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zzyzx97@earthlink.net wrote:
+13-14-15-16-17-18-+---+19-20-21-22-23-24-+
| O X | | O O O O O |
| X | | O O O O O |
| | | O O O |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | X X |
| | | X X X X X |
| O X | | X X X X X |[2]
+12-11-10--9--8--7-+---+-6--5--4--3--2--1-+
> Here are the rest of the facts:
>
> Abe is X and he leads 10-6 in the 17 point match. I am O and I have
> the cube on 2. It is my roll. Abe has 61 pips to go and I have 74. My
> doubling window opens at 45.6% (.456). His take point is .277. So I
> passed the window test. But its not often you should double toward
> the short end of the window. So let's look at what is likely to
> happen.In a non contact race with these pips, I only win 31%. However,
> there is contact. He has 2 blots and I have 16 shots. I let Jellyfish
> play 1296 games on level 5, and left the settlement value at .550.
>
> If I hold the cube I win 57.6% and we both get 1.3% gammons for a
> positive equity of .153
This is a very complex problem, due to the potential of doubling later if
you hold the cube. Here is how I would work it out at the table, using
as much simplifying assumptions as possible.
First: Using Neil's numbers (which give a close approximation of my
match equity table):
6-14: 10% equity
6-12: 20% equity
8-10: 39% equity
10-10: 50% equity
So if it were a now or never situation you would be getting 11 to 10 odds
on the double. However it isn't a now or never situation -- if you hold
the cube, you may get to double later and win some games which might
have been lost if you had to play them to conclusion.
First simplifying assumption: I assume that O will never redouble to 8
(since he can't redouble except as a huge favorite, this isn't far off).
Let's suppose that X hits a shot. If he doesn't double, I assume he
always wins the game, since unless O rolls a joker X will be able to
claim with the cube next turn. This doesn't take O's jokers into
account, but they are somewhat counterbalanced by the potential of X
playing for, and getting, a gammon when O flunks.
If X does double and hits a shot, he has to play to conclusion. I
estimate that X will win 88% of the time when this happens (actually I
think it is a bit lower, but to compensate X will win a few gammons).
Let's suppose that X misses. He is a clear underdog, but has some racing
and some hitting chances. Also, he will clearly win more often if he
hangs onto the cube, since he may get an efficient double later and not
have to play to conclusion a game he might lose. I will guess that X
will win 35% of the time if he let's the cube go, and 40% of the time if
he hangs onto the cube.
So, what does all this mean? Roughly speaking, it looks like about 1/8
of the games X would win if he hangs onto the cube will turn into losses
if he doubles.
How does this figure into the match equities? If X loses a game he would
have won if he hadn't doubled, this costs him 29% equity (difference
between 39% and 10%). Thus, of the games X would have won if he hadn't
doubled, he gains 11% 7/8 of the time and costs 29% 1/8 of the time.
This comes to an average gain of about 6%. As we have seen, X loses 10%
if he doubles and is wrong. Thus, when we take the cube value into
account it looks like X is actually giving 10 to 6 odds with his
redouble. He clearly isn't close to being worth this in this position.
So, if you accept my estimates, the redouble is incorrect. Of course my
estimates may well be quite wrong, and different estimates might lead to
a different conclusion.
Kit
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