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

From: Havirrion (havirrion_at_NOblueDAMNyonderSPAM.co.uk)
Date: 07/11/04 

Date: Sun, 11 Jul 2004 19:50:19 GMT

Archimedes Plutonium wrote:

> 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
>

The intellectual midget doth return.

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