Congruence Relations on a Choice Space: Revealed Indiscernibility

A choice space is a nite set of alternatives endowed with a contractive map, which associates to each menu the nonempty subset of its selected items. We employ the usual point of view that a choice space models an economic agent's observed behavior in a given environment. A congruence relation on a choice space is an equivalence relation on the ground set satisfying the following properties: (i) it induces a well-defined choice correspondence on the quotient set of equivalence classes; (ii) all axioms of choice consistency are preserved in the passage to the quotient, in particular the (quasi-transitive, transitive) rationalizability of the original choice; (iii) the primitive behavior can be retrieved from the quotient choice, regardless of any assumption of rationality. For any choice space, we single out a special congruence relation, called revealed indiscernibility : two items are revealed to be indiscernible whenever they are behaviorally indistinguishable, and the process of collapsing each equivalence class into an item preserves the original choice structure. The quotient choice space induced by revealed indiscernibility exhibits the true skeleton of an economic agent's behavior by consistently eliminating all structural redundancies of items. We show that the family of congruence relations on a choice space forms a lattice under set-containment, having equality as its minimum, and revealed indiscernibility as its maximum. Our approach has strong links with the theory of revealed preferences [5], since well-known relations of revealed similarity of items — revealed indifference, revealed universal indifference [1], and behavioral indifference [3] — coincide with revealed indiscernibility under assumptions of choice rationality. Further, revealed indiscernibility displays connections with the notion of indiscernibility in the framework of rough set theory [4], as well as with the codification of Leibniz's principle of the identity of indiscernibles within quantum mechanics [2].