Stable Sets and Stable Points of Set-Valued Dynamic Systems with Applications to Game Theory

Let X be a metric space. A dynamic system on X is a set-valued function $\varphi $ from X to X which satisfies $\varphi (x) \ne \emptyset $ for $x \in X$. It generates $\varphi $-sequences: \[x^{(t + 1)} \in \varphi (x^{(t)} ),\quad t = 0,1,2, \cdots ,x^{(0)} \in X.\]We study the stability properties of such dynamic systems. Necessary and sufficient criteria for stability of sets and points are given. The main result is, essentially, that a subset of X is stable iff it is an inverse image of a Pareto minimal point of a vector-valued function which decreases along $\varphi $-sequences.As a corollary we obtain a characterization of all stable sets and points of Stearns’ transfer schemes as generalized nucleoli. In particular, the “lexicographic kernel”, is always a stable set of the bargaining sets which may not include the nucleolus. All nonempty $\varepsilon $-cores are also stable sets of the bargaining sets.