, O can make fairly normal initial doubles.
Owning a 2-cube, O can't use the cube very effectively.
Fourth, because of the match score, most of the market losing
sequences that Brian mentioned are big MLs, although they
would be small MLs for money.
Fifth, there are additional MLs. A hit with a 3-1 is usually
an ML even if O comes in. There is a decent chance of getting
an ML after a hit if O enters with an ace. I think many players
would not play a 5-1 from the roof for O bar/24 21/16, and any
other play loses X's market (per JF evaluation).
Sixth, there are many double next turn anyway (DNTA) sequences,
where you lose nothing by cubing, since the positions transpose.
Finally, although no one posted to the newsgroup that they would
pass, there is still some chance in practice. It's a lot different
at the table than when doing a post mortem. Also, the players
I have discussed this with have mostly liked the double when first
looking at the position, but shied away from it seeing the rollout
or evaluation. What this means is that perhaps many players would
tend to overrate X's position a bit, making a pass more probable.
I still lean to no double, but I don't think doubling can be
criticized much.
David Montgomery
monty on FIBS
mo...@cs.umd.edu
Chuck Bower
In article <341ca5ce...@news.erols.com>,
James H. Cochrane <mail.erols.com> wrote:
>
> +13-14-15-16-17-18-+---+19-20-21-22-23-24-+
> | X O O | | O O O X |
> | X O O | | O O O |
> | O | | |
> | | | |64
> | X | | X |
> | X X | | X X O X |
> | O X X | | X X O X O |
> +-12-11-10-9--8--7-+---+-6--5--4--3--2--1-+
>X 5-away, O 3-away. X on roll. Cube action?
(snip a whole bunch of interesting analysis)
>These estimates say that it still a double (35.9% compared to 35.4%).
>Yet the JF evaluation function said no double! It seems to me that the
>jury is still out. Do my allowances for the fractions f and g unduly
>favor the double decision? Is the JF evaluation function unreliable in
>assessing the recube vig for O? Have I erred above?
>
>I'd love to see an even more extensive JF analysis. But at this point
>I am tempted to conclude that, theoretically, the double is a coin
>flip, except for Kit Woolsey's point that the opponent might pass, so
>in practice, double. (In the actual game however, opponent severely
>criticized the double and snatched the cube).
I didn't quite follow all of James's calculation, but it looked
reasonable. However, there are a few things which one of us doesn't
understand.
1) I'm pretty sure that when JF recommended "no double" it was
evaluating as if a $ game. This agrees with virtually everyone else.
2) I believe there is a problem with the way you interpreted Donald
Kahn's rollout results. I explain that in some details in what
follows:
When using level-5 limited cube rollouts, you are allowed to set a
"settlement limit". With rare exceptions, when a player with access
to the cube is on roll and has cubeless equity greater than this
(in the eyes of JF level-5) then the game is cashed.
For money play, it is usually (but not always) a reasonable
assumption that both players have the same settlement limit, and that
0.55 is the actual limit which should be set. When one player or
the other has a large gammon fraction (say more than 1/3) then 0.55
is too low. (I'm talking generalities here, so please pardon me
for not giving more specifics.)
At match play, as most players know, cube handling can be much
different than at money play as the end of the match is approached.
In particular, the players will have DIFFERENT CASH POINTS. One
settlement limit does not work for BOTH players.
In deciding whether it is proper to cube, you need to compare
two scenarios: cube is turned this time, and cube is NOT turned
this time. WHEN A PLAYER OWNS THE CUBE, THEN SETTLEMENT LIMIT ONLY
APPLIES TO HIM/HER, since the other player can't use the cube. So
here, with careful selection of settlement limit, level-5 cubeless
can be used (but you need two different rollouts, each with its
proper settlement limit).
The real problem comes when the cube is centered. Whose
settlement limit do you use? If they are different, you're outa luck.
That is the case in the above problem. Primarily because of the
match score, but also because of the gammon fractions, O (match leader)
must drop "earlier" than X. Thus they have different settlement limits.
I also have looked into this problem. Here is what I conclude,
based on my own JFv3.0 level-6 rollout results which I list now:
total g + bg's bg's
X wins 57.6 16.6 0.6
O wins 42.4 8.2 0.4
I don't remember how many level-6 rollouts I did, but the "equivalent"
number of level-5 rollouts is around 11,600. The cubeless money
equity for X is +0.238 with a standard deviation of 0.012. Using
Rick Janowski's semi-continuous model for money play, the statistical
significance for the following decisions is:
beaver? NO, at the 7 s.d. level.
double? NO, at the 17 s.d. level.
take? YES, at the 27 s.d. level.
O's take point for these gammon fractions (and money play) is 0.59
in equity units, and 28% (cubeless) game winning chances. Again this
is from the above rollout and Rick's cube model.
I doubt that any of these qualitative results surprises anyone!
But now let's dig a bit deeper and address the question which Brian
and Kit argued about: is the cube handling different here than for
money? The answer is "YES". Without listing my calculations, I
give X's doubling windows for money play and this match score WITH
THE ABOVE GAMMON FRACTIONS INCLUDED:
money play: 50% < W < 72%
this match score: 45% < W < 66%
Note that the windows have about the same size (22% vs. 21%) but
that the match trailer's window is shifted downward, making for
earlier doubles (and earlier passes by match leader).
Looks like "TAKE" is right, but what about the double? Well
into the window, yes, but you still have to look at the market
losers.
Using JF level-7 evaluation, I conclude the following: hitting
is not a big market loser. In fact, X will have a rather efficient
double next time EVEN IF O FANS! Hitting argues for holding onto
the cube.
The real market losers are X's 3's. If X rolls ANY 3 and is not
hit back, JF says that X's winning chances will then be in the mid
to high 70's (%). Roughly that's 11/36 * 2/3 or about 20% of the time
X will have a substantial market loss. Is that enough? I don't know.
I believe (but don't quote me, it's just hearsay and my poor memory)
that Kleinman and Ortega use a canonical number in the 25-30% range
for market losers in order to double. If that's true, then by their
rule of thumb, this may not be a double, but it is CLOSE! And my
20% number was just ballpark.
I did one other check. I looked at the match equities under the
following simplified cases:
a) X doubles now, and O takes but then NEVER redoubles. (This result
is biased in X's favor a bit, since O can use the cube.)
b) X holds now and the game is played until one player or the other
has A PERFECTLY EFFICIENT CASH. (This simplification is known as
the "Continuous Cube" model. I believe this case also favors O
a bit, since s/he is more likely to reach a cash than X, and thus
get's to use the efficient cube more often.)
Under these conditions, X wins the match bwtween 39% and 40%, regardless
of whether s/he doubles this turn or NOT!
My conclusion is that given the current state of backgammon analysis,
this problem is just too close to the double/no-double line. Technically
it looks like a coin flip. BUT, it only takes a small chance that O will
drop to make turning the cube correct in the practical sense. And all of
this is EXACTLY what James concluded above!
Chuck
bo...@bigbang.astro.indiana.edu
c_ray on FIBS
Tom Keith
Chuck Bower wrote:
> ...
> But now let's dig a bit deeper and address the question which Brian
> and Kit argued about: is the cube handling different here than for
> money? The answer is "YES". Without listing my calculations, I
> give X's doubling windows for money play and this match score WITH
> THE ABOVE GAMMON FRACTIONS INCLUDED:
>
> money play: 50% < W < 72%
> this match score: 45% < W < 66%
Lest this debate be drawing to a resolution :-),
let me inject a point I haven't seen anyone else make.
Leader's drop point at 3-away, 5-away is quite sensitive
to the match equity you assume for 3-away, 4-away.
The higher the value of Trailer's equity at 3-away, 4-away,
the more reluctant Leader will be to give up one point when
the score is 3-away, 5-away.
Kit Woolsey's table lists Trailer's 3-away, 4-away match
equity at 41%. But there are other tables (e.g., Norman
Zadeh's) which place the value for this score as high as 43%.
Each one point increase in the assumed value of Trailer's
equity at 3-away, 4-away produces a two or three point rise
in Leader's drop point at 3-away, 5-away.
So, as much as anything else, the position of Leader's drop
point at 3-away, 5-away depends on which match equity table
you use.
Tom
Kit Woolsey
Tom Keith (t...@bkgm.com) wrote:
: Lest this debate be drawing to a resolution :-),
: let me inject a point I haven't seen anyone else make.
: Leader's drop point at 3-away, 5-away is quite sensitive
: to the match equity you assume for 3-away, 4-away.
: The higher the value of Trailer's equity at 3-away, 4-away,
: the more reluctant Leader will be to give up one point when
: the score is 3-away, 5-away.
: Kit Woolsey's table lists Trailer's 3-away, 4-away match
: equity at 41%. But there are other tables (e.g., Norman
: Zadeh's) which place the value for this score as high as 43%.
: Each one point increase in the assumed value of Trailer's
: equity at 3-away, 4-away produces a two or three point rise
: in Leader's drop point at 3-away, 5-away.
: So, as much as anything else, the position of Leader's drop
: point at 3-away, 5-away depends on which match equity table
: you use.
: Tom
Good point, Tom. Let's see what this means if we use Zadeh's 43%. I
don't know what his figure is for 1 away, 5 away Crawford, so let's
assume it is the same as mine (85% for the leader). Now, let's look at
the leader's drop point, assuming no gammons:
Leader passes: He is ahead 4 away, 3 away for 57% equity.
Leader takes and wins: He is ahead 1 away, 5 away for 85% equity.
Leader takes and loses: He is even, for 50% equity.
Thus he would be risking 7% to gain 28%, so he would be getting 4 to 1
odds on his take. Therefore, he could take with 20% winning chances,
ignoring possible recube vig.
This just doesn't feel right, does it? All our intuition tells us that
the leader should be more cautious in taking the cube than he should be
in money (except possibly where the double puts him out exactly), but
these figures say he can take positions which are clear money passes. To
me, this indicates that Zadeh's 43% figure has to be off.
When I constructed my match equity table, one of the things I did was to
introduce smoothing effects so anomalies such as the one above would be
avoided. Thus I believe my table will lead to more practical results in
one's match equity calculations.
Kit
Tom Keith
I wrote:
: Leader's drop point at 3-away, 5-away is quite sensitive
: to the match equity you assume for 3-away, 4-away.
: The higher the value of Trailer's equity at 3-away, 4-away,
: the more reluctant Leader will be to give up one point when
: the score is 3-away, 5-away.
:
: Kit Woolsey's table lists Trailer's 3-away, 4-away match
: equity at 41%. But there are other tables (e.g., Norman
: Zadeh's) which place the value for this score as high as 43%.
: Each one point increase in the assumed value of Trailer's
: equity at 3-away, 4-away produces a two or three point rise
: in Leader's drop point at 3-away, 5-away.
:
: So, as much as anything else, the position of Leader's drop
: point at 3-away, 5-away depends on which match equity table
: you use.
Kit Woolsey wrote:
> Good point, Tom. Let's see what this means if we use Zadeh's 43%. I
> don't know what his figure is for 1 away, 5 away Crawford, so let's
> assume it is the same as mine (85% for the leader). Now, let's look
at
> the leader's drop point, assuming no gammons:
>
> Leader passes: He is ahead 4 away, 3 away for 57% equity.
> Leader takes and wins: He is ahead 1 away, 5 away for 85% equity.
> Leader takes and loses: He is even, for 50% equity.
>
> Thus he would be risking 7% to gain 28%, so he would be getting 4 to
1
> odds on his take. Therefore, he could take with 20% winning chances,
> ignoring possible recube vig.
>
> This just doesn't feel right, does it? All our intuition tells us
that
> the leader should be more cautious in taking the cube than he should
be
> in money (except possibly where the double puts him out exactly), but
> these figures say he can take positions which are clear money
passes. To
> me, this indicates that Zadeh's 43% figure has to be off.
Maybe not. There is an "odd-even" effect on match scores that can
produce surprising results. Since points in backgammon often come
in twos and fours, the opponent's match-winning chances take a big
step forward when he goes from 5-away to 4-away. At 4-away he can
win the match with one doubled gammon or two 2-point games.
Because a one-point loss is so expensive when the opponent is 5-away,
and a two-point loss not that much worse, the leader is more willing
than usual to try turning the game around rather than give up exactly
one point.
> When I constructed my match equity table, one of the things I did was to
> introduce smoothing effects so anomalies such as the one above would be
> avoided. Thus I believe my table will lead to more practical results in
> one's match equity calculations.
Here's an interesting question: Which is better, leading a match at
at 3-away/4-away or leading at 4-away/5-away?
Tom
bob koca
bo...@bigbang.astro.indiana.edu (Chuck Bower) wrote:
> The real market losers are X's 3's. If X rolls ANY 3 and is not
>hit back, JF says that X's winning chances will then be in the mid
>to high 70's (%). Roughly that's 11/36 * 2/3 or about 20% of the time
>X will have a substantial market loss. Is that enough? I don't know.
>I believe (but don't quote me, it's just hearsay and my poor memory)
>that Kleinman and Ortega use a canonical number in the 25-30% range
>for market losers in order to double. If that's true, then by their
>rule of thumb, this may not be a double, but it is CLOSE! And my
>20% number was just ballpark.
I think the 25-30% range rule is too simplified to be of much use.
For one thing the size of market loss is important. 1 sequence
which loses market by 20% for instance has as much influence
as 5 sequences each of which lose it by 4%. Secondly how close
one is to the cash point is important, if it is close to a cash
already then very few market losers are needed to warrant the
double.
,Bob Koca
bobk on FIBS
James H. Cochrane
On 19 Sep 1997 23:03:49 GMT, bo...@bigbang.astro.indiana.edu (Chuck
Bower) wrote:
>In article <341ca5ce...@news.erols.com>,
>James H. Cochrane <mail.erols.com> wrote:
>>
>> +13-14-15-16-17-18-+---+19-20-21-22-23-24-+
>> | X O O | | O O O X |
>> | X O O | | O O O |
>> | O | | |
>> | | | |64
>> | X | | X |
>> | X X | | X X O X |
>> | O X X | | X X O X O |
>> +-12-11-10-9--8--7-+---+-6--5--4--3--2--1-+
>>X 5-away, O 3-away. X on roll. Cube action?
>
Thank you Chuck, and again to all who commented. I learned from this
problem and the analysis by all. My error in feeling that the double
was clled for at the time is in over-valuing the hit with a 1.
Although not as important as the safe 3, I felt that many hit
sequences would be market losers (generating more chances to roll
3-out or heightening the gammon), but apparently this is not the case.
So while I thought it was definitely a double and my opponent thought
it was a "bad" double, the answer lies in between: a borderline OK
double.