Refinements of Stationary Points with Applications to Noncooperative Games and Economics

It is well known that any continuous function f defined on a nonempty compact and convex set X has a stationary point. In many circumstances there may exist multiple stationary points and some of them may be undesirable from the viewpoint of stability. In this paper we introduce a new method of eliminating those undesirable stationary points while at the same time retaining some desirable stationary points. The main idea of refining the concept of stationary point is to perturb simultaneously both the domain set X, by taking a sequence of sets in the (relative) interior of X converging to X, and the solution concept, by replacing the concept of stationary point by a coincidence point with some well-defined mapping. If a stationary point is the limit of a sequence of coincidence points, we say that the stationary point is stable with respect to this sequence of subsets of X and the coincidence mapping. It is shown that stable stationary points exist for a large class of perturbations. A stable point is said to be normal-stable if we take the normal cone as the coincidence mapping, implying that any coincidence point on a subset in the sequence is a stationary point of f on this subset. It is shown that a normal-stable stationary point always exists for any sequence of subsets which starts from an interior point and converges to X in a continuous way. Special cases of normal-stability are perfect stationary points and robust stationary points. In addition, several practical applications of these new concepts are provided. © 2006 Society for Industrial and Applied Mathematics.