This article is from the Backgammon FAQ, by Mark Damish damish@ll.mit.edu with numerous contributions by others.

Volume 1: Bearoff Equities and Backgame Probabilities

INTRODUCTION:

This CD-ROM disk contains two large databases of equities and

probabilities. One database has the exact equities for all bearoff

positions when each side has 9 men or less. The other database

contains the probabilities for getting hit or hitting a man when one

side has a backgame of 4 men or less.

The bearoff database is over 400 MB and contains the answers to over

100,000,000 bearoff problems! In addition, you can do a MONTE CARLO

simulation on any bearoff position with more than 9 men on a side. The

program also has a feature which finds the best move given a specific

position and dice roll.

The backgame database is 90 MB and contains the answers to over

22,000,000 backgame positions (some of them are not legal positions).

There is a feature which finds the best move given a dice roll and

position.

These databases will help settle many questions about fair settlement

and the best moves.

BEAROFF:

Once the program has been loaded you can enter any bearoff position,

assuming that all men are in their home board. If both positions you

enter have 9 men or less, the program will give you the EXACT equities

for all 4 cube positions - NO CUBE, ROLLER'S CUBE, CENTER CUBE, AND

NON ROLLER'S CUBE. In addition, it will display the proper cube

decision. These cube equities, as throughout this bearoff program, are

the equities if you roll with the cube in that position.

To compute the proper settlement in any bearoff position(assuming you

are going to roll with the cube in that position) you would multiply

the equity times the value of the cube.

After the equities are displayed, you are given a chance to find the

best move for the position you have just entered. Keep in mind your

best move in the bearoff is the one that leaves your opponent with the

smallest equity. The best move will often be different depending on

the cube position.

If you enter a bearoff position where one or both side have more than

9 men you will enter the MONTE CARLO subroutine. You will first be

asked how many simulations you want to run. There is a limit of

30,000. This Monte Carlo subroutine rolls the dice and moves the men

until each side has 9 men or less. Then it looks up the exact answers

from the database. With a few hundred simulations the NO CUBE result

should be very close to exact. The other cube positions results will

be distorted by the fact that this subroutine assumes that there is no

doubling until the positions are back in the database (ie each side

has 9 men or less). For example, if you have 12 men and the cube is in

the center and your opponent has 12 men, the cube will stay in the

center until each side has 9 men or less. At that point, the

subroutine will retrieve the exact equity from the database.

BACKGAME:

Once the program has finished loading you can enter any backgame

position provided the position not bearing off has 4 men or less. The

side bearing off can have from 1 to 15 men. The program will return

two probabilities: one for each side being on roll. These results are

the probabilities that the side bearing off will have a man hit.

After the probabilities are displayed, you are given a chance to find

the best move for the position you have just entered. Keep in mind

that the best move for the position bearing off is the one that leaves

the other side with the smallest probability of hitting a man. The

best move for the side in the backgame is the one that gives it the

highest probability of hitting a man.

This program assumes that the side playing the backgame has infinite

timing; ie, he will never be forced to leave his opponent's home

board. It follows from this assumption that the backgame side can take

some, part or none of any roll.

Sony's MMCD Player:

This disc works on an MMCD player also. Sony makes this very small

CD-ROM player which takes special exe files. This disc works both for

dos and MMCD. The MMCD version is the same as the PC version except in

a few places. To start the program place the disc in the MMCD player

and turn the power on. If you plan to use the bestmove feature and the

MONTE CARLO routine you will need to put the Volume #1 disc in the

player after the program loads. The MONTE CARLO simulations are

limited to 20,000 games.

_________________________________________________________________

Volume 2: Bearoff Equities for 4 Points and 15 Men

INTRODUCTION:

This CD-ROM disk contains two databases of equities for the bearoff.

The first database(4X15) has the exact equities for all bearoff

positions when each side has 15 men or less on the first 4 points. The

second database(3X15) contains the exact equities for all bearoff

positions when each side has 15 men on the first 3 points. The second

database is a subset of the first. The smaller database(3X15) is

included for a number of reasons, which will be explained later.

Everything that follows applies to the 3X15 database when the 4's are

change to 3's etc.

The bearoff database for 4 points and 15 men is over 240 MB and

contains the answers to over 60,000,000 bearoff problems! In addition,

you can do a MONTE CARLO simulation for any bearoff position where one

or both sides have men on the 5 or 6 points(4,5 or 6 in the case of

the 3X15 database). The program also has a feature which finds the

best move given a specific position and dice roll.

These databases can help settle many questions about fair settlement

and the best moves.

BEAROFF EQUITIES:

Once the program has loaded you can enter any bearoff position,

assuming that all men are in their home board. If both positions you

enter have all their men on the first 4 points, the program will give

you the EXACT equities for all 4 cube positions - NO CUBE, ROLLER'S

CUBE, CENTER CUBE, and NON ROLLER'S CUBE. In addition, it will display

the proper cube decision and probability for winning in the no cube

case. These cube equities, as throughout this bearoff program, are the

equities if you roll with the cube in that position. If you find an

equity greater than 1.000 or less than -1.000 this is due to the fact

that a position with 15 men can still lose a gammon.

To compute the proper settlement in any bearoff position(assuming you

are going to roll with the cube in that position) you would multiply

the equity times the value of the cube.

After the equities are displayed, you have an opportunity to find the

best move for the position you have just entered. Keep in mind your

best move in the bearoff is the one that leaves your opponent with the

smallest equity. The best move will often be different depending on

the cube position.

If you enter a bearoff position where one or both side have men on the

5 or 6 points you will automatically enter the MONTE CARLO subroutine.

You will first be asked how many simulations you want to run. There is

a limit of 30,000. This Monte Carlo subroutine rolls the dice and

moves the men until each side has all men on the first 4 points. Then

it looks up the exact answers from the database. With a few hundred

simulations the NO CUBE result should be very close to exact. The

other cube positions results will be distorted by the fact that this

subroutine assumes that there is no doubling until the positions are

back in the database (ie each side has all men on the first 4 points).

For example, if you have 12 men on the one and 3 men on the 5 point

versus the same, the cube stays in the current position until both

sides have all their men on the first four points. At that point, the

subroutine will retrieve the exact equity from the database.

THE 3X15 DATABASE:

There are several reasons that this database is included. First, it is

small(only 10 MB!). This will allow it, if you desire, to be copied to

your hard disk(be sure to copy BEQT4X15.EXE, helvb.fon and tmsrb.fon).

Secondly, the 3X15 database has an advantage in speed when doing MONTE

CARLO simulations. Simulations will run faster at the expense of some

accuracy. If you have copied the 3X15 database to your hard disk it

will run MONTE CARLO simulations substantially faster.

_________________________________________________________________

Volume 3: Bearoff Equities for 6 Points and 10 Men

INTRODUCTION:

This CD-ROM disk contains a database of equities for the bearoff. The

database has the exact equities for all bearoff positions where one

side has 10 men in the home board and the other side has 10 men or

less in the home board.

The bearoff database for 6 points and 10 men is over 625 MB and

contains the answers to over 156,000,000 bearoff problems! In

addition, you can do a MONTE CARLO simulation for any bearoff position

where one or both sides have more than 10 men. The results of these

simulations will be more accurate than the results from Volume #1. The

program also has a feature which finds the best move given a specific

position and dice roll. You will need Volume #1 to use the MONTE CARLO

feature and you may need Volume #1 for the best move feature.

These databases can help settle many questions about fair settlement

and the best moves.

BEAROFF EQUITIES:

Once the program has loaded you can enter any bearoff position,

assuming that all men are in their home board. If you enter a position

where both positions have 9 men or less you will get an error message

because all these positions are on Volume #1. For positions where one

side has 10 men and the other side has 10 men or less, the program

will give you the EXACT equities for all 4 cube positions - NO CUBE,

ROLLER'S CUBE, CENTER CUBE, and NON ROLLER'S CUBE. In addition, it

will display the proper cube decision and probability for winning in

the no cube case. These cube equities, as throughout this bearoff

program, are the equities if you roll with the cube in that position.

To compute the proper settlement in any bearoff position (assuming you

are going to roll with the cube in that position) you would multiply

the equity times the value of the cube.

After the equities are displayed, you have an opportunity to find the

best move for the position you have just entered. Keep in mind your

best move in the bearoff is the one that leaves your opponent with the

smallest equity. The best move will often be different depending on

the cube position. For some positions you will need Volume #1.

If you enter a bearoff position where one or both sides have more than

10 men you will automatically enter the MONTE CARLO subroutine. You

will need Volume #1 to use this subroutine. You will first be asked

how many simulations you want to run. There is a limit of 30,000. This

Monte Carlo subroutine rolls the dice and moves the men until both

sides have 10 men on less. Then it looks up the exact answers from the

database. With a few hundred simulations the NO CUBE result should be

very close to exact. The other cube positions results will be

distorted by the fact that this subroutine assumes that there is no

doubling until the positions are back in the database (ie each side

has all men on the first 4 points). For example, if you have 12 men on

the six point and 3 men on the 5 point versus the same, the cube stays

in the current position until both sides have 10 men or less. At that

point, the subroutine stores that position and later retrieves the

exact equity from the database on Volume #3 or Volume #1.

Sony's MMCD Player:

This disc works on an MMCD player also. Sony makes this very small

CD-ROM player which takes special exe files. This disc works both for

dos and MMCD. The MMCD version is the same as the PC version except in

a few places. To start the program place the disc in the MMCD player

and turn the power on. If you plan to use the bestmove feature and the

MONTE CARLO routine you will need to put the Volume #1 disc in the

player after the program loads. The MONTE CARLO simulations are

limited to 20,000 games.

Available from Carol Joy Cole, The GAMMON PRESS and The Backgammon

Shop for $99 per volume.

If you have any comments or questions, please forward them to the

Hugh Sconyers

sconyers@bga.com

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