FAX message to Lloyd Shapley c/o Prof. Dubey
  
 
       Economics Dept.
       SUNY at Stony Brook
       Stony Brook, NY

     (at FAX # 1-631-632-7516)

Dear Lloyd,

   On Friday evening, after I had failed to meet you at a convenient time to go
to supper and after I had walked back to 3 V Inn and watched some TV I went to
the nearby Brook House for a sandwich supper. And as I was waiting I studied the
implications, for a game of three players, of a characteristic function derived
on "the Harsanyi basis". (You had suggested to me earlier that this HCF could be
used analogously to the VNMCF in the formula for the calculation of a Shapley
value.)
   (I was looking for a sort of characteristic function that could be used to
simplify the models of games with the election of agencies as the means of
"coalescence".) (When coalescence has failed what are the payoffs?)
   Now the HCF clearly removes some of the information that is otherwise
available in a full game description. Is this O.K. and do we retain what is
needed (for value purposes)? I realized that it would be very simplifying if 
this could be assumed to be true.
   So I studied the 3-person case on a napkin while at Brook House. And suddenly
I realized that a value was implicit by symmetry. (This argument does not make
any "linearity" assumption and DOES NOT extend to 4 players.)
   So it seemed, if this HCF can be accepted as adequately descriptive, that the
value theory for 3-person cooperative games is solved. That would mean that the
real challenge of my project of the agencies, etc. would transform into for 4 or
more players (!!).
   And of course I should mention that if one starts with a "CF game" (my
terminology) where the VNMCF can be regarded as an adequate description of the
game then that the derived value (through going to the HCF) is just the Shapley
value as calculated from the CF.
   This raises the interesting question of what other existing theoretical value
concepts might happen to coincide with the SV for three person "CF" games but not
coincide for higher numbers of players.
   (I had talked a few years ago, in Bergamo, about a 4-person game example where
I felt that the SV evaluation could be improved upon because of its apparent
insensitivity to a natural alliance of two of the players. And I had NOTED then
that this sort of example COULD NOT be reduced to appear in a 3-player context.)
(But now I have no confidence in what I was thinking then and I feel that I would
need to restudy that class of examples to have a really reliable opinion about
them.)
   ///////////////
   Well I hope that the weather doesn't become a lot worse there in SB as you go
through the time of the next meeting, etc. We really had luckily very nice
weather, comparatively, for this year's "Summer Festival".

Yours Truly,

John
(John/JFNJ)