Late Loose Hits

The most serious errors in backgammon are not necessarily the largest. Just as more equity is given up on checker play errors than cube blunders, common errors may contribute more to your error rate than blunders in rare positions. According to Walter Trice, failing to hit loose in the late game is one of the most common errors. In fact, his June 2006 column covered a variety of these positions. In this column, we'll look at one situation: Loose hits late in the game when you have a 5 point board. Opportunities such as these commonly occur after your opponent breaks a high anchor.

Any simple rule is going to be perfect in a restricted context, and wrong when applied out of context. We'll investigate some of the factors which affect the decision.

When ahead in the race, don't race?

Hitting while ahead in the race disagrees with the guideline, "When ahead in the race, race." That helps to explain why people err so frequently with these decisions. Failing to hit is easy to justify. While that guideline is useful, there are a lot of exceptions. For example, it can be right to run into an unfavorable race, even a hopeless race, if staying back will let your opponent gain from attacking you more than you gain from the chance to hit. It is more accurate but less memorable to say, "When racing is better than the alternative, race." When you get a chance to hit loose with a 5 point board, hitting leaves you a favorite, so you need a significant racing lead to prefer racing, particularly if your opponent is impeding your race if you do not hit.

Not hitting means the race is very important. It's still possible for each side to hit the other, but there is a good chance that neither side will hit or block the other significantly, and your winning chances are highly correlated with the race.

When you hit loose with a 5 point board, the race is no longer important. Almost all of your losses come from getting hit, immediately or later. So, hitting is worth about the same amount regardless of the status of the race. When the gap is deeper in your board, a closer race is a sign that some entering rolls crack, so the equity from hitting isn't perfectly constant, but that is a minor effect.

Thus, this decision is unusually sensitive to the race, just like when you might break contact. When you are considering a late loose hit is one of the times an exact pip count is necessary for proper checker play.

Let's consider a sequence of positions close to the following. We'll move White's spares to produce related positions with different racing leads.

Red to play 5-1.

Should Red hit with 8/3 6/5*, or race with 8/2?

Lead after no hit Hit Race Difference
 62-76   0.241   0.305   -.064
 62-75   0.237   0.254   -.017
 62-74   0.236   0.204   0.032
 62-73   0.225   0.129   0.096
 62-72   0.223   0.083   0.140
 62-71   0.218   0.028   0.190
 62-70   0.219   0.003   0.216

In this case, the closest decision was at a difference of 13 pips after moving. The equities from hitting were all in a tight range 0.218-0.237. However, the equities from racing changed by about 0.050 per pip. Getting this type of decision wrong by 2 pips can be a 0.100 blunder.

Thirteen pips may seem like a lot, but it's not a crushing lead. If we were on roll in a race, leading 60-68 would be a close take/pass decision. A lead of 60-73 would mean our opponent would have a huge pass, with about a 14% chance to win. However, we are considering a racing lead when our opponent is on roll, not when we are. Leading by 13 pips after rolling is a real lead of about 9 pips, or 5 pips and the roll.

Other open points

How about loose hits on blots left from other high anchors?

Red to play 6-5.

Red to play 6-5.

These positions may look similar, but the closest decisions are at leads of 11 and 9 pips, respectively. The equity from hitting doesn't change much. The main reason that Red is less eager to hit is that White has decreased chances if Red does not hit. A high double is blocked, and blocked lower rolls may cause wastage and cause White to crack, making later hits more common and more appetizing. Also, a gap on the 3 point does not harm Red's racing chances as much as a gap on the 4 point, which is better than a gap on the 5 point.

Cube location

After hitting, there will often be some easy cube decisions. In the positions we are considering, if you have access to the cube, then a hit followed by a dance is a huge market loser. Doubling when you win over 90% of the time doesn't give you much value from the cube. Similarly, most return hits lead to overkill if your opponent has cube access. On the other hand, racing tends to lead to efficient doubles, giving more value to a player who owns the cube, or to the race leader when the cube is centered.

Since the decision to hit or not affects the cube efficiency so much, the cube location often determines which plays are correct. Given two plays that would be equally strong without the doubling cube in play, you should prefer the play which gives your opponent overkill, and avoids overkill for yourself. This will also make your take/pass decisions easier.

If your opponent owns the cube, you should prefer hitting in positions where it would be slightly too aggressive at DMP. The difference is largest for higher open points, since the return hits are crushing. Entering on the 3 point still leaves your opponent with the task of escaping a small prime, so there isn't as much overkill.

If you own the cube, you should be averse to hitting, preferring to race when it would be slightly better to hit at DMP. When the cube is centered, this is closest to owning the cube, since your opponent is favored to dance if you hit. However, the return hits give your opponent some overkill, too, so you should hit more readily with a centered cube than when you own the cube.

Here are the thresholds for each open point and each cube location plus DMP, according to rollouts. These are the closest decisions, including some close hits and some close non-hits.

  3 point 4 point 5 point
  Oppt's   9   11   13
  DMP   8   10   11
  Centered   5   8   11
  Owned   4   7   9

Other factors

Factors to consider when evaluating the racing play include the pip counts, wastage, and the ability to attack later if the blot stays. If there is a gap, not hitting now many mean leaving a shot later.

Factors to consider when evaluating the hits include the quality of your opponent's offense, how many rolls cover or lift the blot when it is missed, whether some return hits crack, and whether the straggler may have difficulty escaping after a hit.

Since the race is so important, a good way to study these factors systematically may be to try to quantify the number of pips they are worth. This is easily done with wastage, since we normally measure that in pips. A flaw such as an open 5 point for White is worth about 3 pips, since it both introduces wastage and makes some return hits survivable.

Lamford's rule

Paul Lamford introduced a 10% rule for some late hits in one of his books, Improve Your Backgammon. The basic idea is as follows: If you are leading by 10% after moving, don't hit. If you aren't leading by 10%, hit.

I don't know the exact context of this rule, but it is quite different since 10% does not agree with most of the thresholds encountered here. Lamford considered hits with a 4 point board. These hits are not as strong, since they leave additional return shots, and there is a lot more equity from late hits, so it takes less of a racing lead to prefer a race. On the other hand, the location of the cube does not make as much of a difference, since there isn't as much overkill.

I'd like to thank several contributors to the GammonU bulletin board for helpful discussions of Lamford's 10% rule, particularly Tansley, Fabrice Liardet, Marty Storer, and Steve Mellen.


Race when it is better than the alternative.

A common type of error is failing to hit loose with a racing lead that is not overwhelming. Even when your opponent's offense is perfect, it can be right to hit instead of racing when you lead by up to 13 pips after rolling.

The pip lead needed to prefer racing over hitting is sensitive to the cube location. When both hit/dance and return hits lead to crushing positions, the situation most conducive to hitting is when your opponent owns the cube, followed by DMP, then a centered cube. Owning the cube makes racing the most attractive, with marginal decisions 4-5 pips different from when your opponent owns the cube.

When your opponent owns the cube, you may want to hit loose on the 5 point instead of leading by 13 pips after moving, hit loose on the 4 point instead of leading by 11 pips, and hit loose on the 3 point instead of leading by 9 pips. These guidelines assume that you have a 5 point board, and the only outside point is the 8 point.

There is still room to study more classes of late loose hits. We haven't considered hits which add significant gammon chances to either side, or hits on deeper points which might move a blot forward. I hope this is a helpful approach, and encourages you to think about hitting even when you have a racing lead.

Copyright Douglas Zare and GammonVillage.