Advanced-Level Backgammon: An Outsider's Guide

Problem #4

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Black had the good fortune of opening with 31, letting them make their 5-point and start the game with a strong advantage. White rolled 41 in reply, and opted to split their back men (a minor split, to be specific) and bring down a builder onto the 9-point. Now Black's rolled 61, and is faced with a dilemma: Should they play offensively or defensively?

Candidate Moves

Some of Black's more viable options include:

13/6

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13/6, despite being completely risk-free, should be rejected immediately as too passive. Although Black has a lead in development, White, between their 9-point (shown as the 16-point) builder and split back men, is poised to catch up in the next turn or so by making a strong point of their own. If Black plays passively now, they'll effectively be giving away their advantage. That's saying nothing, of course, of the uselessness of the play itself: The extra checker on the 6-point does absolutely nothing for Black here.

13/7 8/7

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13/7 8/7 is also no good. While Black might like to have the bar point for blocking and priming, they'd have to give up the 8-point to make it here, leaving a direct shot in the process! If White manages to hit (19 rolls hit: 66 65 64 63 62 61 52 51 43 42; 52.8% chance), they'll no doubt equalize—and if they miss, they'd still have an excellent builder at their disposal, not to mention the possibility of anchoring. If anything, rather than diminishing White's prospects, this move adds to them!

24/23 24/18

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24/23 24/18 is an unorthodox move, splitting to White's bar point then using the remaining ace to bring Black's second back man forward. There are situations where a play like this one is reasonable, particularly those where Black is ahead in the match and White has stripped their 8-point (as this move doesn't risk very many pips for Black, and White would be unable to break their 8-point safely with two men bearing on it); but this position is not such a situation, so a better play is needed.

8/2*/1*

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8/2*/1* abandons any pretense of positional play in favor of going on the offensive, burying a man on the ace-point in order to hit White twice. This move is the favorite of the older theory, which cites this type of double-hit reply as one of the main weaknesses of a minor split in the opening.[1]

To be sure, it's a strong move—it arrests White's board development, since the hits deprive them of the next roll or two (turning their builder on the 9-point into a liability in the meantime); and especially in the cases where White fails to enter (4 rolls will fan: 66 65 55; 11.1% chance), Black has a solid double.[2] Indeed, it has the highest gammon win chance out of any of the candidate moves for this position!

Playing this move automatically, however, is ill-advised in match play, as the current score should be taken into consideration: While it's by far the best choice if Black needs to win a gammon, picking 8/2*/1* would be dubious in a DMP situation where gammons make no difference, or if Black already had a healthy lead in the match and couldn't justify the risk from the return shots (and shots at expensive blots, no less).

24/18 6/5

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24/18 6/5, in contrast to the previous move, is far more positional, splitting to White's bar point while unstacking the 6-point for a second inner board builder. Bringing a back man up to White's bar point certainly looks questionable: It leaves a triple direct shot! However, it's not quite so dangerous to Black as it appears:

  1. The blot on the bar point is a cheap blot: Black would only lose 7 pips if hit.
  2. Because White only has one inner board point closed, Black is an overwhelming favorite to re-enter immediately if hit (only 66 stays out; 97.2% chance to enter).
  3. Since the blot is sitting in White's outer board, White has no good way to hit and also make a desirable point[3] without lucky doubles (which you should never bank on)—indeed, unless White makes their bar point (begrudgingly) or rolls doubles, they'll have to leave a direct return shot for Black!

Weighted Blot Cost

The unfortunate reality for Black is that, outside of the timidly passive 13/6, there are no viable moves here that don't leave a blot. The race is pretty much tied at the moment, so Black would like to avoid falling behind from a costly hit if possible. In particular, playing the six is troublesome; assuming Black uses the ace to play 6/5, can we determine which of the available plays (namely, 24/18, 13/7, and 8/2*) is the least dangerous for Black?

For starters, let's start by counting the number of shots each option leaves:

Interestingly, for as dangerous as the triple direct shot against 24/18 looks, it only leaves one more shot than the double direct shot against 13/7! That aside, 8/2* leaves the fewest shots, but is it really the best choice? Why not simply play 8/2*/1* instead?

The problem with basic shot counting is that it doesn't account for how cheap or expensive the blot is. Thus, I present the Weighted Blot Cost (WBC) metric. It's a simple enough concept: If you multiply the odds of a blot being hit by the cost of that blot, you end up with the average number of pips the player can expect to lose by leaving that blot. This value doubles as a loose risk estimate: The higher the WBC, the riskier the move.[4] Of course, we can take it further by multiplying this average pip loss by the ratio of the opponent's pipcount to our pipcount (since the further ahead we are in the race, the more the hit will hurt), but in this case, the race is tied and thus won't affect our calculations.

So then, what's the WBC for each option?

Despite leaving a triple direct shot and the greatest number of shots overall, 24/18 rates to be far less dangerous overall for Black than 13/7 or 8/2*! Clearly, the pip cost of a blot is just as important to consider as the number of shots.

In conclusion, 24/18 6/5 is a strong and relatively low-risk move for Black, one that's especially appealing should Black be leading or at DMP—but even in money games, it's still a solid play.

Answer

Key points

Rollouts

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Money game rollout equity table
Rollout, money game
Basic match rollout equity table
Rollout, 7-point match, score 0-0
Black-leading match rollout equity table
Rollout, 7-point match, score 5-0 (Black leading)
White-leading match rollout equity table
Rollout, 7-point match, score 0-5 (White leading)
Double match point rollout equity table
Rollout, 7-point match, score 6-6 (DMP)

Footnotes

[1] As has been discussed previously in this guide, the older theory much prefers aggressive slotting in the player's own inner board (rather than splitting your back men), doubly so in response to your opponent opening with 31 or 42—but newer theory, analysis, and play has shown time after time that splitting provides stronger positional advantages than previously thought, not to mention the risks of your inner board slot being hit when your opponent has more inner board points closed than you!

[2] Incidentally, while Black should absolutely double if White stays out here, it by no means guarantees an immediate win. On the contrary, in most cases, White should take!

[3] You might be tempted to think that White would be satisfied with making their bar point on Black's head, but consider how the position looks for them afterward: Their 6-point and midpoint are still stacked, their bar point and 8-point are both stripped, and they only have one inner board point closed to Black's two. Their attacking and priming prospects alike are rather limited for the next few turns, during which time Black could anchor, escape, extend their prime, or some combination of the three!

[4] I'll be the first to admit that it's not really practical to calculate the exact WBC over the board, especially in cases where the race isn't even. Rather, the main point of it is to underscore how some blots are greater liabilities than others, and that you should be mindful of more than just the number of shots when considering candidate moves for an awkward roll.

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