Cubing At 4-Away, 2-Away

by Gerry Tansey

Consolation - Gerry Tansey

Suppose that you are trailing in a 5-point match by the score of 3-1.  The bad news is that you only have about a 33 percent chance of winning the match against an opponent of equal strength.  The good news is that your side is a lot more fun to play.

For instance, suppose you win the opening roll with a 31 and make the 5-point.  Your opponent rolls a pretty good response with 42, and makes his 4-point.  In a money game, you would never think of doubling here, but at this crazy score of 4-away, 2-away, the decision of whether or not to double is very close.  This rollout indicates that you probably should go ahead and send it over, even though I only rolled out the position long enough to give XG 93.5% confidence in its verdict.

gERRY1

Analyzed in Rollout No double Double/Take
Player Winning Chances: 54.45% (G:17.21% B:1.83%) 54.66% (G:17.42% B:2.01%)
Opponent Winning Chances: 45.55% (G:14.94% B:2.03%) 45.34% (G:15.70% B:2.24%)
Cubeless Equities +0.021 +0.626
Cubeful Equities
No double: +0.617 (-0.010) ±0.009 (+0.608..+0.625)
Double/Take: +0.626 ±0.009 (+0.617..+0.635)
Double/Pass: +1.000 (+0.374)
Best Cube action: Double / Take

So if it is a close decision whether to double if you have made a slightly better point than your opponent on the opening roll, it must be a much bigger double if you have made a point and your opponent has not.

What is going on here?  Well, the trailer has an extra incentive to get the cube into play if winning a gammon is a possibility, since a gammon win on a 2-cube gives the trailer precisely the four points he needs to win.  Further, the normal deterrent to early doubling, namely the fact that one’s opponent can give an uncomfortable redouble later if the game turns around, is out of play at this score.  When the 4-away player doubles the 2-away player, the cube is dead.  The leader can never redouble.

Here I would like to be a little bit more quantitative and tell you about “Neil’s Rule of 80.”  The “Neil” here is Neil Kazaross, one of the greatest players of all time.  The rule is as follows:  At the score of 4-away, 2-away, take the percentage of games that the trailer wins, and add the percentage of games in which the trailer wins a gammon.  If this number is 80 or more, the leader should pass the trailer’s double.

Think about what this Rule of 80 means.  A position in which one side wins 60 percent of the games with 20 percent gammon wins is not usually a correct double for money, but at this special score, not only is it a correct cube, the leader should pass.  Thus the following early 55 blitz position, which is not even a proper cube for money, is a monster pass at the score.

gERRY2

Analyzed in XG Roller++ No double Double/Take
Player Winning Chances: 59.79% (G:27.59% B:0.55%) 59.63% (G:29.36% B:0.50%)
Opponent Winning Chances: 40.21% (G:9.75% B:0.47%) 40.37% (G:10.33% B:0.59%)
Cubeless Equities +0.292 +1.413
Cubeful Equities
No double: +0.854 (-0.146)
Double/Take: +1.413 (+0.413)
Double/Pass: +1.000
Best Cube action: Double / Pass

Let’s consider the opening position, before anyone has moved.  The trailer wins 50 percent of the games, and about 13 percent gammons (some have estimated that about 26 percent of all games played, if played to conclusion, would result in a gammon for one of the players).  So before anyone has moved a checker, the “wins + gammons” number is at 63.  Now, after the sequence of 31 followed by 42, the trailer’s winning percentage and gammon percentage have both jumped, so the “wins + gammons” number is in the neighborhood of 72.  One more jump like that and the leader would have to pass a double, so now is the time to cube.

However, there is another consequence to Neil’s Rule of 80.  If the trailer cannot win a gammon, the leader can take a double if he has a 20 percent chance to win the game.  This 20 percent figure is actually lower than it is for money, where in a typical long-to-medium length race one can take with around a 22 percent chance of winning.  Often it is a good idea for the leader to seize an advanced anchor at the expense of a more offensive play, as this makes it very difficult for the trailer to offer a scary double.  When the gammon threat is nonexistent, the trailer probably shouldn’t offer any cubes that are not proper money doubles.

Finally, let’s consider a last-roll situation.  What percentage of the time does the trailer need to win in order for a last roll double to be correct.  For money, the answer is 50 percent – one should double if one is the favorite, and refrain from doubling otherwise.  At the score, it is a little bit more complicated.

If the trailer does not double and loses, the score will be 4-away, 1-away Crawford, from which the trailer will win about 18.5 percent of the time.  If the trailer doubles and loses, the match is over and the trailer wins 0 percent of the time.  So the trailer risks 18.5 percent match equity by doubling.

If the trailer does not double and wins, the score will be 3-away, 2-away, from which the trailer wins about 40 percent of the time.  If the trailer doubles and wins, the score is tied at 2-away, and the trailer wins 50 percent of the time.  Thus the trailer potentially gains 10 percent match equity by doubling.

So the minimum winning chances the trailer needs to double a last-roll position is given by

Risk/(Risk+Gain) = 18.5/(18.5+10) = 18.5/28.5, which is around 65 percent!  The extra point put at stake by doubling is much more valuable to the leader than the trailer, since it allows the leader to win the match.  This means that the trailer can’t cube positions in some positions in which he clearly should for money, such as this one:

Gerry3

Analyzed in 4-ply No double Double/Take
Player Winning Chances: 63.89% (G:0.00% B:0.00%) 63.89% (G:0.00% B:0.00%)
Opponent Winning Chances: 36.11% (G:0.00% B:0.00%) 36.11% (G:0.00% B:0.00%)
Cubeless Equities +0.278 +0.245
Cubeful Equities
No double: +0.278
Double/Take: +0.245 (-0.032)
Double/Pass: +1.000 (+0.722)
Best Cube action: No double / Take
Percentage of wrong pass needed to make the double decision right: 4.1%

At some point in the future, I may write about the leader’s cube strategy at this score.  For now, I’ll summarize it as follows: “If you can win a gammon, don’t double.  If you think your opponent has a small pass, don’t double.  If your opponent has a huge pass, okay, think about doubling.”

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