How to contain a checker

[Timothy Chow]

XGID=---aBCBB-B---C---aAddc--b-:1:-1:1:65:0:0:0:0:10

Score is X:0 O:0. Unlimited Game
+13-14-15-16-17-18------19-20-21-22-23-24-+
| X O X | | O O O O | +---+
| X | | O O O O | | 2 |
| X | | O O O | +---+
| | | O O |
| | | |
| |BAR| |
| | | |
| | | |
| | | X |
| X X | | X X X |
| X X | | X X X O |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 124 O: 88 X-O: 0-0
Cube: 2, O own cube
X to play 65

---
Tim Chow

[ah....Clem]

X is outboarded, so the loose hit seems wrong. And keeping the nine
point seems thematic since it means O only escapes with a five.

Leaving the blot on the 18 might be right since O is stuck behind a
formidable blockade and a trip to the bar might cause her board to
crunch. But only if she can't shake a five, which is far from certain.

So, keep the nine point and move the blot to safety. That implies 18/7,
which must be wrong due to QF. But that's my play.



--
Ah....Clem
The future is fun, the future is fair.

[Timothy Chow]

X has the opportunity to set up what Chris Bray would call a
"one-roll proposition." By slotting 13/8 18/12, he challenges
O to roll an immediate 5. If O fails, then X is likely to
complete a six-prime and lock up the game. By contrast, if
X plays quietly with 18/7, O will likely have multiple turns
for a good escaping roll, since O has such a big lead in the
pip count.

Of course, if O *does* roll that immediate 5, then X will be
up the creek without a paddle. But a good first approximation
is that X will win if O doesn't roll a 5, which means that
18/12 13/8 will win about 70% of the time. It's not as easy
to judge how often 18/7 will win, but X is only a slight
favorite. The remaining factor to consider is gammon losses,
which are much higher after 18/12 13/8, but they're nowhere
near enough to offset the extra wins, according to the rollout.

To get a contrasting variant, I had to make some major changes
to the position---giving O a killer board, and moving a
checker from X's midpoint to his 6pt. Even then, the bot
slots at DMP.

1. Rollout¹ 18/12 13/8 eq:+0.265
Player: 69.05% (G:5.70% B:0.18%)
Opponent: 30.95% (G:13.22% B:0.13%)
Confidence: ±0.005 (+0.260..+0.270) - [100.0%]

2. Rollout¹ 18/7 eq:+0.050 (-0.214)
Player: 57.32% (G:3.46% B:0.10%)
Opponent: 42.68% (G:4.79% B:0.05%)
Confidence: ±0.005 (+0.045..+0.055) - [0.0%]

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

eXtreme Gammon Version: 2.19.211.pre-release

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

XGID=---aBCCB-B---B---aAcccbb--:1:-1:1:65:0:0:0:0:10

Score is X:0 O:0. Unlimited Game
+13-14-15-16-17-18------19-20-21-22-23-24-+

| X O X | | O O O O O | +---+
| X | | O O O O O | | 2 |
| | | O O O | +---+

| | | |
| | | |
| |BAR| |
| | | |
| | | |
| | | X X |
| X X | | X X X |
| X X | | X X X O |
+12-11-10--9--8--7-------6--5--4--3--2--1-+

Pip count X: 117 O: 85 X-O: 0-0

Cube: 2, O own cube
X to play 65

1. Rollout¹ 18/7 eq:+0.098
Player: 59.98% (G:2.66% B:0.07%)
Opponent: 40.02% (G:3.66% B:0.05%)
Confidence: ±0.005 (+0.093..+0.102) - [68.7%]

2. Rollout¹ 18/12 13/8 eq:+0.096 (-0.002)
Player: 64.67% (G:4.29% B:0.13%)
Opponent: 35.33% (G:20.66% B:0.49%)
Confidence: ±0.005 (+0.091..+0.101) - [31.3%]

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

eXtreme Gammon Version: 2.19.211.pre-release

---
Tim Chow

[ah...Clem]

On 10/4/2023 9:03 AM, Timothy Chow wrote:

> X has the opportunity to set up what Chris Bray would call a
> "one-roll proposition."  By slotting 13/8 18/12, he challenges
> O to roll an immediate 5. 

Fascinating. I was unaware of this stratagem. I'd be interested in
seeing more examples, including some where it's not the right approach.

[Timothy Chow]

I don't know Chris Bray's exact definition of "one-roll proposition,"
but for example, I suspect he would include positions like the one
below. The loose hit loses immediately if O hits back, and if O hits
back...well, it doesn't win the game for X on the spot, but it gives
him something like 90% winning chances. So it's easy to approximate
X's chances after the "one-roll proposition" play.

What's not always as easy to assess are X's chances after the quiet
play. In this case, there's not that much contact after the quiet
play, so you can get a reasonable approximation by pretending that
it's a straight race. Whether the "one-roll proposition" play is
right therefore depends on whether it offers better chances than a race.

The position is harder to assess if O's straggler is partially primed.
The quiet alternative can no longer be treated as a race. I'll dig up
some positions of this type and post them as a series.

XGID=-ACCCaBA---B----a--bbbbcb-:1:-1:1:32:0:0:0:0:10

Score is X:0 O:0. Unlimited Game
+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 | | 2 |

| | | O | +---+
| | | |
| | | |
| |BAR| |
| | | |
| | | |
| | | X X X |
| X | | X X X X |

| X X | | X O X X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 69 O: 73 X-O: 0-0

Cube: 2, O own cube

X to play 32

1. Rollout¹ 7/5* 4/1 eq:+0.191
Player: 62.16% (G:1.61% B:0.01%)
Opponent: 37.84% (G:4.44% B:0.03%)
Confidence: ±0.002 (+0.189..+0.193) - [100.0%]

2. Rollout¹ 7/4 3/1 eq:+0.136 (-0.056)
Player: 62.53% (G:0.26% B:0.00%)
Opponent: 37.47% (G:0.60% B:0.01%)
Confidence: ±0.005 (+0.131..+0.140) - [0.0%]

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

eXtreme Gammon Version: 2.19.211.pre-release

---

Tim Chow

[Timothy Chow]

On 10/6/2023 8:20 PM, I wrote:

> I don't know Chris Bray's exact definition of "one-roll proposition,"
> but for example, I suspect he would include positions like the one
> below.  The loose hit loses immediately if O hits back, and if O hits
> back...well, it doesn't win the game for X on the spot, but it gives
> him something like 90% winning chances.

Sorry...of course I meant, "and if O *doesn't* hit back" then it gives
X about 90% winning chances.

---
Tim Chow