Stability of fixed points of set-valued mappings and strategic stability of Nash equilibria

Abstract

In this paper, we mainly focus on the stability of Nash equilibria to any perturbation of strategy sets. A larger perturbation, strong δ-perturbation, will be proposed for set-valued mapping. The class of perturbed games considered in the definition of strong δ-perturbation is richer than those considered in many other definitions of stability of Nash equilibria. The strong δ-perturbation of the best reply correspondence will be used to define an appropriate stable set for Nash equilibria, called SBR-stable set. As an SBR-stable set is stable to any strong δ-perturbation and, various perturbations of strategy sets are not beyond the range of strong δ-perturbation, it has the stability that various stable sets possess, such as fully stable set, stable set, quasistable set, and essential set. An SBR-stable set is stable to any perturbation of strategy sets, so it will provide convenience for study in strategic stability, which is even used to study any noncooperative game.

On stable fixed points under several kinds of strong perturbations

“…The results, in relation to the existence of a minimal strong stable set in fixed point sets for correspondences, generalize the corresponding results in [28] and [27] from N neighbors to N co (N c , N co ) neighbors, from the above second kind of correspondence to the first kind of correspondence, and from no perturbation of the domain X to the perturbation set K(X). The results also generalize the corresponding results from several aspects in [22] such that, the defined space is generalized from an Euclidean space to a normed linear space; the perturbation of a correspondence is enlarged from an N X neighbor to a strong perturbed N co (N c , N co ) neighbor; the perturbation of the domain X is extended from K(int(X)) to K(X).…”

“…For the deep study of strong stability of fixed point set fix(F) for a correspondence F ∈ U co (X), in the papers [27] and [28], Xiang et al introduce an interesting δ neighbor of F as the following:…”

“…where co(A) is the convex hull of the set A, and B δ (0) = {x ∈ E : x < δ}. The papers [27] and [28] prove some strong stable results related to the subsets of fix(F), and show that a strong stable subset of fix(F) can eliminate some abnormal fixed points. If we only consider correspondences in U co (X), from the references, it holds that…”

“…For strong stable equilibria, the existence condition and discreteness is studied in [15], where a strong stable equilibrium is more stable than an essential equilibrium. Recently, Xiang et al introduce a strong perturbation class of correspondences and obtain strong stable sets for fixed points and Nash equilibira in [27], and further results are obtained in [28]. A strong stable set of fixed points can resist a larger perturbation of correspondences than an essential fixed point set.…”