Cube Theory 
The Doubling Cube in Money Backgammon
Three basic considerations should govern your cube handling in money backgammon. Most importantly, the backgammon position itself. Secondly, the location of the cube. And thirdly, the nature of your opponents.
“Playing your opponent” divides further into three aspects. One is his skill, particularly his ability to play this type of position. Your opponent’s skill is just one more factor determining your winning chances, akin to your pip count in a race or the strength of your home board in a blothitting contest.
The second is your opponent’s morale. The effect of your winning this game may extend beyond just a certain number of points entered on the score sheet. If you can turn a back game around, your opponent may start to steam in future games, especially when he is stuck with a big minus on the score sheet. Sometimes you can use this to help you make close cube decisions.
But the third and most vital aspect of playing your opponent comes when you are contemplating doubling. Then you must adapt your doubling to the liberality of your opponent in taking the cube.
Considerations That Cost You Money
Sometimes you will correctly take a double which you should theoretically pass. You do this because demoralizing your opponent may gain for you over the long haul. But you should recognize that you pay a price in any individual game when you do this. Other things you may let influence your cube handling also rate to cost you money, but without providing compensating advantages during the rest of the session of backgammon.
Steam doubles and scoreboard passes always figure to hurt you. So does letting yourself become intimidated by the size of the cube. If a 32cube makes you uncomfortable, you’re playing for too high a stake. And if you’re worried that an opponent won’t pay off if he loses a huge game, then you should be playing at a club where collecting is guaranteed.
Perhaps you want to take the cube just to see how a game like this one can turn out. I forgive you — so long as you understand that your financial loss is the tuition you pay for this education.
Perhaps you want to take the cube because you consider yourself the strongest player in your chouette, so that you are willing to pay a premium for a chance to be in the box. For this I will not forgive you so easily. You may be overrating your advantage in the chouette. Part of that advantage, moreover, lies in your superior cube handling — which your incautious take has just dissipated. But if you’re really willing to pay a premium for a chance at the box, note that this premium is several times greater when you’re already in the box than when you’re the captain.
But perhaps you kid yourself in justifying your poor takes by your desire to win the box. Maybe you’re just giving into your irrational gambling impulses. A rational desire for the box dictates great reluctance to turn the cube just as much as it allows loose takes. For by waiting until your opponents will pass your doubles, you assure yourself of the box. If you really mean it about wanting the box, you will often play games as captain where your partners in the crew plead in vain for you to turn the cube. Never will we see you offering to buy your partners out for “halves.”
How High May the Cube Get?
Just in case you don’t realize how quickly the cube can bounce around, even in expert games, let’s consider an actual ending in an expert game a few years ago (Diagram 1):

Diagram 1 Black on shake 
So far, the cube action has been tame. White was a strong favorite in the bearoff, and doubled. Now white has rolled poorly, and black has finally become the favorite. Realizing this, black redoubles.
White, however, appreciates that black’s advantage rested solely in owning the cube; without the cube, black is a slight underdog. So white beavers to 8!
Now black rolls 61, bearing off two men to reach Diagram 2:

Diagram 2 White on shake 
White redoubles, and black accepts the cube at 16. White rolls badly, a 31.
He quickly moves 5/2 with his 3 and studies his 1 for three minutes, finally moving 5/4. Then he turns to his kibitzer and asks, “Did I play it right?”
“There is no right or wrong way to play your 31,” answers the kibitzer. Ha! What do kibitzers know, anyway?
The position is now Diagram 3:

Diagram 3 Black on shake 
Black redoubles again, and white takes the cube at 32. Black “misses” with a 62, and back comes the cube at 64 in this position (Diagram 4):

Diagram 4 White on shake redoubles to 64 
What does white roll now? Why, a 42, of course!
Didn’t I tell you? White is Alan Martin!
The only mistake in this sequence of cube returns is black’s first redouble in Diagram 1. All the other cube actions are mandatory.
Your Take Point
Most backgammon authorities offer an oversimple criterion for taking the cube. They say you need a 25% chance to win the game (ignoring the cases where gammons and backgammons can occur). But the commonly accepted 25% take point applies only to decisive shakes, that is, shakes which determine the final outcome of the game, shakes after which the doubling cube will no longer be useful. If you get doubled on an indecisive shake, your exclusive ownership of the cube will give you equity enabling you to take despite winning chances somewhat under 25%.
In a sense, the phrase “winning chances” is meaningless without specification of the location of the cube. When your opponent doubles you and you take, he has reduced his winning chances, for he will no longer win all of the games he would have won by waiting to double you out later on. And if his cube turn was a redouble, he reduces his winning chances even further. For you may be able to double him out later on, keeping him from finishing out many of the games he might have won if rolled to the bitter end.
Thus winning chances which can be calculated to be below 25% with your opponent owning the cube (or even with the cube centered) sometimes turn out above 25% once you get the cube. Even if you can never use the cube to double your opponent out, it can still have a different kind of value: in controlling the stakes. You can sometimes take despite 1 less than 25% winning chances because all your wins will come at redoubled stakes but few of your losses will.
A simple twoman versus twoman bearoff illustrates this (Diagram 5):

Diagram 5 White doubles. Should black take? 
Black will win 65 games in 324, or 20.06%.
Black needs white to roll one of the 10 numbers (any ace except double aces), and then to roll one of 26 numbers himself (any nonace or double aces) in order to win. That’s 260 chances in 1296, or just over 20%.
But black will win all 260 games redoubled, while only 100 of white’s 1036 wins will come at redoubled stakes. This is the equivalent of winning an extra 80 games in 1296. Black has a clear take.
Most positions do not lend themselves to such easy exact calculation. You know only that your take point is somewhat below 25%. Just how much below depends on your chances for turning the game around sufficiently to redouble. Our example demonstrates that the value of the cube for even a 4to1 underdog can be as much as 5%. Whatever you estimate the worth of the cube to be must be added to your percentage equity. Only then should you use 25% as your guide in deciding whether to take doubles.
Gammons and Backgammons
When you needn’t consider gammon possibilities, you may express your estimate of your equity as a simple percentage: your winning chances, adjusted for the worth of cube ownership. The accuracy of this percentage, of course, depends on your skill in appraising the backgammon board. But once you make your estimate, you have a single number to compare against your take point of 25% in deciding whether to accept a double.
Gammon threats complicate your decision. You need good backgammon judgment to gauge gammon chances, of course. But then even after you have exercised judgment to make an estimate, you have four additional numbers complicating your rating of the position: your chances for (1) gammoning and (2) backgammoning your opponent, and for being (3) gammoned and (4) backgammoned yourself.
You can use the Kleinman gammon adjustment to incorporate these four new numbers into a singlepercentage equity. Starting with your original winning percentage, add for your gammon threats and subtract for threats against you. Count gammon percentages at half value and backgammon percentages at full value. (Note that backgammons are not treated as a species of gammons.)
Let W = Probability of winning
Let G = Probability of winning a gammon, but not backgammon
Let g = Probability of losing a gammon, but not backgammon
Let B = Probability of winning a backgammon
Let b = Probability of losing a backgammon
The gammonadjusted winning equivalent is
G − g 
2 
To illustrate this procedure, suppose you think your winning chances are 40 in 100. But you estimate that of your 60 losses, only 40 will be single games; the other 20 will consist of 14 gammons and 6 backgammons. The 14 gammons against you reduce your raw winning percentage of 40 by 7, to 33. The 6 backgammons reduce this further, to 27%. Even without the few percentage points you may want to add for owning the cube, you have a clear take (25% or better) if doubled.
Uses of the Cube
If you are familiar with the stock market, you may compare owning the cube with owning a call option, which is an option to buy a particular stock at a particular price until a stated expiry date. The worth of a call fluctuates with the price of the stock. A call may diminish in value as the expiry date approaches. And a call may be valuable if the stock is a highly volatile one, whose price can rise or fall rapidly.
A call may be exercised to take a sure profit on the stock, buying at the exercise price to sell immediately at the higher current trading price. Or a call may be exercised with the intention of retaining the stock as an investment for the long haul.
In backgammon, the option to double may be exercised in two ways also. You can take a sure profit by doubling your opponent out when he must pass, or invest in higher stakes by doubling while your opponent will still take.
If you wish to exercise your call in the stock market for investment purposes, then it pays you to wait until the last possible moment (i.e., the day just prior to expiry) before buying. By waiting, you protect yourself against the stock taking a nose dive in the time remaining before expiry. If the market price drops below your exercise price, you will refrain from buying.
Likewise if you wish to increase the stakes at backgammon by doubling in the expectation that your opponent will take, it pays you to wait until the last possible moment to do so. by waiting, you protect yourself against the game turning around to the extent that you regret the higher stakes.
If you wish to exercise your call in order to realize a sure profit by selling, you must decide at every point in time whether to be satisfied with the profit you can take immediately or wait for the price of the stock to rise further, risking a possible drop in price, which reduces or eliminates your profit.
It’s different with the doubling cube. As soon as you can double your opponent out, you have achieved the utmost profit possible. Delaying doubling now can only result in your losing the ability to double your opponent out, never in any larger profit.
Thus you have two different doubling strategies, which we can express as the point of last take and the point of first pass.
The value of owning the cube varies with the likelihood that the owner will be able to use it, and thus with the value of his position. It also varies with the volatility of the game, with how easily the value of the position can change. Finally the cube becomes worthless when the game ends.
The Last Shake
Actual last shakes occur only during the final stages of the bearoff, when the entire game will be decided by the outcome of your roll. If you fail to bear all your pieces off, your opponent is certain to bear all his off. Regardless of the location of the cube, you should double whenever you are the favorite in the game. It is simply a matter of increasing the stakes.
Do not confuse quasi last shakes with actual last shakes. A quasi last shake is your last shake, but if you miss your opponent has the actual last shake; you do not have a concession. When the cube is centered, you should treat quasi last shakes as though they were actual. The option you had to turn the cube has expired if you do not use it now, while your opponent still has this option with the centered cube even if you do not double.
But if you own the cube, beware of giving it up on a quasi last shake. Though the cube will no longer have any value to you, your opponent may still be able to use it against you — provided you surrender it to him by doubling now.
Let’s illustrate these distinctions with some more simple twoagainsttwo bearoffs.

Diagram 6 Black is on shake. Should black double? 
In Diagram 6, black has just a tiny advantage, but he must double. Since it’s the last shake, black simply doubles his equity by turning the cube.

Diagram 7 Black is on shake. Should black double? 
In Diagram 7, black must redouble with the same tiny advantage.

Diagram 8 Black is on shake. Should black double? 
Diagram 8 shows a virtual last shake. True, white will have a turn if black fails to bear off. But white will never shake. Instead, white will double black out. So black must double.

Diagram 9 Black is on shake. Should black double? 
In Diagram 9, it is not a virtual last shake for black. If black misses, white may miss too. These misses for white make black almost a 62% favorite. But it is an error for black to redouble. For then he becomes only about a 53% favorite, as in Diagram 8. This reduction in winning chances is far too high a price to pay for the increase in stakes.
The contrast between Diagrams 7 and 9 is known as the Jacoby paradox. Black’s game in Diagram 9 is superior to his game in Diagram 7; yet Diagram 7 warrants a redouble while Diagram 9 does not.
The air of paradox dissolves once you recognize that in Diagram 7 black has the last shake, so that the barest edge dictates turning the cube. But in Diagram 9, white may have another shake.
In Diagrams 10 and 11, black is a cubeless favorite. He figures to win over 57% of the time if the game is rolled out. Yet, except for certain tournament matchscore situations, black must neither double nor redouble. For, if the cube is available, white will double black out whenever black misses.

Diagram 10 Black is on shake. Should black double? 
In Diagram 10, white is the favorite because of this.

Diagram 11 Black is on shake. Should black double? 
But in Diagram 11, black’s ownership of the cube gives him the same advantages he enjoys in a cubeless game.

Diagram 12 Black is on shake. Should black double? 
Despite the fact that if black misses, he will have to take the cube from white, black’s advantage in Diagram 12 warrants a double.

Diagram 13 Black is on shake. Should black double? 
In Diagram 13, however, black should refrain from redoubling. If he makes the mistake of turning the cube to 4, he will still get to finish out the game, but he will have to take the cube back at 8. Even though black is almost a 63% favorite to win the game, giving white the chance to control the stakes with a cube return when black misses would prove far too costly.
Virtual Last Shakes
Sometimes a shake right smack in the middle of the game, in a contact position, resembles a last shake in an important way. This occurs when your current shake will be decisive.
Suppose, for example, that you have shots at an opposing blot. If you hit, your board or prime is strong enough almost to guarantee victory, so that you will be able to double your opponent out comfortably. But if you miss, you may as well concede the game, either because this is your last shot and you trail hopelessly in the race, or because the cube is still centered and your opponent will double you out. To all intents and purposes, then, the game ends with your shake.
If no gammon threats exist, your doubling criterion on these virtual last shakes remains exactly as on actual last shakes: the slightest edge calls for a cube turn. When gammon threats exist, you should still turn the cube if you are the favorite to win the game — provided the gammon threats against you do not exceed your chances to gammon your opponent.
When the gammon favorite is the game underdog on a virtual last shake, you may computer and evaluate the Kleinman ratio W/G as a guide to your doubling. W is the advantage in sinning chances of any sort. G is the advantage in gammon chances. If there are real backgammon chances, we may count backgammons with the gammons but add the edge in backgammon chances to G.
With the cube already turned, the critical value of W/G is 1. Above 1, the game favorite should beaver the gammon favorite’s double! This anomaly (black has a good double and white has a good beaver) may be called the Kauder paradox. It stems from the Jacoby rule, a rule applying only to money games, which excludes gammons with a centered cube.
James Kauder discovered this paradox in a position which looked like Diagram 14:

Diagram 14 Black is on shake with a man on the bar 
Jimmy judged that if he missed this shot, his opponent would double him out. For then his opponent would be certain to cover the slot on the 20point with one of the four builders in the outfield. But Jimmy also figured that hitting the shot would allow him to pick up at least one other blot and gammon his opponent. Jimmy’s 11 hits and 25 misses of the direct 5shot produced a Kleinman ratio W/G = 14/11.
Jimmy doubled! If his opponent knew enough to beaver, Jimmy could expect to win 8 points 11 times and lose 4 points 25 times out of 36 games, achieving an equity of −12/36. But this negative equity was still better for Jimmy than his equity from not doubling, −14/36.*
* But suppose Jimmy figures to win only 101⁄2 gammons in 11 games after hitting, losing 1⁄4 and winning just a plain game in another 1⁄4. Then the Kleinman ratio W/G = 141⁄2/101⁄2 is too high to double. And Jimmy’s equity when beavered becomes (84 − 100)/36, or −16/36, less than without doubling. We have yet to see a true example of the Kauder paradox.
Your Doubling Point
So far, we’ve discussed lastshake doubles, both actual and virtual. Quasilastshake doubles require substantial advantages which vary form position to position. The gain from doubling the stakes must outweigh the loss from surrendering the cube. Each case must be computed separately. Still, your doubling points never reach nearly so high as 75%.
All other doubles — that is, cube turns prior to the nexttolast shake — revolved not around any fixed percentage of winning chances for you, but around your opponents’ take points. Let us treat as normal those players whose take points hover around 25% or dip only slightly below 25%.
But few players tend to be normal. Most are liberals, with take points dipping below 20%, or conservatives with take points noticeably above 25%. Neither liberalism nor conservatism is merely a variation in style. Each is an aberration resulting in moneylosing errors with the cube.
When you estimate your winning percentage in considering a cube turn, you should remember to do two things. First, you should take gammons into account by using the Kleinman gammon adjustment. Second, you should decrease your estimate somewhat (up to 5%) to reflect your opponent’s extra equity from his exclusive ownership of the cube after he takes.
If you can double with a game you rate at 75%, you have achieved perfect efficiency with the cube. Your money expectation is exactly the prior stakes, whether your opponent takes or passes. You should not care which he does, for neither passing nor taking such a perfect double is a mistake.
Against normal opponents, excluding games in which you play for the gammon instead of doubling, the prior value of the cube is the most you can expect to win in the game. Thus anything better than 75% equity is wasted. You may compute your money equity (when the cube is available for you to double) by taking the excess of your percentage equity over 50% and multiplying by twice the value of the cube. But your opponent can limit your money equity to the value of the cube simply by passing your double.
When your game reaches 75%, and further improvement adds nothing to your money equity (still assuming normal opponents). But any deterioration in your position reduces your money equity. You must double now.
Theoretically, you would like to wait until your game reaches 75% exactly before doubling. But in practice such exactness is impossible to achieve, and beyond our ability to recognize even when it does occur. At each turn while your game is improving and approaching 75%, you risk the wasted improvement beyond 75%. In delaying your decision to turn the cube, you face diminishing returns.
Suppose you turn the cube to 2 with a 74% game. Your money equity becomes 0.96. Suppose, on the other hand, that you decide to wait a shake; and that during that extra turn your game has a 50% chance to improve by 4% and an equal chance to deteriorate by 4%. Your money equity then becomes the average of 1 and .80, or .90. Waiting figures to cost you 6% of the value of the cube. This is a very high price to pay for keeping your options open one more turn. Your best policy (against normal opponents) is doubling as soon as your equity threatens seriously to rise above 75%, at the point of last take.
Liberal Opponents
Your very liberal opponents err by taking doubles they should pass. All their passes are proper. When you double right at your liberal opponent’s take point, your money equity actually becomes greater than the prior value of the cube, and in proportion to the liberality of your opponent.
It is merely unfortunate when you wait too long against a normal opponent and wind up doubling him out. You lose but little, and you often should risk this in an effort to edge up closer to his take point before doubling. But waiting too long to double a liberal opponent can be a costly disaster. You want to keep liberal opponents in when you double, since your big gains come when you catch them in their loose takes.
Since you cannot always determine your liberal opponent’s points of last take exactly, double them when you feel close to it. But note that in all such cases, you equity should be comfortably above 75%.
Conservative Opponents
Your conservative opponents err by passing doubles they should take. All their takes are proper. In contrast to your policy against normal and liberal opponents, you best strategy is to double conservative opponents at the point of first pass. You want to be sure to reap the benefit of their cowardly passes. It is a disaster when your conservative opponent takes a double. Your big gains come through doubling conservative opponents out while your equity is still well below 75%.
Chouettes
As captain in a chouette, you have the option of buying the games of unwilling crew members when you decide to double. Except for last shake doubles, half the value of the cube is a bargain price to pay for an extra game. The only reason for you not to snatch such a bargain is the suspicion that the unwilling crew member may have a more accurate evaluation of the position than you do.
Sometimes you will be in the position of the crew member reluctant to double. Should you accept a half point for your game? I suggest you do so whenever you rate your side’s equity below 621⁄2%. Above 621⁄2%, urge the captain to refrain from doubling, but go along with the double if he insists. Your expected gain from going along will be smaller than your gain if you persuade the captain to wait, but still more than 1⁄2 point.
As box player, you cannot implement your optimum policy, which revolves around your opponent’s point of last take or point of first pass, for you have different opponents with different take points. If you are playing against a mixture of conservatives and liberals, you should double when you expect a mixed response, some passes and some takes. Ideally, the mixture of responses should correspond exactly to the division between conservatives and liberals.
When you play the box against a group consisting of all conservatives or all liberals, you should still double when you expect a mixed response. Against conservatives, whom you otherwise want to double out, you want one take. Against liberals, whom you otherwise want to keep in the game, you want one pass. A double prompt enough to keep all the liberals in the game is premature for all but the least liberal, just as the double which all the conservatives pass has allowed all but the least conservative a “free shake.”
This impossibility of tailoring your doubling strategy to each opponent individually is one reason why the best player in a chouette, while playing in the box, cannot expect to win as much as he would if he played each of the others separately. None of the above considerations apply to last shake doubles. These are still exactly calculable, and you should act accordingly.
Too Good to Double
Because of the Jacoby rule, your gammon threats can never make your game too strong for an initial double. After you have been doubled, however, you may turn the game around to the extent that you can surely double your opponent out, but may figure to win more than the value of the cube by playing for the gammon instead. For this to happen, your game must improve radically in a single shake; else, somewhere along the way you would already have redoubled.
A typical toogoodtodouble situation arises when you hit an opposing blot after you have closed your board and you are moving your remaining pieces around the board freely preparing to bear off. While you are doing this, you equity remains constant. Therefore you may postpone your decision. But at some point you run the risk of leaving a shot, getting hit, and losing the option of doubling your opponent out. As you bring your pieces around, you may find yourself stopping on every move to figure out whether double 6’s can damage you on your next shake.
The Kleinman count gives you a quick and accurate arithmetic trick for guarding against double 6’s. Count the number of quadrants you must cross to bear all your men in. Add to this the number of men on the most retarded point in each quadrant (i.e., your 6, 12, and 18 points). The result is your Kleinman count.
A Kleinman count of 5 represents danger on the next shake. You must try to avoid a 5 count. A 7 count represents a smaller, less immediate, danger; sometimes a 7 count leads to a 5 count.
In moving, it is easy to note the effect of moving a piece on your Kleinman count. Crossing a quadrant lowers your count by 1, except when you land on the most retarded point in a quadrant by advancing fewer than 6. Moving within a quadrant lowers your count by 1 only when you move off the most retarded point in that quadrant.
Many players abandon their gammon attempts as soon as any immediate danger appears. This is sometimes an error. Using the Kleinman gammon adjustment, weigh your gammon chances against the risk of losing. An equity of more than 100% indicates that you should still prefer playing for the gammon to doubling your opponent out.
The Freeman Coup
A very different situation in which you may be too good to double occurs only against very liberal opponents. Despite your desire to double a liberal opponent at his point of last take, you may find that a single shake has suddenly improved your game well beyond that point. Most players make the mistake of automatically doubling their liberal opponents out in such cases. But optimum strategy dictates asking yourself: Can the next shake reduce my equity below 75%? If so, doubling is still in order.
But if not, you may attempt a Freeman Coup. As long as your game cannot deteriorate below 75%, you continue to play. You hope your game will deteriorate moderately, just below your liberal opponent’s take point. Then you will complete the Free Coup by doubling. The take makes your money equity greater than the prior value of the cube you could have achieved earlier by doubling your opponent out.
Statistical Checkups
You can collect statistics on the results of your games as a partial check on your cube handling. First, note that lastshake and quasilastshake doubles should be omitted. For these may be calculated exactly. Second, you can only obtain statistics on games you pass in chouettes where some other crew member takes.
You seek a loss quotient. For each take, record the doubler’s net result divided by the size of the cube prior to the double. The doubler’s wins count as positive, losses negative. The results, normalized by the prior size of the cube, should be accumulated as the loss numerator. The loss denominator is simply the number of games counted. Then the loss quotient is the loss numerator divided by the loss denominator.
accumulated results normalized by cube size 
number of games 
Against liberal takes, your doubles should produce a loss quotient above 1. Against quick passers, you should have too few takes of your doubles to allow meaningful statistics; but in a chouette against opponents who are all conservative in varying degrees, the loss quotient on your doubles should fall shy of 1 if you are doubling correctly.