From: dseal@armltd.co.uk (David Seal)
Newsgroups: sci.math
Subject: Re: "Solvability" in Chess
Date: 30 Sep 92 14:16:26 GMT
Distribution: sci
Organization: A.R.M. Ltd, Swaffham Bulbeck, Cambs, UK

In article <1992Sep29.152849.74877@Cookie.secapl.com>
frank@Cookie.secapl.com (Frank Adams) writes:

>In article <7468@armltd.uucp> dseal@armltd.co.uk (David Seal) writes:
>>In article <1992Sep28.155640.152096@Cookie.secapl.com>
>>frank@Cookie.secapl.com (Frank Adams) writes:
>>>In article <1992Sep25.014035.1269@cbnewsl.cb.att.com> ed@cbnewsl.cb.att.com (edward.m.hummel) writes:
>>>>I think this argument rest on the assumption that chess is a zero sum game.
>>>>How do you prove that?  Suppose the value of an infinite game is less than
>>>>a draw for each player (but more than a loss), then neither player
>>>>would choose a draw *with best play*.
>>>
>>>As noted, this was intended to read "more than a draw for each player (but
>>>less than a win)".  However, it is actually closer to correct as written:
>>>the value of an infinite game *is* less than a draw; not as specified by the
>>>rules, but in fact: no one wants to spend the rest of their life playing a
>>>single game of chess.  This isn't *in* the rules for chess; it is *assumed*
>>>by them.
>>
>>In normal circumstances, I would agree with you. But it isn't provable, and
>>isn't even true in abnormal circumstances. Imagine a game of chess with the
>>following playing conditions:
>>
>>[...]
>
>This is chess; this is chess plus strange external conditions.

Sorry, that should be:

  This is chess plus fairly normal external conditions; this is chess plus
  strange external conditions.

Chess *always* has external conditions. The stranger they are, the more it
gets distorted from what we know and love. I chose the silly playing
conditions simply to show that the values of the outcomes are affected by
the external conditions. It's an extreme example, but less extreme forms of
the same effect can distort the game in practical situations - e.g. when the
tournament score means that a player will value a draw just as much as a
win.

My main point is that the rules of the game don't say what the values of the
outcomes are - the external conditions do. For any set of these values, the
game is (in principle) solvable: the solution is not necessarily the same as
that for another set of values, but the game is solvable for *any* set of
values.

>My key point is the last sentence: that infinite games are draws (and that
>chess is a zero-sum game) are not *in* the rules for chess, but are
>*assumed* by those rules.

I don't believe the chess rules make either of those assumptions. IMHO they
do make two related ones:

(a) That players don't want infinite games. This is NOT the same as that
    assuming infinite games *are* draws: they simply offer both players the
    normally more attractive option of a draw when an infinite game looms.

(b) That players rate a win more highly than a draw, and a draw more highly
    than a loss. This is NOT the same as assuming the game is a zero-sum
    game. But being a zero-sum game is a red herring: you don't need it to
    make the min/max technique work. All you need to know to make the
    min/max technique work unmodified is what the preference order of the
    outcomes is, and that an infinite game is rated less highly than a draw.
    A modified form of the min/max technique will show that this last
    requirement is not necessary for solvability, though the algorithm to
    solve it becomes more complicated.

David Seal
dseal@armltd.co.uk

All opinions are mine only...
Newsgroups: sci.math