Doubling strategy in match play differs somewhat from doubling strategy in money play. In a typical tournament match, most cube decisions will be the same as the corresponding money decision. A small number will be different; typically these will include doubles beyond the two level, and doubles when one or both players are within four points of victory.
Of the 35 games in this match [Genud vs. Dwek, Monte Carlo, 1981], only the last three contained cube decisions that were predicated on the match score. Not wishing to leave the area unexplored, I’ve included this introductory chapter to touch briefly on some of the more interesting topics of tournament cube strategy:
- Calculating doubling equities in match situations
- Preserving a lead
- Catching up
- Strategies at particular scores
This is hardly an exhaustive list, and much more could be said about all of these topics. That, however, is the subject matter for another book.
While proper checker play can be taught almost entirely through intuitive positional concepts, proper doubling strategy in match play, regrettably, cannot be. The subject is inherently mathematical. I will attempt to keep this discussion on as simple a level as possible; however, readers who are utterly repelled by numbers and tables should probably skip this section and move directly to the match itself, where calculations are kept to a minimum.
Calculating Doubling Equities in Match Situations
Psychological considerations aside, proper use of the cube in money play involves only an estimation of the probabilities of the various outcomes in a given position. How frequently will one side win a gammon? How often a single game? How often can the defender turn the game around and redouble?
Each game in money play is independent of the games before or after, so no other factors need to be considered. In money play there is only one goal: to make the choice which maximizes equity in this particular game.
In tournament play the goal is quite different: to maximize the chance of winning the match. A correct cube decision increases one’s chances of winning the match; a mistake decreases them. A correct cube decision in tournament play may be identical to a money decision, or it may be the exact opposite.
It is easy to find positions where automatic money plays become senseless when a match score must be factored into the decision. Consider Position 1, for example:
Should black redouble?
Black, on roll, owns a 4 cube and is considering a redouble. He is just slightly more than a 3–1 favorite in the position (78–22, to be exact). For money, no thought is required: Black doubles, white passes.
But now suppose this is a tournament and black currently leads 10–0 in a match to 15 points. A double on black’s part is suddenly answered by a take and a redouble to 16! In one stroke, black’s huge lead has been nullified and the entire match now rests on the outcome of this game alone.
Black is still a healthy favorite, to be sure. But if he never doubles, either this turn or the next, his chances of winning the match would be better than 95%. An automatic money decision has become a colossal blunder when the match score is taken into consideration.
Match Equity Table
The basic tool for studying match play is Table 1 given below. This table shows the probability that the player in the lead will eventually win a 15-point match for any given score. The table was computed by extrapolating backwards from scores near the end of a match, when probabilities are easily calculated, to scores earlier in the match.
Probability of Victory in a 15-Point Match
Table 1 is ideal for working out problems, but not easily remembered for over-the-board situations. Table 2 is an approximation of the information in Table 1, in a form more useful for actual play. It shows the average value of a lead of a given size, both in percentage terms and expressed as an odds ratio.
Average Significance of a Lead in a 15-Point Match
| Size of
| Probability of victory
for leading player
| Odds in favor
Let’s look at two examples, one from an endgame position, one from a middlegame position, to see how these tables can be used to solve actual problems.
Black leads 12–4 in a 15-point match
Black, on roll, owns a 2 cube in a 2-roll position. In order for white to win, white must roll a double after black does not. In pure percentage terms, black is an 86–14 favorite. For money, of course, black doubles and white passes. Let’s calculate exactly what black (and white) should do in this situation.
- Case 1: Black doesn’t double
- 86% of the time, black wins two points and takes a 14–4 lead, with the Crawford rule in effect. Table 1 tells us that black then has a 97% chance of winning.
- 14% of the time, black loses two points, but still leads 12–6. His winning chances are then 87%.
Elementary probability theory tells us that black’s total winning probability is the sum of his probability of winning after each event times the probability of the event itself. His total winning probability is therefore
86% × 97% = 83.4% 14% × 77% = 12.2% 95.6%
- Case 2: Black doubles
- White now has a choice of dropping or taking.
- White passes. White now trails 14–4, and black has a 97% chance of winning.
- White takes, and redoubles to 8 if black does not throw doublets. Now, 86% of the time black wins the match outright, while 14% of the time white wins 8 points and ties the match at 12–12.
Black’s total probability of winning if white takes and redoubles is
86% × 100% = 86% 14% × 50% = 7% 93%
To summarize, black’s winning chances are
- 95.6%, if he doesn’t double
- 97% if he doubles and white drops
- 93% if he doubles and white takes
White’s proper strategy is to accept and redouble if doubled. Black’s proper strategy is not to redouble.
Black leads 9–7 in a 15-point match.
White doubles to 2.
This is the final position from Game 18 of this match, although for the sake of illustration I’ve attached a different score to the position. Black leads in the match but, with two men on the bar, is faced with a somewhat gammonish double. Should he take?
A simulation yielded the following probabilities for black and white:
|White wins a gammon||25%|
(If black’s winning probability seems high, notice that he becomes a favorite in the position as soon as he either hits a blot or makes an anchor. With his five point already made and an excellent distribution of builders, black is in position to win the game with the cube as soon as white’s blitz fails.)
In a money game, particularly with the Jacoby rule in effect, this position is both a double and a take. White must double because he is a clear money favorite and he runs a real risk of losing his market this turn. Black must take since his average loss after taking is only one half point, compared to the one point he would lose by dropping.
Although Position 3 is a clear take for money, many authorities recommend extra caution when taking potentially gammonish doubles in tournament situations. (And, in fact, Dwek dropped when doubled in this match.) Let’s see whether a take is justified given a 9–7 lead in a match to 15.
- Case 1: Black drops.
- Black now leads 9–8 in a 15-point match. Table 1 shows that black’s winning chances are 57%.
- Case 2: Black takes.
- 25% of the time, black loses a gammon and trails 9–11. His winning chances are then 34%.
- 25% of the time, black loses a single game and the match is tied 9–9. Black’s chances are 50%.
- 50% of the time, black wins and leads 11–7. His winning chances are 78%.
Black’s total probability of winning if he takes is
25% × 34% = 8.5% 25% × 50% = 12.5% 50% × 78% = 39.0% 60.0%
So by taking, black improves his winning chances from 57% to 60%. Despite the gammonish nature of the position, a clear take for money is also a clear take in a tournament position with a healthy lead. As verification, here is a chart of the probabilities of winning a match for the same position but different match scores:
if he drops
if he takes
By comparing the last two columns, we can see that regardless of the size of black’s lead, he always improves his chances by taking the double, although the error associated with dropping becomes less significant as he lead increases. This is not an atypical result.
In general, for initial doubles, a proper take in money play is also a proper take in tournament play, even if the gammon chances are significant.
For redoubles, there are not clear-cut generalizations. Taking strategy depends on the size of the lead, the amount of overage (wasted points) involved in the take, and the raw probability of winning the position.
Match winning chances
One question immediately arises after looking at this last table. Why are the differences between dropping and taking so small? With a 9–5 lead, for instance, the discrepancy between a drop and a take is only 1.5%. Whichever decision is correct, shouldn’t the differences be more decisive?
The answer is no, and for a simple reason. Consider the case where black holds a 9–5 lead in a match to 15. At the time of the cube turn, 14 points have already been decided. Disregarding gammons, a cube turn from 1 to 2 certainly brings a 15th and may bring a 16th point into play. These extra points, however, represent only a small fraction of the total points decided in the match so far. Most of black’s winning chances are associated with the points that have already been won; the one or two points that hinge on this cube have a comparatively small effect on the outcome of the match.
To see this more clearly, let’s examine the effect of an inconceivably serious blunder on the initial cube turn. Suppose black led 9–5 in a 15-point match and reached a position he was certain to win. Instead of doubling out his opponent, however, he hallucinated and allowed himself to be doubled out the following turn. How big an effect would an error of this magnitude have on his winning chances?
- Had he won the game, he would have led 10–5. His winning chances would have been 80%.
- By dropping, he led 9–6. His winning chances were then 70%.
Even an error of this magnitude only affected his winning chances by 10%. In the light of this example, we can see that a difference of 1.5% in winning chances between taking and dropping is more significant than might appear. It represents, in some sense, 15% of the maximum possible swing in the position.
Of course, the fact remains that such small differences are not directly calculable over the board, although players with some skill in mental arithmetic can come remarkably close. The value of these techniques lies in testing theories and developing generalizations which are, in themselves, useful.
Preserving a Lead
The most serious cube errors in tournament play are made by the player enjoying a moderate to substantial lead. The reason is not hard to find. Most players tend to be conservative cube handlers in match play even when the score is close. Since players are aware that some degree of extra caution is appropriate when nursing a lead, their natural tendencies to conservatism become exaggerated, sometimes to ludicrous proportions.
Let’s consider the problem of preserving a lead from three aspects:
Checker play: The ideal type of position to reach when leading is a straight race. No gammons or unexpected swings are possible, and cube decisions can be calculated precisely. The next best position is a mutual holding game, which tends to become a race as soon as one player throws a large double. The worst type of position by far is a battle of mutual primes; like a taut rubber band, a prime-vs.-prime game can snap with one bad roll into a hopeless gammon — a very bad situation if the cube has crept its way up to 4 in the process.
I don’t advocate any radical deviation in checker style to try to reach racing positions. Weak plays remain weak even when leading in a match, and often the dice leave no real leeway in the type of game that will be played. Some intelligent choices can be made, however. Consider an opening roll of 4-3:
Both these opening plays of 4-3 have their adherents and are thought to be roughly equal in overall strength. Notice, however, that they tend to lead to quite different types of positions.
In Position 4, white is forced to hit on the 20 point if possible. The most likely sequence is then a series of return hits, with both sides eventually establishing an anchor — a mutual holding game.
In Position 5, white, if he fails to hit a blot, will pull men down from his mid point. Black is then a favorite to make an inner point, with white perhaps reciprocating next turn. Position 5 is much more likely to lead to a battle of mutual ace-point games or mutual primes, much more gammonish positions than those reached from Position 4.
With a lead in the match, playing 24/20, 13/10 with an opening 4-3 is clearly correct. Black can steer toward the kinds of positions he wants while making a play objectively no weaker than the alternative.
Offering doubles: Initial doubles should be made on a more or less normal basis. Redoubles need to be very carefully considered, however, since an accepted redouble probably means the cube will finish at the 8 level. Consider the following position, from a match between Chris Peterson and Malcolm Davis:
Leading 16–7 to 21, should black redouble?
Black leads 16–7 in a match to 21. His raw winning probability if the game were played to a finish is 84%. Should he double?
For money, obviously yes, and white would pass. Here, however, we have a match situation where the notion of overage comes into play. White has the option of taking and immediately redoubling to the 8 level. All 8 points are useful to white, but only 5 of the 8 points are useful to black. The other 3 points are wasted for him, since he already has 16. When overage happens, the odds change rapidly from normal money considerations.
By using the methods explained earlier, the following probabilities can be calculated:
- Black doubles, white drops. Black’s chances are 97%.
- Black doubles, white takes and redoubles to 8. Black’s chances are 93%.
- Black never doubles. Black’s chances are 95.3%.
Calculating what happens if black never doubles is easier than the herculean job of calculating his chances assuming optimal doubling strategy in all possible variations. Obviously, his real chances if he holds onto the cube next turn and doubles properly at some later point must be slightly higher than 95.3%.
It’s clear that black should not redouble. The reader can verify that, with the cube in the middle, black could properly offer an initial double. Note that in the case of an initial double, no overage would be involved on the recube to 4.
Accepting doubles: The decision to accept a double when leading hinges on two criteria:
- Is the double a take for money, and by how much?
- What is the level of the cube compared to the number of points I need to win the match?
Unlike money play, where the level of the cube matters not at all in the decision to take, in match play the level of the cube can be the deciding factor.
Let’s look closely at three endgame positions, in which the leading player has to decide whether or not to take a double. In Position 7, he is a very slight underdog; in Position 8, he is a bit less than a 2–1 underdog; and in Position 9, he has rather a close take — he is almost a 70–30 underdog.
For each position, using the methods described at the beginning of the chapter, black’s chances of winning the match after both dropping and taking have been calculated for several different match scores and for three different cube turns: an initial double, a redouble to 4, and a redouble to 8. The columns labeled “Action” indicate black’s correct decision — drop or take. A dash indicates no difference.
White, on roll, is a 53–47 favorite.
|White doubles to 2||White doubles to 4||White doubles to 8|
White, on roll, is a 64–36 favorite.
|White doubles to 2||White doubles to 4||White doubles to 8|
White, on roll, is a 70–30 favorite.
|White doubles to 2||White doubles to 4||White doubles to 8|
- The initial doubles to the 2 level are all takes, just as in money play. Neither the size of black’s lead nor the relative unfavorableness of the position affects black’s decision (although in Position 9 with a 12–4 lead, black gains almost nothing from his decision to take).
- The redoubles to the 8 level are all passes unless black can utilize all 8 points. The overage involved is so significant that black even has to pass Position 7 with a large lead. (In fact, with a 12–4 lead, black would have to pass a redouble to 8 unless he was better than a 58–42 favorite in the position.)
- The redoubles to the 4 level are not so easy to generalize. Black can always take Position 7, overage or no. In Position 8, black can take with 11 points or less, but should pass (although it’s close) with 12 points. In Position 9, even the ability to utilize all the points doesn’t guarantee a take. Only if dropping would cost him his entire lead should he grab the cube.
The strategy of a player trying to catch up in a match is, in some sense, the opposite of the player in the lead. Whereas the leading player strives for simple positions, the trailing player should seek out complications.
Prime vs. prime and mutual backgames are ideal positions to play from a trailing position. With an opening 4-3, for instance, the trailing player should unhesitatingly play 13/9, 13/10. With an opening 6-5, a very reasonable alternative to lover's leap is 24/18, 13/8.
As before, however, this advice only applies to choices between more or less equal plays. A simple strong move is always to be preferred to a complex weak one.
The most common cube mistake made by trailing players is doubling too soon. As we have seen from the analysis in the last section, the leading player’s taking points for an initial double are virtually the same as his take points in ordinary money play. This implies that, in a theoretical sense, the trailing player’s doubling point should be about the same as in money play.
In a practical sense, the trailing player can do better, actually considerably better, by utilizing the tendencies of players with a lead to pass doubles prematurely. He must be careful, however, to offer doubles which seem to pose some potential gammon danger. As an example, suppose white leads 12–4 in a match to 15. Black opens with a 3-1, making his 5 point, and white rolls a 3-2, pulling two mean down from the mid point.
Black on roll.
Although I’ve seen many players turn the cube as black in such positions, a double this turn is a clear error. Why? Because there is virtually no chance that white will blunder and pass. Black has very real possibilities of stealing a free point at this score if he waits until he has some sort of concrete threat. By doubling now, he squanders that potential.
Suppose, for instance, that the next two rolls are: black, 5-4: 24/15*; white, 6-2: bar/23, 11/5.
Black on roll.
White still has a very easy take if doubled, even with a big lead. Yet many, if not most, players would panic and drop a double in this position, frightened by the, as yet, nebulous possibilities of a gammon.
The key skill in overcoming a big deficit is sensing when the opponent is ready, incorrectly, to pass, and pilfering a free point by doubling at precisely that moment. It’s far easier to win a game by having the opponent resign than by having to win over the board. In this match, we’ll see Dwek steal two points with timely doubles in Games 33 and 34, then make a crucial misplay with a slightly early double in Game 35.
Strategies at Particular Match Scores
11 to 11
When both sides are within 4 points of the match, certain types of gammonish doubles become senseless. Consider the following position, from a match between Paul Magriel and Roger Low:
Black on roll.
The approximate probability distribution for this position is
|Black wins a backgammon||5%|
|Black wins a gammon||25%|
Some players might think white’s winning equity overstated. His position has two main strengths: (1) With his anchor, he will always have the equity of an ace-point game. (2) His 3-point board could turn the game around quickly if he can hit a blot in the next few rolls.
It’s easy to verify both that black has a very strong money double and white has a clear take. Notice, however, that at an 11–11 score in a 15-pont match, black couldn’t consider doubling. If white merely adopted the mindless strategy of accepting and then redoubling, no matter what black’s next roll was, he would have placed the whole match on the line with gammons not counting, and would be even money to win. In order to squeeze any advantage out of the position, black must play for an undoubled gammon.
With the end of the match in sight, gammonish positions will generally be played to completion with the cube still at the one level.
13 to 13
With both sides exactly two points from match, a peculiar situation arises. The cube has no value to either player after the initial cube turn. Consequently, the player with an initial advantage has no reason to be shy about doubling. In fact, virtually any chance of losing one’s market justifies a double. For instance, consider the sequence: black, 3-1: 8/5, 6/5; white, 5-2: 13/8, 13/11.
Black on roll.
Black can lose his market with rolls of 6-6 or 4-4; many other sequences guarantee him a long-term initiative, with constant subsequent threats of rolling a gin number. If he leaves the cube in the middle and the game turns around, white will be looking for the first opportunity to double him in. The likelihood that this will not be the last game of the match is extremely small. With no realistic way to gain by waiting, black is virtually forced to double now.
The Crawford game
The Crawford rule, now universal in tournament play, stipulates that when one player reaches match point, the cube may not be used in the next game only.
In a theoretical sense, the dominant feature of the Crawford game is the importance (or unimportance) of a potential gammon. If the trailing player is at an odd number of points, winning a gammon can save him an entire game.
Suppose, for instance, white trails 14–9 in a 15-point match. Winning a single game brings him to 14–10. He still needs to win three doubled games to win the match. Winning a gammon, however, vaults him to 14–11 and he now need only two wins to win the match. Winning a gammon thus becomes a major objective for the trailing player, second only to the objective of winning the game itself, while avoiding the gammon becomes very important for the player in the lead.
If, however, the trailing player has an even number of points, the gammon is inconsequential. The trailing player will subsequently need to win the same number of games in any case. (A backgammon could be crucial, but this is a very rare occurrence.)
In a practical sense, these observations don’t matter very much. Particularly in the early and middle game, the play to maximize gammon chances will also be the right play by any criteria. And from the leader’s point of view, the best way to avoid losing a gammon is to make the objectively strongest moves.
There are many situations in backgammon where a little knowledge is a dangerous thing: The case of trying to avoid losing a gammon in the Crawford game is a prime example. I have seen players with a lead roll an opening 4-1 and move 13/8, or play an opening 2-1 by moving 13/10! Curiously, these “safe” plays involved more, not less, risk of a gammon.
Moving 13/11, 6/5 on an opening 2-1 is actually the least gammonish way of playing the roll. If the blot on the 5 point is missed, the opening player is almost guaranteed a strong initiative; if hit, the third checker back will be useful in making an advanced anchor.
Don’t be too clever in the Crawford game. A good move is a good move.
After the Crawford game
When the Crawford game has passed, the trailing player should of course double at his first opportunity. By so doing, he ensures that the leading player will have the minimal amount of information on which to base his decision to take or drop.
The leading player will find himself in one of two situations. If his opponent is at an even score, he will have a mandatory take. If his opponent is at an odd score, he will have the option of a free drop.
The free drop: Suppose that white leads 14–13 in a match to 15 and the Crawford game has passed. Black begins with a 4-2 and makes his 4 point. White then rolls 4-3 and plays 13/9, 13/10.
Black on roll.
Black, quite properly, doubles. White must pass. If he takes, this game is for the match and he is an underdog to win. By dropping, the next game will decide the match and white is 50–50, sight unseen, to win that game.
At any smaller odd number, the same principle applies. If white is doubled and accepts with a 14–9 lead, black needs only to win that game and two others to win the match. If white drops, the score becomes 14–10 and black still needs to win three games (disregarding gammons) to win the match.
One word of caution: The free drop, obviously, can only be used once. The more games to go, the bigger a disadvantage white needs to face in order to justify utilizing his free drop. With one game to go, any disadvantage at all is sufficient to pass.
With a lead like 14–3, however, possibly as many as 6 games remain to be played. Now white should take a double even with a small disadvantage, since he is likely to face a worse situation sometime before the end of the match.
The mandatory take: Suppose white leads 14–12 in a 15-point match. Black rolls a 3-1, making his 5 point, and white rolls 6-3, playing 13/4. (Notice that this is about as big a disadvantage as white can face after one roll.)
Black on roll.
Black doubles. White must take. If he passes, the next game is for the match and white’s chances are only 50%. If he takes, his opponent must win this game, then the last game, to win the match. White has essentially a free shot at winning the match in this game. Of course, losing a gammon would be disastrous, but white is as yet in no realistic gammon danger.
The same reasoning applies if black has any lower even score.