Back to "The Zare Formula" written by Douglas Zare

The Last Rolls - Part 1

by Douglas Zare
25 December 2003

Douglas Zare

Since high cubes are often exchanged at the end of the game, it is valuable to be able to judge the late bearoff accurately. There are a few simple and well-known reference positions, but over the board more complicated positions appear instead. It's not the most creative part of the game, but it is relatively easy to get these right.

In this column, we will review the theory of 2-roll, 3-roll, and 4-roll positions. We will determine the effects of each nonworking double or misses. Finally, we will look at applications to match play.

In a future column, I will analyze some close take/pass decisions as well as some close doubling decisions.

Tools

Exact calculations are extremely laborious for all but the simplest bearoffs, and it's not much more fun to roll out a bearoff position a few thousand times.

When Gerry Tesauro started training artificial neural nets to play backgammon, he started with the bearoff, but his results were poor, unlike his spectacular success with TD-Gammon, the first modern backgammon bot. It's natural to assume that the end of the game would be the simplest, with no primes or hitting or timing or gammons. In fact, that is backwards: Neural nets have an easier time figuring out the opening than the last few rolls of the game

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Subject: Re: The Last Rolls - Part 1

From: DBrthrtn Member has a photo available

Date: 29 Dec 2003 18:58 EST

Doug, do you mind publishing the EMG take equities for your four roll position 5-pt match table so we can see how close or how clear some of the decisions are? It may not be as enlightening on the No Double decisions, maybe we could see the EMG D/ND equities for those since three were close. I dimly recall the 5-away 2-away ND result from somewhere (maybe a bearoff Kit Woolsey wrote about?), and equally dimly the 2-away 4-away ND result (maybe Ortega / Kleinman?), but the 3-away 4-away and 3-away 5-away results are new, at least for me.

Please clarify the following bit:

"At 3-away 4-away, the leader has to pass a 3-roll versus 3-roll position, but the early-late ratio does not encourage an early double by the trailer and the potential redouble is efficient."

I took the first part after the second comma to mean trailer should not offer an initial double a 4-roll vs. 4- roll position at this score even though the sequence "non- doublet/non-doublet" would be a tiny market loser trailer. Also, leader's redouble isn't automatic so I'm assuming that the very last part means the sequence "wrong initial double/correct take/non-doublet/doublet/non-doublet" is an efficient recube to 4 for the leader, which is part of the reason the initial double is wrong to begin with. Let me know if I'm reading too much into this....

Btw, thanks for another great article.

Subject: Re: The Last Rolls - Part 1

From: zare

Date: 01 Jan 2004 05:22 EST

You unpacked that sentence accurately. However, I miscalculated before: The trailer should double in a pure 4- roll versus 4-roll position at 3-away 4-away. D/T is worth 0.783 and ND is worth 0.760 EMG. I think I erroneously assumed that the leader would take a 3-roll versus 3-roll position in part of my calculations, but the dead-cube take point is about 22.8%, more than the 21.2% of a 3-roll versus 3-roll position.

Another correction: It appears that the trailer may have a tiny pass at 6-away 7-away, as Double/Take is worth about 1.003 EMG. Before, my roundoff errors were larger, and I thought it was a 0.998 take.

Here are the equities for Double/Take in EMG for a pure 4- roll versus 4-roll position. The rows represent the number of points for the player on roll. The first column is also 2-away.

2: 1.326 0.980 0.342 0.804 0.767 0.820 3: 1.085 1.206 1.526 1.188 1.187 1.071 4: 0.705 0.783 1.051 1.100 1.082 1.122 5: 0.627 0.739 0.843 0.904 0.906 0.928 6: 0.808 0.863 0.919 0.964 0.971 1.003 7: 0.800 0.921 0.897 0.934 0.930 0.947

Here are the differences between Double/Take and No Double in EMG. If your opponent will take, this tells you by how much it is right to double. Of course, at many scores it is less valuable to double because your opponent will pass.

2: 0.565 0.219 -.258 0.044 0.006 0.059 3: 0.324 0.445 0.765 0.427 0.426 0.310 4: 0.018 0.022 0.290 0.339 0.321 0.361 5: -.009 -.012 0.082 0.143 0.145 0.167 6: 0.053 0.102 0.158 0.203 0.210 0.242 7: 0.043 0.160 0.136 0.173 0.169 0.187

By the way, it appears to be a pass trailing 6-away 8-away (1.021) and 6-away 9-away (1.025), and no double leading 2- away 8-away (-.074) and 2-away 9-away (-.054). Otherwise, within a 9-point match the patterns continue. It is Double/Take if the roller needs 8 or 9 points, and you must pass if the roller needs 3 or 4 (or 6) points and you need 8 or 9.

I performed these calculations using Mathematica, so the main difficulty was copying the match equity table. It's a real pain to do the calculations by hand.

Douglas Zare

Subject: Re: The Last Rolls - Part 1

From: zare

Date: 01 Jan 2004 19:56 EST

I thought I might mention a paradoxical result: It is wrong to double a pure 4-roll vs. 4-roll postion trailing 3-away 5-away, but it is right to double a 5-roll vs. 5- roll position, or even an 8-roll vs. 8-roll position!

Normally, it is more right to double with a larger advantage. However, in an n-roll vs. n-roll position, rolling a double followed by a nondouble is always a huge market loser with a centered cube. The question is how much use your opponent will get out of the cube after you roll a nondouble and your opponent rolls a double. At 3- away 5-away, the leader would get a very efficient redouble out in a 2-roll vs. 2-roll position, but can't redouble a 3-roll vs. 3-roll position. On the other hand, the leader's initial double would be more efficient in a 3- roll vs. 3-roll position than a 2-roll vs. 2-roll position. The potential lemon equity is much greater (1.020 EMG rather than .685) from a 4-roll vs. 4-roll position than from a 5-roll vs. 5-roll position, and this outweighs the difference in the size of the market losers, even though if neither side rolls doubles the trailer will regret doubling.

It is right to double a 6-roll vs. 6-roll position but not a 5-roll vs. 5-roll position trailing 6-away 9-away.

It is right to double a 7-roll vs. 7-roll position but not a 6-roll vs. 6-roll position trailing 6-away 7-away or leading 7-away 8-away.

It is right to double trailing 3-away 4-away with an 8- roll vs. 8-roll position, but not a 7-roll vs. 7-roll position. The leader will rarely get an efficient redouble, but may get an efficient initial double, particularly with a double out in a 6-roll vs. 6-roll position. This can be avoided by doubling first.

Douglas Zare

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