(m,n)-rationalizable choices

Rationalizability has been a main topic in individual choice theory since the seminal paper of Samuelson (1938). The rationalization of a multi-valued choice is classically obtained by maximizing the binary relation of revealed preference, which is fully informative of the primitive choice as long as suitable axioms of choice consistency hold. In line with this tradition, we give a purely axiomatic treatment of the topic of choice rationalization. In fact, we introduce a new class of properties of choice coherence, called axioms of replacement consistency, which examine how the addition of an item to a menu may cause a substitution in the selected set. These axioms are used to uniformly characterize rationalizable choices such that their revealed preferences are quasi-transitive, Ferrers, semitransitive, and transitive. Further, regardless of rationalizability, we study the relationship of these new axioms with some classical properties of choice consistency, such as standard contraction, standard expansion, and WARP. To complete our analysis of the transitive structure of rationalizable choices, we examine the case of revealed preferences satisfying weak (m,n)-Ferrers properties in the sense of Giarlotta and Watson (2014). Originally introduced with the purpose of extending the notions of interval orders and semiorders, these Ferrers properties give a descriptive taxonomy of binary relations displaying a transitive strict preference but an intransitive indifference. Here we suggest a possible economic interpretation of weak (m,n)-Ferrers properties, showing that, in a suitable model of transactions, they provide a way of controlling phenomena of money-pump due to the presence of mixed cycles of preference/indifference. Finally, we define (m,n)-rationalizable choices as those having a weakly (m,n)-Ferrers revealed preference, and characterize these choices by means of additional axioms of replacement consistency.