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Feb 23, 1998, 3:00:00â€¯AM2/23/98

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> 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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Feb 24, 1998, 3:00:00â€¯AM2/24/98

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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

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.

regards,

Hugh Sconyers

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

Feb 25, 1998, 3:00:00â€¯AM2/25/98

to

>

>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

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 PII-333 and looks like it will only take 2

years ..... lol

Hugh

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