Discussion:
Pay now?
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Tim Chow
2020-07-19 19:11:12 UTC
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XGID=-aB-BbC-B--BcD---b-dab----:1:-1:1:62:0:0:0:0:10

X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
+13-14-15-16-17-18------19-20-21-22-23-24-+
| X O | | O O O | +---+
| X O | | O O | | 2 |
| X | | O | +---+
| X | | O |
| | | |
| |BAR| |
| | | |
| | | |
| O | | X |
| O X X | | X O X X |
| O X X | | X O X X O |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 120 O: 156 X-O: 0-0
Cube: 2, O own cube
X to play 62

---
Tim Chow
Paul Epstein
2020-07-19 20:33:28 UTC
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Post by Tim Chow
XGID=-aB-BbC-B--BcD---b-dab----:1:-1:1:62:0:0:0:0:10
X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
+13-14-15-16-17-18------19-20-21-22-23-24-+
| X O | | O O O | +---+
| X O | | O O | | 2 |
| X | | O | +---+
| X | | O |
| | | |
| |BAR| |
| | | |
| | | |
| O | | X |
| O X X | | X O X X |
| O X X | | X O X X O |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 120 O: 156 X-O: 0-0
Cube: 2, O own cube
X to play 62
I like this problem.
My candidates are:
8/2 8/6, 8/2 13/11, 11/3, 13/11 13/7

8/2 8/6 is obviously bad. It doesn't help our position
and being hit is not so disastrous that we need to
go to such extremes to avoid paying.

11/3 and 13/11 13/7 diversify for the opponent too much:
the numbers that hit are different to the numbers O needs
to cover O's 5 point. The problem with that is that
it gives too many rolls for the opponent to both hit and cover.

Initially, I was a major superfan of the 13/11 13/7 play but
after an extra seven minutes thought, I just ain't that way no more,
no more. Just ain't that way no more.

I'll play 13/11 8/2 with 71% confidence.
The three to hit duplicates O's three to cover.
Some might complain about me stacking up on the two point but
it's less than the other evils and it does allow me to attack on
the ace sometimes.

I'll do a deal with Tim. I'll agree that Tim's friend knew that
24/23 was correct at DMP if, given that the bot approves my play,
Tim will say that I knew the play here.
Just so that it's not a one-way bet, if I'm wrong I will learn
the theory that says that busy beaver numbers are larger than Graham's
numbers.

Paul
Peter
2020-07-20 05:38:46 UTC
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Post by Tim Chow
XGID=-aB-BbC-B--BcD---b-dab----:1:-1:1:62:0:0:0:0:10
X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
+13-14-15-16-17-18------19-20-21-22-23-24-+
| X O | | O O O | +---+
| X O | | O O | | 2 |
| X | | O | +---+
| X | | O |
| | | |
| |BAR| |
| | | |
| | | |
| O | | X |
| O X X | | X O X X |
| O X X | | X O X X O |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 120 O: 156 X-O: 0-0
Cube: 2, O own cube
X to play 62
---
Tim Chow
13/11 13/7
Tim Chow
2020-07-23 14:39:56 UTC
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XGID=-aB-BbC-B--BcD---b-dab----:1:-1:1:62:0:0:0:0:10

X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
+13-14-15-16-17-18------19-20-21-22-23-24-+
| X O | | O O O | +---+
| X O | | O O | | 2 |
| X | | O | +---+
| X | | O |
| | | |
| |BAR| |
| | | |
| | | |
| O | | X |
| O X X | | X O X X |
| O X X | | X O X X O |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 120 O: 156 X-O: 0-0
Cube: 2, O own cube
X to play 62

I'm still not good at judging whether the opponent's weak board justifies
this kind of early pay-a-double-shot-now decision. See also the variant.

1. Rollout¹ 13/11 13/7 eq:+0.151
Player: 61.42% (G:15.22% B:0.57%)
Opponent: 38.58% (G:9.06% B:0.30%)
Confidence: ±0.009 (+0.141..+0.160) - [100.0%]

2. Rollout¹ 13/7 6/4 eq:+0.112 (-0.039)
Player: 59.97% (G:15.47% B:0.58%)
Opponent: 40.03% (G:10.09% B:0.35%)
Confidence: ±0.009 (+0.103..+0.121) - [0.0%]

3. Rollout¹ 8/6 8/2 eq:+0.083 (-0.068)
Player: 59.57% (G:12.79% B:0.37%)
Opponent: 40.43% (G:9.67% B:0.32%)
Confidence: ±0.008 (+0.075..+0.090) - [0.0%]

4. Rollout¹ 13/11 8/2 eq:+0.074 (-0.077)
Player: 58.67% (G:14.50% B:0.54%)
Opponent: 41.33% (G:10.24% B:0.35%)
Confidence: ±0.008 (+0.066..+0.082) - [0.0%]

¹ 1296 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves: 3-ply, cube decisions: XG Roller

eXtreme Gammon Version: 2.19.207.pre-release

-------
Variant
-------

XGID=-aB-BbC-B--BcD---bbc-b----:1:-1:1:62:0:0:0:0:10

X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
+13-14-15-16-17-18------19-20-21-22-23-24-+
| X O O | | O O | +---+
| X O O | | O O | | 2 |
| X | | O | +---+
| X | | |
| | | |
| |BAR| |
| | | |
| | | |
| O | | X |
| O X X | | X O X X |
| O X X | | X O X X O |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 120 O: 159 X-O: 0-0
Cube: 2, O own cube
X to play 62

1. Rollout¹ 8/6 8/2 eq:+0.105
Player: 59.52% (G:14.29% B:0.41%)
Opponent: 40.48% (G:9.12% B:0.34%)
Confidence: ±0.004 (+0.100..+0.109) - [49.9%]

2. Rollout¹ 13/11 13/7 eq:+0.105
Player: 59.83% (G:14.35% B:0.50%)
Opponent: 40.17% (G:8.78% B:0.34%)
Confidence: ±0.004 (+0.100..+0.109) - [50.1%]

3. Rollout¹ 13/7 6/4 eq:+0.079 (-0.026)
Player: 58.69% (G:15.17% B:0.61%)
Opponent: 41.31% (G:9.63% B:0.37%)
Confidence: ±0.005 (+0.074..+0.083) - [0.0%]

¹ 5184 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves: 3-ply, cube decisions: XG Roller

eXtreme Gammon Version: 2.19.207.pre-release

---
Tim Chow
Paul Epstein
2020-07-24 06:02:31 UTC
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Permalink
Post by Tim Chow
XGID=-aB-BbC-B--BcD---b-dab----:1:-1:1:62:0:0:0:0:10
X:Player 1 O:Player 2
Score is X:0 O:0. Unlimited Game
+13-14-15-16-17-18------19-20-21-22-23-24-+
| X O | | O O O | +---+
| X O | | O O | | 2 |
| X | | O | +---+
| X | | O |
| | | |
| |BAR| |
| | | |
| | | |
| O | | X |
| O X X | | X O X X |
| O X X | | X O X X O |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 120 O: 156 X-O: 0-0
Cube: 2, O own cube
X to play 62
I'm still not good at judging whether the opponent's weak board justifies
this kind of early pay-a-double-shot-now decision.
I think the phrase "a double shot" is a bad way of framing the decision.
A major reason that 13/7 works is that O has 20 shots which is the least
number of hits possible from two directs. This means that the combined parlay
of O hitting and X failing to return-hit is much less than 50%.
In fact that combined parlay isn't much more likely than the standard 31%
probability with a single direct. So the "double shot" formulation is very
misleading (although literally true).
When you variantize to make the double shot more genuine, we no longer pay now.

Paul

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