On Magriel's theorem concerning aces in bearoffs.
bob
gnubg". The thread was getting cluttered so I started this thread.
Magriel's theorem states that it is never incorrect to use a loose
ace in the bearoff if there is no contact. A couple points should be
made. First this is not the same as saying that doing otherwise is
incorrect. It is quite possible that two plays have equal equity. For
example if you have a checker on the 3 point and ace point and have an
ace to play then both aces are equally good. Secondly if there is more
than one ace to play there may be
a non bearing off play which is better. Suppose you have 4 checkers on
the 4 point and have double aces to play. If the opponent has 4
checkers on his ace point, then 4/3 (4) setting up for double 3's is a
better play.
Has the thereom been proved? No. There are some partial results
though. If you look at a one sided bearoff with all checkers borne off
or in the homeboard and play to minimize the expected rolls to bearoff
then using a loose ace to bearoff is always correct. Walter Trice
wrote a computer program to check. This ignores issues of the other
side's position, the cube, and match play though.
In 1994 I posted a result. to this group whoch would be applicable
to any opponent position, cube position and match score. If one can
prove that in a contactless bearoff you would never want to move a
checker from the bar to the ace point then the result follows. Look at
http://groups.google.com/group/rec.games.backgammon/browse_thread/thread/19cdff35033d3a42/fc9bab048ffbe7b1?hl=en&lnk=gst&q=koca+magriel#fc9bab048ffbe7b1
No one has given a proof of that assertion though. Note also that the
assertion being false does not imply that the theorem is false.
Sometimes bearoff plays can be shown to be wrong by showing them to
be a "nullo play" meaning there is no sequence for which it turns out
better. This is not the case though for using an ace to bearoff as was
shown by myself. In a letter to the chicago point around 1997 I gave
a position where if you played an ace by bearing off and I played
something else, then I would bearoff in fewer moves if I was allowed
to call the rolls. Danny Kleinmann termed it Koca's paradox.
Bob Koca
David C. Ullrich
wrote:
> This came up in the recent thread "What is X's best move according to
>gnubg". The thread was getting cluttered so I started this thread.
>
> Magriel's theorem
Has anyone ever asked Magriel if he had an actual proof?
>states that it is never incorrect to use a loose
>ace in the bearoff if there is no contact. A couple points should be
>made. First this is not the same as saying that doing otherwise is
>incorrect. It is quite possible that two plays have equal equity. For
>example if you have a checker on the 3 point and ace point and have an
>ace to play then both aces are equally good. Secondly if there is more
>than one ace to play there may be
>a non bearing off play which is better. Suppose you have 4 checkers on
>the 4 point and have double aces to play. If the opponent has 4
>checkers on his ace point, then 4/3 (4) setting up for double 3's is a
>better play.
To put the same point another way,
Instead of saying this has something to do with double aces
one could simply say it doesn't contradict a careful statement
of the result:
(*) If you have an ace to play and a man on the one point
then bearing the man off is never wrong.
The fact that playing four men to the three point in your
example is better is sort of orthogonal to (*) - in the
course of playing those four aces it never happens
that "you have an ace to play and a man on the one point".
(_If_ you've already played three aces by moving a man
to the one point _then_ (*) correctly says that taking
a man off with the last ace doesn't make things any
worse. But (*) doesn't say anything about whether it
was correct to play the first three aces that way, and
of course it's not.)
> Has the thereom been proved? No. There are some partial results
>though. If you look at a one sided bearoff with all checkers borne off
>or in the homeboard and play to minimize the expected rolls to bearoff
>then using a loose ace to bearoff is always correct. Walter Trice
>wrote a computer program to check. This ignores issues of the other
>side's position, the cube, and match play though.
>
> In 1994 I posted a result. to this group whoch would be applicable
>to any opponent position, cube position and match score. If one can
>prove that in a contactless bearoff you would never want to move a
>checker from the bar to the ace point then the result follows. Look at
>http://groups.google.com/group/rec.games.backgammon/browse_thread/thread/19cdff35033d3a42/fc9bab048ffbe7b1?hl=en&lnk=gst&q=koca+magriel#fc9bab048ffbe7b1
>No one has given a proof of that assertion though. Note also that the
>assertion being false does not imply that the theorem is false.
>
> Sometimes bearoff plays can be shown to be wrong by showing them to
>be a "nullo play" meaning there is no sequence for which it turns out
>better. This is not the case though for using an ace to bearoff as was
>shown by myself. In a letter to the chicago point around 1997 I gave
>a position where if you played an ace by bearing off and I played
>something else, then I would bearoff in fewer moves if I was allowed
>to call the rolls. Danny Kleinmann termed it Koca's paradox.
Can you reproduce this position? I'm very curious.
>Bob Koca
David C. Ullrich
Raccoon
> This came up in the recent thread "What is X's best move according to
> gnubg". The thread was getting cluttered so I started this thread.
>
> Magriel's theorem states that it is never incorrect to use a loose
> ace in the bearoff if there is no contact.
>
> though. If you look at a one sided bearoff with all checkers borne off
> or in the homeboard and play to minimize the expected rolls to bearoff
> then using a loose ace to bearoff is always correct. Walter Trice
> wrote a computer program to check.
Not bearing off with an ace that could be used to bear off a checker
can save at most one-half roll in the bearoff, which negates the one-
half roll lost by not bearing off with the ace. Not bearing off with a
2, 3 or 4 can (although rarely) save more than one-half roll.
I suppose that's an assertion. But it explains why it's never
incorrect to bear off with an ace. Perhaps the proof you seek lies in
demonstrating the truth of that assertion.
t...@bkgm.com
Midnight" (http://www.bkgm.com/books/Kleinman-
TheOtherSideOfMidnight.html). He writes:
"Why is it never wrong to bear a man off with a 1? Essentially,
because a 1 is the most flexible number you can roll. If bearing a
man off with a 1 now results in an empty 1-point later, you'll always
be able to play a later 1 constructively anyhow.
The proof of Magriel's Theorem is more complex. I'll sketch it
informally. For 1/off to be wrong, it must lead to two subsequent
misses. Obviously, one of these misses can be on a later 1. The
other must be on a later n, where n is the point to which the original
1 could otherwise be moved. But a later 1 can always be moved from n
+1 to n; and a later n from n+1 to 1. At worst, the position can be
transposed to the same position that results from an original play of n
+1 to n. But then an original play of n+1 to n would either have
resulted in another miss (when a point higher than n+1 is occupied),
or emptied the n-point as well, on the play of the roll of n+1 which
empties that point."
Tom
David C. Ullrich
If this is a proof (note I'm not taking sides on the question of
whether it is) then it seems to be a proof of the "stronger
result" that there's no sequence of later rolls which would
make bearing a man off from the one point worse than
moving a one. Bob says that he gave a counterexample
to this stronger result years ago - I wish he could find it for us.
>Tom
David C. Ullrich
paulde...@att.net
I agree fully with your analysis. It's not a proof. Also, if this
proof-attempt worked, it would also work if you kept the basic rules
of backgammon the same but had dice numbered from 1 to n (with n not
necesarily 6) and where the number of slots in the inner table was > 6
and the number of checkers > 15. In this "generalgammon", can you
think of any counterexamples?
Paul Epstein
Raccoon
Chew on these:
GNU Backgammon Position ID: vx8AAGD+EgAAAA
Match ID : cAknAAAAAAAA
+24-23-22-21-20-19------18-17-16-15-14-13-+ O:
O | O O | | |
O | O O | | |
O | O O | | |
| O O | | |
| 6 6 | | |
| |BAR| |v DMP
| 7 | | |
X | X | | |
X | X | | |
X | X X | | | Rolled 61
X | X X X X | | |
+-1--2--3--4--5--6-------7--8--9-10-11-12-+ X:
1. 6/off 1/off wins 42.81%
2. 6/off 3/2 wins 41.69% (-1.12%)
GNU Backgammon Position ID: vx8AAKD+EgAAAA
Match ID : cAknAAAAAAAA
+24-23-22-21-20-19------18-17-16-15-14-13-+ O:
O | O O | | |
O | O O | | |
O | O O | | |
| O O | | |
| 6 6 | | |
| |BAR| |v DMP
| 7 | | |
X | X | | |
X | X | | |
X | X | | | Rolled 61
X | X X X X X | | |
+-1--2--3--4--5--6-------7--8--9-10-11-12-+ X:
1. 6/off 1/off wins 40.4%
2. 6/off 3/2 wins 40.1% (-0.3%)
GNU Backgammon Position ID: vx8AACD/EgAAAA
Match ID : cAknAAAAAAAA
+24-23-22-21-20-19------18-17-16-15-14-13-+ O:
O | O O | | |
O | O O | | |
O | O O | | |
| O O | | |
| 6 6 | | |
| |BAR| |v DMP
| 8 | | |
X | X | | |
X | X | | |
X | X | | | Rolled 61
X | X X X X | | |
+-1--2--3--4--5--6-------7--8--9-10-11-12-+ X:
1. 6/off 1/off wins 33.3584%
2. 6/off 3/2 wins 33.3507% (-0.0077%)
GNU Backgammon Position ID: vx8AACD/EQAAAA
Match ID : cAknAAAAAAAA
+24-23-22-21-20-19------18-17-16-15-14-13-+ O:
O | O O | | |
O | O O | | |
O | O O | | |
| O O | | |
| 6 6 | | |
| |BAR| |v DMP
| 9 | | |
X | X | | |
X | X | | |
X | X | | | Rolled 61
X | X X X | | |
+-1--2--3--4--5--6-------7--8--9-10-11-12-+ X:
1. 6/off 1/off wins 32.9907%
2. 6/off 3/2 wins 32.9907% (no difference)