From: xeno@iastate.edu (Gary L Snethen)
Newsgroups: sci.logic,sci.math,rec.games.chess
Subject: Re: "Solvability" in Chess
Date: 28 Sep 92 04:59:35 GMT
Organization: Iowa State University, Ames IA

In <1992Sep28.010335.27141@cbnewsl.cb.att.com> ed@cbnewsl.cb.att.com (edward.m.hummel) writes:

>xeno@iastate.edu (Gary L Snethen) writes:
>>But I propose that we will always either take the draw option or continue
>>out of the loop using the following reasoning:
>>One of the assumptions we make is that our opponent, when given the
>>chance to force a win, will, and if not, will force a draw if possible...

>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...  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.  Only GOD could do this,
however, as it involves ideal analysis of your opponent's current (and future)
psychology to know what mistakes he WILL make and what 'mistakes' you can
make without him/her catching on and forcing a win against you.

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 agree that it IS an assumption.  There IS BETTER PLAY than game theory can
provide us with...  And again, I can revert to tic-tac-toe to prove my point.
Tic-tac-toe is a drawn game from the beginning...  I can never beat a computer,
and a computer can never beat me.  However, I can beat a human opponent more
often than a computer can (unless it has been programmed with my strategy,
which it wasn't capable of determining for itself using standard game theory.)

But 'solvability' does not mean that the computer will DEFINITELY beat an
average player more than a grand master will...  The game is not transitive.

As an example, let's start the game up in a forced-draw position.

*  Our GM will NEVER beat the computer, but the computer MAY beat the GM
   if he makes a mistake.
*  An average player will NEVER beat the computer, but the computer MAY beat
   the player if he makes mistakes.
*  A GM will use psychology and confuse the position as much as possible to
   beat the average player.  He will more than likely win more often against
   average players than the computer if the original position was fairly
   straight-forward, because the computer will not intentionally pick a path
   that would offer 'attractive' (by human standards) mistakes to the player.
   Of course, the average player MIGHT beat the grand master, although this
   is very very doubtful.  But it is possible.


>>Consensus has everything to do with it...  We must have rules before we
>>may prove or disprove the solvability of chess...  and since the rules
>>of FIDE and USCF (to my knowledge) do not cover what happens in an
>>'infinite' game, we should set this rule down before we can continue with
>>the analysis.  Since humans are interested in finite-length games of
>>chess (when playing each other), I think we could arrive at a consensus
>>that any position that is repeated infinitely constitutes a forced draw.

>Perhaps you are giving up too easily.  There is still a lot of interesting
>analysis that can be done without adding to the set of rules.  Once you
>decide to add to the rules, there are lots of ways to do it.  Whether
>a consensus is used, or an edict from a Federation, it doesn't much
>matter.  The point has very little practical value anyway.  The 
>argument has been about whether the current rules allow for chess to
>be solved.  I think it has been proven well enough that the game
>is solvable.  My point has been that the rigorous solution is not as
>"simple" as some people have been claiming.

>>But I'll try to contain my analysis to the case where this rule does not
>>apply, since I think I can show solvability on your ground.

>Please do.  If you look through the previous posts you will find
>the proof.  I have not been claiming that chess is insolvable, just that
>the solution is a function of a large (but finite) number of
>undetermined parameters.


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'.

  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.

Interesting side-effects:

   If the opening position is a draw, then it's possible that good chess
players will win more games against a human opponent than the computer will.
   If the opening position is a loss, then good players may draw or even win
more games against the human opponent than the computer will.
   If the opening position is a win, the computer will always win, and the
good chess player will make mistakes and draw or even lose some games against
his/her opponent.

************
*** NOTE ***
************

The most interesting aspect of this is that regardless of the value of the
first position, a human player may always possess a better rating than a
computer, when both play in a large pool of chess players.

This is because the computer may lose a greater number of it's 'forced-loss'
positions than a human, and may win fewer of it's force-drawn positions than
a human.  (Although it will ALWAYS win it's forced-win positions).

In THIS regard, humans may indefinitely retain their superiority over
computers in chess.  If pitted in direct competition, however, the
human would ALWAYS come out of the series of games even or under (assuming
50% of the first moves go to the computer).

So, just because chess is 'solvable', it doesn't mean that humans can't play
the game 'better' against a human opponent than a computer can against a
human opponent.


---Xeno

    
Newsgroups: sci.logic