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The Aggression Coefficient Formula for Safe or Bold Bear-Ins
by François Tardieu - 10 August 2006
François Tardieu
The purpose of this article is to derive a formula which indicates when to play “bold” or “safe” in positions with the following characteristics:

 - White (the top player) has been hit as he was bearing off and is sitting on the bar.

- Red (the bottom player) is bearing in and has one or more open points in his home board.

- Red rolls a number that offers the choice to bear off one checker leaving a shot and gaining a roll in the bear off process, or to just play it safe.

 

Sample position:

 

Here Red has two choices: Bold: 10/6 3/0 or Safe: 10/6 6/2

In this article, we will refer to these choices as “the bold play” (leaving a shot) and “the safe play” (move the checker to an inner point) and we will use the following five parameters in the formula:

Parameter 1 - The number of crossovers for each player.

Here's how a crossover is counted in this article - When a checker is not in the homeboard, a crossover is the movement of a checker from one quadrant of the board to the next. However, when all checkers are in homeboard, crossovers count as the number of rolls it takes to bear off a checker without a miss.

This is defined by the number of rolls in which a player bears off all his checkers after the safe play, considering he will not miss.

There are number of sample positions further down in this article. For example, in Position 1:

 

Red will need eight crossovers to bear off his 15 checkers without a miss and White needs nine; four to reach his inner board with the straggler and another five to bear off the nine checkers.

To indicate the number of crossover for player “on roll” or “not on roll”, we will use the following abbreviations as part of our formula:

COOR = number of Cross Overs for the player On Roll (Red)

CONOR = number of Cross Overs for the player Not On Roll (White)

The bigger the difference between COOR and CONOR makes Red inclined to play “bold”.

Parameter 2 - The number of hits after the “bold” play.

According the number of open points in Red’s home board he may leave between 11 and 16 shots. The lesser amount of shots he leaves encourages him to play “bold”.

Parameter 3 - The strength of Red’s board may be an asset to make his opponent waste some time on the bar.

With only a three-point board, Red most often has rolls with doublets that gain a roll but three points keeps White on the bar only 25% of the time. Therefore, with a three-point board Red is inclined to play “bold”.

Parameter 4 - Gaps.

In Positions 7 and 8 (down below), White might not use small doubles efficiently during bear off, so this kind of configuration inclines Red to play “safe”.

Parameter 5 - The parity of the number of remaining checkers on the board from the player who is on the bar.

In Position 3 and 3a (further down in this article), White has the same number of crossovers but he may reach positions such as in this sample…

….in which he has an odd number of checkers remaining. An even number is not as good because he may find himself in a four-roll position with a roll containing a 1, 2 or a 3 during the bear in process instead of a five-roll position.

Therefore, when White has an odd number of checkers remaining, Red is more inclined to play “bold”.

Others parameters might be taken into account but those we will not be considered here. Ones like:

- The distribution after the “safe” or the “bold” play. Poor distribution of the spare in Red’s board might leave shot equity for the opponent.

- The presence of a blot inside White’s home board.

Further analysis will be needed to take account of these factors in the formula.

Once we have our five parameters we compute an “Aggression Coefficient” indicated as AC to help us to determine which play gives the best chance to win the game.

AC = COOR – CONOR + H + B + G + P

Where,

H = +1.1 if player on roll leaves 11 hits with a four-point or five-point board, -1.1 if player on roll leaves 15 hits, 0 otherwise.

B = +1 if we have only a three-point board, 0 otherwise.

G = -1 if opponent has a gap in his home board with at least four checkers behind, 0 otherwise.

P = +1 if opponent’s number of remaining checkers is odd, 0 otherwise.

If AC is > 0 the right play is the “bold” one

IF AC is < 0 or = 0 the right play is the safe one

We used GNUBG 4-ply for all evaluations.

Position 1

 

AC for Position 1

COOR

CONOR

H

B

G

P

AC

8

9

+1.1

0

-1

+1

+0.1

             


 

AC

Decision according AC

+0.1

bold

Correct play according GNU 4Ply

bold

after “bold” play

%

0.580

Equity

+0.453

after “safe” play

%

0.573

Equity

+0.351

Difference between plays

%

0.007

Equity

-0.102

 

Below follows an explanation of the data seen in the tables accompanying positions.

First table below each position using above Position 1 as an example: 

AC for Position 1

COOR

CONOR

H

B

G

P

AC

8

9

+1.1

0

-1

+1

+0.1

             

- After 4-3 played safe (10/3) Red has 8 crossovers so COOR = 8

- White has 4 crossovers to get into his home board and 5 to bear off his 9 checkers so CONOR = 9.

- If Red plays “bold” leaving a shot in his home board, he leaves only 11 shots so H = +1.1

- Red has a 5-point board so B = 0

- White has a gap on his 1 point which will probably cause some wastage during the bear off process so G = -1

- White has 9 checkers remaining on the board which is an odd number so P = +1

So the formula becomes:

AC = COOR – CONOR + H + B + G + P

AC = 8 -9 +1.1 + 0 -1 +1 = +0.1

The AC result is a positive number so Red should play “bold” and take a checker off (10/6 3/0).

The second table in each position gives the decision according to the value of AC and the evaluation of GNU 4-ply:

 

AC

Decision according AC

+0.1

bold

Correct play according GNU 4Ply

bold

after “bold” play

%

0.580

Equity

+0.453

after “safe” play

%

0.573

Equity

+0.351

Difference between plays

%

0.007

Equity

-0.102

The table indicates the raw winning chances after each play, the cubeful equity, and the difference between plays as well. Although the difference in cubeless chances are sometimes small we can reason that the equity difference is significant because of the extra checker out.

In these types of positions, GNUBG’s 4-ply assessment is usually better than Snowie 4.5.


Here are more positions we tested with the formula to verify its accuracy:

Position 2

 

AC for Position 2

COOR

CONOR

H

B

G

P

AC

8

9

-1.1

1

0

+1

-0.1

 

 

AC

decision

-0.1

safe

Correct play according GNU 4Ply

safe

after “bold” play

%

0.324

Equity

-0.247

after “safe” play

%

0.360

Equity

-0.168

Difference between plays

%

0.036

Equity

-0.079

 


 

Position 3

 

 

AC for Position 3

COOR

CONOR

H

B

G

P

AC

8

9

0

0

0

+1

0

 

 

AC

decision

0

safe

Correct play according GNU 4Ply

safe

after “bold” play

%

0.409

Equity

-0.040

after “safe” play

%

0.420

Equity

-0.012

Difference between plays

%

0.011

Equity

-0.028

 


 

Position 3a

 

AC for Position 3a

COOR

CONOR

H

B

G

P

AC

8

9

0

0

0

0

-1

 

 

AC

decision

-1

safe

Correct play according GNU 4Ply

safe

after “bold” play

%

0.504

Equity

+0.097

after “safe” play

%

0.563

Equity

+0.281

Difference between plays

%

0.059

Equity

+0.184

 


 

 Position 4

 

AC for Position 4

COOR

CONOR

H

B

G

P

AC

8

9

0

+1

0

+1

+1

 

 

AC

decision

+1

bold

Correct play according GNU 4Ply

bold

after “bold” play

%

0.380

Equity

-0.102

after “safe” play

%

0.363

Equity

-0.160

Difference between plays

%

0.017

Equity

-0.058

 


 

Position 5

 

AC for Position 5

COOR

CONOR

H

B

G

P

AC

8

9

+1.1

0

0

+1

+1.1

 

 

AC

decision

+1.1

bold

Correct play according GNU 4Ply

bold

after “bold” play

%

0.453

Equity

+0.070

after “safe” play

%

0.421

Equity

-0.011

Difference between plays

%

0.032

Equity

-0.081

 


 

Position 6

 

AC for Position 6

COOR

CONOR

H

B

G

P

AC

8

9

0

0

0

+1

        0

 

 

AC

decision

0

safe

Correct play according GNU 4Ply

safe

after “bold” play

%

0.401

Equity

-0.060

after “safe” play

%

0.417

Equity

-0.020

Difference between plays

%

0.016

Equity

-0.040

 


 

Position 7

 

AC for Position 7

COOR

CONOR

H

B

G

P

AC

8

8

+1.1

0

-1

0

+0.1

 

 

AC

decision

+0.1

bold

Correct play according GNU 4Ply

bold

after “bold” play

%

0.427

Equity

+0.006

after “safe” play

%

0.400

Equity

-0.051

Difference between plays

%

0.027

Equity

-0.057

 


 

Position 8

 

AC for Position 8

COOR

CONOR

H

B

G

P

AC

8

8

0

0

-1

0

-1

 

 

AC

decision

-1

safe

Correct play according GNU 4Ply

safe

after “bold” play

%

0.387

Equity

-0.084

after “safe” play

%

0.400

Equity

-0.051

Difference between plays

%

0.013

Equity

-0.033

.



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