is there a Great Attractor in Gametheory of VonNeumann?? Re: There exists a Nim version that is a "draw" OS

From: Archimedes Plutonium (a_plutonium_at_iw.net)
Date: 07/11/04 

Date: Sun, 11 Jul 2004 13:55:07 -0500

Sat, 10 Jul 2004 15:24:42 -0500 Archimedes Plutonium wrote:

> Sat, 10 Jul 2004 13:16:51 -0500 Archimedes Plutonium wrote:
>
> > 10 Jul 2004 00:25:53 -0700 Jan Kristian Haugland wrote:
> >
> > > A combinatorial game such as Nim can not be a draw.
> > > Here is a distant relative that can be a draw:
> > >
> > > http://home.no.net/zamunda/split.htm
> >
> > I beg to differ.
> >
> > Yesterday I was working on a game of Nim, a morph of Nim where there are
> > no draws in the game itself but where either player can win in the OS
> > and not automatically that one player always wins the OS. Call it a
> > pseudodraw.
>
> The minimax theorem says a singular point. Thus a pseudodraw is
> nonexistent.
>
> Unless there is a draw within the game itself can the OS be a draw.
>
> >
> >
> > Secondly, I was looking for another Nim morph where it actually has a
> > draw within the game itself and the OS is a draw.
> >
> > Thirdly I was looking for a Tictactoe morph that was _not_ a draw in the
> > OS and where either X or O can win in the OS. Call it a pseudodraw.
> >
> > Here is what I come up with:
> >
> > Nim-morph with pseudodraw OS: Let me call the person with first move as
> > white and let me call the person with second move as black. The first
> > move in this game is not the removal of any matchsticks but is the
> > actual layout of the number of rows and the number of matchsticks within
> > each row. Black then proceeds as in normal nim. I contend, thence, that
> > this nim morph will end up as a win for one of the players but not
> > automatically the black player (provided regular nim is considered a
> > loss for the one who is forced to pick up the last matchstick).
>
> This is a erroneous claim. Even if I added the rule that only one or two
> matchsticks can be removed per move.
>
> >
> >
> > Nim-morph with a Draw in the game itself: This is where white with first
> > move determines the number of rows of matchsticks and the number of
> > matchsticks in each row. And finally, determines that at least one row
> > is a "Draw row" so that if this row or any of its matchsticks is picked
> > up last then the entire game is a draw.
>
> This is possible. It perhaps needs the rule of only one or two matchsticks
> removed per move.
>
> >
> >
> > TicTacToe-morph with pseudodraw OS: this one was a tough one to work out
> > last night. I would have thought that Nim was going to be the tougher
> > challenge. We have several rule changes to normal tictactoe. Call the
> > first mover as X and the second mover as O. In this morph, O gets two
> > first moves so that at the end of the game there will be five O on the
> > board to four X. And the other change in rule is that if there are no
> > three-in-a-row for a outright win then the win goes to the person who
> > has the most two-in-a-row. Now I have not fully played out all the
> > consequences. But I suspect, not sure of this suspection, that the OS of
> > this morph tictactoe is a win for either X or O or a pseudodraw. And
> > that every game played of this morph will produce a winner whether it be
> > X or O.
> >
>
> Trouble with whether "end row middles" would count as 2-in-a-row rather
> than having only "shortened 3-in-a-rows" count as 2-in-a-row. When X makes
> first move with placing an X in center square then X has the most
> 2-in-a-row unless we count end-row-middles as 2 in a row for O.
>
> Here again, the concept of Pseudodraw is erroneous, and that unless a draw
> exists in the game itself can the OS be a draw. And the minimax theorem
> says as much.
>
> >
> > Now, the most important aspect of the above, if true, implies that there
> > exists a Pseudodraw for the games of checkers and chess, but more
> > importantly, that those games OS is a draw with their current and
> > present rules.
>
> But the above is not all lost and wasted. I can salvage the idea that to
> make Nim a draw is to add the rule that the player with first move decides
> the arrangement of how many rows and number of matchsticks per row and
> which row is the "Draw" row.
>
> The implications for chess and checkers still remain. That if a game has a
> draw possibility, then the OS of that game ends up into that draw play.
>
> Nim OS is a win for one of the players always, well, because there is no
> draw possibility while playing the game.
>
> I never played Go. I suspect it has a draw possibility. If it does, then
> that is its OS-- a draw. Chess has a draw possibility, thus chess OS is a
> draw.
> This claim can be made into a assertion and then a theorem.
>
> Devise a game that is a VonNeumann game which has a draw possibility but
> has a nonDraw OS. Nim has a nondraw OS but nim has no draw within the game
> itself. So when we inject a draw possibility into Nim then does the one
> player always win the OS??????
>

Initially I was tempted to call a draw game in any VonNeumann game as a
gravity attractor such as gravity equilibrium or gravitational center so that
if you introduce a "draw game inside of Nim" that the OS of Nim shifts and
then becomes something different from its automatic win for second player.
That the moment you introduce a possible draw game that the entire OS of Nim
shifts and becomes that draw end result.

But there is another concept in physics that is like gravitational
attraction. And I suppose a "good physicist" not the usual run of the mill
sort can tell you the conceptual difference between gravity attraction and
Great Attractor in chaos theory.

I like to think of Great Attractors in EM of electricity and magnetism.

Anyway, Nim is VonNeumann gametheory and the OS is a certain victory for
second player. But introduce just one possibility of a draw outcome, then,
does the entire OS of this Nim change to the draw outcome? As like a Great
Attractor, the draw outcome forces itself as the Optimal Strategy.

Archimedes Plutonium
www.archimedesplutonium.com
www.iw.net/~a_plutonium
whole entire Universe is just one big atom where dots
of the electron-dot-cloud are galaxies

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