A difficult one to Axelise

[peps...@gmail.com]

This plays nothing like money because the opponent
has no recube vig.
Apparently, my only advantage is being on-roll.
It's a big double and a big take.
The take was clear to me but I wasn't confident in my
double even though holding would be a huge blunder.
I did double but not with any confidence.
I suppose the Axelisation formulas for winning prob
combined with the theory of 4A 4A would have helped.

Paul
XGID=-BEABCB------------caebba-:1:1:1:00:7:7:0:11:10

X:Daniel O:XG Roller+
Score is X:7 O:7 11 pt.(s) match.
+13-14-15-16-17-18------19-20-21-22-23-24-+
| | | O O O O O O |
| | | O O O O |
| | | O O |
| | | O |
| | | O |
| |BAR| |
| | | X |
| | | X |
| | | X X | +---+
| | | X X X X X | | 2 |
| | | X X X X X X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 50 O: 54 X-O: 7-7/11
Cube: 2, X own cube
X on roll, cube action

Analyzed in 4-ply
Player Winning Chances: 64.54% (G:0.00% B:0.00%)
Opponent Winning Chances: 35.46% (G:0.00% B:0.00%)

Cubeless Equities: No Double=+0.291, Double=+0.862

Cubeful Equities:
No redouble: +0.743 (-0.118)
Redouble/Take: +0.862
Redouble/Pass: +1.000 (+0.138)

Best Cube action: Redouble / Take

eXtreme Gammon Version: 2.10, MET: Kazaross XG2

[Axel Reichert]

"peps...@gmail.com" <peps...@gmail.com> writes:

> XGID=-BEABCB------------caebba-:1:1:1:00:7:7:0:11:10
>
> X:Daniel O:XG Roller+
> Score is X:7 O:7 11 pt.(s) match.
> +13-14-15-16-17-18------19-20-21-22-23-24-+
> | | | O O O O O O |
> | | | O O O O |
> | | | O O |
> | | | O |
> | | | O |
> | |BAR| |
> | | | X |
> | | | X |
> | | | X X | +---+
> | | | X X X X X | | 2 |
> | | | X X X X X X | +---+
> +12-11-10--9--8--7-------6--5--4--3--2--1-+
> Pip count X: 50 O: 54 X-O: 7-7/11
> Cube: 2, X own cube
> X on roll, cube action

Let's see. If you redouble, it will be for the match for both
players. In this case the take point is the match equity after a pass
(one of the most important rules to memorize in match play), so we need
to calculate the match equity at -4/-2. Turner formula says

50 - (24/4 + 3)*(4-2) = 32 %

Pip count is 50 versus 54, adjusted 50 + 3 (stack on 2) + 1 (less
checkers off) = 54 versus 54 (no adjustments needed). The Isight method
gives

80 - 54/3 + 2*(54 - 54) = 62 %

winning chances for you, so 38 % for your opponent, whose take point is
32 %. Normally you want to double close to your opponents take point,
especially if you are far away from a last roll situation. Not
sure. Calculating the doubling point may help to see where we are in the
window. The doubling point is

risk/(risk+reward)

and risk is match equity at -4/-2 minus match equity at -4/0. At -4/-2
you have 32 %, see above. At -4/0 you have 0 %. So you risk 32 %. Now
the reward, which is the match equity an 0/-4 minus match equity at
-2/-4. At 0/-4 you have 100 %, at -2/-4 you have 68 %, see above. So the
reward is 32 %. Your doubling point is 32/(32+32) = 50 %. I would
double, and the take is clear.

Best regards

Axel

[peps...@gmail.com]

Thanks for this analysis. However, the doubling point, in sharp contrast
to the take point is not at all tractable. Everything depends on the tree
of how the probabilities are likely to move. This is different in each position
and impossible for a human to handle. Suppose that we did the calculations correctly
to arrive at your conclusion that the take point was 32% but the opponent's probability
is 38%. Should we double? I prefer the following heuristic.
Well, how far are we from the take/drop border? From an additive perspective, we are 6%
away. And from a multiplicative perspective, we are 20% away in the sense that 120% of 32
is approx 38%.
So I would ask: Is it usually correct to double when we are this close to the take/pass border?
I see no reason why the distance between doubling point and take point here should be
significantly different to the average distance.
For money, if you bear in mind recube vig, the typical take/pass border is between 22% and 23%.
And would a double be correct if our opponent's chances were 6% higher than that -- 28 to 29%.
Almost definitely yes, we should double in that position.
So we double now, too.

Paul