On the solution of w-stable sets

Rational stability of choice functions

“…$$V$$ is a $$w \mbox{-} $$ stable set (Han & Van Deemen, 2016) if for all $$x,y\in V,not(x\,⪼\,y)$$; and for all $$x\in V,z\in X\setminus V$$, if $$z\,⪼\,x$$ then $$x\,⪼\,z$$.…”

“…As mentioned, our aim is to introduce a new stability concept that choice functions should satisfy. At a first sight, $$r \mbox{-} $$stability is directly linked to the notion of von Neumann–Morgenstern stable sets ( vNM ‐stable, in what follows), or its generalizations: the generalized stable set (Harsanyi, 1974; Van Deemen, 1991), $$m \mbox{-} $$stable set (Peris & Subiza, 2013), or $$w \mbox{-} $$stable set (Han & Van Deemen, 2016). These notions, as the classical stability, are based on internal/external conditions.…”

A note on the relationship between the core and stable sets in three-sided markets

“…Since the core always satisfies internal stability, it is included in any stable set; and if the core is externally stable, then it is the only stable set. Other notions of stability are analyzed in Roth (1976), Peris and Subiza (2013), and Han and van Deemen (2016).…”