ON THE CORES OF SCALAR MEASURE GAMES

Abstract

A $CVM(k)$ game is a game of the form $f\circ \lambda $, where $\lambda $ is a $k$-dimensional non-atomic measure and $f$ is a continuously differentiable function on $R^k$. For a convex $CVM(1)$ game, we characterize the ``least upper bound'' and ``greatest lower bound'' of the core elements in terms of the distribution function. We also show that the core of a convex $CVM(1)$ game expands as the underlying measure $\lambda$ changes in a ``convex manner''. These results provide a partial geometric picture for the core and its variations of a convex $CVM(1)$ game.