Potentials and weighted values of nonatomic games

The ‘potential approach’ to value theory for finite games was introduced by Hart and Mas-Colell. 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 value. As a corollary we obtain the uniquiness of the Aumann–Shapley value on certain subspaces of games. Next, the potential approach is applied to the weighted case, leading to ‘weighted non-automatic 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.