Outline

       (of talk/lecture/colloquium at Stony Brook in July 1999)


                   Remarks on Cooperation as a Concept

      It is taken as obvious, particularly when it is a matter of human 
relations, that there can be cooperation and that this can be "high 
level cooperation" or "high grade cooperation". But really there are 
some elements of mystery where all might seem to be simple and obvious. 
Instincts, culture, and tradition are factors leading the cooperative 
behavior of humans or animals.
     Civilization is a key to the context of the cooperative 
possibilities for corporations. Civil law provides the means by which 
contracts between or among enterprises can be binding and enforceable.
     Also, a cooperative game has the characteristic that it is not war 
and a player cannot simply be conquered and deprived of the right of 
separate existence.


       Test Example for the Approach Described Last Year at Stony Brook

    I have been doing work since talking here last year to apply the 
theory or approach that was presented then to an example of a 3-person 
game.
    Already then (when I spoke in 1998) I had been able to treat 
bargaining-type games of 2 persons and to get results consistent with 
the solution point of Zeuthen and Nash.
    This topic of previous presentations was "Reduction of Coalitions 
to Agencies, a Scheme for the Analysis of Cooperative Games". (The TEXT 
VERSION of that (without the graphically presented illustrative 
examples) is available on my "web page" under the topic of AGENCIES.)
    The basic idea of this approach or theory is that a game can be 
studied in the repeated game context where the repeating can help to 
facilitate cooperation in a context otherwise unfavorable. This is like 
when cooperation can be found to be quite stable in a repeated 
prisoner's dilemma game.


                  Transfiguring Coalitions into Agencies

      The idea of agencies is a means for reducing a game where the idea 
of cooperation is quite essential to the structure of the game to a game 
which is, in form at least, entirely a  non-cooperative game where the 
players simply have specified possible "moves" (or strategic actions) 
which they carry through in the formal context of a non-cooperative 
game. But we will only be studying this derived game under the 
presumption of indefinite repetition so that we will not be considering 
it as a "one shot" non-cooperative game would be considered.
      Thus cooperation is expected to arise or derive, but this is 
expected to depend essentially on the context of repetition.
     And the 'agencies" are the analogue of the chairpersons of 
committees, as if every coalition be regarded as equivalent to a 
committee with such a chairperson. Indeed our agents have ABSOLUTE 
power, to the extent that they are like powerful chairpersons of 
committees. But the power of any player that he might exercise as a 
powerful agent is limited by the concept of the repetition of the game 
and by the convention the he/she must be "elected" (or chosen, or 
accepted) by at least one other player and by the concept that each 
player will build up a reputation that will be known to all the other 
players.


                     Election of Agents or Agencies

      An agent is like someone with a complete "power of attorney" 
given by another person or persons. In the simplest case of a game 
of 2 persons, where we wish to look for the ideal possibilities of 
cooperation but also to take into account the conflicting or competitive 
interests of the 2 players, it would be that either of the two players 
could accept the other as his agent. And THEN, after one of the two had 
become thus the "general agent", the idea is that that player could 
then COORDINATE all the possible uses of the pure strategies a priori 
available to the two separate players. (And with more than 2 players 
a "general agent" would be one that had been given agency power to 
represent all of the players of the game.)
      Our study of the specific very simple 3-person game that we 
studied made use of a specific convention on the election of agents. 
The various possibilities for such election rules are discussed more 
broadly in the earlier text on "AGENCIES' that is also available on 
the web site where this outline is being made available.
    For our model study we used a convention that if a "cycle" was 
elected, where a string of voters would be connected in a loop by 
the fact of each having voted for the next voter of the loop, then the 
"cycle" would be resolved by making a RANDOM removal of one of the votes 
so that the cycle member whose vote was thus removed would become the 
player elected by all the other members of the loop to be their agent.
    And we also used the "chain rule" so that if A votes for B and at 
the same time (same election stage) B votes for C then this elects C to 
be the agent for both A and B (and himself). Of course a use of this 
rule is implicit in the description just given of how the election of 
a "cycle" leads, after a random choice, to the election of one member of 
the "cycle" as the agent elected to represent all of the cycle's members 
(including himself).
   A fuller discussion of election rules for agents is presented in the
text called "agencies".
   For the specific model game (or for any game of only three players) 
the election rules we used achieve the effect that ONLY TWO stages of 
election are needed. Then either a "general agent" has been election 
or there is given the "verdict" that cooperation (or coalescence) has 
failed.
   The first stage of election may reduce the game to one of just two
parties or also may have achieved at one step the election of a 
"general agent", one player entitled to act "in behalf" of all players. 
If there is a reduction to two parties then one of these will be a 
player that has been elected to be the agent for one other player 
(besides himself). Then the second stage of election follows and this 
is, in form, the same as it would be if we were beginning with just a 
game of 2 players. And if with two players (or 2 parties) voting, if 
neither of them accepts the other as agent, then the "verdict" is that 
cooperation has failed. Then, in a general game, the 2 parties would 
be expected to then act non-cooperatively, utilizing the strategic 
possibilities available to them for non-cooperative action. (One of 
the parties would be an elected agent representing himself and one 
other of the original trio of players. This agent would, in the 
non-cooperative play, have the potential of using ALL of the pure 
strategies of BOTH of the players represented, with full coordination 
of these pure strategies.)
   But in our model game of three players there is nothing to do in the
case that full coalescence (full cooperation) is not achieved. In the
specific case that the coalition of players P1 and P2 has been achieved
but that general agency or full cooperation has not been achieved then
this amounts to that one or the other of P1 and P2 has become the agent
of both and our specification of the model game is just that in this 
case that P1 and P2 are given the smaller bonus of b to share between 
them (with a payment of b/2 for each as we have set up the game's payoff
function in terms of votes and "allocations"). And if coalescence fails
at the first election stage or if it fails with either the coalition
of P1 and P3 or that of P2 and P3 having been formed then the payoff
rule is that all players are given zero.
   On the other hand, if a general agency has been elected and we can
consider that the triple coalition (1,2,3) has been formed or achieved,
then the rule is that that coalition is awarded the sum of +1 in payoff
to be divided among the members (all three players) with the use of
"transferable utility". This means, in our context of agencies, that
the elected "general agent" is given the payoff of +1 which he is able
to divide, using transferable utility, as if it were a sum of money,
among the three players (including himself).


                        Allocations

     For the model game study and calculations which I had done by the
time of speaking in July 1999 at Stony Brook I had SIMPLIFIED the game
by restricting the strategic choices that a priori would be available
to players as agents by requiring that no matter HOW a specific player
might have happened to become elected "general agent" that he would
choose THE SAME "allocation" in every case.
     In principle, a "general agent" has the right to do whatever 
he/she pleases and this operating with all the pure strategies of all 
the individual players. Thus, for a game of 3 players and transferable
utility, an elected general agent has the right to dispose of all the
resources of v(1,2,3) (from the characteristic function). And in our
model game we had  v(1,2,3) = +1  and we assumed that a general agent
was constrained to divide up that total amount of transferable utility
in such a fashion that the amount thus "allocated" to any individual
player would be non-negative and that all 3 of the amounts would sum
to  +1  which is the quantity of v(1,2,3).
    But in general, in principle, if we are looking for "stable behavior"
strategies of players operating as electors of agents and as agents then
whenever a player arrives at the circumstance of having been elected to
be general agent then he/she CAN DISTINGUISH BETWEEN all the possible
various routes by which he/she happened to become so elected. And it is
like in politics, where a politician may choose to differently reward
voters depending on WHEN, in the stages of election, a voter happened
to vote for the politician. With a game of 2 players there would be only
one or two alternative possible "routes of election" by which a player
might become elected to be general agent. But with 3 players there can
be many many more, depending on the precise election rules.
    In the calculations that I had done before speaking at Stony Brook
in Summer 1999 I had chosen to SIMPLIFY by requiring that any player
must "behave" with a uniform formula of allocations (that he/she would
impose whenever elected, by whatever election route, to be general
agent). Thus if P1 was thus final player-agent (general agent) (like
the general partner in a partnership in which all but one partner would
be limited partners) the P1 would allocate according to (u11,u21,u31)
with  u11 + u21 + u31 = +1  (and this was the notation that we used in
the calculations (using MATHEMATICA)). Thus u21, for example, means the
amount of utility allocated to player 2 by player 1 whenever P1 would
have become the final player-agent.
    (Subsequently, since the calculations seemed to indicate that 
the coalition of (P1,P2) was failing to be "properly appreciated" 
or properly effective, and since this had the effect that the results 
seemed not to favor the Shapley value at all, we/I came to the opinion 
that it will be necessary to use a more elaborated modeling and to give 
the quasi-robotic players of the modeling more strategic potential so 
that, like politicians, they can give more rewards to supportive voters 
in PRIMARY ELECTIONS than they would give to supportive voters in FINAL 
ELECTIONS (or vice versa).)

 
        Returning to Remarks at Stony Brook 1999

    The text above contains some remarks that represent my views as they
have evolved subsequent to the time of the 1999 meeting at Stony Brook. 
Now I continue to write more or less what was said at that time.
    I remarked that I had hoped to be able to present fully worked out
calculations of a model with reactive strategies for the repeated game
based on the simple 3-person example that has been discussed. But as it
happened, I had had difficulties with the calculations and I had not
found equilibria that were really adequately persuasive that they were
indicating a good analysis of the game. So I presented some examples
of calculation results as they were obtained (by that time) and I went
into a discussion of how such results could relate to the comparison
of the Shapley value and the nucleolus or any other game theoretic
concepts that would assign a value vector to a cooperative game in the
way that this is done by the Shapley value or by the nucleolus.
    A few transparencies were exhibited included one that presented the
results of calculations using MATHEMATICA to find equilibria arising
from the model (restricted to what are pure strategies in terms of
continuous parameters). The modeling had the quantity b described
above which describes the special advantage of the coalition (P1,P2)
of players P1 and P2; and it also had a quantity which was originally
handwritten as "epsilon" and which became "e" for the formulae to which
the machinery of MATHEMATICA would be applied. This quantity "epsilon"
effectively measured the precision with which a player could "demand"
that he should be getting favorable allocations of utility when another
player would become general agent as a precondition for electing to
"accept" that other player as agent (in the elections which elect the
agency structures leading to a choice of a general agent or to a 
failure of the achievement of (full) coalescence.
    A calculation with epsilon = 1/200 and with b = 1/4 led to the 
finding of an apparent equilibrium with the property than P3 was 
actually getting the best payoff. Thus this equilibrium (in the model,
not in "ideal" modeling, presumably) could not be reasonably 
interpreted since the effect of the advantage of the coalition (1,2)
SHOULD be advantageous, value-wise, to players P1 and P2 rather than 
to player P3.
    Similar calculations done with epsilon larger, at 1/30, gave 
results seeming more reasonable, and either with b = 1/4 or with 
b = 1/10 an advantage to players P1 and P2 was indicated. But I was
surprised by the fact that the advantage of the favored players came
out as rather small. But then this seemed to correspond well, at least
for b < 1/3, with the fact that the "nucleolus value" has that the 
advantage given to P1 and P2 by  v(1,2) = b  is ineffectual  and that,
so long as  b < 1/3 , the value of the game is simply (1/3,1/3,1/3).
And it was clear that before we would have epsilon passing to a limit
value of zero so that we would consider the limiting results for the
"evaluation" of the game under that asymptotic hypothesis that before
that we should not expect precise results but rather approximate 
results so that it seemed that, if we COULD properly get a result for
the limiting case that it might be that as epsilon would tend towards
zero that the SMALL advantage of players P1 and P2 would entirely
disappear and that the "nucleolus value" would be the derivation.
    This approach, with "epsilon" or "e" tending to zero was suggested
by what had proved to be successful for examples of bargaining games 
of just two players. And in those studies it was found that if a 
bargaining game was of the sort that would have the value of (1,1) by
the "solution" of Nash or Zeuthen but with an advantage to player P1
by the Kalai-Smorodinsky solution (and with the Pareto boundary 
happening to be smooth and convex) then that as "e" (epsilon) would
decrease towards zero that the advantage found for P1 over P2 would
decrease parallelwise and that the "Pareto efficiency" would increase
towards 100%.
    With 3 players the results of calculations seemed to indicate that
the first efforts at modeling were too primitive but that indeed it
WOULD also result for 3 players that as "e" would tend towards zero
that the Pareto efficiency would tend towards 100%.
    (A separate text is being prepared to describe the way in which
the players' strategies were so defined in the modeling that they 
would choose "demands" relating to their (behavioral) "acceptance
probabilities" rather than choosing those acceptance rates directly.
For example, at the first stage of voting for agents ("accepting"
agencies) player P1 was modeled as choosing to accept player P2 as
his agent with the probability  a1f2  (in a context of repeated play).
And  a1f2  was not a quantity that P1 would choose as a strategic 
choice, but RATHER player P1 would choose  d1f2 , a "demand" relating
to his acceptance rate  a1f2  and  d1f2  would enter into a formula
determining  a1f2  which also involved  u12  which was the amount of
utility that player P2 would allocate to player P1 IF it developed that 
P2 became general agent.)(These details were not actually gone into
in the limited time of my talk in July 1999.)

     
       Conclusions in Relation to Further Model Development

   As I said above, for smaller values of the parameter b at least, the 
calculations and the equilibria (to the extent that I could find them)
seemed to suggest that the "nucleolus value" of (1/3,1/3,13) was better
than the Shapley value of ([b+2]/6,[b+2]/6,[1-b]/3). But of course, as
b increases to an ultimate possible value of +1 then the "nucleolus
value" must become ([b+1]/4,[b+1]/4,[1-b]/2) when b increases above 1/3
and when b increases to +1 then these two alternative value vectors
must become the same, (1/2,1/2,0). But calculations failed to indicate
that the modeling was adequate, for larger values of b, to fit either
the Shapley or the nucleolus evaluation. In effect, the modeling of the
cooperative potentialities of the players seemed to be weak with regard
to its effective appreciation of the influence of the strengths of the
potential alliances of the favored players, P1 and P2.
   So it seemed indicated that the modeling should be refined and THEN,
with that done, then possibly the strong benefits to players P1 and P2
that are indicated by the Shapley value might be more effectively
appreciated by the outcome of calculations.
   In the equilibria that could actually be apparently well calculated,
with  e = 1/30  and b = 1/4 or b = 1/10  it was found that in the cases
where they players (in the repeated game) would fail to achieve general
coalescence (or the election of a general agent) then that more often
they would fail to achieve any coalescence at all and less often a
coalition of two players would be formed (or an agent representing only
himself and one other player would be elected). Then thinking about
this I began to realize that it would be necessary for the outcomes of
incomplete coalescence to occur more frequently (compared with the
frequency of total non-coalescence) in order for the special advantage
of players P1 and P2 to make itself felt with more strength. The 
calculations had the players voting (accepting agencies) with such
probabilities that once any agent had been elected at the first voting
stage that then it was almost certain that a general agency (agent)
would be elected at the second stage. So the value of the coalition
(P1,P2) was effectively entering very little into the resulting 
payoffs since that coalition was so rarely not superseded by a general
coalition with  v(1,2,3) = +1 .
   Of course, according to the nucleolus evaluation, the (1,2) 
coalition SHOULD actually fail to be effective unless b rises and 
becomes more than 1/3. But I failed to find the right sort of outcome,
by calculations, for  b = 2/3 , for example. This led me to think that
the modeling should be refined so that the players can more 
effectively seek to attract desirable acceptance results (from other
players) particularly at the time of the first stage of voting.
   

               Program for Further Work

    So we can continue the study by calculations (using MATHEMATICA)
for these simplest non-trivial 3-person bargaining games. It seems that
we should have 5 different varieties of allocation strategy for each
player so that he/she, if elected to be general agent, allocates the
gain to the players according to these cases:

   (Illustrated for player P1)

(1): P1 was first elected by P2 in stage 1, then by P3 in stage 2.
(2): P1 was first elected by P3 in stage 1, then by P2 in stage 3.
(3): P1 was elected in stage 1 by both of P2 and P3.
(4): P1 was elected in stage 2 by P2 as agent for (P2,P3).
(5): P1 was elected in stage 2 by P3 as agent for (P3,P2).

    This, obviously, will greatly increase the number of parameters
(strategic parameters) for which we will be seeking numerical
solutions. Is this practical?
    Here I think there is a very fundamental principle relevant to the
current status of the world's technological civilization. It is 
becoming more and more practical to solve problems by calculation
procedures on a scale that was previously outside the range of feasible
practicality. Large weather and geological calculations can be done in
the area of PDE or PDES theory. The 4 color problem was successfully
studied 20+ years ago but in the heavy computing era.
   The large scale calculations in "theoretical biology" are analogous,
that area COULD study even more complex evolutionary issues involving
game contexts relating to repeated asymmetrical prisoner's dilemma
games. Etc. And there is, with the computation resources that are or
are becoming available, the possibility of "experimental mathematics"
research relating to game theory. And that experimental variety or side
of research is where I think that the approach in terms of agents as
the means of cooperative coalescence can lead to a genuinely scientific
approach to the challenge of "evaluating" cooperative games.
   It is not quite obvious, a priori, why there should be a function
defined for ALL cooperative games that would give a value vector for
the game as a function of its description. And there are indeed very
degenerate example NTU games where the concept of a value vector seems
inappropriate. On the other hand, it IS reasonable that MOST such games
might be assignable a reasonable value concept. But what is reasonable?
A scientific approach hopefully can illuminate this question.