  The range of a payoff function for an $n$-player finite strategic game is
investigated using a novel approach, the notion of extreme points of a
non-convex set. The shape of a noncooperative payoff region can be estimated
using extreme points and supporting hyperplanes of the cooperative payoff
region. A basic structural characteristic of a noncooperative payoff region is
that any of its subregions must be non-strictly convex if the subregion
contains a relative neighborhood of a point on its boundary. Besides, applying
the properties of extreme points of a noncooperative payoff region is a simple
and effective way to prove some results about Pareto efficiency and social
efficiency in game theory.
