Backgammon Rollouts

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

Did you find the information in this article useful?

Do you have any comments you'd like to add?