Potentials and Weighted Values of Nonatomic Games

Abstract

The “potential approach” to value theory for finite games was introduced by Hart and Mas-Colell (Hart, S., A. Mas-Colell. 1989. Potential, value, and consistency. Econometrica 57 589–614.). Here this approach is extended to non-atomic games. On appropriate spaces of differentiable games there is a unique potential operator, that generates the Aumann and Shapley (Aumann, R. J., L. S. Shapley. 1974. Values of nonatomic games. Princeton University Press, Princeton, New Jersey.) value. As a corollary we obtain the uniqueness of the Aumann-Shapley value on certain subspaces of games. Next, the potential approach is applied to the weighted case, leading to “weighted non-atomic values.” It is further shown that the asymptotic weighted value is well-defined, and that it coincides with the weighted value generated by the potential.