Theory

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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

ystein 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,
ystein 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

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Inconsistencies in how EMG equity is calculated  (Jeremy Bagai, Nov 2007)  [GammOnLine forum]
Janowski's formulas  (Joern Thyssen+, Aug 2000) 
Janowski's formulas  (Stig Eide, Sept 1999) 
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