Author: kjfnj
Company: Princeton University

Currently I am very much into some computational work for
a model which would progress from the previously studied model
which led to the publication in the IGTR journal.
    I have no assistants right now and no NSF project grant is
applicable at this time (late 2010); so I am limited to doing
the calculations myself and attempting to check the formal
accuracy of my work myself.
    The earlier model intrinsically involved 39 "demand" strategies (for three players for which we seek (or sought)
to find a cooperative game equilibrium. (Note that players 
of a REPEATED game of "prisoners' dilemma" can have a good
cooperative equilibrium (in an infinitely repeated game context) 
where each player has, strategically, a DEMAND that the other 
player should play in an approved cooperative fashion (thus 
a fashion not betraying either prisoner to the adverse judgment 
of the police and magistrates). 
    The earlier model involved 39 strategic dimensions deriving
from 13 demand strategy parameters controllable by each player. 
    The projected new model would first add three new strategies
by allowing each player to operate, at the first level for votes
of acceptance, with two demand numbers rather than just one.
And thus a player at that stage would decide freely how much
to "demand", in payoff expectation, from each of the other two
players. (The prior model did not allow full choice freedom at
this stage of the elective coalescence mechanism (leading to
the effective cooperation).)
    Then 18 new strategic dimensions would be added by having,
whenever a player would have the option of voting to accept
the leadership (or "agency") of another player, of the potentially
accepting player modulating his probability of acceptance by
having a "demand" relating to the observed probability of action
by the other party to accept conversely the leadership of the
party making the "demand".
    And with the general move towards a more elaborated model
to be studied (with the use of modern resources and technology
for actual computations!) we have also the idea of setting up
a game model with properly transferable utility. Thus when
cooperative coalescence is imperfect and a coalition of just
two players remains, unaccepted and unaccepting, then its
leading player is given a strategic choice of how to divide
the resources (determined by a "characteristic function")
among himself/herself and the other player in that small
coalition. This adds 6 strategies of the "utility allocation"
type (of which type there were 24 in all in the earlier model).
   So the total number of strategic parameters will be
39 + 3 + 18 + 6 = 66 in all.
   The idea of refining the prior model by added the optional
demands applying to the acceptance behavior of the counterparties
being considered for acceptance was stimulated by the computing study that I attempted of Game 9 of a set of 10 experimental games. In this investigation it was found that when one tried 
to find a good theoretical equilibrium in Game 9 it was as if,
in approximating game examples tending towards Game 9 (in their 
2-player coalition strengths) that Player 3 was beginning to reduce his/her probability of “accepting” a presented coalition 
of players P1 and P2 to zero, or even to a negative number (although that would be impossible for an actual probability).
    With the modeling change the formed two player coalition
of P1 and P2 would be able to DEMAND something relating to 
the acceptance behavior by P3 (as a condition regulating their
probability of accepting P3 as a final leader. 
    (It is not very easy to give here a fully elaborated explanation of all the relevant considerations.)
    (But study of a variant modeling will surely shed some light
on the general question of the effectiveness of this sort of study
for games where cooperation is expected.)