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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 E P X a unique element c A in A we confine ourselves to choice functions and not correspondences.

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. Special thanks are due to Uzi Segal for improving the constructive part of the proof of Proposition 3. Kalai and Spiegler would like to thank the Institute for Advanced Study at Princeton for its hospitality.

The DM'S behavior is rationalized after the state of the world is added to the description of the alternatives. Clearly, every choice function has an RMR, since every choice set can be said to have its own relevant rationale.

Among the many RMR's that can be given to a choice func- tion, we propose to focus on those that employ the minimal number of orderings. The meaningfulness of this explanation method for any particular choice function c can thus be assessed by r c -the minimal number of orderings among the RMR's of c.

The larger r c , the less meaningful is the rationalization by multiple rationales that can be given to c. A customer chooses a main course from a restaurant's menu.

He chooses chicken when the menu consists of steak and chicken only, yet goes for the steak when the menu consists of steak, chicken, and frog's legs.

IIA is violated in this example. Note that the description of the choice procedure already uses the terminology of RMR. Therefore, it is clear that the minimal RMR requires two orderings: one is applied when frog's legs are on the menu and another is applied when they are not.

This minimal RMR is consistent with Sen's attribution of the violation of IIA to the "epistemic value of the menu": the presence of frog's legs in the menu conveys information about the quality of the other items.

An individual aims to be as moral as possible. However, if pursuing this goal leads to harmful consequences from an egotistic point of view, he abandons morality and adopts the best selfish alternative.

Typically, such a choice violates IIA. This element may belong to A and yet be different from the u-maximal element in A.

Once again, the description of the procedure already uses the terminology of RMR with two orderings, induced by u and v. Like the original u, v procedure, the verbal description of the "twin" pro- cedure employs two rationales.

A recruiting officer has an ordering over all possible candidates for an academic job. He classifies the candidates according to their field.

Given a set of candidates, he chooses the best one from the most popular field in the set. We now turn to our analytical task in this paper: obtaining general bounds on r c ,and computing it for a variety of choice procedures.

A trivial upper bound on the minimal number of rationales that are needed to provide an RMR for any choice function is N, the cardinality of X.

Whatever the choice a is, it is explained "retrospectively" by the rationale pa. Our first proposition employs a variant on this construction to establish a tighter upper bound on r c :.

For every a b, let pa be an ordering, in which a is the top element and b is the second-best element.

Consider a set A E P X. Our next result shows that "almost all" choice functions need N -1 orderings for an RMR. Continue modifying piin this way until its top element appears on the top of no other ordering.

This procedure is iterated for any other pair of orderings having the same top element, until the K orderings have distinct top elements.

We will then show that this upper bound tends to 0 as N tends to infinity. By Lemma 0, c has an RMR whose r c orderings have distinct top elements.

Thus, to every c with r c 5 N -2, we can attach a triple a, b, T E 2,such that there exists an RMR of c, for which i a and b are not top elements of any ordering and ii T is the set of top elements in those RMR orderings, in which a is ranked above b.

What is the proportion of choice functions c with r c 5 N -2, to which a particular triple a, b, T E Z can be attached in this way?

The cardinality of Z is smaller than N2. It is quite hard to construct such a choice function for an arbitrary N. Details will be provided upon request.

In contrast to the examples presented in the introduction, we will now analyze two choice procedures, whose description does not explicitly involve multiple rationales: the "second-best" and the "median" procedures.

For axiomatic characterizations of these procedures, see Baigent and Gaertner and Gaertner and Xu We will see that providing an RMR for these procedures requires a "large number" of orderings.

In particular, r c is not independent of N. The detailed calculations illustrate the type of analytical arguments involved in the application of the RMR method.

We will show that there is no way to rationalize the second-best procedure with a "small" and independent of N number of rationales. The construction will be inductive.

Consider a choice set A. It remains to be shown that if is an RMR of c, then N 2". Assume the contrary, i.

Thus, for every element x z, there is a unique proper subset of orderings, in which x is ranked above z. The number of proper subsets of the n orderings is 2" Therefore, N -1 52" The literature contains a number of anecdotal references to the second-best procedure.

In Sen , a college tea party participant consistently chooses the second-largest slice of cake. McFadden mentions people's tendency to choose the second-cheapest wine on the wine list in restaurants.

In these examples, choice behavior violates IIA when alternatives are described by their size or price alone. However, in both cases, the context suggests that the DM's have additional considerations.

In Sen's example, the DM may be concerned with his perceived greediness, while in McFadden's example, he may believe that the cheapest wine on the wine list is not drinkable.

Thus, in each case, rationality of choice behavior is restored if the description of alternatives is extended, so as to include perceived greediness or wine quality.

It should be stressed that in both anecdotes, the appeal of the rationalizations derives from the additional information concerning the DM's considerations, provided by the choice problem's description.

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A trivial upper bound on the minimal number of rationales that are needed to provide an RMR for any choice function is N, the cardinality of X.

Whatever the choice a is, it is explained "retrospectively" by the rationale pa. Our first proposition employs a variant on this construction to establish a tighter upper bound on r c :.

For every a b, let pa be an ordering, in which a is the top element and b is the second-best element. Consider a set A E P X.

Our next result shows that "almost all" choice functions need N -1 orderings for an RMR. Continue modifying piin this way until its top element appears on the top of no other ordering.

This procedure is iterated for any other pair of orderings having the same top element, until the K orderings have distinct top elements.

We will then show that this upper bound tends to 0 as N tends to infinity. By Lemma 0, c has an RMR whose r c orderings have distinct top elements.

Thus, to every c with r c 5 N -2, we can attach a triple a, b, T E 2,such that there exists an RMR of c, for which i a and b are not top elements of any ordering and ii T is the set of top elements in those RMR orderings, in which a is ranked above b.

What is the proportion of choice functions c with r c 5 N -2, to which a particular triple a, b, T E Z can be attached in this way?

The cardinality of Z is smaller than N2. It is quite hard to construct such a choice function for an arbitrary N. Details will be provided upon request.

In contrast to the examples presented in the introduction, we will now analyze two choice procedures, whose description does not explicitly involve multiple rationales: the "second-best" and the "median" procedures.

For axiomatic characterizations of these procedures, see Baigent and Gaertner and Gaertner and Xu We will see that providing an RMR for these procedures requires a "large number" of orderings.

In particular, r c is not independent of N. The detailed calculations illustrate the type of analytical arguments involved in the application of the RMR method.

We will show that there is no way to rationalize the second-best procedure with a "small" and independent of N number of rationales.

The construction will be inductive. Consider a choice set A. It remains to be shown that if is an RMR of c, then N 2".

Assume the contrary, i. Thus, for every element x z, there is a unique proper subset of orderings, in which x is ranked above z.

The number of proper subsets of the n orderings is 2" Therefore, N -1 52" The literature contains a number of anecdotal references to the second-best procedure.

In Sen , a college tea party participant consistently chooses the second-largest slice of cake. McFadden mentions people's tendency to choose the second-cheapest wine on the wine list in restaurants.

In these examples, choice behavior violates IIA when alternatives are described by their size or price alone.

However, in both cases, the context suggests that the DM's have additional considerations. In Sen's example, the DM may be concerned with his perceived greediness, while in McFadden's example, he may believe that the cheapest wine on the wine list is not drinkable.

Thus, in each case, rationality of choice behavior is restored if the description of alternatives is extended, so as to include perceived greediness or wine quality.

It should be stressed that in both anecdotes, the appeal of the rationalizations derives from the additional information concerning the DM's considerations, provided by the choice problem's description.

In contrast, our rationalization method is "context-freen- it does not make use of any information other than choice behavior. It follows from Proposition 3 that if we were given information about the DM's choices, without being given any information on the content of the alternatives, then searching for the minimal G.

RMR would lead us to a rationalization, which is quite far from those rationalizations that intuitively fit the two anecdotes. This is in contrast to the examples of Section 1, in which the minimal RMR's seem to coincide with the intuitive rationalizations inspired by the context or by the procedure behind the choices.

A voter orders all potential political candidates along a left-right axis. Given a set of candidates, he chooses the "median" candidate.

For every set A c X consisting of an odd number of elements, let med A be the median element in A, with respect to the natural ordering.

We do not define the choice function for sets consisting of an even number of elements. We will show that r med is quite close to N Since the number of orderings is less than N, there exists z E Z that is ranked above all other members of Z in no more than fiorderings.

Denote the set of top elements in those orderings by TI. Of course, IT'l 5 fi. The proof is by contradiction. We need to verify that we can rationalize the choice from every set A with the new profile of orderings.

We are left with the case of x E A. If med A x,then there are two possibilities:. Note that by Proposition 4, r med is "close" to the upper bound given by Proposition 1, as well as significantly larger than r second-best.

As with the second-best choice proce- dure, searching for the minimal RMR of the median procedure leads to a rationalization, which does not seem to fit any intuitive explanation for this procedure.

Under RMR, the DM always applies a single rationale to any particular choice problem, although different rationales may be applied to different choice problems.

The median procedure may be better explained by the existence of two fixed, diametrically opposed rationales, such that the DM tries to balance between them in the face of any choice problem.

The economist's standard way of explaining choice behavior is by seeking an order- ing, whose maximization is consistent with the behavior.

We proposed a variant of this approach: seeking a "small" number of orderings, such that the choice in every "case" a choice set is consistent with maximization with respect to one of those orderings.

We extend the standard approach to choice functions that violate IIA by following the premise that the appeal of the explanation depends on the number of orderings involved.

We fully acknowledge the crudeness of this approach. The appeal of the RMR proposed for "Luce and Raiffa's dinner" does not emanate only from its small number of orderings, but also from the simplicity of describing in which cases each of them is applied.

More research is needed to define and investigate "structured" forms of rationalization. As emphasized in the introduction, our approach is "context-pee.

For example, if we knew that when the DM picks a slice of cake, he also cares about his perceived greediness, we would find it fitting to explain his choice behavior by adjusting the consequence space, so as to accommodate this additional motive and we would find.

Thus, although we accept that "there is no way of determining whether a choice function is con- sistent or not without referring to something external to choice behavior" Sen , we believe that rationalization by a minimal number of rationales is an appropriate extension of the economist's standard approach--especially in the absence of information about anything "external to choice behavior.

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