A characterization of lexicographic preferences

This paper characterizes lexicographic preferences over alternatives that are identified by a finite number of attributes. We say two alternatives are 'totally different' if they are different with respect to every attribute. Our characterization is based on two key concepts: a weaker notion of continuity called 'mild continuity' (strict preference order between any two totally different alternatives is preserved around their small neighborhoods) and an 'unhappy set' (any alternative outside the set is preferred to all alternatives inside).