Weakened WARP and top-cycle choice rules

We propose the following weakened version of WARP: if the decision maker selects an alternative x and rejects another alternative y in some context, he cannot select y and reject x in another context. This axiom is consistent with cyclic choices. It is necessary and sufficient for the choice from every subset A of a (finite) universal set X to coincide with the weak upper-contour set of the transitive closure of some fixed complete relation at some alternative in A. Adding further simple axioms forces the choice from each subset to coincide with the top cycle (in that subset) of some fixed tournament over the universal set.