I was very unsure about what to do here.
Do I win the game or do I go for more gammons?
To me it seemed like a coin toss and I chose the gammon-hungry play.
The analysis rates this as very slightly wrong.

Paul

XGID=-CCBbB-----------bagc-----:1:-1:1:32:0:4:3:0:10

X:Daniel O:XG Roller+
Score is X:0 O:4. Unlimited Game, Jacoby Beaver
+13-14-15-16-17-18------19-20-21-22-23-24-+
| O O | | O O | +---+
| O | | O O | | 2 |
| | | O O | +---+
| | | O |
| | | 7 |
| |BAR| |
| | | |
| | | |
| | | X X |
| | | X O X X X |
| | | X O X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+
Pip count X: 25 O: 122 X-O: 0-4
Cube: 2, O own cube
X to play 32

1. 4-ply 5/3 5/2 eq:+1.717
Player: 99.98% (G:71.75% B:0.00%)
Opponent: 0.02% (G:0.00% B:0.00%)

2. 4-ply 3/Off 2/Off eq:+1.707 (-0.010)
Player: 98.34% (G:74.36% B:0.24%)
Opponent: 1.66% (G:0.00% B:0.00%)

3. 3-ply 3/1 3/Off eq:+1.596 (-0.121)
Player: 98.07% (G:64.08% B:0.05%)
Opponent: 1.93% (G:0.00% B:0.00%)

4. 1-ply 5/Off eq:+1.299 (-0.418)
Player: 88.80% (G:56.25% B:0.02%)
Opponent: 11.20% (G:0.00% B:0.00%)

5. 1-ply 5/2 3/1 eq:+1.156 (-0.561)
Player: 86.34% (G:47.77% B:0.01%)
Opponent: 13.66% (G:0.00% B:0.00%)

eXtreme Gammon Version: 2.10

On Tuesday, January 16, 2024 at 7:44:55 PM UTC, peps...@gmail.com wrote:
> I was very unsure about what to do here.
> Do I win the game or do I go for more gammons?
> To me it seemed like a coin toss and I chose the gammon-hungry play.
> The analysis rates this as very slightly wrong.
>
> Paul
>
> XGID=-CCBbB-----------bagc-----:1:-1:1:32:0:4:3:0:10
>
> X:Daniel O:XG Roller+
> Score is X:0 O:4. Unlimited Game, Jacoby Beaver
> +13-14-15-16-17-18------19-20-21-22-23-24-+
> | O O | | O O | +---+
> | O | | O O | | 2 |
> | | | O O | +---+
> | | | O |
> | | | 7 |
> | |BAR| |
> | | | |
> | | | |
> | | | X X |
> | | | X O X X X |
> | | | X O X X X |
> +12-11-10--9--8--7-------6--5--4--3--2--1-+
> Pip count X: 25 O: 122 X-O: 0-4
> Cube: 2, O own cube
> X to play 32
>
> 1. 4-ply 5/3 5/2 eq:+1.717
> Player: 99.98% (G:71.75% B:0.00%)
> Opponent: 0.02% (G:0.00% B:0.00%)
>
> 2. 4-ply 3/Off 2/Off eq:+1.707 (-0.010)
> Player: 98.34% (G:74.36% B:0.24%)
> Opponent: 1.66% (G:0.00% B:0.00%)
>
> 3. 3-ply 3/1 3/Off eq:+1.596 (-0.121)
> Player: 98.07% (G:64.08% B:0.05%)
> Opponent: 1.93% (G:0.00% B:0.00%)
>
> 4. 1-ply 5/Off eq:+1.299 (-0.418)
> Player: 88.80% (G:56.25% B:0.02%)
> Opponent: 11.20% (G:0.00% B:0.00%)
>
> 5. 1-ply 5/2 3/1 eq:+1.156 (-0.561)
> Player: 86.34% (G:47.77% B:0.01%)
> Opponent: 13.66% (G:0.00% B:0.00%)
>
>
> eXtreme Gammon Version: 2.10
BTW, XG's estimate that we lose with the safe play
0.02% seems like a massive overestimate to me.
Of course I can't calculate this exactly and, with this event
so rare, a rollout might not help.
But heuristics are available.
The safe play puts XG on roll with 24 crossovers needed against 10.
So even if XG gets a 66 on every single roll, XG is still far more likely to
lose than to win. XG would need me to fail to get doubles and for me to get at least
one roll which fails to take two off.

On the other hand, XG can also win without always rolling a 66.
I would say that a win for XG is less likely than XG rolling 44 or better
from its next shake until the end of the game.
Assuming generously, that each shake of 44 or better always gets 4 crossovers,
and assuming very generously that I somehow fail to bear off in 6 shakes, we arrive at
a near upper bound for the probability. It's not quite a strict upper bound because 33 might
sometimes work for XG too. But I think that this is a reasonable heuristic and that XG's winning
chances are actually much less than this.
44 or better has a 1/12 probability. This needs to happen 6 times for a probability of 12 ^ (-6) which
is much less than one in a million which is only 10 ^ (-6)

0.02% indeed!

Paul

On 1/16/2024 3:00 PM, peps...@gmail.com wrote:
> BTW, XG's estimate that we lose with the safe play
> 0.02% seems like a massive overestimate to me.

Yes, I agree. I just did a 1 million game rollout and X won every
game. An interesting feature of the rollout is that X won a backgammon
22 times! Of course there's no guarantee that XG is playing correctly
but I think this makes sense. If O rolls 21 then 8/6 7/6 should yield
the best chances of running off the gammon, and that should offset the
slight increase in the risk of losing a backgammon.

As for your original position, these decisions can be tough to figure
out OTB, but if your chances of winning a gammon are over 70% (and
getting hit seriously damages your winning chances) then you should
probably play safe. You can estimate your gammon winning chances by
using standard racing formulas. Running off the gammon is not exactly
the same as winning a normal race, but it's close enough for the
formulas to be useful.

---
Tim Chow

On Thursday, January 18, 2024 at 8:26:26 AM UTC-5, Timothy Chow wrote:
> On 1/16/2024 3:00 PM, peps...@gmail.com wrote:
> > BTW, XG's estimate that we lose with the safe play
> > 0.02% seems like a massive overestimate to me.
> Yes, I agree. I just did a 1 million game rollout and X won every
> game. An interesting feature of the rollout is that X won a backgammon
> 22 times! Of course there's no guarantee that XG is playing correctly
> but I think this makes sense. If O rolls 21 then 8/6 7/6 should yield
> the best chances of running off the gammon, and that should offset the
> slight increase in the risk of losing a backgammon.
>
> As for your original position, these decisions can be tough to figure
> out OTB, but if your chances of winning a gammon are over 70% (and
> getting hit seriously damages your winning chances) then you should
> probably play safe. You can estimate your gammon winning chances by
> using standard racing formulas. Running off the gammon is not exactly
> the same as winning a normal race, but it's close enough for the
> formulas to be useful.
>
> ---
> Tim Chow

There is zero chance of the opponent losing a backgammon after we make the safe play in the original position. Even after an immediate roll of [21] and we play both to the six point the opponent is on roll with a 3 roll position. No matter what we roll over the course of two rolls [21 followed by 21 for eg] we get off the backgammon. Even 3 ply knows this.

Stick

On 1/18/2024 3:36 PM, Stick Rice wrote:
> There is zero chance of the opponent losing a backgammon after we make the safe play in the original position. Even after an immediate roll of [21] and we play both to the six point the opponent is on roll with a 3 roll position. No matter what we roll over the course of two rolls [21 followed by 21 for eg] we get off the backgammon. Even 3 ply knows this.

It took me a while, but I finally figured out what XG was doing.
As you suggested, we start by giving the opponent a roll of 21,
which is played 8/6 7/6. That gives us the following position
(discussion continues below diagram).

XGID=-CDCb------------bagc-----:1:-1:-1:12:0:0:3:0:10

Score is X:0 O:0. Unlimited Game, Jacoby Beaver
+12-11-10--9--8--7-------6--5--4--3--2--1-+
| O O | | O O | +---+
| O | | O O | | 2 |
| | | O O | +---+
| | | O |
| | | 7 |
| |BAR| |
| | | |
| | | X |
| | | X X X |
| | | O X X X |
| | | O X X X |
+13-14-15-16-17-18------19-20-21-22-23-24-+
Pip count X: 20 O: 122 X-O: 0-0
Cube: 2, O own cube
O to play 12

1. 4-ply 8/6 7/6 eq:-1.839
Player: 0.00% (G:0.00% B:0.00%)
Opponent: 100.00% (G:83.91% B:0.01%)

2. 4-ply 21/20 8/6 eq:-1.862 (-0.023)
Player: 0.00% (G:0.00% B:0.00%)
Opponent: 100.00% (G:86.19% B:0.01%)

3. 4-ply 21/20 21/19 eq:-1.883 (-0.044)
Player: 0.00% (G:0.00% B:0.00%)
Opponent: 100.00% (G:88.27% B:0.00%)

eXtreme Gammon Version: 2.19.211.pre-release

Now suppose X rolls something like 65, taking two checkers off, and
then O rolls 21 again. Evidently, 21/20 and 8/6 and 21/18 have exactly
the same equity, but for whatever reason, XG chooses 21/20 8/6, which
loses some backgammons but gives it better chances of getting off the
gammon.

XGID=-CDAb------------a-ic-----:1:-1:-1:21:0:0:3:0:10

Score is X:0 O:0. Unlimited Game, Jacoby Beaver
+12-11-10--9--8--7-------6--5--4--3--2--1-+
| O | | O O | +---+
| | | O O | | 2 |
| | | O O | +---+
| | | O |
| | | 9 |
| |BAR| |
| | | |
| | | X |
| | | X X |
| | | O X X |
| | | O X X X |
+13-14-15-16-17-18------19-20-21-22-23-24-+
Pip count X: 14 O: 119 X-O: 0-0
Cube: 2, O own cube
O to play 21

1. 4-ply 21/20 8/6 eq:-1.929
Player: 0.00% (G:0.00% B:0.00%)
Opponent: 100.00% (G:92.36% B:0.57%)

2. 4-ply 21/18 eq:-1.929
Player: 0.00% (G:0.00% B:0.00%)
Opponent: 100.00% (G:92.95% B:0.00%)

eXtreme Gammon Version: 2.19.211.pre-release

---
Tim Chow

On 1/19/2024 8:19 AM, I wrote:
> Now suppose X rolls something like 65, taking two checkers off, and
> then O rolls 21 again. Evidently, 21/20 and 8/6 and 21/18 have exactly
> the same equity

Correction: 21/20 8/6 is apparently slightly better.

XGID=-CDAb------------a-ic-----:1:-1:-1:21:0:0:3:0:10

Score is X:0 O:0. Unlimited Game, Jacoby Beaver
+12-11-10--9--8--7-------6--5--4--3--2--1-+
| O | | O O | +---+
| | | O O | | 2 |
| | | O O | +---+
| | | O |
| | | 9 |
| |BAR| |
| | | |
| | | X |
| | | X X |
| | | O X X |
| | | O X X X |
+13-14-15-16-17-18------19-20-21-22-23-24-+
Pip count X: 14 O: 119 X-O: 0-0
Cube: 2, O own cube
O to play 21

1. Rollout¹ 21/20 8/6 eq:-1.9246
Player: 0.00% (G:0.00% B:0.00%)
Opponent: 100.00% (G:91.89% B:0.57%)
Confidence: ±0.0001 (-1.9247..-1.9244) - [100.0%]

2. Rollout¹ 21/18 eq:-1.9252 (-0.0007)
Player: 0.00% (G:0.00% B:0.00%)
Opponent: 100.00% (G:92.52% B:0.00%)
Confidence: ±0.0001 (-1.9253..-1.9251) - [0.0%]

¹ 5184 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves and cube decisions: XG Roller++
Search interval: Gigantic

eXtreme Gammon Version: 2.19.211.pre-release

---
Tim Chow

On Friday, January 19, 2024 at 1:27:09 PM UTC, Timothy Chow wrote:
> On 1/19/2024 8:19 AM, I wrote:
> > Now suppose X rolls something like 65, taking two checkers off, and
> > then O rolls 21 again. Evidently, 21/20 and 8/6 and 21/18 have exactly
> > the same equity
> Correction: 21/20 8/6 is apparently slightly better.
> XGID=-CDAb------------a-ic-----:1:-1:-1:21:0:0:3:0:10
>

Stick's reasoning is obviously faulty and I would not expect fewer backgammons than
they were. The point is that 21/18 only achieves one crossover, and retains the possibility
that the 18 point checker might incur wastage in going to the inner board.
Obviously, if the leader is on a last-roll position, we do what we can to run off the backgammon.
But 21/18 isn't a particularly efficient play and it isn't correct to minimise backgammon loss at
the cost of all other considerations.
Besides making an error in your previous post, your logic (as well as Stick's) is hard to follow.
You say that "evidently ... and ... have exactly the same equity." However, such an assertion can't
possibly be "evident" unless we can show that all future sequences (of any possible ply) lead to
the same outcome. It's not like you attempted to show this and made an error in your demonstration:
you didn't attempt to show it.

Paul

On 1/20/2024 5:32 AM, peps...@gmail.com wrote:
> Besides making an error in your previous post, your logic (as well as Stick's) is hard to follow.
> You say that "evidently ... and ... have exactly the same equity." However, such an assertion can't
> possibly be "evident" unless we can show that all future sequences (of any possible ply) lead to
> the same outcome. It's not like you attempted to show this and made an error in your demonstration:
> you didn't attempt to show it.

I attempted to "show" it by quoting the bot as an oracle, but
of course the problem was that I wasn't looking at enough decimal
places.

---
Tim Chow

On Saturday, January 20, 2024 at 7:09:58 PM UTC, Timothy Chow wrote:
> On 1/20/2024 5:32 AM, peps...@gmail.com wrote:
> > Besides making an error in your previous post, your logic (as well as Stick's) is hard to follow.
> > You say that "evidently ... and ... have exactly the same equity." However, such an assertion can't
> > possibly be "evident" unless we can show that all future sequences (of any possible ply) lead to
> > the same outcome. It's not like you attempted to show this and made an error in your demonstration:
> > you didn't attempt to show it.
> I attempted to "show" it by quoting the bot as an oracle, but
> of course the problem was that I wasn't looking at enough decimal
> places.

Oh, ok. Thanks. I was thoroughly confused. I had no idea (until now) the reason why you initially
saw the two plays as being equivalent.

Paul