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 Janowski's formulas

 From: Joern Thyssen Address: jth@chem.ou.dk Date: 2 August 2000 Subject: Janowski's formulas for cubeful equity Forum: rec.games.backgammon Google: 39881471.6E3AB38B@chem.ou.dk

```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
```

 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 ```

 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 ```

 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 ```

 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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