Rational stability of choice functions

Two independent approaches have been used to analyze choices. A prominent notion is rationalizability: individuals choose maximizing binary relations. An alternative choices in terms of standards behavior with the von Neumann–Morgenstern (vNM)-stability. We introduce a new concept ( r - $r \mbox{-} $ stability) that turn extends stability and rationality. Our main result establishes every rationalizable choice function stable vNM-stable has an selection. appealing property well-known solution concepts (top cycle, uncovered set, …) are stable, while they neither nor vNM-stable.