# Doubling in a LONG race

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### bshe...@hasbro.com

Feb 23, 1998, 3:00:00â€¯AM2/23/98
to

> I think that for a given LONG race , in a money game against a PERFECT
> opponent, in which we can know the exact cubeless winning chances (and
> also the volatility) we should define a mathematical double (redouble) point.
>
> Am I in error?

In this post I hope to describe the state of the art in racing models, so you
can see where we stand. Possibly you can supply some of the missing
pieces in our understanding.

It should be clear that in a short race (say once we get to a bearoff)
no simple formula will work. An easy way to see this is that there will be
some cutoff involved (e.g. CWP > 0.4) whereas we know that bearoff
positions exist in which doubling is correct with a CWP almost equal to 0.

But you specified a LONG race, and that changes things quite a bit.
In a LONG race the usual caveats and exceptions (listed by other
respondents to this thread) apply with greatly reduced force.

The usual rule of thumb (i.e. take if your pip deficit is less than 10%
of the opponent's total) are derived from exhaustively enumerating a
simplified racing game in which each player has just one checker, and
advances his man in accordance with the rules of backgammon. This
game doesn't feature the bearoff tactics of backgammon (e.g. wastage),
but it is a terrific approximation to actual LONG races.

In fact, if you could reasonably approximate the "wastage" involved in
bearing off, then this model would be correct for any practical purpose.
And this is where complicated models of pip-counting come in.

The problem is to adjust the actual pip count by some amount intended to
account for inefficient bearoffs. The Thorp count, for example, adds 1 for
every man on the ace point, since such men are likely to waste pips, and
it adds 1 for every empty point in the inner table, since missing involves
wastage, and it adds 2 for every man that you have to bear off, since races
with high pip counts and fewer men to bear off are most efficient. Every pip
counting method makes adjustments of this nature, with the goal
of estimating wastage.

My backgammon program pushes this approach to the limit with
a more-or-less exhaustive enumeration of wastage. The estimator
can estimate pip counts with an average error less than 0.1 pips,
and a worst-case error of 1.0 pips in the extreme position (which for this
method is to have 15 men on the 7 point). So it is possible to push this
approach a very long way.

The next problem is volatility. The 10% rule is predicated on races that
have the usual volatility. But the volatility of some positions differs
because of "speed boards." A speed board is is crunched on the
lower points. Any doublet bears off 4 men, so this board is the cause
of many surprising come-from-behind victories. Other positions
(i.e. those maximally spread out) have lower volatility than usual, since
few doublets bear off 4 men.

To my knowledge, no published rules adjust for volatility in any way, let
alone a theoretically satisfying way. So we are theoretically
far away from having the practically perfect rule that you seek.

To some extent, LONG races preclude speed boards.
Yet it is still possible for volatility to affect equity. For example, I
believe that JellyFish spreads its men too much when bearing them in
prior to a race. The ideal is to have men on the 4, 5, and 6 points in ratio
3 to 5 to 7 men respectively. I consistently see men on the 1, 2,
and 3 points when JF prepares for a race. (Particularly when
JF is playing an "almost race," like midpoint-only contact.)

So rollouts are the only theoretically sound method of determining
whether to double in a race. In practice even rollouts are not
theoretically sound, since in theory checker play is imperfect, and
doubling during the rollout is imperfect. But that level of accuracy is not
required for practical understanding of a particular position.

I hope that this post has shown the gaps in our understanding. If you want to
extend the state-of-the-art, you can work on two areas. Probably most
important is to model racing equity in bearoffs. Another opportunity is to
tell us how to model volatility.

Warm Regards,
Brian Sheppard

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### Hugh Sconyers

Feb 24, 1998, 3:00:00â€¯AM2/24/98
to

bshe...@hasbro.com wrote in message <6cs6ph\$umr\$1...@nnrp2.dejanews.com>...

>
>> I think that for a given LONG race , in a money game against a PERFECT
>> opponent, in which we can know the exact cubeless winning chances (and
>> also the volatility) we should define a mathematical double (redouble)
point.
>>
>> Am I in error?
>
>In this post I hope to describe the state of the art in racing models, so
you
>can see where we stand. Possibly you can supply some of the missing
>pieces in our understanding.
>
>It should be clear that in a short race (say once we get to a bearoff)
>no simple formula will work. An easy way to see this is that there will be
>some cutoff involved (e.g. CWP > 0.4) whereas we know that bearoff
>positions exist in which doubling is correct with a CWP almost equal to 0.
>
>But you specified a LONG race, and that changes things quite a bit.
>In a LONG race the usual caveats and exceptions (listed by other
>respondents to this thread) apply with greatly reduced force.
>
>The usual rule of thumb (i.e. take if your pip deficit is less than 10%
>of the opponent's total) are derived from exhaustively enumerating a
>simplified racing game in which each player has just one checker, and
>advances his man in accordance with the rules of backgammon. This
>game doesn't feature the bearoff tactics of backgammon (e.g. wastage),
>but it is a terrific approximation to actual LONG races.

>>>>> Anyone interested in the equities for the one checker race should
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.

(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.

regards,

Hugh Sconyers

### Kit Woolsey

Feb 24, 1998, 3:00:00â€¯AM2/24/98
to

Hugh Sconyers (scon...@bga.com) wrote:

: Anyone interested in the equities for the one checker race should

: CPW .779192

: ---------------------------

: CPW .794301

: ----------------------

: TAKES:

: for ex:

: 75 vs 85

: __________________________

: DOUBLES from CENTER:

: for ex:

: -----------------------------------------
: REDOUBLES:

: 46 vs 51

: -------------------------

: ex:

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.

Kit

### Hugh Sconyers

Feb 25, 1998, 3:00:00â€¯AM2/25/98
to
Kit Woolsey wrote in message ...

>
>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.

>>>> 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 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.

>>>> 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 CD-ROM 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 4-6 3-5 2-4 1-3 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 non-roller position the
one checker model OVER-estimates 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.

>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.

>>>> Again these positions don't have the same character.

>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.

>>>> 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.

>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

>>>>>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