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Hi
I've got hold of three issues for Hoosier BG Club magazine from
1993-1994 (Volumn X, no. 6 (1993); Volumn XI, no. 1 (1994); Volume XI,
no. 2 (1994)) where Rick Janowski writes about take points in money
game.
I've tried to reproduce Rick Janowski's formulas but I have some
trouble.
W is the average points won pr. game, L is the average loss pr. game.
All the formulas are for money game without the Jacoby rule [Janowski
does give formulas for play with Jacoby w/o beavers, but let's not
complicate things more than necesary].
Take points:
Eq 1: dead cube take point: TP = (L-0.5)/(W+L)
Eq 2: live cube take point: TP = (L-0.5)/(W+L+0.5)
Eq 3: general take point: TP = (L-0.5)/(W+L+0.5x), where 0<=x<=1
(x=0 gives dead cube model, x=1 gives live cube model).
Cubeless take equity:
Eq 4: E_{take} = TP ( W + L ) - L
Equation 4 is just a special case of the cubeless equity:
E = p W - ( 1 - p ) L
= p ( W + L ) - L
with p = TP, you get Eq. 4.
Cubeful equity:
Eq 5: E_O = E_{I own cube} = Cv [ p (W+L+0.5x) - L]
Eq 6: E_{Opp own cube} = Cv [ p (W+L+0.5x) - L - 0.5x ]
Eq 7: E_{cent. cube} = 4 Cv / (4-x) [p (W+L+0.5x) - L - 0.25]
(Cv is the current value of the cube, p is the probability that you'll
win this game).
But Eqs. 5-7 is giving me trouble. Well, so far only Eq. 5, because I
didn't want to check the others before I could reproduce Eq. 5 [
Actually, Eq.
6 is easily derived from Eq. 5 (just interchance W <-> L, p <-> 1-p, and
voila!].
I would derive Eq. 5 as follows:
At p = 0: my cubeful equity is -L,
at p = TP: my cubeful equity is +1,
and do a linear interpolation inbetween.
Then I arrive at:
E_O = C_V [ p ( 1 + L ) / ( L + 0.5 + 0.5 x ) ( W + L + 0.5 x ) - L ]
whereas Janowski gets
Eq. 5: E_O = C_V [ p ( W + L + 0.5x ) - L ]
If I plug in p = 0 in Janowski's formula E_O = -L. Fine!
But with p = TP = 1 - (W-0.5)/(W+L+0.5x) (I'm using my opponents
takepoint, since I own the cube) I get:
E_O = 0.5x + 0.5 (Janowski) != 1 (for x != 1 ).
E_O = 1 ("my" formula).
So, does any of you know how Janowski derives his Eq. 5?
I appreciate any help.
Joern Thyssen
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Phill Skelton writes:
Part of the problem is that your cube efficiency and your opponents may
well not be the same, or even close to each other. Sticking x = 0 into
the equations evidently implies that you can't double at all as it
reproduces the cubeless equity. OTOH if you are able to double, your
opponent has a dead cube and p = 0.75 (i.e. you are at his dead cube
take-point, when W = L = 1) then your equity is clearly 1, and you are
offering a perfectly efficient double. The assumptions Jankowski
uses evidently means that the formula is not applicable to this
situation.
OTOH the approach taken by Joern makes a different assumption (that the
player holding the cube has a perfectly efficient double and that the
parameter x only applies to your opponent). In the real world x is
different for each player and varies depending on the situation - you
might try to model it as a function of the volatility and the distance
from the ideal doubling point, but that begins to introduce even more
variables into the equations.
I think that the bottom line is that it's just not possible to come up
with a formula for cubeful equities that is generally applicable and
sufficiently simple to be useful, but perhaps the 'special case'
equities may be good enough for many situations.
Phill
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Øystein Johansen writes:
Phill Skelton wrote:
> In the real world x is different for each player and varies depending on
> the situation - you might try to model it as a function of the volatility
> and the distance from the ideal doubling point, but that begins to
> introduce even more variables into the equations.
This is the real interesting part. Joern, does your article say anything
about this?
I have done some small experimets to calculate a good value for x, as
function off the position. This following values are based on Jellyfish
evaluations and conversations with a strong player.
Pure race: x ~ 0.65
Holding games: x ~ 0.5
Deep backgames: x ~ 0.4
Faceing an attack: x ~ 0.9
Last roll positions: x = 0.0 (obviously)
It is hard to estimate a good value for x in late bearoff positions. I
think maybe it should be increasing with the expected number of rolls
left for the accepting/droping player. Take a "~one-roll-each" position.
Then if this player expects to get off in 1.25 rolls, (27 good rolls - 9
bad rolls), the value of x should be 1.0.
Of course you can try to make this values more sophisticated. They're
far for perfect.
Best regards,
Øystein Johansen
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Joern Thyssen writes:
> Joern, does the article say anything about this?
I quote:
"Finding accurate values for x is a difficult, almost impossible task.
However, we can make estimates of /typical/ values for /typical/
situations. In my opinion, for the majority of /typical/ positions, x
will commonly be between 1/2 and 3/4, with 2/3 being a /normal/ value."
> Joern could you define more clearly the W and L? Are they the
> possibilities for you winning and losing the game? I 'm afraid I don't
> understand that "W is the average points won pr. game, L is the average
> loss pr. game". An arithmetic example would be of great help.
W = p(win) + p(win gammon/backgammon) + p(win backgammon)
-----------------------------------------------------
p(win)
Analogous for L.
For example, in a pure race (with no contact):
W = p(win) = 1
------
p(win)
Suppose a evaluation gives:
win 50%
g/bg 11%
bg 0%
(a typical opening situation)
W = 50 + 11 = 1.22
-------
50
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Joern Thyssen writes:
> Cubeful equity:
>
> Eq 5: E_O = E_{I own cube} = Cv [ p (W+L+0.5x) - L]
> Eq 6: E_{Opp own cube} = Cv [ p (W+L+0.5x) - L - 0.5x ]
> Eq 7: E_{cent. cube} = 4 Cv / (4-x) [p (W+L+0.5x) - L - 0.25]
>
> (Cv is the current value of the cube, p is the probability that you'll
> win this game).
After personal communication with Rick Janowski I now know how to derive
these formulae:
E = (1-x) (E_dead) + x (E_live)
where
E_dead = p * (W+L) - L (equal to the cubeless equity)
E_live = p * (W+L+0.5) - L
(by doing linear interpolation between (p=0,E=-L) and (p=opponent take
point=1-(W-0.5)/(W+L+0.5)=(L+1)/(W+L+0.5), E=+1).
This gives
E_O = (1-x) ( p (W+L) - L ) + x ( p (W+L+0.5) - L )
= p (W+L+0.5x) - L
Joern
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Theory
- Derivation of drop points (Michael J. Zehr, Apr 1998)
- Double/take/drop rates (Gary Wong, June 1999)
- Drop rate on initial doubles (Gary Wong, July 1998)
- Error rate--Why count forced moves? (Ian Shaw+, Apr 2009)
- Error rates--Repeated ND errors (Joe Russell+, July 2009)
- Inconsistencies in how EMG equity is calculated (Jeremy Bagai, Nov 2007)
- Janowski's formulas (Joern Thyssen+, Aug 2000)
- Janowski's formulas (Stig Eide, Sept 1999)
- Jump Model for money game cube decisions (Mark Higgins+, Mar 2012)
- Number of distinct positions (Walter Trice, June 1997)
- Number of no-contact positions (Darse Billings+, Mar 2004)
- Optimal strategy? (Gary Wong, July 1998)
- Proof that backgammon terminates (Robert Koca+, May 1994)
- Solvability of backgammon (Gary Wong, June 1998)
- Undefined equity (Paul Tanenbaum+, Aug 1997)
- Under-doubling dice (Bill Taylor, Dec 1997)
- Variance reduction (Oliver Riordan, July 2003)
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