35 44

Leonid Riskin 4 15 point match Kit Woolsey 0

Interesting. At an even match score this wouldn't be close to a redouble. It might not seem as though a 4-0 lead would make that much difference in a 15 point match, since both sides have a lot of points to go. That would be true for an initial cube, and for some extent a redouble to 4. However, we are talking about a potential 8-cube, and that makes a lot of difference. First of all, if he takes his recube vig will be very small. My choice on a recube to 16 would be to pass and be behind 12-0 or to play for the match, so I would need very little to justify a take. Consequently, he would be able to redouble only if he were nearly gin. Secondly, I can use those eight points a lot more than he can. That would catapult me into the lead in the match. Yes, I can afford to be much friskier with my redoubles here than I could for money or at an even match score. Let's see how it looks from his point of view:

He passes: 4-4, 50% equity.
He takes and wins. Ahead 12-0 (3 away, 15 away), 96% equity.
He takes and loses. Behind 8-4 (7 away, 11 away), 30% equity.

He will be risking 20% in order to gain 46%, far worse than the 3 to 1 odds he would be getting at an even match score. His take point is about 30%. This means that he will be dropping much faster than normal, so I have to be redoubling much quicker than usual in order to avoid losing my market.

How does the actual position look. Let's suppose I roll something like 6-3, and he gets on with his first roll -- say he rolls 6-3. I will have 12 men left so I may get off in 6 rolls, but I might not -- I could roll too many aces along the way. Can he get off in five rolls? Yes, he can quite easily. His first roll brings him to his outer board, his second roll might be in and a man off, and then he would have only five checkers left. This means that if he enters at his first opportunity I probably won't even be the favorite. I'm certainly not close to the 70% mark. He has a trivial take. Can I lose my market? Of course I can if I roll 6-6, but what about on normal sequences. Suppose I roll that 6-3 and he flunks. Will I have lost my market? Not so clear. I will have those same 12 checkers left, but since he is on the bar I will have to worry about playing safe next turn, so I probably won't be able to take two checkers off. If he enters after I have taken only one checker off at that point, the race looks to be a photo from the previous analysis. This means that on the sequence 6-3 by me, flunk by him, it is far from clear that I will have lost my market. Since that is a very favorable sequence for me, it looks right to hold off doubling. I will probably get another chance later.

I rolled 4-3. Safety is not the prime consideration here. The race is the big thing. Every checker off counts. Also, blocking sixes is significant. I played 4/1, 4/0. This looked better than clearing the six point, even though it is not safer.

My opponent flunked. This left the following position:

35 37

Leonid Riskin 4 15 point match Kit Woolsey 0

Now what? I have survived his first attempt at entering and taken a man off, but otherwise my position isn't so great. If I roll my 6-3 I will be down to 11 checkers. It isn't hard to imagine him entering and zipping around and off in six rolls. In addition, I might be missing due to the gap on the four point. If he flunks next turn I will have a good advantage, but even then I may not have lost my market or at least not lost it by much. And we are talking about a pretty good roll, 6-3, for me. Other rolls such as 4-3 are much more awkward. Another point is opened, and I don't even get to take a checker off. Once again I thought it was right to hold off doubling for a roll, depsite the match score.

I rolled boxes! So much for not losing my market. I took four men off and reached for my pencil to enter the score.

And he rolled 5-5!! Oh my gosh! Now things looked like:

15 15

Leonid Riskin 4 15 point match Kit Woolsey 0

It sure is a good thing I took that checker off with the 4-3. Now I am down to 8 checkers rather than 9, and that could be very crucial. What is happening now? Is it a double? Is it a take?

It would be hard to imagine that it isn't a take. He has only six checkers left. That means that if he rolls a six or a five next roll and avoids rolling an ace, he will get off in three rolls. In addition, he might roll doubles along the way. Of course, so might I. However, it is clear that his winning chances must be well over 30%. He has a trivial take. But do I have a double?

I am certainly the favorite. There is a good chance that he will miss on his next turn. If he doesn't miss, he still has to survive the following two rolls without rolling an ace. How about market loss potential? He misses on all aces except 1-1. Tack on 3-2, 4-2, and 4-3, and and that comes to a total of 16 missing numbers. If he does miss we will be in a three-roll position, and he will have winning chances of around 21%. We have determined that his take point is 30%. That means that almost half the time I will lose my market by a country mile. It sure looks right to double.

One other consideration. Suppose we roll along with nobody rolling doubles but him not missing. He will come down to two checkers on his two point versus two checkers on my ace point. At that point, he would have an extremely efficient redouble for money. However, we are not playing for money, and the match score is very significant. Let's examine what his odds on redoubling to 16 look like:

He doesn't redouble and wins: Ahead 12-0 (3 away, 15 away), 96% equity.
He redoubles and wins: Wins match, 100% equity.
He doesn't redouble and loses: Behind 8-4 (7 away, 11 away), 30% equity.
He redoubles and loses: Loses match, 0% equity.

Therefore he would be risking 30% in order to gain a mere 4%. If this situation is reached, he won't be close to a redouble. I don't have to worry about that cube coming back going into that last roll at all. For money that would be a serious consideration when deciding whether or not to redouble, but here it is not even a factor.

I redoubled to 8. He took, of course. I rolled 5-4, taking two checkers off. Now it was up to him to roll a five or a six.

The good news was that he didn't roll a five or a six. The bad news (the very bad news) was that he rolled 2-2! Yikes! I hadn't thought of that possibility (not that it would have changed my action). He took four men off. I rolled another non-doubles. That left this position with him on roll:

7 5

Leonid Riskin 4 15 point match Kit Woolsey 0

Is it a double? Is it a take? Well, what are my winning chances? He needs to fail to get off, and he has 19 rolls which get him off. I then need to roll double twos or better. At a glance it is easy to see that my winning chances are less than 1 in 12. As we have already determined, he is laying about 30 to 4 odds if he redoubles. That makes his redouble to 16 very clear. As expected, the cube was soon flying across the table.

Now what? Do I have a take? This looks like it might be close. Time to fine tune the pencil for this one.

For me to win, first of all he has to miss. That will happen 17 times out of 36. Then I have to roll 2-2 or better. That's 5 out of 36. Multiplying these fractions together, we come up with 85/1296.

There is one more piece of vig. He might roll 2-1 twice in a row. Then I can win without rolling doubles. A single 2-1 comes 2 in 36, so he rolls it twice in a row 4 times in 1296. That will make a difference when I don't roll doubles, so we can tack on about another 3 1/2 times out of 1296. That comes to 88 1/2 out of 1296. How much is that?

Let's go for a ballpark figure. We know that 129 out of 1296 is about 10%, and 65 out of 1296 is about 5%. 88 1/2 is between these, roughly 2/3 of the way closer to 65, so it looks to be about 6 1/2%. For those who have a slide rule for a brain, they can compute the actual figure to be 6.83%. That is too tough for me -- a rough estimate is fine.

This is one position where we can calculate the winning chances excatly, yet it isn't clear what to do with the results. My match equity table gives me only 4% winning chances at 3 away, 15 away, so if I go by that alone it becomes an easy take. However, there is a definite bias in favor of the leader with distorted scores such as this. The match equity table is low in the gammon estimates, and it tends to undervalue cube leverage of the trailer. The true figure is probably around 5 or 6%. At least that's what it feels like to me -- I would think I am better than 4% to win behind 3 away, 15 away.

Another factor is the opponent factor (or the fish factor, as Jake Jacobs would call it from his book which discusses this topic of varying your cube strategy depending upon the strength of the opponent). If the opponent is clearly a weaker player, then my winning chances from behind 3 away, 15 away are way better than the match equity table would indicate, and I should pass. I had never played my current opponent before. However, from the two games I had played him it was clear that he was quite competent. I had little if any skill advantage. Of course, my ego had to make it that I had some advantage, which would push my take point a bit higher -- maybe to around 6 1/2 or 7%.

So, after all these calculations, match equity tables, weighing in the strength of the opponent, and anything else I could think of, the decision looked like a photo. Was there something else to go on?

There is one further factor. We had started play at 11:00 in the morning, and I had not had a chance to eat breakfast before game time. I was hungry! When all else fails, that had to be the deciding factor. If I took then win or lose the match would be over with, and I could get some breakfast. I took. He rolled 6-5, and that was that.

Incidentally, computer analysis appears to confirm the correctness of all of these cube decisions. However, when the cube gets to 8 or 16, that isn't good enough. You have to be right!