In this paper we consider choice problems which are bounded both above and below and provide a new axiomatic characterization of the equal loss choice function. We subsequently turn to a study of various properties implied by shifts in the choice problem, one of which was used in characterizing the equal loss solution. Then we characterize rational choice behavour when a decision maker is confronted with a choice problem. Finally we turn to an axiomatic characterization of a particular rationalizable choice function – the utilitarian choice function – using a shift invariance property.