Congruence relations on a choice space

A choice space is a finite set of alternatives endowed with a map associating to each menu a nonempty subset of selected items. A congruence on a choice space is an equivalence relation that preserves its structure. Intuitively, two alternatives are congruent if the agent is indifferent between them, and, in addition, her choice is influenced by them in exactly the same way. We give an axiomatic characterization of the notion of congruence in terms of three natural conditions: binary fungibility, common destiny, and repetition irrelevance. Further, we show that any congruence satisfies the following desirable properties: (hereditariness) it induces a well-defined choice on the quotient set of equivalence classes; (reflectivity) the primitive behavior can be always retrieved from the quotient choice, regardless of any feature of rationality; (consistency) all basic axioms of choice consistency are preserved back and forth by passing to the quotient. We also prove that the family of all congruences on a choice space forms a lattice under set-inclusion, having equality as a minimum, and a unique maximum, called revealed indiscernibility. The latter relation can be seen as a limit form of revealed similarity as the agent’s rationality increases.