Rationalizing Choice Functions by Multiple Rationales

Imagine that you receive information on the choices made by a decision maker (DM) from all subsets of some set X. You know nothing about the context of these choices. You look for an explanation for the DM’s behavior. You would probably look first for a single rationale explaining the behavior. Specifically, you would seek a rationalizing ordering—that is, a linear ordering on X, such that for every choice set A⊆X, the DM’s choice from A is the best element in A according to the ordering. You recall that the “Independence of Irrelevant Alternatives” Axiom (IIA)—which requires that the chosen element from a set also be chosen from every subset that contains it—is a necessary and sufficient condition for the existence of such an explanation. If you had more information about the context of the DM’s choices—that is, the content of the alternatives and his possible considerations—you might assess this explanation further. For example, you would probably be somewhat skeptical towards this explanation if you discovered that given the context, the rationalizing ordering minimizes what you clearly perceive as the DM’s well being. However, in the absence of information about the context of the DM’s behavior, you are likely to find an explanation by a rationalizing ordering persuasive. Real-life choice procedures often violate IIA. When confronted with such a procedure, we tend not to give up on explanation by a rationalizing ordering so quickly. We search for ways to argue that the procedure does not “really” violate rationality. We have different ways of doing this. One way is to argue that the DM’s choices originated not from a single rationale, but from several, each rationale being appropriate for a subset of choice problems. Formally, let X be a (finite) set of alternatives. Denote its cardinality by N . Let P X be the set of all nonempty subsets of X. A choice function on X assigns to every A ∈ P X a unique element c A in A (we confine ourselves to choice functions and not correspondences). Our central new concept is the following: a K-tuple of strict preference relations k k=1 K on X is a rationalization by multiple rationales (RMR) of the choice function c if for every A, the element c A is k-maximal in A for some k. One possible interpretation of this explanation method is that the choice set conveys information about its constituent elements and given this information, the DM chooses what he thinks is the best alternative. In other words, the DM has in mind a partition of P X and he applies one ordering to each cell in the partition. A cell is like a state of