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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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