Cube Handling in Matches

A Killer Problem
Bob Floyd, 1983

From Backgammon Times, Volume 3, Number 2, Summer 1983.

13 14 15 16 17 18 19 20 21 22 23 24
12 11 10 9 8 7 6 5 4 3 2 1
White (4 away)
Should Black double?
Black (2 away)
This artificial-looking position arose for San Jose's Killer Joe Glazier in a recent Louisville tournament. In a money game, it is a double, and only a bare take because of the redouble equity; White's winning chance if the game is played to completion is 24.96%. At the match score, though, the redouble is a wicked weapon.

Joe doubled, missed, and dropped the redouble. Surprisingly, this is the right strategy even though it gives him only 53% chance to win the game, while waiting would give him a 75% chance for the game.

First a little background. If Black wins one point, his chance to win the match is 82%; White must win three consecutive games, unless he wins a gammon on the first or the second. If Black wins two points, he wins the match. If White wins one point, Black has a 59% chance for the match. (There is argument about this figure; some place it as low as 55%.) If White wins two points, the chances are equal. If White wins four points, Black loses the match.

If Black decides not to double, and then misses, White must not double; to do so gives up 18% match equity when he loses the game, and gains no more than 9% match equity when he wins. White should not give even a last-roll double unless his winning chance is more than 2-to-1. White could barely double on the last roll with checkers on his 2 and 3 points.

So if Black does not double, the game will be played for one point, Black will win almost exactly 3/4 of the time, and his match chance will be 3/4 × .82 + 1/4 × .59 = 76.25%.

If Black doubles and misses, he must drop the redouble; it is better to even the match score at two-away/two-away than to play on for the match as a slight underdog. So if Black doubles, his match chance will be 19/36 + 17/36 × .50 = 76.39%. Joe's double gained him an extra 1/7 of one percent chance of winning the match. With that kind of precision in his doubling, is it any wonder tha they call him Killer?

When you are two points away from match and your opponent is four or more away, a last-roll double is a powerful weapon. You should double with even a 36% chance to win the game, as with checkers on the 2 and 6 points, or the 3 and 5 points. Your opponent should drop if your chance is more than 64%; checkers on the 1 and 5 points, or the 2 and 4 points, are marginal drops.

On the other hand, before your last roll, because your opponent can give the cube back and put the match on the line, your opponent will take even if you have an 82% chance; you should not even consider doubling and taking the redouble unless your chance is 77%. If each player has eight checkers on the 1-point, a bare take in a money game, you should not double at a two-away/four-away match score; in fact, mathematically you should wait until each player has just four checkers on the ace point. In practice, even a 5% chance that your opponent will drop justifies a double with six checkers on each ace point.

Bob Floyd, a professor of computer science at Stanford University, is a regular contributor to Backgammon Times.

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