🚀 go-pugleaf

ed@cbnewsl.cb.att.com (edward.m.hummel) writes:
>Consider the game 1. Nf3 Nf6 2. Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 5.Nf3 Nf6
>and so on, forever; with neither player choosing to claim a draw.

And, presumably, neither player choosing to break out of the loop by doing
something else.

>I am asking for a proof that when collapsing the game tree to determine the
>value of the initial position, this sequence must have the value of a draw.
>As far as I can see, such a sequence (or the node corresponding to
>one of the positions in the sequence) has indeterminable value.  This
>does not mean that chess is unsolvable.

Under the assumption that both players are playing optimally (under the
assumption that their opponents are playing optimally, under the
assumption...), I can prove this.  If either player could force a win by
breaking out of the loop, one or the other would have done so before the
loop completed EVEN ONCE.  If neither player can force a win, and if
neither player is daring enough to break out of the loop into a possibly-
losing situation on the assumption that the opponent will make a mistake,
the game is effectively drawn whether either player "claims" it or not.

If you insist on a formal claim, I have to make the additional assumption
that at least one of the players prefers a draw to an infinite game.  If
you grant me that, said player will follow the above reasoning, conclude
that the best (s)he can hope for is a draw, claim a draw, and get on with
the rest of life.

-- 
					Stephen Bloch
				sbloch@silver.cs.umanitoba.ca