The core of a class of non-atomic games which arise in economic applications

We prove a representation theorem for the core of a non-atomic game of the form v = fOil, where Il is a finite dimensional vector of non-atomic measures and f is a non-decreasing continuous concave function on the range of Il. The theorem is stated in terms of the sub gradients of the function f. As a consequence of this theorem we show that the game v is balanced (i. e., has a non-empty core) iff the function f is homogeneous of degree one along the diagonal of the range of Il, and it is totally balanced (i.e., every subgame of v has a non-empty core) iff the function f is homogeneous of degree one in the entire range of Il. We also apply our results to some non-atomic games which occur in economic applications