On 21 Feb 2007 12:43:08 +0000 (GMT), David Richerby
wrote:

Ray Johnstone wrote:
David Richerby wrote:
Ray Johnstone wrote:
wrote:
Is the initial position in chess a mutual Zugswang?

We will probably never know. See:
http://members.iinet.net.au/~ray/Chessgames.htm

Just because chess is likely impossible to brute-force doesn't mean we
can never know the outcome of theoretical best play. [...]

I agree, which is why I said "probably".

OK. Your web page seems a little more certain than that, though.


I can't imagine any method other than brute force but I couldn't
have imagined calculus, Newton's laws...

:-) Strategy-stealing (as used in chomp) is the usual way to produce
a proof that one player can force a win without knowing how to do it.
But that doesn't apply to chess because there's no first move that
white can make that is equivalent to passing.

One possibility would be to come up with a proof along the following
lines. Somehow, classify positions as `good' or `bad' and produce a
score for each position that is a positive integer, such that every
position where White has given checkmate is scored 1 and all other
positions have scores greater than one. If you could then show that
the initial position is `good' and that, furthermore, whenever White
is in a `good' position, he has at least one move such that every one
of Black's replies leaves the game in another `good' position with a
strictly lower score, you would have shown that, with best play, chess
is a forced win for White.
I don't see how this helps. You would still need to examine a vast
number of positions. Even an enormous sample would not be adequate.

(The point of the `good' positions is that the scoring function can
behave arbitrarily on bad positions without affecting the argument.)


Also, there is an error in your web page. You write,

``Suppose White wins a particular game [with perfect play] which
began with say a3. Consider Black's last unforced move. All
other moves at that branch point must also lead to mate in as many
moves or fewer or Black would have chosen one of them in
preference. This argument applies at every branch point so all
games starting with a3 must then be wins for White. Games
starting with the other 19 possible moves are of little
consequence. They could all be wins for the unfortunate Black,
who would never get to play them.

``Black's prospects are therefore rather gloomier: to win any game
it is necessary that every game be a win for Black.''
-- http://members.iinet.net.au/~ray/Chessgames.htm

First, there's the unimportant point that, in perfect play, the
concept of a move being forced or not doesn't really exist: the whole
point of saying that all perfect games are won for White is that it
really doesn't matter what Black plays. Since he has no hope of
winning, it doesn't really make much sense to assert that avoiding a
mate in one to allow a mate in ten is `forced'. But that doesn't
really matter.
"Forced" is generally used in chess to describe a move which must be
played to avoid a disastrous loss of material or position but I use it
here to describe a move which is the only legal move. Perhaps I
shoould have made this clearer.

There are two copies of the same problem in these two paragraphs. You
say that ``... all games starting with a3 must then be wins for
White''. This is not true. All you can conclude is that all games in
which White plays a3 and then continues to play perfectly are wins for
White. There are plenty of games starting with 1.a3 that are losses
for White. For example, 1.a3 a6 2.f3 e5 3.g4?? Qh4#.
I think the context makes it clear that "game" as used here refers to
"perfect games".

The same problem occurs in the second paragraph. Black only has to
win the games in which he plays perfectly: the other games are, as you
have observed, of no relevance. This means that he must have a
winning reply to each of White's 20 possible first moves but that's
not really so much worse than White's situation because, if chess is a
win for White, he must have a winning reply to each of Black's 20
possible first moves, too!
"Black’s prospects are therefore rather gloomier" is my weakly
humorous attempt to describe Black's situation. In the hypothetical
case I describe only games beginning 1 a3 are perfect games.

Apologies if this all sounds rather pedantic -- I'm reading
rec.games.chess.* while I'm supposed to be writing an academic paper
on game theory. ;-)


Dave.