"Dr A. N. Walker" wrote in message
...
In article ,
David Richerby wrote:
Even though there are more chess games than atoms in the universe.

We don't know *that*. "Visible" universe perhaps?

True. My chess partner an engineer is much worried by the Special Theorum,
where is this universe with all its mass? He asks.

For the purposes of the theory of games of perfect information, any
finite number of players is equivalent to two. [...]
Why does it suffice to consider two-player games? Suppose we have n2
players. We can analyze the game from the point of view of player 1
by assuming that some super-human plays the moves for players 2, ...,
n himself. If player 1 can beat the super-human, he can also beat any
collection of ordinary people acting individually. Likewise for each
of the other players.

Yes, but when one player can beat all the others combined is the
uninteresting [and unusual] case. Interesting games in this sense are
those when each player could be defeated by a coalition of the others
[which will "almost always" happen if the game is in reasonable balance,
in some sense]. In this case, the outcome of the game is determined as
much by social factors [which players are "friends"] as by the theory.
Try 3-player Nim some time ....

Note also that if a game is not zero sum [eg "Prisoner's Dilemma"],
then you can make it so by adding an extra player who sole role is to act
as "banker". [Or "society" in PD.] But this will turn "most" 2-player
non-ZS games into unstable 3-player games [even if they are PI].

Further note: even if a game is mathematically ZS, it may not be
so in terms of the utility of the result. This is more relevant to
probabilistic games, however. But even in deterministic games, you are
not required to assume that your opponent will play perfectly, and you
may rationally prefer a move that wins #1000 unless your opponent finds
exactly the right response,

Just to interject a moment, since ordinary citizens may be confused, these
are all asides and suppositions to the 2 plain questions, no? Alternate
universe questions, not chess answers.

but then wins only #0.99, to a move that wins
#1.00 no matter what the response. Scope also for an essay on the topic
of "randomising" vs "simplification" by which a strong player aims to
confuse a position which is drawn [or even losing] in order to give an
opponent a chance to go wrong, vs [eg] sacrificing back almost all of a
large advantage in order to reach a K&P ending which is a guaranteed win
vs an easily won position in which the opponent has fiddling chances.
It can be hard to determine what is "perfect play" in such cases. This
is a concept which is ill-understood, as yet, by the computers.

Andf not to indulge, but...

[Technical note to any mathematicians reading: yes, I'm aware that not
all infinite ordinals are limit ordinals but introducing the idea of
games of length omega+1 in the current discussion will just confuse
everyone.]

Not *everyone*! But it's perhaps worth noting that there are
perfectly playable games with infinite [in various senses] values, and
["of course"] with infinitesimal values. Some of these are relevant
to chess [eg there are K&P endings most easily understood in terms
of infinitesimal, but strictly positive, games].

But neither answer, answerable, Andy?

Phil

--
Andy Walker, School of MathSci., Univ. of Nott'm, UK.