Forum Archive : Terminology

"Kauder paradox"

From:   Carl Tait
Address:   tait@news.cs.columbia.edu
Date:   27 November 1995
Subject:   Kauder Paradox - Robertie's discussion
Forum:   rec.games.backgammon
Google:   49bnho$s56@ground.cs.columbia.edu

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
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

Jeremy Bagai  writes:

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

Did you find the information in this article useful?          

Do you have any comments you'd like to add?     



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