Rollouts

 Settlement limit in races

 From: Alexander Nitschke Address: alexander.nitschke@ww.tu-berlin.de Date: 18 December 1997 Subject: Re: Middle game position Forum: rec.games.backgammon Google: 34996C64.C15903D9@ww.tu-berlin.de

```Kit Woolsey wrote:
> So, what should that certain point be?  Analysis of many "known"
> positions has indicated that if we use .550 as the settlement point, then
> the results appear to be accurate.  If we use a higher or lower number,
> the results appear to be distorted.  This is just judgment, but many good
> players have analyzed this problem and everybody has pretty much come up
> with the same result.  So, that is what we mean when we say that .550 is
> the settlement point for rollouts.

Fredrik Dahl wrote:
> Right. In races 0.55 is too low, however, particularly in short races.
> The idea behind the settlement procedure is that sometimes the doubler
> gets to cash takable positions, and other times he loses good chances to
> double his opponent in. We choose the settlement point in the hope that
> these effects will cancel out. For positions with gammon chances and
> plenty of contact 0.55 appears right, but in short races the drop point
> will often be below 0.55, so the doubler never gets the benefit of
> cashing takes. I would use 0.5 in races. In bearoffs it should be even
> lower, and for very short bearoffs the procedure of settlements is
> probably too crude to give accurate results.

Daniel Murphy wrote:
> What's the best settlement limit for rolling out a bearoff position
> like this one?
> Rephrased (since the equity for this position can be calculated
> precisely for all cube states), which settlement level makes the
> rollout agree most closely with the real values?
>
>   +24-23-22-21-20-19-+---+18-17-16-15-14-13-+
>   |    O  O  O  O  O |   |                  |
>   |       O  O  O    |   |                  |
>   |                  |   |                  |
>   |                  |   |                  |
>   |                  |   |                  |
>   |                  |   |                  |
>   |                  |   |                  |
>   |                  |   |                  |
>   |                  |   |                  |
>   |             X    |   |                  |
>   | X  X        X    |   |                  | 4
>   | X  X        X  X |   |                  |
>   +-1--2--3--4--5--6-+---+-7--8--9-10-11-12-+
>
> X on roll, holds cube at 4 (in the event, X redoubled and was punished
> by eventually winning 16 points)

I think there is no such thing as a general settlement limit for bear
off positions, the recube possibilities vary too much from position to
position.

Nevertheless, I found the idea of an empirically derived settlement
limit appealing, so I sought for an answer to this question.

First, the exact equities for this position:
Cubeless play.
Equity:                    0.415 (70.77%)
Cube in center.
Equity (no double):        0.643
Equity (double accepted):  0.698
Equity (double declined):  1.000
Cube with player 1.
Equity (no double):        0.682
Equity (double accepted):  0.698
Equity (double declined):  1.000
Cube with player 2. (Player 1 cannot double.)
Equity:                    0.349
Player 1 has a redouble, which player 2 should take.

Now, I had to try with JF level 5 rollouts to find the settlement limit,
which gives equities nearest to the exact equities above.

Settlement level 0.400:
Cubeless play:                0.414
Cube in center:               0.672
Cube with player on roll:     0.713
Cube with player not on roll: 0.355

Settlement level 0.450:
Cubeless play:                0.414
Cube in center:               0.639
Cube with player on roll:     0.680
Cube with player not on roll: 0.361

Settlement level 0.500:
Cubeless play:                0.414
Cube in center:               0.610
Cube with player on roll:     0.648
Cube with player not on roll: 0.366

Settlement level 0.550:
Cubeless play:                0.414
Cube in center:               0.577
Cube with player on roll:     0.612
Cube with player not on roll: 0.373

Summary:

For centered cube and cube with the player on roll a settlement limit of
0.450 seems about right. The results diverge only very little from the
exact results.
For cube with player not on roll the settlement limit in this position
must be lower than 0.400 to give correct results. I assume this is only
partly due to the settlement but also due to the perfect checker play
from the underdog which is required to obtain maximum equity. The
optimal checker play regarding to the cube location in such positions
can quite differ from optimal cubeless checker play (which is applied in
JF rollouts). And my observations are that the underdog must differ more
than the favorite.

In general the correct settlement limit for this position seems to be
0.450. I'd like to test some more reference positions though to make
some more experiments to get deeper insight into this topic. The
positions may not have more than roughly 9 checkers / 35 pips on each
side to determine the exact equities.

--
Alexander
```

### Rollouts

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Combining rollouts  (Gregg Cattanach+, Dec 2003)
Confidence intervals  (Bob Koca, Nov 2010)
Confidence intervals  (Timothy Chow, May 2010)
Confidence intervals  (Gerry Tesauro, Feb 1994)
Cubeless vs centered-cube rollouts  (Ron Karr, Dec 1997)
Duplicate dice  (David Montgomery, June 1998)
How reliable are rollouts?  (David Montgomery, Aug 1999)
Level-5 versus level-6 rollouts  (Michael J. Zehr, June 1998)
Level-5 versus level-6 rollouts  (Chuck Bower, Aug 1997)
Positions with inaccurate rollouts  (Douglas Zare, Oct 2002)
Reporting results of rollouts  (David Montgomery, June 1995)
Rollout settings  (Lokicol+, Apr 2010)
Settlement limit  (Michael J. Zehr, Apr 1998)
Settlement limit  (Kit Woolsey, Dec 1997)
Settlement limit in races  (Alexander Nitschke, Dec 1997)
Some guidelines  (Kit Woolsey, Apr 1996)
Standard error and JSD  (rambiz+, Feb 2011)
Standard error and JSD  (Stick+, Oct 2007)
Systematic error  (Chuck Bower, Oct 1996)
Tips for doing rollouts  (Douglas Zare, June 2002)
Truncated rollouts  (Gregg Cattanach, Oct 2002)
Truncated rollouts: pros and cons  (Jason Lee+, Jan 2006)
What is a rollout?  (Gregg Cattanach, Dec 1999)