Lowest equity contact position that's double/take (money)

backg...@email.com

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May 27, 2006, 6:03:42 AM5/27/06

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Hi
What's the lowest equity in money play in a contact position that's
double/take?.

Brad

David C. Ullrich

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May 27, 2006, 6:40:03 AM5/27/06

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-1.

>Brad

************************

David C. Ullrich

backg...@email.com

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May 27, 2006, 7:04:34 AM5/27/06

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Ok that's a correct double and correct take....

Hardy Hübener

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May 27, 2006, 8:12:57 AM5/27/06

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David C. Ullrich schrieb:

>>What's the lowest equity in money play in a contact position that's
>>double/take?.
>
>
> -1.
>

I don't think I would double if I had a equity of -1.

Hardy ;-)

--
Hardy's Backgammon Pages: www.hardyhuebener.de (Last update: May 2006)

Adam Tansley

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May 27, 2006, 10:08:16 AM5/27/06

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Brad:

"What's the lowest equity in money play in a contact position that's
double/take?"

It's not clear whether you mean lowest equity to double, or lowest
equity to take.

What's the lowest equity to take?
I'm sure you're familiar with the oft-quoted 25% gwc required to take,
but this ignores (i) the value of owning the cube and (ii) the effect
that gammons have on equity.
For the purposes of finding the lowest equity to take, we want to
remove gammons from the equation (since these raise the take point), so
we are looking at races, or since you specify a contact position,
holding games and the like.
Owning the cube means that we can take with less than 25% gwc. In
theory, we need only 20% gwc to take, provided the cube remains fully
live (we can recube when the opportunity arises). This is because
owning the cube means that when our gwc reach 80%, say, we in fact win
100% of the time by cashing. Thus the fully live cube adds 25% to our
gwc (that is 100% instead of 80%), raising our 20% gwc to 25%, as
required.
This is often referred to as the continuous model.
In practice, you won't find a contact position with a take point of 20%
gwc because a) the gammon rate won't be zero in a contact position, and
b) our recube vig only ever seems to get close (but not reach) the
theoretical possible amount.
Something like the following is probably about as low as you can get in
practice (this is a take with -0.5523 equity):

Rollout cubeless equity +0.5523
Cubeful equities:
1. Double, take +0.9902
2. Double, pass +1.0000 ( +0.0098)
3. No double +0.9268 ( -0.0634)
Proper cube action: Double, take
Rollout details:
Centered 1-cube:
0.7728 0.0101 0.0002 - 0.2272 0.0035 0.0001 CL +0.5523 CF +0.9268
[0.0008 0.0004 0.0000 - 0.0008 0.0002 0.0000 CL 0.0016 CF 0.0043]
Player Gnu owns 2-cube:
0.7784 0.0092 0.0001 - 0.2216 0.0029 0.0000 CL +1.1263 CF +0.9902
[0.0010 0.0005 0.0000 - 0.0010 0.0003 0.0001 CL 0.0040 CF 0.0054]
Full cubeful rollout with var.redn.1296 games, Mersenne Twister dice
gen. with seed 1028786508 and quasi-random dice
Play/Cube: 0-ply cubeful prune [expert]

What's the lowest equity to double?
This is trickier to answer, but I suspect that we are probably looking
at Kauder paradox situations. That is, in money play with the Jacoby
rule in force, positions where the correct cube action is
double/beaver. These arise because doubling activates our gammons, and
raises our negative equity.
In Advanced Backgammon, Robertie gives the following example (although
there are many others):

X wins when he rolls a 6, and loses to O's cube when he doesn't.
Winning just 30.56% of games, he should double to activate his gammons.
As Robertie explains, over a cross-section of 36 games, this reduces
X's loss from 14 points to 6 if O takes, or 12 if O beavers. X's equity
is thus -0.389 before cubing, and -0.333 after double/beaver.

tansley

backg...@email.com

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May 27, 2006, 11:35:40 AM5/27/06

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Thanks for the Kauder example which I'm aware of but I'll try and be
more precise with my question.
What's the lowest cubeless equity example someone can find that's a
correct initial double in a money game in a contact position NO
Jacoby.
Thanks

Brad

David Levy

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May 27, 2006, 11:44:29 AM5/27/06

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"Adam Tansley" <ad...@tansley.wanadoo.co.uk> wrote in message
news:1148738896.3...@j33g2000cwa.googlegroups.com...

> What's the lowest equity to take?

> In
> theory, we need only 20% gwc to take, provided the cube remains fully
> live (we can recube when the opportunity arises). This is because
> owning the cube means that when our gwc reach 80%, say, we in fact win
> 100% of the time by cashing. Thus the fully live cube adds 25% to our
> gwc (that is 100% instead of 80%), raising our 20% gwc to 25%, as
> required.

The theoretic minimum for an optional take is 18.75%.

To redo the math, the fully live cube increases our gwc by one-third from
75% to 100%. So in theory, 18.75% gwc, increased by one-third to 25%, can be
an optional take.

But enough theory. In practice give each side a single checker on the six
point. Non-roller has an optional take of roller's double with 18.75% game
winning chances.

Compare this to the position where each side has two checkers on the three
point (same pip count). Here non-roller must drop roller's double with
24.92% gwc because the cube ownership is worthless.

Grunty

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May 27, 2006, 1:23:10 PM5/27/06

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Hardy Hübener wrote:

> David C. Ullrich schrieb:

> > backg...@email.com wrote:

> >>What's the lowest equity in money play in a contact position that's
> >>double/take?.
> >
> > -1.
> >
>
> I don't think I would double if I had a equity of -1.

Yes, if you have a clue that your opponent will drop.

Raccoon

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May 28, 2006, 11:36:25 AM5/28/06

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Did you see the 1999 thread on this topic called "Lowest equity that's
still a take?"?

backg...@email.com

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May 28, 2006, 5:38:56 PM5/28/06

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I'm after the lowest equity that's a double.
I've seen examples where the cubeless equity is around 0.28 but it must
be lower.

bob_...@hotmail.com

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May 29, 2006, 2:42:27 AM5/29/06

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Try this one: X 1 checker on 17 point, 1 checker borne off and a
prime from 4 point to 10 point with two spares.
O 11 checkers on ace point, 1 on 7 point, 1 on 9
point, 1 on 10 point, 1 borne off.

This is essentially a last roll position. Cubeless X becomes a huge
(> 95%) favorite with a hit and if he misses he is an even huger dog
with a later hit and the race as the only possibilities. If it
actually were the case that X wins with 20 rolls and loses with 16 then
it is a correct initial double with equity of only .11

By adjusting the prime and spares placement we can probably cut down
on X's wins to just barely over half while keeping the double correct.
Note that if X misses he would be cubed out regardless of cube position
so the only thing keeping it from being a true last roll position is
those few losses after a hit.

Bob Koca

Raccoon

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May 29, 2006, 2:59:16 AM5/29/06

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A contact position with low equity for the DOUBLER that is a proper
double?

Then something like this:

X on roll
+13-14-15-16-17-18------19-20-21-22-23-24-+ O:

| X O O | | X O O O |
| X | | O O O |

| | | O O O |
| | | O O O |
| | | O |
v| |BAR| | (Cube: 1)
| | | |
| | | |
| | | |
| | | X X X X X X |
| | | X X X X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+ X:

or this:

X on roll
+13-14-15-16-17-18------19-20-21-22-23-24-+ O:

| X | | O O X O O O |
| X | | O O O |

| | | O O O |
| | | O O O |
| | | O |
v| |BAR| | (Cube: 1)
| | | |
| | | |
| | | |
| | | X X X X X X |
| | | X X X X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+ X:

or this:

X on roll
+13-14-15-16-17-18------19-20-21-22-23-24-+ O:

| O X | | X O O |
| | | O O |

| | | O O |
| | | O O |
| | | 9 O |
v| |BAR| | (Cube: 1)
| | | |
| | | |
| | | |
| | | X X X X X X |
| X | | X X X X X X |
+12-11-10--9--8--7-------6--5--4--3--2--1-+

backg...@email.com

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May 29, 2006, 6:29:35 AM5/29/06

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...Try this one: X 1 checker on 17 point, 1 checker borne off and a

prime from 4 point to 10 point with two spares.

I make that 18 checkers ???

O 11 checkers on ace point, 1 on 7 point, 1 on 9

point, 1 on 10 point, 1 borne off....

Any chance of a diagram?

Although I didn't mention it, I was actually looking for a position
with no checkers off but yours is interesting.

Thanks

Brad

backg...@email.com

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May 29, 2006, 6:32:45 AM5/29/06

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Ok looks good. So according to Snowie 3-ply position B wins it 0.198
cubeless.

Thanks

Brad

paulde...@att.net

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May 29, 2006, 12:39:23 PM5/29/06

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....

Raccoon,

Both these doubling suggestions seem to be blunders on your part. It's
a double if O cashes when X misses. But these are actually beavers
because O is too good when X misses.

Not 100% sure about the above, hence the word "seems". What do the
bots say?

Surely, the 2nd position you give is a beaver.

Hmmm, X has gammons too by twice rolling aces. Perhaps I'm wrong.

Paul Epstein

bob_...@hotmail.com

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May 29, 2006, 12:56:41 PM5/29/06

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I think I meant X has prime from 4 to 9 point with 1 spare.

Bob Koca

Raccoon

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May 29, 2006, 1:36:05 PM5/29/06

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paulde...@att.net wrote:
> Raccoon wrote:

> Raccoon,

>
> Both these doubling suggestions seem to be blunders on your part. It's
> a double if O cashes when X misses. But these are actually beavers
> because O is too good when X misses.

Just because O is too good to redouble if X misses doesn't make either
position a beaver. Yes, O can play on for gammon when X misses, but he
won't win many. And X wins some gammons, too. In both positions X must
be a favorite to win the game, with 20/36 hitting numbers, and X, with
two blots to shoot at, ought to win more gammons than O does.

> Surely, the 2nd position you give is a beaver.

Surely not! Besides X's gammon wins, also worth a mention is that
almost all of X's nonhitting numbers are better than average racing
numbers, which makes it less likely than it might seem that X fails to
run off the gammon. The average nonhitting number is 11.25 pips! So if
X misses X will only need 29 pips on average -- less than 4 rolls -- to
run off the gammon.

paulde...@att.net

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May 30, 2006, 9:02:36 AM5/30/06

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Raccoon,

I'd really appreciate it if you could tell me what the bots say.

Thanks,

Paul

Raccoon

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May 30, 2006, 12:41:45 PM5/30/06

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paulde...@att.net wrote:
> Raccoon,
>
> I'd really appreciate it if you could tell me what the bots say.

Did you misplace yours?

Raccoon

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May 30, 2006, 1:06:44 PM5/30/06

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backg...@email.com wrote:
> Ok looks good. So according to Snowie 3-ply position B wins it 0.198
> cubeless.

Two more positions with this theme:

GNU Backgammon Position ID: 3/cIAADbtgMABQ
Match ID : cAkAAAAAAAAA
+24-23-22-21-20-19------18-17-16-15-14-13-+ O: White
| O O O X X O | | | 0 points

| O O O | | |
| O O O | | |
| O O O | | |

Above, X's equity is quite low. 20 numbers hit and they don't all win
(if X has cubed), while 16 numbers miss and lose, with gammons hardly
worth worrying about.

Below, a position X can double (but not redouble) with only 19 hitting
numbers.

GNU Backgammon Position ID: /v4AAgDbtgNABA
Match ID : cAkAAAAAAAAA
+24-23-22-21-20-19------18-17-16-15-14-13-+ O: White
| O O X | | X O | 0 points

| O O | | |
| O O | | |
| O O | | |

In the second position X's equity isn't actually very low. X wins many
more games in the second position than in the first position, since X
can (and should) take O's redouble, if X misses.