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The Importance of Match Score

There are three layers of challenge in backgammon, and all are important when playing competitively.

  1. The first layer is to solve the problem of what move should be made. This one is tough enough to keep you occupied for a lifetime, but that's a discussion for another time.
  2. The second layer is the cube decision. Instead of considering the next move to make, the player now has to picture the entire game unfolding to its conclusion, and estimate the chances of the various outcomes.
  3. The third layer takes the cube decision and puts it in the context of a match. Now you have match winning chances to consider, the Crawford Rule, free drops, and the rest of it.

While a full treatment of the subject would take too long, a position came up during a friendly game on Monday that quite nicely illustrates the type of considerations match-play requires. First, here is the position in question. Frank has the green pieces (home for him is in the bottom left of the diagram), and is on roll, considering whether to offer the cube. Clive is playing with the white pieces. Should Frank (green) double?

Frank had in fact already decided to offer the cube, and was wondering whether the position was too strong and should have instead played on for the gammon. Considering that white has an anchor (even if it is only on the ace point), is not ridiculously behind in the race, and might have quite a few opportunities to roll some lucky numbers, I felt that it was not good enough to go for the gammon and that Frank should double. The computer confirms a money equity of about 0.67 so it's Double/Drop.

However, as I was leaving, I realised that I had forgotten to ask what the match score was, and was told that Frank was leading 3-1 in a 5-point match. That changed eveything, turning a Double/Drop position into No Double/Take. Why had the situation changed so much?

Here's the short version: a large cube value favours the trailer more than the leader, because the score for the leader overshoots the target score in the match more readily than when the trailer wins. In other words, the leader is playing for points he does not need.

Here's the long version. The key point about being 4 points away from winning versus 2 points away is that if Frank doubles, then Clive will have the cube on 2, but since Frank needs 2 points to win the match and Clive needs 4, he is immediately going to send the cube back across the table and play out the game for the match.

To analyse this properly, we need to look at the winning chances of various scorelines. Much work has been done on this, and I have made use of Kit Woolsey's Match Equity Table in what follows. We find that If Frank holds on to the cube, then a win (4-1 Crawford) makes him an 83% favourite (a gammon wins the match of course) and losing (3-2) makes him 60% favourite.

So what are Clive's chances of winning from the above position? Does about 25% look reasonable? If Frank holds onto the cube, that's a 25% chance of getting a 40% chance of winning the match, and a 75% chance of getting a 17% wining chance (actually less because of the gammon risk). With the gammon risk, those combine to about a 20% chance of winning if the cube is not turned. But turn the cube, and that 25% is Clive's chance of winning the entire match! He would dearly want Frank to offer the cube so he could send it back, so at this scoreline the position is No Double/Take (followed by an immediate redouble).

You're not going to be able to perform this type of analysis over the board, but it illustrates the following central ideas:

  1. Never forget the match score when considering the cube. Every time.
  2. Don't forget those automatic redoubles: if you have access to the cube and it's high enough for your opponent to win the match with a single win, don't delay! Send that cube over as soon as you can. (This is the same principle that demands an automatic double by the trailer after the Crawford game.)
  3. Be cautious with the cube if you're ahead in the match, because your lead could evaporate dramatically. Conversely, you can take the cube in more marginal positions when you're losing, because the option to redouble is so powerful.

Tony Lezard
18th March 2007