Let's first look at a fastest double/drop example in
gamblegammon:

=========================================
Player1 rolls 52, position equity +0.106, roll equity +0.030
Player1 moves 24/22 13/8, equity gained +0.014
Luck total EMG (Points), Player1 +0.030, Player2 +0.000

Player2 rolls 55, position equity -0.013, roll equity +0.492
Player2 moves 8/3*(2) 6/1*(2), equity gained +0.535
Luck total EMG (Points), Player1 +0.030, Player2 +0.505

Player1 dances, position equity -0.549, roll equity -1.000
Player1 can't move
Luck total EMG (Points), Player1 -0.422, Player2 +0.505

Player2 Proper cube action: Double, pass
=========================================

Now let's look at the worst case scenario starting
with the same 52 roll in levelgammon:

=========================================
Player1 rolls 52, position equity +0.106, roll equity +0.030
Player1 moves 24/22 13/8, equity gained +0.014
Luck total EMG (Points), Player1 +0.030, Player2 +0.000

Player2 will be given a "calculated roll" looking at the
temperature map and picking the roll with the nearest
equity, which is 61 in this case.

Player2 rolls 61, position equity -0.013, roll equity -0.019
Player2 moves 13/7 8/7, equity gained -0.016
Luck total EMG (Points), Player1 +0.030, Player2 -0.006

Looking at the temperature map, the *worst* dice that
Player1 can then randomly roll is 51, which will result in:

Player1 rolls 51, position equity +0.112, roll equity -0.141
Player1 moves22/16, equity gained -0.167
Luck total EMG (Points), Player1 -0.124, Player2 -0.006

Player2 Proper cube action: No double, beaver (22.7%)
=========================================

Since we don't have a bot tool to run long experiments
in levelgammon, I don't know what would happen after
1,000 or 10,000 games but most likely the majority of
the games will last much longer on the average than in
gamblegammon, they will be played out to the last rolls
and with fewer cube actions.

Surely there won't be any 3-roll double/drop sequences!
(nor any short, i.e. "dropped", games in general).

I expect that the "points per game" will be much lower
but "points per move" will be even more drasticlly lower
(I don't know if such a stats is kept but I think it would
be useful to see how much of the so-called "cube skill"
is a product of luck).

BTW: could anyone be good to explain the -1.000 equity
for "dancing rolls" in Gnubg's temperature map? Thanks.

MK

On 2023-01-12, MK <murat@compuplus.net> wrote:

> Let's first look at a fastest double/drop example

[52, split / 55 / dances]

>=========================================
> Player1 rolls 52, position equity +0.106, roll equity +0.030

> Player2 rolls 55, position equity -0.013, roll equity +0.492

> Player1 dances, position equity -0.549, roll equity -1.000

> BTW: could anyone be good to explain the -1.000 equity
> for "dancing rolls" in Gnubg's temperature map? Thanks.

After Player1 dances, Player2 will double and Player1's best choice will
be to pass. Even a 0 ply evaluation sees this. Hence his equity is -1.000.

On January 12, 2023 at 2:44:31 PM UTC-7, Philippe Michel wrote:

> On 2023-01-12, MK <mu...@compuplus.net> wrote:

>> BTW: could anyone be good to explain the -1.000 equity
>> for "dancing rolls" in Gnubg's temperature map? Thanks.

> After Player1 dances, Player2 will double and Player1's
> best choice will be to pass. Even a 0 ply evaluation sees
> this. Hence his equity is -1.000.

Ah, okay. I thought all dancing rolls were marked -1.000

What about dancing against a closed board? All rolls and
the average equity are -0.969 so you break even on luck
rate and not lose any equity while falling further behind
but the player bearing off is gaining equity with each roll??

I'm trying to understand how to use things like this in the
temperature map to level the luck...

MK

On 2023-01-13, MK <murat@compuplus.net> wrote:

> What about dancing against a closed board? All rolls and
> the average equity are -0.969 so you break even on luck
> rate and not lose any equity while falling further behind
> but the player bearing off is gaining equity with each roll??

The rolls of the closed-out player being all equally lucky should be
obvious. In fact, in live play, he wouldn't even bother to roll.

The other player doesn't necessarily gain equity though. It seems to be
the case in relatively normal positions, for instance in:

GNU Backgammon Position ID: dncHAEDbtg8AAA
Match ID : QQkXAAAAAAAA
+24-23-22-21-20-19------18-17-16-15-14-13-+ O: GNUbg (Cube: 2)
| O O O O O | O | | 0 points
| O O O O O | | |
| O O O O | | |
| | | |
| | | |
| |BAR| |v
| X | | |
| X | | |
| X | | |
| X X X X X X | | | Rolled 65
| X X X X X X | | | 0 points
+-1--2--3--4--5--6-------7--8--9-10-11-12-+ X: You
Pip counts: O 83, X 60

65 is slightly above average (that was a surprise to me) because 66, 55
and 44 are much worse.

On the other hand, in:

GNU Backgammon Position ID: 3wcAAHzbtg8AAA
Match ID : QQkXAAAAAAAA
+24-23-22-21-20-19------18-17-16-15-14-13-+ O: GNUbg (Cube: 2)
| O O | O | | 0 points
| O O | O | |
| O O | O | |
| O O | O | |
| O O | O | |
| |BAR| |v
| X | | |
| X | | |
| X | | |
| X X X X X X | | | Rolled 65
| X X X X X X | | | 0 points
+-1--2--3--4--5--6-------7--8--9-10-11-12-+ X: You
Pip counts: O 140, X 60

it is unlucky (according to 0 ply but the margin looks large enough to
stand in a deeper evaluation) and X loses some equity while still
keeping a closed board.

On January 14, 2023 at 4:38:29 PM UTC-7, Philippe Michel wrote:

> On 2023-01-13, MK <mu...@compuplus.net> wrote:
>> What about dancing against a closed board?
>> All rolls and the average equity are -0.969 so
>> you break even on luck rate and not lose any
>> equity while falling further behind but the player
>> bearing off is gaining equity with each roll??

> The rolls of the closed-out player being all
> equally lucky should be obvious.

Yes, I'm not basing any argument on that.

> The other player doesn't necessarily gain equity
> though.

Okay, but I won't dwell on rarities, (i.e. your second
example), since X's gains/losses are accounted for.

I'm questioning what happens to O's equity.

Based on your example, lets go back a little to:

Gnubg ID: dncHAEDbtgHgAA:QQkAAAAAAAAA
X's average: +0.862 O's average: -0.809

Gnubg ID: dncHAEDbth0AAA:QQkAAAAAAAAA
X's average: +0.893
-0.880

This is your example:
Gnubg ID: dncHAEDbtg8AAA:QQkAAAAAAAAA
X's average: +0.917 O's average: -0.919

After X rolls 65:
Gnubg ID: dncHAEC3bQcAAA:QQkAAAAAAAAA
X's average: +0.759 O's average: -0843

After X rolls 61:
Gnubg ID: dncHAEDbtgEAAA:QQkAAAAAAAAA
X's average: +0983 O's average: -0.924

Playing from first position above, X rolled/moved
4 times 66, 33, 65, 61 and O danced 5 times. Game
analysis shows X gained +0.265 but O lost +0.000

If I want to compensate O for the 5 times that it
danced by giving it proportionately lucky dice when
it can enter after X opens its board, can I somehow
figure it out from the averages of the positions?

While X gained +0.121 in four rolls on the average,
O lost -0.115 in five rolls.

From these numbers, can we derive the real equity
loss for O that is not accounted for?

Any other ideas?

MK