From: mathwft@math.canterbury.ac.nz (Bill Taylor)
Subject: Re: 0-sum N-Person Game Theory
Date: 29 Sep 1999 07:34:32 GMT
Newsgroups: sci.math
Keywords: 2-player non-zero-sum games
Michael Jørgensen writes:
|> Go over that again, please:
|>
|> Is there no such thing as 'Optimal Strategi' for N-person games with N>2?
More or less, that is so.
For two-person non-0 sum games, and for n-player games, there still IS
a minimax strategy for each player; that is, a mixed strategy which will
obtain his personal "minimax value". Unfortunately these various values
no longer have very much relation to one another. The minimax strategy
for a player is that which he will adopt by firmly fixing his attention
merely on his *own* payoffs, for each of his *own* options and each of
the all-of-the-others-together's options, treating all opposition as
just one big nasty single opponent.
As he can frequently do far better than this, by using craft and guile,
and more pertinently by using negotiating skills, his minimax strategy
is hardly "optimum" in any meaningful sense of the word.
Please ask for more clarification if necessary.
----------------------------------------------------------------------------
Bill Taylor W.Taylor@math.canterbury.ac.nz
----------------------------------------------------------------------------
Cry HADDOCK, and let loose the cods of war!
----------------------------------------------------------------------------
P.S. "Minimax" strategies would better be called "maximin" strategies,
as far as game theory goes. Amusingly, there was once a Roman
emperor by the name of Maximin. He was assasinated after only
two years at the top. Not a good exemplar of negotiating skills,
craft, guile, or game theory!
==============================================================================
From: mathwft@math.canterbury.ac.nz (Bill Taylor)
Subject: Re: 0-sum N-Person Game Theory
Date: 30 Sep 1999 05:46:26 GMT
Newsgroups: sci.math
Jon Haugsand writes:
|> What do you mean here by "craft and guile"? Psychology? Exploitation
|> of the others weaknesses? Exploitation of the fact that the others compete?
All of that and more. Probably political/business books have more useful
things to say than math books do!
Here's a simple example:
Oppo
1 2
.-----------.
1 | 5 | 1 |
You |-----+-----| [payoffs to "you", as usual]
2 | 3 | 2 |
`-----------'
This game has a trivial minimax-and-optimum strategy of always playing
option 2, and being content with payoff 2, (with maybe occasionally 3).
This is fine, until you have the craft to recall that this is part of
a non-zero sum game with these overall payoffs...
Oppo
1 2
.-----------.
1 | 5/3 | 1/0 |
You |-----+-----| [right-payoffs positive to opponent]
2 | 3/5 | 2/2 |
`-----------'
...and the guile to notice that the oppo is always going to pick option 1,
regardless of what you do; so you should also pick 1 and win 5. YAY!
So your option 2 is minimax, but hardly optimal!
There are copious ways to alter this, involving announcements, threats,
sacrifices, negotiation etc. And if it's to be played more than once... OY.
And once you get into 3-player games... OY, MOY!
-------------------------------------------------------------------------------
Bill Taylor W.Taylor@math.canterbury.ac.nz
-------------------------------------------------------------------------------
"Everybody loves my baby,
"But my baby don't love nobody but me. (The Russell love song)
-------------------------------------------------------------------------------
==============================================================================
From: kramsay@aol.commangled (Keith Ramsay)
Subject: Re: 0-sum N-Person Game Theory
Date: 01 Oct 1999 04:14:34 GMT
Newsgroups: sci.math
One of my favorite types of 2-person nonzero sum games is the
"forebearance" game:
Right
A B
A (10, 10) (-10, 9)
Left
B (10.01, -10) (0, 0)
The only Nash equilibrium is for both players to play B. Regardless
of what Right plays, Left gets a greater reward by playing B. If Left
plays B, then Right is much better off also playing B.
However, in practice (especially if there's a chance for the players
to get acquainted, or repeat the game) people have been experimentally
found often opt both to play A, which leaves them both better off.
It's Left's job to persuade Right that s/he doesn't automatically
follow the dominant strategy, and has enough forbearance to give up
a small bonus, because it's fairer or some other reason.
It seems to me that real life has situations something like this,
where people have to voluntarily give up something like a fraction of
a minute of lost time, or a chance to get a dollar or two and suchlike
in order to keep playing A as Left. It seems also to be sort of the
defining characteristic of weakly civilized societies that members
would not be inclined to play A as Left to begin with, together with
the fact that to play A as Right is now "being a chump" or "being a
loser" and considered stupid, which makes playing A as Left into
"being a naive idealist who doesn't understand the realities of the
situation". And who can blame them? Circle the wagons and get ready
for an unpleasant time.
Keith Ramsay