Quasitransitive Rational Choice

We consider a finite universal set of alternatives and the set of all feasible sets are simply the set of all non-empty subsets of this universal set. A choice function assigns to each feasible set a non-empty subset of it. An interesting problem in such a context is to explore the possibility of the choice function coinciding with the best elements with respect to a binary relation. This is precisely the problem of rational choice theory. There is a large literature today on this topic. In this paper, we propose three new axioms which are used to fully characterize all choice functions which are rationalized by quasi-transitive, semi-transitive and a third kind of ‘almost’ transitive (:the property is called intervality in the literature) binary relations. These ‘almost’ transitive (:but not exactly so!) binary relations, which are now quite popular in the literature (:see Yu [1985]), have the rather interesting feature of revealing intransitive indifference for single valued choice functions. This phenomena has been dealt with rather elegantly by Kim [1987]. Our purpose, is to shed new light on the problem in the absence of the single-valuedness assumption. We, propose axiomatic characterizations which are minimal. Several examples are provided, to show that the assumptions we use are logically independent. While characterizing choice functions which coincide with the best elements with respect to a binary relation satisfying intervality, we invoke a property due to Fishburn [1971],which we refer to in the paper as Fishburn’s Intervality Axiom. In Aizerman and Aleskerov [1995], can be found an axiom called Functional Acyclicity, which generalizes Fishburn’s Intervality Axiom. It is correctly claimed in Aizerman and Aleskerov[1995], that satisfaction of Functional Acyclicity is equivalent to the existence of two real valued functions, one with domain being the finite universal set and the other with domain being the set of all finite subsets of the universal set, such that given a feasible set, only those alternatives are chosen whose value corresponding to the first function is at-least as much as the value assigned to the feasible set by the second function. Such choice functions are called threshold rationalizable. In a final section to the paper, we provide a correct proof of this result, in view of obvious logical discrepencies in the proof available in Aixerman and Aleskerov [1995]