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BACKGAMMON BY THE BAY Problem #5 solution		Match to 11 Blue 0, White 1
		Blue to play. Cube action? Candidate plays: Roll Double / Accept Double / Pass

RK:
Blue is on roll and clearly has a big lead. Does White have enough winning chances to take the cube? And if so, should Blue double now or wait for a better time?

Blue has a big racing lead (45 pips, plus extra for White's checker on the bar). Therefore, for White to win, he must hit a shot, for all practical purposes. After hitting, he doesn't necessarily have to close out the hit checker, but he needs to contain it at least long enough to catch up in the race.

The good news for White is that he has an anchor on Blue's 4 point, so he can't be closed out. Blue has the checker on 16 to bring home, and then she needs to clear the midpoint and the 8 point. Once she's done that, she's in good shape, but there are even some chances to leave a shot on the 6 and 5 points. In addition, White might be able to make a second anchor, which makes it harder for Blue to clear her points, and results in more shots if she leaves a blot.

As far as containing a hit checker, things look pretty good for White. He's got the best possible four-point board, and shouldn't have much trouble moving his other checkers to get more ammunition in place. Cube ownership should be worth a lot to White: he's unlikely to lose his market by a lot after hitting, but should be able to double efficiently when the time is right.

The bad news for White is that he has four checkers back, with one on the bar. If Blue can get home safely, she has decent gammon-winning chances. For example, she has several numbers this time (55, 33 and 53) to put a second checker in the air and make a four-point board. These variations are extremely strong; if White fanned, Blue's gammon chances would be around 44%. Therefore, Blue must double now or risk losing her market by a lot if she rolls one of her best numbers.

The rollout results indicate that it's a fairly clear take for White. One way to use Jellyfish to get a feel for a position is to modify it slightly to see how certain features affect the equity. For example:

Move Blue checker from 6 point to 5 point:	.599, double/drop
Move Blue checker from 16 point to 13 point:	.623, double/drop
Move White checker from bar to 21 point:	.439, double/take
Move White checker from bar to 22 point:	.274, no double/take

(The above figures are Jellyfish level 7 evaluations. The fact that the cubeless rollout results of the initial position are close to the level 7 numbers indicates that Jellyfish "understands" this position pretty well.)

JF:   Double / Accept (level 7 evaluation)

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Pip counts:          Blue   112                     White  157

                     Blue                           White              
Level 7 Evaluation   Equity   Win    G/BG    BG     Win    G/BG    BG  
Cubeless             +0.528   66.8%  24.3%   0.9%   33.2%   5.8%   0.2%

Level 5 Rollouts     Equity   Win    G/BG    BG     Win    G/BG    BG  
Cube centered        +0.734   84.5%   4.8%   0.1%   15.5%   0.5%   0.0%
Blue owns cube       +0.779   88.1%   5.1%   0.1%   11.9%   3.4%   0.1%
White owns cube      +0.425   60.2%  22.6%   0.4%   39.8%   0.8%   0.0%
Cubeless             +0.533   67.5%  24.0%   0.5%   32.5%   6.0%   0.2%

Level 6 Rollouts     Equity   Win    G/BG    BG     Win    G/BG    BG  
Cubeless             +0.531   67.4%  23.5%   0.5%   32.6%   5.6%   0.2%

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