BGonline.org Forums

4-roll position cube action with funny dice

Posted By: Daniel Murphy
Date: Thursday, 13 September 2012, at 4:58 a.m.

This post is just a repost correcting the errors that Casper pointed out that I made here. Elsewhere in that thread are other examples of how skewing the frequency of doublets also skews the frequency of nondoublets, and changes the game in many ways that, as I said, I doubt are obvious to players using the skewed online dice.

Unlimited match, Blue on roll.

White	8
Blue	8
Position ID: /wAAAP4BAAAAAA Match ID: cAkAAAAAAAAE

This backgammon reference position is, as we all know, a double/take and redouble/take.

In the thread about XG Mobile marketing which veered into a discussion of dice at Safe Harbor Games, I posed the question: what is the correct cube action when doublets occur less frequently than they should, as they do in the Safe Harbor Games "red room." Before answering that, I thought I would try to calculate, in the most direct way, how often White wins holding a two-cube, assuming normal dice. Since the underdog's GWC is the smaller number, that should be the quickest to find.

Since Blue is off in at most four rolls, to win, White must bear off in three rolls or two (or be able to redouble Blue out), roll at least one doublet, and roll one more doublet than Blue. With normal dice, 1/6 of all rolls are doublets and 5/6 are not. There are four winning sequences for White:

(1) Blue rolls no doublets, White rolls a doublet on his third turn.	5/6 * 5/6 * 5/6 * 5/6 * 5/6 * 1/6 =	0.066979
(2) Blue rolls no doublets, White rolls a doublet on his second turn.	5/6 * 5/6 * 5/6 * 1/6 * 5/6 =	0.080375
(3) Blue rolls no doublets, White rolls a doublet on his first turn (and redoubles Blue out).	5/6 * 1/6 * 5/6 =	0.115740
(4) Blue rolls a doublet on either his first or second turn, White rolls two doublets.	(5/6 * 1/6 * 1/6 * 1/6) + (1/6 * 1/6 * 5/6 * 1/6) =	0.007716
	total:	0.270812

Holding a 2-cube, White wins 27.08%. Cubeless, White wins only 25.47%. The difference comes from being able to double Blue out as in (3), above. This gains 5/6 * 1/6 * 5/6 * 5/6 * 1/6 = 1.61%.

---

What is the correct cube action in the Safe Harbor Games rooms marked with the red dice that don't roll a doublet one time in six? That depends on the probability of doublets there. Estimates offered in this forum vary a bit. I'm going to rely on a post by Michael Petch in April 2012 where he wrote: "Red Dice = Incorrect dice that generate about 9.2-9.6% doubles. Affectionately called 'social dice.'"

So I will use 9.4% as the frequency of doublets and, perforce, 90.6% as the frequency of nondoublets. Now how often does White win?

(1) Blue rolls no doublets, White rolls a doublet on his third turn.	.906 * .906 * .906 * .906 * .906 * .094 =	0.057381
(2) Blue rolls no doublets, White rolls a doublet on his second turn.	.906 * .906 * .906 * .094 * .906 =	0.063334
(3) Blue rolls no doublets, White rolls a doublet on his first turn (and redoubles Blue out).	.906 * .094 * .906 =	0.077158
(4) Blue rolls a doublet on either his first or second turn, White rolls two doublets.	(.906 * .094 * .094 * .094) + (.094 * .094 * .906 * .094) =	0.001505
	total:	0.199379

Holding a 2-cube, White wins 19.94%. Correct cube action is double/pass. Cubeless, White wins only 19.28%. The difference, as with normal dice, comes from being able to double Blue out as in (3), above. This gains .906 * .094 * .906 * .906 * .094 = 0.66%.

---

Real backgammon:

White	8
Blue	8
Position ID: /wAAAP4BAAAAAA Match ID: cAkAAAAAAAAE

Cube decision
Rollout cubeless equity		+0.49051
Cubeful equities:
1.	Double, take	+0.91671
2.	Double, pass	+1.00000	+0.08329
3.	No double	+0.76080	-0.15590
Proper cube action:		Double, take
Rollout details
Win W g W bg Lose L g L bg Cubeless Cubeful Centered 1-cube 0.74525 0.00000 0.00000 - 0.25475 0.00000 0.00000 +0.49051 +0.76080 Standard error 0.00000 0.00000 0.00000 - 0.00000 0.00000 0.00000 0.00000 0.00000 Player White owns 2-cube 0.74525 0.00000 0.00000 - 0.25475 0.00000 0.00000 +0.98101 +0.91671 Standard error 0.00000 0.00000 0.00000 - 0.00000 0.00000 0.00000 0.00000 0.00001
Full cubeful rollout with var.redn.
46656 games, Mersenne Twister dice gen. with seed 694778273 and quasi-random dice
Play: world class 2-ply cubeful prune [world class]
keep the first 0 0-ply moves and up to 8 more moves within equity 0.16
Skip pruning for 1-ply moves.
Cube: 2-ply cubeful prune [world class]

White	8
Blue	8
Position ID: /wAAAP4BAAAAAA Match ID: UQkAAAAAAAAE

Cube decision
Rollout cubeless equity		+0.49051
Cubeful equities:
1.	Double, take	+0.91671
2.	Double, pass	+1.00000	+0.08329
3.	No double	+0.79296	-0.12375
Proper cube action:		Redouble, take
Rollout details
Win W g W bg Lose L g L bg Cubeless Cubeful Player Blue owns 2-cube 0.74525 0.00000 0.00000 - 0.25475 0.00000 0.00000 +0.49051 +0.79296 Standard error 0.00000 0.00000 0.00000 - 0.00000 0.00000 0.00000 0.00000 0.00000 Player White owns 4-cube 0.74525 0.00000 0.00000 - 0.25475 0.00000 0.00000 +0.98101 +0.91671 Standard error 0.00000 0.00000 0.00000 - 0.00000 0.00000 0.00000 0.00000 0.00001
Full cubeful rollout with var.redn.
46656 games, Mersenne Twister dice gen. with seed 694778273 and quasi-random dice
Play: world class 2-ply cubeful prune [world class]
keep the first 0 0-ply moves and up to 8 more moves within equity 0.16
Skip pruning for 1-ply moves.
Cube: 2-ply cubeful prune [world class]