This article originally appeared in the December 1999 issue of GammOnLine.
Thank you to Kit Woolsey for his kind permission to reproduce it here.

Initial Doubles

By Hal Heinrich
Kit's article on doubling theory in the September issue inspired me to do a little digging in that area. I'm largely interested in the question of when an initial double is correct—especially in the early going where we encounter the same positions repeatedly.

To decide if it's right to double, we need to look ahead two rolls to our next opportunity to double. At that point, we are either glad, indifferent, or sorry for whatever cube decision we chose. If the resulting position is double/take, it doesn't matter whether we doubled or waited—we can reach the same situation by doubling now. If the position is now double/pass, we're glad if we doubled and regret having waited. The equity swing is the difference between our equity with our opponent holding a two cube and one—which we get now when our opponent passes. Correspondingly, if the resulting position is no-double, we wish we hadn't cubed and are happy if we waited. The equity swing is the difference between the equity with the cube in the middle and the equity with our opponent holding a two cube. To decide if an initial double is proper, average the equity swings over your possible rolls, properly played, combined with the same for your opponent. Refer to Kit's article for a more complete explanation.

Hands-on experience tells us that a cubeless equity of 0.39 typically corresponds with a borderline initial double. A volatile position should be doubled with a lower equity, while a stable position requires a higher equity to justify a double. The interesting thing about cube turns is that you can gain a lot of equity without being sorry you waited because your opponent still has a take!

I've put together some code that generates Jellyfish 3-ply cubeless equities for the positions occurring two rolls down the road from an initial position. These equities are then averaged to reach an estimate of the gain or loss from doubling. Note that this also provides a 5-ply cubeless equity for the initial position. 

One of the commonest early doubling opportunities starts with an opening split of the back men, followed by a 5-5 attack making the three and one points, and then a poor roll by the attackee. The result is a pass after the following sequences: 

                                                      5-ply
                                                      Equity
1. 5-2 24/22 13/8   5-5 8/3(2)* 6/1(2)*  2. Dances    0.679
1. 6-3 24/18 13/10  5-5 8/3(2) 6/1(2)*   2. Dances    0.658
1. 6-4 24/18 13/9   5-5 8/3(2) 6/1(2)*   2. Dances    0.656
1. 6-2 24/18 13/11  5-5 8/3(2) 6/1(2)*   2. Dances    0.653
1. 5-1 24/18        5-5 8/3(2) 6/1(2)*   2. Dances    0.617
There's nothing unusual here—this matches common knowledge, except perhaps for the 5-1 which is an unusual opening.

Here are the doubling equities when the defender fans. After breaking the twenty-four anchor on the opening shake, there will be one, two, or three blots on the board. The positions are grouped by the number of blots left and sorted by the cubeless equity. 

                                                      5-ply    Doubling
3 blots                                               Equity   Gain
1. 4-3 24/20 13/10  5-5 8/3(2) 6/1(2)*   2. Dances    0.515    0.221
1. 4-3 24/21 13/9   5-5 8/3(2) 6/1(2)*   2. Dances    0.475    0.143
1. 3-2 24/21 13/11  5-5 8/3(2) 6/1(2)*   2. Dances    0.474    0.148
1. 4-1 24/23 13/9   5-5 8/3(2) 6/1(2)*   2. Dances    0.448    0.084
1. 2-1 24/23 13/11  5-5 8/3(2) 6/1(2)*   2. Dances    0.442    0.082

2 blots
1. 6-3 24/15        5-5 8/3(2) 6/1(2)*   2. Dances    0.548    0.208
1. 6-2 24/16        5-5 8/3(2) 6/1(2)*   2. Dances    0.538    0.195
1. 6-4 24/14        5-5 8/3(2) 6/1(2)*   2. Dances    0.509    0.156
1. 5-4 24/20 13/8   5-5 8/3(2) 6/1(2)*   2. Dances    0.450    0.106
1. 5-3 24/21 13/8   5-5 8/3(2) 6/1(2)*   2. Dances    0.417    0.051
1. 5-1 24/23 13/8   5-5 8/3(2) 6/1(2)*   2. Dances    0.397    0.009

1 blot
1. 6-5 24/13        5-5 8/3(2) 6/1(2)*   2. Dances    0.371   -0.051
These numbers tell us we should double if there's another blot exposed anywhere on the board. Within each grouping, the gain from doubling rises with the cubeless equity in a relatively smooth way. Doubling gains more for a given cubeless equity if two blots are exposed instead of just one blot—this is the effect of volatility. Compare the first entries in the two and three blot categories. Our cubeless equity is 0.033 lower vs. the extra blot position, but we gain an additional 0.018 equity by doubling. You can see the same effect if you compare positions with equities near 0.445 in the above list. 

Note that the gain from doubling after 5-1 23, 8 sequence is quite small -- so that you can tailor your cube action to suit your temperament and situation with this position.

If you think your opponent might drop after running with 6-5 and then dancing, how often do you have to be right to justify doubling? You're risking 0.051 to gain 0.629, so if you're right once out of 13.33 = (0.051 + 0.629) / 0.051, you'll break even. Being wrong twelve times costs you 0.612, while being right once earns 0.629 for a net gain of 0.017.

Now let's consider the situations where the defender enters, but does so badly. The most important category here occurs after an opening split to the eighteen point, followed by the 5-5 attack and then a 6-4 or 6-5. After entering, there are two or three plausible options for moving the six: 

                                                           5-ply    Doubling
                                                           Equity   Gain
1. 6-3 24/18 13/10  5-5 8/3(2) 6/1(2)*   2. 6-4 B/15       0.418    0.095
                                                B/21 13/7  0.444    0.162
                                                B/21 10/4  0.457    0.143

1. 6-2 24/18 13/11  5-5 8/3(2) 6/1(2)*   2. 6-4 B/15       0.407    0.082
                                                B/21 11/5  0.409    0.098
                                                B/21 13/7  0.442    0.166

1. 5-1 24/18        5-5 8/3(2) 6/1(2)*   2. 6-4 B/21 13/7  0.402    0.102
                                                B/15       0.403    0.042

1. 6-3 24/18 13/10  5-5 8/3(2) 6/1(2)*   2. 6-5 B/14       0.398    0.073
                                                B/20 10/4  0.431    0.137
                                                B/20 13/7  0.464    0.218

1. 6-4 24/18 13/9   5-5 8/3(2) 6/1(2)*   2. 6-4 B/15       0.398    0.020
                                                B/21 9/3   0.429    0.117
                                                B/21 13/7  0.448    0.175

1. 6-4 24/18 13/9   5-5 8/3(2) 6/1(2)*   2. 6-5 B/14       0.397    0.076
                                                B/20 13/7  0.437    0.182
                                                B/20 9/3   0.472    0.191

1. 5-1 24/18        5-5 8/3(2) 6/1(2)*   2. 6-5 B/14       0.385    0.017
                                                B/20 13/7  0.412    0.135

1. 6-2 24/18 13/11  5-5 8/3(2) 6/1(2)*   2. 6-5 B/14       0.381    0.043
                                                B/20 13/7  0.455    0.209
                                                B/20 11/5  0.468    0.193
The results here are consistent—the defender should run all the way with the man on the roof, followed by a double and a take. Some points of interest:

  • After the 5-1 18 and 6-4 sequence, both plays yield the same cubeless equity, but slotting the seven point more than doubles the gain from a cube turn! If the position goes double/take, it doesn't matter which play was made—but if the cube isn't turned, slotting the seven point is the right play by a significant margin. Because a drop is available when the slotting play goes sour, a discount is achieved—but any gains from the slot are paid in full! If you suspect your opponent will double one play but not the other—choose the play that won't get doubled.

  • Slotting inside is better than slotting the seven point if the inside hitting numbers are duplicated, but worse otherwise. 

  • After the 6-2 18, 11 and 6-4 sequence, slotting the five point is as good as running out.

  • In two sequences, doubling only gains 0.02—noticeably less than in the other variations. The common denominator seems to be that a subsequent six by the defender safeties the outfield blot leading to a one blot position.

Still looking at the opening split to the eighteen point, let's see what happens when the defender comes in with 3-2 or 4-2. 

                                                           5-ply    Doubling
                                                           Equity   Gain
1. 5-1 24/18        5-5 8/3(2) 6/1(2)*   2. 3-2 B/23 18/15 0.291   -0.083
                                                B/23 13/10 0.301   -0.049

1. 6-4 24/18 13/9   5-5 8/3(2) 6/1(2)*   2. 3-2 B/23 18/15 0.266   -0.086
                                                B/23 9/6   0.283   -0.086
                                                B/23 13/10 0.326    0.008

1. 5-1 24/18        5-5 8/3(2) 6/1(2)*   2. 4-2 B/23 18/14 0.265   -0.108
                                                B/21 18/16 0.265   -0.087
                                                B/23 13/9  0.308   -0.038

1. 6-2 24/18 13/11  5-5 8/3(2) 6/1(2)*   2. 3-2 B/23 11/8  0.264   -0.104
                                                B/23 18/15 0.266   -0.092

1. 6-3 24/18 13/10  5-5 8/3(2) 6/1(2)*   2. 4-2 B/21 18/16 0.246   -0.084
                                                B/23 18/14 0.253   -0.100
                                                B/23 10/6  0.283   -0.086

1. 6-2 24/18 13/11  5-5 8/3(2) 6/1(2)*   2. 4-2 B/23 18/14 0.235   -0.123
                                                B/21 13/11 0.252   -0.109
                                                B/21 18/16 0.255   -0.082

1. 6-3 24/18 13/10  5-5 8/3(2) 6/1(2)*   2. 3-2 B/23 13/10 0.151   -0.204
                                                B/23 18/15 0.260   -0.097

1. 6-4 24/18 13/9   5-5 8/3(2) 6/1(2)*   2. 4-2 B/23 13/9  0.113   -0.235
                                                B/23 18/14 0.237   -0.111
Don't touch that cube! Doubling here turns out to be a costly error. The checker plays are getting subtler—I don't see a rule that covers all the above cases. However hanging back on the twenty-three is desirable, as is getting off the eighteen, not to mention making a point in the outfield. A couple of points:

  • In the 6-3 18,10 and 4-2 variation, the power of duplication makes it right to enter on the higher point.

  • If our opponent errs after 6-4 18, 9 and 3-2 by playing the loose 23, 10 we are in the doubling zone even though our equity is 0.065 below the typical 0.39 cubeless equity needed for a cube turn. That's how much volatility four blots can produce!

Finally, let's examine some other positions where the attackee comes in without accomplishing much: 

                                                           5-ply    Doubling
                                                           Equity   Gain
1. 6-5 24/13        5-5 8/3(2) 6/1(2)*   2. 5-1 B/20 6/5   0.263   -0.086
                                                B/20 8/7   0.333   -0.012

1. 5-1 24/23 13/8   5-5 8/3(2) 6/1(2)*   2. 5-1 B/20 6/5   0.255   -0.089
                                                B/20 8/7   0.287   -0.058

1. 5-4 24/20 13/8   5-5 8/3(2) 6/1(2)*   2. 2-1 B/23 6/5   0.255   -0.089
                                                B/23 8/7   0.287   -0.058

1. 4-3 24/20 13/10  5-5 8/3(2) 6/1(2)*   2. 6-4 B/21 20/14 0.225   -0.114
                                                B/15       0.252   -0.068
                                                B/21 10/4  0.267   -0.074

1. 3-2 24/21 13/11  5-5 8/3(2) 6/1(2)*   2. 6-5 B/14       0.222   -0.120
                                                B/20 11/5  0.243   -0.094
                                                B/20 21/15 0.260   -0.055

1. 4-3 24/21 13/9   5-5 8/3(2) 6/1(2)*   2. 6-5 B/14       0.215   -0.118
                                                B/20 21/15 0.228   -0.085
                                                B/20 9/3   0.290   -0.046

1. 6-5 24/13        5-5 8/3(2) 6/1(2)*   2. 2-1 B/23 6/5   0.209   -0.158
                                                B/23 8/7   0.273   -0.099

1. 4-3 24/21 13/9   5-5 8/3(2) 6/1(2)*   2. 3-2 B/23 13/10 0.196   -0.150
                                                B/20       0.201   -0.152
                                                B/23 9/6   0.201   -0.166

1. 3-2 24/21 13/11  5-5 8/3(2) 6/1(2)*   2. 3-2 B/23 11/8  0.178   -0.186
                                                B/23 13/10 0.194   -0.149
                                                B/20       0.210   -0.140

1. 4-3 24/20 13/10  5-5 8/3(2) 6/1(2)*   2. 4-2 B/21 10/8  0.173   -0.187
                                                B/23 10/6  0.216   -0.146
                                                B/23 13/9  0.218   -0.124
                                                B/23 20/16 0.253   -0.099
I don't see much of interest here—all of these positions fall into the category of no-double/take.

In summary, after hitting with 5-5 8/3(2), 6/1(2) on move 2, the proper cube play is as follows:

  • Double/pass
    • - if the defender has two men up
    - if the defender splits to the eighteen point and then dances

  • Double/take
    • - if the defender has one man up and a blot anywhere but the eighteen point
    - if the defender splits to the eighteen point and comes in with 6-4 or 6-5
    - note that the 6-4 and 6-5 should be played my moving the checker from the roof to the outfield.

  • No-double
    • - all the rest.

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