Could someone explain why it is "clearly correct" for blue to double here (per Woosly's article)



"Obviously this is a take, since White will win 17/36 of the time. However, the position couldn't be more volatile, since the whole game will be decided by Blue's next roll. Consequently, it is clearly correct for Blue to double. Note the famous Jacoby paradox here -- if White's men were on the one and four points then it would still be correct to make an initial double but not to redouble. The difference is that here the next roll doesn't decide everything -- Blue still has some life after death if he holds the cube. "


Ok I don't know how to make this image show up, or how to attach images from my own hard drive. The position is a bear off in which blue has two men left, one on the 2 point and one on the 5 point. White also has two men left, both are on his 1 point, blue is on roll with the cube in the center.


??

OK, I'll take a crack at it.

Consider a cross-section of 36 games, one for each possible dice roll.

If Blue doesn't double, he wins 19 games with the cube on 1, and loses 17. His profit over the 36 games = 2 points.

If Blue doubles and White takes, he wins 19 games with the cube on 2, and loses 17. His profit over the 36 games = 4 points.

4 > 2, so I double.

Bill wins

Ok that seems pretty straightforward. I don't know why I thought there was more to it than that.

So for all these '1 roll left' positions, you can simply break it down into an expected value problem, since 'owning' the cube has no real value?

And also, does this hold for all match scores (assume 5 pt match) in which doubling is possible? Am I correct in assuming it would not be wise to double in a match near match point and push such a small edge, especially if you thought you were the better player?

I'd be very cautious with arguments like "I know it's technically a double but I'm the better player and I don't want to give him a chance to get back in the match and blah blah blah..."

It doesn't take a lot of mistakes like this before the "better" player suddenly isn't a favorite anymore. In all games, aggressive play dominates passive play. If you think you have a technically correct double at a given match score, just double.

This is rather complex to answer generally, but often it would be even clearer to double when you're close to winning the match than at an even score (in last roll positions that is)

Here's a classic:



Blue wins 39% of the time, and not doubling is a huge blunder. This is both because the point blue wins from leading 4-0 to winning the match is very valuable and the point white wins from trailing 3-1 to trailing 3-2 is not all that valuable (because he will be playing aggressively for a gammon at both scores).

Ok,

If it is correct for blue to double in this spot based on the expected value, then why is it correct for white to TAKE?

If white takes, then out of a 36 game sample he is -4 points ((17*2) - (19*2))

Since he has no re-cube equity wouldn't this constitute a drop?

And how much does he lose if he drops all 36 games?

Right but I'm assuming its correct for him to take, and so he does take every time.

So is it NOT correct for white to take? That's what it looks like to me?

If you count him dropping every time its even lower, far lower........

Exactly. He has two choices: take or drop.

If he takes as described ealier he will lose (19-17)*2/36=.111 point per game.

If he drops he will lose 1 point per game.

I think it's starting to make sense. So the reason taking is correct, is that he loses LESS than he would if he just straight dropped.? Kind of like splitting 8's against a face card in black jack?

It just seems counter intuitive because by taking, he is DOUBLING the amount he loses when he is an underdog.

It's just like calling all-in on a decent draw.