Values of smooth nonatomic games : the method of multilinear approximation

In their book Values of Non-Atomic Games, Aumann and Shapley [1] define the value for spaces of nonatomic games as a map from the space of games into bounded finitely additive games that satisfies a list of plausible axioms: linearity, symmetry, positivity, and efficiency. One ofthe themes of the theory of values is to demonstrate that on given spaces of games this list of plausible axioms determines the value uniquely. One of the spaces of games that have been extensively studied is pNA, which is the closure of the linear space generated by the polynomials of nonatomic measures. Theorem B of [1] asserts that a unique value ~ exists on pNA and that II~II = 1. This chapter introduces a canonical way to approxi1)1ate games in pNA by games in pNA that are "identified" with finite games. These are the multilinear nonatomic gamesthat is, games v of the form v = F 0 (111,112, . . . ,l1n),where F is a multilinear function and 111,112,. . . ,l1n are mutually singular nonatomic measures. The approximation theorem yields short proofs to classic results, such as the uniqueness of the Aumann Shapley value on pNA and the existence of the asymptotic value on pNA (see [1, Theorem F]), as well as short proofs for some newer results such as the uniqueness of the 11value on pNA(I1) (see [4D. We also demonstrate the usefulness of our method by proving a generalization to pNA of Young's characterization [6 and Chapter 17 this volume] of the Shapley value without the linearity axiom, and by generalizing Young's characterization [7] of the Aumann-Shapley price mechanism. In the last chapter we use the ideas behind the multilinear approximation in order to supply an elementary proof to a classic result in analysis: the Weierstrass approximation theorem.