Kauder Paradox - Robertie's discussion
Carl Tait
in Problems 126 and 127 of _Advanced Backgammon_.
(Summary for those who aren't familiar with the paradox: with the
Jacoby Rule in effect, a position can theoretically be both a
proper double and a proper beaver. By doubling, the underdog gets
full value for his potential gammons, thus raising his equity.
However, as long as this equity remains negative, the doubler's
opponent should naturally beaver.)
First, there seems to be an error in the discussion of Problem 127.
There are a few perverse cases in which Black wins but does not win
a gammon. For example, if he starts with 61 followed by seven 21s,
he is still 6 pips short of getting all his checkers off. White
gets an eighth roll, and if he has rolled eight consecutive 66s,
he saves the gammon. Ludicrously unlikely, of course, but it
violates one of the conditions in Problem 126: not *all* of Black's
wins are gammons.
But I think this doesn't matter. The catch is that Black has
better than the lower-bound 30% chance of winning (30.6%). This
almost certainly compensates for his incredibly rare single wins.
So my question is twofold:
1. Is this analysis correct?
2. Can someone point me to a more detailed discussion of the paradox?
Carl Tait
Morten Daugbjerg Hansen
>I have a question about Robertie's discussion of the Kauder Paradox
>in Problems 126 and 127 of _Advanced Backgammon_.
>First, there seems to be an error in the discussion of Problem 127.
>There are a few perverse cases in which Black wins but does not win
>a gammon. For example, if he starts with 61 followed by seven 21s,
>he is still 6 pips short of getting all his checkers off. White
>gets an eighth roll, and if he has rolled eight consecutive 66s,
>he saves the gammon. Ludicrously unlikely, of course, but it
>violates one of the conditions in Problem 126: not *all* of Black's
>wins are gammons.
I cant remember the excact position, but I noticed that Robertie wrote some-
thing like: White now needs xxx pips and cant get that in 7 rolls.
I thought about the eight roll too, and you CAN get xxx pips, BUT
you will have a tremendous waste when bearing into your homeboard,
and therefore you cannot save the gammon even in this rare case.
Try set it up for yourself.
(this is all from free memory so forgive me if i remembered wrong)
Morten Daugbjerg
Jeremy Bagai
In article <49bnho$s...@ground.cs.columbia.edu>,
Carl Tait <ta...@news.cs.columbia.edu> wrote:
>I have a question about Robertie's discussion of the Kauder Paradox
>in Problems 126 and 127 of _Advanced Backgammon_.
>
>(Summary for those who aren't familiar with the paradox: with the
>Jacoby Rule in effect, a position can theoretically be both a
>However, as long as this equity remains negative, the doubler's
>opponent should naturally beaver.)
>
*Analysis of Robertie's example -- which seemed correct but which I
didn't really read*
>
>1. Is this analysis correct?
>2. Can someone point me to a more detailed discussion of the paradox?
>
Yes, In Robertie's magazine Inside Backgammon, Volume 3:Number 6
(November/December 1993). Very thorough article by Paul Lamford called
The Famous Kauder Paradox -- Revisited! Shows that while Robertie's
criteria were sufficient, they were by no means necessary, which means
that there are many more Kauder positions than one might think. Many
examples.
Jeremy
Dave Biggs
>An e-mail correspondent pointed out an article on the paradox in
>an old issue of "Inside Backgammon." It turns out that it's *not*
>necessary for all of the underdog's wins to be gammons: there's a
>function of winning rolls, gammon chances, and losing chances that
>must fall into a certain window for the paradox to apply.
Sorry to butt in on your conversation but I have never heard of the
mag "Inside Backgammon". Could you please pass on an address or
phone number so I could subscribe.
Dave Biggs
dbi...@magicnet.net
Stephen Turner
> >in Problems 126 and 127 of _Advanced Backgammon_.
>
> >There are a few perverse cases in which Black wins but does not win
> >a gammon. For example, if he starts with 61 followed by seven 21s,
> >he is still 6 pips short of getting all his checkers off. White
> >gets an eighth roll, and if he has rolled eight consecutive 66s,
> >he saves the gammon. Ludicrously unlikely, of course, but it
> >violates one of the conditions in Problem 126: not *all* of Black's
> >wins are gammons.
>
> I cant remember the excact position, but I noticed that Robertie wrote some-
> thing like: White now needs xxx pips and cant get that in 7 rolls.
>
> I thought about the eight roll too, and you CAN get xxx pips, BUT
> you will have a tremendous waste when bearing into your homeboard,
> and therefore you cannot save the gammon even in this rare case.
> Try set it up for yourself.
>
> (this is all from free memory so forgive me if i remembered wrong)
>
From memory too, I think you can move the prime one place forward and construct
an equivalent example that makes a simple count work.
Why do I remember more about this position than about all the ones that might
come up in games? Sigh...
--
Stephen R. E. Turner
Stochastic Networks Group, Statistical Laboratory, University of Cambridge
e-mail: sr...@cam.ac.uk WWW: http://www.statslab.cam.ac.uk/~sret1/home.html
"I always keep one big file in case I run out of space." A colleague of mine