Match Play Cubes
 And the Trailer Doubles Anthony Patz April 2001

Examining the weekly club and the monthly team score sheets, I notice a lot of matches that go into the Crawford game from a tied score of 2-away, 2-away. The theory is that, between two strong players, the 2-away, 2-away game should be the last. This is because it always pays the leader to offer the cube and play for the match, however slim his lead, and as one of the players will always be the leader, the game should always be doubled. Thus it should only go to Crawford if the leader erred and missed his market, by waiting unnecessarily.

Winning a single point, the leader's match-winning chance (at 1-away, 2-away) is 70%. So if his game winning probability is p (ignoring gammons), his match winning equity is:

0.7p + 0.3(1 − p) = 0.3 + 0.4p

If he cubes, his match winning probability is exactly that of this game, p. Now p > 0.3 + 0.4p when p > 50%; i.e., whenever the player has any lead at all.

A lot of players know this, but they only don't wish to double prematurely with a small lead, in case their fortunes are reversed and they find themselves trailing. As the theory goes this is an unnecessary risk, as a knowledgeable opponent will not let you get below your drop point, so the only risk you run is of losing your market. At this score this is much higher than usual, as your opponent's drop point is 30%, compared to an early match or money game where it could be as low as 20% due to recube vig.

So while most of us are wondering just how small an advantage is enough to double, the fact is, that it can sometimes pay to cube even when you are trailing in the face of even a smallish market loser! Take the following bearoff position as an example.

 Position 1: 21 point match Black 19, White 19 Black on roll. Cube action?

If black does not cube his chances are as follows: 1/6th of the time he will roll a double and go into the lead (20 to 19) with a 70% match equity. On those 5/6th of the time that he fails, white should cube and black will have to drop, leaving him trailing (19 to 20) with a 30% match equity. (With white on roll and only 1 man left, black actually has a 25% chance of winning, but as his drop point at this score is 30% he cannot take). Combined, this gives black a total match equity, before rolling, of 36.7%.

If black does cube, however, 1/6th of the time he will win the match. If he fails he still has a 25% chance of winning as there are nine rolls which fail to get white off. White cannot cash once black has doubled, as the cube will be dead, so in 5/6th × 1/4 cases black will still win. This combined probability is 37.5% which is higher than his uncubed chances.

Admittedly the above position is unlikely to come up very often, if at all, in a player's lifetime, as the score has to be 2-away, 2-away and the cube unturned as well. However it does appear in different guises throughout the game, such as in the following position. Assume again that the score is 19-all playing to 21, black on roll:

 Position 2: 21 point match Black 19, White 19 Black on roll. Cube action?

Here again black is trailing; in Position 1 he had to roll one of six doubles to win, here black has six indirect shots to do the same. If black failed in Position 1 he would have to drop the inevitable cube from white.

In Position 2 his racing chances are slightly better than 25% which still means a cash because of his 30% drop point. (If he does not hit or clear the white checker on this roll black does have a few chances of a second shot, but most of this advantage is lost in burying his checkers, so his overall chance if he rolls and fails is just off 28.5%). As his chances are even better than in Position 1, black knows he is better off cubing without having to do a lot of fancy calculations.

These are black's probabilities. If he fails to cube, then, just as in Position 1, he has a 1/6th chance to hit and win, or he will be cubed out for an overall match winning probability of 36.7%. If he does cube , 1/6th of the time he will win the match and 5/6th of the time he will be left with a 28.5% survival chance for an overall 41% match winning chance.

So, even though he trails with a 41% game winning chance, it pays black to cube and make that his match winning probability, a big improvement over his uncubed 36.7%.