Who remembers the backgammon craze of the 1970's? I do. In America and beyond, the game's popularity grew in great proportions. With this upsurge in popularity came increased interest in backgammon technique. There was money to be made, after all. Some of the game's best players produced serious works of theory: Jacoby and Crawford's The Backgammon Book (1970); Joe Dwek's Backgammon for Profit (1975); Paul Magriel's groundbreaking text Backgammon (1976); Barclay Cooke's Paradoxes and Probabilities (1978); and more. The 70's produced Bill Robertie, who established himself in the 1980's as a premier player and theoretician, and continues as such up to the present day. Robertie's Lee Genud vs Joe Dwek (1982), Reno 1986 (1987), and particularly Advanced Backgammon (1st edition 1984; 2nd edition 1991), remain classics.

In the 1990's, human players were joined by the bots. To the delight and consternation of players everywhere, three strong neural-net-based backgammon programs burst upon the scene: Gerald Tesauro's TD-Gammon, Frederik Dahl's JellyFish, and André Nicoulin and Olivier Egger's Snowie. With the help of Internet backgammon servers like FIBS and GamesGrid, the ideas of the bots spread like wildfire and met with great success. Backgammon will never be the same again. Many human conceptions about the game—some vigorously articulated in the above-mentioned books—were shown by the bots to be nonsense, or at best oversimple. The result is that backgammon at the start of the millennium looks very little like backgammon of two or three decades ago.

However, we carbon-based life forms are adapting well to the silicon invasion. We are integrating bot insights into our own praxis. In fact, bots have become an indispensable tool for analysis and study; time and again, they recommend moves that seem counterintuitive or even bizarre, and time and again their choices prove right. Of course they're not perfect, but the strongest of them, Snowie 3, rivals the world's best humans in playing strength. With enough work, we may yet arrive at a theoretical synthesis surpassing even the knowledge of the strongest bot. Some would argue that the world's best humans already have.

With New Ideas in Backgammon (1996), Hal Heinrich and Kit Woolsey introduced the first important work in a new theoretical genre, that of analysis significantly aided by the insights of bots. It's time to give the new genre a name: I call it BEST, for Bot-Enhanced and Synthesized Theory. (An alternative name is BEAST—giving the "and" a place—which may be preferred by human-racial purists, or "bot-dissers.") The name reflects my view that in the 21st Century, using bots as an analytical aid is truly a must. Heinrich and Woolsey showed how powerful bot-aided analysis can be.

In New Ideas, Heinrich and Woolsey examined problem positions mostly solved incorrectly by very strong human players. They used JellyFish evaluations and rollout results to help locate and analyze the correct plays. They succeeded so well that for many players, New Ideas helped impart a radical change in vision. That book is one of the finest advanced works existing; now it is joined in its genre by another marvelous book, Jeremy Paul Bagai's Classic Backgammon Revisited.

Though Bagai's approach is a bit different from Heinrich and Woolsey's, it's definitely BEST. With the help of Snowie 3, Bagai analyzes 120 problems from some of the old classics: The Backgammon Book (Jacoby and Crawford), Backgammon for Profit (Dwek), Backgammon (Magriel), Paradoxes and Probabilities (Cooke), and Advanced Backgammon (Robertie). The problems he examines were all gotten wrong by the authors of these books. Bagai tells why, giving Snowie's rollout results to support his own analysis. Although Bagai uses Snowie to help find the correct play—or at worst a group of close plays clearly stronger than the move recommended by the original author—the analysis is Bagai's own, and it's first-rate.

Questions come to mind: Why reanalyze positions from old books, many of which are now considered obsolete? Why not leave the quaint and dusty tomes on the shelf, and use the bots to break completely new ground? What's to be gained by trashing the hard work of authors who labored, even made great advances, without benefit of bots?

The answer is that countless players formed their conception of backgammon under the influence of the ideas in those five books. The old classics were considered quite accurate in their day, and have their adherents even now. It's very useful to debunk some of these ideas; even though they're on the way out, many were so popular that they're still influential. Bagai also shows how many useful ideas had been wrongly implemented; he points out many instances of The Inflated Comparison. Thus the charge of "trashing," particularly in the face of Bagai's incisive and accurate analysis, can't survive serious examination. Bagai knows that he stands on the shoulders of giants, and he doesn't hesitate to say so.

Take Paradoxes and Probabilities as an example. We now know that Barclay Cooke fundamentally misunderstood many key concepts of racing, timing, blocking, doubling, and match play. As Bagai says, over half the problems in Cooke's book are wrong, and many are not just wrong but so wrong as to seem ridiculous by today's standards. Why does Bagai spend so much effort bashing over 40 of Cooke's solutions?

In fact, Paradoxes and Probabilities is quite an interesting book. Cooke obviously put much effort into analyzing his problems. He writes succinctly and his reasoning is very logical; it's not always easy to find fault with it. Although most good players now understand that Cooke often starts with the wrong premises, or gives valid ideas the wrong emphasis, many people are unaware of those flaws. Many are still struggling to break free of pernicious misconceptions dispersed into the environment by Cooke's three books. Those misconceptions have been particularly insidious because Cooke did get many of his problems right, analyzing them creatively and well. It's therefore very easy to take his word as gospel, on faith in his analytical ability as well as his experience and renown. As a result, many of Cooke's misconceptions have a long half-life.

For example: I pull from my shelf Le Backgammon, a French-language beginner-intermediate text (Hatier, Paris, 1989) by Frank Lohéac-Ammoun ("l'un des meilleurs spécialistes"), and I see Position 160 from Paradoxes and Probabilities on the front cover, unattributed. (Cooke got that one right.) Lohéac-Ammoun's Exercice 16 (p. 76) is a modification of Cooke's problem 44, with Cooke's analysis recapitulated very closely. As Cooke misanalyzed his problem 44, so Lohéac-Ammoun, in virtual echo of Cooke, misanalyzes a similar problem. Désastre! Let's hope Bagai's book will have a wide audience, and will stop the spread of such classically bad analysis; we don't want Mad Cooke Disease to invade France or anywhere else.

If you understand Bagai's analysis of Cooke's problem 44, you won't be fooled by Lohéac-Ammoun's Exercice 16. But in fairness to Cooke, and to Bagai's credit as an analyst, Bagai does his best not only to determine why Cooke gets this problem wrong, but to show under what circumstances Cooke's reasoning might be valid. He gives a slight modification to Cooke's position, in which Cooke's emphasis on timing over racing would be well motivated, and where Cooke would therefore have recommended the right play for the right reasons.

Such balanced, useful treatment abounds in Bagai's fine book. Not only does he illustrate his points by modifying problem positions to show the relative influence of various factors—he's using BEST practices there—but he cites many instances of other authors' relevant analysis, thus making comprehensive study much easier. Very few writers give as many or as helpful citations.

Of course, Cooke isn't the only guilty party who gets revisitation privileges. Other authors have done their part to propagate fundamental misconceptions. Though all five classics covered by Bagai are chock-full of excellent analysis, they also contain many errors. As Bagai says, and as I know from experience, "...bad reference positions in classic books can be so devastating to your game." Classic Backgammon Revisited does a great deal to correct bad reference positions from all five books, and that's a very valuable service to the world of backgammon.

How good is Bagai's analysis? Again, it's first-rate; it's original, accurate, entertaining, and terse. Bagai says a lot in few words, and not many authors can do that. In that respect he's on the level of Robertie and Magriel. More importantly, almost every solution gives useful insight into an important facet of the game. Bagai does not fall into the "bot-kisser's" trap of blindly taking Snowie's word; his insights are his own.

Problem 8, Magriel #18-5, is Bagai at his best:

Bagai #8 (Magriel #18-5)

	138
	$$
	161
	Board image courtesy of GO-Figure

This position and the next illustrate a theme that has plagued generations of intermediates, all (quite reasonably) following the gospel according to Magriel and Dwek. Black should make the 5 point. The "action play" of stepping out to the bar is very wrong.

To win, Black needs first to hit a shot and then to contain it. 23/18 6/5 is worse on both counts: it's hard to hit checkers from the roof; it's hard to contain them with a board full of blots. Magriel says that 23/18 will "prevent White from making a key inner-board point." But White doesn't need key inner-board points—he needs to maintain his 23-pip racing advantage without getting hit. If he can make a prime, so much the better, but as long as he doesn't get hit he wins ....

... Moving Black's front checkers at random is better than stepping out to the bar. His back checkers are already optimally placed for hitting a shot: they are split, away from immediate attack, and give White no room to play behind them. Even were we to put both of Black's back checkers on the 24 point and give him a 61 to play, making the 5 point would still come out on top.

There you have it: short; sweet; accurate in every respect. Bagai captures the essentials of the position with no wasted words. Back in the Mesozoic Era, I followed Magriel and Dwek, and repeatedly made moves similar to 23/18 6/5, no doubt with predictable results.

Except for Bagai's explanation of Snowie rollouts, and except for one position—Bagai's problem 52 and Magriel #12–1, which I believe Bagai misanalyzes somewhat—my negative feelings about anything in the book are in the nature of quibbles. Before we get to problem 52, I'll give my take on the book's other shortcomings.

I wish Bagai had broken the positions down into more categories than "Openings," "Middle Games," "Bearing In," "Backgames," and "End Contact." This is a quibble, as Bagai's analysis does cover a lot of ground. I may really be asking for a book two or three times as big, and that wouldn't be a fair criticism.

Bagai should not have included problem 70 (Robertie #142). (In fairness to Jeremy, I don't think I made that suggestion when I reviewed some of his book in progress.) He admits that Robertie's recommended play is

A simple oversight, the like of which appear more and more often as the session extends into the dawn. They become more costly, though, when they make their way into print.

More costly in print? Not very much. Almost everyone knew it was a simple oversight as soon as Advanced Backgammon came out. As Bagai says, Neil Kazaross has already dealt with this one in an early review of Advanced Backgammon; Bagai adds nothing to Kazaross's analysis of the position. Furthermore, the only theme illustrated is basic technique to retain a board while trying to keep your opponent on the bar long enough to escape a back checker. There are more interesting illustrations of such technique available, and Bagai could easily have substituted one of these, or something similar.

Bagai #84 (Robertie #107)

	172
	$$
	101
	Board image courtesy of GO-Figure

Unlike some authors who don't even say whether or how they use bots, Bagai tells how he used Snowie to help select candidate moves, and gives reasonable justifications for his rollout parameters. That's good, but his explanation of Snowie's rollout parameters is confusing for readers who don't own Snowie. In the Introduction, he says all his rollouts were "cubeful" (a term coined by the creators of Snowie). We are then sent to Appendix A, "Rollout Methodology," where we are informed in boldface type that "[a]ll positions in this book were rolled out cubeless." Wait—shouldn't "cubeless" be the opposite of "cubeful"? In his preceding paragraph, we read:

In a cubeless rollout, each game is played to completion, resulting in a cubeless equity that is subsequently transformed into a cubeful equity.

How does a "cubeful equity" take the cube into account? The only thing we're told is that there's some transformation applied. Never does Bagai tell exactly what "cubeful" means. Although it has something to do with the cube—"Snowie 3 fully integrates the doubling cube into its analysis of positions"—it's not a "live-cube" rollout. A live-cube rollout is one in which Snowie uses the cube, and where resulting equity is based on outcome multiplied by cube level, as in a real money or match game. A cubeful rollout is supposedly more "procedurally legitimate" than a live-cube rollout; "Snowie's creators [say] that, as of Snowie 3, the live cube introduces more bias than the cubeful-to-cubeless transformation." It seems we must take on faith the statement that Snowie "fully integrates" the cube into its analysis. In fact, knowledgeable people have told me that the "cubeful-to-cubeless transformation" is proprietary information that Snowie's creators won't share with the public.

The Snowie-less reader—I'm one such—ought to be as confused about this as I've been. Even Chuck Bower, who has done much study of Snowie rollout methods, has a natural question: Why do Snowie's creators include the live-cube feature at all? If cubeful rollouts are more reliable, why should anyone ever settle for anything less?

In a private email, Bagai informs me that Olivier Egger says Snowie's handling of the cube at 3 ply can sometimes lead to big inaccuracies. Bagai's opinion is that a live-cube rollout can be more accurate if the result strongly depends on short-term cube actions, and if Snowie gets those particular decisions right. This is a matter of judgment that must be applied case by case. Given that live-cube rollouts may be problematic, Bagai's decision to stick to cubeful rollouts seems very reasonable.

Occasionally Bagai focuses more on what makes an author's recommendation wrong than what makes the best play right. In such cases, extending the rollouts would have been a welcome addition. Most of his rollouts were 2-ply; he might have tried 3-ply for further illumination. For instance, in problem 84, Robertie #107, why do rollouts put 24/22 13/10 .030 ahead of 24/22 11/8?

Bagai #29
(Jacoby and Crawford #71)

	133
	$$
	182
	Board image courtesy of GO-Figure

What about problem 29, Jacoby and Crawford #71: If bar/23 6/5* is as automatic as Bagai says it is, why is making the 23 point so close in equity? However, Bagai's analysis is so good overall that an occasional paucity of rollouts is a minor deficiency.

Finally, though I generally support Bagai's treatment of Cooke, I think he slightly overemphasizes Paradoxes and Probabilities. The book might deserve a quarter or a fifth of Bagai's attention, but not over a third, as is the case.

Now let's revisit problem 52.

Bagai:

Bagai #52 (Magriel #12–1)

	103
	$$
	127
	Board image courtesy of GO-Figure

This one is quite surprising. Indeed, Magriel's analysis seems so clear and compelling: White is likely to run soon, forced or not, so Black wants to have the maximum number of builders trained on the straggler left on his 3 point. Thus 7/6 7/4, which creates three active builders, rather than 8/7 8/5, or 8/6(2), which each create two.

So why does 8/7 8/5 come out the clear favorite in the rollouts, followed by 8/6(2)? The difference here seems to be the long-term value of the bar point once Black begins to attack. Unless he rolls perfectly, he may have a difficult time completing a full closeout. In those cases, he'd much prefer to have a semisolid five- or six-prime to constrain White while he moves his other checkers into the outfield. Losing the bar point means that, until closed out, White will always be threatening an immediate escape.

Shift Black's deuce point to the ace...and Black can never hope to form that constraining prime. Thus, tactics prevail and 7/4 7/6 is best.

In other words, the main reason for breaking the 8 point in preference to the bar is primarily structural, not tactical: The 8 point doesn't cooperate with the 2 point as part of a full prime, and the 7 point does, so it's structurally better to break the 8 point. The better structure does have tactical ramifications; when Black attacks, the bar point is better for constraint, working well against persistent threats by White to enter and escape. This is critically important, Bagai asserts, even considering the extra builder given by Magriel's recommended 7/4 7/6.

Bagai is right that these factors play a role. But I think he gets problem 52 right for essentially the wrong reason. In this problem I think tactical factors are more important than structural. Let's see why.

The first thing to note about problem 52 is that the 2 point is valuable no matter which outside point Black chooses to break. Black's 2 point can't be part of a six-point prime including the 8 point, but it can serve as part of a powerful five-prime from the 2 through 6 points. Although a six-prime is better than a five-prime, the threat to make that five-prime is still very strong, and an extra builder augments the threat in conjunction with the live cube. It's probably for this reason more than any other that Magriel recommended 7/4 7/6.

Since Black is a clear favorite owing to his better blocking position and potential for outfield control, it seems that a live cube will help Black more than White. It certainly looks that way in the short-term attacking variations. This makes the case for 7/4 7/6 more compelling than Bagai and his "cubeful" rollouts would have us believe. My own cubeless JellyFish Level 6 (2-ply) rollouts give somewhat different results than Bagai's cubeful Snowie 2-ply rollouts. The ordering is the same: 8/5 8/7 best, then 8/6(2), last 7/4 7/6. Perhaps because of the different types of rollouts, the equities are different: Bagai's are +.398, +.357, and +.341, respectively; mine (7776 rollouts apiece, time factor 1000; individual standard deviations of .003) are +.265, +.261, and +.246. Can Magriel's play be correct?

Let's look at two tactical factors in favor of 8/5 8/7. I mean short-term constraint of White's numbers, and diversification of Black's. With 8/5 8/7, Black constrains White's 42 and 44, and diversifies his own good numbers when White runs. When White runs and leaves one checker behind, Black has 4's and 2's to attack, numbers that are blocked on White's side of the board. Black's 44, 22 and 42 play very well instead of very poorly after the alternative. Black's 6's, 5's and 3's play from the 21 point, so keeping the 7 point works well on numbers like 54, 52, 34 and 32.

In contrast to such useful constraint and diversification, 7/4 7/6 diversifies White's numbers by giving him a good 4, 22/18. That's bad. It's less useful to keep the 8 point and constrain 5's, than it is to keep the 7 and constrain 4's. White's blocked 5's hurt him less than would blocked 4's, because he has both 8/3 and 6/1 on his side of the board; his only good 4 on his side is 6/2. Also, 7/4 7/6 duplicates Black's 5's and 3's to escape and to attack.

Though Bagai doesn't mention them, short-term constraint and diversification appear to be important factors. Are they critically important, or are they secondary to the value of keeping the bar point? After all, with a spare on the bar point, Black can often attack and threaten to make a winning six-prime. Not only that, the bar point blocks a White checker that may enter on Black's 1 point, whereas the 8 point doesn't.

52A

	101
	$$
	127
	Board image courtesy of GO-Figure

One way to get an idea of the effects of short-term tactics versus long-term constraint is to change the numbers constrained by White and by Black on White's side of the board. I did long JellyFish Level 6 cubeless rollouts of two modified positions; both are the same as Bagai's position 52, except that White's 8 point shifts to his 7 point (52A) or to his 9 point (52B). In both positions, rollouts clearly favored 7/4 7/6. In 52A, 7776 rollouts apiece, time factor 1000, put 7/4 7/6 in front with cubeless money equity of +.302, ahead of 8/5 8/7 at +.283, while 8/6(2) was third at +.279. In 52B, with the same rollout parameters, 7/4 7/6 won out at +.314; 8/6(2) was second at +.240; and 8/5 8/7 was third at +.233. Each individual standard deviation was .003.

In 52A, Black has less need to constrain White's 4's than in Magriel's original position. Black has a good 4 on White's side of the board, 21/17, so he has less reason to create attacking 4's in case White runs. Finally, White constrains Black's 3's and 1's on White's side of the board, so Black has more reason to give himself good attacking 3's and 1's when White breaks anchor immediately. Therefore, Black plays 7/4 7/6, giving himself three active builders for the 3 point, diversifying in the process.

52B

	105
	$$
	127
	Board image courtesy of GO-Figure

In 52B, the story is the very similar. Black's 5's are constrained on White's side of the board; White's 5's on his own side are also somewhat constrained because the only good 5 is 6/1. Accordingly, Black does not give White a good 5 on the other side, and keeps the 8 point so that he himself has a good attacking 5 should White break anchor. Again he plays 7/4 7/6. Creating three builders for the 3 point is a bonus.

What happens in the modified position Bagai discusses, where everything is the same as in the problem position, but Black's 2 point is shifted to his 1 point? Bagai correctly picks 7/4 7/6; my JellyFish 2-ply rollouts agree (7776 trials apiece, time factor 1000; individual standard deviations of .003). They give +.213, +.143, and +.133 for 7/4 7/6, 8/6(2), and 8/5 8/7, respectively.

Why does 8/7 8/5 show up so badly when Black's 2 point is shifted to the 1 point? Diversification and constraint factors appear similar whether Black has the 7 and 1 or the 8 and 2, but the most important thing is the weakness of the 7-1 combination. That combination is much worse than the combination of the 8 and 2 points. Owning the 7 and 1 points, Black badly needs to attack as soon as he can; therefore, tactics do indeed prevail. Black's 8 point better restrains both White checkers, and it's optimally placed to attack both open points in Black's inner board. Black thus keeps his 8 point and creates an extra builder.

Analyzing his position 12–1, Magriel correctly notes that "the entire game depends largely on what happens on the next roll." Magriel therefore recommends creating three builders rather than two, but I think the short-term tactical considerations have to be analyzed further than that. I think short-term constraint and diversification, with the long-term priming power of the bar point, combine to favor Bagai's recommended 8/5 8/7. The value of the bar point in attacking variations seems to be the least important factor; rollout data seem to support that statement. At any rate, there's much more going on in problem 52 than Bagai says, and it's still possible that Magriel is right. Will live-cube rollouts tell?

I'll answer my own question: Late-breaking live-cube rollouts support Bagai's play, but they bring Magriel's up to second place. David McKenzie let Snowie 3 run for days in order to perform 2412 full live-cube rollouts of each of four plays: 8/5 8/7; 7/6 7/4; 8/6(2); and 7/6(2) 2/1(2). The checker-play search settings were 3 ply, huge, 100%; the cube settings were 3-ply, with automatic settlement performed at equity of 0.550 at 16 points. I believe these are about the most exhaustive lookahead settings possible.

The results are as follows: Bagai's 8/5 8/7 was first at +.399, with 95% confidence interval of +/-.017. Magriel's 7/4 7/6 was second at +.368, with 95% confidence interval of +/-.019. Third was 8/6(2) at +.348, 95% confidence interval +/-.018. Well back in fourth place was 7/6(2) 2/1(2), at +.295, 95% confidence interval +/-.018.

Though the live-cube results for the top two plays don't quite give statistical significance at the 95% confidence level, Bagai's move is further supported. It's interesting that when Snowie uses the cube during rollouts, Magriel's play improves relative to 8/6(2). This supports my statement that an active cube should be a factor in favor of Magriel's play.

There, I'm done with my lengthy revisitation of that problem. That's the biggest disagreement I have with any of the book's analysis, and it's really not very big. If Bagai gets it right for mostly the wrong reason, no big deal; any flaws in his analysis of that position comprise one of the book's very few and very minor defects. Bagai's fine treatment of the other positions can carry them all.

The backgammon world can wish Jeremy Paul Bagai well as he joins the ranks of the best theoreticians. Classic Backgammon Revisited is a momentous start.