Potentials and weighted values of nonatomic games

The `potential approach' to value theory for finite games was introduced by Hart and Mas-Colell (1989). 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 (1974) 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.