Answers to Match Doubling Quiz
Kit Woolsey, 1980
Las Vegas Backgammon Magazine, December 1980
light bulb These are the answers to the Match Doubling Quiz, which is presented here.

Problem 1

15 13
Trailing 9 to 13,
should Black double?
I had this position against Paul Magriel in a match a few years ago. Even though White would have a beaver in a money game, since White has good chances of escaping and gammoning Black, it is proper for Black to double at this score. There are two likely scenarios:

  1. White scampers around safely while Black is trying to build his prime. If this happens, Black will almost surely be gammoned, so the double won't cost.

  2. Black succeeds in building the prime. In this case, White will now have a pass, so Black would wish that he had doubled earlier (even though he might still lose the game.)

Consequently, the double cannot cost. In addition, if Black rolls the 1-1 joker and White does not move up White would now have a pass at this score, so Black does risk losing his market. Therefore, it is correct to double. In actual play I did in fact double but failed to contain the loose checker and so lost the match, but the double didn't cost as I would have been gammoned anyway.

Problem 2

15 10
Tied at 10, should
Black redouble to 8?
It seems pretty hopeless if Black doesn't hit the shot, so perhaps Black should double on the come, as my opponent did in a recent match. However, careful analysis shows that doubling is a mistake. Let's suppose that Black does, in fact, hit the shot. Will White still take the double? Clearly yes, since White is only a 3 or 4 to 1 underdog in the position, while his chances of coming back from 14–10 are 6 or 7 to 1 against.

Consequently, there is no reason to double, since Black won't lose his market for quite a while. Black should wait until he has started his bearoff and White fails to enter at his first opportunity; White will still have a take at this score. At it happened, my opponent hit the shot and successfully closed his board, but I entered at my first opportunity and won the race by a roll. At no point would I have refused a double, so my opponent gave away his chance to recover from 14–10 for no reason. Note that had the score been, say, 8–7 my favor his double would be correct, for if I am hit I would rather play behind 11–8 than play the game for the match, so he would be risking losing his market by not doubling.

Problem 3

15 9
Leading 11 to 9, should
Black redouble to 4?
This came up in a world championship match against Chuck Papazian. I had just rolled an extremely lucky 5-5 to escape my men, and now I could offer to put the match up for grabs. I have 10 completely safe rolls, 3-3 and 1-1 are pretty good, while anything else leaves a shot. Consequently, I will be hit about one quarter of the time. So if I double, I will win the match 75% of the time (assuming he takes). If I don't double, I have a 13–9 lead 75% of the time, and an even match 25% of the time. If we assume that a 13-9 lead is 80% to win the match, an estimate which intuitively seems high to me, then my probability of winning by not doubling is .80 × .75 + .50 × .25 = .725, so it is correct to double.

A more difficult problem is whether or not White should take the double (Chuck chose to pass the double, but I won the match anyway). Note how a slightly different score can change things: suppose I had been ahead 11–7 instead of 11–9. Now doubling would be foolish and taking would be clear-cut, since 13-7 is a big lead, and even if I lose I will still be ahead 11–9, while if I double the whole match will be up for grabs.

Problem 4

15 12
Trailing 11 to 12, should Black double?
Pip count 51–57, cube in the middle, normally an easy double. But not at this score! The reason is that a 13–12 lead in a 15 point match is only slightly better than an even match (about 55–45 according to most experts) since the player with 12 has all the cube leverage, while 14–11 is much more serious than 13–11. Consequently White needs the point more than Black does, so Black should wait a roll.

Now, let's change things a bit — same score, but Black owns a 2 cube. In a money game it would be a close double and an easy take, but at this score it is a clear double and an easy pass, since White would rather play behind 13–12 than play this position for the match.

Problem 5

15 12
Tied at 12,
should Black double?
Surprsingly this is a double at this score, although it would be premature in a money game. The reason is that 13–12 is not a significant lead, so it is worth it to push for the extra point. Also, since being behind 13-12 is not a serious deficit, White will be quick to pass the double if Black's position improves at all on the next roll, so Black stands a good chance of losing his market. In fact, at this score, it might well be correct for White to fold now!

Problem 6

15 11
Leading 13 to 11,
should Black double?
This is quite a paradox. Black is leading in the match, he is an underdog in the position, yet it is correct for him to double! The reason is that the extra point is not too important for White, but it is extremely important for Black since it puts him out. Black chances are:

Double:    Win match    17/36
Even match    19/36
Don't double:    Have 14–11 lead    17/36
Have 13–12 lead    19/36

If we assume a 13–12 lead is a 60% favorite (a very high assumption in the opinion of most experts), and a 14–11 lead is an 85% favorite, we have:

Double:    Win match    1.00 ×  17/36 =   0.472  }  0.736
Even match    0.50 ×  19/36 =   0.264
Don't double:    Have 14–11 lead    0.85 ×  17/36 =   0.401  }  0.718
Have 13–12 lead    0.60 ×  19/36 =   0.317

So doubling is clearly correct.

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