Puzzles

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Zero equity positions

From:   Kit Woolsey
Address:   kwoolsey@netcom.com
Date:   17 April 1995
Subject:   Re: doubles, redoubles, re-redoubles, etc.
Forum:   rec.games.backgammon
Google:   kwoolseyD760MF.35t@netcom.com

Christopher Yep wrote:
> This leaves open questions which I have wondered about for awhile:
> Are there any backgammon positions (other than the initial position,
> before either player has rolled) where:
>
> 1. The equity is 0? [money play]
> 2. The player on roll has a 50% chance to win the game? [1 pt. match
>    (no cube)]

1) I know of a couple.  One which is very easy to see is as follows:

X has one checker on 9 point, O has one checker on 6 point, X owns cube,
X is on roll.  X wins outright on 12 of his rolls.  Since all of his
rolls get him to at least the 6 point (where he has a claim or break-even
on O's decision whether or not to take a redouble), in essence X wins 1/4
of the 24 rolls he fails to get off immediately (since O gets off on 3/4
of his rolls), which comes to the desired 18 out of 36.

Another example which is more complex but my trusty computer says is so is:

X has two checkers on 4 point and one checker on 6 point, O has three (or
four) checkers on ace point, X owns cube, and X is on roll.  Check it out
if you want to spend some pencil and paper time.

An example of a contact position might be as follows:

X has a closed board -- his other three checkers are two on O's 3 point
and one on O's 7 point.  O's position:  3 on 1 point, 3 on 2 point, 4 on 4
point, 4 on 5 point, 1 on 12 point.  X is on roll, cube is in center
(money game, Jacoby rule in use).  Note that X has exactly 18 hitting
numbers, and if he misses O certainly has a claim with the cube.  If we
assume that after hitting X's gammon chances are less than half his losing
chances (which seems like a proper assumption to me) then it is not
correct for X to double now.  Therefore X wins when he hits, loses when he
misses, for equity of 0.

2) With no cube in play, I don't know any zero equity positions.  If there
were one in a race I'm sure our regurgitating computer programs would
have spit it out by now and I would have heard of it.  I guess in theory
there could be one in a contact position, but that would be awfully hard
to prove -- keep in mind that "gin" positions just aren't 100% gin.

    Kit
 
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