Recent Research Studies on Cooperative Games In the recent years since around 2003 I have been doing research following up on a specific idea for the finding of a cooperative solution for a simple type of game through the means of solving a game problem that is the analogue of a "Prisoner's Dilemma" game problem where the investigator seeks to find means for understanding how the prisoners might be able to cooperate, if their dilemma (or bad situation) occurs repeatedly (because of the overall game being defined as a "repeated game”). Other game theorists, including R. J. Aumann, have studied the repeated game version of the "Prisoners' Dilemma" context and studied how a form of cooperation can emerge and change a "losers' situation" into a "winners' situation". This natural phenomenon became the enabling support for my idea of the "method of acceptances" which made it possible to put the actual process of the formation of coalitions into a simple form that could be studied mathematically (for a suitable basic cooperative game). The specific idea of "the method of acceptances" occurred to me at the time when I was contributing at a summer "national science camp" and lecturing on evolution and on how it is natural for forms of cooperation (between or among species) to evolve in Nature. The Project at Princeton I started by doing some calculations myself applying the “Method of Acceptances” to simple games of two players where it could be seen that a suitable form of equilibrium would exist (as the players would be strategically deciding on how they would behave strategic- ally in reaction to the behavior of another player and to the intrinsic structure of the game). This form of equilibrium that I was finding was parallel to the sort of equilibrium that I had earlier found and exploited in “Two-Person Cooperative Games” which appeared in 1953 in the journal Econometrica. After “the method of acceptances” appeared to lead to the same sort of results, for comparable games of two players or parties, as I had found earlier by converting the cooperative game of two players into a non-cooperative form in which the two players would interact with “demands” and “threats”, then I was encouraged to study games of three persons using the newly found “method of acceptances”.

After “the method of acceptances” appeared to lead to the same sort of results, for comparable games of two players or parties, as I had found earlier by converting the cooperative game of two players into a non-cooperative form in which the two players would interact with “demands” and “threats”, then I was encouraged to study games of three persons using the newly found “method of acceptances”. I began the study of three person games (oriented towards achieving a good level of cooperation of the players) by allowing them to form alliances (and thus cooperate) ONLY by the means of “actions of acceptance”). In such an “action of acceptance” the “accepting player” would be permitted to simply, as if, choose to give to some other player his “irrevocable power of attorney” and then the game would continue from that point (with the player who had “accepted” another player no longer having any executive power in the immediate run of the game). Note, as an aid to generally understanding the idea of the acceptance actions, that with a 3-person game, after two actions of acceptance have occurred, then one player will have become authorized, by having been “accepted” by the other two players), to act for all three of the players. And if the game is a 3-person cooperative game with transferable utility then this elected player will have at his or her disposition the handling of the entire utility amount of v(1,2,3) where v(i), v(i,j), and v(1,2,3) are the numbers defining the Von-Neumann-Morgenstern “characteristic function” for the game. My and our research has, relating to games of three players, been concerned only with games that are completely describable by their “characteristic function” data. (These can be called “CF games”.) Usually the CF is required to be “superadditive” so that if M is the set of players formed by combining two non-overlapping” sets A and B (of players) then v(M) >= (v(A) + v(B)). It needed to be arranged, for my idea, that the three players would each have a strategy of continuous action, with each player having also reactions to the observed behavior of the other players. (Each player has full knowledge of the behavior of the other players since the game (for mathematical solution purposes) is regarded as continuous and infin- ite.) This continuous action by the players became represented ultimately, in the mathem- atical modeling, by sets of probabilities which described the likelihood of the event that a particular player would act with a specific action when in a specific situation. Thus a1b2 could mean “the probability, in any stage of an infinitely or indefinitely repeated game, that Player 1 would be “accepted” by Player 2 (among the players P1, P2, and P3 of the game).

Equilibria with Cooperation in Repeated Games Game theorists, by now, have long studied repeated game contexts where the game played repeatedly is something with dangerous possibilities for negative payoffs (like, in particular, the classic “prisoners’ dilemma”). And by having appropriately “reactive” strategies the players can escape from the negative “dilemma”. A simplest version of this, for the simplest sort of a model PD game, is a repeated-game strategy of “Tit-for-Tat” form, where each player plays his “good” C strategy (instead of his “bad” D strategy) so long as he observes (in the repeated context) that the other player is also playing a “good” C strategy. (It is perhaps well to remark here that if SIMPLE “Tit-for-Tat” strategies are presumed in use then a single error in play would condemn the erring player forever, to play always with the other player simply playing his or her “D” strategy.) In the case of games of more than two players the possibilities for the evolving of cooperative forms of equilibria seem as good as for games of two parties. And what I have been studying recently is the challenge of how to relate model games of three players, of the repeated-game variety, to the search for a good understanding of the general category of three-player cooperative games (to the level of generality of games definable by the stating of the “characteristic function” for the game). It is the “repeatedness” of an originally non-cooperative game of two or three players that makes it possible for cooperative forms of behavior of the players to emerge as equilibrium strategies (in the repeated game context but not in the original). In these equilibria of the repeated game the players are adopting strategies that encourage “cooperative” behavior of the other players that they confront and which discourage less favorable uncooperative actions of those other players. We can illustrate how in the current work the strategies assumed for the repeating play of the players are structured so that each player is effectively “demanding” a level of favorableness of the play of each of the other players backed up by the threat of not ever “accepting” such an other player as an “accepted agent”. But now I want to continue the description of the work that has been done on this research by looking at the pdf file versions of the two published papers (which files are freely available through “open access”).