6 6

White 2 7 point match Blue 5

A straightforward position. White's winning chances are 21.2%. Thiscan be calculated as follows:

In order for White to have a chance, he will need for Blue to notroll a set of doubles on one of his next two rolls. The chance that Bluewill not roll a set on his first roll is 5/6, and the chance that he willnot roll a set on his second roll is also 5/6. Therefore, the chance thathe will not roll a set on either of his next two rolls is 5/6 X 5/6 = 25/36.

Given that Blue doesn't roll a set, White will still need to roll a seton one of his next two rolls in order to win. By the same calculations,the chance that White fails to roll a set on one of his next two rollsis 25/36. Therefore, the chance that he rolls the needed set is 11/36.

For White to win, he needs the parlay of Blue not rolling a set and Whiterolling a set. From what we have calculated, the probabality of this is(25/36) X (11/36) = 275/1296 = 21.2%.

It is clear that White has a big money pass. However, it might be adifferent story at the match score. Keep in mind that White will beredoubling to 4 immediately, so when he wins he will win 4 points.For my calculations, I will be using my match equity table. Click HERE to see the table.

White passes: Behind 6-2 Crawford (1 away, 5 away), 15% equity.
White takes and wins: Ahead 6-5 Crawford (1 away, 2 away), 70% equity.
White takes and loses: Loses match, 0% equity.

Therefore, White is risking 15% in order to gain 55%. His takepoint is15/70 = 21.4%. White has a pass, but it is a very slim pass instead ofthe monster pass it would be for money. If Blue's position were slightlyweaker (say 3 checkers on each of the ace and two points so 1-1 isn'teffective for Blue), that would be sufficient to give White a take.

The odds can change considerably when the last roll of the game (or theequivilant) is reached. The key is that, while the trailer will makethe automatic redouble, the leader is not required to take! The cubeis frozen, and will stay there. For example:

2 6

White 2 7 point match Blue 5

A quick count shows that Blue has 23 rolls which get him off and 13 whichdon't. Obviously White has a huge money take, with about 36.1% winning chances. Since White is behind in thematch, it would appear that White would have an even easier take at the matchscore.

Appearances can be deceiving. The key is that the cube is frozen on 2 forall practical intents and purposes. White will never get to make ameaningful redouble to 4, since Blue has an automatic pass which doesn't costany equity. If we look at the odds on White's take:

White passes: Behind 6-2 Crawford (1 away, 5 away), 15% equity.
White takes and wins: Behind 5-4, 40% equity.
White takes and loses: Loses match, 0% equity.

White is risking 15% equity in order to gain 25%. He is getting muchworse than 2 to 1 odds, compared to the 3 to 1 odds on a money take.In fact, his takepoint under these circumstances is 37.5%. Since he onlyhas 36.1% winning chances, the surprising conclusion is that his correctaction is to pass the double.

This is quite a remarkable situation. At identical scores, we saw in theprevious position that Blue had a borderline take/pass with only around21% winning chances, while here he has a proper pass with over 36% winningchances. It all depended upon whether or not the cube was frozen afterthe double.

Since such unusual skewing is involved, one would think that there mightbe positions where it is proper to double as a clear Underdog even whenahead in the match. That is correct. Consider the following:

2 8

White 2 7 point match Blue 5

Blue gets off with 14 out of his 36 rolls. This makes him quite an underdog.Can it possibly be correct for him to double? Let's see.

Blue doesn't double and wins: Ahead 6-2 Crawford (1 away, 5 away), 85% equity.
Blue doubles and wins: Wins match, 100% equity.
Blue doesn't double and loses: Ahead 5-3, 68% equity.
Blue doubles and loses: Ahead 5-4, 60% equity.

Blue is risking 8% (68% - 60%) in order to gain 15% (100% - 85%). He isgetting nearly 2 to 1 odds on his double. He only needs to win 34.8% ofthe time to justify doubling. With 14 out of 36 rolls to get off, Blue'sactual winning chances are 38.9%. Thus, doubling is very clear.

The concept of the frozen cube exists for middle-game positions also.For example, suppose you are behind 8 away, 3 away. Your opponent willbe cautious about doubling, of course. However, he still may have someeffective cubes, particularly in straight races. He can use the full 2points and some of the 4 points (if you should take and recube), so witha decent advantage he may well have a double.

Now, let's suppose the same score, but that you have already doubled him to2 and he has taken. What will his cube strategy be? If he ever redoublesyou only need 6% winning chances to take (since you can redouble to 8 forthe match), because that would be your match equity behind 8 away, 1 away.Your opponent will never redouble if there is any contact, and hewon't double in a straight race until he is virtually gin. Thus, by yourinitial double you didn't really give him the cube. In fact, you took it awayfrom him, since there would be positions where he would double from 1 to 2but not from 2 to 4. At this match score, you don't need much of an excuseto double. Any advantage combined with a couple of market losers shouldbe sufficient. Giving him the cube doesn't cost you anything, because hecan't make use of it the way he could if the cube were in the center.

Let's see how this concept might be used in a real live position.

69 84

White 8 11 point match Blue 3

Blue has 11 aces which virtually claim the game, but if he misses he willbe well behind in the race. Assuming Blue rolls about 9 pips with his miss,that will put him six pips behind. The crossover situation make Blue'srace even worse. It looks likely that if Blue doesn't hit White willhave a quite efficient cube (even at the match score), and Blue probablywill have a take. Let's see about what Blue's takepoint is:

Assuming Blue does take, he will be very quick on the trigger with therecube. Let's suppose that Blue makes sure he never loses his market(which is close to his proper strategy anyway), so whenever Blue winsthe game he will win 4 points. Of course this aggressive strategy willcost Blue some 4-point losses also. Let's estimate that 1/3 of Blue'slosses will be with the cube on 4 due to this agressive redoubling.

Blue passes: Behind 9-3 (8 away, 2 away), 12% equity
Blue takes and wins: Behind 8-7 (4 away, 3 away), 41% equity (remember, Bluealways wins 4 points when he wins the game).
Blue takes and loses: By our assumptions he will lose 2 points andbe behind 10-3 (8 away, 1 away) for 6% equity 2/3 of the time, and hewill lose 4 points and lose the match for 0% equity 1/3 of the time. Thiscomes out to a weighted average equity of 4%.

Therefore, Blue is risking 8% (12% - 4%) in order to gain 29% (41% - 29%). Thiscomes to 21.6% winning chances Blue needs. I would estimate that Bluewill be slightly better than this on most of his missing rolls, so thelikely cube action if Blue doesn't double and misses the shot is double/take.

Suppose Blue turns the cube! If he hits, obviously he is a happy camper.What if he misses? As we have seen, White won't be close to a recube, sinceBlue's takepoint is around 6%. White will virtually never redouble.

Thus, for all practical intents and purposes, if Blue misses the shot the cube will wind up on 2and the position played out whether or not Blue doubles. Given that,Blue might as well double. The double will gain tremendously if he hitsthe shot, while it costs very little if he misses. By turning the cubeBlue freezes it, but if he doesn't turn it White has full access.

How far can this concept be carried. Quite far if the circumstances areright. Jake Jacobs told me about the following situation. First, a fewpreliminary cube decisions:

2 6

White 3 7 point match Blue 4

Offhand, double and take appear to be the proper cube actions. Blue has23 rolls which get off. Let's check it out.

From White's point of view if Blue doubles:

White passes: Behind 5-3 (4 away, 2 away), 32% equity.
White takes and wins: Ahead 5-4 (3 away, 2 away), 60% equity.
White takes and loses: Behind 6-3 Crawford (4 away, 1 away), 17% equity.

White is risking 15% in order to gain 28%, which comes to a drop pointof 34.8%. He wins 13/36 of the time, which is 36.4%. Therefore White hasa take, although it is surprisingly close.

From Blue's point of view (although we can be pretty sure of the answerfrom the closeness of the take).

Blue doesn't double and wins: Ahead 5-3 (4 away, 2 away), 68% equity.
Blue doubles and wins: Ahead 6-3 Crawford (4 away, 1 away), 83% equity.
Blue doesn't double and loses: Tied at 4-4, 50% equity.
Blue doubles and loses: Behind 5-4 (3 away, 2 away), 40% equity.

Blue is risking 10% to gain 15%. Since he is a favorite in the positionand there is no recube involved, his double is very clear.

Now, let's keep the same score and position, but change the cube situation:

2 6

White 3 7 point match Blue 4

We can see at a glance that if Blue doubles, White has a trivial take.If White passes he is behind 6-3 (1 away, 4 away) with 17% equity, whileif he takes it is for the match. White's winning chances are 36.4%, sothe pass/take decision isn't close. But what about the double?

Blue doesn't double and wins: Ahead 6-3 Crawford (1 away, 4 away), 83% equity.
Blue doubles and wins: Wins match, 100% equity.
Blue doesn't double and loses: Behind 5-4 (2 away, 3 away), 40% equity.
Blue doubles and loses: Loses match, 0% equity.

By doubling, Blue is risking 40% (40 - 0) in order to gain 17% (100 - 83),so he is getting worse than 2 to 1 odds on the double. Since Blue fails toget off more than 1/3 of the time, it is clear that Blue should not double.

With these two results in mind, we are now prepared to examine Jake's position.

6 5

White 4 7 point match Blue 3

Can it possibly be right for Blue to double? He needs 2-2 or better to getoff this roll, and if he doesn't roll them he is a sizable underdog. Blueis only behind one point in the match, so the match score doesn't seem toorelevant. If Blue were to double for money, this would be a monster beaver.

Let's look at the problem a little more objectively. Obviously if Blue doesroll 2-2 or better he will be happy he doubled. Suppose Blue doesn't roll 2-2or better. Then:

1) If Blue didn't double, White would then double and Blue would take. Whitewould be rolling for a 5-1 with the cube on 2.
2) If Blue did double, White would then not have a redouble. Again, Whitewould be rolling for a 5-1 with the cube on 2.

Therefore, the value of the cube going into White's roll (assuming Bluedoesn't roll 2-2 or better) will be the same regardless of whether or notBlue doubles now. The only difference will be who owns the 2-cube, but sinceWhite's next roll will be the last roll of the game cube ownership doesn'tmake any difference. The startling conclusion is that it is clear for Blueto turn the cube. He has everything to gain and nothing to lose. Doublingis a heads Blue wins and tails Blue breaks even proposition. By doublingBlue froze the cube for the following roll, which is all that mattered.

The concept of the frozen cube comes up more often than one might imaginein match play. It is important to look ahead and see what the futureramifications of doubling or not doubling are. Sometimes the conclusionscan be very unintuitive.