Backgammon Articles

The '30s Understanding of Doubling
by Vic Morawski

70s View of the Origin of Doubling
When they looked back on its origins, '70s backgammon authors tended to view the doubling cube as the invention of a lone genius. Jacoby and Crawford dedicate their 1970 work The Backgammon Book "To the genius who invented the doubling cube and made backgammon the game it is." (p. 4). Later in the book they expand upon this theme:

"In the United States and Europe interest in backgammon greatly revived in the 1920s when some unknown genius playing in one of the American clubs came up with a revolutionary idea. He proposed that, at his turn to play, a player might insist on doubling the stakes. His opponent would have the right to refuse, in which case the game would then end and be scored at the original stake." (The Backgammon Book, p. 51.)

Bruce Becker, in his later 1974 book, Backgammon for Blood, carries on the tradition:

"Some fifty years or so ago some genius who wanted more action than he was getting with the double and triple game invented the doubling cube." (p. 23).

If we assume that Jacoby and Crawford and Becker mean the same person in both places quoted above, the picture we get from the two accounts is consistently the same: a single individual invented the cube and simultaneously introduced and began to popularize the notion of offering voluntary, mid-game doubles in backgammon.

Forgive me, while I know that the inventor(s?) of the doubling cube was most likely a New Yorker, I cannot at this point keep myself from imagining a scene between those two little Monty-Python style characters from the beer ads as one of them emerges from his basement laboratory holding history's first doubling cube.

British Guy 1:   What's that you've got there?
British Guy 2: It's my new invention—a large die with numbers imprinted on it in powers of 2.
British Guy 1: Brilliant! What does it do?
British Guy 2: It enables you to double the stakes in backgammon.
British Guy 1: Really? Brilliant! And what is it called?
British Guy 2: I call it a Doubling Cube.
British Guy 1: Brilliant!

Those little guys amuse us because we know we share the common understanding that things didn't really happen that way and we naturally find funny any suggestion that they might have. But such may be the same in the case of the doubling cube.

70s Conception Wrong?
One thing that Bill Davis' online article, "Backgammon and Doubling: What we Know", makes abundantly clear is that the doubling cube, if it had a single inventor, was invented to more easily facilitate and track doubles in what was already an ongoing practice of offering voluntary mid-game doubles in backgammon—not the other way around. Bill Davis shows that it was proposed as an improvement over already existing methods of keeping track of doubles using matches or pointers. (See his Statement 3, pp. 3-6.)

Evidence that it was in fact viewed early on as an improvement over these methods comes in part from its being recommended as such in an early book of the period, Vanity Fair's Backgammon to Win, written by Egyptian-born New York resident George Mabardi (copyright 1930). In a discussion of backgammon equipment near the end of his book, Mabardi says, "It will be found a saving of time energy and often argument to use a doubling cube for doubling purposes, rather than matches." (p. 155). Also, we can assume that this recommendation had a hand in promoting the popularity of the doubling cube nationally as we can easily believe a book backed by such a prominent magazine as Vanity Fair to have had national circulation and significant impact.

But this now raises the question: If the invention of the cube did not give rise to the practice of mid-game doubling, then what did?

A Logical Extension of Automatic Doubles?
Conjecture: Could mid-game voluntary doubling have started as the logical extension of pre-Game mandatory doubles?

Bill Davis highlights '30s author, Grosvenor Nicholas' conjecture in Modern Backgammon to the effect that, "It is said that doubling, which has so greatly increased the possibilities of the game, as well as its popularity, was first originated on the continent of Europe in connection with the game of golf" (p. 23). Now, I personally would like to know a little more about just how doubling worked in the game of golf. Did two players suddenly shoulder their bags of clubs and walk off the course at, say, the thirteenth hole because one player, at that point significantly behind, declined a double? It is hard to imagine this happening in golf yet this obviously happens in backgammon all the time.

It seems reasonable and a bit more realistic to me to think that the practice of offering voluntary mid-game doubles in backgammon simply arose as a logical extension of the practice of allowing automatic doubles at the start of a game when two players happened to roll the same number. If this were so, then there would be a metaphorical sense in which the dice themselves would be the first "doubling cubes" as they are what determine a mandatory double. Again, if this were the case then we might imagine that perhaps several persons, not just one single person, noticed that doubling could be extended to mid-game situations in this way and so there would be no one person who invented the practice as '70s books have suggested. Longacre in fact speaks of those who initiated the practice of voluntary mid-game doubling in the plural:

"Within recent years, some of its devotees, appreciating the present-day urge to inject a financial interest into everything, realized that this game ... was eminently adaptable to this modern requirement. They thereupon evolved what is known as the 'Doubling Game.'" (Backgammon of Today, pp. 4-5, boldface mine).

In fact, an early statement of the rules of backgammon for the New York clubs, (which I have as a separate booklet but which appears word-for-word in Longacre's book (pp. 124-129), along with a notice that it is copyrighted by himself and Nicholas), includes a rule for mandatory pre-game doubles but none for mid-game voluntary doubles. The latter are mentioned only in connection with Chouette. We can take from this that the practice of mandatory pre-game doubles gained a foothold before that of mid-game voluntary doubles, which did not merit receiving their own rule until the 1931 revision of the rules. If this was the case, then it is not hard to conceive of the one practice as arising out of the other.

Along with their speculation that a solitary individual invented both the doubling cube and the practice of mid-game doubling, '70s authors Jacoby and Crawford add the further conjecture that, "this same inventive genius, or perhaps a second one, added the redoubling feature which allows a player who has been doubled to redouble his opponent in the same manner whenever it is his turn to roll the dice." (The Backgammon Book, p. 54).

I don't think anyone needed to invent the redoubling feature. I think that it would have been easy enough for the person who had the advantage and had doubled initially to see that it was also in his/her own best interest to redouble on the next roll and, for that matter, on each successive roll as long as that person still retained a significant advantage. What required invention was a mechanism to keep this from happening. Longacre shows that, right from the beginning, it was recognized that, in the interest of fairness, such a mechanism was needed, and why:

"The option of redoubling rests with the player who has last been doubled; the original doubler, no matter how much his advantage may have increased, may not again, successively raise the stake. Thus the player who is at a temporary disadvantage cannot be 'raised out': he can at least play the game out, and have a run for his money, at no greater expense than he has already assumed." (Backgammon of Today, pp. 85-86, boldface mine.)

Thus was born the convention that the person initially doubled now alone owns the right to redouble.

The first proponents of mid-game voluntary doubling were not necessarily viewed by all in a favorable light. After having already characterized them as persons who simply must interject a financial interest into everything, Longacre then refers to them as "habitual seekers after 'action'" (p. 87). Now, he at least does recognize that doubling has added more than just an extra element of action to the game; he realizes that it has also added to the game a new necessity for the successful player "to correctly appraise the relative positions of the opposing sides for doubling purposes and that skill in the treatment of the double is an important test of a player's personal skill in general at the game." (p. 87).

George Mabardi, however, is not so kind. After admitting that doubling does add an element of excitement to the game, he also says that it adds to it a "gambling color" that is in fundamental opposition to an approach which emphasizes knowledge of the game's fundamental principles and strategy. So he recommends that, "If you want to gamble, by all means play the Double; if you are more interested in the strategy of the game, omit it." (Vanity Fair's Backgammon to Win, p. 120.)

So, how did they arrive at these opposing positions and what changed mid-game doubling in backgammon from something that would interest only adventurous gamblers to something that would also capture the interest of serious students of the game?

The 3-to-1 Principle
Expert acceptance of voluntary mid-game doubling rose in parallel with acceptance of the 3-to-1 principle. This principle, called such and explained by Paul Magriel in his monumental work, Backgammon, says that, "If your opponent doubles when he is better than a 3-to-1 favorite—if he is going to win the game in question more than 75% of the time—you should pass. When his edge is less than 3 to 1, you should accept." (p. 269)

Covered in nearly every backgammon book from the '70s on, the 3-to-1 principle is the most basic principle in doubling. In fact, so engrained in our thinking is it that it must be hard for contemporary players either to imagine a time when it was not generally known or to imagine that anyone, upon hearing it explained, would not immediately accept it as obviously true and legitimate. Yet, in the 1930s this principle was neither universally known nor universally accepted among backgammon players, even among those who knew of it. Some examples below will illustrate this.

The importance of the 3-to-1 principle for the legitimacy of mid-game doubling is that it allows a double both to be correctly offered and correctly accepted. Without it, we should be in the situation described by George Mabardi who claims that, "if two absolutely perfect players engaged in a match, there would never be an accepted double. In other words, a correct double is one that is made only when the doubler is definitely ahead; and a correct acceptance, likewise, is made only when the accepter knows that he is not definitely behind. Consequently, if every double made were perfect, none would be accepted, if every acceptance were perfect." (Vanity Fair's Backgammon to Win, p. 121.)

Then Mabardi gets to the bottom line: "We must conclude, then, that accepted doubles can only be the result of imperfect play on the part of one adversary or the other. The author's chief objection to the double is precisely this—its only raison d'etre [i.e., reason for existence] is the incompetent play of those who use it." (Ibid.)

Obviously not a very high view of mid-game doubling or of those who adopt it as a practice! After some remarks to the effect that two experts would always agree on the relative value of the position before them on the board and so, if one made a correct double the other would never accept, Mabardi concedes that, "as long as the vast majority of players are not expert, the double is with us to stay." (Ibid., p. 121-122.) He then goes on to offer some practical advice on doubling for ordinary players, not all of which is bad.

He warns players to use the double sparingly. He lets them know that they put themselves at a bit of a "tactical disadvantage" when they pass the privilege to double from the middle to their opponent, who now owns the right to redouble. He warns them against doubling too early as their opponent may have too many subsequent opportunities to catch up or their own position may improve so much that they will realize they should have played for the gammon.

But Mabardi's cardinal rule for the making or acceptance of doubles, "Never depend on the dice" (p. 123), stems directly from his apparent lack of knowledge of the 3-to-1 principle (nowhere is it mentioned in his book) and leads to what we now know is obviously bad advice. He views the acceptance point for a double as anything even or better than even and when he says that we should never depend on the dice, he means that we should never, when considering the acceptance of a double, presuppose that our dice will be at all better than the dice of our opponent. In any situation where our opponent will always win, the dice being equal, we must always resign (p. 124-125). Contrast this with the acceptance point of a double based on the 3-to-1 principle.

John Longacre, another 30s author, understands the 3-to-1 principle but thinks that it is based on faulty reasoning:

"Some players accept an unfavorable double on the theory that they are thereby obtaining attractive odds. That is, if Black doubles, White's original stake is forfeited unless he accepts. But, by putting up an additional stake, he has a chance to win Black's two, plus the one he has already lost. Therefore, he is now getting odds of 3-1. Ingenious but fallacious." (Backgammon of Today, p. 93, copyright 1930, reprinted 1973, italics mine.)

And just why does Longacre think that this reasoning is fallacious? He goes on to explain:

"White can win nothing unless he puts up an extra stake, in which case he has a chance to win two from Black. But he still stands to lose his original stake, and to win nothing more. He will now either win two, or lose two. He has merely doubled his liability on a disadvantageous proposition. In other words, instead of taking the odds, he has paid, with an even-money bet, to keep alive an odds-against chance." (Ibid.)

While he mentions above that White will lose his original stake unless he accepts, he omits reference to this fact in his explanation of why the reasoning behind the 3-to-1 principle is supposedly faulty—yet this fact is critically important to the legitimacy of the principle. It is not just that "(w)hite can win nothing unless he puts up an extra stake" or that he still might lose his original one even if he takes, but that he definitely stands to lose it if he does not. This is what makes all the difference, here.

Another way of looking at it is that, if white were allowed to reject the double and yet have the game continue out to its natural conclusion at the original stake, then by accepting the double when behind White would indeed be doubling his liability on a disadvantageous proposition, as Longacre claims. But we know that this is not a live option for White. Therefore, Longacre's reservations concerning the 3-to-1 Principle were unjustified. Yet his rejection of the principle leads to his offering bad doubling advice, especially on the acceptance of doubles.

His view as to when we should accept a double is, "Early in the game or not at all—provided the double is based on an indisputably advantageous position." (Backgammon of Today, p. 92.) He then goes on to explain why, using a racing analogy: "Twenty yards handicap in a half-mile race may be overcome, in the hundred it is hopeless." (Ibid.)

Readers will recognize that Longacre's advice holds good only regarding those positions for which his analogy holds, that is, long racing positions. As across-the-board advice on doubling in general, however, it is off the mark. And it falls short specifically because of his rejection of the 3-to-1 principle. While he does not go as far as Mabardi in claiming that we never correctly accept a double if we are behind, we had better not be very far behind and we had better give ourselves enough time to catch up if we are to accept correctly. Both Mabardi and Longacre view the take point of a double as even or something close to even.

Walter L. Richard, in his book Complete Backgammon (copyright 1931), shows that while doubling is an even proposition for the doubler, it is not for the person being offered the double.

"It is plain that when a player doubles, he is undertaking an additional even proposition, and therefore requires a clear-cut advantage over his opponent to make the double sound. But the player accepting a double is not actually accepting an even proposition, he is really receiving 3 to 1 odds from his opponent." (Complete Backgammon, p. 17.)

After noting the above, Richard then goes on to give as clear and lucid an explanation of the 3-to-1 principle as one will find anywhere:

"This is not difficult to understand once it is explained. If the player refuses the double and gives up the game he immediately loses 1. Should he however, accept and win, he wins 2, which makes him 3 better off than he would have been if he had refused the double in the first place. Thus, in exchange for an opportunity to be 3 richer, he risks only an additional 1, so that he is accepting a 3-to-1 proposition." (Ibid., p. 18.)

Gone from Richard is Longacre's recommendation to accept a double early or not at all. He asks us merely to be aware that, "the farther the game has progressed the warier one must be in accepting a double, as the time grows shorter and there are less chances of catching up." (p. 56). We can now accept correctly at many points in the game.

For Richard, the odds should be our guide and we can correctly take a double even on the last one or two rolls of the game if we are mathematically justified in doing so. He especially discusses doubling in end game situations where both players have only two checkers left and even gives us a table of the odds for these (p. 16-17, 57). Now accepting a double requires some detailed, specialized knowledge of the sort an advanced player would have and an inexperienced novice would not. No longer is mid-game doubling perceived to exist only because of player incompetence.

Doubling in Tournaments
Even in 1930 the issue of whether or not to allow doubling in tournaments was around. Mabardi, whose perspective is national by virtue of the fact that he was also backgammon columnist for Vanity Fair, observes that, as of that year, "There are no Authorized Rules for conducting tournaments ... These vary in different locales and clubs." (Vanity Fair's Backgammon to Win, p. 150). Concerning doubling he writes that, "Many tournament committees make a ruling that outside of the gammon, or backgammon, there should be no automatic or natural doubling, nor, indeed, doubling of any kind." (p. 150).

Not surprisingly, he then voices his personal view that "doubling should be omitted from all tournaments." Permitting it, he thinks, would undermine the "real purpose" of a tournament, namely to find the best player among those competing--this player, due to doubling, may be knocked out early "by an incredibly lucky throw of an opponents dice." (p. 150-151). Not an altogether irrelevant objection. And even in 1930, he observes that tournament committees saw the wisdom of forbidding automatic doubles (p. 150).

In the introduction to Complete Backgammon, it is said of Walter L. Richard not only that he "conducted the first regular tournament in this country" but that he even formed an inter-club league among members of the leading New York clubs (p. ix). When we, however, look at a section of his book discussing various tournament formats—North-South, Elimination, Round Robin—we find that he suggests to his readers that no automatic or other doubling be allowed in any of them (p. 101, 104, 105, and 108). My copy of his book is the fourth printing, dated 1937.

Here we must ask, was this the policy followed by Richard for the New York clubs or was he merely sparing less experienced tournament directors nationwide, who would read his book, the additional complication of dealing with the issue of doubling in their tournaments? If the former, then this brings to light the surprising fact that doubling may have been forbidden from tournaments in the city of its birth well into the 1930s! So we leave with this question: When did doubling first gain widespread tournament respectability?

Vic Morawski
January 2008

A semi-retired Philosophy professor, Vic Morawski now manages a retail store.
He is Director of the newly-formed Baltimore Backgammon Club and can
be reached for comments and criticism at:

See:  Other articles on Backgammon History

Return to:  Backgammon Galore