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R/O - money game equities

Posted By: Daniel Murphy
Date: Thursday, 8 March 2012, at 1:29 p.m.

In Response To: R/O (Snua)

Money game, Blue has cube, Blue on roll.

White	19
Blue	13
Position ID: ZAEA4DIAAAAAAA Match ID: UQkAAAAAAAAE

Cube decision
Rollout cubeless equity		+0.3922
Cubeful equities:
1.	Double, take	+0.6676
2.	Double, pass	+1.0000	+0.3324
3.	No double	+0.5532	-0.1144
Proper cube action:		Redouble, take
Rollout details
Win W g W bg Lose L g L bg Cubeless Cubeful Player Blue owns 2-cube 0.6961 0.0000 0.0000 - 0.3039 0.0000 0.0000 +0.3922 +0.5532 Standard error 0.0000 0.0000 0.0000 - 0.0000 0.0000 0.0000 0.0000 0.0000 Player White owns 4-cube 0.6961 0.0000 0.0000 - 0.3039 0.0000 0.0000 +0.7844 +0.6676 Standard error 0.0000 0.0000 0.0000 - 0.0000 0.0000 0.0000 0.0000 0.0000
Full cubeful rollout with var.redn.
46656 games, Mersenne Twister dice gen. with seed 914008424 and quasi-random dice
Play: supremo 2-ply cubeful prune [world class]
keep the first 0 0-ply moves and up to 16 more moves within equity 0.32
Skip pruning for 1-ply moves.
Cube: 2-ply cubeful prune [world class]

Cubeless Blue would win 69.61%. But cubeful, Blue wins 74.80% after no redouble, 68.15% after redouble/take. And, after redouble/take, White's cube vig reduces Blue's GWC to the equivalent of 66.69%, if all games were worth 4 points.

[Cubeless equity = 0.7844. Divide that by 4 to get 0.1961. Add 0.5 to that to get 0.6961. That's Blue's cubeless GWC, 69.61%. Cubeful equity = 0.6676. Divide that by 4 and add 0.5 to the result to get 66.69%. I don't know if this number has a name. Cube-adjusted cubeless GWC? The main point is, if the game were played out cubeless, Blue would win 69.61%. Blue wins less than that (68.15%) after giving the cube to White because White can double Blue out sometimes. And because White can also sometimes redouble Blue in, and gains equity in the games that end with the cube on 8 or higher, the total impact of White's cube ownership is as if Blue won 66.69% of all games with the cube on 4.]

After redouble/take:

Game-end cube value	%	White wins	Blue wins
4	87.34%	23.74%	63.60%	(Blue wins 72.80% of the 4-cubes)
8	10.89%	7.45%	3.44%	(White wins 68.39% of the 8-cubes)
16	1.11%	0.33%	0.78%	(Blue wins 70.38% of the 16-cubes)
32	0.66%	0.33%	0.33%	(White wins 50.49% of the 32-cubes)

Money game, centered cube, Blue on roll.

White	19
Blue	13
Position ID: ZAEA4DIAAAAAAA Match ID: cAkAAAAAAAAE

Cube decision
Rollout cubeless equity		+0.3922
Cubeful equities:
1.	Double, take	+0.6676
2.	Double, pass	+1.0000	+0.3324
3.	No double	+0.5024	-0.1652
Proper cube action:		Double, take
Rollout details
Win W g W bg Lose L g L bg Cubeless Cubeful Centered 1-cube 0.6961 0.0000 0.0000 - 0.3039 0.0000 0.0000 +0.3922 +0.5024 Standard error 0.0000 0.0000 0.0000 - 0.0000 0.0000 0.0000 0.0000 0.0000 Player White owns 2-cube 0.6961 0.0000 0.0000 - 0.3039 0.0000 0.0000 +0.7844 +0.6676 Standard error 0.0000 0.0000 0.0000 - 0.0000 0.0000 0.0000 0.0000 0.0000
Full cubeful rollout with var.redn.
46656 games, Mersenne Twister dice gen. with seed 914008424 and quasi-random dice
Play: supremo 2-ply cubeful prune [world class]
keep the first 0 0-ply moves and up to 16 more moves within equity 0.32
Skip pruning for 1-ply moves.
Cube: 2-ply cubeful prune [world class]

Note that even if Blue rolls his worst number, 3-1, in the position above, White does not have a double:

Money game, centered cube, White on roll.

White	19
Blue	9
Position ID: lwAAAFkAAAAAAA Match ID: MAEAAAAAAAAE

Alert: wrong double ( -0.0742)! [bad]

Cube decision
Rollout cubeless equity		+0.1701
Cubeful equities:
1.	No double	+0.2555
2.	Double, pass	+1.0000	+0.7445
3.	Double, take	+0.1814	-0.0742
Proper cube action:		No double, take (9.1%)
Rollout details
Win W g W bg Lose L g L bg Cubeless Cubeful Centered 1-cube 0.5851 0.0000 0.0000 - 0.4149 0.0000 0.0000 +0.1701 +0.2555 Standard error 0.0000 0.0000 0.0000 - 0.0000 0.0000 0.0000 0.0000 0.0000 Player Blue owns 2-cube 0.5849 0.0000 0.0000 - 0.4151 0.0000 0.0000 +0.3398 +0.1814 Standard error 0.0000 0.0000 0.0000 - 0.0000 0.0000 0.0000 0.0000 0.0000
Full cubeful rollout with var.redn.
15139 games, Mersenne Twister dice gen. with seed 914008424 and quasi-random dice
Play: supremo 2-ply cubeful prune [world class]
keep the first 0 0-ply moves and up to 16 more moves within equity 0.32
Skip pruning for 1-ply moves.
Cube: 2-ply cubeful prune [world class]

Messages In This Thread

• Cube action?
  Snua -- Wednesday, 7 March 2012, at 9:22 a.m.
  • Cube action?
    Timothy Chow -- Thursday, 8 March 2012, at 1:21 a.m.
    • R/O
      Snua -- Thursday, 8 March 2012, at 11:43 a.m.
      • R/O - money game equities
        Daniel Murphy -- Thursday, 8 March 2012, at 1:29 p.m.
        • R/O - money game equities
          Snua -- Thursday, 8 March 2012, at 6:30 p.m.