On randomized stopping points and perfect graphs

Randomized stopping points form a convex set associated with the information structure that arises in the context of the optimal stopping problem for two-parameter processes. We study combinatorial properties of this structure when the underlying space is finite, in which case this convex set can be identified with a bounded polyhedron. Study of the extreme points of this polytope motivates the definition of an apparently new class of perfectly orderable graphs. Properties of this class of graphs are examined. For this setting, it is shown that under a classical hypothesis of the probabilistic model, the extremal elements of the set of randomized stopping points are precisely ordinary stopping points.