Cube Handling in Match Play

Backgammon match play is tricky since the points do not have the same value. It gets particularly interesting when the doubling cube is on 2 or higher, and those decisions are often very far from decisions in a money session. In this column, we will look at a few chances for the leader to redouble at 3-away 7-away.

If you play 7-point matches, you will encounter 3-away 7-away often, perhaps after a player wins a doubled gammon or redoubled game to start the match. Then the trailer often doubles early while the leader can still take, and then the leader needs to understand when to redouble.

A common beginner’s mistake is to redouble too early in a position which would be a large pass for money, but which lets the trailer back into the match. Woolsey's Rule on doubling says that if you are not 100% sure a position is a take, you must double. However, it is very dangerous to apply this to redoubles when you lead in the match, particularly if the source of your uncertainty is that you don’t know what the take point is.

White can win from this position about 15% of the time from a combination of chances to race and to hit. This would be a large pass for money, but it would be a blunder to redouble leading 3-away 7-away. White should happily take and recube for the match, and redouble/take is worth about 0.567 EMG.

If black puts the cube aside and forgets about it, that is worth 0.695. However, black can do a little better by waiting to get very close to white’s take point. That is worth about 0.815, so redoubling would cost 0.248.

The difference between 0.815 and 0.695 means black expects to get about 0.120 worth of use out of the ability to redouble (cube vigorish). Because of the trailer’s ability to redouble to 8, it is hard to find positions where the leader can redouble the trailer in. However, it is not hard to find positions where the leader can get value from the ability to redouble.

Black should not be afraid of the whip to 8 here. If black redoubles, white has the choice of taking and redoubling to 8 for the match, or passing and going to Crawford 7-away, which is worth about 9%-9.3% according to some match equity tables. If white can win more often from this position, he should play out this game for the match. This position is only worth about 8.0% wins, so white should pass. Black’s cube vigorish from redoubling here is 8.0% of 2 points, or 0.160.

Failing to redouble here would cost about 0.070. Errors of that size shouldn’t be ignored, but at most scores, failing to double a volatile close pass would cost much more.

Races

In money play, cube decisions require us to evaluate positions accurately which are worth about 20%–25%. The leader’s redoubles at 3-away 7-away force us to evaluate positions which are much more lopsided, where the trailer wins around 10%. Almost any type of position contains examples that lopsided. For simplicity let’s see what close decisions mean in races.

To evaluate lopsided races where both sides are efficient, you may want to use the Kleinman count, (difference + 4)²/(sum − 4). The take point of 9.3% corresponds to a Kleinman count of about 3.6, and 11% corresponds to 3.0. This isn’t perfect, but it can get you close.

Black leads 49–64 in the race. The Kleinman count is

suggesting that white wins about 10%. The actual value is about 8.7%, making this a small 1.040 pass.

If we push black back a pip, black leads 50–64, and white wins 9.9%. This is a 0.950 redouble/take.

Pushing black back one more pip makes it 51–64. White wins 11.2%, and now it would be slightly wrong to redouble. Redouble/take is worth 0.843 while not redoubling is worth 0.867. It appears that the window for redoubling the trailer in has a width a little wider than 1 pip.

A reasonable approximation of the leader’s strategy is to wait until the trailer has to pass. There is a thin window of positions which are correct redoubles and takes, but you don’t lose much on average if you wait to redouble the trailer out.

Last-Roll Redoubles

The main reason the leader must be careful about redoubling is the recube to 8 on the next roll. In a last-roll situation, the game may be decided before the trailer can redouble. This means the leader can redouble much earlier, and the trailer’s take point is doubled from 9.3% to 18.6%.

If black redoubles in a last-roll situation, black stands to gain the difference between leading Crawford 7-away and winning (9.3%), while black risks the difference between leading 3-away 5-away and being tied at 3-away (15.3%). Black needs to win

Black wins on 23 rolls out of 36, or 63.9%. This is more than the 62.2% needed to redouble in a last roll position. Redouble/take is worth 0.324 EMG, while not redoubling is worth 0.278.

If white’s position were slightly worse, then white could get some use out of the doubling cube to avoid missing. Normally, as black’s winning chances increase, redoubles get stronger. That is not the case if we weaken white’s position from this last-roll position.

White’s potential misses with 2-1 only matter if black does not redouble. The value of redoubling stays at 0.324, but the value of not redoubling increases to 0.318, so this is a very close redouble even though black now wins 65.9%.

If we weaken white’s position more, then black’s cubeless winning chances increase to 68.9%. However, redoubling would become a mistake since white would get more value out of the ability to redouble to 8 and avoid 5 misses. Not redoubling would be worth 0.378, while redouble/take is still 0.324.

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