Forum Archive :
Cube Handling in Races
Hugh Sconyers wrote:
> Anyone interested in the equities for the one checker race should download
> BG Interface from www.realtime.com/~sconyers . The interface allows you to
> look up one checker races for all pipcounts from 1 to 300. The interface
> is free. The 10% mentioned above is conservative, as you can see from the
> following examples from BG Interface:
>
> Roller's Pipcount 70
> Opponent's Pipcount 80
>
> CPW .779192
>
> Equities:
> Cubeless .558384
> Roller's Cube .872846
> Centered Cube .855556
> Opponent's Cube .494483
>
> Double from Center
> Redouble
> Take
>
> 
>
> Roller's Pipcount 70
> Opponent's Pipcount 81
>
> CPW .794301
>
> Equities:
> Cubeless .588602
> Roller's Cube .900131
> Centered Cube .886554
> Opponent's Cube .529958
>
> Double from Center
> Redouble
> Take
>
> 
>
> For the medium and long races of one checker the following works really
> well for Takes, Doubles and Redoubles, but not perfectly:
>
> TAKES:
>
> X = Roller's Pipcount  Opponent's Pipcount
> Y = 10% of Roller's Pipcount (Round Y to nearest integer ex: for 79 Y =
> 8, for 75 Y = 7)
>
> if X <= Y + 3 then Take
> if X > Y + 3 Pass
>
> for ex:
>
> 75 vs 85
>
> X = 85 75 = 10
> Y = 7
> X <= 7 + 3 hence Take
>
> 75 vs 86
> X = 86 75 = 11
> Y = 7
> X > 7 + 3 hence Pass
>
> 79 vs 90
> X = 90  79 = 11
> Y = 8
> X <= 8 + 3 hence Take
>
> 79 vs 91
> X = 91  79 = 12
> Y = 8
> X > 8 + 3 hence Pass
>
> __________________________
>
> DOUBLES from CENTER:
>
> X = Roller's Pipcount  Opponent's Pipcount
> Y = 10% of Roller's Pipcount (Y NOT rounded: for 79 Y = 7, for 73 Y = 7)
> if X >= Y  1 then double from center
>
> for ex:
>
> 90 vs 97
> X = 97 90 = 7
> Y = 9
> X < 9  1 hence do not Double from center
>
> 90 vs 98
> X = 98  90 = 8
> Y = 9
> X >= 9 1 hence do Double from center
>
> 71 vs 76
> X = 76  71 = 5
> Y = 7
> X < 7  1 hence do not Double from center
>
> 71 vs 77
> X = 77  71 = 6
> Y = 7
> X >= 7 1 hence do Double from center
>
> 
> REDOUBLES:
>
> X = Roller's Pipcount  Opponent's Pipcount
> Y = 10% of Roller's Pipcount (Y NOT rounded: for 79 Y = 7, for 73 Y = 7)
> if X > Y then redouble
>
> 46 vs 50
> X = 50  46 = 4
> Y = 4
> X <= Y hence do not Redouble
>
> 46 vs 51
>
> X = 51  46 = 5
> Y = 4
> X > 4 hence Redouble
>
> 
>
> For people who like rules in english:
>
> If your lead in a race is greater than or equal to 10% of your pipcount
> less 1 then Double from the center.
>
> If your lead in race is greater than 10% of your pipcount then Redouble.
> (The above percentages are not rounded.)
>
> If you are behind no more than 10% of your opponent's pipcount plus 3
> pips then Take. (This percentage is rounded and .5 is rounded down.)
>
> ex:
>
> roller's pipcount = 80
>
> opponent's pipcount: is between 1 and 86, then no Double
> opponent's pipcount: is 87, then Double from center but don't Redouble and
> is a take
> opponent's pipcount: > 88 and < 91, then Double and Redouble and is a take
> opponent's pipcount: > 92, then Double and Redouble and Pass
>
> These are the rules I use in money play to decide on the proper cube
> action. In matches you have to make adjustments for the match score.
> They seem to work well. Keep in mind these racing rules are for race
> with no hitting chances. If there are hitting chances then doubling
> ,redoubling and take points change. Also, these rules come from looking
> at races with one only checker, but imo one checker races are close to
> backgammon races.
An excellent discussion of the one checker model. However, I do not
agree that the one checker race is close to a normal backgammon race.
For the one checker race, obviously there is no wastage. Not so in a
normal backgammon race. When you roll those double 4's, 5's, or 6's,
while they are certainly nice numbers to roll, they do lead to some
wastage. Since the trailer often needs those big doubles to overcome the
racing deficit, the wastage when he rolls the big doubles will hurt him.
The leader will also waste when he rolls the big doubles, but it won't
matter to him since the big doubles will only extend his lead.
Consequently, I believe the one checker model makes things rosier for the
trailer than they actually are.
As a test, I examined the following position. X (on roll), has 5 men on
the 6 point, 4 on the 5 point, 3 on the 4 point, 2 on the 3 point, and 1
on the 2 point. 70 pips exactly, and the optimal bearoff structure for
70 pips and 15 checkers. O has the same position, except the checker on
the 2 point was moved back to the 12 point, giving him 80 pips.
I then gave this to jellyfish to roll out, 12960 times. It should be
noted that on level 5 (where the rollouts were done) jellyfish won't
always play O's first roll correctly (for example, it plays a 42 12/8,
4/2 rather than the proper 12/6), but the cost of this error is
relatively small. Once the outfield checker is home, the program will
play each side equally well or badly.
Results: X won 80% of the games. This is far from the 77.9% in the one
checker variation, and makes what would be a borderline take (in the one
checker game) a big pass in real life. I'm sure the difference is
because of the wastage for O when he rolls big doubles.
Moving the outfield checker to the 11 point cut X's win percentage down
to 78.5%, and moving it to the 10 point cut it further to 76.5%  now a
take, according the jellyfish using the cube (and it would also appear to
be a take from Sconyer's results). Thus, for races of this length it
appears that the 1 checker model makes the trailer appear to be about
1 1/2 or 2 pips better than he actually is.
What about longer races? Here the wastage effect shouldn't be so serious,
since if O rolls his big doubles early he can play them with no wastage.
I tried tossing a couple of the checkers on the low points into the
outfield, making the pip count 96 to 85 (increasing the pip difference
from 10 to 11 to compensate for the longer race). Here, the rollout gave
X 79.2% wins. Moving O one pip closer (95 to 85) the rollout gave X
77.7% wins, which is probably a borderline take/pass.
My conclusion is that the one checker model favors the trailer, and the
shorter the race the more it favors the trailer. It can be used
profitably, but only if you make a 1 or 2 pip adjustment (depending on
the length of the race). As always, other factors such as smoothness,
men off, crossovers, etc. play an important role  this model only works
if the other factors are equal or have already been appropriately
compensated for.
Kit


Hugh Sconyers writes:
This does not seem like a 'fair' test, since one side has a checker in
the outerboard. In addition, the one checker model and the rules listed
above are NOT trying to predict equities, but ARE trying to predict cube
decisions.
I rolled this out 12960 times using Jellyfish on level 7 got X winning
79.2% of the games with a sd of .002. I don't consider either the 1.2%
difference I got or Kit 2.1% a major difference given the position. I ran a
simulation of the same position 30,000 times using my CDROM databases of
10 checkers vs 10 checkers ( needs 9 vs 9 also). The following where the
results:
CPW .791737
Equities:
Cubeless .583476
Roller cube .764600
Center cube .721350
Opponent cube .517574
Lets look at a similar positions where the EXACT answer is know for all
cube position:
side on roll has 46 35 24 13 pipcount of 50 and his opponent has the
same:
EXACT answer: one checker 50 vs 50: one checker 50 vs 51
CPW .618189 .608351 .633297
Equities:
Cubeless .236377 .216702 .266594
Roller cube .455154 .425768 .491951
Center cube .363817 .331952 .411079
Opponent cube .124417 .104259 .162208
Clearly, for this position the EXACT equities results are the same as the
one checker equities, but close imo. But the cube decisions are the same
for this position and one checker of 50 vs 50 (and for 50 vs 51). Also,
note that for the one checker model this position understates the advantage
of the side on roll, but by adding just ONE pip to the nonroller position
the one checker model OVERestimates the chances of side on roll!!!!!! In
addition, for this position 50 vs 50 is much closer to right than 50 vs 51
and the rules listed predicted the correct cube decisions for this
position.
imo and also many others, one Jellyfish's weakest part is in races.
Jellyfish makes plays that also no 'expert' would agree with. The arguement
that these plays cancel out is just not correct.
I agree with Kit that is most cases the one checker model favors the
trailer when looking at equity, but when trying to decide takes, passes,
double and redoubles I think the rules listed above get very close to the
correct answers.
I have a program running now that will find the 'best rules' for takes,
passes, doubles and redoubles by looking at know exact results and the one
checker model. It is running on PII333 and looks like it will only take 2
years ..... lol
Hugh




Cube Handling in Races
 Bower's modified Thorp count (Chuck Bower, July 1997)
 Calculating winning chances (Raccoon, Jan 2007)
 Calculating winning chances (OpenWheel+, Nov 2005)
 Doubling formulas (Michael J. Zehr, Jan 1995)
 Doubling in a long race (Brian Sheppard, Feb 1998)
 EPC example (adambulldog+, Jan 2011)
 EPC example: stack and straggler (neilkaz+, Jan 2009)
 EPC examples: stack and straggler (Carlo Melzi+, Dec 2008)
 Effective pipcount (Douglas Zare, Sept 2003)
 Effective pipcount and type of position (Douglas Zare, Jan 2004)
 Kleinman count (Øystein Johansen+, Feb 2001)
 Kleinman count (André Nicoulin, Sept 1998)
 Kleinman count (Chuck Bower, Mar 1998)
 Lamford's race forumla (Michael Schell, Aug 2001)
 Nroll vs nroll bearoff (David Rubin+, July 2008)
 Nroll vs nroll bearoff (Gregg Cattanach, Nov 2002)
 Nroll vs nroll bearoff (Chuck Bower+, Dec 1997)
 Near end of game (Daniel Murphy, Mar 1997)
 Near end of game (David Montgomery, Feb 1997)
 Near end of game (Ron Karr, Feb 1997)
 One checker model (Kit Woolsey+, Feb 1998)
 Pip count percentage (Jeff Mogath+, Feb 2001)
 Pipcount formulas (Tom Keith+, June 2004)
 Thorp count (Chuck Bower, Jan 1997)
 Thorp count (Simon Woodhead, Sept 1991)
 Thorp count questions (Chuck Bower, Sept 1999)
 Value of a pip (Tom Keith, June 2004)
 Ward's racing formula (Marty Storer, Jan 1992)
 What's your favorite formula? (Timothy Chow+, Aug 2012)
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