# Rollout: Vision laughs at counting

On Oct 26, 9:56 am, I wrote:
XGID=aBBBCBB----------------bc-:1:1:1:32:0:0:3:0:10

X:Player 2   O:Player 1
Score is X:0 O:0. Money session
+24-23-22-21-20-19------18-17-16-15-14-13-+
| O  O             |   |                  |
| O  O             |   |                  |
| O                |   |                  |
|                  |   |                  |
|                  |   |                  |
|                  |BAR|                  |
|                  | O |                  |
|                  |   |                  |
|          X       |   |                  | +---+
| X  X  X  X  X  X |   |                  | | 2 |
| X  X  X  X  X  X |   |                  | +---+
+-1--2--3--4--5--6-------7--8--9-10-11-12-+
Pipcount  X: 46  O: 32 X-O: 0-0
Cube  : 2, X own cube
X to play 32

This sort of position is deceptively simple-looking, but there are a
lot of things to consider.

Let's start by arguing that 3/off 2/off can't be better than 4/1 2/
off. Suppose we play 2/off and have to decide between 3/off and 4/1.
If O rolls 63 53 43 33, would you prefer that O dance or would you
prefer that O hit you? It should be clear that we would prefer O to
dance. Getting hit accomplishes nothing other than making us an
underdog to hit O as she comes around the board; the best-case
scenario is to put O back on the bar again, where she would be anyway
if she danced. Of course O's 31 from the bar plays better for us
after 3/off, but this is canceled out by the fact that O's 21 from the
bar plays better for us after 4/1. A play like 3/off 2/off is usually
correct only if O *already* has a blot in her home board that we're
trying to pick up. The only real advantage of 3/off 2/off is that it
takes off another checker, but since we're so far behind in the race
anyway, one extra checker off can't compensate for the extra lost
point. When we're behind like this, it's rarely correct to reduce to
a four-point board voluntarily when we can keep a five-point board.

Now for the more subtle question. Given that we are going to break a
point, which point should we break? Breaking a high point has the
following advantages: It typically uses pips more efficiently for the
race, and it typically allows us to keep a strong board a little
longer. Breaking a low point has the following advantages: If O
enters then we have a better chance of sending her back to the bar
with a hit, or, failing that, of blocking some of her rolls, perhaps
forcing O to play 2/1 with an ace. Also, keeping the high points
closed blocks some of our opponent's best racing rolls.

Generally, the rule of thumb is that if we are underdogs, then we
should play to use pips efficiently and keep our board longer, not
worrying so much about what to do if O enters since we're probably
going to be hosed anyway in that case. That means breaking a high
point and not worrying about leaving a blot on the broken point if we
can use the pips efficiently otherwise. On the other hand, if we're
slight favorites, then usually we should open up a lower point,
increasing our counterchances if O does enter. Finally, of course, if
we're big favorites (but gammons don't count or are very unlikely),
then we should just clear from the rear and avoid getting hit.

In the position I posted, Kleinman advocated 3/1 3/off, but the
rollout indicates that we're far enough behind that 5/off is better.
If we give O an extra checker, then it's pretty much a toss-up. If we
give O yet another checker, then 3/1 3/off comes out on top.

=======
Rollout
=======

1. Rollout: 5/0ff eq:-0.079
Player : 38.55% (G:0.00% B:0.00%)
Opponent: 61.45% (G:0.00% B:0.00%)
Confidence: ±0.006 (-0.085<E<-0.073)

2. Rollout: 4/1 2/0ff eq:-0.123 (-0.044)
Player : 36.77% (G:0.00% B:0.00%)
Opponent: 63.23% (G:0.00% B:0.00%)
Confidence: ±0.008 (-0.131<E<-0.115)

3. Rollout: 3/1 3/0ff eq:-0.136 (-0.057)
Player : 36.83% (G:0.00% B:0.00%)
Opponent: 63.17% (G:0.00% B:0.00%)
Confidence: ±0.006 (-0.142<E<-0.130)

4. Rollout: 4/2 3/0ff eq:-0.168 (-0.089)
Player : 35.05% (G:0.00% B:0.00%)
Opponent: 64.95% (G:0.00% B:0.00%)
Confidence: ±0.007 (-0.175<E<-0.161)

5. Rollout: 6/4 6/3 eq:-0.184 (-0.105)
Player : 34.37% (G:0.00% B:0.00%)
Opponent: 65.63% (G:0.00% B:0.00%)
Confidence: ±0.005 (-0.189<E<-0.179)

1296 Games rolled with Variance Reduction.
Dice Seed: 2
Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.21

=========
Variant 1
=========

XGID=aBBBCBB----------------cc-:1:1:1:32:0:0:3:0:10

X:Player 2 O:Player 1
Score is X:0 O:0. Money session
+24-23-22-21-20-19------18-17-16-15-14-13-+
| O O | | |
| O O | | |
| O O | | |
| | | |
| | | |
| |BAR| |
| | O | |
| | | |
| X | | | +---+
| X X X X X X | | | | 2 |
| X X X X X X | | | +---+
+-1--2--3--4--5--6-------7--8--9-10-11-12-+
Pipcount X: 46 O: 34 X-O: 0-0
Cube : 2, X own cube
X to play 32

1. Rollout: 5/0ff eq:+0.069
Player : 44.80% (G:0.00% B:0.00%)
Opponent: 55.20% (G:0.00% B:0.00%)
Confidence: ±0.003 (+0.066<E<+0.072)

2. Rollout: 3/1 3/0ff eq:+0.064 (-0.005)
Player : 45.43% (G:0.00% B:0.00%)
Opponent: 54.57% (G:0.00% B:0.00%)
Confidence: ±0.003 (+0.061<E<+0.067)

3. Rollout: 6/4 6/3 eq:+0.034 (-0.035)
Player : 43.66% (G:0.00% B:0.00%)
Opponent: 56.34% (G:0.00% B:0.00%)
Confidence: ±0.007 (+0.027<E<+0.041)

4. Rollout: 4/1 2/0ff eq:+0.006 (-0.063)
Player : 42.23% (G:0.00% B:0.00%)
Opponent: 57.77% (G:0.00% B:0.00%)
Confidence: ±0.007 (-0.001<E<+0.013)

Candidates 1 and 2: 5184 Games rolled with Variance Reduction.
Candidates 3 and 4: 1296 Games rolled with Variance Reduction.
Dice Seed: 2
Moves and cube decisions: 3 ply

=========
Variant 2
=========

XGID=aBBBCBB----------------cd-:1:1:1:32:0:0:3:0:10

X:Player 2 O:Player 1
Score is X:0 O:0. Money session
+24-23-22-21-20-19------18-17-16-15-14-13-+
| O O | | |
| O O | | |
| O O | | |
| O | | |
| | | |
| |BAR| |
| | O | |
| | | |
| X | | | +---+
| X X X X X X | | | | 2 |
| X X X X X X | | | +---+
+-1--2--3--4--5--6-------7--8--9-10-11-12-+
Pipcount X: 46 O: 35 X-O: 0-0
Cube : 2, X own cube
X to play 32

1. Rollout: 3/1 3/0ff eq:+0.352
Player : 57.19% (G:0.00% B:0.00%)
Opponent: 42.81% (G:0.00% B:0.00%)
Confidence: ±0.007 (+0.345<E<+0.359)

2. Rollout: 6/4 6/3 eq:+0.319 (-0.034)
Player : 55.19% (G:0.00% B:0.00%)
Opponent: 44.81% (G:0.00% B:0.00%)
Confidence: ±0.005 (+0.314<E<+0.324)

3. Rollout: 5/0ff eq:+0.275 (-0.078)
Player : 53.01% (G:0.00% B:0.00%)
Opponent: 46.99% (G:0.00% B:0.00%)
Confidence: ±0.005 (+0.270<E<+0.280)

4. Rollout: 4/1 2/0ff eq:+0.231 (-0.121)
Player : 51.12% (G:0.00% B:0.00%)
Opponent: 48.88% (G:0.00% B:0.00%)
Confidence: ±0.007 (+0.224<E<+0.238)

1296 Games rolled with Variance Reduction.
Dice Seed: 2
Moves and cube decisions: 3 ply

---
Tim Chow
.

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