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James Eibisch wrote:
> Would some experts like to comment on this -2 -2 game? Neither of us
> doubled immediately, and it took some time for a doubling situation to
> arise, as far as I can see.
>
> I don't double immediately at -2 -2 as I want to exploit any error my
> opponent might make by not doubling at this score when he gains an
> advantage.
>
> I'm X. I haven't re-played this match, and perhaps O should have doubled
> before I did, but I wasn't happy doubling until I did.
A fascinating game at this score, James. I often wonder about similar
situations. I went through it with the aid of Jellyfish to see what was
going on.
First, I'll risk boring those who have been through the endless
discussions on this with a summary of the theory behind doubling at this
match score:
If the reaches 1-away, 2-away Crawford, the trailer's match-winning
chances are generally agreed to be 30%, assuming equal players.
Therefore in order to justify taking you need to have a 30% chance to
win the game (and there's no extra penalty for being gammoned).
In the cases where either player doubles and the other takes, the cube
is dead. Therefore, in all of those cases it really doesn't matter who
doubled, since no one gets any value from redoubling.
Suppose X doubles and O passes. Assuming it's truly a drop, then X
"lost his market". This means X would have been better off having
doubled earlier when O still had a take, so X made a mistake by not
doubling. But, you may ask, suppose X had rolled poorly instead of
well; wouldn't he have been glad he didn't double? No, the theory goes,
because then O would double at a point where X still had a take, so it
wouldn't matter if X had doubled first!
So, technically, if you know your opponent will never lose his market,
then it's never correct to risk losing your market by even a little.
Therefore, you may as well double at your first opportunity and get it
over with.
However, this doesn't take into account the fact that quite often we
play actual human beings (as opposed to computers), and they don't
always judge the situation correctly. As James said, he wanted to give
his opponent a chance to make an error, and this is usually a good
idea! If you double immediately (as Jellyfish does, even as a huge
underdog), your opponent has zero chance to make an error. If you wait
til you are closer to his take point, at some risk of losing your
market, he may erroneously pass, or even if you do lose your market, he
may erroneously take, thus turning your error into a great play.
So let's see what happened in the match. In all cases I looked at what
I thought was the best possible sequence for the player on roll to see
if he could possibly lose his market, using Jellyfish's Level 7
evaluations..
> Score is 3-3 in a 5 point match.
>
> X: (4 1) 12-16 1-2
With X's back men split, can O possibly lose his market? According to
Jellyfish, if O rolls 44 and X fans, X still has 32% winning chances!
Therefore there's no reason to double here.
> O: (4 1) 6-2 2-1
O's best: 11, coming in with both checkers, hitting, and making his 5
point. O then has 30.3 winning chances, still a take.
> X: (2 4) bar-4 bar-2
Best: O 44 and X fans; X wins 32%.
> O: (2 2) 6-4 6-4 24-22 22-20
Best: X 11, O fans; O wins 29.7%.
> X: (3 2) bar-2 17-20
O 55, X fans: X wins 32%
> O: (2 6) bar-23 24-18
In all the above cases, both players seem right to wait since at best
they can lose their market by a minuscule amount; surely the chance of
an opponent error is better.
X 44, O fans: O wins 28.2%. This is the first significant market loser
possible. In fact, if O had fanned after X's actual 22, O would have
had 28.4%. So it's still pretty close to a take; and there's always the
chance that O would take.
> X: (2 2) 16-18 18-20 19-21 19-21
> O: (2 3) bar-23 13-10
OK, now here's the first situation where a double seems clearly correct.
According to JF, O's winning chances are only 35% before the roll, so
it's getting pretty close to a drop. X has a clearly superior blockade,
O's blot on the ace point is a serious drawback, and there are some
market losers. Escaping with 55 results in O having around 25% chances
(depending on his reply); 53, hitting, followed by a fan, results in 23%
chances. At this point, the best chance for O to make an error is to
double him NOW and hope he drops.
> X: (5 2) 12-17 12-14
> O: (3 1) 10-7 8-7
Now, with O having made his bar and leaving no shots, it's probably not
necessary to double anymore.
> X: (6 5) 14-20 17-22
> O: (4 5) 13-8 13-9
Now X has some shots again. If X hits and O fans, O's chances will only
be 23.8%.
> X: (4 6) 12-18 18-22
> O: (6 2) 13-7 13-11
Now, if X hits and O fans, O's chances are only 17%!,
> X: (3 4) 2-5 5-9
> O: (1 2) bar-24 11-9
Again, we're talking serious market losers here, if X hits.
> X: (3 3) bar-3 2-5 19-22 12-15
BTW, Jellyfish likes shifting points with 21/24(2) about 9% better.
> O: (5 2) 7-5 8-3
> X: doubles
> O: accepts
Interestingly, Jellyfish liked O's "desperation" play of hitting twice
and leaving 3 home-board blots, and estimates O with 32% winning
chances, so a correct take at this score! (Not for money, of course,
because X wins tons of gammons.)
So: there were some early rolls where it seemed definitely right for
both players to wait, from a tactical point of view. Then there were a
few rolls where it seemed that X was definitely risking too much by
waiting. And after all that: the final position was: correct double,
correct take!
Whew!
Ron
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