acepoint’s home

Checker distribution

We start with an easy example and compare a few positions with an equal pipcount of the player but with different checker distribution on one side:

Pip counts: gnubg 11, ace 13

Position ID: UgAAEAwAAAAAAA Match ID: UQkAAAAAAAAA

• ace doubles

Cube decision
2-ply cubeless equity		+0.119
	0.559 0.000 0.000 – 0.441 0.000 0.000
Cubeful equities:
1.	No double	+0.188
2.	Double, pass	+1.000	+0.812
3.	Double, take	-0.097	-0.285
Proper cube action:		No redouble, beaver (25.9%)

We now change the distribution of the green checkers:

	CL	ND	NR	D/RD
661	0.119	0.016	0.188	-0.097
553	0.265	0.266	0.359	0.325
544	0.299	0.346	0.413	0.445
643	0.351	0.402	0.456	0.580

We can see, that between the worst distribution 661 (no double/beaver) and the best 643 (redouble/take) is a cubless equity difference of 0.232 (0.351-0.119).

Pip counts: gnubg 12, ace 14

Position ID: jAAAIAwAAAAAAA Match ID: UQkAAAAAAAAA

• ace doubles

Cube decision
2-ply cubeless equity		+0.145
	0.572 0.000 0.000 – 0.428 0.000 0.000
Cubeful equities:
1.	No double	+0.277
2.	Double, pass	+1.000	+0.723
3.	Double, take	+0.191	-0.087
Proper cube action:		No redouble, take (10.7%)

Again we change the distribution of the green checkers:

	CL	ND	NR	D/RD
662	0.145	0.208	0.277	0.191
644	0.242	0.323	0.372	0.394
554	0.248	0.337	0.381	0.408
653	0.294	0.401	0.431	0.525

Between the worst distribution (662) and the best (653) is a cubeless equity difference of 0.149.

Pip counts: gnubg 3, ace 12

Position ID: BwAAgAMAAAAAAA Match ID: UQkAAAAAAAAA

• ace doubles

Cube decision
2-ply cubeless equity		+0.117
	0.559 0.000 0.000 – 0.441 0.000 0.000
Cubeful equities:
1.	No double	+0.228
2.	Double, pass	+1.000	+0.772
3.	Double, take	+0.035	-0.193
Proper cube action:		No redouble, take (20.0%)

And again we’ll change the distribution of the green checkers:

	CL	ND	NR	D/RD
444	0.117	0.128	0.228	0.035
552	0.163	0.207	0.276	0.197
633	0.202	0.141	0.256	0.174
651	0.256	0.249	0.338	0.338
642	0.291	0.335	0.409	0.439
543	0.293	0.335	0.399	0.470
443	0.354	0.429	0.475	0.617

The cubeless equity difference between the worst (444) and the best distributions (443) is now 0.237.

Some conclusions:

• It looks like the best distribution on three different points is “2-away” + “1-away”.On future pages I’ll call this “gd” (good distribution).
• Next best seems to be 2 checkers on one point and the remaining checker “1-away” or three checkers in a row. This is also a gd.
• The further away two checkers are, the worse a position is.
• Two checkers on the same point with a distance of at least 3 to the third checker is just bad. I’ll call it “bd” (bad distribution).
• Three checkers on the 4-, 5- or 6-points (no distribution) on greens side of the board is also a bd.
• All checker distributions which don’t fall into category gd or bd are “ad” (average distribution).
• The opponent usually needs doubles, so three checkers on one point doesn’t hurt him so much.
• The cube position (centered or owned by the roller) usually doesn’t make a difference. There are only few “no redouble but initial double” positions.

Legend
CL:	cubeless
ND:	no double
NR:	no redouble
D:	double
RD:	redouble
Green:	redouble/take
Orange:	initial double/take
Red:	no double/beaver