Problem of the Week #130: December 11


Black on roll owning a 2-cube, leading 11-3 in a 15-point match.




Should Black double? If Black doubles, what should White do?

Cube, reship, win = match. Cube, reship, lose = 11-11 50%. Cube/drop = 94.9%. You have a 26-pip lead, so you win.. a lot. I don't even know what this lead is supposed to translate to since it's so much better (more than 10 pips) than decisions normally worth contemplating. 90% game win/95% MWC if he takes and reships seems like a minimum here, and there are no gammon considerations, so it looks like you should just claim your two points.

Double/drop.

Over the board, Black should analyze this as a straight race in which there are no gammons for either side. Black leads by 26 pips, 100 to 126.

In a cash game, this position is an easy pass. One way to see this is to add 10% + 2 pips to the leaders pip count, and round up to the nearest whole number. The result gives the final take point for the trailer. By this method, White's take point is 100 + 10 + 2 = 112. Alternatively, the more complex Keith Count gives the same result. Adding 1/7 to the leader's count, and rounding down, gives 114. By the Keith Count, White can take when his count is 112 or less. Thus, the two racing formulas give the same take point in a money game.

Is Black too good? Probably not, because his gammon wins are so few.

Match Equity Calculations
In a match, things may be quite different. After a double and a take, White has an auto-recube, so Black must understand that when White takes, the game will be played with the cube at 8. From the Kazaross XG2 MET, we find these Match Winning Chances (MWC) for the trailer:

Double, pass, Black wins 2: At 13-3 (2-away, 12-away), the trailer's MWC is 4.75%.
Double, take, auto-recube, Black wins 8: At 21-3, (match over), the trailer's MWC is 0%.
Double, take, auto-recube, White wins 8: At 11-11 (4-away, 4-away), the trailer's MWC is 50%.

For the trailer, the take point is found as:

Gain by taking = MWC( take, recube, win ) - MWC( pass ) = 50% - 4.75% = 45.25%
Loss by taking = MWC( pass ) - MWC( take, recube, lose ) = 4.75%
Take point = Loss / ( Gain + Loss ) = 4.75% / (45.25% + 4.75% ) = 4.75% / 50% = 9.5%

This means White can take any time his Game Winning Chance (GWC) is better than 9.5%. My guess is that White is still better than that in this race, so Black should not double, and if doubled, White should take, and recube on the next turn.

Kleinman Count
At home, we can be a bit more precise about this. Danny Keinman gives the following technique for estimating the GWC in a race.

Start by adding 4 pips to the leader's pip count, to take into account the fact that he is on roll.

L = 104 = leader's pip count + 4
T = 126 = trailer's pip count
D = 22 = difference = T - L
S = 230 = sum = T + L
K = 2.1 = Kleinman count = D * D / S

For K > 1.0, Øystein Johansen gives us:

GWC = 84.48% = 0.76 + 0.114 * ln(K), where ln(K) is the natural logarithm of K

As expected, Black is still way short of a double.

My solution: No double, take, auto-recube

For the Record
I am so often wrong that I like to post my record in these messages. It's kind of a truth-in-advertising thing. Grunch: I have been answering these problems without the use of a bot, and before checking the excellent solutions of others, since Problem 28. My record at this writing is 51%.

100-126

i guess white has 10% race chance, maybe the extra crossovers for white makes this a bit lower.
Say 8% race.

when losing normal (13-3) white probably has to win 6 times in a row.
Chance is 0,5 ^6 = 1/64, let's give him 2%.

So when doubled taking is clear for white.

When black doesn't double he has 92 % race * 98% match and 8% to (11-5) say 80% match

when he doubles he has 92% race * 100% match and 8% * 50% match

Doubling gives +92%*2% - 8%*30% = negative

So: no double

White's match equity looks about 5% either way. I wait as black and take as white, hoping to win and tilt black. As black hope white takes incorrectly later.

No double

Black leads 126-100 in what is basically a pure race. Let's start with the take/drop decision, because White will have an auto-reship with a cube now at 8. But if White takes, it would be for his tournament life.

Basically, does he have better chances now, or at 2away-12away?

My feeling (I'm too lazy to consult a MET, so I treat this as an OTB problem!) I think he has better chances now, so I would take/auto-reship as White.

Therefore, I would hold the cube as Black.

No redouble/Take + auto-reship

Seems like you are penalising the guy who is on roll. The formula says to add 4 to the difference between leaders and trailers pipcount (and then subtract 4 from the sum of the two pipcounts).

Thanks, Mute. You are right!

I should have been subtracting, so that:

LRAW = leader's raw pip count
L = LRAW - 4
T = trailer's raw pip count

According to Øystein's post at Backgammon Galore!:

It is my recollection that Danny used T and L in his original formula.

D = T - LRAW + 4 = T - (LRAW - 4) = T - L
S = T + LRAW - 4 = T + (LRAW - 4) = T + L

For the position in Problem of the Week #130, this gives:

L = 96 = 100 - 4
T = 126
D = 30 = difference = T - L
S = 222 = sum = T + L
K = 4.05 = Kleinman count = D * D / S

GWC for leader = 92% = 0.76 + 0.114 * ln(K), where ln(K) is the natural logarithm of K.

This is a very different result. By this calculation, White's GWC is only 8%, and Black is already beyond White's take point of 9.5%. If these numbers hold up, then my solution above is incorrect.

Tom has it right. Proper play is double, drop.

Grunch.

“Given the same amount of intelligence, timidity will do a thousand times more damage than audacity” - Karl von Clausewitz


I have just discovered a nice resource. OTB, i would use neil's numbers and Kit Woolseys advices to come up with the take point. Here i look at the score card of Fabrice Liardet, and voila, a take point, given an automatic redouble, of 9,51%. By the way, if you look on this nice score card and the doubling windows, gammon prices, gammon rates and the growing take points for the leader, you get a feeling for the very different cube play in a match.

And now we are forced to go beyond simple strategies for evaluating our race equity. We know, that we are a huge favorite. But doubling with only 90% would be a fault. The audacity must have the right amount. Chuck Bower posted a table for an approximately K. This is a little bit more to learn. But if you make a Kleinman Count DxD/2=K and get a K of 4,05, the table gives you 92%. Ha! No time for timidity. Ship it over. Your opponent has to drop. Otherwise you will gain.

Redouble, and Drop.

Here's my quick and dirty way of figuring out these kind of things. I count crossovers until bearoff, black has 8, white has 13. White is getting destroyed and loses ALOT of the time. Even if white rolls boxes he's still behind in crossovers.

If white takes it's an auto reship unless he's a total donkey....

It looks like a good practical double to me, even at the score... not sure of the correct action for white but I think most guys would drop here.

No Cube/ Instant Reship

Black has a gammon opportunity here and if he gets it, he wins the game. He's also very far ahead in match score and race. When white takes, he's playing for the match so he may as well instant reship to get the most bang for his buck if he wins.

White is 26 pips behind in 100-pip race. That's very much. For example, consider this:
If black throws 2-1 and white throws 6-6, black is still 5 pips ahead and on roll. That translates to more than 65% GWC for black - and this is a worst case scenario.


As guys before me calculated, white needs to have more than 9,5% GWC to take this.


In a pure race, if pipcount were 113-126, white's GWC would be around 22%.

I'm sure that 13 more pips of lead are worth more than 13% (only 1% per pip).
I'm not sure how many % is one pip worth here, but even if it is as low as 1,2%/pip (but it is probably higher), this would be 15,6% less than 22%, making it 6,4% GWC for white.
So based on race, this is big pass even at the score.

But this is not a pure race. If black throws X-1 on his first roll, he cannot clear the midpoint.
Is this enough to put white into taking territory? I think it isn't.


Redouble, pass

"Would" is a fault. I wanted to write "could". If Fabrice Liardet is right, 90% is in the doubling window.

I agree with the bolded part.

Gammons must be super low, less than 10%. I don't think playing on for the gammon makes any sense here and shouldn't really factor in to your decision making here.

Black has almost no gammon chances- <1%. Maybe <.1%. Playing on for that chance is fine if you can't blow the game- if every sequence is still double-drop- but here that's clearly not the case. Black can blow the game easily if he lets white hang around to roll big doubles next turn.

But white has 2 loose checkers, 4 trapped on black's 1 point and black has a good chance to build a 5 or 6 prime.

Also, being a n00b. How do you calculate gammon %? I've just been going on number of checkers behind and if you have a prime.

They're not going that way. Look at the numbers.

There's no real good way except by knowing some reference positions.

To paraphrase Aaron: Black moves clockwise; White moves counterclockwise.

Whoops, I misread the board. When I play, I set mine so my homeboard's on the right. Rookie mistake. My bad.

In this case Double/Reship.

Black has a lead may as well go for the match and he probably has a low percentage of losing this game. White has to instant reship because if he drops it's practically game over, if he loses, it's definitely game over, but black will be forced to take and on the off chance white wins, it's an even game again.

I really doubt it can be Redouble/Take with such a lopsided score. I would guess it's either No Double / Take or Double / Drop.

Every Cube decision is based on an evaluation of the takepoint. I don't understand why only a few readers give a takepoint as a starter for the cube decision. Also the race can be estimated. Gammons are not a question. So it is easy to come up with a near bot solution. A little more sportsmanship, guys.

Who will show us the use of neil's numbers and then the calculation of the odds?!

Using Neil's numbers at 2-away scores usually won't give you very good results. Neil himself has said that you are better off just memorizing them.

For instance for 12-away the number is 4.75, which would give 97.5% at 2-away, 12-away.

Respect. You saw the ?!

Kit Woolsey wrote (see the link): "Neil's numbers are incredibly
accurate if the leader has 3 or more points to go, but tend to break down
when the leader has 2 or 1 points to go. Thus it is best to memorize
that part of my equity table (even I was able to do that), and you're all
set."

If you are going to a tournament with 15 point matches or more, you have to make a little home work more.

Ah, didn't realize it was a trap!

I have all 1-away 2-away and 3-away equities memorized to about 10-away and can reasonably eyeball it from further away.

I do use Neils (new) numbers for the rest when necessary.