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storm's position analyzed (but something still puzzles me...)

Posted By: Timothy Chow
Date: Tuesday, 13 April 2010, at 11:45 p.m.

About a month ago, storm posted a safe versus bold problem that I found quite intriguing. In this post I share some rollouts that have helped me understand the position, but something still puzzles me about it, as I will explain below.

First, here is the position again.

The score (after 0 games) is: gnubg 0, jose 1 (match to 5 points)

gnubg	99
jose	122
Position ID: 2+YMABRjN4MIAw Match ID: QYmpAAAACAAA

The obvious question is whether Blue should hit or play safe (with 13/8 let's say). The score does matter somewhat here, but from this point on I will pretend that it's an unlimited game because I find it helpful to understand the unlimited case first and then adjust for the score if necessary afterwards.

Daniel Murphy rolled the position out (ATS), and my rollout (unlimited) confirms that in the given position, playing safe beats out hitting. (Rollouts appear at the end of this post.) We might think that this is a simple consequence of safe/bold criteria. White's board is stronger so getting hit back is very costly and could lead to a gammon. Containing White's rear checkers by playing safely is a better bet even though Blue is behind in the race.

That's partly true, but it's not the whole story. I tried rolling out the position giving Blue a deuce-point anchor instead of a five-point anchor. The rollout then prefers hitting by about 0.04. My explanation for this is if Blue is stuck with a deuce-point anchor then Blue is in serious trouble already. His anchor doesn't give him any outfield coverage and it's hard for him to escape those back checkers safely. So he loses less (compared with the original five-point anchor position) by getting hit back and he stands to gain a lot.

This all sound plausible, but now comes the part that I don't understand. I decided to switch the location of the cube to see if that made any difference. I found that owning the cube, and with a deuce-point anchor, hitting and playing safe were virtually a toss-up. This sort of makes sense too, I think, because usually when you're holding the cube, making a risky loose hit damages your recube efficiency. You'd prefer to creep up on an efficient recube than to go for an all-or-nothing hit that will make your cube worthless if things go poorly and will net you only one point if things go well. The funny thing, though, is that with a five-point anchor and owning the cube, the rollout slightly prefers to hit! I can't quite make sense of this. I would have predicted that the hit would be even more wrong than in the original position. Anyone have an explanation? I think I am not correctly understanding the value of cube ownership in these positions.

Here are the rollouts. As stated, they are all unlimited games.

---

Original position

gnubg	99
jose	122
Position ID: 2+YMABRjN4MIAw Match ID: QYkJAAAACAAA

#		Ply	Move	Equity
•	1	R	13/8	+0.194
			0.614 0.172 0.004 - 0.386 0.094 0.002 +0.308 +0.194 0.001 0.002 0.000 - 0.001 0.001 0.000 0.003 0.004
			Full cubeful rollout with var.redn.
			1296 games, Mersenne Twister dice gen. with seed 853693477 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]
	2	R	16/13 5/3*	+0.152 ( -0.042)
			0.617 0.180 0.004 - 0.383 0.127 0.002 +0.288 +0.152 0.001 0.002 0.000 - 0.001 0.001 0.000 0.003 0.006
			Full cubeful rollout with var.redn.
			1296 games, Mersenne Twister dice gen. with seed 853693477 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]

---

Deuce-point anchor

gnubg	105
jose	128
Position ID: s+0MABRjN4MIGA Match ID: QYkJAAAACAAA

#		Ply	Move	Equity
•	1	R	16/13 5/3*	-0.391
			0.450 0.122 0.003 - 0.550 0.213 0.010 -0.199 -0.391 0.001 0.002 0.000 - 0.001 0.001 0.000 0.004 0.006
			Full cubeful rollout with var.redn.
			1296 games, Mersenne Twister dice gen. with seed 853683931 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]
	2	R	13/8	-0.430 ( -0.039)
			0.420 0.108 0.003 - 0.580 0.160 0.007 -0.217 -0.430 0.001 0.002 0.000 - 0.001 0.002 0.001 0.004 0.006
			Full cubeful rollout with var.redn.
			1296 games, Mersenne Twister dice gen. with seed 853683931 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]

---

Deuce-point anchor, owning the cube

gnubg	105
jose	128
Position ID: s+0MABRjN4MIGA Match ID: UYmpAAAACAAA

#		Ply	Move	Equity
•	1	R	13/8	-0.154
			0.407 0.110 0.003 - 0.593 0.152 0.007 -0.267 -0.154 0.002 0.002 0.000 - 0.002 0.002 0.000 0.005 0.005
			Full cubeful rollout with var.redn.
			1296 games, Mersenne Twister dice gen. with seed 854066340 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]
	2	R	16/13 5/3*	-0.158 ( -0.004)
			0.436 0.123 0.004 - 0.564 0.211 0.009 -0.294 -0.158 0.002 0.003 0.000 - 0.002 0.002 0.001 0.005 0.006
			Full cubeful rollout with var.redn.
			1296 games, Mersenne Twister dice gen. with seed 854066340 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]

---

Original position, but owning the cube

gnubg	99
jose	122
Position ID: 2+YMABRjN4MIAw Match ID: UYkJAAAACAAA

#		Ply	Move	Equity
•	1	R	16/13 5/3*	+0.553
			0.598 0.190 0.004 - 0.402 0.135 0.001 +0.254 +0.553 0.001 0.001 0.000 - 0.001 0.001 0.000 0.003 0.005
			Full cubeful rollout with var.redn.
			1296 games, Mersenne Twister dice gen. with seed 853752760 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]
	2	R	13/8	+0.532 ( -0.021)
			0.602 0.175 0.004 - 0.398 0.092 0.001 +0.291 +0.532 0.001 0.002 0.000 - 0.001 0.001 0.000 0.003 0.006
			Full cubeful rollout with var.redn.
			1296 games, Mersenne Twister dice gen. with seed 853752760 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]