Values of Non-Atomic Games, III

Values and Derivatives

by Robert J. Aumann, Lloyd S. Shapley


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The third study in a series concerned with the value of participation in a nonatomic game. The value of a particular [n]-person game is expressed as a function that associates with each player, intuitively speaking, a number representing an a priori opinion of what it is worth to him to play in the game. Coalitions can form among the members; the worth of such a coalition is the total payoff that results for the members. In this Memorandum, the characteristic function of the game is extended to a space that includes not only the ordinary coalitions (measurable sets) of players, but also certain ideal coalitions — sets to which a player can "belong" at any intensity between 0 and 1. The value of the game is then redefined in terms of the Frechet derivatives of this extended function. Relationships are discussed between this definition and previous approaches (axiomatic, asymptotic, and mixing). (See also RM-5468, RM-5842.)

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