# Double the Trouble

## Advanced Backgammon Strategy: Playing with the Doubling Cube

#### Estimating Your Chances in a Game

Douglas Fairbanks, Jr. and Joan Crawford at their home in 1931 using a scoring dial (an early version of the doubling cube). The writer Chris Bray owns perhaps the only known remaining scoring dial, which he bought from eBay 20 years ago.

The champion backgammon player Kit Woolsey says, ‘A player who is savvy with the cube will have an advantage over an opponent who moves the checkers slightly better but makes inferior cube decisions. Thus, it is vital to understand cube decisions in order to be a winning backgammon player.’ Players the world over underestimate the importance of being able to use the doubling cube competently. As you get more into advanced backgammon strategy, you will see the importance of the doubling cube. Kent Goulding says, ‘Every turn is a new cube decision. Always. Never forget. The biggest error you can make is to fail to double at the appropriate time.’

However, many players do forget the vital role the doubling cube plays. Former world champion Tim Holland wrote a backgammon book in the 1970s of more than 200 pages with barely a mention of doubling.

There are just two questions you need to know the answer to as every game progresses: when to accept a double and when to offer a double. Any backgammon pro would tell you your answers to these questions will determine much of your success in the game.

To make good doubling decisions, you must get to grips with how to calculate or at least estimate what is your percentage chance of winning the game from that point. (A simple way of understanding this is to think, if you played 100 games that were identical up to the point at which you are calculating your chances, how many games would you – statistically – go on to win after that point? If the answer is 50, your chances at that point are 50%. If it was 36, your chances at that stage are 36% etc.)

#### When Should You Accept a Double?

Advanced backgammon strategy would tell you that there is a great question you can ask yourself to help you work out whether or not to accept a double. If you feel your chance of winning the game is more than 25%, you should accept the double. Many people would assume that this figure should be 50%. Instinctively it feels wrong to agree to up the stakes if you are losing at all, but to work toward being a backgammon pro you should. You may feel that to accept a double when you are losing by any degree would be madness. However, there is a reason why your take point should actually be 25%. I will explain this reason anon.

Remember, your opponent will offer the double when he feels he is ahead in the game and assumes he will win. But don’t make a decision until you have really worked out your chances. If you can see that just a little bit of luck could turn the game in your favour, take the risk and keep playing. Remember, if you win, you get double the points. Plus you are in control of the cube, which means you can double back and win even more points if your opponent accepts.

Don’t worry if a doubling decision proves costly in the short run; the dice can be cruel. Statistically, accepting the double when you still have a more than 25% chance of winning will work out in your favour once luck and bad luck cancel each other out.

This 25% threshold is approximate (and is worked out before factoring in the possibility of gammons, the match score, the value in holding the cube and other possibilities), but it does stand up to mathematical testing. If you always accept a double when you have a 25% chance of winning a game at the time it is offered, you will, on average, come out ahead.

How can that be when logically it feels as though the figure should be closer to 50%?

Here’s the explanation. If you aren’t keen on maths, just trust me on this and skip along. Imagine you could play four games that were identical games up to the stage where you were offered a double, and at that stage you had a 25% chance of winning. If you refused the double offer all four times, you would definitely lose 4 points. If you accepted the double offer all four times, the odds say that you will win one of those four games (you have a 25% chance, remember). This game is worth 2 points. The other three games you would lose (if you accepted all four times) would lose you a total of 6 points. Thus your net loss will be 4 points. This is exactly the same as if you had refused the offer all four times. Thus, when you are at 25% you are at a break-even point. Statistically, you have exactly the same chance of losing 4 points whichever way you go. Thus you should take the chance of winning 2 points.

You can also think of it in the following way. Just as offering doubles is a way of maximising your gains in the games you are winning, accepting doubles is a way of limiting your losses in the games you are losing (as long as you still have a more than 25% chance of winning). Also remember that whenever you accept the doubling cube, you have the advantage of having exclusive use of the cube from that point on. If things do turn around, you (and only you) can double the stakes once again.

Let’s look at how this 25% rule works in the simplest scenario. In the example below, it is Black’s turn and he offers you a double. Should you take it?

Obviously, Black has an excellent chance of winning with the next throw. But if he throws any dice combination that includes a 1 (except for a double 1), he can’t win. Then you are guaranteed to win on your next turn.

So what are the chances of Black winning? There are 12 out of 36 possible throws (anything with a 1 including a double 1) that will lead to Black being unable to win on his next throw. 12 out of 36 is about 33%; thus there is a 33% chance he will not win. That is well over the 25% threshold, so you should chance the odds and take the double.

With a 31% chance of winning, you should accept the double.

Now let’s look at a slightly different position and consider what you should do if offered the double. In the example below, the only throws that will prevent Black from winning are 1+1, 2+1, 1+2, 3+1, 1+3, 3+2 and 2+3. There are only 7 throws that would not enable Black to win. 7 out of 36 is around 19%, which is lower than your 25% threshold. In this scenario you should cut your losses and decline the double, letting Black win.

In this position there are only 7 rolls that will prevent Black from winning, so you only have around a 19% chance of winning. You should decline the double.

You can see how important it is to try to work out the statistics because, on face value, you might have thought that the first scenario gave you a lower chance of winning than the second, simply because Black seemed closer to home. Always try to work out the actual probability of winning at any stage, rather than relying on sight of the board alone.

#### When should you offer a double?

The thought process behind offering a double is similar to that behind accepting a double. But because the ball is in your court with the offering of doubles, you need to consider it on every move in order to best develop your advanced backgammon strategy. Remember, every roll is a new potential doubling opportunity. The worst mistake many players make is to miss great opportunities to offer a double because they are not thinking about it before every throw. Missing an opportunity to double can cost a player the whole match.

It’s vital that you offer a double before you ‘lose your market’, i.e. before you are so likely to win that your opponent automatically refuses. The trick is to offer a double in that window where he can be lured into accepting because he thinks he still has a chance.

Remember, if you offer a double that is accepted and you then go on to win a gammon, you win four points rather than two, so it is well worth getting it right and building it into your advanced backgammon strategy!

Timing is everything when it comes to offering the doubling cube. Offer it too soon and it’s a powerful tool (and potential gift) for your opponent; offer it too late and end your chances of winning both a doubled game and a gammon because your opponent will likely decline, forfeiting the game for only one point.

As Philip Martyn famously (to us backgammon nerds, that is) said, ‘You are very easy to play against if you double either too early or too late. You may be moving your men well, but you will find yourself losing money.’ And, according to Chris Bray, ‘The fat lady is already singing before many doubles are offered.’ If you miss that crucial moment when you can offer a double, the point at which your opponent might accept it, on the next turn it will be likely be too late. You will be too far ahead and if your opponent is a strong player, he will know not to accept the double.

Before every turn, as part of advanced backgammon strategy,  ask yourself the following two questions in order to determine whether you should offer a double. 1) Are you winning by enough? If your chances aren’t exactly 75%, are they at least close? And 2) Is there a chance that you will throw something so good that it will be too late to double on your next turn (because your opponent is sure to decline)?

If you’re a beginner, you probably couldn’t answer those questions without a great deal of laborious calculation, but as you play more, study the game and become more of a backgammon pro, you will get a better feel for where you stand. Understanding your chances of winning from a particular position is something you can only learn through study and experience; it takes time.

Let’s look at a couple of scenarios where you have the opportunity to offer a double.

In the scenario shown in the diagram below, you clearly have a chance of winning on this turn as long as you get a double (other than double 1) or you throw a 6 with one of your dice. There are 11 ways to throw a 6. And there are 5 doubles that will allow you to finish. So that’s 16 out of 36 throws will give you a win, which is around 44%. If you don’t win, Black has the same odds with his throw, so Black has a 44% times 44% chance of winning. Black thus has a 19% chance of winning at this stage in the game. If he doesn’t win on his next throw (after you fail to win), you will almost certainly win on your next throw (unless you are unlucky enough to throw a 2+1 twice). Thus your overall chance of winning from this position is almost 81% and Black’s is a little over 19%. When you ask yourself the question do you have a 75% chance of winning, the answer is yes, you do!

Figure 110: To double or not to double?

Now let’s look at the second question. If you wait one more throw, will you lose your market? Well, if you don’t win on this throw and Black doesn’t win on his, there is no way he would then accept a double on your third throw after this stage, because your chances of winning would have increased considerably, giving him much less than a 25% chance of winning.

So when you ask yourself the question: might I throw something so good that it will be too late to double on my next turn? The answer is also yes.

In summary, now is the right time to offer a double.

I assume that by now you are beginning to understand why so many of the top players are outstanding mathematicians.

In more complicated race positions Chris Bray suggests an 8 – 9 – 12 formula for working out whether you should offer a double. If you are 8% or more ahead on the pip count, (which equates to approximately a 75% chance of winning) you should offer a double but you should be 9% ahead if redoubling (where your opponent has already doubled in the game). And if you are no more than 12% behind and you are offered a double, you should take it.

#### The value of holding the live cube

As long as the game is still in play, the cube is ‘live’. The person holding the live cube can use it again to recube. If your opponent offers you a double, you also want to factor into your advanced backgammon strategy the value of possessing that live cube. You are then the only player who can offer the cube. You could use it to close out the game early or to double your potential points haul if your luck improves and you turn the game around. Kit Woolsey says, ‘In general, when the decision is otherwise very close, you should lean towards a race when you own the cube, but lean towards contact positions when your opponent owns the cube.’

#### Basic doubling opportunities

There are three main scenarios where you should definitely offer or accept a double. Each takes into account your position in the game and the timing of when you would lose your market.

1) When you have an advantage but are still in contact (i.e. you and your opponent have not passed each other completely).

Figure 112: Here it is Black’s throw. The players are very much still in contact and the pip count is close.

Although the pip count is close in the example shown, you are stuck at the back and Black has a good chance of freeing one or both of his backmarkers, and even of hitting you, which would lead you to have three checkers in his home board when you re-enter. If Black threw something really good on this turn, he could lose his market. So Black’s best option is to offer the double now. You still have a more than 25% chance of winning, and because it’s early in the game there are many possible scenarios for Black to get into trouble, so you should accept the double.

In the example shown below, you are in a strong position in the race and it is your throw; there really is little chance of you not getting past Black, so if you offer the double, Black would be best advised to decline. Whenever you are offered a double, always look at the position and ask yourself, ‘What’s in it for me? Do I have a good play here?’

Figure 113: If you offer the double, Black should decline.

2) When you are past each other and racing to finish.

In the game shown below, the outcome isn’t certain, but you are in the lead by 6 pips and it’s your throw. Remember that the value of a pair of dice is on average around 8.2, so if you add that to your lead of 6 in total that’s a lead of approximately 14 pips.

If you don’t offer the double now and then throw something good, Black will not accept your double on your next turn and you’ll miss the chance of 2 points. So you must offer now or you’ll lose your market. This is probably the latest stage in this particular game where you are likely to tempt Black into accepting your double. So this kind of position, a 10% lead plus your throw, is roughly where you are likely to tempt Black into accepting, and then you’ve a good chance of winning 2 points.

Figure 114: Double before you lose your market.

3) When you want to guarantee your win.

In the scenario shown in the example below, you can use the cube to avoid bad luck. Without the cube you would have to play the game to the bitter end. Let’s go back to statistics (for those of you who can stomach it!) If you have an 80% chance of winning, this means that if you were to play five identical games up to this stage, going forward you would lose 1 in 5. Say you offered a double in each of those games (at this stage). Because your opponent only has a 20% chance of winning, he should refuse every time, i.e. in 5 out of 5 of the games. Thus, offering the doubling cube increases your odds of winning to 100%. Statistically, whenever you are at the point of having an 80% chance of winning, you should offer the double to guarantee your win. Even if Black accepts, you still have that 80% chance of winning, only this time with double the points.

At a high level, very few games are played to the end.

In the example shown below, the game has been pretty close but you have recently thrown a good double and have found yourself a couple of pips ahead. But it is your turn, which is an advantage at this late stage. In fact, in this scenario, Black’s best option would be to decline and forfeit the game, but he may not choose to do so as it’s a close call.

Figure 115: Your odds of winning are always better if you offer the double when you have around an 80% chance of winning.

#### A more complex doubling opportunity

The example below shows a more involved situation and more advanced backgammon strategy for improving your game and becoming more of a backgammon pro.

You have a huge lead on pips, the cube is in the center and you roll a 5-1.  Of course you hit, but then you should continue with that checker to move to safety so that 6-2 does not hit back.  If you hit and play safe, then if your opponent rolls anything but 6-2 it will be a double and a pass, so why take the chance of getting hit?  Now, if the cube had already been turned, you would leave the checker on the 8 point because making that point on your next roll would be important to make a 6-prime and fully contain your opponent’s backmarker.  There is a reason to risk getting hit with 6-2 because there is a reward on the other 34 out of 36 rolls where he doesn’t roll that number.

It’s impossible to learn all the potential positions you could find yourself in with the cube. At the end of the day it mostly boils down to experience and study. Get some help from books, articles and mentors to help you learn. The more you play and study doubling situations, the better you will become.

If only the backgammon books on doubling were nearly as exciting!

And remember, both your offering decisions and your accepting decisions must be correct. Even if both your checker play and your decision-making over accepting double offers are better than a world champion’s, if you regularly offer doubles at the wrong time (too early and/or too late), you will lose more often than not.

#### Using the score to determine whether or not to accept or offer a double

Before each doubling decision, you should also think about the overall score in terms of the whole match and develop your advanced backgammon strategy accordingly. The example below shows how your decision on whether or not to accept or offer a double changes depending on the overall match score.

This is a well-known position where we know the taker wins just under 25 percent of the time.  At some scores it is right to take a double or redouble when you have just under 25 percent—simply put, the risk you take to try to win is worth the reward you get if you do win.  But at some scores it is too big a risk.  For example, in this position you should pass an initial double at most scores in a 5 point match.

But what if you were losing 3-0?  Then it would be correct to accept the cube, immediately redouble (because your opponent only needs 2 points to win the match, why not push the cube to 4?) and hope to roll doubles.  You have a better chance that way than if you pass the double and go down 4-1 in the match.  Remember, when the score goes to 4-1 the Crawford Rule comes into play and you can’t double for one game, so at that point you would have to win 3 games in a row (or win the first game, double the next game and win a gammon).

Often in positions like this, which we call “end game” positions, you already have the cube and are considering a redouble.  Without going into great detail here, at some scores it is correct to redouble this position and it would be a pass, and at other scores it is a take and not even correct to redouble.  Again, it’s all about risk and reward…comparing your chances of winning against just passing the cube compared with where you stand in the match if you take and win or if you take and lose.  This example helps to illustrate how much more complicated match play is compared to money play.

Of course in money games there is no upper limit, you just play games for whatever the stakes are until someone folds, so the score doesn’t really affect your doubling decisions.

To work out if your timing is right and decide on your advanced backgammon strategy, you may find it useful to put yourself in your opponent’s shoes. How would you react to the offering of a double if you were in your opponent’s position? If you’re indecisive, if you think you’d be on the cusp of accepting or not accepting, then this is the perfect time to double. If means you’ve got the timing just right. If, from your opponent’s perspective you’d definitely take your double, you’ve offered too soon. If you’d definitely decline the offer, you’ve offered too late.

In their book, Oswald Jacoby and John Crawford say, ‘In all gambling games there is a loser’s syndrome which keeps people continuing on when they’re behind and going further into the hole. The doubling feature in backgammon makes this syndrome far more dangerous.

Remember, doubling too early is one of the worst mistakes you can make in backgammon. If the game turns against you (as can often happen), your opponent has all the power because he controls the cube and can thus double you back at his whim, making it a 4-point match. If you lose a gammon as well, that’s 8 points against you. When you offer the cube too early, it is usually an easy decision for your opponent to accept it.

#### The gammon effect

You should always be aware of your chances, and your opponent’s chances, of winning a gammon. If, for example, you accept a double when you are only two points away from winning the match, the value of a winning a gammon to you becomes zero; you don’t need to win a gammon to win the match because you will be getting 2 points (and thus winning the match) if you win this game.

But say you are on 3-1 in a game to 5 points. Although you only need 2 points to win the match, your opponent can win the match with 4 points. If your opponent offers you the cube and you accept, but then he goes on to win a gammon, you will lose the entire match. The price of losing a gammon in this scenario is very high. In fact, in this situation, the value of a gammon (to your opponent) is equal to the value of winning a game. If you win the game you win the match; if you lose a gammon you lose the match. At this score, 3-1, consider your advanced backgammon strategy carefully. You have to be much more careful about accepting a cube if you are at risk of losing a gammon.

It may seem a daunting task to try to estimate your odds of winning and losing a gammon, but with some practice and the help of a computer programme (that will automatically tell you your odds until you get used to recognizing certain patterns), you’ll soon get the hang of it. There are some obvious general patterns. If you have more backmarkers, you get gammoned more; if you have an anchor, you get gammoned less. If you have a high anchor, you get gammoned even less. Skilled players can often do these estimates within a percentage point or two for most positions.

In the example shown below, you have a clear lead. But you also have a good chance of getting a gammon because Black has three backmarkers in your home board.

Figure 146: You have a good chance of getting a gammon.

Even though Black still has around a 30% chance of winning, now is a good time for you to offer the double before it’s too late (because Black will soon be too worried about losing a gammon to accept).

#### A free drop

After a Crawford game (where no doubles are allowed), if the match hasn’t ended, at least one player remains within one point of winning. In this scenario, for the losing player there is no downside to offering a double as soon as possible. In fact, it’s silly not to turn the cube to 2 when your opponent only needs 1 point because this is the only way you can leapfrog him.

But is it correct always to accept this double at the start of the game? Would a backgammon pro accept? Not if your opponent needs an even number of points to win.

If the match is to seven points and you are winning 6-5, then the match will be decided either on this game if you accept the double or on the next game if you refuse it. If your opponent has a slight lead (for example, he has started with an excellent throw like a 3+1) then at this point he has more than a 50% chance of winning this game and the match (if you accept the double and continue playing). In this scenario, you would be better off refusing a double, dropping the game, letting the score go to 6-6 and taking your chances in the next game where no one will need to offer the doubling cube anyway. This is called a  ‘free drop’.

But say the score was 6-4 in a match to seven points. Now you must accept the double, even if your opponent is leading in the current game. He would still have to win two games (unless he wins a gammon, in which case it’s all over for you). If you drop this game, on the next game your opponent offers you the double on his first roll and only needs to win one game to win the match.

Many people get confused and either don’t use their free drop when they should or drop when they think they have a free drop and don’t. Making either of these mistakes will cost you winning chances, so really familiarize yourself with this tactic of advanced backgammon strategy.

#### Doubling right back (‘The Beaver’)

If you are offered a double that you are very happy to accept because you feel your chances of winning are particularly high, you can actually double again immediately and still retain control of the cube. When you accept the double, you say, ‘I beaver,’ and turn the cube over again. You would only do this if your opponent has made a huge mistake (from his standpoint in the game) in offering you the double and you are absolutely sure that the odds are very much in your favour. The key point to remember here is that the cube stays on your side of the board. This is a highly unusual move and most tournaments don’t allow beavers. The tactic is usually only used in matches for money.