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3-away 4-away cube action - Analysis

Posted By: Daniel Murphy
Date: Sunday, 9 October 2011, at 8:00 p.m.

In Response To: 3-away 4-away cube action (Daniel Murphy)

Repeating the position, which Jake Jacobs wrote up in 1995 (do see http://www.jakejacobs.org/kurt.htm for Jake's 1995 analysis and humor):

Blue on roll leads 4-3/7, that is, Blue leads 3-away 4-away.

White	10
Blue	16
Position ID: /wIAALwfAAAAAA Match ID: cAngADAAIAAE

The first question should be: is it a take?

If we use the Rockwell-Kazaross Match equity Table, White can drop to trail 4-away 2-away with 33.14% match equity. Or White can take. White's CUBELESS probability of winning = 30.4848%. This, by the way, is 2.23% better than a pure 5-roll position with all checkers on the ace point, where CPW = 28.25%.

Correct cube action is MET dependent. So is recube action on Blue's next roll. Yes, White often has an immediate redouble as an underdog!

With the Rockwell-Kazaross Match Equity Table, White can pass to trail 4-away 2-away for 33.1414% Match Equity, and this is a double and close pass.

Action			Recube?	CPW		ME =>
Pass	->	Trail 4-away 2-away			->	ME =	33.1414%
OR
Take	->	Blue rolls 66 55 44 33 22	No RD	5.2583	->	ME = 5/36 * 20.79327% =	2.8880%
	->	Blue rolls 11	No RD	25.587	->	ME = 1/36 * 31.30085% =	0.8695%
	->	Blue rolls other non-ace	No RD	34.338	->	ME = 20/36 * 34.50329%=	19.1685%
	->	Blue rolls other ace	REDOUBLE	35.881	->	ME = 10/36 * 35.88123%=	9.9670%
						TOTAL	32.8930%

Since 33.14% is more than 32.8935%, Pass is correct by 0.248% ME.

A note for Gnubg users: 2-ply is wrong on the initial double and on the one redouble. 2-ply and 4-ply analysis usually agree, but not here.

Other Match Equity Tables:

With the Woolsey-Heinrich MET, Double/Take.

Action			Recube?	CPW		ME =>
Pass	->	Trail 4-away 2-away			->	ME =	32.0000%
OR
Take	->	Blue rolls 66 55 44 33 22	No RD	5.258	->	ME = 5/36 * 19.26110% =	2.6752%
	->	Blue rolls 11	No RD	25.587	->	ME = 1/36 * 30.37182% =	0.8436%
	->	Blue rolls other non-ace	REDOUBLE	34.338	->	ME = 20/36 * 34.33781%=	19.0766%
	->	Blue rolls other ace	REDOUBLE	35.881	->	ME = 10/36 * 35.88123%=	9.9670%
						TOTAL	32.5624%

Gnubg 2-ply gets both of the redoubles wrong.

With the Snowie MET, Double/Pass.

Action			Recube?	CPW		ME =>
Pass	->	Trail 4-away 2-away			->	ME =	33.6000%
OR
	->	Blue rolls 66 55 44 33 22	No RD	5.258	->	ME = 5/36 * 20.27169% =	2.8155%
	->	Blue rolls 11	No RD	25.587	->	ME = 1/36 * 30.85824% =	0.8572%
	->	Blue rolls other non-ace	REDOUBLE	34.338	->	ME = 20/36 * 34.33782%=	19.0766%
	->	Blue rolls other ace	REDOUBLE	35.881	->	ME = 10/36 * 35.88123%=	9.9670%
						TOTAL	32.743%

Gnubg 2-ply gets the initial double and both of the redoubles wrong.

With the G11 MET, Double/Pass.

Action			Recube?	CPW		ME =>
Pass	->	Trail 4-away 2-away			->	ME =	33.6000%
OR
	->	Blue rolls 66 55 44 33 22	No RD	5.258	->	ME = 5/36 * 20.42642% =	2.8155%
	->	Blue rolls 11	No RD	25.587	->	ME = 1/36 * 31.05721% =	0.8572%
	->	Blue rolls other non-ace	REDOUBLE	34.338	->	ME = 20/36 * 34.33782%=	19.0766%
	->	Blue rolls other ace	REDOUBLE	35.881	->	ME = 10/36 * 35.88123%=	9.9670%
						TOTAL	32.7160%

Gnubg 2-ply gets the initial double and both of the redoubles wrong.

---

I wrote this up some weeks ago but neglected to post it. That was before Gnubg had the XG MET. With the XG MET, this is a close pass:

Cube decision
Rollout cubeless MWC		61.727%	(Money: +0.3903)
Cubeful equities:
1.	Double, pass	66.870%
2.	Double, take	67.125%	0.255%
3.	No double	64.788%	-2.082%
Proper cube action:		Double, pass
Rollout details
Win W g W bg Lose L g L bg Cubeless Cubeful Centered 1-cube 0.6952 0.0000 0.0000 - 0.3048 0.0000 0.0000 61.727% 64.788% Standard error 0.0000 0.0000 0.0000 - 0.0000 0.0000 0.0000 0.000% 0.002% Player White owns 2-cube 0.6952 0.0000 0.0000 - 0.3048 0.0000 0.0000 68.821% 67.125% Standard error 0.0000 0.0000 0.0000 - 0.0000 0.0000 0.0000 0.000% 0.001%
Full cubeful rollout with var.redn.
1296 games, Mersenne Twister dice gen. with seed 934941877 and quasi-random dice
Play: 4ply 4-ply cubeful prune [4ply]
keep the first 0 0-ply moves and up to 16 more moves within equity 0.32
Skip pruning for 1-ply moves.
keep the first 0 2-ply moves and up to 4 more moves within equity 0.08
Skip pruning for 3-ply moves.
Cube: 4-ply cubeful prune [4ply]

Messages In This Thread

• 3-away 4-away cube action
  Daniel Murphy -- Saturday, 20 August 2011, at 2:49 p.m.
  • 3-away 4-away cube action
    neilkaz -- Saturday, 20 August 2011, at 3:27 p.m.
    • 3-away 4-away cube action
      Fabrice Liardet -- Saturday, 20 August 2011, at 9:15 p.m.
  • 3-away 4-away cube action
    Fabrice Liardet -- Saturday, 20 August 2011, at 9:28 p.m.
  • 3-away 4-away cube action - Analysis
    Daniel Murphy -- Sunday, 9 October 2011, at 8:00 p.m.