Quasitransitive rationalization and the superset property (Q798241)

\textit{G. Bordes, D. H. Blair, J. S. Kelly} and \textit{K. Suzumura} [J. Econ. Theory 13, 361-379 (1976; Zbl 0361.90005)] assumed that the choice correspondence has a domain which includes all nonempty finite subsets of the universal set of alternatives. They also take the superset property to be a necessary condition for a choice correspondence to have a quasitransitive rationalization. The author presents an example of a choice correspondence which satisfies the domain condition mentioned above and has a quasitransitive rationalization but does not satisfy the superset property. Then, two restrictions on the domain of the choice correspondence are introduced, either of which is sufficient to make the superset property a necessary condition for quasitransitive rationality.