So You Think Backgammon is Easy?

LOL a georgian friend of mine told me if u put the checker on the 6 point u should cube since u have a take if u do not roll 52 or better.....

I agree with the analysis. In both cases the equity after doubling is 8/36 since white cashes anyway after Black misses. On the other hand, the equity after not doubling is clearly worse in the second case, since White has no cube access and has to play the game to the end. I have both equities as 13.44/36 and 7.77/36, which makes position 2 a very thin double (I could be wrong with the numbers). Nice example, but the math is more convincing than putting the position into a bot.

From: tobaksa   

This is old material (the fact that it has a paradox named for a player who has been dead twenty-five years hits at how old). Nothing wrong with old material; we all need a reminder, and the new generation needs a heads up. But it points to a limitation of relying on the bots: they don't explain why something is right or wrong. The math is easy, and should have been included. A ten-cent #2 pencil will provide a more lucid explanation than a four-hundred dollar software program.

Jake Jacobs

This is indeed very old material and I agree it would have been better to present the maths (not difficult). There are five Jacoby Paradoxes where it is correct to double but not redouble. In each case the player on roll has checkers on 5pt and 2pt. The opponent has a checker on [5pt] or [4pt, 1pt] resulting in a recube to 8 which has to be dropped; or [6pt] with an optional take or drop of recube to 8; or [2pt, 2pt] or [2pt, 3pt] with a recube to 8 which should be taken.

Rich: can you give an example of the more complex positions to which you refer ? Thanks.

Ray Kershaw

From: PhilSimborg   

In response to Jake's comments, what's a "No. 2 pencil?" Many of us are not that old.

As for this being old material, the fact is that I discovered this myself, over the board. (A few weeks ago I also came up with a great idea for putting wheels on suitcases.)

As for the math, I agree with Coolray...if you do it yourself instead of having someone else do it for you, you will understand it better and it will be more meaningful.

But there is another answer that is less mathmetical...if you don't double one position you will win more because of the extra misses your opponent has, but he doesn't have those extra misses if he can cube you out.

Phil

Kamyar

In position 1, if black does not redouble, (a) black bears off 19 times out of 36 and wins 2; (b) black does not bear off 17 times out of 36 but then white misses 5 times out of 36 and black wins 2 (black doubles white out if black's previous roll was 21); (c) black does not bear off 17 times out of 36 but white does bear off 31 times out of 36 and black loses 2. On average, black wins [(19/36) x 2] + [(17/36) x (5/36) x 2] - [(17/36) x (31/36) x 2] = 0.373 points per game.

In position 1, if black does redouble, (a) black bears off 19 times out of 36 and wins 4; (b) black does not bear off 17 times out of 36 but then white doubles black out and black loses 4. On average, black wins [(19/36) x 4] - [(17/36) x 4] = 0.222 points per game.

In 1296 games, by redoubling black wins 4 instead of 2 points 684 times (19 x 36), loses 4 instead of 2 points 527 times (17 x 31) and loses 4 instead of winning 2 points 85 times (17 x 5).

Ray Kershaw

Actually, putting it in any good program reveals that not recognizing the first position as a no redouble is a real mistake (about 0.07 mistake),

while not recognizing the second position as a redouble can hardly be categorized as a mistake, since it's only 0.003 off from no redouble to redouble/take.

Not understanding the second position will have little statistical significance on your play.

It is a very nice illustration of a very nice idea though.