In a Jacoby money game with the cube at 2, find a TG position where
taking is as small an error as possible for the opponent.

Paul

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On Saturday, March 5, 2022 at 3:07:24 PM UTC-5, peps...@gmail.com wrote:

In a Jacoby money game with the cube at 2, find a TG position where
taking is as small an error as possible for the opponent.

Paul

Does it really matter if it's Jacoby or not if the cube is already at 2?

Stick

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On Sunday, March 6, 2022 at 1:46:05 AM UTC, Stick Rice wrote:

On Saturday, March 5, 2022 at 3:07:24 PM UTC-5, peps...@gmail.com wrote:

In a Jacoby money game with the cube at 2, find a TG position where
taking is as small an error as possible for the opponent.

Paul

Does it really matter if it's Jacoby or not if the cube is already at 2?

I mentioned Jacoby because, if I didn't mention it, the question could be asked:
"Why is the problem posed with the cube at 2? The units would be simpler if the value was 1."

Paul

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On Sunday, March 6, 2022 at 8:47:37 AM UTC, peps...@gmail.com wrote:

On Sunday, March 6, 2022 at 1:46:05 AM UTC, Stick Rice wrote:

On Saturday, March 5, 2022 at 3:07:24 PM UTC-5, peps...@gmail.com wrote:

In a Jacoby money game with the cube at 2, find a TG position where taking is as small an error as possible for the opponent.

Paul

Does it really matter if it's Jacoby or not if the cube is already at 2?

I mentioned Jacoby because, if I didn't mention it, the question could be asked:
"Why is the problem posed with the cube at 2? The units would be simpler if the value was 1."

Paul

I'll make my case by analogy. Suppose we are planning to entertain friends with dinner and
dessert. We might say (something like): "I think I'll provide a dessert of chocolate fudge too. The guests
would enjoy that. I think they have a sweet tooth."

If we've established that they would enjoy the dessert, why does it matter whether they have a sweet
tooth or not? It could be that they'd enjoy it for other reasons. They might enjoy fudge because of the texture,
rather than the sweetness. Or maybe they strongly dislike sweets in general, with fudge being the unique
exception.
The answer is that the enjoyment statement provided extra background explaining the situation of providing the fudge.
My mention of Jacoby provided extra background in explaining why I put the cube at 2.

Paul

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On 3/5/2022 3:07 PM, peps...@gmail.com wrote:

In a Jacoby money game with the cube at 2, find a TG position where
taking is as small an error as possible for the opponent.

Interesting challenge. Here's a straw man to get the ball rolling.

XGID=-C-------------bbbbcca---A:1:1:1:00:0:0:0:0:10

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-+
| O O O O | | O O O |
| O O O O | | O O |
| | | O O |
| | | |
| | X | |
| |BAR| |
| | | |
| | | |
| | | X | +---+
| | | X | | 2 |
| | | X | +---+
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 28 O: 105 X-O: 0-0
Cube: 2, X own cube
X on roll, cube action

Analyzed in Rollout
No redouble
Player Winning Chances: 79.48% (G:3.20% B:0.36%)
Opponent Winning Chances: 20.52% (G:0.00% B:0.00%)
Redouble/Take
Player Winning Chances: 78.83% (G:2.47% B:0.35%)
Opponent Winning Chances: 21.17% (G:0.00% B:0.00%)

Cubeful Equities:
No redouble: +1.003
Redouble/Take: +1.067 (+0.064)
Redouble/Pass: +1.000 (-0.003)

Best Cube action: Too good to redouble / Pass
Percentage of wrong take needed to make the double decision right: 4.0%

Rollout:
5184 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves: 3-ply, cube decisions: XG Roller
Confidence No Double: ± 0.002 (+1.001..+1.004)
Confidence Double: ± 0.009 (+1.058..+1.075)

eXtreme Gammon Version: 2.19.207.pre-release

---
Tim Chow

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I don't know the solution to this problem, but it seems interesting (to me).
I am playing an optimal computer in a Jacoby money session (think of XG but ignore
any weaknesses XG may have).
XG doesn't know this but I have a side bet with a friend as to whether or not
I win the current game. Furthermore, this win/lose side bet is the only
thing that concerns me.
Would it ever be correct for me to offer the cube when XG has a take?

Thanks,
Paul

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On 7/27/2022 6:46 AM, peps...@gmail.com wrote:

I don't know the solution to this problem, but it seems interesting (to me). I am playing an optimal computer in a Jacoby money session (think of XG but ignore
any weaknesses XG may have).
XG doesn't know this but I have a side bet with a friend as to whether or not I win the current game. Furthermore, this win/lose side bet is the only thing that concerns me.
Would it ever be correct for me to offer the cube when XG has a take?

Let's say you're playing XG for $1 a point and your side bet is $100.
You don't really care about gammon wins or losses, and you're going to
take any cube offered to you.

Seems to me the optimal strategy would be to wait until XG has a pass
and cash the game. Perhaps there's some example where it would be
correct to offer a cube that's a take, but my hunch is that if there is
one then upping the side bet to $1000 or $10,000 would negate it.

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On 7/27/2022 2:41 PM, ah...Clem wrote:

On 7/27/2022 6:46 AM, peps...@gmail.com wrote:

I don't know the solution to this problem, but it seems interesting (to me). >> I am playing an optimal computer in a Jacoby money session (think of XG but ignore
any weaknesses XG may have).
XG doesn't know this but I have a side bet with a friend as to whether or not
I win the current game. Furthermore, this win/lose side bet is the only
thing that concerns me.
Would it ever be correct for me to offer the cube when XG has a take?

[...]

Seems to me the optimal strategy would be to wait until XG has a pass
and cash the game.

It certainly seems this way, but when I tried to prove it, I discovered
that the problem is more subtle than it seems at first.

First of all, it seems to me that Paul left open the possibility that he
is betting against himself; i.e., he wins the side bet if he loses
against XG. Even in this case, it's not clear to me what the answer is.
But let's assume that Paul is betting that he'll win the game. One
could imagine that D/T could change XG's subsequent checker play, in a
way that increases Paul's winning chances. It's hard to imagine, but it
also seems hard to rule out rigorously.

---
Tim Chow

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On Thursday, July 28, 2022 at 2:09:22 AM UTC+1, Tim Chow wrote:

On 7/27/2022 2:41 PM, ah...Clem wrote:

On 7/27/2022 6:46 AM, peps...@gmail.com wrote:

I don't know the solution to this problem, but it seems interesting (to me).
I am playing an optimal computer in a Jacoby money session (think of XG but ignore
any weaknesses XG may have).
XG doesn't know this but I have a side bet with a friend as to whether or not
I win the current game. Furthermore, this win/lose side bet is the only
thing that concerns me.
Would it ever be correct for me to offer the cube when XG has a take? [...]

Seems to me the optimal strategy would be to wait until XG has a pass
and cash the game.

It certainly seems this way, but when I tried to prove it, I discovered
that the problem is more subtle than it seems at first.

First of all, it seems to me that Paul left open the possibility that he
is betting against himself; i.e., he wins the side bet if he loses
against XG. Even in this case, it's not clear to me what the answer is.
But let's assume that Paul is betting that he'll win the game. One
could imagine that D/T could change XG's subsequent checker play, in a
way that increases Paul's winning chances. It's hard to imagine, but it
also seems hard to rule out rigorously.

Thanks for your interest.
I'm not betting against myself -- I meant to rule that out.
My intuition is that some clever funfair-style construction tricks can be done to make D/T optimal but I also can't imagine what this might be.

Paul

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On Thursday, July 28, 2022 at 8:11:27 AM UTC+1, peps...@gmail.com wrote:

On Thursday, July 28, 2022 at 2:09:22 AM UTC+1, Tim Chow wrote:

On 7/27/2022 2:41 PM, ah...Clem wrote:

On 7/27/2022 6:46 AM, peps...@gmail.com wrote:

I don't know the solution to this problem, but it seems interesting (to me).
I am playing an optimal computer in a Jacoby money session (think of XG but ignore
any weaknesses XG may have).
XG doesn't know this but I have a side bet with a friend as to whether or not
I win the current game. Furthermore, this win/lose side bet is the only >> thing that concerns me.
Would it ever be correct for me to offer the cube when XG has a take? [...]

Seems to me the optimal strategy would be to wait until XG has a pass
and cash the game.

It certainly seems this way, but when I tried to prove it, I discovered that the problem is more subtle than it seems at first.

First of all, it seems to me that Paul left open the possibility that he
is betting against himself; i.e., he wins the side bet if he loses
against XG. Even in this case, it's not clear to me what the answer is.
But let's assume that Paul is betting that he'll win the game. One
could imagine that D/T could change XG's subsequent checker play, in a
way that increases Paul's winning chances. It's hard to imagine, but it also seems hard to rule out rigorously.

Thanks for your interest.
I'm not betting against myself -- I meant to rule that out.
My intuition is that some clever funfair-style construction tricks can be done
to make D/T optimal but I also can't imagine what this might be.

Paul

I can solve my own problem as below. When you see the basic story, it will
be utterly trivial to construct games that meet the demands of the problem.

I have a good game where I'm solidly ahead but never by so much that XG should pass.
Then we get a sequence that dramatically swings the game so that XG has great gammon chances
and few losing chances.

D/T is very likely to be my best play now (under the side bet hypothesis). When you have good gammon chances, you often need
to sacrifice your winning chances to max your equity.

Paul

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On 7/28/2022 6:06 AM, peps...@gmail.com wrote:

I have a good game where I'm solidly ahead but never by so much that XG should pass.
Then we get a sequence that dramatically swings the game so that XG has great gammon chances
and few losing chances.

D/T is very likely to be my best play now (under the side bet hypothesis). When you have good gammon chances, you often need
to sacrifice your winning chances to max your equity.

I don't follow your reasoning. I guess you're saying that with Jacoby,
XG will double you if you leave the cube in the center, whereas if you
double first, XG will hang on to the cube because it will be TG. I can
see that XG's checker play could differ in the two scenarios, but:

- in both scenarios---i.e., whether or not XG owns the cube---XG will
probably be sacrificing some winning chances for gammon chances; and

- even if XG is sacrificing *more* winning chances when it owns the
cube, it's not clear that this will be enough to compensate for your
inability to cash the game if the equity suddenly swings in your favor
again.

---
Tim Chow

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