Cash game, center cube.

White to play 3-2.

Suppose you’re an intermediate-level backgammon player (a little vague, to be sure, but you get the idea) and you’d like to improve. What’s the best way to study the game in a systematic manner?

Different players have different approaches. Here’s what I like to do as a training regimen:

1) Play a daily practice match against Extreme Gammon (XG). There are other bots, but XG is the best. I like five or ten-game cash sessions, since even a short session will usually produce a good amount of interesting study material.

2) Let the bot analyze the session when finished. The bot will highlight a number of errors.

3) Perform rollouts on the errors to make sure they are, in fact, errors. I like 1296 trials, 3-ply. It’s not perfect, but it’s relatively quick and will produce more accurate results than the bot’s raw evaluation. You can always do longer and more rigorous rollouts later if the position seems to warrant extra attention.

4) When you’ve found a real error, try to categorize the position. Think about why you liked your original play, and try to see why the bot’s play might be better. Keep a notebook of different categories of positions. Print out your new position and stick it in the notebook. Your goal is to find categories where you’re making a lot of errors, and try to refine your understanding of that group of positions.

5) Keep track of your error rate for the session in a spreadsheet. Individual sessions will have huge variance, so average your results over 20-session samples. Over time, you’d like that error rate to drop slowly.

6) Repeat.

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A key skill in this whole process is the ability to categorize positions effectively. Weak players tend to see positions in terms of broad, mushy categories: ‘middle games’, ‘back games’, ‘endings’, and so forth. Better players see the game in terms of large numbers of narrow but well-defined categories: ‘1-3 backgame with adequate timing’, or ‘5-point anchor game with a third checker back’. Being able to see the game in terms of such narrow categories is advantageous because narrow categories may have strategies and heuristics which apply well within the category but don’t apply once we change the position slightly. The doubling strategies which govern a pure 5-point anchor position, for instance, don’t apply well once the defender has a third checker back.

I like to categorize positions is two broad ways:

1) By position type. “Position type” just refers to the broad outline of the structure of the position. Typical examples are “2-4 backgame” or “5-prime versus 4-prime”.

2) By tactics. A tactical category is just a description of the choices available. “Slot versus split”, “run out or build point”, and “escape prime or extend prime” are examples of tactical categories.

Sometimes we may have to combine both a positional type and a tactical type to make a useful category. For instance, “slotting versus splitting” tends to obey one set of rules in the very early game and a different set somewhat later. So we might have an “Early game: slotting versus splitting” category and a “Middle game: slotting versus splitting” category, with different strategies and guidelines.

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Now let’s look at this position. Before we start discussing the pros and cons of the possible plays, our first job is to categorize the position. It’s a very common type that I call “One Man Back”. One side (Black in this case) has escaped both his back checkers, while the other side (White) still has one checker back in a deep position.

In “One Man Back” positions, Black generally has a small lead in the race, something like 5-10 pips. Black won’t have a big lead in the race because generally no hitting has occurred yet, but he should have some racing edge because he’s rolled big enough to get his back checkers out, while White hasn’t.

What sort of interesting problems do we get in “One Man Back” positions? While many plays are routine, the interesting ones generally fall into one of three groups:

1) White’s checker plays. White’s problems arise when he can move his back checker but isn’t sure if he should. If he can run into the outfield and try to disengage, he should almost always do so. If all he can do is advance in Black’s board, then we get positions like this one. Should he move up where Black can point on him, thereby getting a chance to escape, or not?

2) Black’s checker plays. In general Black is bringing down builders and trying to make inner points. His good checker play problems tend to be one of three types:

a) Leaving indirect shots: If Black can leave an indirect shot but get an extra builder, should he do so?

b) Make a point or pick and pass: Suppose White plays 24/21 in this position and Black then rolls 2-1 or 3-1. Should he make the 5-point or pick and pass?

c) Hit loose: If White advances and Black can play safe or hit loose, should he hit?

3) Cube decisions. When should Black be doubling? In general, he needs less of a race lead for doubling than required in a straight race, since he also has the vigorish of winning with a prime.

Now let’s look at this position, and see just what White should do with a 3-2.

White has three choices. If he wants to advance his back checker with 24/21, then his deuce will be 6/4. If he leaves his back checker alone, he can play either 6/3 5/3, making the 3-point, or 6/4 6/3, leaving two blots.

Let’s look at the last two plays first. They’re close, but making the point is slightly better. Its advantage comes in the variations where White runs into the outfield next turn and Black hits. In those variations, White will sometimes have return shots, and if he hits a shot, the inner board blots are a real liability. For instance, here’s a typical such sequence:

White 3-2: Plays 6/4 6/3

Black doubles

White takes

Black 5-2: Plays 13/6

White 6-4: Runs with 24/14

Black 2-1: Hits with 13/11*/10.

While these sequences are unlikely, they should be very strong for White. They’re much weaker, however, with two blots floating around in the home board.

Eliminating 6/4 6/3 leaves us with a simple choice: stepping up in the board or making the 3-point. The race is relatively close (White trails by six, 120-114 after the roll) and that fact points to the solution to the problem. White should move up and try to escape, even though he’ll be escaping into a race where he’s an underdog. Consider these arguments:

(1) If White can get to a race, Black has only one way to win, namely the race. If White stays back, Black has two ways to win: either in the race, or by building a prime. Unfortunately for White, the small numbers that work poorly for Black in the race allow Black to fill in his 4-point and 5-point, building a prime and making the race essentially irrelevant.

(2) Getting pointed on after moving up hardly hurts at all, because the pointing numbers are crushing in any event. Suppose you knew that Black’s next roll was going to be 3-2. If you play 24/21 6/4 and Black points on you, you’re about 20% to win from the bar. If you stay back and make the 3-point instead, you’re still only about 22% to win. Better to be sure, but not by enough to matter.

(3) Moving up gains enormously if Black’s next roll is 2-1 or 3-1. If you stay back, Black makes his 5-point with these numbers and you’re about 20% to win. If you come up, Black’s best play is to pick and pass with each number (6/4*/3 or 7/4*/3), but then you’re about 30% to win. That’s a big gain on 11% of Black’s possible throws.

(4) Moving up or staying back doesn’t affect the cube action next turn. In either case Black has a clear double and White has a big take. His take is easier if he moves up, but it’s very easy in any case.

This is a key position because this type of decision (move up under pressure or stay back) arises frequently and is often misplayed. Moving up to reach an inferior race seems counter-intuitive to many players, but if the race isn’t too bad it’s mostly the right play. And in these “one man back” positions, the race is almost never too bad because of the way the positions come about.