"A New Method for Studying the Coalescence Process and the Natural Realization of Cooperation in Cooperative Games"

John Nash, Jr.
Princeton University

By modeling a cooperative game as a game that is to be played repeatedly and in which the players actually achieve cooperation by specifically allowed processes it becomes possible to study the results of their cooperative efforts as a problem in large scale computation. In our studied case (or cases, since we study three person CF games having various amounts of specified obtainable value for the three two-player coalitions) we find that there are 39 strategy parameters in all, as the result of our modeling, and we arrive at a system of 42 equations in 42 variables to be solved. This system of equations proved to be quite challenging, in terms of finding efficient methods for finding numerical solutions. We used Mathematica both for the development and verifications of the equations and also for obtaining the numerical solutions which made possible the interpretation of the predictions by the model. And these output results, for cases where the two-player coalitions were comparatively weak, lay on a curve that was between the graphs (a line and a broken line) representing the payoff predictions for the players according to the Shapley value or the nucleolus.

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