Hugh  Sconyers (scony...@bga.com) 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 4-2 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.