Bearing In Against 2 Point Anchor (Part 1)

	is Professional score: 0 pip: 117 Money session Jacoby Beaver pip: 90 score: 0 is Me_v_Bot
XGID=–bBBBCBBB–a-a—abbbbb–:1:-1:1:53:0:0:3:0:10
to play 53

eXtreme Gammon Version: 1.21

The natural play here seems to be 9/6 9/4, which gives

	is Professional score: 0 pip: 112 Money session Jacoby Beaver pip: 82 score: 0 is Me_v_Bot
XGID=–bBCBDBB—a—–acbbbb–:1:-1:1:00:0:0:3:0:10
on roll, cube action?

eXtreme Gammon Version: 1.21

Now the issue with clearing from the 9 is apparent, a 65 forces a blot on the 7 point!

The correct play is the awkward looking 8/5 8/3

	is Professional score: 0 pip: 107 Money session Jacoby Beaver pip: 82 score: 0 is Player 1
XGID=–bCBCCB-B—a—–cbbcb–:1:-1:1:00:0:0:3:0:10
on roll, cube action?

eXtreme Gammon Version: 1.21

Now the worst role for white is 55, this still forces a blot but it is an indirect that can only be hit with 4 rolls rather than a direct.

I was hoping this could be generalized into a rule such as never leave 2 checkers each on the 7 and 8 points when bearing in against a two point board, but it is not that simple.

	is Professional score: 0 pip: 107 Money session Jacoby Beaver pip: 88 score: 0 is Player 1
XGID=–bBBBCCBA—a—–cbbcb–:1:-1:1:62:0:0:3:0:10
to play 62

* 1.	XG Roller+	9/3 7/5	eq: +0.724
	Player: Opponent: 85.87% (G:5.65% B:0.09%) 14.13% (G:0.45% B:0.01%)
2.	XG Roller+	9/1	eq: +0.704 (-0.020)
	Player: Opponent: 85.00% (G:5.38% B:0.08%) 15.00% (G:0.52% B:0.01%)
3.	XG Roller+	7/1 6/4	eq: +0.666 (-0.058)
	Player: Opponent: 83.29% (G:5.57% B:0.11%) 16.71% (G:0.73% B:0.02%)
4.	XG Roller+	9/3 6/4	eq: +0.634 (-0.090)
	Player: Opponent: 82.25% (G:4.77% B:0.09%) 17.75% (G:0.83% B:0.02%)
5.	XG Roller+	7/1 3/1	eq: +0.384 (-0.340)
	Player: Opponent: 73.09% (G:7.05% B:0.19%) 26.91% (G:6.20% B:0.15%)

eXtreme Gammon Version: 1.21

Here the answer is to do just that! To see why let’s examine the alternatives

	is Professional score: 0 pip: 100 Money session Jacoby Beaver pip: 80 score: 0 is Player 1
XGID=–bCBCCBB———-ccbcb–:1:-1:1:00:0:0:3:0:10
on roll, cube action?

eXtreme Gammon Version: 1.21

This is the position after the best play. There are 3 bad rolls for white in this position, 65 as in the position above leaves a blot on the 7, 55 is even worse in this case as it leaves a blot on the 6 point.  But what are the options?

9/1 is the second best option, after 9/1 we have:

 

	is Professional score: 0 pip: 96 Money session Jacoby Beaver pip: 80 score: 0 is Player 1
XGID=-AbBBBCCB———-cbbcba-:1:-1:1:52:0:0:3:0:10
to play 52

eXtreme Gammon Version: 1.21

At first pass this looks about the same as 9/3 7/5, 55 blots, this time on the 8 instead of the 6 and 64. Both plays leave 3 blot rolls this time, so the difference seems to lie in what happens with the none blotting rolls.  After 9/3 7/5 the non blotting rolls play pretty well. But after 9/1 some of the non-blotting rolls are problematic. For example to avoid leaving a blot with 44 will force a gap on the 5, 66, 65 and 63 all leads to blotting numbers on the next roll

 

One has to look even deeper to see way 7/1, 6/4 falls so far behind. It leaves only 65 as an immediate blotting number, which would entice one to make the play on a quick analysis.  However, after 7/1 6/4 playing 6s creates a lot of problems for future rolls.

 

It is relatively easy to analyze this type of position with the aid of XG or GNUBG, the dice distribution feature in XG and Temperature Map in GNUBG aid in determining the problematic rolls after a play. The real issue is seeing these problems OTB, and doing it fast enough to be able to make the right play. I have a long way to go on that, so it is time to practice, practice, practice.

Advertisement

Chat Test

http://backgammonchat.chatango.com/group
[ Copy this | Start New | Full Size ]

Rollouts from Magriel’s Backgammon

 

NOTE: THIS IS A WORK IN PROGRESS, ADDITIONAL ROLLOUTS ARE STILL BEING CONDUCTED AND WILL BE ADDED AS THEY BECOME AVAILABLE.

I recently started reading Paul Magriel’s classic Backgammon. I had avoid this book because I knew that it was prebot and contained some errors. However I was struggling—even with reading Trice—to put to get clear in my mind some of the guiding principles that were being used to drive plays in various positions. After reading some comments online about Magriel’s book I decided to give it a try. I wanted to know – beyond the openings—were Magriel was not in agreement with the bots. My search turned up the following link at Hardy’s Backgammon pages:  http://www.hardyhuebener.de/engl/magriels_irrtuemer.html.  This is excellent information, however the rollouts are all cubeless and GNUBG 0-ply. So I decided to use this list as a starting point and reroll these positions in XG.

I choose money play, no Jacoby, cubeful, 3-ply for the rollout parameters. Money play without Jacoby was chosen because I believe it will give the best overall information for checker play in most money situations as well as most match play situations where the checker play is not score dependent.  Using 3-ply was a practical consideration for doing the rollouts in a reasonable amount of time and still providing a high degree of accuracy. I screened the candidate plays using XG Roller + and included at least all the plays that were better than Margriel’s in the longer rollout. I used 1296 as the base number of games, but in some cases extended until the play either reached statistical significance (i.e. the confidence intervals between the top play and Magriel’s did not overlap) or until it was pretty clear that the plays were to close to call. For example in Position 12.1 the equity difference between Magriel’s play and the top play is 0.003, but both the top play and Magriel’s have confidence intervals of ±0.003 after 41,472 games, so this is a too close to call situation. Magriel’s plays are shown in bold italics in the rollout results.

There are some significant differences between these results and the GNUBG published at Hardy’s Backgammon Pages, undoubtedly due to differences in bots, plys, and choice of settings. This work is not intended as a criticism of the work done by the author of that site, therefore I will not go into a position by position comparison of the results from the two projects.

Position 12.1

1.	Rollout1	8/7 8/5	eq: +0.364
	Player: Opponent: 58.83% (G:15.71% B:0.59%) 41.17% (G:7.65% B:0.17%) Conf: ± 0.003 (+0.361…+0.367) Duration: 5 hours 49 minutes
2.	Rollout1	7/6 7/4	eq: +0.361 (-0.003)
	Player: Opponent: 57.97% (G:16.67% B:0.62%) 42.03% (G:8.56% B:0.23%) Conf: ± 0.003 (+0.358…+0.364) Duration: 5 hours 20 minutes
3.	Rollout1	8/6(2)	eq: +0.354 (-0.010)
	Player: Opponent: 58.09% (G:15.96% B:0.59%) 41.91% (G:7.94% B:0.19%) Conf: ± 0.003 (+0.351…+0.357) Duration: 4 hours 37 minutes
1 41472 Games rolled with Variance Reduction. Dice Seed: Random Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.13

Position 12.4

1.	Rollout1	10/9 6/4	eq: +1.000
	Player: Opponent: 78.72% (G:3.05% B:0.06%) 21.28% (G:1.31% B:0.01%) Conf: ± 0.000 (+1.000…+1.000) Duration: 1.9 seconds
2.	Rollout1	9/8 6/4	eq: +0.982 (-0.018)
	Player: Opponent: 78.29% (G:2.57% B:0.06%) 21.71% (G:0.94% B:0.01%) Conf: ± 0.006 (+0.976…+0.988) Duration: 45.1 seconds
3.	Rollout1	9/7 6/5	eq: +0.965 (-0.035)
	Player: Opponent: 78.94% (G:2.77% B:0.05%) 21.06% (G:0.88% B:0.01%) Conf: ± 0.005 (+0.960…+0.970) Duration: 1 minute 25 seconds
4.	Rollout1	10/8 6/5	eq: +0.893 (-0.107)
	Player: Opponent: 76.18% (G:3.03% B:0.06%) 23.82% (G:1.43% B:0.02%) Conf: ± 0.001 (+0.892…+0.894) Duration: 1.9 seconds
5.	2 ply	6/5 6/4	eq: +0.889 (-0.111)
	Player: Opponent: 74.90% (G:2.98% B:0.07%) 25.10% (G:3.13% B:0.04%)
1 1296 Games rolled with Variance Reduction. Dice Seed: Random Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.13

Position 13.20

1.	Rollout1	24/18 13/11	eq: -0.281
	Player: Opponent: 43.91% (G:7.68% B:0.35%) 56.09% (G:13.56% B:0.64%) Conf: ± 0.020 (-0.301…-0.261) Duration: 1 hour 06 minutes
2.	Rollout1	24/22 13/7	eq: -0.306 (-0.026)
	Player: Opponent: 42.86% (G:6.83% B:0.28%) 57.14% (G:12.20% B:0.45%) Conf: ± 0.018 (-0.324…-0.288) Duration: 58 minutes 32 seconds
3.	Rollout1	24/22 24/18	eq: -0.317 (-0.036)
	Player: Opponent: 42.83% (G:7.07% B:0.32%) 57.17% (G:13.23% B:0.51%) Conf: ± 0.021 (-0.338…-0.296) Duration: 1 hour 00 minute
4.	Rollout1	13/11 13/7	eq: -0.411 (-0.130)
	Player: Opponent: 40.29% (G:7.82% B:0.36%) 59.71% (G:14.03% B:0.68%) Conf: ± 0.019 (-0.430…-0.392) Duration: 59 minutes 14 seconds
1 1296 Games rolled with Variance Reduction. Dice Seed: Random Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.13

Position 13.25B

1.	Rollout1	20/16 10/4	eq: +1.421
	Player: Opponent: 88.52% (G:60.19% B:1.39%) 11.48% (G:2.48% B:0.23%) Conf: ± 0.009 (+1.412…+1.430) Duration: 7 minutes 36 seconds
2.	Rollout1	20/14 10/6	eq: +1.408 (-0.013)
	Player: Opponent: 88.36% (G:59.80% B:1.37%) 11.64% (G:2.46% B:0.23%) Conf: ± 0.008 (+1.400…+1.416) Duration: 6 minutes 46 seconds
3.	Rollout1	10/4 9/5	eq: +1.405 (-0.017)
	Player: Opponent: 87.65% (G:57.48% B:1.24%) 12.35% (G:1.98% B:0.11%) Conf: ± 0.008 (+1.397…+1.413) Duration: 5 minutes 19 seconds
4.	Rollout1	20/14 9/5	eq: +1.404 (-0.018)
	Player: Opponent: 88.21% (G:59.58% B:1.37%) 11.79% (G:2.55% B:0.23%) Conf: ± 0.008 (+1.396…+1.412) Duration: 4 minutes 19 seconds
5.	Rollout1	20/16 9/3	eq: +1.402 (-0.020)
	Player: Opponent: 88.25% (G:59.56% B:1.38%) 11.75% (G:2.62% B:0.24%) Conf: ± 0.009 (+1.393…+1.411) Duration: 4 minutes 51 seconds
1 2592 Games rolled with Variance Reduction. Dice Seed: Random Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.13

Position 14.16

1.	Rollout1	8/4 6/4	eq: +0.447
	Player: Opponent: 63.05% (G:9.98% B:0.60%) 36.95% (G:8.63% B:0.30%) Conf: ± 0.015 (+0.432…+0.462) Duration: 22 minutes 52 seconds
2.	Rollout1	14/8	eq: +0.305 (-0.142)
	Player: Opponent: 58.90% (G:8.41% B:0.44%) 41.10% (G:9.82% B:0.27%) Conf: ± 0.016 (+0.289…+0.321) Duration: 20 minutes 27 seconds
3.	Rollout1	7/1*	eq: +0.302 (-0.145)
	Player: Opponent: 59.80% (G:7.72% B:0.29%) 40.20% (G:10.83% B:0.34%) Conf: ± 0.017 (+0.285…+0.319) Duration: 25 minutes 49 seconds
1 1296 Games rolled with Variance Reduction. Dice Seed: Random Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.13

Position 16.9

Magriel only shows the 3, I placed a checker on the bar and made the other dice roll a 4 to arrive the position in Magriel.

1.	Rollout1	Bar/21 13/10	eq: -0.178
	Player: Opponent: 45.45% (G:10.68% B:0.51%) 54.55% (G:13.25% B:0.62%) Conf: ± 0.022 (-0.200…-0.156) Duration: 52 minutes 54 seconds
2.	Rollout1	Bar/21 8/5*	eq: -0.272 (-0.094)
	Player: Opponent: 45.20% (G:12.42% B:0.67%) 54.80% (G:20.24% B:1.41%) Conf: ± 0.028 (-0.300…-0.244) Duration: 1 hour 03 minutes
1 1296 Games rolled with Variance Reduction. Dice Seed: Random Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.13

Position 17.8

1.	Rollout1	13/10	eq: -0.606
	Player: Opponent: 33.70% (G:7.81% B:0.28%) 66.30% (G:15.57% B:0.80%) Conf: ± 0.018 (-0.624…-0.588) Duration: 26 minutes 41 seconds
2.	Rollout1	24/23 13/11	eq: -0.690 (-0.084)
	Player: Opponent: 31.83% (G:6.83% B:0.27%) 68.17% (G:16.65% B:0.72%) Conf: ± 0.022 (-0.712…-0.668) Duration: 44 minutes 54 seconds
3.	Rollout1	24/23 8/6	eq: -0.720 (-0.114)
	Player: Opponent: 31.03% (G:6.22% B:0.19%) 68.97% (G:13.85% B:0.58%) Conf: ± 0.024 (-0.744…-0.696) Duration: 29 minutes 22 seconds
4.	Rollout1	8/5	eq: -0.731 (-0.125)
	Player: Opponent: 31.70% (G:6.66% B:0.22%) 68.30% (G:18.70% B:0.95%) Conf: ± 0.025 (-0.756…-0.706) Duration: 47 minutes 01 second
5.	Rollout2	13/11 8/7	eq: -0.740 (-0.134)
	Player: Opponent: 32.52% (G:7.07% B:0.26%) 67.48% (G:21.43% B:1.20%) Conf: ± 0.018 (-0.758…-0.722) Duration: 1 hour 34 minutes
1 1296 Games rolled with Variance Reduction. Dice Seed: Random Moves and cube decisions: 3 ply 2 2592 Games rolled with Variance Reduction. Dice Seed: Random Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.13

Position 18.3

1.	Rollout1	Bar/24 11/6	eq: -0.810
	Player: Opponent: 29.52% (G:7.39% B:0.21%) 70.48% (G:16.60% B:1.15%) Conf: ± 0.022 (-0.832…-0.788) Duration: 34 minutes 54 seconds
2.	Rollout1	Bar/24 23/18	eq: -0.891 (-0.082)
	Player: Opponent: 30.66% (G:9.52% B:0.51%) 69.34% (G:25.53% B:2.58%) Conf: ± 0.025 (-0.916…-0.866) Duration: 38 minutes 47 seconds
3.	XG Roller+	Bar/24 13/8	eq: -1.000 (-0.190)
	Player: Opponent: 27.86% (G:6.81% B:0.17%) 72.14% (G:22.82% B:2.47%)
4.	XG Roller+	Bar/24 6/1	eq: -1.000 (-0.190)
	Player: Opponent: 26.02% (G:4.99% B:0.09%) 73.98% (G:21.62% B:1.53%)
1 1296 Games rolled with Variance Reduction. Dice Seed: Random Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.13

Title Change

 

You may have noticed I have changed the title of this blog, I have also changed the “What is this Blog About” page.  I am still struggling with finding time to write much here, and I have certainly lost the original theme of the blog. Hopefully what I do post will still be of benefit, but all I can say right now is the blog is “about Backgammon.”

Opening 21

Yes, I know, I am not being consistent at all with what I am posting here based on the stated theme, but part of me is still trying to get a handle on the best way to learn this game, so I am going all over the place. For those few readers I have hopefully you will understand and bear with me, otherwise by blog becomes simply a place for me to try to organize my own study which is okay as well.

Opening Roll Source Material

Most of the information I am using for my study of opening rolls is coming from the opening information at http://www.bgonline.org/., these are often referred to as “Stick’s” rollouts because bgonline is maintained by Stick Rice. Timothy Chow, who comments here frequently, has published a very handy summary of 2nd roll information which can be found at http://www-math.mit.edu/~tchow/2ndmove.shtml. You will also find on Timothy’s page links to information on nactation, a link to a similar summary of opening moves using standard notation, and a link to the opening information on Tom Keith’s Backgammon Galore site. I will not bother to duplicate those links here at the moment.

These charts are very handy, but I am finding that to get a grasp on the material I need more than that, so I am attempting to group and offer some reasoning for the plays in an attempt to more easily learn what to do rather than rely on brute force memorization. I am also trying to get more comfortable with using nactation, so I am using it mixed with traditional notation and simple descriptive text (eg, make the 20pt) below.

Opening 21

There are basically two options for playing the opening 21, slotting (21$, 13/11 6/5) and splitting (21S, 13/11 24/23).

My tendency has been to play 21S except at gammon-go scores where I tended to the slot. The rollouts indicate that 21$ is a slightly better play than 21S. One thing I did find surprising is that according to Stick’s rollouts 21$ is best even at gammon save scores, Keith’s rollouts differ from this but those are cubeless rollouts. However the difference in slot and split at GS in Stick’s rollout is very small, probably not statistically significant.

I have started playing 21$ instead of 21S as my typical 21 opening. I am still struggling with this in a couple of respects, one is mentally dealing with getting hit, especially after a roll of 64 where you end up with two checkers on the bar. More difficult for me at this point is playing my second roll when I am unable to cover the slotted checker. This is an area I need to work on.

Replies to 21$

Hit the slotted checker if possible except with 11.  This is pretty easy to remember as it seems quite logical. The only question that remains is what do with the rest of the roll, of course with 31 you don’t have a rest of the roll after hitting.

11: Is the one exception to the hit the slot if you can rule. The best play with 11 is to make your 5 and 7 points. (nactation: N). Hitting here is around a 0.03 error. Without doing a lot of work to try and understand this one it seems that the value of making the 3 prime and having a good shot at a quick 4 prime must outweigh the value of hitting. Still this seems like a less than obvious best play to me.

22: This one can gives me pause OTB as it is very tempting to make the 20pt – given all we hear about the value of doing that, but the best option here (based on Keith’s data and bot evaluations) is to hit the slot and make the 4 point. (nactation: e)

41: Play 24/23 with the second checker, the two other options of slotting the 5 or 7 pts leave too many return shots. (nactation: U)

42: Play the second checker down to the 11 pt, it gives you a builder and can only be hit with 65. (nactation: S)

43: Stick’s rollouts (which for 2nd rolls do not have any at score information) gives a very slight edge to playing 24/21 with the second checker, Keith’s are the same except at GG Keith’s play is down (13/10). This makes sense as you need to contain checkers for the gammon and getting the extra builder down improves you chances of doing this. (U, except at GG S)

44: Pretty easy, make the 20 pt and the 9pt. (B)

54: A very slight edge goes to bringing a checker down to the 8pt (S), running the checker out after the hit ( R) is not far behind although I find it surprising that R is not better at least at GS scores)

64: Easy, run your checker to the 14pt hitting both blots in the process. (K)

Split Against The Slot.  At first blush splitting when you cannot hit the slotted checker may not seem the best thing to do since you are giving your opponent two blots to shoot out and the slot gives them more ways to hit. However, some of the hitting plays that present themselves after these splits are not necessarily all that good for your opponent as they either bury checkers, force them off their 5pt, and/or leave good return shots.

32, 51, 52: S

21: Timothy’s chart gives both S and $ as options here, recent extensive 4ply XG rollouts by Neil Kazaross posted on the BGonline forum make it pretty clear that the $ is best, but the difference is still only 0.005 http://www.bgonline.org/forums/webbbs_config.pl?noframes;read=59007

62: Has 3 viable options S (24/18 13/11), Z (13/7 24/22) and N (13/7, 6/4). Running out with a back checker (R) and bringing two down from the midpoint (D) have also gotten consideration. Based on recent rollout data S is best with Z being close and N not totally out of the question. http://www.bgonline.org/forums/webbbs_config.pl?noframes;read=58683

63: Running a checker out (R) is the best play here rather than splitting, and the reverse split (24/22 13/7, Z) is slightly better than the normal split. None of the plays are huge errors (S being about –0.015 worse than R)

Rather Obvious Plays. 53, 55 both make your 3 pt, 65 always runs a checker on the 2nd roll except for against 61P where you can’t, and 66 is forced 61 make your bar point.

And Finally.  33: Make your 5 pt and the 22pt. This play is harder to remember than it seems because in many cases the 2nd 33 is not used to make the 5 pt even though that might seem like the logical play at first glance. My instinct with this play is to make my 5 and 3 pt because having two home board points seems like such a good idea. But that is behind the best play and the more typical 33 second roll play of making the 22 and the 10 (B).

Replies to 21S

Play Like First Roll.  Many of the replies to 21S are played liked the first roll—or at least like of the main options for the first roll. These are:

31P, 51S, 61P, 32S, 42P, 52S, 54S, 62S, 63S (or 63R), 65R

Other Non-doubles

21: Playing 24/21 (21U) has a very slight edge over 21S, but both are viable plays.

41K: Hit both blots in your home board.

43:

53P or 53S: 53P gets an extremely small edge, but splitting is very close which is why I put it here rather than in the play like first roll group.

64H: Hit the checker on the 15 with 24/15

Doubles

11e (6/5(2),24/22) – Against the slot 11 is played as N, but with the split playing N gives shots with 6s and 7s, so this requires a safer play

22N, make the 4 and 11 points.

33B, make the 21 and 10 points.

44B, make the 20 and 9 points.

33 and 44 are both difficult for me to keep in mind because I have to fight the desire to make the 5 point with them. Of course with 44 this is moving two from the 13, with 33 you still have 2 3s to play. While making the 5pt with a 33 on the second roll is right at times, making it with a 2nd roll 44 is almost never right. 

55A, Hit on the 1 and make the 3, giving you a good start on a blitz.

66B Same thing for 66 on all 2nd rolls except after 61P, where you can’t, B with a 66 makes both bar points.

Things to Remember – Handy Rules and Guides

One of the things that I am struggling with is remembering all of the rules of thumbs, guidelines, etc. that I am coming across as I attempt to improve my game. So I decided to create a post that will expand as I collect more information, and hopefully find that it is helpful to me and others. Some of the items have been discussed in other places, but this is sort of a “cheat sheet” for trying to keep things in a handy reference. (And yes, I know, I am not posting much here as I seem to always be behind on my backgammon study)

Safe/Bold Criteria

Trice discusses this and I think I dealt with it in an earlier post, It is a guide for deciding whether to play safe or take more risk to gain a positional advantage.

The follow description of safe/bold criteria is taken from http://www.bkgm.com/articles/Townsend/BasicsOfBackgammon1.html

Here are some additional criteria for bold vs. safe play and how to interpret them:

1. Home board points. Relative strength in home board points allows for more aggressive play.
2. Anchors. One anchor vs. none allows for more aggressive play, this extends to:
   • Multiple anchors vs. single anchor.
   • A higher anchor vs. a lower anchor.
3. Blots in your home board discourage aggressive play (if you hit, they may hit you as they re-enter). Conversely enemy blots in his home board invite an exchange of hits.
4. More back men allow for more aggressive play, while counterintuitive, this is based on sound logic:
   • More than one man back can combine to form an anchor.
   • A single man back can only escape or be attacked.
   • A hit with only one man back may critically reduce your racing lead.

Other factors to be evaluated include outer board blocking points (especially points that block the escape of an enemy back man with 5’s or 6’s), blots exposed (the side with more exposed blots will wish to clean them up rather than expose more in many cases), and racing lead (lead discourages bold play).

The Last Two Checkers

Trice gives a chart of the number of the rolls that bears off one or two checkers at the end of game. Trice discourages memorization of the chart and supports learning to calculate the rolls. I find that – at least in online play—taking the time to calculate the rolls is a bit of an issue and my memory is not what it used to be.  While you have to know the number of rolls for doubling decisions there is a handy rule for determining how to place you checkers when playing. This will usually be where you have born off the 13th checker with half the roll and then must decided how to play the other half of the roll, for example if the have checkers on the 5 and 3 with a 1 to play should you move a checker from the 5 to the 4 or from the 3 to the 2.  To determine the best play use 3, 2, 4, 1, 5, 0 as a guide to determine how far apart to place you checkers, 3 apart being the best if possible, followed by 2 apart and so on. [Note: In a comment on this article Timothy Chow points out that an easier way to remember this is to place your two checkers as close to 2.7 pips apart as possible.]

So in the above example moving 3/2 is better than moving 5/4 because it places the checkers 3 apart rather than 1 apart.

For a nice chart of 1 and 2 checker positions and rolls to bear off see http://www.bkgm.com/articles/Koca/CuringYourShortBearoffBlues.html.

N-Roll Positions

N-Roll positions where the number of rolls to bear off without a double can be readily determined. This would be situations where there are checkers only the ace and deuce point, N being the number of checkers on the points divided by 2 and rounded up. For N roll positions the player on roll has the following chances of winning the game (rounded, cubeless):

2 – 86%

3 – 79%

4 – 75%

5 – 72%

Some positions that are not true N-roll positions will be very close, for example if you have 1 checker on the 3pt, 2 on the 2 pt, and 3 on the ace point you do not have a true N-roll position, but unless you roll 2-1 three times in a roll it is not going to matter.

Race Formulas

When there is not longer contact there are a variety of formulas that can be used as a guide to determine if you should double. None is perfect, but most are better than just guess. The two I use are:

Walter Trice’s Method which he discusses in Boot Camp:  I have rearranged it some to make it more formulaic, but this is the same logic:

If leaders pip count is > 62 Trice Number= leader count/10 + 1 round up

If leaders pip count < 62 Trice Number = (leader count-5)/7 round down

Count Difference = Trailer Count – Leader Count

If Count Difference < Trice Number  — Take

If Count Difference +3 > Trice Number – Double

If Count Difference + 2 > Trice Number – Redouble

Trice applies this formula to low wastage positions. In positions with higher wastage Trice applies the above to an adjusted pip count. In Boot Camp he uses the Ward count to make adjustments to pip counts but applies the decision criteria above.

Keith Count – Is the adjusted pip count method I use. The Keith Count works as follows:

Start with the raw pip count

add 2 for each checker more than 1 on the ace point

add 1 for each checker more than 1 on the 2 point

add 1 for each checker more than 3 on the 3 point

add 1 for each gap on the 4, 5 and 6

For player on roll add 1/7 of pip count (round down)

Once the above is calculated for each side

Keith Number = Adjusted Count for Player on Roll – Adjusted count for opponent

If Keith Number < 4 double

If Keith Number < 3 Redouble

If Keith Number > 2 Take

For an excellent discussion of the Keith Count as well as other formulas and race theory in general see http://www.bkgm.com/articles/CubeHandlingInRaces/

Bearing Off with Opponent on Bar But With Checkers Off

This type of position arises from a backgame, ace point game, or similar situation. You managed to hit your opponent and send them to the bar after they started bearing off. Now you are bearing off with the opponent on the bar and need to decided whether to bear off two checkers leaving a blot in your home board or to move to avoid the risk. Bill Robertie gives the following guidance for these situations:

The first metric we want to calculate is the crossover count. A crossover is simply a move of a checker from one quadrant to another, or from the bar to the opponent’s inner board, or from the inner board to the bearoff.

Next we employ the following rule of thumb:
>If you trail by two or less in the crossover count, play safe. You’re doing well enough in the race that there’s no need to take additional risks.
>If you trail by five or more in the crossover count, take two checkers off and leave a blot. You’re a big underdog in the race, and you need the extra checker speed.
>If you trail by three or four, you’re in a grey area.

In the grey area, decisions depend very much on the exact arrangement and count of checkers in the inner board. You next want to look at all of the following considerations and see if they point toward one play or another.
(1) If you trail by three crossovers, tend to play safe. If you trail by four, tend to bear off.
(2) If White has a blot in his board, tend to bear off. If no blot, tend to play safe.
(3) If taking two checkers off brings you to an even number of checkers, tend to bear off, otherwise tend to play safe.
(4) If you have a speed board, tend to play safe, otherwise tend to bear off. A speed board is one where Black’s home board spares are heavily concentrated on the one and two points, which implies that small doubles are more likely to bear off four checkers through the bearoff. With a slower board, where the checkers are spread evenly across points, small doubles often won’t save a roll.

The above is taken from a problem solution on the Backgammon Forum at 2+2 Poker Server. http://forumserver.twoplustwo.com/138/backgammon/problem-week-39-solution-651194/

Game Winning Chances When You Have One Checker on the Bar and Checkers Off

This is another bearoff situation where you have borne off several checkers and end up with one checker closed out on the bar. The following assumes that Red has a fast board (all checkers on the lower points) and white has the idea bear off position with a closed board and the spares on the 6, 5, and 4 pts. Then the cubeless winning percentages are:

% to win in reference positions
N off	1	2	3	4	5	6	7	8	9	10	11	12
Red	4.5	7	11	15	22	30	39.5	51	58	69	76.5	84
White	95.5	93	89	84	78	70	60.5	49	42	31	23.5	16
d		2.5	4	4.5	7	8	9.5	11.5	7	11	7.5	7.5

The last line shows the difference for red (the player on the bar) by having one more checker off than is in the top column.

This table can be used not only to aid in doubling decision but in  making decisions about the best way to play certain rolls that force you to leave a blot in the bearoff. It is taken from an article by François Tardieu in which he discusses using this information to make these types of decisions. The complete article can be found at http://www.gammonlife.com/writers/07tardieuart1.htm.

I See Said the Blind Man

 

Despite my best intentions of posting regularly and working through 501 Essential Problems along with Boot Camp my blog is going largely unattended. I am playing a lot of online tourneys, but not spending the in study and analysis I should be. And writing things for a blog as opposed to notes for myself takes more time. But I wanted to post a position that I was just looking at from the analysis of one of my matches. Maybe it is overly simple, but it took be quite a while to see what would have lead me to the right answer so maybe someone else will find it of interest.

My thought process when I played 20/17 with the three was to reduce the chances of getting hit.  Playing 20/17 as opposed to the correct play of 8/5 with the 3 leaves 9 shots as opposed to 11 shots.

When I first looked at this error in the analysis I was left scratching my head about what I would have seen during the match that would have lead me to the correct the play. My play actually gives a fractionally high winning percentage, but is a blunder based on equity due to the extra gammons (and backgammons) picked up with the correct play. 

I was about to give up on ever being able to see this in the course of a match, when suddenly the answer hit my, and it is remarkable simple. While leaving the blot on the 5 gives black 2 more shots at hitting it, it also gives white potentially 3 more shots at picking up the blot next turn if black is unable to move it.  The moral of the story for me is to not only think defensively, but think about how to give yourself the best chance of picking up the blot. 

	is Player 2 score: 1 pip: 100 11 point match pip: 119 score: 0 is Player 1
XGID=-b–CCBCA—-B–a–cabbbbA:0:0:1:53:0:1:0:11:10
to play 53

1.	Rollout1	Bar/20* 8/5	eq: +1.051
	Player: Opponent: 80.01% (G:26.77% B:2.06%) 19.99% (G:4.55% B:0.06%) Conf: ± 0.012 (+1.039…+1.063) Duration: 8 minutes 32 seconds
2.	Rollout1	Bar/20* 20/17	eq: +0.925 (-0.126)
	Player: Opponent: 80.52% (G:25.51% B:1.77%) 19.48% (G:7.29% B:0.12%) Conf: ± 0.011 (+0.914…+0.936) Duration: 11 minutes 25 seconds
1 1296 Games rolled with Variance Reduction. Dice Seed: 25 Moves and cube decisions: 3 ply

eXtreme Gammon Version: 1.11, MET: eXtremeGammon

Problem 64 From 501 (Part 3)

 

Continuing our discussion from Part 2, lets say that red got their worst roll of 42 and played it as best as possible, followed by blue fanning we arrive at the following position.

Pip counts: Blue 132, Red 72
Position ID: m20BwFDuzRkCAA Match ID: QQkAAAAAAAAA

Now looking at the temperature map we see:

There are several rolls here that force red to leave a shot, which is getting to the bottom the issue. Of course we could continue with all the other possibilities for a complete analysis, but we have enough to see why the 8/4 move is coming out ahead.

Equities and GNUBG

 

Trice’s introduction to equity gives a good basic overview of what equity is, but if you are like me you are not often settling short bearoffs in money games. The other place that most of us will encounter equities in our bot evaluations. I would have thought everyone interested enough in backgammon to be reading this blog would have at least learned a little about using a bot, but I am finding that is no the case. Below is example of a GNUBG move analysis, without all of the rollout details. The position and details of the rollout are not important for our purposes.

#		Ply	Move	Equity
•	1	R	9/4 6/4	+1.029
			0.774 0.068 0.002 – 0.226 0.023 0.001 +0.595 +1.029 0.003 0.002 0.000 – 0.003 0.001 0.000 0.007 0.021
	2	R	8/1	+0.723 ( -0.306)
			0.699 0.073 0.002 – 0.301 0.041 0.001 +0.430 +0.723 0.003 0.002 0.000 – 0.003 0.002 0.000 0.007 0.017
	3	0	14/9 6/4	+0.338 ( -0.691)
			0.611 0.104 0.004 – 0.389 0.089 0.002
			0-ply cubeful prune [expert]

The first two moves are from a rollout, as the R in the ply indicates. You can get all the details of the rollout if you want, ie. number of games, playing levels, etc. I did the rollout because I wanted to be able to look at the cubeless equities – more on that in a moment.

The first row of numbers under the move are the winning and losing probablities and equities, as follows:

wins, win gammons, win backgammons, lose, lose gammons, lose backgammons, cubeless equity, and cubeful equity.

The numbers under those are the standard deviations for those values, you could use those to calculate confidence intervals and statistical significance if you wanted to.

The first thing to keep in mind is that the wins and losses includes gammons and backgammons, and the gammons include backgammons. So to calculate the cubeless equity you need to correct for those, so to get the cubeless equity:

((.774-.068)+(.068-.002)*2+.002*3)-((.226-.023)+(.023-.001)*2+.001*3)

If you run this calculation you will get 0.594, the differences between this and the 0.595 is likely due to rounding as GNUBG will carry more decimal places than shown in the calculation.

So if we compare the cubeless equity for the first and second play we see that the first play is better by 0.595-0.430 or 0.165. Another way to think about this is that if you played 100 games for $1 a game from this position you would win $16.50 more by playing the first move instead of the second. (Of course this is only the expected amount, even if you played the 100 games out perfectly you would not likely get that exact result due to all the dice variables, if you played it out a million times you would get closer because you would begin to take into account all the possible dice rolls).

The main point is the first play is significantly better than the second. The last number is cubeful equity, this is the number that really matters in most cases since you will likely be playing with the cube. Getting to the cubeful equity is a lot harder. If you want to know more about calculation of this value you the gnubg documentation discusses it (http://www.gnubg.org/documentation/doku.php?id=appendix#8).

It is enough to know, however, that the value of the cube and doubles in included in this number. 

Once word of caution when looking at the best play in match is that the best play may change based on the current score. For example, you will find situations where the play with the highest equity is not the play that wins the most games but wins enough more gammons to make up for losing a few more games. But, if you evaluate that play in a match context where the person making the play only needs one point to win the match then the play that wins the most games will come out best because winning a gammon has no value in that situation.

Safe Harbor Games Dice

UPDATE3

Mike Petch collected over a million rolls using the new dice and tells me that everything was well within statistical expectations, however I have not see his final publication. I don’t have the programming ability Mike does, so I am not going to attempt to analyze the matches I collected for him, but since I have switched to Extreme Gammon for my match analysis I can use its dice analysis feature. All of the following are all from rooms using the corrected dice (rooms marked with yellow dice), mostly Doubling Cube and Gammon Zone

 

So as you can see the frequency of doubles, distribution of doubles, and number of doubles in a row are all within what would be expected.

UPDATE2

There is a group that continues to complain about the new dice code at SHG. While data is still being collecting for version .35 it is clear that all the versions from .28 produced the correct amount of doubles, or at least something very close to it. I keep hearing complaints of “one-sided dice” “6 doubles in a row” “20 doubles in a game” and players losing “time and time and time again” to multiple doubles at the end of the game. However,  as far as I can tell no one complaining about this has produced even one match log, not less the hundreds that should be out their, to show that this is happening. I would love to see all of these match logs, if it is happening with anywhere near the frequency some people claim there should be hundreds of these kinds of matches. So please, attach you mat files to an email and send them to showmethelogs@gmail.com.

UPDATE

The follow a couple of recent developments on this check out the following threads on the BGONLINE.ORG forum:

http://www.bgonline.org/forums/webbbs_config.pl?noframes;read=51451

http://www.bgonline.org/forums/webbbs_config.pl?noframes;read=51473

For those of you that don’t know Neilkaz is Neil Kazaross, the all times points leader on the American Backgammon Tour and #5 on the 2007 Giants of Backgammon List.

Original Post

This post does not really related to the subject of the blog, but i am putting it here so people that ask me about it on SHG can have a reference for what is going on.

If you see me (BigWill) kibitzing a lot of games on SHG it is because I am collecting data to continue a study of the dice at SHG. Mike Petch, who does a lot of gnubg development, is doing the analysis at this point, I am helping collect data.

Here is the whole story: 

A few weeks ago in the bgonline.org forum there was a comment from Neil Robins that his XG gammons dice data from SHG showed less than the expected number of doubles.  I doubted that could be true, so I analyzed my own matches. Doubles should occur 16.67% of the time, but on SHG they were only coming up about 9.2% of the time. If you want to read the whole thread that kicked off the whole thing you can find it here http://www.bgonline.org/forums/webbbs_config.pl?noframes;read=49530#Responses.

Mike’s data for the old dice can be found at http://www.capp-sysware.com/analysis/shgold_stats.txt.

Mike is a programmer and knows the people at SHG – he does not work for them, nor do I – so he let them know that there was an issue and shared our data with them.

As of this writing the corrected dice code is in the Cove, Doubling Cube, and Lagoon rooms on SHG.

The data we are collecting on the new dice is available here: http://www.capp-sysware.com/analysis/shgnewdc_stats.txt.

This data shows that doubles are now as you would expect. A lot of people are complaining about the number of doubles, but the data speaks for itself. Mike is continuing to look for other anomalies such as runs in the data, but runs can occur in truly random data sets.  A good introduction to the issue of statistically proving randomness can be found here: http://www.random.org/analysis/.

Now if you are thinking that is a lot of crap to wade through I would say if you are not willing to spend the time to understand that then shut up about the dice, you don’t have a leg to stand on. For the rest of you, you now know as much as the story as I do.

 

Some Notes on Random Number Sets

Mike and I have taken quite a beating in the room lobbies, particularly the Cove, over our "ruining" the play at SHG. Of course, all we have done is analyze the data before and after the change (actually Mike has done all of the after change analysis I have just been helping collect data), SHG has made the changes based on seeing the problem. However, one of the things that we have learned in the course of those lobby chats is that many people do not have a good grasp of probabilities or "randomness." For example one person told us that our claim that doubles should occur every 1 in 6 rolls was "subjective." Another person e-mailed Mike and told him that their understanding was that if the dice were correct than if you got a roll of 12, for example, you should not expect to see 12 again for 18 rolls. 
Neither of these, or many of the other things we are hearing from those convinced that the new code is generating too many doubles is correct. I am including this brief discussion to hopefully clear up some of the confusion people have with what they should expect to see in random dice.

There are numerous places on the internet to find the basic dice probabilities for backgammon, so I am not going to repeat that here. The simple fact is there are 36 possible rolls, 6 of which are doubles. Therefore the probability of getting any double is 6/36, or 1 in 6. There is nothing subjective about this.
The next thing to keep in mind is that every roll of the dice is independent of the previous or subsequent rolls. The dice don’t remember what was rolled, so on any given roll the probability of a double is 1/6, the probability of any specific nondouble 1/18 and so on. Those that think that they are more likely to get a double this roll because they "have not seen one in while" are committing one of the classic gambler’s fallacies.
Now, just because the probabilities of each roll are independent does not mean you cannot say anything about the probabilities of these independent events following one another. The math is very simple in this case, the probabilities of two independent events occurring is simply the probability of event 1 * the probability of event 2. So, if I ask the simple question what is the probability of throwing 2 doubles in a row the answer is simply 1/6 * 1/6, or 1 in 36.
This is all pretty simple, but now things get a little trickier, and this is where a lot of people, I think, struggle with the dice behavior. Consider the following two sequences of rolls: 21 21 21 21 and 54 63 31 42.  Most people look at the first sequence and see non-random and the second sequence and see random. However the probability of those two sequences of four rolls occurring from random rolls of the dice is exactly the same. Of course if we saw 12 repeated say 15 times in a sequence of supposedly random numbers we would have reason to question if the numbers were truly random because this has a probability of only about 1.4e-19 times, or 1 time in about 67,000,000,000,000,000,000. Notice I said question, not say for sure, we would need more evidence before reaching a conclusion. On the other hand, if I had a set of 100,000,000,000,000,000,000 random dice rolls the fact that I found a run of 15 12s should not be too surprising.
The problem is that with smaller number sets it gets harder to say when a repeated sequence is evidence of non-randomness. There are various statistical test for runs in sets of data, but it has been shown that truly random numbers will sometimes produce runs that fail these test.
Sometimes it easier to get a handle on some of the statistical oddities by experimenting. To illustrate this I simulated 216 matches (216 was chosen simply because I was using Excel to handle the data and this was the biggest set that easily fit) of 300 rolls each to see what kind of things came up. 300 rolls for one player is a realistic number of rolls for many 7-11 point matches, of course the real world number will be highly variable, but for the purposes of this experiment it works well.
I downloaded simulated dice data from random.org. Random.org uses atmospheric noise to generate true random numbers and their numbers have passed a battery of statistical test. Short of rolling real dice multiple thousands of time this is as close as we are going to get to simulating true, random dice rolls.
So, what can we learn from this experiment? Well here are a couple of items of interest:

Repeated Sequences of Doubles:  This is a complaint we hear a lot, "my opponent rolled 4 doubles in a row that never happens with real dice", for example.  Well looking at the data from the simulation a sequence of 4 doubles in a row came up 38 times, and in 3 of the simulated matches this happened twice!  Now from a pure probability standpoint a sequence of 4 doubles occurs only 1 time in every 1296 rolls. People see this and think that means that you will rarely see it, yet in our simulation it happened 38 times!  (Actually 38 times is lower than what the probability would predict, which would be 50 times.

And what about an even longer sequence of doubles? The longest sequence in the simulation is 7 doubles in row! Which would, based on the probabilities occur only 1 time in 279,936 rolls, but here it is a data set of only 64,800 random rolls. There were also 5 sets of 5 doubles in row, which a lot of people might question but is fewer than what one would expect.

Rolling Multiples of the Same Double in a Game or Match:  We hear people say something like "I got 13 double sixes in one game."  In the few cases we have been able to get the match log we have found these types of claims to be greatly exaggerated. However, getting more, or less, of one double than expected is not all that unusual in these short sets of random numbers. I had one simulated match in which there were 17 double 6s in the 300 rolls, more than twice the number one would expect. More surprisingly, perhaps, I had  a simulated match with only 1 double 1, another with only 1 double 2, and yet another with only 1 double 4. The probability of getting only 1 of a specific double in 300 rolls is 0.000006, or 1 time in 163,834 rolls. Yet here we have this low probability event occurring 3 times in a small set of simulated matches!  Now rarely is anyone going to complain that I only got 1 double 1, yet the absence of these is just as powerful of an indicator of how far off from the "average" values short samplings of random numbers can be.

Hopefully these couple of examples will help you to see that what some might think are evidence of non-random dice are actually events that occur with some regularity in short sets of data from a random source. I will end with one additional story.  A couple of days ago I won a game in 15 rolls (I should have doubled and gotten out of it a lot earlier probably), of which 7 were doubles. This included back to back 66s, followed a few rolls later by back to back 55s.  Now I am sure that some people well read that and say, see I told you there was a problem with the SHG dice!
Only thing is, I was playing GNUBG!