| Cube Theory |
The Doubling Cube in Tournament Matches
Match play injects several new variables into cube handling. You must still evaluate positions and react to your opponent’s psychology. But the match score and the cube size assume paramount importance. Skill differences affect the cube. The worth of owning the cube ranges from nothing to almost everything. Last shake doubling points vary wildly from 50%, and take points seldom approximate the 25% of money backgammon.
Since the Jacoby rule does not apply to matches, gammons may occur in undoubled games. The Kauder paradox never arises. The Kleinman gammon adjustment changes from a flat .50 to a variable ranging from 0 to more than 2.
Only the principle of last-take doubles against liberal opponents but first-pass doubles against conservatives carries over from money backgammon. Even here, you must apply this principle differently. For opponents unfamiliar with the strategy discussed here act liberally in some situations but timidly in others. In particular, most tournament players double too quickly when behind in a match, and pass too readily when ahead.
I hope the strategy I will recommend appeals to your reason. But it is rooted in calculations whose results are summarized in the appended charts, calculations based on certain assumptions.
Unless otherwise stated, the match is between opponents of equal skill. Of all games played out, 20% tend to be gammons. Though in any given position, the game favorite may be the gammon underdog, on the average I assume, following a suggestion of Paul Magriel, that each player’s gammoning chances are proportional to his game-winning chances.
We call the first game after match point the Crawford game, after the Crawford rule, which eliminates the cube from this one game. For convenience, we consider the 9-point match. But what matters is really the distance of each player from the goal of 9, not the points scored. Thus our 6–3 score is identical to a 4–1 score in a 7-point match or a 10–7 score in a 13-point match.
Different Values for Different Points
The central concept of the tournament cube, making it very different from money games, is the worthlessness of points beyond 9. Thus when a win will bring the leader to 9 points or more, the trailer in the match can turn the cube “automatically,” unless a gammon threat renders him too good to double.
After the Crawford game, the trailer should often double at his first turn. And when the leader has 7, his strong gammon threat should provoke the trailer into doubling before hitting a single game-winning shot.
The ninth point weighs more heavily than any other, since it clinches the match. As a rule, each higher point is worth more than an earlier point, just as in football, yards closer to the goal line become more precious. As the leader approaches 9, however, the odds in each game come to favor the trailer in the match.
At match point, neither gammons nor the cube work for the leader. The Crawford rule mitigates this by taking the cube away from the trailer for one game. At 7 points, leader can use gammons or the cube but not both. To a lesser extent, this applies to 6 points also. Thus at these totals, the leader’s advantage may often be not as great as may appear on the surface.
Match Point
After the Crawford game, the trailer should normally double as soon as possible. Delay helps the leader by allowing him to observe how the game develops before deciding whether to take. The leader will pass if he sees a gammon danger. But in practice the trailer doubles before the gammon threat looms.
Paradoxically, when the leader has a “free pass” to give, the trailer may refrain from doubling if he jumps off to a strong advantage early in the game. His game may improve to a possible gammon if he waits, while he can reserve the ability to double if his game deteriorates to about even.
Likewise, when the leader has a “free take,” the trailer may try to exploit ignorance of this by waiting until he has a strong position with a gammon threat before doubling, in the hope of getting a foolish pass.
Normally, games at match point get played out with the cube at 2. Thus the trailer must win one game for every two points he needs to reach 9. Each advance to an odd score brings him a game closer to victory, giving the leader a free pass when the trailer is already at an odd score. Each advance to an even score gains practically nothing for the trailer, giving the leader a free take when the trailer is already at an even score.
These odd-even considerations influence proper cube handling as the match progresses. The trailer prefers to be on odd after the Crawford game, therefore on even entering the Crawford game. This distinction affects strategy whenever match point is a possible outcome of a game.
At 8–7, the leader should pass a double if even the slightest underdog in the game. His free pass expires unless he uses it now. But with a larger lead, such as 8–5 or 8–1, the leader should save his free pass for a game where the trailer begins with a truly superior initial roll.
Even Matches
In money games, the slightest advantage on the last shake dictates a cube turn. This principle applies to all even scores in matches, but only to a few uneven scores. At all other scores, winning chances above 50% by a certain amount are needed for a last-shake double by one of the players while winning chances below 50% by that same amount will justify a last-shake double by the other player.
Since each succeeding point in a match tends to be increasingly valuable, you should be slightly conservative in taking the cube at 2 or 4 in even matches. Losing 2 points is a tiny bit more than twice as bad as losing 1. Losing 4 points is slightly more than twice as bad as losing 2. But since you cannot really lose 8 points if your opponent already has more than 1, as scores above 1–1 you should become progressively more liberal in taking 8-cubes.
Two Points Away
As a general guideline, the leader should become more conservative and the trailer more liberal in rough proportion to the gap in their scores. Leading with 7, you should become somewhat reluctant to turn the cube in the middle of the game. If you do, gammons will no longer be of any worth to you; and you can expect your opponent to redouble immediately at no risk to himself. You therefore gain one extra point if you win the game, but lose three extra points if you lose the game.
In taking the cube at 2, however, you don’t need to be so conservative. How the cube stays at 2, and just one extra point is available for both you and your opponent. That point — your ninth — is extremely valuable to you. Only when the second point is equally valuable for your opponent — when winning a 2-game would bring hime to 8 or 9 points — should you become conservative. At other scores, you can afford to be liberal. But you must remember to adjust for your opponent’s gammon threats while ignoring your own.
You have a special reason for taking liberally with a 7–3 or 7–1 lead. You figure to retain the lead even by losing, and with the Crawford game on the horizon, you should be very reluctant to let your opponent advance to the next even point.
If your opponent leads with 7, your ability to return the cube at 4 immediately dictates that you take quite liberally. This doesn’t apply at a 7–6 score because only the third point, not the fourth, will be working for you when you redouble; and if you pass, you will enter the Crawford game trailing 8–6, and even point. In contrast, trailing 7–5 you can take very loosely indeed. For when you redouble, the fourth point you put at stake works for you with maximum efficiency, winning the match exactly.
Two things — the ability of the trailer to redouble immediately, plust the gammon-killing effect of a cube turn — render the seventh point somewhat less valuable to the leader in the match.
But on the very last shake, paradoxically, you must reverse all your normal tendencies. For now redouble is possible any more. If you lead in the match, double even as an underdog sometimes. If you trail, however, you should become timid in doubling, supercautious in taking. The two points at stake with a double and a take win the match exactly for the leader.
Three Points Away
When the leader has 6 points, there is another seeming paradox. The leader should be less conservative with the cube than the trailer! A single point for the leader brings him to 7, where the cube works against him; therefore the trailer should be somewhat more willing than ordinarily to concede a single point by passing rather than giving the leader a good chance to reach 8.
The leader’s take strategy should be influenced by odd-even considerations. If the trailer has an odd number, the leader should be quick to take the cube. For, if he wins two points, he enter the Crawford game with the trailer on odd, a favoriable situation.
In contrast, if the trailer has an even number, the leader should be slightly reluctant to take. By passing, he puts the trailer on odd. By taking, even if he wins the two points, he enters the Crawford game with the trailer on even, an unfavorable situation.
In considering cube turns to 4, however, the leader should be timid and the trailer bold. The two extra points put at stake by the redouble both work for the trailer. But only one of them works for the leader. The other constitutes his tenth point, utterly worthless.
Four Points Away
With victory fairly far away for both players, a regular, normal cube strategy emerges for both players. With a point total of 5, the leaders uses the cube and gammons just as effectively as the trailer. And gammons at the 2 level work with with perfect efficiency for the leader, winning the match for him exactly. Since the Crawford game is seldom on the horizon, odd-even considerations have little weight here.
For cube turns to 2, the leader should be only slightly more conservative than the trailer. But the automatically redoubling possibilities for the trailer should make him quick to take the cube if redoubled to 4.
More Than Four Points Away
With victory so far off, the tournament cube idffers little from the money cube. The leader in the match should tighten up his takes only ever so slightly. He should beware the 8-cube, however.
At all lower levels of the cube, win or lose, the points already scored still count in the leader’s favor. An 8-cube, however, drowns out the results of all prior games in the match, essentially putting the match on the line for both players most of the time. And more of the last four points put at stake by the cube turn to 8 work for the trailer than for the leader.
Shortcuts in Estimating
Few players can memorize the tournament cube-handling charts. But some simple techniques permit reasonable estimates of last-shake doubling points. Count the new points introduced for each player by a cube turn, then compute the odds from these.
For example, suppose you own the cube at 2 trailing 6–5. By redoubling to 4, you introduce two new points — your eighth and ninth — for yourself, but only one working points — his ninth — for your opponent. You get 2-to-1 odds by redoubling. Your last-shake doubling point is thus 33%, or 12 winning numbers in 36.
You may refine this estimate by taking odd-even considerations into account. If you lose a 2-game, you trail 8–5, an odd score, bad for your. This is an extra incentive for you to redouble to 4. Count it as worth about 3%, or one extra number in 36. With this refinement, you can guage your last-shake doubling point at 30% — exactly what the chart shows!
Eleven winning numbers in 36 justify a last-shake redouble. This could be a single, final direct shot in a holding game which will otherwise become a hopeless losing race.
Unequal Players
A difference in the skills of the opponents in a match makes the better palyer the favorite to win the match just as if he already had a lead in a match against an equal opponent. This makes the optimal cube strategy obvious. The better player should become slightly more conservative than against an equal opponent. Just one 4-game or 8-game can drown out the results of several 1-games or 2-games won by the better player.
Even if we could measure skill difference numerically we would need separate charts to show proper cube strategy for every different assumption we might make about the degree of inequality. But we can illustrate the effect of skill differences with a cube turn at a score of 7–7.
Between equal players, 30% winning chances justify a take. If one player is a 55–45 favorite to win each game, however, the weaker player should take with barely over a 25% chance, while the stronger player should require better than 36% chances in order to take.
The longer a match runs, the truer to form. The more points at stake per game, the fewer games in the match. Liberal cube handling helps the weaker player; conservative cube handling the stronger.