Danny Kleinman, 1983
How Can I Keep from Dancing, © 1983 Danny Kleinman

In November of 1976, my brief career as a backgammon teacher almost came to an end. I found Paul Magriel’s textbook Backgammon and told my students: “Read this book. It contains almost everything I have to teach you. Mark each page with your penciled questions and comments, which we will discuss in our future lessons together.”

Bless you, Paul Magriel. You saved me a lot of work. I quickly finished up with all my students except one, who requested a lesson on the doubling cube in tournament backgammon, of which I knew nothing. My frantic search of the backgammon literature turned up nothing either.

Desperately, I set out to develop my own theory. I spent two weeks calculating equities at various scores in a 9-point match and deriving cube strategies from these. The result was the first version of the “Tournament Match” section of “The Care and Feeding of the Doubling Cube” (later a chapter, after many revisions, in Vision Laughs at Counting. This was the first thing I ever wrote about backgammon.

I would have been spared all this labor had I known about Norman Zadeh’s paper, On Doubling in Tournament Backgammon, which had been submitted for publication one year earlier. Unfortunately, the journal to which Zadeh sent his paper, Management Science, delayed its publication for a year and a half — until May of 1977. Besides, I had never heard of that journal. It was only recently that a friend gave me a copy of Zadeh’s paper.

I admire the power and elegance of Zadeh’s mathematics. Zadeh’s 1975 thinking follows the same lines as my own, and encompasses even more variables than my latest (1980) calculations of tournament cube strategies. Zadeh introduces a factor which he represents by the Greek letter delta (Δ) to account for the discontinuities in backgammon, and even gives an estimate delta. I omit this.

Zadeh’s recursive formula takes what I call “cube bias” into account for doubles and redoubles alike. I incorporate cube bias into my assumptions for initial doubles only. Zadeh sets his gammon factor at 25%, a bit higher than the 20% I assume, but lowers it to 15% for the player who is being doubled.

I haven’t checked out Zadeh’s mathematics; that might consume months of my time. Nor have I verified whether his assumptions are more accurate than mine; I wouldn’t know how to begin such a momentous empirical task.

Zadeh’s match equity chart differs from mine in many places, but never by more than 2%. Sometimes there is no difference, but usually the difference is 1%. The direction of the differences is always the same. Zadeh’s more thoroughgoing use of cube bias and higher gammon frequencies combine to give the trailer slightly better winning chances at most match scores.

Zadeh’s chart showing take points for initial doubles differs sharply from mine. There are at least two reasons for the discrepancies. When redoubles are still possible, essentially, Zadeh’s take points represent the winning chances already adjusted to reflect the value of owning the cube. My take points (except where redoubles are “free”), in contrast, require the player to add his own adjustments (again, except in the “free redouble” situations) for the extra winning chances flowing from cube ownership.

I likewise require the player to evaluate the gammon chances in a particular position and adjust the percentages according to a gammon-price list I supply. Zadeh incorporates gammon threats in his figures by setting D (his figure for the winning chances of the doubling player) = 2.4 D − 1.15 (where D does not include a gammon threat). This makes Zadeh’s chart inappropriate for nongammonish positions such as races or holding games.

I think Zadeh’s chart exaggerates the looseness with which a trailing player can afford to take the leader’s double. Consider, for example the 6% break-even point Zadeh gives for taking the cube at 2 when trailing 6-0 in a 9-point match. Zadeh gives 10% as your winning chances if you pass and start the next game trailing 7-0. (This is very close to my own figure of 912%.)

If you take and lose, then you trail 8-0 entering the Crawford game (giving you 5% chances according to Zadeh’s chart, very close to my own 4.8%). If you take and win 4 points (a very optimistic hypothesis, since your opponent may well pass your redouble if you succeed in turning the game around), you still trail 6-4 (with 35% chances according to Zadeh’s chart, very close to my own 34.3%).

If you take and win only 2 points (presumably, by a judicious redouble just at the point where your opponent does as well to pass as to take), you still trail 6-2 (with 24% chances according to Zadeh’s chart, but only 22% according to mine).

I can’t for the life of me see how he gets his 6% figure. I haven’t got his program listing to check for bugs, so I don’t know why his elegant mathematics yield absurd results. (Zadeh did write a computer program to apply his formulae — the pencil-and-paper calculations would be prohibitive.)

Norman Zadeh also modifies his formulae to derive charts for lopsided matches in which one player enjoys an enormous advantage in skill over his opponent. As in the simpler case of equal opponents (which we had assumed in our previous discussion), Zadeh uses an intelligent approach. He assumes a skill difference so sharp as to make the stronger player a 60-40 favorite in a 1-point match, and develops the rest of his figures from there. The cube strategy he derives conforms to our expectations: The weaker player should be more liberal with the cube, the stronger player more conservative.

Naturally, this tendency must be integrated with the strategies indicated by the score. Zadeh cites the 18% takes he advocates for the stronger player who trails 6-5 in a 9-point match, which are looser than the 24% takes appropriate for his weaker opponent who leads 6-5.

As a rule, the stronger player should be more liberal than the weaker with the doubling cube in money games. His skill advantage lets him win games that he might lose to his peer. Thus Zadeh’s “18%” take does not represent the stronger player’s chances of winning this 18% game against this weaker opponent. Says Zadeh, “His actual chances are 39.7%” in a “position that would give him 18% chance against a good player (with no cube).” Presumably, a player so weak as to have only 40% chance in a 1-point match plays badly enough to blow an “82%” game almost 40% of the time.

This follows from Zadeh’s elegant assumptions about each player’s “ease of movement” from positions with given winning probabilities to other positions with even higher winning porbabilities. Indeed, some such assumptions are needed to be able to begin calculating at all; and in my aborted attempt to derive charts like Zadeh’s for lop-sided matches, I too followed this approach.

But what do “82%” games look like in practice? They are often races with huge leads, or bear-offs against enemy men closed out on the bar, or bear-ins against holding games. I don’t think these are games where the errors of weak players hurt them very often. Weak players may misplay them, all right, creating extra bad numbers for themselves. But then their opponents must still roll well to profit (i.e., by hitting the shot left needlessly by the weak player). More often, I think, weak players make costly errors in 30% to 70% positions.

Curse you, Norman Zadeh. First for not getting your paper into my hands seven years ago, when it would have saved me much work. And second for giving me so much analysis of your charts to do now, when I know what to do with them.