2. THE BASIC NOTATION AND DEFINITIONS

An agent is described by a triple <X, , C>, where X is a given non-empty set of options, is a non-empty class of non-empty subsets of X, and C is a function, which, for every A ∈ , specifies exactly one non-empty subset of A, to be denoted by C(A). is to be interpreted as the different non-empty sets of feasible alternatives (menus or opportunity sets) with which the agent under consideration may be confronted, and, given a menu, A, C(A) is to be interpreted as the set of options that the agent chooses from A. We call C the choice function of the agent.

Definition 2.1. Consider an agent described by <X, , C>. The choice function C satisfies:

1. (i) Chernoff's condition iff, for all A, B ∈ and all x, y ∈ X, if A ⊆ B and [x ∈ C(A) and y ∈ A − C(A)], then not [y ∈ C(B)];

2. (ii) the weak axiom of revealed preference iff, for all A, B ∈ and all x, y ∈ X, if [x ∈ C(A) and y ∈ A − C(A)], then not [y ∈ C(B) and x ∈ B];

3. (iii) the congruence axiom iff there do not exist a positive integer n, menus A ₁, . . ., A_n, A _n+1 ∈ , and x ₁, x ₂, . . ., x_n ∈ X, such that, for all t ∈ {1, 2, . . ., n}, [x_t ∈ C(A_t) and x _t+1 ∈ A_t] and [x _n+1 ∈ C(A _n+1) and x ₁∈(A _n+1−C(A _n+1))];

4. (iv) rationalizability in terms of an ordering iff there exists an ordering, , defined over X, such that, for all A ∈ , C(A) is the set of -greatest elements in A.

The properties introduced in Definition 2.1 are very familiar in the literature.Footnote ³ The first three of these properties, Chernoff's condition, the weak axiom of revealed preference, and the congruence axiom are in ascending order of logical strength, and the congruence axiom is a necessary and sufficient condition for the choice function C to be rationalizable in terms of an ordering (see Richter Reference Richter1966). Chernoff's condition, which requires that, if an option x is revealed to be strictly better than an option y in a set A, then, when the set A is enlarged to a set B, y cannot be revealed to be at least as good as x in the set B, is the weakest of the four properties and seems highly plausible. As we shall, however, see in Examples 3.1 through 3.4 below, when choices are menu-dependent, this condition becomes immediately questionable. The weak axiom of revealed preference requires that, whenever an option x is revealed to be better than an option y in a set A, y cannot be revealed to be at least as good as x in another set B. The congruence axiom says that if an option x is revealed to be at least as good as an option y through possibly a chain of feasible sets, then y cannot be revealed to be better than x in any set. Rationalizability of C in terms of an ordering, which is logically equivalent to the congruence axiom for C, requires that the agent's choices should be compatible with the standard notion of ‘preference optimization’, i.e. there should exist an ordering such that the set of options chosen by the agent from any given admissible menu would coincide with the set of best alternatives defined for that menu by that ordering.

3. EXAMPLES

Example 3.1 (SenReference Sen1993). In a party, when the fruit tray comes to an individual, there are several pears and one apple in the tray. He chooses a pear. If, however, the tray had an additional apple, he would have chosen an apple. The individual's choices violate Chernoff's condition. The explanation lies in the fact that, when the tray contains only one apple, choosing it violates social norms (‘a polite person does not pick up a fruit if it happens to be the single fruit of its type in the tray’).

Example 3.2. In a firm, there has been a long-standing rule that all customers’ queries must be responded to within a week. A given employee, say, i, of the firm, however, disposes of all such queries directed to him in 4 days. A new manager comes in and introduces a rule saying that all customers’ queries must be dealt with in no more than 4 days. After the rule is introduced, i responds to all queries in 2 days. The choice behaviour of the employee violates Chernoff's condition. A possible reason may be how i wants to be seen. Given the rule that the job must be done in no more than x number of days, i does not want to be seen as someone who works at the margin of the rules by finishing the job exactly in x days. At the same time, he does not want to jump up and finish the job immediately because, if he does so, he is likely to be seen as the ‘management's poodle’. So he settles for some safe point between the two extremes, namely, x days and 0 days, and, in the process, violates Chernoff's condition.

Note that, in Examples 3.1 and 3.2, in judging an option, the agent is using a criterion (possibly, with other criteria) that depends on the set of feasible alternatives from which the option is being chosen. In Example 3.1, the criterion of fulfilling social norms depends on whether the chosen fruit is the single fruit of its type in the feasible set. In Example 3.2, the criterion is one of taking the safe ‘middle path’ between the maximum possible days allowed by the manager to respond to customer queries and 0 days. The ‘middle path’ constitutes a menu-dependent evaluative criterion. Examples similar to Example 3.2 can be found in empirical studies of consumers' behaviour (see, for example, Simonson Reference Simonson1989). Examples 3.1 and 3.2 illustrate what we shall call the agent's menu-dependent criteria for judging options.

Example 3.3 (Luce and RaiffaReference Luce and Raiffa1957). The waiter in a restaurant gives a customer a menu for the day's dishes, which has two items: steak and fish. The customer orders fish. The waiter subsequently reports that, because of a mistake, frog's legs and fried snails have been omitted from the day's menu but they are available. The customer then orders steak. Again, this is a violation of Chernoff's condition. The intuition is that the customer would choose steak rather than fish or fried snails or frog's legs if he has some assurance that the restaurant is good (‘an indifferent restaurant would not know how to handle steak’) and the customer's experience tells him that fried snails and frog's legs are served only by good restaurants.

Example 3.4. The following is a true story of a university professor. The professor spent at least 4 hours each week to discuss with his two research students their academic problems. Then the chair of his department imposed a rule, which required that each supervisor of doctoral dissertations must devote no less than one hour each week to each of his Ph.D. students. After the rule was introduced, our professor devoted exactly one hour each week to discussions with each of his Ph.D. students. The professor's choices obviously violated Chernoff's condition. There can be several alternative explanations of the professor's behaviour. But the one, which the professor himself gave, was the following: ‘If the university does not trust me to do a good job of supervising my students on my own, then I am going to follow exactly the rules introduced by the university.’ In this case, the feasible set for the professor acted as an indicator of the chair's trust, and trust and reciprocation for trust were important considerations for the professor. The professor's behaviour cannot be dismissed as the eccentric behaviour ‘typical of academics’. There seems to be some evidence that too many rules and regulations governing the behaviour of workers on the shop floor may actually adversely affect the work that they put in. In a laboratory setting, Fehr et al. (Reference Fehr, Klein and Schmidt2007) study the effects of two different contracts, ‘bonus contract’ with fewer rules and relying on trust, and ‘incentive contract’ with detailed rules regarding workers’ efforts and relying on verifications by a third party, and find that bonus contracts can outperform incentive contracts in inducing workers’ effort.

In Examples 3.3 and 3.4, the menu from which the agent makes his choice gives him more information about the options. They constitute examples of what we shall call menu-dependent information.Footnote ⁴ Note that, Examples 3.3 and 3.4 share one feature of Examples 3.1 and 3.2. In Examples 3.3 and 3.4, as in Examples 3.1 and 3.2, the agent has concerns that are not captured in the original descriptions of the options: the description ‘steak’, in itself, does not capture the quality of the steak with which the agent is concerned in Example 3.3; nor does the description, ‘spending x hours each week for discussions with a student’ say anything about the chair's trust in the professor – a criterion or concern of the agent in Example 3.4. There is, however, one difference between Examples 3.1 and 3.2 on the one hand and Examples 3.3 and 3.4 on the other. Neither the criterion represented by the social norm of not picking up an only fruit of its type (see Example 3.1) nor the criterion of the ‘middle path’ (see Example 3.2) can be articulated without referring to a menu. In Example 3.3, however, the criterion of quality can, in principle, be articulated without any reference to the menu where ‘steak’ figures; similarly, in Example 3.4, the criterion of appropriate reciprocation for trust also can be stated without any reference to the menu where the option ‘spending x hours each week for discussions with a student’ figures. Thus, the criteria under consideration in Examples 3.3 and 3.4 are definable without any reference to a menu; what seems to be menu-dependent in Example 3.3 (resp. Example 3.4) is the agent's (imperfect) information, at the time of choosing, about the fulfilment of the relevant criterion when he chooses the option ‘steak’ (resp. the option ‘spending x hours each week for discussions with a student’).Footnote ⁵ This difference provides the basis of our distinction between situations of ‘menu-dependent criteria’ and situations of ‘menu-dependent information’.

Examples 3.1 through 3.4 show that, in certain fairly plausible situations, the agent's choice function can violate Chernoff's condition, which constitutes one of the weakest of consistency conditions discussed in the literature. Yet, in none of these examples, the agent acts in a way that one can reasonably call ‘irrational’. The above examples show how reasonable agents may violate internal consistency conditions for choice. More importantly, they all illustrate a broader methodological point, forcefully made by Sen, that the reasonableness of the internal consistency properties of choice cannot be determined without considering the criteria or motives behind the agent's choices. In Examples 3.1 and 3.2, as well as in Examples 3.3 and 3.4, the agent's violation of Chernoff's condition does not seem irrational at all once we know the reasons behind the agent's choices. The conclusion that Sen sought to draw from his examples was that it was not possible to formulate the theory of rational choice exclusively in terms of internal consistency of choices without going beyond choice as such to explore the criteria or motives guiding the agent's choices.

Examples 3.1 through 3.4 naturally raise the following issues.

Issue 1. Can the phenomena described in these examples be accommodated in the standard framework of revealed preference theory by plausibly reformulating the model so that the consistency conditions will not be violated in the reformulated model?

Issue 2. If at all it is possible to reformulate the model so as to accommodate the phenomena described in the examples without any violation of internal consistency conditions in the reformulated model, then what are the implications of such reformulation for the basic methodological point raised by Sen, to wit, whether one can discuss the reasonableness of internal consistency conditions without going beyond the concept of choice to look into the objectives guiding the agent's choices?

Issue 3. The problems raised by Examples 3.1 through 3.4 have been typically discussed in the context of the theory of revealed preference where one starts with the primitive notion of choice rather than with preferences of the agent. Do similar problems arise in models where one starts with the primitive notion of preference?

Issue 4. Finally, what happens if we treat the consistency conditions for choice as empirically testable hypotheses rather than as normative conditions for the agent's ‘rationality’? In Section 4, we take up the first issue. We show that it is indeed possible to adapt the basic model so that the agent's behaviour, when considered in the adapted version of the original model, will not violate the relevant internal consistency condition for choice. In Section 6, we take up the other three issues noted above.

4. A GENERAL FRAMEWORK TO HANDLE THE PHENOMENA OF MENU-DEPENDENT CRITERIA AND MENU-DEPENDENT INFORMATION

Faced with the violation of internal consistency conditions in situations involving menu-dependent criteria, one of the typical responses of theorists has been to reformulate some or all the characterizing features (the universal set of alternatives, the set of potential menus, and the choice function) of the agent in a way that gets rid of such violation. In this section, we present a reformulation of the conventional theory to accommodate Examples 3.1 and 3.2 discussed earlier. Though, for the purpose of our discussion, we focus on the case of menu-dependent criteria, at the end of this section we indicate how similar reformulations can deal with menu-dependent information (Examples 3.3 and 3.4).

In Example 3.1, as well as in Example 3.2, the agent uses a menu-dependent criterionFootnote ⁶ to assess the options available in alternative menus. Every such criterion refers to some features of the choice of a given alternative from a given menu. The analytical strategy that we adopt is to introduce the notion of indistinguishability/distinguishability, in terms of the relevant features, between the choice of an option from one menu and the choice of the same option from another menu.

Consider a case of menu-dependent criteria, where <X, , C> is the ‘initial’ description of the agent, and C violates some internal consistency condition, say, Chernoff's condition. For ease of presentation and without loss of generality, we assume that, for all x in X, there exists some A ∈ such that x ∈ A. For all x ∈ X, let ~_x be a reflexive, symmetric and transitive binary relation defined over the set Z_x, where Z_x is defined as the set of all ordered pairs (x, A) such that x ∈ A ∈ ; thus, ~_x is an equivalence relation defined over Z_x. Our intended intuitive interpretation of (x, A) ~_x (x, B) is that, in terms of the descriptive features referred to by the menu-dependent criteria (or, criterion) under consideration, choosing x from the set A is indistinguishable from choosing x from the set B. Thus, for every x ∈ X, ~_x is induced by the menu-dependent criteria under consideration. In Example 3.1, where the menu-dependent criterion is that of not choosing a fruit from a fruit tray if the fruit happens to be the only one of its type in the tray, if x is a particular apple that belongs to two different sets, A and B, of fruits, then (x, A) ~_x (x, B) if and only if either x is the only apple in A as well as in B or x is not the only apple in either A or B. Note that, for distinct alternatives x, y ∈ X, we have not introduced any notion of distinguishability or otherwise between choosing x from a set A ∈ and choosing y from a set B ∈ (irrespective of whether or not A and B are identical) because we do not need this additional information for our purpose.

It may be worth clarifying the intuitive content of the equivalence relations ~_x, ~_y, etc. Our notion of equivalence relations such as ~_x, ~_y, etc. is different from the concept of an ordering over {(a, A): a ∈ A ∈ }, that makes comparisons of the type ‘the choice of x from the menu A fulfils the menu-dependent criteria at least as much as the choice of b from the menu B’. Not only is it true that our equivalence relations, ~_x, ~_y, . . ., do not presuppose, either formally or intuitively, any ordering over {(a, A): a ∈ A ∈ }, but the intuitive interpretation that we have for these equivalence relations is also very different from the notion of fulfilling the menu-based criteria to the same extent. Given the descriptive features relevant for the menu-dependent criteria, our notion of indistinguishability between (x, A) and (x, B) in Z_x refers to indistinguishability in terms of those descriptive features rather than in terms of identical degrees of fulfilment of the menu-dependent criteria. Indistinguishability in terms of the relevant descriptive features intuitively entails identical degrees of fulfilment of the menu-dependent criteria, but the converse is not necessarily true. Note that, though we have chosen to interpret our equivalence relations as relations indicating indistinguishability in terms of the descriptive features relevant for the menu-dependent criteria and not as relations indicating identical degrees of fulfilment of the menu-dependent criteria, it may be noted that much of what we say below applies to both these interpretations.

Let {~_x}_{x ∈ X} be a given set of equivalence relations interpreted as above. For all x ∈ X, we say that ~_x is trivial if and only if, for all A, B ∈ , (x, A) ~_x (x, B) implies A = B. For all (x, A) ∈ Z_x, let E[x, A] be the equivalence class of (x, A) defined by ~_x, i.e. E[x, A] is the class of all (x, B) ∈ Z_x, such that (x, B) ~_x (x, A). Clearly, E[x, A] is a singleton if ~_x is a trivial equivalence relation. Note that, though formally one can consider any arbitrarily specified set, {~_x}_{x ∈ X}, of equivalence relations, to make intuitive sense these equivalence relations need to be suitably interpreted in terms of the menu-dependent criteria under consideration.

Having introduced the relevant equivalence relations {~_x}_{x ∈ X} and the corresponding equivalence classes, we can now transform the initial description, <X, , C>, of the agent in our example of menu-dependent criteria into a new description <X _*, _*, C _*> specified as follows:

(4.1)

(4.2)

(4.3)

It is clear that, given the set, {~_x}_{x ∈ X}, of equivalence relations, there exists a unique triple <X _*, _*, C _*> satisfying (4.1), (4.2) and (4.3). Essentially, what is involved in the transition from the description of the agent in terms of <X, , C> to the description in terms of <X _*, _*, C _*> is the replacement of the original notion of an option by an ‘extended’ notion of an option to capture an aspect that is relevant for the agent's choices but does not figure in the original description of an option. The concept of the class of potential menus, as well as the concept of options chosen from a menu, is then adjusted accordingly. It may be noted that the transformed description of an agent by <X _*, _*, C _*> depends on the underlying class of equivalence relations {~_x}_{x ∈ X}, which, in turn, are determined by the given menu-dependent criteria.

How does our reformulated framework tackle the violation of Chernoff's condition considered in Examples 3.1 and 3.2? To answer this question, we first note the following result. Its proof can be found in the appendix.

Proposition 4.1. Let <X, , C> be any given initial description of the agent and let {~_x}_{x ∈ X} be a given class of equivalence relations. Let <X _*, _*, C _*> be the modified description of the agent satisfying (4.1), (4.2) and (4.3).

1. (i) If C satisfies any of the properties (Chernoff's condition, the weak axiom of revealed preference, the congruence axiom, and rationalizability in terms of an ordering) introduced in Definition 2.1, then C _* must satisfy the same condition.

2. (ii) If, for all x ∈ X, ~_x is trivial, then C _* must satisfy all the properties (Chernoff's condition, the weak axiom of revealed preference, the congruence axiom, and rationalizability in terms of an ordering) introduced in Definition 2.1.

3. (iii) It is possible for C _* to satisfy all the properties introduced in Definition 2.1 even when, for all x ∈ X, ~_x is non-trivial and C violates Chernoff's condition, the weakest of the properties introduced in Definition 2.1.

Propositions 4.1 (i) and 4.1 (iii) show that the requirement of any of the four properties, namely, Chernoff's condition, the weak axiom of revealed preference, the congruence axiom, and rationalizability in terms of an ordering, for the choice function C _* figuring in the new description <X _*, _*, C _*> is weaker than the requirement of the same property for the choice function C figuring in the initial description <X, , C>. Therefore, when C _* satisfies, say, Chernoff's condition but C does not, we can say that, though the agent, described by <X, , C>, violates Chernoff's condition, the violation is really due to the fact that he regards the choice of an option x from some feasible set A to be distinguishable from the choice of the same option x from some subset B of A, and when the problem is re-formulated to take into account such ‘distinguishability’ of acts of choosing in addition to the features of the initial options, the violation of Chernoff's condition disappears in the re-formulated choice problem. Proposition 4.1 (ii) shows that when all the equivalence relations in {~_x}_{x ∈ X} are trivial, C _* must satisfy all our rationality conditions irrespective of whether C satisfies any of the rationality property. Proposition 4.1 (ii), therefore, shows that, if, instead of being determined by exogenously given menu-dependent criteria, the equivalence relations can be specified in any way one likes, then one can always specify the equivalence relations in such a way that, when the model of the agent's choices is reformulated with reference to such specification, the agent's choice function in the reformulated model will necessarily satisfy all our rationality conditions.

Having reformulated the initial description, <X, , C>, of the agent so as to derive the new description, <X _*, _*, C _*>, which satisfies (4.1), (4.2) and (4.3), we can now define a modified notion of rationalizability of the initial choice function C in terms of an ordering. The choice function C is said to satisfy <X _*, _*, C _*>-based rationalizability in terms of an ordering iff C _* is rationalizable in terms of an ordering.

The following proposition follows immediately from the definition of <X _*, _*, C _*>-based rationalizability in terms of an ordering and the fact that, for every choice function, the congruence axiom is equivalent to rationalizability of that choice function in terms of an ordering (see Richter Reference Richter1966).

Proposition 4.2. Let <X, , C> be the initial description of the agent, and let <X _*, _*, C _*> be the modified description satisfying (4.1), (4.2) and (4.3). C satisfies <X _*, _*, C _*>-based rationalizability in terms of an ordering iff C _* satisfies the congruence axiom.

It may be noted that, when, for all x ∈ X and all (x, A), (x, B) ∈ Z_x, (x, A) ~_x (x, B), <X _*, _*, C _*>-based rationalizability of C in terms of an ordering becomes equivalent to the standard rationalizability of C in terms of an ordering. On the other hand, by Proposition 4.1 (ii), when, for all x ∈ X, ~_x is trivial, any choice function C satisfies < X _*, _*, C _*>-based rationalizability in terms of an ordering.Footnote ⁷ Given this observation, it is clear that every choice function C satisfies <X _*, _*, C _*> -based rationalizability in terms of an ordering for some class of equivalence relations {~_x}_{x ∈ X}. This, by itself, is, however, of little intuitive interest unless the equivalence class {~_x}_{x ∈ X} underlying the <X _*, _*, C _*> -based rationalizability of C is ‘induced’ by some intuitively plausible menu-dependent criteria. Before concluding this section, we indicate briefly how the case of menu-dependent information can be handled by a variant of the modelling strategy discussed above. For convenience in exposition, we concentrate on Example 3.3. Here the choice of a dish can be intuitively thought of as being associated with an uncertain prospect; for some dishes, the uncertain prospects may be trivial (i.e. they may really be certain prospects), but for others, such as steak, the uncertain prospect is non-trivial. The uncertainty involved may be probabilistic (for example, the uncertain prospect may be a lottery with probability p for getting high-quality steak and probability 1−p for getting low-quality steak) or non-probabilistic. The important point is that the uncertain prospect, however conceived, that is associated with the choice of a dish, such as steak, can change, depending on the menu from which the option is chosen. Thus, assuming for the moment that the uncertainty involved is probabilistic, the agent's (subjective) probability for high-quality steak can go up when he learns that the menu also includes frogs’ legs and fried snail. Given this, for all x ∈ X, we can now introduce an equivalence relation ~_x over Z_x with the following interpretation: (x, A) ~_x (x, B) means that the uncertain prospect that the agent associates with x when the menu is A is the same as the uncertain prospect that the agent associates with x when the menu is B. We can then suitably transform the original description of the agent into a new description, where the equivalence classes, E[x, A], E[y, B], etc. constitute the re-specified options (one can think of a different interpretation of ~_x, under which the specification of the new options as (x, E[x, A]), (y, E[y, B]), etc. would make sense, but we find the interpretation of ~_x given above more direct and natural).

5. SOME ALTERNATIVE REFORMULATIONS

As we mentioned earlier, faced with the violation of internal consistency conditions in situations involving menu-dependent criteria, a typical response of theorists is to reformulate some or all of the characterizing features, including the universal set of alternatives, the set of potential menus, and the choice function, of the agent in such a way that such violation no longer occurs in the new formulation. In what follows, we consider a few alternative strategies for such reformulation to be found in the existing literature.Footnote ⁸

5.1 Re-specification of the set of potential menus

In an interesting paper, Bossert and Suzumura (Reference Bossert and Suzumura2009a) present a formulation that re-constructs the set of potential menus. They start with Example 3.1 due to Sen, where the ‘anomalous’ choice behaviour of the agent arises from the agent's concern about a menu-dependent criterion. Following the terminology of Sen (Reference Sen1993), Bossert and Suzumura call such menu-dependent criteria ‘external norms’ and suggest a formal device through which external norms can be incorporated in the model.Footnote ⁹

Without going into the details of Bossert and Suzumura's analysis, we outline here the intuition underlying their framework with specific reference to Example 3.1. In Example 3.1, the violation of Chernoff's condition occurs in a framework where the agent i is described by <X, , C>, such that the universal set of options, X, is the set of all possible fruits, and the class of potential menus, , is the class of alternative sets of fruits that the agent may have to choose from. The menu-dependent criterion or external norm involved in Example 3.1 happens to be the social norm that forbids the agent to choose a fruit if it is the only fruit of its type in the menu. Essentially, the analysis of Bossert and Suzumura can be interpreted as a transformation of the original description, <X, , C>, of the agent into another description where the universal set of options continues to be X, but the set of potential menus and the choice function are re-specified. The main components of their analysis can be described as follows.Footnote ¹⁰

(i) First, the external norms are formally introduced as a function, g, which, for every A ∈ , specifies exactly one (possibly empty) subset g(A) of A, g(A) being interpreted as the set of options in A, which the external norms forbid the agent to choose from A. It is assumed that, for all A ∈ , A − g(A) is non-empty.

(ii) Next, one identifies a triple <X ₊, ₊, C ₊>, such that

(5.1)

(5.2)

and

(5.3)

It can be checked that a triple <X ₊, ₊, C ₊> satisfying (5.1), (5.2) and (5.3) will exist if and only if the following two conditions, (5.4) and (5.5), hold:

(5.4)

(5.5)

Further, it can also be checked that there cannot exist more than one triple <X ₊, ₊, C ₊> satisfying (5.1), (5.2) and (5.3).

(iii) Finally, one introduces the notion of ‘norm-conditional’ consistency: for every consistency condition σ, one says that the choice function C figuring in the original description, <X, , C>, of the agent satisfies σ subject to the constraints imposed by the menu-dependent criteria or external norms represented by g if and only if there exists a triple <X ₊, ₊, C ₊> satisfying (5.1), (5.2) and (5.3), such that C ₊ satisfies σ.

Thus, intuitively, one constructs another description of the agent, such that, for every possible menu A in the initial description, the set of options that the agent chooses from A in the initial description is the set of options that the agent chooses from the menu (A − g(A)) in the new description. If the choice function in the original description violates the internal consistency condition σ, but the choice function in the new description satisfies σ, we can say that, though the agent described by <X, , C>, may be violating σ, the violation is due to the fact that, given a menu, he treats the options forbidden by the external norms as ‘inadmissible for choice under the external norms’, and, that when the menus are re-specified to take into account such norm-based inadmissibility in addition to physical infeasibility, the violation of σ disappears in the re-specified choice problem.

It may be noted that the framework proposed by Bossert and Suzumura (Reference Bossert and Suzumura2009a) is a special case of our general framework in the following sense. Suppose we have an initial description, <X, , C>, of the agent and a menu-dependent criterion or external norm represented by a function g as described earlier. Suppose, for every x ∈ X, ~_x is an equivalence relation defined over Z_x, such that, for all (x, A), (x, B) ∈ Z_x, (x, B) ~_x (x, A) iff either x ∈ (A − g(A)) ∩ (B − g(B)) or x ∈ (g(A) ∩ g(B)). Then,

(5.6)

and

(5.7)

The proof of (5.6) is straightforward. To see that (5.7) is true, an example will suffice. In the spirit of Sen's example involving the choice of a fruit in a party, let the initial description, <X, , C>, of the agent be such that X = {x ₁, x ₂, y ₁, y ₂, z}, where x ₁ and x ₂ are two apples, y ₁ and y ₂ are two pears, and z is a mango; = {A, B, G, H}, where A = {x ₁, x ₂, y ₁}, B = {x ₂, y ₁, y ₂}, G = {y ₁, y ₂, z}, and H = {x ₁, x ₂, z}; and [C(A) = {x ₁, x ₂}, C(B) = {y ₁, y ₂}, and C(G) = C(H) = {z}]. Let the menu-dependent criterion be the norm of not choosing from a set of fruits the last fruit of its type, so that g(A) = {y ₁}, g(B) = {x ₂}, and g(G) = g(H) = {z}. Let σ be Chernoff's condition. It is clear that, by definition, C cannot satisfy Chernoff's condition subject to the constraints imposed by external norm represented by g since, given [C(G) = g(G)] and [C(H) = g(H)], (5.4) is violated and, hence, no triple <X ₊,₊, C ₊> can possibly satisfy (5.1), (5.2) and (5.3) simultaneously. It is, however, easy to check that there exists a triple <X _*, _*, C _*> satisfying (4.1), (4.2) and (4.3), such that C _* satisfies Chernoff's condition.

6. A REASSESSMENT OF SEN'S ARGUMENT

In Section 4, we have developed a rather general framework in which the choice problems are reformulated to encompass issues relating to menu-based criteria without making many of the assumptions that are typically made in the existing approaches discussed in Section 5. What, however, is the relevance of our reconstruction in Section 4, as well as the reconstructions discussed in Section 5, for Sen's criticism of the interpretation of internal consistency conditions as rationality properties that can stand on their own without any reference to the objectives of the agent under consideration? We now take up this issue and we reach a conclusion very different from that often reached in the existing literature.

It seems to us that, while our reformulation of the type of choice problems illustrated in Examples 3.1 through 3.4 as well as other reformulations including that of Bossert and Suzumura (Reference Bossert and Suzumura2009a, Reference Bossert and Suzumura2009b) are of interest, they do not address the central problem that Sen raised; nor do they reduce in any way the impact of Sen's criticism of the status the internal consistency conditions have sometimes been accorded in the literature on the theory of revealed preference. To see this, consider again the contention of Sen in this context. For convenience, we concentrate on Example 3.1, though the discussion can be readily extended to all the other examples in Section 3. At the risk of being over-elaborate, let us spell out explicitly the different strands of Sen's reasoning.

Let <X, , C> be the initial description of the agent, where the universal set of options, X, is simply the set of all possible fruits. Sen's argument then seems to proceed in three distinct steps. The first step consists of the observation that, when we characterize the agent in terms of <X, , C>, the agent's choice function C violates Chernoff's condition. The second step consists of the observation that, when we know that, not only does the agent care about the fruit that he eats, but he also cares about the social norms under consideration, his choice behaviour with respect to fruits does not seem irrational or bizarre at all. The third step consists of the conclusion that Sen draws from the two observations. The conclusion is that the appeal of Chernoff's condition as a property of rational choice in the choice problem described by <X, , C> depends on our information about the objectives or motives of the agent: if we are not aware that, not only does the agent care about what fruit he eats, but he also cares about conforming to certain social norms about the choice of fruits in a party, then, given the information that we have, the violation of Chernoff's condition would seem to us to be an indication of ‘irrational’ choice; on the other hand, if we know that the agent cares about the social norms under consideration in addition to caring about what fruit he eats, then the agent would seem reasonable to us despite the observed violation of Chernoff's condition by the choice function C.

Now consider what our analysis in Section 4 above shows. What it shows is that, if the theorist modelling the agent's choice behaviour knows that the agent cares about the social norm, in addition to caring about what fruit he (i.e. the agent) eats, then the theorist can plausibly transform the description given by <X, , C> into another description <X _*, _*, C _*> where C _* satisfies Chernoff's condition. While this is of interest, does it intuitively contradict in any way Sen's conclusion? We do not think so. Indeed, it seems to us that our analysis only serves to reinforce the point that Sen is making. It is true that, if the theorist knows about the agent's concern about social norms, he can plausibly transform the original description of the agent so as to get rid of the problem of violation of Chernoff's condition in the reformulated description. But such a plausible formal transformation of the original description will be possible only if the theorist knows that the agent cares about certain social norms and also knows what these norms are (this latter piece of information is necessary for the theorist to decide whether the choice of an option x from a set A is, in terms of the social norms the agent cares about, distinguishable from the choice of x from another set B). The decision between the alternative formulations, <X, , C> and <X _*, _*, C _*>, itself will depend on the theorist's information and beliefs about the objectives or motivations of the agent. If we do not know anything about the relevance of social norms for the situation described in Example 3.1, then there is no way of formulating the choice problem in terms of <X _*, _*, C _*>. Then we would presumably formulate the choice problem in terms <X, , C>, and, in that case, the agent's behaviour would seem to violate Chernoff's condition. What Sen's analysis emphasizes is the importance of our information about the agent's concerns in assessing the intuitive appeal of internal consistency condition as criteria for ‘rational choice’ in any given model of choice. What our analysis demonstrates is that, given suitable information about the agent's concerns, it may be possible to reformulate the choice problem plausibly in such a way that an internal consistency condition will be satisfied in the reformulated version though it was violated in the original formulation. But, of course, the very possibility of such reformulation will depend on our information and/or belief about the agent's concerns. The issue can be stated slightly differently. One can distinguish between two distinct aspects of the intuitive notion of rationality. The first is the rationality of an agent's goals (goal rationality). For example, it is possible to argue that the agent has irrational goals if, other things remaining the same, the agent would choose to torture more of the animals around him. In general, positive economics has scrupulously avoided the issue of goal rationality. It has exclusively focused on what may be called structural rationality, i.e. the issue of whether the choices that the agent makes are coherent given the goals that the agent has. It is this notion of coherence of choice, given the goals of the agent, that the internal consistency conditions are intended to capture. What Sen's argument shows is that, even when we identify the notion of rational choice with this limited notion of structural rationality or coherence of choices given whatever goals the agent may have, what constitutes coherent choice given one set of goals may not be coherent for a different set of goals. Therefore, whether or not the agent is being coherent in his choices would depend crucially on our intuition about what constitute the goals of the agent. Where do the agent's goals enter into our formal models of an agent's choices? Note that, before we can even formally define the internal consistency conditions, we have to describe the choice situation, namely the universal set of options, the class of opportunity sets that the agents may choose from, and the notion of the agent's choice function. The very first step here is, of course, to specify the universal set of options which embodies our conception of the type of objects that the agent is really concerned with.Footnote ¹² If the universal set of options is specified as eating a mango, eating an apple,. . ., then implicitly we are taking the view that the agent is concerned only with what fruit he eats. If, in a model of choice specified in this fashion, the agent's choices violate Chernoff's condition, then it may be because the options as we have specified them (in the course of specifying the universal set which reflects the goals of the agent) completely and correctly capture what the agent cares about, but the choices of the agent are incoherent given the agent's concerns and goals, or it may be because we have specified the options in a way that does not reflect the concerns of the agent appropriately. Therefore, even when we limit ourselves, as we typically do in positive economics, to the notion of structural rationality as the only conception of rationality, whether the choice behaviour of an agent is structurally irrational (i.e. incoherent, given the agent's goals) cannot be decided independently of what we consider to be the goals or objectives of the agent. It seems to us that Sen's criticism of the view of internal consistency conditions as properties of rational choice that can stand on their own without any reference to the agent's concerns is valid in a fundamental sense and no formal reconstruction of the agent's description detracts from its impact in any way. Thus, our assessment of Sen's arguments regarding the status of internal consistency conditions as properties of rational choice is very different from that of Bossert and Suzumura (Reference Bossert and Suzumura2009a) who, after demonstrating that some of Sen's examples can be accommodated in their modified version of the conventional framework,Footnote ¹³ conclude that their analysis ‘builds a bridge between rationalizability theory and Sen's criticism’ and that what emerges from their analysis is ‘the possibility of a peaceful coexistence of a norm-conditional rationalizability theory and Sen's elaborate criticism against the internal consistency of choice’.

We would like to clarify three other related points. First, note that so far we have taken a normative interpretation of the internal consistency conditions by treating them as conditions that a rational individual's choice behaviour will satisfy. Sen focused on this interpretation which is often adopted in the literature. What if we treat the internal consistency conditions as testable empirical hypotheses regarding an agent's choice behaviour? It is easy to see that the problem to which Sen drew our attention in the context of the normative interpretation of internal consistency conditions arises again, though in a somewhat different form, when we treat those conditions as empirically testable hypotheses. We can introduce the internal consistency conditions only after we specify the universal set of options, i.e. only after we commit ourselves to a particular view of the alternatives that the agent chooses. If, given a particular specification of the choice problem, the choice function of the agent turns out to violate some internal consistency condition, say Chernoff's condition, then such violation falsifies the conjunction of two hypotheses. The first hypothesis is that our specification of the universal set of options correctly embodies what the agent is concerned with in making his choices and the second hypothesis is that the agent's choice function satisfies Chernoff's condition when the options are specified in a way that correctly captures the agent's goals. Given the falsification of the conjunction of these two hypotheses (i.e. given that the choice function figuring in a given description of the agent violates Chernoff's condition), we cannot decide whether to reject the second hypothesis without committing ourselves to a position about whether or not the options, as we have specified them in our formal model, capture the objectives of the agent.Footnote ¹⁴

The second point is this. Sen's original argument was formulated with reference to the theory of revealed preference where we start with the primitive concept of choice rather than preference. What happens if we start with the primitive notion of preference and formulate, as usual, the notion of structural rationality in terms of the requirement that the agent's preferences be an ordering, that is, the requirements that the agent's preferences satisfy reflexivity, connectedness and transitivity? It is easy to see that this would make little difference to the validity or impact of the basic point of Sen. Again, before we can even introduce the notion of the agent's preferences, we need to specify the universal set of options over which the preferences are to be defined. If the preferences violate, say, transitivity,Footnote ¹⁵ then again we have to face the problem of deciding whether, given our chosen formulation of the choice problem, the requirement of transitivity has much appeal in light of what we know about the agent's goals.

Finally, we consider a point that sometimes comes up in the context of Sen's critique of the treatment of internal consistency conditions as conditions for rational choice. It is sometimes argued that: (1) the theory of revealed preference and the associated internal consistency conditions for choice were never intended to be applied to situations involving menu-dependent criteria or menu-dependent information; and (2) that there has been an implicit understanding among revealed preference theorists that the scope of the theory must be restricted to exclude such situations. Whether revealed preference theorists, such as Samuelson (Reference Samuelson1947, Reference Samuelson1948), Little (Reference Little1949), Houthakker (Reference Houthakker1950) and Richter (Reference Richter1966), had in mind such a scope-restricted theory of revealed preference is an interesting issue in the history of economic thought, but it is beyond the scope of our paper. What, however, is germane for the purpose of this paper is the implications, for Sen's critique, of explicitly restricting the applications of internal consistence conditions, as properties of rational choice, to situations where neither the criteria for choice nor the information relevant for choice are menu-dependent. For convenience of exposition, let us focus on Chernoff's condition. What the proposed restriction of the applicability of Chernoff's condition as a criterion of rationality of choice essentially does is to replace the intuitive position represented by (6.1) below by the intuitive position represented by (6.2):

(6.1)

(6.2)

Though the examples in Section 3 can be used as intuitive counter-examples against the position represented by (6.1), clearly they cannot be used in that fashion against the position represented by (6.2) since all those examples involve either menu-dependent criteria or menu-dependent information. Does this in any way work against the substance of Sen's argument that, without referring to an agent's objectives and concerns, one cannot treat the internal consistency conditions for choice, on their own, as tests of the rationality of the agent's choices? We do not think so. Suppose we take position (6.2) and we find that the agent's choices violate Chernoff's condition. Then we cannot say anything about the rationality of the agent's choices unless we commit ourselves to a position about whether the agent's choice situation involves menu-dependent criteria or menu-dependent information relevant for the agent, and, of course, we cannot take a position about whether the agent's choice situation involves menu-dependent criteria or menu-dependent information without considering what the agent's goals/objectives/concerns may be. We are then back to the core of Sen's argument, namely, that without considering the objectives and concerns of the agent, one cannot say much about the rationality of the agent's choices exclusively on the basis of internal consistency conditions.

We have considered above the proposal to restrict the application of internal consistency conditions for choice, when they are interpreted as conditions of rationality. Exactly similar reasoning can, however, be given if, interpreting the internal consistency conditions as empirically testable hypotheses, we replace hypothesis (6.3) below by hypothesis (6.4):

(6.3)

(6.4)