2. SEN'S CRITICISM OF INTERNAL CONSISTENCY OF CHOICE

Let C be a choice function that specifies, for any admissible non-empty set S of feasible alternatives, a non-empty subset C(S) of S. In the standard case, there are no further restrictions on the choice set C(S); in the norm-respecting cases to be considered in this paper, the choice set has to be consistent with the external norms imposed. A more precise definition is provided shortly. Note that the standard case is the special case of the norm-respecting framework in which there are no norms. For that reason, we simply refer to C as a choice function without specifying the assumption that it is norm-respecting: there is no need to make a distinction because the traditional setting is a special case.

Sen (Reference Sen1993: 500) poses the following question: ‘[C]an a set of choices really be seen as consistent or inconsistent on purely internal grounds without bringing in something external to choice, such as the underlying objectives or values that are pursued or acknowledged by choice?’ To bring his point into clear relief, Sen invites us to examine the following two choices:

As Sen rightly points out, this pair of choices violates most of the standard choice consistency conditions including the weak and the strong axioms of revealed preference, Arrow's axiom of choice consistency, and Sen's condition α. It is arguable and indeed Sen (Reference Sen1993: 501) argues that this seeming inconsistency can be easily resolved if only we know more about the person's choice situation: as mentioned in the Introduction, the person making these choices may simply respect the external norm ‘do not choose the last available apple’.

On the face of it, Sen's argument to this effect may seem to go squarely against the theory of rationalizability à la Arrow (Reference Arrow1959), Richter (Reference Richter1966, Reference Richter, Chipman, Hurwicz, Richter and Sonnenschein1971), Hansson (Reference Hansson1968), Sen (Reference Sen1971), Suzumura (Reference Suzumura1976a) and many others, where the weak axiom of revealed preference is a necessary condition for rationalizability.

The standard theory of revealed preference has its origins in the analysis of consumer behaviour where feasible sets are given by budget sets, pioneered by authors such as Samuelson (Reference Samuelson1938, Reference Samuelson1947, Reference Samuelson1948, Reference Samuelson1950), Houthakker (Reference Houthakker1950) and many others. However, much more general approaches have been developed in the meantime, particularly those due to Richter (Reference Richter1966, Reference Richter, Chipman, Hurwicz, Richter and Sonnenschein1971), Hansson (Reference Hansson1968), Suzumura (Reference Suzumura1976a, Reference Suzumura1977, Reference Suzumura1983) and others who explored the traditional notion of rationalizability without any domain restrictions whatsoever. It is this general theory of rationalizability that we modify so as to develop a new concept of norm-conditional rationalizability and build a bridge between rationalizability and Sen's criticism. In essence, what emerges from this modification is the peaceful co-existence of a norm-conditional notion of rationalizability and Sen's elaborate criticism of internal consistency conditions. Although Sen's suggestion to the effect that the rationality of choice behaviour, or the lack thereof, cannot be judged only by means of the internal structure of choices made is well taken, it turns out that we can modify the axioms of revealed preference theory in such a way as to provide an axiomatization of choices under external norms in terms of suitably modified revealed preference axioms.

As an auxiliary step, we introduce a model of choice where external norms are taken into consideration by specifying all pairs consisting of a feasible set and an element of this set with the interpretation that this element is prohibited from being chosen from this set by the relevant system of external norms. Norm-conditional rationalizability then requires the existence of a preference relation such that, for each feasible set in the domain of the choice function, the chosen elements are at least as good as all elements in the set except for those that are prohibited by the external norm. This approach is very general because no restrictions are imposed on how the system of external norms comes about – any specification of a set of pairs as described above is possible. Needless to say, we do not by any means suggest that any arbitrary system of norms thus specified is desirable; clearly, what we advocate is a method to incorporate any norm into a model of choice without completely eschewing all notions of traditional rationality altogether. In fact, the traditional model of rational choice is included as a special case – the case that obtains if the set of prohibited pairs is empty.

We note in passing that the ‘do not choose the last available apple’ example is not the only one Sen (Reference Sen1993) uses to criticize internal choice consistency properties. A second example he uses to call into question the imposition of internal choice consistency conditions focuses on the epistemic value of a feasible set. We do not pursure this issue further in this paper and refer the reader to Bossert (Reference Bossert2001) for a detailed discussion and an attempt to link this phenomenon to standard rationalizability properties within a model of decision making under complete uncertainty.

3. MODELLING EXTERNAL NORMS

A choice situation is described by a feasible set S of alternatives, where S is a non-empty subset of the non-empty universal set X. External norms such as those discussed in the Introduction can be expressed by identifying feasible sets and alternatives that are not to be chosen from these feasible sets. For example, suppose there is a feasible set S={x, y}, where x stands for selecting nothing and y stands for selecting a single apple. Now consider the feasible set T={x, y, z} where there are two (identical) apples y and z available. The external norm not to take the last apple can easily and intuitively be expressed by requiring that the choice of y from S is excluded, whereas the choice of y (or z) from T is perfectly acceptable. In general, norms of that nature can be expressed by identifying all pairs (S, w), where w∈S, such that w is not supposed to be chosen from the feasible set S. To that end, we use a set , to be interpreted as the set of all pairs (S, w) of a feasible set S and an element w of S such that the choice of w from S is prevented by the external norm under consideration.

More formally, suppose is the power set of X excluding the empty set. A choice function with a non-empty domain Σ ⊆ is a mapping C: Σ → such that, for all S∈Σ, C(S) ⊆ S\{z∈S|(S, z)∈}. Let C(Σ) denote the image of Σ under C, that is, C(Σ)=∪_S∈ΣC(S). To ensure that the non-emptiness of the choice set does not conflict with the restrictions imposed by the norm , we require to be such that, for all S∈Σ, there exists x∈S satisfying (S, x) ∉. The set of all possible norms satisfying this restriction is denoted by N. If is empty (that is, there are no external norms), we obtain the standard model of choice.

Returning to Sen's example involving the norm ‘do not choose the last available apple’, we can, for instance, define the universal set X={x, y, z}, the domain with S={x, y} and T={x, y, z}, and the external norm described by the set = {(S, y)}. Thus, the external norm requires that y∉C(S) but no restrictions are imposed on the choice C(T) from the set T – that is, this external norm represents the requirement that the last available apple should not be chosen.

The notion of rationality explored in this paper is conditional on a system of external norms ∈ N as introduced above. In contrast with the classical model of rational choice, an element x that is chosen by a choice function C from a feasible set S∈Σ need not be considered at least as good as all elements of S by a rationalizing relation, but merely at least as good as all elements y∈S such that (S, y) ∉ . That is, if the choice of y from S is already prohibited by the norm, there is no need that x dominates such an element y according to the rationalization. Needless to say, the chosen element x itself must be admissible in the presence of the prevailing system of external norms.

To make this concept of norm-conditional rationalizability precise, let a system of external norms ∈N and a feasible set S∈Σ be given. An -admissible set for , S), (S) ⊆ S, is defined by letting, for all x∈S,

Note that, by assumption, (S) ≠ ∅ for all ∈ N and for all S∈Σ.

4. NORM-CONDITIONAL RATIONALIZABILITY

We now introduce our analytical framework and define the basic concepts of norm-conditional rationalizability. Let R⊆X×X be a (binary) relation on the non-empty universal set X. The asymmetric factor P(R) of R is given by (x, y)∈P(R) if and only if (x, y)∈R and (y, x) ∉ R for all x, y∈X, and the symmetric factor I(R) of R is defined by (x, y)∈I(R) if and only if (x, y)∈R and (y, x)∈R for all x, y∈X.

The transitive closure tc(R) of a relation R is defined by letting, for all x, y∈X,

For any binary relation R, tc(R) is the smallest transitive superset of R.

A relation R⊆X×X is reflexive if, for all x∈X,

and R is complete if, for all x, y∈X such that x≠y,

R is transitive if, for all x, y, z∈X,

It is clear that R is transitive if and only if R=tc(R). A quasi-ordering is a reflexive and transitive relation and an ordering is a complete quasi-ordering.

R is consistent if, for all x, y∈X,

This notion of consistency is due to Suzumura (Reference Suzumura1976b) and it is equivalent to the requirement that any cycle must be such that all relations involved in this cycle are instances of indifference – strict preference cannot occur. To facilitate the understanding of this concept, we may define the consistent closure cc(R) of R as the smallest consistent superset of R. This is the concept coined by Bossert et al. (Reference Bossert, Sprumont and Suzumura2005), which may be written explicitly as follows. For all x, y∈X,

Clearly, for any binary relation R, we have R⊆cc(R)⊆tc(R) and R is consistent if and only if R=cc(R). It is easy to verify that consistency implies (but is not implied by) the well-known acyclicity axiom which rules out the existence of strict preference cycles (cycles composed entirely of instances of strict preference). Consistency and quasi-transitivity, which requires that P(R) is transitive, are independent. Transitivity implies consistency but the reverse implication is not true in general. However, if R is reflexive and complete, consistency and transitivity are equivalent.

A relation R* is an extension of R if and only if R⊆R* and P(R)⊆P(R*). If an extension R* of R is an ordering, we refer to R* as an ordering extension of R. One of the most fundamental results on extensions of binary relations is due to Szpilrajn (Reference Szpilrajn1930) who showed that any transitive and asymmetric relation has a transitive, asymmetric and complete extension. The result remains true if asymmetry is replaced with reflexivity, that is, any quasi-ordering has an ordering extension. Arrow (Reference Arrow1951: 64) stated this generalization of Szpilrajn's theorem without a proof and Hansson (Reference Hansson1968) provided a proof on the basis of Szpilrajn's original theorem. While the property of being a quasi-ordering is sufficient for the existence of an ordering extension of a relation, this is not necessary. As shown by Suzumura (Reference Suzumura1976b), consistency is necessary and sufficient for the existence of an ordering extension; see Suzumura (Reference Suzumura1976b: 389f).

We say that a choice function C on Σ is -rationalizable if and only if there exists a binary relation ⊆ X × X such that, for all S∈Σ and for all x∈S,

In this case, we say that -rationalizes C, or is an -rationalization of C.

To facilitate our analysis of -rationalizability, a generalization of the notion of the direct revealed preference relation R _C⊆X×X of a choice function C is of use. For all x, y∈X,

The (indirect) revealed preference relation of C is the transitive closure tc(R_C) of the direct revealed preference relation R_C.

We consider three basic versions of norm-conditional rationalizability. The first is -rationalizability by itself, where an -rationalization does not have to possess any additional property (such as reflexivity, completeness, consistency, or transitivity). This notion of rationalizability is equivalent to -rationalizability by a reflexive relation (this is also true for the standard definition of rationalizability without external norms; see Richter Reference Richter, Chipman, Hurwicz, Richter and Sonnenschein1971). The second is -rationalizability by a consistent relation (again, reflexivity can be added and an equivalent condition is obtained; see Bossert et al. Reference Bossert, Sprumont and Suzumura2005). Finally, we consider -rationalizability by a transitive relation which, again as in the classical case, turns out to be equivalent to -rationalizability by an ordering; see Richter (Reference Richter1966, Reference Richter, Chipman, Hurwicz, Richter and Sonnenschein1971).

We are now ready to identify a necessary and sufficient condition for each one of these notions of -rationalizability of a choice function. To obtain a necessary and sufficient condition for simple -rationalizability (that is, -rationalizability by a binary relation that does not have to possess any further property), we follow Richter (Reference Richter, Chipman, Hurwicz, Richter and Sonnenschein1971) by generalizing the relevant axiom in his approach in order to accommodate an externally imposed system of norms . This leads us to the following axiom. -conditional direct-revelation coherence: For all S∈Σ and for all x ∈ (S),

Our first result establishes that this property is indeed necessary and sufficient for -rationalizability.

Theorem 1Let ∈ Nbe a system of external norms and let C be a choice function. C is -rationalizable if and only if C satisfies -conditional direct-revelation coherence.

As is the case for the traditional model of rational choice on general domains, it is straightforward to verify that -rationalizability by a reflexive relation is equivalent to -rationalizability without any further properties of an -rationalization; this can be verified analogously to Richter (Reference Richter, Chipman, Hurwicz, Richter and Sonnenschein1971). However, adding completeness as a requirement leads to a stronger notion of -rationalizability; see again Richter (Reference Richter, Chipman, Hurwicz, Richter and Sonnenschein1971).

Next, we examine -rationalizability by a consistent relation, which is equivalent to -rationalizability by a reflexive and consistent relation. As in the traditional case, adding completeness, however, leads to a stronger property, namely, one that is equivalent to -rationalizability by an ordering; see Bossert et al. (Reference Bossert, Sprumont and Suzumura2005) for an analogous observation in the traditional model.

The requisite necessary and sufficient condition is obtained from -conditional direct-revelation coherence by replacing R_C with its consistent closure cc(R_C).

-conditional consistent-closure coherence: For all S∈Σ and for all x ∈ (S),

We obtain

Theorem 2Let ∈ Nbe a system of external norms and let C be a choice function. C is -rationalizable by a consistent relation if and only if C satisfies -conditional consistent-closure coherence.

Our final result establishes a necessary and sufficient condition for -rationalizability by a transitive relation which is equivalent to -rationalizability by an ordering. We leave it to the reader to verify that the proof strategy employed by Richter (Reference Richter1966, Reference Richter, Chipman, Hurwicz, Richter and Sonnenschein1971) in the traditional case generalizes in a straightforward manner to the norm-dependent model when establishing that transitive -rationalizability is equivalent to -rationalizability by an ordering.

The requisite necessary and sufficient condition is obtained from -conditional direct-revelation coherence by replacing R_C with its transitive closure tc(R_C).

-conditional transitive-closure coherence: For all S∈Σ and for all x ∈ (S),

We obtain

Theorem 3Let ∈Nbe a system of external norms and let C be a choice function. C is -rationalizable by a transitive relation if and only if C satisfies -conditional transitive-closure coherence.

5. ALTERNATIVE FORMULATIONS

Our model of norm-conditional choice may appear somewhat restrictive at first sight because it specifies pairs of a feasible set and a single object not to be chosen from that set. One might want to consider the following seeming generalization of this approach: instead of only including pairs of the form (S, x) with x∈S when defining a system of norms, one could include pairs such as (S, S′) with S'⊆S, thus postulating that the subset S′ should not be chosen from S. Contrary to first appearance, this does not really provide a more general model of norm-conditional rationalizability because, in order to formulate our notion of norm-conditional rationality, we require that a chosen element x∈C(S) has to be at least as good as all feasible elements except those that are already excluded by the external norm according to a norm-conditional rationalization – that is, x has to be at least as good as all y∈S except for those y∈S such that (S, y) ∈ . Allowing for pairs (S, S′) does not provide a more general notion of norm-conditional rationalizability because the subset of S, the elements of which have to be dominated by a chosen object, can be obtained in any arbitrary way from the subsets S′ such that S′ cannot be selected from S according to the external norm. For simplicity of exposition, we work with the simpler version of our model but note that this formulation does not involve any loss of generality when it comes to the definition of norm-conditional rationality employed in this paper.

An alternative way of modelling external norms is based on a suggestion of one of the referees. It employs a norm-embodying relation R ⊆ (X × ) × (X × ), where a pair (x, S) with x∈S⊆X stands for the act of choosing x from a choice environment S given an underlying norm. The statement ((x, S), (y, T))∈R can then be interpreted to mean that choosing x from the admissible choice environment S is at least as good as choosing y from T in terms of conformity with the requisite external norm. The basic difference between such an approach and ours is that the former works with a comparative notion of norm satisfaction, whereas our model treats external norms as constraints on acceptable choice behaviour. We do not view one setup as being superior to the other but we believe that our formulation in terms of constraints is more in the spirit of Sen's (Reference Sen1993) example because it allows us to rule out choice behaviour that is not permitted in the sense of violating the requisite norm constraints. With an alternative model that is based on the relation ‘is at least as norm-conforming as’, it seems to us that it is more difficult to capture the situations Sen had in mind.

This paper does not constitute the first attempt to accommodate Sen's (Reference Sen1993) criticism within a suitably modified theory of rational choice. In response to Sen (Reference Sen1993), Baigent and Gaertner (Reference Baigent and Gaertner1996) employ a non-standard notion of rationalizability that obeys the restriction imposed by the external norm not to choose the uniquely greatest element according to some relation but behaves as the traditional version of rationalizability when the set of greatest objects contains at least two elements. A choice function that is rationalizable in their sense selects all second-greatest elements according to a rationalizing relation if there is a unique greatest element; if there are several greatest elements, C chooses all of these greatest elements. Baigent and Gaertner (Reference Baigent and Gaertner1996: 241) state that they axiomatize the maxim ‘always choose the second largest except in those cases where there are at least two pieces which are largest, being of equal size. In that case, either may be chosen’. However, this informal maxim appears to be in conflict with the formal definition and characterization provided by Baigent and Gaertner (Reference Baigent and Gaertner1996: 243): if there is no unique greatest element, all greatest elements are chosen and not just one of them. Thus, there is a gap between their formal axiomatization and the informal maxim, the axiomatization of the latter being left unaccomplished so far. Moreover, their model is restricted to a rather narrow class of choice problems. If, instead of having two apples in the feasible set, the decision-maker faces a fruit basket containing one apple and one orange, picking the second-greatest element according to some rationalizing relation no longer seems to represent reasonable behaviour: the fact that I as the decision-maker prefer apples to oranges, say, does not mean other people have the same preferences and, as a consequence, norms of politeness and decency do not dictate the choice of the orange – what if everyone else at the table prefers oranges to apples?

Gaertner and Xu (Reference Gaertner and Xu1999a) discuss an alternative approach covering cases where external norms may lead to the choice of the median alternative(s) according to some antisymmetric relation on X. This approach is compared with the traditional rational choice setup and with the Baigent and Gaertner (Reference Baigent and Gaertner1996) framework in the antisymmetric case in Gaertner and Xu (Reference Gaertner and Xu1997) and in a more general setting in Gaertner and Xu (Reference Gaertner and Xu1999b).

An alternative type of norm-constrained choice is characterized in Gaertner and Xu (Reference Gaertner and Xu2004). The choice functions analysed in this contribution have a domain that contains the empty set in addition to all non-empty subsets of X and, moreover, choice sets may be empty even if feasible sets are non-empty. The behaviour Gaertner and Xu (Reference Gaertner and Xu2004) attempt to capture is the refusal to make a choice in response to the suppression of alternatives: if there is but a single alternative available, the decision-maker may choose the empty set as a means of expressing his or her displeasure with the suppression of other feasible alternatives. An example put forward by Sen (Reference Sen1997: 755) and used by Gaertner and Xu (Reference Gaertner and Xu2004) as a motivation of their approach is that of a government that outlaws all newspapers but one that is owned by the government itself. They argue that if several papers are available, the government paper may well be the choice of a decision-maker, but the absence of any alternative sources leads the agent to boycott the single available news outlet.

Xu (Reference Xu2007) discusses some special cases of norm-conditional rationalizability, namely, a variant of the Baigent and Gaertner (Reference Baigent and Gaertner1996) ‘never choose the uniquely largest’ rule, the median-based rule (see Gaertner and Xu Reference Gaertner and Xu1999a) and two versions of the ‘protest-based’ norm of Gaertner and Xu (Reference Gaertner and Xu2004). These special cases are obtained by ruling out the choice of unique best elements, elements better than the (bottom) median element, and non-empty choices in the case of single-valued feasible sets. See also Baigent (Reference Baigent2007) for a summary and discussion of the relevant literature.

All of the above responses to Sen's criticism are geared towards the characterization of a specific type of choice behaviour and the notion of rationality that is used is, in turn, specific to the particular external norm under consideration. In contrast, our approach focuses on a general method to incorporate any external norm one might wish to specify.

7. PROOFS

We first provide three preliminary results. The following lemma states that the direct revealed preference relation R_C must be respected by any -rationalization . This observation parallels that of Samuelson (Reference Samuelson1948, Reference Samuelson1950) in the traditional framework; see also Richter (Reference Richter, Chipman, Hurwicz, Richter and Sonnenschein1971).

Lemma 1Let ∈Nbe a system of external norms and let C be a choice function. If is an -rationalization of C, then R _C ⊆ .

Proof. Suppose that is an -rationalization of C and x, y∈X are such that (x, y)∈R _C. By definition of R_C, there exists S∈Σ such that x∈C(S) and y ∈ (S). Because is an -rationalization of C, we obtain (x, y) ∈ . Thus, R _C ⊆ must be true.

Analogously, any consistent -rationalization must respect not only the direct revealed preference relation R_C but also its consistent closure cc(R_C).

Lemma 2Let ∈ Nbe a system of external norms and let C be a choice function. If is a consistent -rationalization of C , then cc(R _C) ⊆ .

Proof. Suppose that is a consistent -rationalization of C and x, y∈X are such that (x, y)∈cc(R _C). By definition of the consistent closure of a binary relation, (x, y)∈R _C or [(x, y)∈tc(R _C)and(y, x)∈R _C] must hold. If (x, y)∈R _C, (x, y) ∈ follows from Lemma 1. If [(x, y)∈tc(R _C)and(y, x)∈R _C], there exist K ∈ ℕ and x ⁰, . . ., x ^K∈X such that x=x ⁰, (x ^k−1, x ^k)∈R _C for all k∈{1, . . ., K} and x ^K=y. By Lemma 1, x ^k−1, x ^k) ∈ for all k∈{1, . . ., K} and, thus, (x, y) ∈ tc(. Furthermore, (y, x)∈R _C implies (y, x) ∈ by Lemma 1 again. If (x, y) ∉ , it follows that (y, x)∈ P( in view of (y, x) ∈ . Because (x, y) ∈ tc(, this contradicts the consistency of . Therefore, (x, y) ∈ . Thus, cc(R _C)⊆ must be true.

Finally, if transitivity is required as a property of an -rationalization , this relation must respect the transitive closure tc(R_C) of R_C.

Lemma 3Let ∈ Nbe a system of external norms and let C be a choice function. If is a transitive -rationalization of C , then tc(R _C) ⊆ .

Proof. Suppose that is a transitive -rationalization of C and x, y∈X are such that (x, y)∈tc(R _C). By definition of the transitive closure of a binary relation R_C, there exist K ∈ ℕ and x ⁰, . . ., x ^K∈X such that x=x ⁰, (x ^k−1, x ^k)∈R _C for all k∈{1, . . ., K} and x ^K=y. By Lemma 1, we obtain x=x ⁰, (x ^k−1, x ^k) ∈ for all k∈{1, . . ., K} and x ^K=y. Repeated application of the transitivity of implies (x, y) ∈ . Thus tc(R _C)⊆ must hold.

Proof of Theorem 1. ‘Only if.’ Suppose is an -rationalization of C. Let S∈Σ and x ∈ (S) be such that (x, y)∈R _C for all y ∈ (S). By Lemma 1, (x, y) ∈ for all y ∈ (S), which implies x∈C(S) because is an -rationalization of C.

‘If.’ Suppose C satisfies -conditional direct-revelation coherence. We complete the proof by showing that = R _C is an -rationalization of C. Let S∈Σ and x ∈ (S).

Suppose first that x∈C(S). By definition, it follows immediately that (x, y) ∈ R _C = for all y ∈ (S).

Conversely, suppose that (x, y) ∈ R _C = for all y ∈ (S. It follows that -conditional direct-revelation coherence immediately implies x∈C(S). Thus, C is -rationalizable by = R _C.

Proof of Theorem 2. The proof is analogous to that of Theorem 1. All that needs to be done is replace R_C with cc(R_C) and invoke Lemma 2 instead of Lemma 1.

Proof of Theorem 3. Again, the proof is analogous to that of Theorem 1. All that needs to be done is replace R_C with tc(R_C) and invoke Lemma 3 instead of Lemma 1.