The Cube Provocation Play
Danny Kleinman, 1980
Vision Laughs at Counting, Vol 2, © 1980 Danny Kleinman

We sometimes refer to an egregious error in backgammon as a cube provocation play. This is not to dignify the bad move, but to condemn it with gentle irony. For what can be worse than to play your roll in such a way as to draw a double from your opponent when an alternate move would restrain him from doubling? In a similar vein, we may call an exceptionally unfavorable roll a cube provocation roll.

Avoiding cube turns actually serves as a heuristic aid for the learner in backgammon. Some backgammon teachers tell their students to reason as follows: “If I make play 1, my opponent will turn the cube. But if I make play 2, he won’t. Therefore I should make play 2.”

Of course there are three trivial kinds of exception to this rule, so obvious that backgammon teachers usually don’t need to state them. First, you don’t want to make the play that keeps the cube away if your opponent thereby gets a chance to play for the gammon instead of doubling you out. Second, you don’t mind getting the cube via your opponent’s error, his misconception that your position is weaker than it is. Third, there are the classical Jacoby Paradox end positions in which your opponent will redouble you when your game is stronger but not when it is weaker. Thus:

Position 1.
Jacoby Paradox:
Black to play 4-1.

By moving (correctly) 4/off, 2/1, black will get the cube back (also correctly) with white being ever so slightly a favorite to win the game on the last shake. By moving (erroneously) 4/3, 3/off, black forestalls white’s cube return but becomes a heavy underdog nonetheless. This difference stems from the continuing usefulness of the doubling cube to black if he makes the wrong play and white fails to bear off. Instead of having a claimer (which makes the cube irrelevant), black then has a redouble and white a take. Far better for black to have the claimer and get the cube.

These exceptions out of the way, can we then formulate the general rule? Avoid making plays which get you the cube. For a long time, I thought so. Until a friend wrote to me about this position:

Position 2.
Black to play 2-1.

Black has two possible plays: 9/6 and 9/7, 9/8. I have deliberately omitted the doubling cube from the diagram because I now wish to introduce it as a variable. In his letter, my friend cited equities (as a fraction of the magnitude of the doubling cube) based on the location of the doubling cube. He claimed to have seen these equities, together with the proper cube actions, in a computer printout.

With black owning the cube, white has an expectation of 0.22034 if black plays 9/7, 9/8, but only 0.21588 if black plays 9/6. Therefore black should minimize his expected loss, moving 9/6.

With the cube centered (if that’s conceivable in a position like this), white has an expectation of 0.44068 if black plays 9/7, 9/8 but only 0.43176 if black plays 9/6. This is obvious from the fact that with the cube centered, white will double and black will take no matter which move black plays now, reaching the same state of affairs as when black already owns the cube, but with twice the equity because of the cube turn. Again, black should minimize his expected loss by moving 9/6.

With white owning the cube, white has an expectation of 0.44068 if black plays 9/7, 9/8 but of 0.44693 if black plays 9/6. White obtains this equity if black plays 9/7, 9/8 by redoubling. But if black plays 9/6, white must not redouble. For his equity of 0.44693, keeping the cube, is more than twice the equity of 0.21588, redoubling. And to warrant a redouble, the equity owning the cube must be no more than twice the equity with the cube given to the opponent.

Notice, however, that black is worse off moving 9/6 despite the fact that this play keeps white from redoubling. This is the Wisecarver Paradox: a proper (and nontrivial) cube provocation play, in which it is mandatory to walk into a correct redouble instead of avoiding it.

At first I couldn’t believe the Wisecarver Paradox. I thought there had to be some mistake, either in the computer program or in my friend’s transcription of the equities from the printout. I decided to check out the equity that seemed most suspect: where white owns the cube and black moves 9/6. By the time I wore out my eraser doing pencil-and-paper calculations, I had indeed found an error. My friend had inadvertently transposed digits, writing 0.44963 instead of the true equity, 0.44693. But this small error didn’t alter the essential result.

So now let’s try to explain the Wisecarver Paradox. First we should note that the differences are rather small. It’s close whether black should move 9/7, 9/8, or 9/6. It’s also close whether white should redouble whichever move black chooses. Thus we can expect small advantages to tip the balance in favor of either choice.

Next let’s see which is the better move in the absence of the doubling cube. This isn’t merely a theoretical question. For in tournament matches, the Crawford Rule and certain match score produce inactive cubes.

We must examine a cross-section of 46,656 games, representing the next two shakes for both white and black. Black has 36 shakes but white may be thought of as having only 6 shakes: 1 doublet and 5 “singlets.” We can see at once that white is off in two shakes 11/36 of the time, off in three shakes 25/36 of the time.

When white is off in two, the only shake for black producing any difference between the two plays is double 4’s on the next shake. This wins only after black has played 9/7. 9/8 and swings 396 games to black. All the other “swing shakes” occur only when white is off in three. Then black’s rolls of 6-4 followed by 3-2 swing 100 games to black only if he plays 9/6. Black’s rolls of 6-3 or 5-4 followed by 6-1 or 5-1 likewise swing 400 games to black only if he plays 9/6, for we may assume that when black plays 9/7, 9/8 he will then use a 5-4 to move 8/4, 7/2. On the other hand, when black rolls 6-2 or 5-3, he can move 7/5, 8/2 or 8/5, 7/2 to reach a 5-2 bearoff. But if he plays 9/6, he must play a 6-2 by moving 9/3, 6/4 and a 5-3 by moving 9/4, 6/3, reaching a 4-3 bearoff. This swings 200 games to black only if he plays 9/7, 9/8.

A roll of 6-1 for black allows him to play 9/3, 6/5 if he has moved 9/6, whereas he must play 8/2, 7/6 if he has moved 9/7, 9/8. But this is exactly balanced by the roll of 5-2, which compels white to play 9/2 if he has moved 9/6 but allows him to play 8/3, 7/5 if he has moved 9/7, 9/8.

Thus the swings favoring 9/6 are 100 and 400, totalling 500 games. The swings favoring 9/7, 9/8 are 396 and 200, totalling 596 games. Without any cube, black should therefore move 9/7, 9/8, which wins 96 more games in 46,656 than the alternate move of 9/6. The decisive factor is the creation of a joker for black, double 4’s, which guarantees victory in those cases where black needs to get off in one shake rather than two. Without that, the greater flexibility of having the two men 3 pips apart instead of just 1 would tip the balance in favor of 9/6.

Why, then, should the presence of the doubling cube — either in the center or in black’s possession — swing the balance in favor of the otherwise inferior move of 9/6? For, no matter which move black takes, the doubling cube will still be available to him. This is the true mystery underlying the Wisecarver Paradox.

The solution lies in the concept introduced in “Cuboffs and the Crippled Cube”: the logic of the doubling cube isn’t a simple “yes or no,” “available or unavailable,” two-valued logic. Its healthiest use occurs when your opponent has an equally bad pass or take. Its most crippled use occurs when you have either a virtual claimer or just the barest of advantages at the time you double.

When the cube is available to black in Position 2, moving 9/6 creates the prospect of a 100% efficient, super-healthy cube turn for black two shakes later. This occurs whenever white fails to roll any doublets while black rolls exactly 9 pips — either a 6-3 or a 5-4 — on his next shake. Then black remains with a lone man on his 6-point and, when he turns the cube, the expected loss for white becomes the full value of the cube whether white passes or takes.

If black plays 9/7, 9/8, however, this super-healthy cube turn cannot become available to him. Most of the swings favoring the play of 9/7, 9/8, in fact, occur without black’s being able to turn the cube at all, in the cases where black rolls an immediate double 4 while white is rolling a doublet.

This explains the real heart of the mystery, why the availability of the doubling cube to black should induce him to move 9/6. It also explains the second puzzle — why white must not redouble after black plays the inferior 9/6. White barely has a redouble after black moves 9/7, 9/8. White gains just slightly more by doubling the stakes than he loses by giving black the ability to turn the cube. But after black moves 9/6, despite the fact that doubling the stakes gains even a bit more for white, giving black the ability to turn the cube loses considerably more; because now black’s redoubles will include the perfectly efficient cube turns possible with just one man left on the 6 point.

Are there other proper cube provocation plays? We may conjecture that there are, in other branches of the “Advanced Bear-In Technique.” How about cube provocation plays in contact positions? Before seeing the Wisecarver Paradox, I would have said, “Of course not!” But now, I’ll guess that there are. Won’t some other friend please write me and show me one? I won’t ask for a copy of your computer printout.

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