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 is to analyze choices in terms of standards of behavior with the notion of von Neumann-Morgenstern (vNM)-stability. We introduce a new concept (r- $r \mbox{-} $stability) that in turn extends the notion of stability and rationality. Our main result establishes that every rationalizable choice function is r- $r \mbox{-} $stable and every vNM-stable choice has an r- $r \mbox{-} $stable selection. An appealing property of r- $r \mbox{-} $stability is that well-known solution concepts (top cycle, uncovered set, horizontal ellipsis ) are r- $r \mbox{-} $stable, while they are neither rationalizable nor vNM-stable.