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7. December 2005, 22:27:43
playBunny 
Subject: Re: Cube numbers
grenv, Hqrls I won't claim to be an expert in this area so here's what I understand myself.

W 72.8%, Wg 29.3%, Wbg 2.6%, L 27.2% Lg 6.4%, Lbg 0.5%

Wins (all), Wins by gammon, Wins by backgammon, Losses ditto
The percentages are for the person cubing.
(That's why I said "nicely judged" to you grenv because Hrqls' chaces were 27.2%)

1. Double, pass _______ +1.000
2. Double, take _______ +1.217 (+0.217)
3. No double __________ +0.964 (-0.036)

The equity figures are how many points you'd gain or lose on average. Drop the cube and you'll lose every match 1.000. Fail to double now and you'll only win .964, ie. delaying for 1 roll will cost 3.6% of the possible points (because of losing the game or the next one or from subsequent cube decisions). If the opponent takes the cube then the gain will be .217 above expectation. That's a huge increase, and a huge additional loss for the loser given that they could have dropped and only lost the single point.

The 75%/25% double and take rule works because it's the break-even point for the taker. if they play 4 nmatches and take, then losing 75% at 2 will cost 6 points but they'll win 2 back with the other 25% for a net loss of 4. That's the same as if they had dropped the cube in all four games.

That rule doesn't take into account the gammon and backgammon wins which, in this example, are considerable. Two men on the wrong side of a 3-prime with good builders. That's the extra factor that makes it such a blunder. The maths now becomes
(72.8 - 29.3 - 2.6 = 40.9) x 2 win points +
(29.3 - 2.6% = 26.7) x 4 gammon points +
2.6 x 8 backgammmon points
versus
27.2 x 2 lost points.

As far as knowing exactly how the equity figure is worked out I can't say. In a single game it's as straightforward as the maths for the 75%/25% example shown but for longer matches I believe the equity value also has the future matches factored in through the use of the match equity table. That's where I get uncertain because to it seems logical that equity for a match can only go as high as 1. Perhaps Alan can explain this aspect.

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