From: Steve Lord <slord@wpi.edu>
Subject: Re: R: Game value of a matrix
Date: Sun, 3 Sep 2000 22:10:24 -0400
Newsgroups: sci.math
Summary: [missing]

[deletia --djr]

On Sat, 2 Sep 2000, ferrante formato wrote:

> 
> > In 2by2 matrices it is:
> > m={{a,b},{c,d}}
> > detm=a*d-b*c
> > game value =detm/(a+d-b-c)

Consider the game where the top row of the game matrix M is all 1's and
the bottom row is all 0's.  Det(M)=0, a+d-b-c=0.  However, the value is
not 0/0.  Player A (the one who chooses the row) will want to maximize the
payoff and will always choose the top row.  No matter which column player
B (who chooses the column, and wants to minimize the payoff) chooses, he
cannot decrease the payoff below 1.  Therefore, the value of the game is
1: the minimum amount of payoff A is guaranteed to receive, and the
maximum amount B is guaranteed to have to pay, if both play optimally.

> Dear professor Bagula
> The definition of value of the game, with respect to a player,  is
> probabilistic in nature, in the sense that it is an expected value. It is
> strange to my opininion that it can be computed
> as you  wrote.
> 
> The idea of value is that It is the  maximum  gain that a player can
> guarantee
> "In any case", i.e. independently from  the strategies followed by the other
> players.

Essentially.

> I am just a beginner in GT , so I can't go any further,
> but suggesting to ask some  estabilished researcher
> in the field. The first name I can think of is professor Shmuel Zamir, of
> Hebrew University of Jerusalem, at present in paris, at Crest
> kindly regards
> Ferrante Formato

Another good reference I have seen is an oldie but a goodie, The Compleat
Strategist.  Check your local library, or if you can't find it there check
with inter-library loan.  I know at least WPI's library has it :)

	Steve L
==============================================================================

From: "glenn" <glenn@axiom.gr>
Subject: Re: Game value of a matrix
Date: Sun, 3 Sep 2000 02:22:36 +0300
Newsgroups: sci.math,sci.fractals,alt.fractals

"ferrante formato" <ferrante_formato@libero.it> wrote in message
news:80ds5.41402$Wi7.557419@news.infostrada.it...
>
> > In 2by2 matrices it is:
> > m={{a,b},{c,d}}
> > detm=a*d-b*c
> > game value =detm/(a+d-b-c)
> Dear professor Bagula
> The definition of value of the game, with respect to a player,  is
> probabilistic in nature, in the sense that it is an expected value. It is
> strange to my opininion that it can be computed
> as you  wrote.
>
> The idea of value is that It is the  maximum  gain that a player can
> guarantee
> "In any case", i.e. independently from  the strategies followed by the other
> players.
>
> I am just a beginner in GT , so I can't go any further,
> but suggesting to ask some  estabilished researcher
> in the field. The first name I can think of is professor Shmuel Zamir, of
> Hebrew University of Jerusalem, at present in paris, at Crest
> kindly regards
> Ferrante Formato

And you sound like a beginner in GT. You got the wrong idea about the value
of a 2x2 Z.S. game. The formula given above is right, provided that the game
is not strictly determined. The value of a 2x2 Z.S. non-strictly determined
game is indeed probabilistic in nature but is NOT the  maximum gain that a
player can guarantee. The players can only guarantee the MaxMin and MinMax
values for themselves, i.e. the conservative values. The value is the
expectation for the gain (loss) of each player, after a large number of
rounds, provided that the players play the game optimally i.e. according to
the calculated optimal mixed strategies. There are other interpretations of
mixed strategies and value e.g. that of Harsanyi, but this goes too far.