Black is on roll leading 8-5 to 11. To decide the proper cube actions – Should Black redouble? Should White take if Black does? – it would be handy to know within 1% Black’s cubeless winning chances in this race. Paul has a method for determining just that.
Paul’s formula is: P = 50% + (900 + R + 400L)/(R + 7L + 25)% where P is the winning percentage, R the leader’s race length, and L the difference in adjusted pips. In the position above, you will be happy to learn, no adjustments are needed, so we can just crank out the answer without breathing hard. I am sure most of you finished before I typed the last sentence, but if not, let’s step through it. Black leads 95-119, so R is 95, and L 24. So 900 + 95 + 400*24 = 10,595, and that must be divided by 95 + 168 + 25, or 288. It comes to about 37%, which, added to 50% brings us up to a cubeless winning probability of 87%. A rollout finds we have come within 1%, as the actual CPW is 88%, and, Paul informs us, confirms that a redouble is marginally correct.
It isn’t clear that Paul would have actually redoubled at the table. A chart much later in the book for “gammonless redoubles” puts the doubling window at 87-89%. It is usually wrong to double at the bottom of the doubling window except in last shake situations – not the case here. So, the formula’s triumph is not unequivocal, but to be fair, missing your redouble in this situation is not critical, and the formula does a good job of confirming the need to take, which is. Another bit of rain on Paul’s parade is that the problem comes directly beneath a table of race advantages keyed to length and lead, and should one use the table instead of the formula for this particular race one would put Black’s winning chances at 91%, a 3% error.
Those quibbles aside, the formula seems quite effective. The question is: who will use it? I’m afraid it might scare the socks off of most beginners, and old timers are often set in their ways. Still, for the advancing intermediate, or any others with ambition, a flair for math, and a good memory (egad, he wrote this book for me!), it is worth their while learning it. I am reminded of a conversation I overheard years ago. A player was talking with Bill Davis one Tuesday night, and said: “I realize I’m an intermediate, and in most situations I can’t compete with the experts. But I think there is one area I can outdo them. When I count a race, I use the Magriel, Ward, Robertie, Thorp, and Kleinman counts, and if they don’t all agree I go with the majority.” His math was questionable, but there was always a chance his opponents might expire while he was counting.
What else will you find here? A lot! The book examines the concept of equity, of how it is calculated, and how money and match equities differ – not just whether a decision changes in a match, but why it does. Among other things, Paul offers his own formula for computing match equities while at the table, and his own match equity charts. These are a lot of wheels to reinvent, especially in a book that runs only 120 pages, some of those devoted to a glossary and lists of books and internet sites. Wherever he does dip his toe in the water though, he dips it deep. It makes me wish that he had picked one area to focus on, and put in the same amount of work, as he might have produced a classic, instead of merely a good book. (He might also have caught two embarrassing goofs. His solutions to parts A and C of problem 13 are both wrong, due solely to poor proofreading would be my guess.)
Here’s an example of the sort of problem that is well handled.