MJR writes:
> In money play using the Jacoby rule, people will double earlier to
> activate their gammons. I am wondering if this notion is correct
> according to doubling theory. The way I see it, the Jacoby Rule doesn't
> affect take decisions, it just means you have to cash cubecentered
> games when you would otherwise play on for the gammon. Given that the
> rule doesn't affect the take decision, wouldn't the doubling point be
> the same as it would be without it? If there aren't enough market
> losing sequences w/o the Jacoby rule to justify doubling, then how can
> it be correct to double the same position with the Jacoby rule?
In my opinion you're right about the take decision being unaffected, but
there can be factors which justify doubling under the Jacoby rule which
wouldn't be normal doubles. In particular, market losers in gammonish
games hurt you more with the Jacoby rule than without  if you are in a
very volatile position where you are reluctant to double because your
overall chances are relatively weak, but you have outside shots at a
gammon, then losing your market under the Jacoby rule is a disaster (you
suddenly find yourself in a position that is too good to double, and you
have big gammon chances that are no use to you). Without the Jacoby rule
 well, at least you can keep playing for 2 points.
So in ungammonish games (races and high anchor holding games), I assert
that cube handling under the Jacoby rule is essentially the same as for
normal money play; for gammonish positions (primevsprime, blitzes, back
games) you might be forced to double early under the Jacoby rule if the
position is volatile enough.
Time for an example, I think: here's a very volatile position where O's
chances aren't huge overall, but if he rolls well he could end up with big
gammon potential:
+131415161718192021222324+
 O X X X   X X O 
 O X X  X  X X 
   X 
   
   
v BAR  O on roll, 1 cube, money game,
    Jacoby rule in effect.
   
 X   
 X O   O O O 
 X O O O   O X O O O O 
+121110987654321+
Let's assume that over the next exchange, there are only two possibilities
(these numbers are contrived to give a result that demonstrates the effect
of the Jacoby rule, I don't give a damn if Jellyfish rollouts show I'm out
by 72% :) 
a) O hits and covers; X fails to anchor or hit: 40%
b) X hits or anchors and ends the blitz: 60%
Under a), let's give O a 70% chance of pulling off the gammon, 20% single
wins, no gammon losses and 10% single losses. For b), we'll say the
numbers are 10%, 20%, 20% and 50% respectively. So, O's cubeless equity
is:
Gammons count Gammons don't count
a) 1.5 0.8
b) 0.5 0.4
For overall cubeless equity of 0.3 (if gammons count) or 0.08 (if they
don't).
If O does not double, his equity is:
when the blitz continues (30%), cubeless equity is 0.8 and he's lost his
market; he can claim with the cube next turn for equity 1.0;
otherwise, his cubeless equity is 0.4, the cube is still centred and
I'll value the cube's worth at 0.2 points to him and 0.45 to X (see my
previous article) for equity 0.65;
0.4 x 1.0 + 0.6 x 0.65 = 0.010 cubeless equity at a 1 cube is 0.010
points.
If O does double, his equity becomes:
when the blitz continues (30%), cubeless equity is 1.5 and X owns the
cube which I'll value at 0.05 points worth of recube vig for 10% wins;
overall equity 1.45;
otherwise, his cubeless equity is 0.5 and X has enormous recube vig
(I'll say 0.45 points) for equity of 0.95.
0.4 x 1.45 + 0.6 x 0.95 = 0.010 cubeless equity at a 2 cube, for 0.020
points.
If the game was played without the Jacoby rule, and O did not double, his
equity would be:
1.5 cubeless equity when the blitz continues, the cube is centred and
almost dead but worth 0.1 points to him and 0.05 to X for equity of
1.55;
if X enters, 0.5 cubeless equity and the cube is again worth 0.2 to O
and 0.45 to X for equity 0.75.
0.4 x 1.55 + 0.6 x 0.75 = 0.170, and the cube is at 1 for 0.170 points.
So under the Jacoby rule, O should double (to win 0.02 points instead of
0.01); without the Jacoby rule, he should not (0.170 points against 0.02).
Note that X has a massive take in each case; it is very close to being a
beaver (the Kauder Paradox).
Cheers,
Gary (GaryW on FIBS).

Gary Wong, Computer Science Department, University of Auckland, New
Zealand
gary@cs.auckland.ac.nz http://www.cs.auckland.ac.nz/~gary/
