Avoid Overkill, and Figure Odds
Paul Magriel, 1977
New York Times, June 23, 1977
The delicate interplay between the movement of the checkers and the position of the doubling cube can lead to unexpected complications. Even an expert player went astray in the diagrammed position, which occurred during the Phillip Morris International Championships last summer in Monte Carlo.

Black to play 4-3.

In this position, Black is far ahead in the race and owns the doubling cube. His sole concern is to get home safely and avoid being hit. He rolls a 4-3, a number which does not bring him past all of White’s men, and must choose among three plays. His most obvious consideration is to leave the minimum number of shots, or ways he can be hit.

Examine his three options and the number of shots each move leaves for White.

1. Black can stay put in White’s outer board and move 5/1, 4/1. Based on 36 possible rolls, Black would leave 18 shots in this position.

2. Black can move part of the way, 18/15, 5/1. This move gives White 22 ways to hit a man.

3. Black can move his man all the way, 18/11. This move gives White 24 ways to hit a man.
Option 1: 5/1, 4/1 Option 2: 18/15, 5/1 Option 3: 18/11
There is another aspect of the position that is critical to consider. Besides the number of immediate shots he is leaving, Black must also evaluate the resulting position if White misses, to determine how close he is to home.

With this in mind, Black must dismiss the first alternate play, 5/1, 4/1, even though it is safest on the next roll. This play does not bring the critical man on the 18-point any closer to home. Even when Black is not hit, he is still dangerously stranded in White’s outer board.

The second alternate play, stopping on the 15-point, leaves Black in a quite favorable position to come home; nevertheless, on his next move, Black could still roll 15 numbers that do not get him past White.

The third alternate play makes Black virtually certain to come home safely next turn if he is missed.

The expert player decided on the third alternative, 18/11. He felt that the assurance of bringing his man almost home was worth the slight increase in immediate risk (two more shots than the second play). And indeed, this would have been the correct play if Black did not own the doubling cube!

The correct play is, surprisingly, 18/15, 5/1 (the second move). The reasoning behind this play is based on Black’s possession of the cube. If Black reaches the 15-point and White misses, Black has the power to redouble. In fact, White cannot even accept the double! Therefore reaching the 15-point safely without being hit effectively wins the game for Black. Thus even a small extra risk to come closer is unnecessary and unjustified.

Rollout

Tom Keith 2013
Money play
Black owns 2-cube
Black rolls 4-3

1296 games with VR
Checker play: 2-ply
Cube play: 3-ply Red

 4-3: Game G BG Equity 1 18/15, 5/1 W L .4641 .5359 .0814 .0252 .0003 .0006 +0.1216 Option 2 2 18/11 W L .5058 .4942 .0715 .0213 .0004 .0004 +0.0920 (0.0296) Option 3 3 18/14, 4/1 W L .4217 .5783 .0597 .0274 .0002 .0007 −0.0264 (0.1480) 4 18/14, 5/2 W L .4184 .5816 .0585 .0263 .0002 .0006 −0.0320 (0.1536) 5 5/1, 4/1 W L .3914 .6086 .1096 .0322 .0005 .0009 −0.0872 (0.2088) Option 1

Note that if White owned the doubling cube, or if the game were to be played out to the end without any doubling, then the expert’s play would have been correct. Without the leverage of the cube, getting to the 15-point safely does not insure certain victory. 18/11 would then give Black the best overall chance to get home safely.

When you own the cube and can reach a favorable position, you must avoid playing too strongly. That is, you need to take only enough risk to reach a decisive doubling position. Any additional risk is overkill.

Rollout

Tom Keith 2013
Money play
White owns 2-cube
Black rolls 4-3

1296 games with VR
Checker play: 2-ply
Cube play: 3-ply Red

 4-3: Game G BG Equity 1 18/11 W L .5054 .4946 .0736 .0204 .0004 .0004 −0.1875 Option 3 2 18/15, 5/1 W L .4664 .5336 .0811 .0247 .0003 .0004 −0.2670 (0.0795) Option 2 3 5/1, 4/1 W L .3976 .6024 .1049 .0313 .0003 .0006 −0.3921 (0.2046) Option 1 4 18/14, 4/1 W L .4257 .5743 .0569 .0255 .0002 .0004 −0.3980 (0.2105) 5 18/14, 5/2 W L .4235 .5765 .0565 .0251 .0002 .0004 −0.4035 (0.2160)

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