1. Our Current Project of Research The Previous Work and the Paper We have a paper that was published in the issue for Dec. 2008 of the journal “International Game Theory Review”. The title of that paper was: “The Agencies method for Modeling Coalitions and Cooperation in Games”. Our current research is a continuation from that, and in particular is on the structure of the “model players” where we are seeking to find refinements. In our method of approach the players are playing an infinitely (or indefinitely) repeated game and they are “reacting” to the behavior of the other players. This makes it POSSIBLE for a game like “the repeated game of “prisoners’ dilemma” to be cooperatively resolved (to the benefit of the players) by the circumstance of the discovery of strategies of a reactive equilibrium where the two prisoners will react discouragingly to any actions by the other player that corrupt the ideal of the cooperation of the prisoners against the police and the authorities of justice.

2. We are now wishing to refine and improve upon the modeling of what the players can actually do, particularly with regard to how they can react to disapproved or approved actions by other players. (A part of the refinement, which itself will simply make the model a little less simple, will be to provide that there is, effectively, fully transferable utility, when the game reduces to a coalition of two, led by one of them (as the boss) and another player who has failed to get in a coalition and will receive payoff zero. Then the boss (of the surviving coalition of two players) can freely choose a partition of the payoff resources of that coalition and assign payoffs to himself/herself on the basis of that.) (Our model previously had this partition always 50%/50%, but that is not realistic generally, especially for a game taken as described by a “characteristic function”.) Random Proposers A method, which we can call "the method of random prop- osers" has been used to reduce the problem of the "evaluation" of 3-person cooperative games to a study of a group of non- cooperative games involving the three players.

3. The history of this approach goes back to I. Staohl and to Ariel Rubinstein. It does have the basic attribute of the "Nash program" for the study of cooperative games in that using the random proposers models enables the reduction of the cooperative game to the study of non-cooperative games where there is a specific designated player whose move in the game is to make a proposal and then the other two players (acting separately) move to accept or reject the proposal. So this is analogous also to an auction situation, where one of the players, chosen at random, is given the position of being the auctioneer and the other two become simply participants in the auction. (But it is not exactly like an auction!) More recent contributors to the analyses of 3-person cooperative games using this method of approach include A. Okada of Hitotsubashi University and Armando Gomes of Washington University. Gomes developed an approach, using a random proposers reduction, which led to the concept of a "CBV" (or coalitional bargaining value) with very nice properties. Depending on the value of a simple linear function of the payoffs assigned by the "characteristic function" to the three two-person coalitions (under conditions where the rest of the charac-

4. teristic function would be naturally specified by having appropriate choices for the utility functions of the three players) the evaluation prediction given by the CBV would be either the "Shapley value" or the "nucleolus" (or both of these when they would become coincident). But, and this is our position, the EQUALIZING aspect of the "random" proposers approach can be questioned for games of more than two players. As it turns out that to be the "proposer" is essentially profitable or relatively favorable then to have this more favorable game position assigned at random is effectively an ordained formula for equalization that could be considered, comparatively, as if it were imposed by an arbitrator. My personal view (or "our view") is that a high goal of the game theoretic study of "cooperative games" is to discover rules or answers that could be applied, justifiably, in situ- ations of arbitration or of counseling. In such situations the parties in a generalized bargaining situation may be helped by good advice or counsel coming from a sufficiently developed game-theoretical understanding of games in general. If the arbitration formula is already arbitrarily put into the game theoretical study then a formula cannot be validly DEDUCED from that study.

5. Contributions from the Area of Experimental Games After the theoretical work, of the paper in the IGTR journal, and the calculations based on that work, there developed an interest in testing out, as much as possible, through experiments, the theory developed to fit players playing in these game situations where cooperation would be necessarily achieved by the means of actions of “acceptance” in which an individual player would unilaterally accept the agency (or leadership) of another player or coalition. (Here, if a coalition were “accepted” this would amount to the acceptance of the previously formed leadership of that coalition as also the leadership of the enlarged coalition.) The players would also, in our larger modeling scheme, be expected to react to the (observed) pattern of behavior of other players so that they could reward or punish favorable or unfavorable behavior patterns observed on the parts of the other players. The experiment worked out with the players reaching a comparatively high level of efficiency in terms of cooperating enough so as to usually obtain all of the benefits obtainable by the the “grand coalition” (which is the ensemble of all three players somehow cooperating).

6. And, as for the general pattern of the observed results, there was a tendency for the division of the profits among the players to be more “equalized” than what would be predicted by many of the existing concepts in game theory which provide “evaluation” “imputations” for these games. Our fully developed mathematical model, which effectively studied the cooperative behavior of a sort of robotic players dealing with a repeated game, turned out not to be calculable for 9 of the 10 experimental games. (The games in the experi- ments were not most favorably chosen to favor the calculab- ility of model solutions; at the time when they were chosen it wasn’t well understood which games would be “calculable” and which wouldn’t.) More on the Experimental Research The experiments were carried out at the behavioral research center of the University of Koeln (Cologne) under the direction of Prof. Axel Ockenfels. And there were three persons involved in the design and motivation of the program for the experiments. These persons were Prof. Ockenfels, myself, Prof. Reinhard Selten, and Prof. Rosemarie Nagel.

7. (Prof. Selten actually composed the specific set of 10 games to be studied and they seem to have been well chosen to explore the general concept of three party cooperation (although they proved to be unfavorable for the actual comput- ability of solutions according to my modeling scheme of that time).) There is a PowerPoint presentation available from a presentation in Barcelona which describes the 10 specific games and the experimental results obtained. Only on Game 10 were the strengths of the 2-player coal- itions weak enough so that my model (as if of robotic players) worked out to give a calculable solution. (The numbers from that compared better with the experimental behavioral results than did the nucleolus or the Shapley value.) Calculations for an equilibrium could be made with the parameters of Game 9 also, BUT it turned out that the result of that was that two numbers, which can be called a3f12 and a3f21, were both negative, which was inconsistent with their proper significances as probabilities. We failed with the attempt to simply set these quantities as zero and to remove the corresponding strategies from the game presentation since other complications seemed to appear with the attempt to find, thusly, an appropriate sort of game-theoretic solution

8. involving dis-used strategies. (It often happens in game theory, that some of the pure strategies available turn out not to be used in minimax or equilibrium strategy solutions. Thus these are “disused” strategies in the actual solutions). Revising the Modeling Structure With Regard to Demands The basic theoretical foundation of our model, from which we wish to obtain “evaluations” for cooperative games, is that of the phenomenon of the possibility of the “evolution of cooperation”. So we are using a model structure which has each player able to explicitly form coalitions only by actions of “acceptance”. But, in addition to that, each player is able to make “demands” and to relate his/her probabilities of accepting to the observable (in the repeated game structure of complete information about all past actions) behavior of the other players. Our model structure in the article published in the International Game Theory Review involved 39 strategic param- eters controlled by the 3 players. Among these were 24 strat- egically chosen values for “allocations” of utility. Such allocations would occur, for example, whenever a player would

9. have been elected, in 2 steps, to be the “final agent” and to control all of the resources of the “grand coalition”. Also involved were 15 “demand strategies” whereby a player, either for himself/herself alone or for a coalition of which he/she had become (by election) the leader, would specify desiderata in relation to whatever would be the expected payoff benefits in case that player would act to accept the agency (and the leadership) of the other party related to a demand. Our new idea is, especially since we don’t want any parameters that represent probabilities to become negative, that we can hope to favorably introduce additional demand parameters that correspond to observable behavior on the part of parties which a player would possibly “accept” (to become the agent/representative) of the accepting elector. Thus, at the beginning of the elections Player 1 could look at Player 2 and ask 2 questions in the form of “demands”: First, “How well can I expect to fare, in terms of payoffs, whenever I “accept” (the leadership of) Player 2.” and Second, “How often is it observed that rather than that Player 1 accepts Player 2 that Player 2 is accepting Player 1?” We are hoping that if the players and the players as agents are enabled to also make demands of this sort that the calculated equilibrium solutions will work out more favorably.

10. Actually, with 2 other refinements of design also included (one of which will provide for fully transferable payoffs within final coalitions of two players) this envisioned elaboration of our model will lead to 66 strategic parameters rather than 39. (And we are hoping that modern computer technology and science will make feasible the game solving calculations needing to be made!) The Induced Theory Issues Applying to the Reduction to Two Persons In relation to the game model structure that we used in our recent publication in the IGJR journal, we had previously initially studied the type of modeling of the players in a reduced form with only two players. This made it possible to verify that there were natural phenomena of equilibration relating to the calculable solutions. Our first lectures given while developing our studies were of 2-person game cases. And now, if we modify the theory by changing the modeling of the players so that they have 66 strategic parameters through which they control their patterns of activity there is, again, the possibility of looking at the reduced case where two players are interacting with each other.

11. Each player will be able to make “demands”, both on how frequently the other player “accepts” the (always available) leadership role of the demanding player and also on how much utility benefit the demanding player is receiving from the other player in those cases of play when the demanding player acts to accept the coalition leadership of the other player. Thus there are 4 distinct demands to be made by the two players and also each player acts as an “allocator” whenever he/she has become elected to become the agent for the other player and to access the resources of the “grand coalition” (which in this reduced case is simply the coalition of the two players). So the result is that there are six strategy parameters and also the two rates of “acceptance” by the two players. This can lead to a system of 8 simultaneous equations to be solved for 8 numbers to find what the two players would be doing at equilibrium. We have already been computationally studying some of the model properties in connection with the reduction to two players, although the precise structure of the modeling scheme is not fully defined yet.

12. Our Recent Publication John F. Nash, Jr. [2008] The Agencies Method for Modeling Coalitions and Cooperation in Games, International Game Theory Review, 10(4), 2008, 539-564.