This paper examines a class of lexicographic compositions of two criteria for social or individual decision problem. As Tadenuma (2002) shows, a lexicographic composition of two binary relations often faces inconsistency problem, e.g. lexicographic rankings violate acyclicity. We consider this inconsistency problem for general environments and prove the followings: (1) in some cases, lexicographic rankings cannot satisfy acyclicity if the first criterion is a quasi-ordering; (2) lexicographic rankings satisfy quasi-transitivity if the first criterion is an ordering; (3) necessary and sufficient conditions are obtained for a lexicographic composition of two relations to be transitive but these conditions seem to be very restrictive on the economic domain.
