Von Neumann–Morgenstern stable set rationalization of choice functions

Rational stability of choice functions

“…Knoblauch (2020) analyzes acyclic vNM‐stable choice functions; that is, choice functions $$F$$ such that for all $$X\in {\mathscr{P}}({\mathscr{S}}),F(X)$$ is a vNM‐stable set and, additionally, the base relation $${\succ }^{b}$$ is acyclic. The following example shows that although we require acyclicity, vNM‐stability and $$r \mbox{-} $$stability do not coincide.…”

“…Knoblauch (2020) proves that if the binary relation that provides rationalizability or vNM‐stability is transitive, then both conditions coincide. As a consequence, the following result is obtained.…”

“…Remark As far as we know, Knoblauch (2020) is the first reference analyzing if a choice function is vNM‐stable (vNM‐rationalizable). Knoblauch (2020) shows that when transitivity of the asymmetric binary relation is required, then the notions of vNM‐stable and rationalizable choice functions coincide.…”

“…Remark As far as we know, Knoblauch (2020) is the first reference analyzing if a choice function is vNM‐stable (vNM‐rationalizable). Knoblauch (2020) shows that when transitivity of the asymmetric binary relation is required, then the notions of vNM‐stable and rationalizable choice functions coincide. We show in Section 5 that this is also true for acyclic binary relations, when complete binary discrimination is fulfilled (this is not the case of the choice functions in Example 1).…”

“…We show in Section 5 that this is also true for acyclic binary relations, when complete binary discrimination is fulfilled (this is not the case of the choice functions in Example 1). A significant difference between our model and the one in Knoblauch (2020) is that we consider every agenda (every $$X\subseteq S$$) as a possible feasible set for the choice function, whereas Knoblauch (2020) allows for a domain containing only some of the possible agendas.…”