Two stage selection procedures are quite common. Decisions arrived at on the basis of the composition of the binary relations in some form, is what is implemented in such two stage choice procedures. The resulting binary relation is referred to as a lexicographic composition of the individual binary relations. In this paper we begin by obtaining a necessary and sufficient condition for a quasi transitive binary relation to be transitive. Then we obtain necessary and sufficient conditions for the lexicographic composition of two quasi transitive binary relations to be quasi transitive. In passing it is noted that the lexicographic composition of two transitive binary relations is always transitive. Finally, we obtain conditions for the lexicographic composition of two binary relations to be acyclic. It is observed that if the second stage binary relation is acyclic, then the lexicographic composition is acyclic if and only if the first stage binary relation is. All our binary relations are assumed to be reflexive and complete. Such binary relations are called abstract games.