This article originally appeared in the September 1999 issue of GammOnLine.
Thank you to Kit Woolsey for his kind permission to reproduce it here.

Doubling Theory

By Kit Woolsey
This is going to be a somewhat mathematical article, which many readers may find difficult to follow. I'll try to keep it as simple as possible. Although it is largely theoretically oriented, I believe that some understanding of the concepts involved will be or practical value for making doubling decisions at the table.

The question we are going to analyze is as follows: How can we determine if it is theoretically correct to double. We are going to forget about all the psychological reasons why it might be right to double, and assume our opponent is infallible and will always make the correct decision. We will also assume that we have access to the knowledge of our equity in the position, and in the positions which will follow after the next exchange of rolls. In other words, we will pretend we are bots.

Suppose we are considering turning the cube in a position where our opponent has a take. How can we decide whether or not it is correct to double? What we must do is look down the road, and examine all possible resulting positions after the next exchange of rolls (we roll, he rolls). Since there are 36 possible dice rolls (yes, I know that 2-1 and 1-2 are the same, but for accounting purposes it is easier to think of them as being different), there are 1296 positions to examine (many of which are duplicated or quadruplicated, of course). For each of these positions, there are three possibilities:

1) It is double and pass. If this is the case we wish we had doubled the previous roll (this is losing our market).
2) It is double and take. If this is the case it didn't matter whether or not we doubled last turn.
3) It is no double. If this is the case we wish we had not doubled the previous roll.

If we already own the cube or if we are playing in a tournament there is a fourth possibility -- too good to double. For the purpose of this discussion we will assume we are playing for money with the cube in the center, so we will not have the option of playing for an undoubled gammon.

In case 1), how much do we cost by not doubling? If we didn't double, our equity is 1.00 (since we now double and he passes). If we did double, our equity is 2 (the value of the cube) X the equity of the position with our opponent owning the cube. For example, if the equity (on a 1-cube) with the cube on the other side is .600, then the cost of not doubling would be 1.200 - 1.00 = .200

In case 3), how much do we cost by doubling? We must compare our equity in position with the cube in the center against twice the equity with our opponent owning the cube. For example, suppose the equity with the cube in the center is .800, but with the opponent owning the cube it is .300. The cost of doubling is .800 - 2 X .300 = .200. Of course this must be a positive number. If it were negative that would mean the we were in case 2), double and take.





money game


Should Blue double? This looks like it might be a fairly close decision. The calculations shouldn't be too difficult, since a lot of Blue's rolls can be grouped together.

Group 1: 6-6, 5-5 (2 numbers)
Blue wins outright. Clearly Blue wishes he had doubled, and the cost of not doubling is a full point.

Group 2: 4-4, 3-3, 6-5, 6-4, 6-3, 5-4, 5-3 (12 numbers)
Here everything is determined by whether or not White rolls doubles. 5/6 of the time Blue will gain a point by doubling, but 1/6 of the time he will cost a point by doubling. This averages out to 2/3 of a point, or .67.

Group 3: 6-2, 5-2 (4 numbers)
If Blue doesn't double he can claim with the cube next turn. Since White will roll doubles 1/6 of the time, Blue's equity when he doesn't double is .67. If Blue does double and White doesn't roll doubles, Blue still might roll 2-1. Thus, Blue's equity after doubling is slightly less then 2 X .67. In fact, it comes to about 1.15. So, Blue gains 1.15 - .67 = .48 by doubling.

Group 4: 6-1, 5-1, 3-2 (6 numbers)
These rolls all lead to positions where Blue will have a double and White will have a take if White fails to roll doubles. Consequently Blue neither gains or loses by doubling if White doesn't roll doubles. The 1/6 of the time White does roll doubles, however, Blue costs himself a full point. Consequently on average Blue loses 1/6 or .17 by doubling.

Group 5: 3-1, 1-1 (3 numbers)
This leads to a known but unusual situation. Even though Blue gets off 17/36 of the time next turn, White is still the favorite because he might roll doubles first. In fact, it turns out that it is barely correct for White to turn the cube, even though Blue will be sending it back if White doesn't roll doubles. Overall (and you'll have to trust me on this or work out the math yourself), White's advantage on a 1-cube is about .15. It is twice that if Blue has doubled, so doubling costs .15.

Group 6: 2-1 (2 numbers)
White will double (or redouble) of course, and Blue has a take. Blue's equity comes out to -.35 on a 1-cube, so of course it is -.70 on a 2-cube. Thus doubling costs .35.

Group 7: 2-2, 4-3, 4-2, 4-1 (7 numbers)
These are the horror rolls. White has a double (or redouble) and Blue has a pass. In these cases, Blue costs himself a full point by doubling.

As you can see, there are 18 rolls where Blue is happy he doubled, and 18 rolls he is unhappy he doubled. Of course that by itself doesn't solve the problem. It is the magnitude of the gains and losses as well as the number of gains and losses which determine whether or not it is correct to double. Let's add them up:

Gains from doubling: 2 X 1.00 + 12 X .67 + 4 X .48 = 11.96
Losses from doubling: 6 X .17 + 3 X .15 + 2 X .35 + 7 X 1.00 = 9.17

Since the amount of gain is greater than the amount of loss, it is correct to double.

The above analysis is all fine and good in end-game positions where we can calculate every possibility exactly. What about middle-game positions? Now things get tricky. The problems come from determining the value of cube ownership.

First of all, if Blue doubles what is White's take point (i.e. what equity does White need in the position to justify a take). If White weren't allowed to recube, the answer would be easy. White would pass if Blue's cubeless equity were greater than .50, since 2 X .50 = 1.00 which is what Blue would win if White passed the double. In real life, White does have access to the cube. This means that White won't have to play the game to conclusion in order to win. He just needs to reach a point where he can redouble and Blue will have to pass. Blue, on the other hand, must play to conclusion. Consequently, White will win some games he would have lost if he were forced to play to conclusion.

The above analysis indicates that White can take a double even if Blue's equity is somewhat above .500. But how much higher can White go? The answer to this is vague, and depends on the type of position. For example:





money game


White is so far behind in the race that he pretty much needs to hit a shot in order to win. If White does hit that shot, his cubeless equity will suddenly shoot to nearly +1.00, since his board is so good. This means that White will get very little value out of his cube ownership, since he will not be able to redouble until the win is a near certainty. Therefore, White's take point for this position can't be very much above .500.





money game


Here White is playing a deep back game, with plenty of shot-hitting potential. If and when White does hit a shot, he will not be able to claim immediately with the cube. What happens is that White slowly puts his prime together, and eventually has a checker trapped behind a partial prime. This slow process will usually lead to a very efficient cube for White, where Blue has a close pass/take decision. Consequently in this sort of position White's take point is quite high.

If we forget about gammons and assume that White will always have a perfectly efficient recube, it is easy to see that White can take if he wins the game 20% of the time (or cubeless equity of .600 for Blue). This is the so-called "continuous model". The idea is that White will only need to reach 80% winning probability in order to "win", and if he starts from a position with 20% winning chances he will get to that 80% point 25% of the time (he has 60% to go while Blue has 20% to go). Of course, real life is not like that. In the holding game position White will suddenly shoot way above the 80% mark in one roll when things go his way. The back game position is closer to the continuous model.

It should also be noted that in positions with high gammon risk the take point is actually higher. Why is this? Let's look at a typical blitz which is a close take/pass decision.





money game


Snowie's estimates for this position (which might be wrong, of course), are:

Blue wins gammon or backgammon: 35.0%
Blue wins: 65.5%
White wins: 34.5%
White wins gammon or backgammon: 6.7%
Blue's cubeless equity: +.603

White gets gammoned a lot, but he also wins a lot of games. The key is that since White has such a high win percentage, his cube leverage is greater than usual because he will get to use the recube more than he would in a close pass/take decision where gammons weren't involved. In this sort of position, White may have a take even if Blue's cubeless equity is greater than .600. Scary, isn't it.

So how does all this help us? It would be too difficult to classify each position type when it comes up. We are trying to arrive at some kind of general formula which we can use. Overall, experience has indicated that for "normal" positions, the take point is about .570. This assumes that White will get some reasonable cube leverage, but it won't be perfectly efficient. Not a perfect way of doing things, but for this analysis we will assume that .570 is White's take point.

Let's think of some arbitrary position where Blue is considering doubling, doesn't double, and rolls a joker to lose his market. How much does this cost him? Suppose his cubeless equity after the joker sequence is .700. By not doubling (and then cashing), he wins 1 point. If he had doubled, it would appear that his equity is 2 X .700 = 1.4, so he would have cost himself .4 points. But it's not that simple. White owns the cube, and that has to be worth something. Blue's equity in the position is less than .700 because of the value of cube ownership to White. If that cube ownership drops Blue's equity to .600, then the cost of not doubling would be .2 rather than .4.

What we need is a magic formula to convert cubeless equity into actual equity when your opponent owns the cube. This is not easy to come by, and it may not be accurate. I have come up with one, but the derivation of it is complex and makes assumptions which won't necessarily fit the conditions of any given position. For those of you who really want to get into this sort of thing, please take a look at Doubling which is a paper I wrote several years ago in order to try to organize my thoughts on this difficult subject. I do not claim that it is mathematically sound, but the results seem reasonable and they appear to work. The formula which comes out of this paper is as follows:

X = 2 * ((((E + 1) / 2) -.14) / .86) - 1

Where E is the cubeless equity, and X is the new equity which takes into account the opponent's cube ownership. If you don't want to plough through the paper you'll just have to take my word for it that this formula is ok. We can make a few simple checks. If we make E = .57 (which we have decided is the usual minimum take point), we get that X = .50, which is exactly what it should be if the position is a borderline take/pass. Also if E = 1 (which means that the doubler has won the game) we get that X = 1. So, for my hypothetical example of a joker which gave you a cubeless equity of .70, plugging that into the formula for E gives x = .651, so the cost of not doubling would be 2 X .651 - 1 = .302.

We also have consider the equity for both sides with the cube in the center, since we have to look at the cost of doubling and being wrong when things go badly. For this, the magic formula I use is:

Y = 2 * ((((E + 1) / 2) - .165) / .67) - 1

Let's see how reasonable this looks as far as when we have an initial double. Suppose we plug E = .390 into these two formulas. We get:

X = .291
Y = .582

Remember that X is our equity if the opponent owns the cube, and Y is our equity if the cube remains in the center. Since Y is twice X, this gives us that it is borderline to double with an equity of .390.

Of course this is not always the case. The necessary equity for turning the cube is a function of the volatility of the position, as well as the actual equity of the position. If the position is very volatile, it is correct to double with equity considerably lower than .390, while if the position is not very volatile we need a much higher equity to justify doubling. Unfortunately it would be too difficult to attempt to measure the volatility of all of the 1296 positions after the next exchange (at least we don't yet have the tools to do this although the bots will be getting there soon), so we have to go with some reasonable number and this seems to work ok.

For example, suppose after we turn the cube we get a poor sequence which drops our cubeless equity to .200. Plugging this into the formulas, we get:

X = .07
Y = .30

Thus our cost of doubling and being wrong is .30 - 2 * .07 = .16.

Going back to our original problem of determining whether or not to double, the three cases can now be defined as follows:

Case 1) Our cubeless equity is greater than .570. We have lost our market. If we plug E (our cubeless equity) into the formula to get X (our equity with the opponent owning the cube), then the cost of failing to double is (2 * X) - 1.

Case 2) Our cubeless equity is between .390 and .570. According to our assessment, this is now double and take. We have neither gained or lost if we doubled the roll before.

Case 3) Our cubeless equity is less than .390. This time we have cost ourselves by doubling. If we plug E (our cubeless equity) into the formulas to get X and Y (the equities when our opponent owns the cube and the equity when the cube is in the center), the cost of doubling is X - 2 * Y.

Let's examine an actual position using this analysis. We will choose a position which has White on the bar against a closed board, so the analysis will be relatively simple.





money game


Blue is on roll. Snowie 3-ply puts Blue's equity at .468, and says it is double and take. This estimate seems a bit high to me, but we'll live with Snowie's estimates for now since they are as good as anything available. Now we will look at all of Blue's possible rolls. We don't have to worry about White's responses. Blue keeps his closed board unless he rolls 4-4 or 5-5, and if he rolls one of those he clearly won't have a double regardless of what White rolls, so we can just take the equity after Blue's roll.

It is obvious that if Blue rolls an ace he will lose his market, while if he rolls anything else he won't have a double. The question, then, is does Blue gain enough on those 11 aces to make up for the loss on the 25 non-aces to justify doubling. Below are Blue's possible rolls, the equity after the play according to Snowie, and the gain from doubling (if an ace) or the loss from doubling (if not an ace).

Let X = equity if opponent owns cube
Let Y = equity if cube in center
These are calculated according to the previously discussed magic formulas.

Roll            Cubeless equity           X                     Gain (2*X - 1)

1-1             .930                    .918                    .837
1-2             .948                    .939                    .879
1-3             .922                    .909                    .817
1-4             .896                    .879                    .758
1-5             .870                    .849                    .677
1-6            1.498                   1.579                   2.158

Roll            Cubeless equity           X       Y             Cost (Y - 2*X)

2-2             .246                    .123    .367            .121
2-3             .320                    .209    .478            .059
2-4             .336                    .228    .501            .046
2-5             .318                    .207    .474            .061
2-6             .246                    .123    .367            .121
3-3            -.044                   -.214   -.066            .362
3-4             .318                    .207    .474            .061
3-5             .246                    .123    .367            .121
3-6             .176                    .041    .263            .179
4-4            -.201                   -.397   -.300            .493
4-5             .176                    .041    .263            .179
4-6             .107                    .038    .160            .236
5-5            -.317                   -.531   -.473            .590
5-6             .082                   -.067    .122            .257
6-6             .336                    .228    .501            .046

As one might expect, the cost of not doubling and rolling an ace is huge, while the cost of doubling and not rolling an ace is usually just moderate. Adding up the costs with the 11 aces and the 25 non-aces, we get:

Total gain from doubling: 11.59
Total cost from doubling: 4.25

This figures indicate that Blue has a very clear double if we accept Snowie's estimates. This is not a surprising result since Snowie estimates the original position as .468 equity and the volatility is clearly very high, so if Blue's equity really is that high it must be a good double. My guess is that Snowie is overestimating Blue's chances, but it still looks like a good double.

The above position is an extreme illustration of a very common theme. When you have a solid position which has a few knockout sequences and you are likely to retain an advantage even in the bad scenarios, it is probably correct to turn the cube. It is important that the good sequences really be killers. If you just lose your market by a small amount on your good sequences, there isn't any reason to double. It is the danger of losing your market by a huge amount a significant portion of the time (such as in this example when Blue rolls an ace) that makes it a good cube.

For my last example, I will analyze a position from the online match where the readers had an interesting decision about whether or not to double. The vote was 28 to 27 in favor of waiting. Let's see how the readers did.





money game


Snowie's 3-ply estimate has Blue's equity at .353, and Snowie says no double. A long cubeful rollout pushed the equity up to .385, and Snowie still says no double but very close.

I think this is a good position to study. It is very typical of a potential cube. Blue has a clear advantage and a few crushers, but White also has some chance to take over the advantage in one roll. On most sequences the equity won't swing too much. Also this is a fairly straightforward position, and Snowie is likely to evaluate the resulting 1296 positions accurately.

Here is a breakdown of the 1296 possible exchanges:

Case 1) Double and pass (Greater than .570): 369
Case 2) Double and take (Between .390 and .570): 330
Case 3) No double (less than .390): 597

Of course these numbers don't necessarily tell the story, as we saw in the previous example. It is the magnitude of the swing as well as the swing which makes the difference. Plugging all the results into the equations, we came out with:

Total gain from doubling: 133.90
Total loss from doubling: 144.76

Since this is over 1296 rolls the difference is very small, less than .01. However waiting turned out to be barely correct, as both Snowie and the majority of the readers thought.

It would be too laborious to list all 1296 exchanges and their equities, but below is a frequency chart which gives us an idea of what is happening:

Range                Frequency

-.65 to -.60            8
-.60 to -.55            0
-.55 to -.50           24
-.50 to -.45           12
-.45 to -.40            4
-.40 to -.35           18
-.35 to -.30            0
-.30 to -.25            9
-.25 to -.20           16
-.20 to -.15           34
-.15 to -.10           38
-.10 to -.05           14
-.05 to 0              24
0 to +.05               8
+.05 to +.10           44
+.10 to +.15           31
+.15 to +.20           13
+.20 to +.25           16
+.25 to +.30           56
+.30 to +.35    `     136
+.35 to +.40           98
+.40 to +.45          139
+.45 to +.50           83
+.50 to +.55           86
+.55 to +.60          103
+.60 to +.65           39
+.65 to +.70           59
+.70 to +.75           76
+.75 to +.80           33
+.80 to +.85           41
+.85 to +.90            5
+.90 to +.95            6
+.95 to +1.00           0
+1.00 to +1.05          0
+1.05 to +1.10          0
+1.10 to +1.15         25
As one would expect 6-6 is the big roll for Blue. The 25 entries in the 1.10 to 1.15 range are the sequences where Blue rolls 6-6 and White flunks, and the next highest 11 sequences are where Blue rolls 6-6 and White enters. If it weren't for the possibility of rolling 6-6, Blue wouldn't have close to a double.

The rest of the quite favorable sequences for Blue (where he loses his market by a good amount) are 3-3 played 13/7(2), 6-1 played 13/7, 8/7 and not being hit, and various sixes where Blue hits loose and White flunks (particularly rolls such as 6-2, where Blue breaks the eight point for the extra builder). The very favorable sequences for White are when he isn't hit and either rolls 5-5 or hits a fly shot in the outfield. When Blue hits loose and White hits back but doesn't escape, that is good for White making the game almost even. Also if White escapes to safety that is quite good for White, and if he escapes to where Blue has just a single shot Blue is only a small favorite. Other sequences don't involve much of a swing. Many of them are in the double-take range, and when they are out of the range it is only by a little bit so they don't affect the figures very much. Always remember it is the sequences which involve huge market loss or which create a big turnaround that make the difference.

So, what have we learned from all this. Obviously any attempt to calculate all 1296 possibilities in actual play is impossible. Only the bots can do that. However we can be on the lookout for whether or not there are many sequences which will cause us to lose our market by a lot, and compare them with the sequences which make us wish we hadn't doubled. If the size of the market loss from the good sequences tends to be bigger than the downside from the bad sequences, it may well be correct to send a speculative cube over. On the other hand if the potential market loss isn't great, it is probably correct to wait even if there are a lot of sequences which lose the market by a little bit, unless the position is already so strong that your opponent is getting close to a pass. Thinking along these lines will sharpen your doubling skills.

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