Earlier this year I had a discussion with Mr JellyFish Fredrik Dahl about
Jelyfish always doubles at 2-away,2-away positions.
The subject was that JF
doubles right after the first move, whether it's ahead or not.
Here is the
discussion:
Michael Bo Hansen Wrote:
Hi F. Dahl
I've been playing a lot
with the JellyFish 3.0 player shareware
version. I have encountered that
when JF and I both are two points
away from victory, and I open with 4-2 (or
3-1 in fact), JF doubles!
I must say I was quite surprised the first time it
happened.
I know that an early double is necesarry when the score is 2-away
2-away, but doubling when you are underdog, that can't be correct.
I'll will
be glad if you can come up with an explanation.
Yours truly
Michael Bo
Hansen, Denmark.
F. Dahl replyed:
This is in the FAQ in the JF users
manual :-).
If you start the game with the plan of doubling right away,
the game
will surely be played to the end with the cube on 2. So the game
will
decide the match. Obviously this means that you should win 50%.
Because
the score 2-away, 2-away is symmetric, it's clearly worth 50%, so
using
the strategy of always doubling right away does not cost any equity,
and
therefor is not wrong.
Another way to look at it:
If it's wrong for
JF to jack up the cube, then it must be right for
you
(because you win what
it loses). So if it doesn't double, you should,
and then it must take. The
result is the same, the cube ends on 2.
A third way to look at it:
If you
say it's wrong to always double, you must think it costs
equity.
Then you
should be willing to pay your opponent some small amount
(less
than the
size of the error) to make this error. But this is clearly
stupid, as you
pay him money to play an even game.
A fourth way:
Always doubling
transforms 2-away,2-away to 1-away,1-away, which
cannot
be wrong.
Michael Bo Hansen:
Hi again Fredrik Dahl.
I've been playing a lot with
the JellyFish 3.0 player shareware
version. I have encountered that when JF
and I both are two points
away from victory, and I open with 4-2 (or 3-1 in
fact), JF doubles
right after! I must say I was quite surprised the first
time it
happened. Here is an explanation of why I thinks it's wrong.
After
I have rolled 31 in the opening roll, JellyFish (JF) is underdog
of winning
the following game (distinguish from winning the total
match).
The
probability of JF wins the game is therefore
P{JF wins game} < 0.5
We
have that the probability for JF winning the total match when
doubling is
P{JF wins match by doubling} = P{JF wins game} < 0.5
because there is only
one game left.
On the other hand, the probability for JF to win the match
when JF
does not double is (while not taking gammons into account)
P{JF
wins match by no doubling} = P{Opponent doubles} × P{JF wins
game}
+P{Opponent doesn't double} × P{JF wins game} × EQ{1-away,2-away}
+P{Opponent doesn't double} × P{JF loses game} × EQ{2-away,1-away}
where
P{Opponent doubles} is the probability that your opponent
doubles right away
(or at least while you still can take the cube),
and EQ is equity of winning
the game at that score.
Of course you have
P{Opponent doubles} = 1 -
P{Opponent doesn't doubles}
P{JF wins game} = 1 - P{JF wins game}
EQ{1-away,2-away} = 1 - EQ{2-away,1-away}
>From Equity tables, such as the
one made by Tom Keith
(
http://www.bkgm.com/articles/met.html):"How to
Compute a Match Equity
Table", we get
EQ{1-away,2-away} = 70 %
Let's
say you are playing against a horrible player, who doesn't know
the doubling
cube. From that
P{Opponent doubles} » 0
and therefore
P{JF wins
match by no doubling}
= P{JF wins game} × 0.7 + (1- P{JF wins game})× 0.3
=
0.3 + 0,4 P{JF wins game}
which, in the interval [0, .5 [, always is
higher than P{JF wins
game} itself.
Therefore
P{JF wins match by
doubling} < P{JF wins match by no doubling}.
The same calculations can be
made for different values of P{Opponent
doubles}, and only for P{Opponent
doubles} = 1 (he always doubles)
it's the same whether you double or not.
But you never know, do you?
Therefore the correct action must be not to
double when you are an
underdog in a 2-away,2-away situation.
Yours Truly
Michael Bo Hansen, M.Sc
Denmark.
F. Dahl:
This sounds plausible, but is
not quite correct. If JF doesn't
double,
then you can. So nothing can be
gained by 'trying to keep the cube
down'.
If you think it's wrong to
promise always to double at this score,
try
playing it as a proposition: I
always double, you don't. Who has the
advantage? Noone, obviously, as all
games will be doubled early. So I
can't ba making any error, ok?
Michael
Bo Hansen:
OK. I know that I will always double after I have rolled 31 or 42
in
my opening roll, because I'm a favorite to win ( Not much...but
enough),
but not all people do. JF should wait until I double, because
there is no
reason for JF to double. If I'm stupid enough NOT to
double, JF should take
advantage of that, not disadvantage.....
F.Dahl:
This is right. Against
an opponent who is willing to risk losing his
market, you can do better by
waiting with the cube. I believe JF does
this if it wins the opening roll
and the opponent responds with a
crushing doublet.
Michael Bo:
I haven't
investigated your proposal, but shouldn't it also wait
doubling when the
opponent rolls 31 and 42?
F. Dahl:
Against a good opponent it makes no
difference, as the cube will be
turned evenually. But you could wait till
the first point where you
risk
losing your market, and I agree that there
is no such risk after 31
or
42.
--
- Fredrik Dahl
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