return to indexThe question we are going to analyze is as follows: How can we determine ifit is theoretically correct to double. We are going to forget about all thepsychological reasons why it might be right to double, and assume ouropponent is infallible and will always make the correct decision. We willalso assume that we have access to the knowledge of our equity in theposition, and in the positions which will follow after the next exchangeof rolls. In other words, we will pretend we are bots.
Suppose we are considering turning the cube in a position where our opponenthas 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 resultingpositions after the next exchange of rolls (we roll, he rolls). Since thereare 36 possible dice rolls (yes, I know that 2-1 and 1-2 are the same, butfor accounting purposes it is easier to think of them as being different),there are 1296 positions to examine (many of which are duplicated orquadruplicated, of course). For each of these positions, there are threepossibilities:
1) It is double and pass. If this is the case we wish we had doubled theprevious roll (this is losing our market).
2) It is double and take. If this is the case it didn't matter whether ornot we doubled last turn.
3) It is no double. If this is the case we wish we had not doubled theprevious roll.
If we already own the cube or if we are playing in a tournament there isa fourth possibility -- too good to double. For the purpose of thisdiscussion we will assume we are playing for money with the cube in thecenter, 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, ourequity 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 withour opponent owning the cube. For example, if the equity (on a 1-cube) withthe cube on the other side is .600, then the cost of not doubling would be1.200 - 1.00 = .200
In case 3), how much do we cost by doubling? We must compare our equity inposition with the cube in the center against twice the equity with ouropponent owning the cube. For example, suppose the equity with the cubein 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 bea positive number. If it were negative that would mean the we were in case 2),double and take.
| 4 11 | White money game Blue |
Should Blue double? This looks like it might be a fairly close decision. Thecalculations shouldn't be too difficult, since a lot of Blue's rolls canbe grouped together.
Group 1: 6-6, 5-5 (2 numbers)
Blue wins outright. Clearly Blue wishes he had doubled, and the costof 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. 5/6of the time Blue will gain a point by doubling, but 1/6 of the time hewill 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 Whitewill roll doubles 1/6 of the time, Blue's equity when he doesn't doubleis .67. If Blue does double and White doesn't roll doubles, Blue stillmight roll 2-1. Thus, Blue's equity after doubling is slightly less then2 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 Whitewill have a take if White fails to roll doubles. Consequently Blue neithergains or loses by doubling if White doesn't roll doubles. The 1/6 of thetime 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 off17/36 of the time next turn, White is still the favorite because he mightroll doubles first. In fact, it turns out that it is barely correct forWhite to turn the cube, even though Blue will be sending it back if Whitedoesn't roll doubles. Overall (and you'll have to trust me on this orwork out the math yourself), White's advantage on a 1-cube is about .15. Itis 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'sequity 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 hasa 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, and18 rolls he is unhappy he doubled. Of course that by itself doesn't solvethe problem. It is the magnitude of the gains and losses as well as thenumber of gains and losses which determine whether or not it is correctto 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 correctto double.
The above analysis is all fine and good in end-game positions where we cancalculate every possibility exactly. What about middle-game positions? Nowthings get tricky. The problems come from determining the value of cubeownership.
First of all, if Blue doubles what is White's take point (i.e. what equitydoes White need in the position to justify a take). If White weren't allowedto recube, the answer would be easy. White would pass if Blue's cubeless equity weregreater than .50, since 2 X .50 = 1.00 which is what Blue would win ifWhite 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 towin. He just needs to reach a point where he can redouble and Blue willhave 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 toconclusion.
The above analysis indicates that White can take a double even if Blue'sequity is somewhat above .500. But how much higher can White go? Theanswer to this is vague, and depends on the type of position. For example:
| 131 86 | White money game Blue |
White is so far behind in the race that he pretty much needs to hit a shotin order to win. If White does hit that shot, his cubeless equity willsuddenly shoot to nearly +1.00, since his board is so good. This means thatWhite will get very little value out of his cube ownership, since he will notbe able to redouble until the win is a near certainty. Therefore, White'stake point for this position can't be very much above .500.
| 190 72 | White money game Blue |
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 immediatelywith the cube. What happens is that White slowly puts his prime together,and eventually has a checker trapped behind a partial prime. This slowprocess will usually lead to a very efficient cube for White, where Bluehas a close pass/take decision. Consequently in this sort of positionWhite's take point is quite high.
If we forget about gammons and assume that White will always have a perfectlyefficient recube, it is easy to see that White can take if he wins the game20% of the time (or cubeless equity of .600 for Blue). This is the so-called "continuous model". The idea isthat 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 that80% point 25% of the time (he has 60% to go while Blue has 20% to go). Ofcourse, real life is not like that. In the holding game position Whitewill suddenly shoot way above the 80% mark in one roll when things go hisway. The back game position is closer to the continuous model.
It should also be noted that in positions with high gammon risk the takepoint is actually higher. Why is this? Let's look at a typical blitz whichis a close take/pass decision.
| 162 147 | White money game Blue |
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 thatsince White has such a high win percentage, his cube leverage is greater thanusual because he will get to use the recube more than he would in a closepass/take decision where gammons weren't involved. In this sort ofposition, White may have a take even if Blue's cubeless equity is greaterthan .600. Scary, isn't it.
So how does all this help us? It would be too difficult to classify eachposition type when it comes up. We are trying to arrive at some kind ofgeneral formula which we can use. Overall, experience has indicated thatfor "normal" positions, the take point is about .570. This assumes thatWhite will get some reasonable cube leverage, but it won't be perfectlyefficient. Not a perfect way of doing things, but for this analysis we willassume 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 thiscost 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 costhimself .4 points. But it's not that simple. White owns the cube, and thathas 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 cubeownership drops Blue's equity to .600, then the cost of not doublingwould be .2 rather than .4.
What we need is a magic formula to convert cubeless equity into actualequity when your opponent owns the cube. This is not easy to come by, andit may not be accurate. I have come up with one, but the derivation of itis complex and makes assumptions which won't necessarily fit the conditionsof any given position. For those of you who really want to get into thissort of thing, please take a look at Doublingwhich is a paper I wrote several years ago in order to try to organize mythoughts on this difficult subject. I do not claim that it is mathematicallysound, but the results seem reasonable and they appear to work. The formulawhich 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 intoaccount the opponent's cube ownership. If you don't want to plough throughthe paper you'll just have to take my word for it that this formula isok. We can make a few simple checks. If we make E = .57 (which we havedecided is the usual minimum take point), we get that X = .50, which isexactly what it should be if the position is a borderline take/pass. Alsoif 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 equityof .70, plugging that into the formula for E gives x = .651, so the costof not doubling would be 2 X .651 - 1 = .302.
We also have consider the equity for both sides with the cube in thecenter, since we have to look at the cost of doubling and being wrong whenthings 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 ourequity if the cube remains in the center. Since Y is twice X, this givesus that it is borderline to double with an equity of .390.
Of course this is not always the case. The necessary equity for turning thecube is a function of the volatility of the position, as well as the actualequity of the position. If the position is very volatile, it is correct todouble with equity considerably lower than .390, while if the position isnot very volatile we need a much higher equity to justify doubling. Unfortunatelyit would be too difficult to attempt to measure the volatility of all of the1296 positions after the next exchange (at least we don't yet have the toolsto do this although the bots will be getting there soon), so we have to gowith some reasonable number and this seems to work ok.
For example, suppose after we turn the cube we get a poor sequence whichdrops our cubeless equity to .200. Plugging this into the formulas, weget:
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 equitywith 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 ourassessment, this is now double and take. We have neither gained or lost ifwe doubled the roll before.
Case 3) Our cubeless equity is less than .390. This time we have costourselves by doubling. If we plug E (our cubeless equity) into the formulasto get X and Y (the equities when our opponent owns the cube and the equitywhen 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 positionwhich has White on the bar against a closed board, so the analysis will berelatively simple.
| 115 84 | White money game Blue |
Blue is on roll. Snowie 3-ply puts Blue's equity at .468, and says it is doubleand take. This estimate seems a bit high to me, but we'll live with Snowie'sestimates for now since they are as good as anything available. Now we willlook at all of Blue's possible rolls. We don't have to worry about White'sresponses. Blue keeps his closed board unless he rolls 4-4 or 5-5, and ifhe rolls one of those he clearly won't have a double regardless of whatWhite 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 herolls anything else he won't have a double. The question, then, is doesBlue gain enough on those 11 aces to make up for the loss on the 25 non-acesto justify doubling. Below are Blue's possible rolls, the equity afterthe play according to Snowie, and the gain from doubling (if an ace) orthe 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 .046As 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'sestimates. This is not a surprising result since Snowie estimates theoriginal 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 isthat Snowie is overestimating Blue's chances, but it still looks like agood double.
The above position is an extreme illustration of a very common theme. Whenyou have a solid position which has a few knockout sequences and you arelikely to retain an advantage even in the bad scenarios, it is probablycorrect to turn the cube. It is important that the good sequences reallybe killers. If you just lose your market by a small amount on your goodsequences, there isn't any reason to double. It is the danger of losingyour market by a huge amount a significant portion of the time (such as inthis 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 wherethe 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.
| 121 128 | White money game Blue |
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 saysno double but very close.
I think this is a good position to study. It is very typical of a potentialcube. Blue has a clear advantage and a few crushers, but White also hassome chance to take over the advantage in one roll. On most sequences theequity 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 theprevious example. It is the magnitude of the swing as well as the swing whichmakes the difference. Plugging all the results into the equations, we cameout 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 themajority of the readers thought.
It would be too laborious to list all 1296 exchanges and their equities, butbelow 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
240 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 the1.10 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 closeto a double.
The rest of the quite favorable sequences for Blue (where he loses his marketby a good amount) are 3-3 played 13/7(2), 6-1 played 13/7, 8/7 and not beinghit, and various sixes where Blue hits loose and White flunks (particularlyrolls 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 eitherrolls 5-5 or hits a fly shot in the outfield. When Blue hits loose andWhite hits back but doesn't escape, that is good for White making thegame almost even. Also if White escapes to safety that is quite good forWhite, and if he escapes to where Blue has just a Single Shot Blue is onlya small favorite. Other sequences don't involve much of a swing. Many ofthem are in the double-take range, and when they are out of the range it isonly by a little bit so they don't affect the figures very much. Alwaysremember it is the sequences which involve huge market loss or which createa big turnaround that make the difference.
So, what have we learned from all this. Obviously any attempt to calculateall 1296 possibilities in actual play is impossible. Only the bots can dothat. However we can be on the lookout for whether or not there are manysequences which will cause us to lose our market by a lot, and compare themwith the sequences which make us wish we hadn't doubled. If the size of themarket loss from the good sequences tends to be bigger than the downsidefrom the bad sequences, it may well be correct to send a speculative cubeover. On the other hand if the potential market loss isn't great, it isprobably correct to wait even if there are a lot of sequences which losethe market by a little bit, unless the position is already so strong thatyour opponent is getting close to a pass. Thinking along these lineswill sharpen your doubling skills.