On Mar 25, 10:18 am, (Torben Ægidius Mogensen)
wrote:
samsloan writes:
One factor to be considered is that the number of possible moves in a
backgammon games is infinite. The players could easily just keeping
hitting each other to infinity.

That doesn't matter, as long as the number of possible board positions
is finite (which it is).

The main difference between Backgammon and, say, Checkers is not the
possibility of infinite play but the fact that Backgammon involves
random elements, so few positions are definitely winning or definitely
losing -- all you can say is the probability of winning with perfect
play (i.e., always picking the move that gives you the best winning
probability after moving).

You can solve Backgammon by for each possible position have edges to
every other position that it is possible to get to in one move, and
label each edge with the dice outcome that allow this move).

This can be translated into a set of equations that you can solve to
find the probability of each possible position being winning or
losing. The set of equations is huge, but finite.

Torben

Even that is not obvious. There are 21 possible rolls of the dice (6!
= 21) and three possible positions of the doubling cube plus 24
possible slots for each checker.

The average chess position has 27 moves and most chess games are over
in 50 moves.

I have written a chess playing computer program and a shogi playing
computer program. However, I once tried to write a backgammon playing
computer program and I quickly gave it up as hopeless. It is much
harder than it looks.

Although backgammon seems to be an easier game than chess, I am not
sure that this is really true.

Sam Sloan

samsloan wrote:

Even that is not obvious. There are 21 possible rolls of the dice (6!
= 21) and three possible positions of the doubling cube plus 24
possible slots for each checker.


BZZZT!

Are the checkers in your backgammon set marked in some way?

--
Kenneth Sloan
Computer and Information Sciences +1-205-932-2213
University of Alabama at Birmingham FAX +1-205-934-5473
Birmingham, AL 35294-1170 http://KennethRSloan.com/

samsloan wrote:

Even that is not obvious. There are 21 possible rolls of the dice (6!
= 21) and three possible positions of the doubling cube plus 24
possible slots for each checker.

I haven't been following this thread so maybe I've missed something...
BUT
from what I read above: What do you mean by "possible rolls of the
dice"?
Combinations or Permutations? There are 21 of the former but 36 of
the
latter. Also 6! (6 factorial) is 720 not 21. Now, where did I go
wrong???

LRB

samsloan writes:

On Mar 25, 10:18 am, (Torben Ægidius Mogensen)
wrote:
samsloan writes:
One factor to be considered is that the number of possible moves in a
backgammon games is infinite. The players could easily just keeping
hitting each other to infinity.

That doesn't matter, as long as the number of possible board positions
is finite (which it is).
[...]
This can be translated into a set of equations that you can solve to
find the probability of each possible position being winning or
losing. The set of equations is huge, but finite.

Torben

Even that is not obvious. There are 21 possible rolls of the dice (6!
= 21)

6! is 720, actually. But you are right that the number of different
rolls is 21 = 6*7/2. This is still finite, though.

and three possible positions of the doubling cube

I can't see how this would affect the winning probability.

plus 24 possible slots for each checker.

You forgot the bar and home, so there are 26 possible positions. But
since the checkers are not distinct, and since black and white pieces
can't coexist (except on the bar and in the home), you get a lot fewer
than the 30^26 different positions you imply.

In any case, my point was that the number of positions is finite (but
huge), so arguing that there are many possible rolls and positions of
doubling cubes and pieces doesn't change that, unless you can show
something is infinute.

The doubling cube is normally limited to 7 possible positions (absent
or 2, 4, ..., 64), but even if you allow unbounded doubling, this
doesn't change the probability of winning.

Torben

On Mar 26, 4:49*pm, (Torben Ægidius Mogensen)
wrote:
...

The doubling cube is normally limited to 7 possible positions (absent
or 2, 4, ..., 64), ....

This is completely false. There is nothing wrong or problematic with
a double to 128. Admittedly, this can't be recorded on a normal
doubling cube but players can just write it down. Also, doubling
"cubes" are available which reach a max value of 256. Doubles beyond
64 are rare among good players, but if you're talking about what is
rare rather than what is legal, you shouldn't include 64 since good
players don't often let the cube get even that high.

Paul Epstein

On Mar 26, 4:49*am, (Torben Ægidius Mogensen)
wrote:

and three possible positions of the doubling cube

I can't see how this would affect the winning probability.

plus 24 possible slots for each checker.

You forgot the bar and home, so there are 26 possible positions. *But
since the checkers are not distinct, and since black and white pieces
can't coexist (except on the bar and in the home), you get a lot fewer
than the 30^26 different positions you imply.

In any case, my point was that the number of positions is finite (but
huge), so arguing that there are many possible rolls and positions of
doubling cubes and pieces doesn't change that, unless you can show
something is infinute.

The doubling cube is normally limited to 7 possible positions (absent
or 2, 4, ..., 64), but even if you allow unbounded doubling, this
doesn't change the probability of winning.

* * * * Torben- Hide quoted text -

- Show quoted text -

If we agree that owning a 2-cube has the same theoretical meaning at
owning a 4-cube or 16-cube (or whatever level), how do you get 7
possible cube positions? I count only three: centered, owned by me,
or owned by the opponent.

--
Gregg C.

Kenneth Sloan wrote:

samsloan wrote:

Even that is not obvious. There are 21 possible rolls of the dice (6!
= 21) and three possible positions of the doubling cube plus 24
possible slots for each checker.


BZZZT!

Are the checkers in your backgammon set marked in some way?


And, the relevance to computer chess is...???


Crosspost to rec.games.board, rec.games.chess.misc,
rec.games.chess.computer, rec.games.chess.politics
and rec.games.backgammon noted and contrasted with:

|
| From: Kenneth Sloan
| Newsgroups: rec.games.chess.computer
| Date: Wed, 16 Apr 2008 16:04:09 -0500
| Message-ID:
|
| Please don't cross-post.
|