The Doubling Cube in the Wonderland of Errors
Technical analysis can generate optimal taking and doubling strategy gainst perfectly rational opponents. But we live in a crazy world. Money backgammon players, especially those who are gamblers, are crazier than most. You can profit from the irrationality and errors of other players by adapting your own cube handling to their idiosyncrasies.
In so doing, you run a risk. Perhaps another player will suddenly stop making the kinds of errors you have been counting on him to make. Then you will find yourself worse off than with pure technique. Nonetheless, it is a risk worth taking. For in cube handling more than anywhere else, the game is to the psychologist, not the technician. It will pay you to become a keen observer and analyst of your opponents’ cube habits.
How often have you urged the captain to turn the cube when you’ve been on the outside among the crew? This is usually a mistake. Ask yourself: Why has the captain not suggested doubling on his own? The answer will be revealing. If a cube turn seems reasonable, the the captain may be in error.
Perhaps the captain fails to appreciate an overwhenlming lead in a pure race. Perhaps he doesn’t realize fully just how many game-winning shots he has. Perhaps he hasn’t noticed how close to crumbling the box's game is in a prime-versus-prime battle. He may be just plain timid with the cube, or he may not know that the box is so timid as to pass a double.
In such an instance, you should urge the captain to double. His failure to evaluate the situation properly doesn’t diminish his chances of winning the game. But much of the time, the captain’s failure to urge a double flows from a misconception of the position. Perhaps he conceives his task a restraining the box’s backward men when converting to a race could assure victory. Perhaps he fails to relize that the box’s lack of inside points can let him move his own laggards around the board freely, leaving blots. In that case, the captain may misguidedly play “safe” now and risk serious difficulties in disengaging later. If so, then the captain has not urged a double because he doesn’t see how to win the game from here. Do not suggest doubling to him.
I learned that lesson long ago in a weak chouette. Early in the game, the captain embarked on a blitz. As the box, I would certainly have passed if doubled. I urged the captain to double. He didn’t really think we should but acceded to my presumably knowledgeable demand.
To my surprise, the box took. Then, with two opposing men on the bar, the captain proceeded to move “safely,” making irrelevant outfield points instead of bringing hitters within range of the box’s blotts as they came in from the bar. As a result, the box anchored on our 4 point.
The captain then had trouble breaking his outside points. He started leaving shots. The box redoubled us, hit a shot, and gammoned us.
This double didn’t backfire because the captain was a weaker player than the box. In fact both played equally poorly. And neither understood closeouts. But only the captain had a chance to misplay this kind of game. For, before anchoring on our 4 point, the box had nothing but forced moves to play. The captain was right in not wanting to double: He didn’t know how to win this game.
Technically, only the location of the cube should influence your actions, never the size. Psychologically, however, the larger the cube, the quicker you should turn it.
If this seems paradoxical, contrary to common sense, that makes it all the more valid. Most backgammon players like to minimize big swings. Many come inadequately bankrolled. Observe how often some backgammon players sacrifice equity by taking the worst of settlements.
Send such players a high cube early. In effect, you are offering to settle the game for the old value of the cube. They will accept to their detriment. With a large cube, most players become conservative. Double them out early.
I learned this inadvertently. I owned the cube at 16, in the box against three opponents. Well into the bearoff, I could see that I had a proper redouble and they had a proper take. If I feared big swings or lacked ample wagering capital, I might have kepts the cube. But I was unafraid, so I turned it to 32. Incredibly, all my opponents passed. I was delighted not to have to roll that game out to the end.
Luck and Momentum
Neither I nor mathematics has anything to say about whether there are truly “good rollers” and “bad rollers.” But you can observe and profit from the habits of those who consider themselves bad rollers.
Convinced of his own bad luck, the “bad roller” waits until victory is virtually assured before doubling you out. He has seen too many games turn sour to double you promptly. As a result, you enjoy several free shakes against him. Sometimes you win a game you would have passed if doubled. Strangely enough, this only confirms his faith in his bad luck and reinforced his reticence in doubling.
In playing against a “bad roller,” you should be careful to take any borderline doubles he inadvertently offers. If you win, you will drive him to further conservatism. Even by making the game close while losing, you may scare him into greater timidity.
You can take maximum advantage of the bad roller’s pessimism in noncontact positions. With only a small lead in the race, you can double him out. In midgame positions, he may take liberally only because he thinks his superior skill compensates for his “bad arm.” But in a race he become an archconservative.
Against opponents who believe in lucky and unlucky streaks (as opposed to good and bad arms), you can modify your strategy accordingly. Loosen up your cube when they betray their fickle conservatism by remaks like: “Ordinarily, I’d take this game. But the way you’ve been rolling lately, I’d better pass!”
Conversely you can afford to wait against them when they’ve been winning. For then they display a temporary liberalism and say, “I know it’s a pass, but I’m hot, so I’ll take!”
Some players react to the momentum not just of a series of games, but of the last one or two rolls. Once I played in the box against a holding game. The captain escaped from my 4 point with a set of double 5’s. At the urging of the rest of the crew, he redoubled. Nobody bother to count the race or they would have seen I was still a few pips ahead. I beavered.
If the positions had been reversed, I suspect they would all have passed. But would I have been astute enough to doubling them out?
In money backgammon, the standard 25% requirement for taking is too strict. In effect owning the cube adds to your winning chances by as much as 5%. Thus we may still take 20%-games in some midgame positions.
In their very mathematical essay, Keeler and Spencer argue that 20% is the theoretical minimum for taking. My own mathematics, however, pushes this down to 18.75%. I will not burden you with the equations proving this, for in his infinite bounty, the backgammon god has given us a simple example of an optional take with exactly 18.75% winning chances.
White doubles, black may take
Since each player has 9 misses with just one man left on the 6 point, each player has a 3⁄4 chance of getting off next shake, so that black figures to win 3 games in 16, an 18.75% chance. But because all of black’s wins and only 1⁄13 of black’s losses will come at redoubled stakes, black may as well take as pass.
In money backgammon, only a special combination of circumstances can drive your take point below 20%. Against a rational opponent, or in the box against a mixture of opponents, you can treat 20% as your rock-bottom minimum take point. In most cases, you should require at least 22% chances. But against a lone, irrational opponent, you can take the cube despite winning chances startlingly below 20%. Here’s why.
Technically, owning the cube compensates for only 5% winning chances. But psychologically, the cube can be worth much more. And how much more depends on how liberal or how conservative your opponent is in taking your redoubles. The direction of your opponent’s deviation from the normal 20% take point does not matter, only the size. Either his liberalism or his conservatism makes your cube more valuable and hence lowers your take point.
Against conservative takers, let us measure the degree of conservatism as the doubling point D for you at which you can successfully double them out. Then the simple formula
Against liberal takers, suppose U represents the point of ultimate take for them at which you double to keep them in. For values of U of 4⁄5 and higher, a complicated quadratic equation yields your proper take point, T. But a simple approximation is
|10U − 4|
Not that while the interpretation of the value of 1 for U may be that your opponent will still take when you are mathematically certain to win, it may also be3 siimply that your expectation is the new value of the cube. This means that your chances of gammoning your opponent are double your chances of losing the game.
In general, you cannot fine-tune your takes according to your opponents’ psychology in chouettes precisely for the reason that you cannot fine-tune your doubles. As a crew member or even as captain, you cannot decide unilaterally when to double. In the box, you cannot fine-tune your doubles because different crew members have different taking psychologies. You can lower your own take point beneither 20% only when all the crew members opposing you deviate from normal taking psychology in the same direction.
If the standard take point for players is 25%, this means most of your opponents are overconservative in their takes. You should therefore be delighted to be able to double them out when your own equity is still below 80%. Since many players tend to double with about 67% equity to keep their opponents in, your proper doubling policy may appear very tight to other players.
When you thus delay doubling, your game may sometimes improve suddenly to the extent that it seems too good to double. Your gammon chances may become more than twice your chances of losing. In the real, crazy world you should often double anyway. Sometimes your opponents will fail to realize how hopeless their game is.
Larry Cohen played a delightful 1-3-5-point back game in which his opponent had to leave three blots simultaneously during the last phase of bearing in.
Black on shake.
Should black double?
Should white take?
A triple shot, with 27 hitting numbers in 36? Hardly. Anyone troubling to count shots could see that all 36 shakes of the dice hit. With those three blots of his opponent’s lying around, Larry had an almost certain win plus excellent gammon chances. He was actually too good to redouble.
Nonetheless, Larry returned the cube. The captain snatched up the cube and some of the crew took as well. Nobody but the kibitizer had bothered to count shots or evaluate gammon chances!
Cohen’s opponent did get lucky, managing to make Cohen’s 1 point after getting all three blots hit and finally rolling large doublets at the end to avoid the gammon.
Playing for the gammon was technically correct. Perhaps a world champion might have been able to move his checkers skillfully enough to lock up the gammon. It was hard to rell. What was easy to tell was that psychology had worked. Ignorance of the nature of the position induced some of the crew to take.
Moral: When your opponent misunderstands the position, you’re never too strong to double.