To double, or not to double...
Phill Skelton
Here are some positions from a recent game I played on FIBS:
1 2 3 4 5 6 7 8 9 10 11 12
+-----------------------------------------+
| O X O | O | O O X |
| X O | | O O X |
| O | | O O |
| O | | |
| | | |
| | | O |
| X X X | | X X O |
| X X X X | | X X O |
+-----------------------------------------+
24 23 22 21 20 19 18 17 16 15 14 13
Cube=1 (centered) O to move
score is 3 away : 3 away
O doubled. Is this correct? I thought it was a trivially easy take,
implying that the double was probably wrong. Even if O hits from the
bar and covers the blot on 4, O doesn't seem to be in a commanding
position.
Next interesting bit:
1 2 3 4 5 6 7 8 9 10 11 12
+-----------------------------------------+
| O O X O | | O O X |
| O O X O | | O O X |
| O | | |
| | | |
| X X X | | |
| O X X X X | | O |
| O X X X X | | O |
+-----------------------------------------+
24 23 22 21 20 19 18 17 16 15 14 13
Cube=2 (owned by X) X to move
score is 3 away : 3 away
Is this position worth a redouble? In retrospect I am not sure.
In the game O had had a man on the bar and I had set up a nice attack,
when O re-entered to make the anchor. Figuring my chances of a gammon
were gone, I decided to try and double my opponent out. He took.
Last position:
1 2 3 4 5 6 7 8 9 10 11 12
+-----------------------------------------+
| O O O X O O | | O |
| O O O X O O | | O |
| O O | | O |
| | | |
| X | | |
| X X | | |
| X X X X X | | |
| X X X X X | | |
+-----------------------------------------+
24 23 22 21 20 19 18 17 16 15 14 13
same game cube=4 (ie cube is dead).
X to play 4:4
pipcounts: X: 85 O: 72
Simple question: should X play 4-8(2) 19-23(2) to try and force a hit,
or play 4-12(2) and turn it into a running game? (I ran and won the
race by virtue of throwing 4:4, 6:6 and 1:1 in my next few throws.
Any comments / suggestions on these welome.
Phill
Peter Bell
In article <32428717...@sun.leeds.ac.uk>, ph...@sun.leeds.ac.uk
(Phill Skelton) wrote:
< Here are some positions from a recent game I played on FIBS:
<
< 1 2 3 4 5 6 7 8 9 10 11 12
< +-----------------------------------------+
< | O X O | O | O O X |
< | X O | | O O X |
< | O | | O O |
< | O | | |
< | | | |
< | | | O |
< | X X X | | X X O |
< | X X X X | | X X O |
< +-----------------------------------------+
< 24 23 22 21 20 19 18 17 16 15 14 13
<
< Cube=1 (centered) O to move
< score is 3 away : 3 away
<
< O doubled. Is this correct? I thought it was a trivially easy take,
< implying that the double was probably wrong. Even if O hits from the
< bar and covers the blot on 4, O doesn't seem to be in a commanding
< position.
<
Instructive positions. I'm a bit rusty, but I'll give these a try.
In my opinion, the double is very, very wrong. I would beaver instantly
for money. The simple analysis: after hit/no return hit, O is clearly
ahead. After hit/return hit, things are still up in the air. After no
hit, X is clearly ahead. There are many more "no hits" than "hit/no
return hits", therefore X is ahead, therefore O has no business doubling.
It's interesting to think about why O would double. The most favorable
variations for O involve hit-cover followed by dance. After this
variation, O will have an excellent double. Let's assume no gammons
throughout. Then, the take point is 70%, quite a bit lower than the 75%
of money play. Therefore, *in this variation*, I would drop such a double
if I were X. In other words, such variations are "market losers" for O.
Sometimes beginner/intermediate players are afraid to "lose their
market". If there even a few market losers in a position, such players
will double. There are two major problems with this approach:
- A "small" market loser (one that barely raises your winning chances
above your take point) should not encourage you to double: be happy to
cash afterwards if such an exchange of rolls occurs. Only "big" market
losers (those that raise your winning chances to near 100%) should
encourage you to double.
- You need many big market losers to double, and the further away your
winning chances are from the take point, the more big market losers you
need.
In this situation, O has a tiny number of small market losers, a very bad
time to double, in general.
< Next interesting bit:
<
< 1 2 3 4 5 6 7 8 9 10 11 12
< +-----------------------------------------+
< | O O X O | | O O X |
< | O O X O | | O O X |
< | O | | |
< | | | |
< | X X X | | |
< | O X X X X | | O |
< | O X X X X | | O |
< +-----------------------------------------+
< 24 23 22 21 20 19 18 17 16 15 14 13
<
< Cube=2 (owned by X) X to move
< score is 3 away : 3 away
<
< Is this position worth a redouble? In retrospect I am not sure.
< In the game O had had a man on the bar and I had set up a nice attack,
< when O re-entered to make the anchor. Figuring my chances of a gammon
< were gone, I decided to try and double my opponent out. He took.
<
The take point here is 75%, just like in money play.
First, if you're not sure whether this is a take or drop, you should
double, by Woolsey's Rule. If a double was "objectively" correct, you are
fine. If a double was "objectively" incorrect, you still may gain if your
opponent drops. You only lose if the double was incorrect and your
opponent takes. (One caveat: be careful with Woolsey's Rule if you are
quite uncertain about a position and you are playing a better player.
Otherwise, use it religiously.) Note that double-take is quite often the
correct cube action: there's generally no need to beat yourself up if your
opponent takes. Also, you should be very happy if you think it's a drop
and your opponent takes, as in this position!
As O, if doubled, I would drop. X's position is a little ahead of itself,
so he could run out of time, and he has to roll first. On the other hand,
I'm not very excited about my home-board configuration: even if X runs out
of time first and I hit a blot, I will have trouble finishing my attack.
Meanwhile, a hit by X will probably devastate me, since he has such a
strong home board.
< Last position:
<
< 1 2 3 4 5 6 7 8 9 10 11 12
< +-----------------------------------------+
< | O O O X O O | | O |
< | O O O X O O | | O |
< | O O | | O |
< | | | |
< | X | | |
< | X X | | |
< | X X X X X | | |
< | X X X X X | | |
< +-----------------------------------------+
< 24 23 22 21 20 19 18 17 16 15 14 13
<
< same game cube=4 (ie cube is dead).
<
< X to play 4:4
<
< pipcounts: X: 85 O: 72
<
< Simple question: should X play 4-8(2) 19-23(2) to try and force a hit,
< or play 4-12(2) and turn it into a running game? (I ran and won the
< race by virtue of throwing 4:4, 6:6 and 1:1 in my next few throws.
<
< Any comments / suggestions on these welome.
<
< Phill
Waiting seems inferior to me. X would already have to waste pips moving
in his homeboard. Then, no roll by O forces a shot. Then, on X's next
roll, he will either have to waste more pips or be forced to run anyway in
a less advantageous position.
Thanks,
Peter Bell (USRobots)
Kit Woolsey
Peter Bell (pb...@aimnet.com) wrote:
: First, if you're not sure whether this is a take or drop, you should
: double, by Woolsey's Rule. If a double was "objectively" correct, you are
: fine. If a double was "objectively" incorrect, you still may gain if your
: opponent drops. You only lose if the double was incorrect and your
: opponent takes. (One caveat: be careful with Woolsey's Rule if you are
: quite uncertain about a position and you are playing a better player.
: Otherwise, use it religiously.) Note that double-take is quite often the
: correct cube action: there's generally no need to beat yourself up if your
: opponent takes. Also, you should be very happy if you think it's a drop
: and your opponent takes, as in this position!
I do not agree that the quality of the opponent should make a
difference. It is perhaps more important to follow the rule if you are
playing a better player. Failure to do so is likely to lead to missed
doubling opportunities.
For those of you who aren't familiar with my rule, it is as follows: If
you aren't ABSOLUTELY sure whether the position is a take or a pass, then
it is ALWAYS correct to double. I'm not kidding! I follow this rule
religiously in actual play, and any player who does likewise will see his
results improve tremendously.
Note that the converse is not necessarily true -- you may be absolutely
sure it is a take and yet it is still correct to double, provided it is a
very volatile position.
Kit
Peter Bell
In article <kwoolseyD...@netcom.com>, kwoo...@netcom.com (Kit
Woolsey) wrote:
< I do not agree that the quality of the opponent should make a
< difference. It is perhaps more important to follow the rule if you are
< playing a better player. Failure to do so is likely to lead to missed
< doubling opportunities.
<
< For those of you who aren't familiar with my rule, it is as follows: If
< you aren't ABSOLUTELY sure whether the position is a take or a pass, then
< it is ALWAYS correct to double. I'm not kidding! I follow this rule
< religiously in actual play, and any player who does likewise will see his
< results improve tremendously.
<
< Note that the converse is not necessarily true -- you may be absolutely
< sure it is a take and yet it is still correct to double, provided it is a
< very volatile position.
<
< Kit
Hi Kit,
We've had this discussion several times in person. I think we've agreed
to disagree at this point! I'd like to state my reasoning, however...
As a beginner and low intermediate player, if I got an advantage in a
game, I was *never* ABSOLUTELY sure whether a given position was a take or
a pass! Therefore, by following your rule, I would have been doubling
like a madman. Since early doubles are big mistakes, my results would
have been horrible, and I would hardly be developing the refined judgement
that I pretend to have today.
In more mathematical terms, my uncertainty in judging my winning chances
might often be on the order of 15-20%! In other words, I might say, "I'm
pretty sure I'm ahead, but I'm pretty sure I'm not close to 75% yet.
However, I COULD see how I might be behind, but I COULD see how it might
be a drop." I'm not kidding about this -- I distinctly remember thinking
this way many times in my early days on FIBS.
If my opponent is the better player, she will have a smaller uncertainty
in her evaluation. I am therefore unlikely to induce a mistake. The best
I can hope for is that the double is actually correct. The most likely
outcome, however, is that my double is premature, and it gets taken. Not
good odds.
In fact, even if my opponent is at my skill level (in other words, is just
as confused as me about the position), the chance of inducing a mistake is
still small, while the chance of early double/take is quite large.
Therefore, Bell's Corollary to Woolsey's Rule says, in essence,
"If you think there is a good chance that your double is premature, do
not follow Woolsey's Rule."
In other words, Woolsey's Rule is most valuable when you are sitting on
the fence about doubling, then you look at the position and say, "Could my
opponent reasonably drop this?" If the answer is "yes", then double.
However, I recommend that you do not go out of the way to apply Woolsey's
rule when you are fairly sure that a double would be premature.
The only real way to settle this is a full mathematical treatment, which I
might whip up someday. Meanwhile, I hope my reasoning makes at least
some sense :-)
Thanks,
Peter Bell (USRobots)
Julian
In article <32428717...@sun.leeds.ac.uk>
ph...@sun.leeds.ac.uk "Phill Skelton" writes:
(position 1)
> O doubled. Is this correct? I thought it was a trivially easy take,
> implying that the double was probably wrong. Even if O hits from the
> bar and covers the blot on 4, O doesn't seem to be in a commanding
> position.
Agreed. It looks to me that if O doesn't hit from the bar then he'll
be on the receiving end of Mr. Cube.
> Next interesting bit:
(position 2)
> Is this position worth a redouble? In retrospect I am not sure.
I think not. If you could safely clear your opponent's 5, then you
would be OK for a double, however this is very tricky. He already
has a 3 point board, and plenty of men hanging around waiting to
attack you.
> Last position:
>
> Simple question: should X play 4-8(2) 19-23(2) to try and force a hit,
> or play 4-12(2) and turn it into a running game? (I ran and won the
> race by virtue of throwing 4:4, 6:6 and 1:1 in my next few throws.
I think you run. The problem with trying to force a hit is that O has
a spare man in the outfield; you have none and also can't play 5's or
6's in your home board. If anyone is going to get forced to leave a
game-winning shot, it is you. If you run, your opponent is 3 pips behind
on roll in a straight race, which is virtually even.
-------------------------------------------------------------------------
Julian Hayward 'Booles' on FIBS jul...@ratbag.demon.co.uk
+44-1344-640656 http://www.ratbag.demon.co.uk/
-------------------------------------------------------------------------
"A week is a long time in politics, and three weeks is twice as long."
- Rosie Barnes MP
-------------------------------------------------------------------------
Craig Connell
kwoo...@netcom.com (Kit Woolsey) wrote:
>Peter Bell (pb...@aimnet.com) wrote:
>: First, if you're not sure whether this is a take or drop, you should
>: double, by Woolsey's Rule. If a double was "objectively" correct, you are
>: fine. If a double was "objectively" incorrect, you still may gain if your
>: opponent drops. You only lose if the double was incorrect and your
>: opponent takes. (One caveat: be careful with Woolsey's Rule if you are
>: quite uncertain about a position and you are playing a better player.
>: Otherwise, use it religiously.) Note that double-take is quite often the
>: correct cube action: there's generally no need to beat yourself up if your
>: opponent takes. Also, you should be very happy if you think it's a drop
>: and your opponent takes, as in this position!
>I do not agree that the quality of the opponent should make a
>difference. It is perhaps more important to follow the rule if you are
>playing a better player. Failure to do so is likely to lead to missed
>doubling opportunities.
>For those of you who aren't familiar with my rule, it is as follows: If
>you aren't ABSOLUTELY sure whether the position is a take or a pass, then
>it is ALWAYS correct to double. I'm not kidding! I follow this rule
>religiously in actual play, and any player who does likewise will see his
>results improve tremendously.
>Note that the converse is not necessarily true -- you may be absolutely
>sure it is a take and yet it is still correct to double, provided it is a
>very volatile position.
>Kit
Could you explain again why it is proper to double in a volatile
situtation. What I don't understand is that if the situation is very
volatile, that probably means the next few rolls will drastically
effect the outcome of the game. Given that, your winning chances
cannot be extremely high (if your opponent is lucky on the next few
rolls your gonna lose). Assuming the match is tied at say 3 away, 3
away (my market window table says the perfect double is at 72% at that
point) why is it proper to double?
Kit Woolsey
Craig Connell (con...@alpha.fdu.edu) wrote:
: Could you explain again why it is proper to double in a volatile
: situation. What I don't understand is that if the situation is very
: volatile, that probably means the next few rolls will drastically
: effect the outcome of the game. Given that, your winning chances
: cannot be extremely high (if your opponent is lucky on the next few
: rolls your gonna lose). Assuming the match is tied at say 3 away, 3
: away (my market window table says the perfect double is at 72% at that
: point) why is it proper to double?
If you think about it, what really happens when you turn the cube is that
you are doubling the stakes you are playing for. Given that, it is
potentially correct to double any time you are the favorite. The only
reason not to double when you are the favorite is if it is likely that
after the next exchange (you roll, he rolls) if things go well for you he
will still have a take.
Let's suppose you are playing a game of backgammon, you are a small
favorite, and you are on roll. Suddenly God comes down and says: I'm
changing the rules this game -- you may double now if you wish, but if
you don't double now I'm removing the doubling cube from the game. What
should you do? Clearly you should double -- you are doubling the stakes
when you are the favorite and this is your last opportunity to do so.
Back to reality. Let's look at the classic position where you have two
men left, one on the five point and one on the two point, and your
opponent has one on the ace point. You probability of winning is 19/36.
Since this is the last roll of the game it is essentially like the
situation described above where you must double now or never, so it is
clearly correct to double now. Note that the volatility could not be any
higher -- everything is riding on the next roll.
Getting back to more complex types of positions, let's suppose you have
some position where you are 65% to win. However this 65% comes out as
follows: After the next exchange of rolls, half the time you are 85% and
the other half you are 45%. Very volatile, that's for sure. Should you
double? Absolutely! If things go well it is very costly for you not to
have doubled -- your opponent will now pass if you double, and since you
are 85% to win this is a very correct pass -- you will have lost
considerable equity. If things go badly you are 45% to win -- so you
have doubled the stakes as a small underdog, not a big deal.
Consequently, the cost of not doubling and being wrong is greater than
the cost of doubling and being wrong.
The above example is artificial, of course -- I doubt if one can come up
with a position which meets these criteria. It is possible to get close,
however, with normal positions. Take the following position, for example:
13 14 15 16 17 18 19 20 21 22 23 24
+------------------------------------------+
| | | X X X X |
| | | X X X X |
| | | X X X |
| | | X |
| | | |
| | | |
| | | |
| | | O |
| | | O O O O O O |
| O O X | | O O O O O O |
+------------------------------------------+
12 11 10 9 8 7 6 5 4 3 2 1
O is on roll. X has two men off. O has 20 numbers out of 36 which hit.
If O hits he is a huge favorite, perhaps 85-90% to win. If O misses it
is a close race with O appearing to be a small underdog. Thus this
position roughly meets the criteria I gave for my artificial example.
Even though O is nowhere near X's drop point, the position is so volatile
that it is correct for O to double.
In theory it would be nice to be able to double right at the opponent's
minimum take point. In practice, backgammon just isn't that way. It is
often too volatile. If there is a significant chance that you may shoot
way beyond your opponent's minimum take point by a lot on the next
exchange of rolls, then it is important to double now even if your
advantage isn't too great. The cost of losing your market by a lot is
very expensive. That is why very volatile positions (that is, positions
where the equity is likely to swing a lot on the next exchange of rolls)
call for early doubles.
Kit
William C. Bitting
Kit Woolsey (kwoo...@netcom.com) wrote:
: Craig Connell (con...@alpha.fdu.edu) wrote:
: : Could you explain again why it is proper to double in a volatile
: : situation. What I don't understand is that if the situation is very
: If you think about it, what really happens when you turn the cube is that
: potentially correct to double any time you are the favorite. The only
: reason not to double when you are the favorite is if it is likely that
: after the next exchange (you roll, he rolls) if things go well for you he
: will still have a take.
: Getting back to more complex types of positions, let's suppose you have
: follows: After the next exchange of rolls, half the time you are 85% and
: the other half you are 45%. Very volatile, that's for sure. Should you
: double? Absolutely! If things go well it is very costly for you not to
: have doubled -- your opponent will now pass if you double, and since you
: are 85% to win this is a very correct pass -- you will have lost
: considerable equity. If things go badly you are 45% to win -- so you
: have doubled the stakes as a small underdog, not a big deal.
: Consequently, the cost of not doubling and being wrong is greater than
: the cost of doubling and being wrong.
: The above example is artificial, of course -- I doubt if one can come up
: with a position which meets these criteria. It is possible to get close,
[cut]
: In theory it would be nice to be able to double right at the opponent's
: minimum take point. In practice, backgammon just isn't that way. It is
: often too volatile. If there is a significant chance that you may shoot
: way beyond your opponent's minimum take point by a lot on the next
: exchange of rolls, then it is important to double now even if your
: advantage isn't too great. The cost of losing your market by a lot is
: very expensive. That is why very volatile positions (that is, positions
: where the equity is likely to swing a lot on the next exchange of rolls)
: call for early doubles. : Kit
Speaking of volatility! Pips: O = 131, X = 110. O on roll. Double?
| . . . . X . | | X X X O X . | X 3-away
| X | | X X X O X Not sure this was a double, X took
| | | 5 O | O 41 (22-18 6-5)
| |BAR| | X 65 (8-2 6-1!! ..forced)
| | | 5 O | sure looks like a giant
| O | | O O O X | market loser! O gammoned. wcb
| . . . . . O | | O O . O . X | O 4-away
Phill Skelton
Peter Bell wrote:
>
> In article <32428717...@sun.leeds.ac.uk>, ph...@sun.leeds.ac.uk
> (Phill Skelton) wrote:
>
> < Here are some positions from a recent game I played on FIBS:
> <
> < 1 2 3 4 5 6 7 8 9 10 11 12
> < +-----------------------------------------+
> < | X O | | O O X |
> < | O | | O O |
> < | O | | |
> < | | | |
> < | | | O |
> < | X X X | | X X O |
> < | X X X X | | X X O |
> < +-----------------------------------------+
> < 24 23 22 21 20 19 18 17 16 15 14 13
> <
> < Cube=1 (centered) O to move
> < score is 3 away : 3 away
> <
> < implying that the double was probably wrong. Even if O hits from the
> < bar and covers the blot on 4, O doesn't seem to be in a commanding
> < position.
> <
>
>
> In my opinion, the double is very, very wrong. I would beaver
> instantly for money. The simple analysis: after hit/no return hit,
> O is clearly ahead. After hit/return hit, things are still up in
> the air. After no hit, X is clearly ahead. There are many more "no
> hits" than "hit/no return hits", therefore X is ahead, therefore O
> has no business doubling.
I'm glad to find that I am not going mad. In the game I had been
considering doubling (as X), but decided that I wasn't quite far
enough ahead yet. Then O doubled me! After I posted this I went home
and had a stab at estimating the equity, and I made it +0.25 to X,
making it a beaver if possible, as you said.
Thanks for responding. I'm glad to find out I do know what I am doing
sometimes.
Andy Latto
Getting back to more complex types of positions, let's suppose you have
some position where you are 65% to win. However this 65% comes out as
follows: After the next exchange of rolls, half the time you are 85% and
the other half you are 45%. Very volatile, that's for sure. Should you
double? Absolutely! If things go well it is very costly for you not to
have doubled -- your opponent will now pass if you double, and since you
are 85% to win this is a very correct pass -- you will have lost
considerable equity. If things go badly you are 45% to win -- so you
have doubled the stakes as a small underdog, not a big deal.
Consequently, the cost of not doubling and being wrong is greater than
the cost of doubling and being wrong.
I think it's not so clear, once you factor in the
possibility of future doubles by both yourself and your opponent.
You don't say whether the 85% and 45% are cubeless numbers, numbers
assuming you hold the cube, or numbers assuming your opponent holds
the cube. Let's assume that these are cubeless winning percentages (CWP).
You don't mention whether the cube is currently in the center, or
on your side, though your use of "double" rather than "redouble" suggests
that the cube is currently in the center. But let's assume for the
moment that you hold a 2-cube, and are considering redoubling.
The naive analysis goes something like this.
If you double, then half the time you are 85% to win, and
half the time you are 45% to win. So your expectation is
1/2(.85(4) + .15(-4)) + 1/2(.45(4) + .55(-4)) = 1.4 - .2 = 1.2
(or more simply, 65% of the time you win 4, and 35% of the time you
lose 4, so your expectation is .65(4) + .35(-4) = 1.2)
If you don't double, then you will double next roll if you have an
85% chance to win, and your opponent will drop. If you have a 45% chance
to win, you will not double, of course, but you will still win 2 points
45% of the time. So your expectation will be
1/2(2) + 1/2(.45(2) + .55(-2)) = 1 - .1 = .9
So doubling is better than waiting.
But while this analysis includes the value of the cube on this roll
and the next (because it includes the fact that if you don't double,
and have an 85% CWP next roll, you will then double your opponent out,
increasing your chance to win from 85% to 100%), it ignores completely
the value of the cube on future rolls. If you roll badly this roll,
you will have a cubeless winning probability of 45%. But your actual
chances of winning the game after this are very different if you hold
the cube and if your opponent holds the cube.
Assuming the position after this roll is one with low volatility, so
that you and your opponent will both have the opportunity to make
efficient doubles in the future, I think you're better off waiting.
If you double, then you have a CWP of 65%. But with your opponent
holding the cube, this doesn't give you a 65% chance of winning the
game. Assuming future doubles on both sides are efficient, your
opponent will be able to double you out if your CWP drops below 20%,
while you have to actually win the game, that is, bring your CWP all
the way up to 100%. So assuming your opponent will have a chance to
make an efficient double in any game in which he would ultimately win,
the continuous model gives you a 7/12 chance to win the game, and a 5/12
chance to be doubled out. So your expectation is
7/12(4) + 5/12(-4) = 2/3
If you don't double, then you will double next roll if you have an 85%
chance to win, and your opponent will drop. If you have a 45% CWP, you
will not double, of course, but holding the cube with a 45% CWP, you
will still win 9/16 of the time, while your opponent will win 7/16 of
the time. You're a favorite, even with a 45% CWP, since it's a longer
distance from 45% to 0%, where your opponent wins, than it is
to 80%, where you can double him out.
So if you don't double, your total expectation will be:
1/2(2) + 1/2(9/16(2) + 7/16(-2)) = 9/8
So if you hold the cube, and the position after next roll is nonvolatile
and likely to remain so, I think you're better off holding the cube.
What if the cube is in the center? Is this position worth an initial
double? If you double now, your equity is
7/12(2) + 5/12(-2) = 1/3
If you wait, then with a centered cube and a 45% CWP, your actual winning
chances, assuming future efficient doubles by both sides, are
5/12. So your total equity is
1/2(1) + 1/2(5/12(1) + 7/12(-1)) = 5/12.
So even with a centered cube, you may have slightly higher equity by
waiting than by doubling now.
In theory it would be nice to be able to double right at the opponent's
minimum take point. In practice, backgammon just isn't that way. It is
often too volatile. If there is a significant chance that you may shoot
way beyond your opponent's minimum take point by a lot on the next
exchange of rolls, then it is important to double now even if your
advantage isn't too great. The cost of losing your market by a lot is
very expensive. That is why very volatile positions (that is, positions
where the equity is likely to swing a lot on the next exchange of rolls)
call for early doubles.
All very true. But in deciding whether to double, it's important to
consider not only the value of the cube this turn, but the value
of the cube on future turns, both to you and to your opponent.
Andy Latto
an...@harlequin.com