From: ed@cbnewsl.cb.att.com (edward.m.hummel)
Subject: Re: "Solvability" in Chess
Organization: AT&T
Date: Mon, 28 Sep 1992 11:45:30 GMT

In article <xeno.717656375@pv34ad.vincent.iastate.edu> xeno@iastate.edu (Gary L Snethen) writes:
>>Well, of course!  Once you ASSUME this, the rest is trivial.  The point
>>was that an additional assumption is NEEDED.
>
>No, an additional assumption is not needed.
>
>Let me rephrase:  If you wish to apply the theorems we have used
>so far from game theory, this assumption has ALREADY been made...  

Speak for yourself.  I had not made this assumption, AND the game
can be proven "solvable" even without making the assumption.  In
the "real" game of chess, the rules do not make the assumption.

>If this assumption
>had not been made, then of course one could assign a unique value to EVERY
>position on the board...  Not only this, but I will extend it to say
>that you can assign a unique value to EVERY position on the board for
>EVEY opponent EVERY game you play with him/her.

Here you may be going overboard, but you are getting closer to the
truth.  In fact, not every position can have a unique value.  There
will be positions where the only legal sequences drive the game to
a conclusion, BUT there will be a large number of values that need
to be supplied.  In fact, the number of parameters that need to be
supplied is finite, and so could be assigned in principle.  Therefore
the game is "solvable".  Once the values are assigned (for a single
game, or a single opponent) the solution is in principle knowable.

>In other words, the assumption is:  We don't take chances and assume our
>opponent will be blind to our weaknesses.  We assume that if he CAN force
>a win, he WILL, and that if he CAN force a draw, he WILL do that
>in preference 
>to allowing you to force a win against him.  This is an assumption we (who
>support 'solvability') have been making all along.

I also "support solvability", and I managed to do it without making
the assumption.

>I agree that it IS an assumption.

That's all I've asked.

>Summary of my position:
>  Making the assumption that our opponent will not make mistakes, chess is
>  solvable and the 'loops' presented do not interfere with the analysis.
>
>  Without this assumption, the version of game theory we are trying to
>  apply does not hold, anyway...  And indeed, I feel it's very
>  provable that optimal play cannot be obtained without insight
>  into your opponent's 'algorithm' or 'psychology'.

I would be interested in such a proof.  I think there is a solution
to the "chess problem" that does not involve psychology or "insight
into your opponents algorithm".

>  It all boils down to:
>    If a computer can force a win, it will, and this is computable.
>    If the computer were in a drawn or forced-loss position, it could not
>       possibly determine 'optimal' play, since that is entirely dependant
>       on your opponent.

Actually, all one needs to do is agree on the value of achieving
all possible "results" of the game.  This can be simply done by going
through the game tree and assigning values to all positions that
are undetermined.  Once this has been done, optimal play is well
defined and psychology has nothing to do with it.
Xref: sparky sci.logic:1611 sci.math:12400 rec.games.chess:8283