John F. Nash Jr.
Title of Presentation: The Agencies Method for Modeling Coalitions and Cooperation in Games
Our work in this research project represents the beginning of an effort to study the game-theoretic phenomenon of cooperation in cooperative games (that is, in games where it is understood that the players MAY cooperate whenever they would naturally desire to do that so as to realize mutually advantageous benefits). (This project of work could be described as work along the lines of the (old) “Nash program”, seeking to reduce the study of “cooperative games” to the area of the (theoretically simpler) study of “equilibrium” in “non-cooperative games”.)
The theoretical key for the inter-linking of these areas is the concept of the “evolution of cooperation” (in Nature) which has been studied both by theoretical biologists and by game theorists.
Our key idea for the reduction of a process realized through cooperation to a process achieved by actions taken independently (and separately) by the Players that are involved in the game context lies in the introduction of moves (or actions) of “acceptance” together with the supposition of an indefinitely repeated game context in which the Players may react (in punishing fashions) against unfavorable actions on the part of other Players. Our model designed for this purpose has the players strategically making “demands” in relation to the behavior of the other players.
And at the same time any Player also chooses, strategically, how he (or she) will allocate the resources available to a coalition if he (or she) has become accepted (through acceptance elections) to have that power. As it happens, in short, the studied model, for a game with three players, involved 39 strategic parameters to be controlled by the players, and we represented it in terms of 42 variables.
There was a substantial challenge of computation to find game theoretic solutions. These were sought in terms of PURE STRATEGIES. And the game example itself, although being described by a characteristic function giving rise to three numerical parameters setting the payoff benefits accessible by specific coalitions, was an “NTU game” rather than a game with “transferable utility”. However we have compared solution results with corresponding indications for the games deriving from the Shapley value or from the nucleolus.
The general context of computations is found to be quite challenging. We have used Mathematica in the connection with the work done up to now. There seem to be possibilities for refinements in the model structure and games of more than three players can also be studied.