Forum Archive : Puzzles

Janowski Paradox

From:   Robert-Jan Veldhuizen
Address:   rjv1112@iname.com
Date:   3 November 2000
Subject:   Paradox puzzle
Forum:   rec.games.backgammon
Google:   4sn40t08ds229tnc6bj0j1gpb3rdvok0kp@4ax.com

Some us might have heard of the Kauder paradox; it's also in Robertie's
AB Vol.II. It exists because of the Jacoby Rule: (back)gammons only
count after a double. There are bg positions that are proper initial
doubles (activating (back)gammons), yet also proper beavers (after
taking, the opponent is the favorite with his cube ownership).

In Janowski's paper, I read about another paradox (I forgot the name),
also existing because of the Jacoby rule: bg positions that are proper
redoubles, but not initial doubles!

Normally, initial doubles can be offered earlier than redoubles, because
with a redouble you give your opponent cube access that he didn't have
before. So, a position that is an initial double might not be a
redouble; a redouble generally needs a bit more.

With the Jacoby rule however, an initial double activates gammons, which
a redouble does not. Simply put, if activating gammons in itself is
negative for the doubler, he might not want to give an initial double;
with gammons already activated he might want to redouble though.

In practice, it doesn't happen too often that you would like to double
when the gammon games are quite clearly *not* in your favour. Still it
doesn't seem at all impossible and it looks more natural to occur than
proper double/beaver positions.

So here's a nice puzzle, especially for the people that can rollout some
play with a bot: construct a position in a money game, Jacoby that is a
proper redouble yet not an initial double! :-)


Mary Hickey  writes:

Sorry it took me so long to get around to reading your post, Zor-bie, but
I thought it might be better to respond better now than never, which is
what it would have been if I'd waited any longer, as your post would have
expired and I'd have never read it at all :0(

           X on roll.  Jacoby in use.  Cube action?
|           +24-23-22-21-20-19-------18-17-16-15-14-13-+
|           | X  O O      O  O |   |                   |
|           | X  O O           |   |                   |
|           |    O O           |   |                   |
|           |                  | X |                   |
|           |                  |   |                   |
|           |                  |BAR|                   |
|           |                  |   |                   |
|           |                  |   |                   |
|           |                  |   |                   |
|           | X  X  X  X  X  X |   |                   |
|           | X  X  X  X  X  X |   |                   |
|           +-1--2--3--4--5--6--------7--8--9-10-11-12-+
|           CubeValue: A.  1 centered, or B. 2 with X owning it

I think this position qualifies as a solution to your "Paradox".  Here is
a 1440 game JellyFish 3.0 Level 6 full-rollout, which it says its variance
reduction makes equal to 32,000 games, and who are we mere humanoids to
question this:

X wins = 67.1%

O wins 32.9%, of which 13.8% are singles, 15.2% are gammons, and 3.9% are

First, the center-double case:

If he waits, X will hit 25/36 times and win with the cube; 11/36 times he
will miss and be doubled out, since O has no reason to play on with
gammons not activated:

25/36 x 1 + 11/36 x -1
  = 69.44 - 30.56 = + 38.88 points in 100 games

If he doubles now, I'll use the win % JF gets even though in reality
there'd be some small adjustment for O being able to turn the cube
sometimes after being hit if X has an accident bringing this home
after hitting.

67.1 x +2 + 13.8 x -2 + 15.2 x -4 + 3.9 x -6
  = 134.2 - 27.6 - 60.8 - 23.4 = + 22.4 points in 100 games

It is apparent that if the cube is centered, X is better off to wait.
Now, about that redouble:

If X holds the cube at 2, his wins will be the 69.44% we saw in the
wait from the center case.  This is higher by 2.34 % than his wins if
the game is played out, and it should come entirely at the expense of
O's single wins, not his gammons or backgammons, so I adjusted the
numbers for the double-from-center by this amount:

2.34 x 2 = swing on this case in X's favor = 4.68 x 2 cube = 9.36 points
in 100 games

22.4 + 9.36 = 31.76 points in 100 games

If instead X redoubles, he ends up with the same result as if he'd
doubled from the center, except now he's on a 4 cube:

22.4 x 2 = 44.8 points in 100 games

So it would appear X should redouble here to get the higher equity,
despite the gammon risk, even though because of Jacoby he could not
double from the center with this position.

Mary Hickey
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