PRASANTA K. PATTANAIK AND YONGSHENG XU

ON PREFERENCE AND FREEDOM

ABSTRACT. We consider the role of preferences in the assessment of an agent’s freedom, visualized as the opportunity for choice. After discussing several possible intuitive approaches to the problem, we explore an approach based on the notion of preference orderings that a reasonable person may possibly have. Using different sets of axioms, we characterize the rules for ranking opportunity sets in terms of freedom. We also show that certain axioms for ranking opportunity sets are incompatible. KEY WORDS: Freedom, Opportunity set, Ranking, Reasonable person’s preferences

1. INTRODUCTION

The purpose of this paper is to explore the role of preferences in the assessment of an agent’s freedom as reflected in his opportunity set. Consider an agent who may be faced with different sets of alternatives. Given a set of alternatives, he has to choose exactly one alternative from the set. Each possible set of alternatives offers him some freedom of choice. How does one rank these different opportunity sets in terms of the degrees of freedom that they offer to the agent? This is the problem that has been considered by a number of writers recently (see, a.o., Arrow (1995), Dutta (1990), Foster (1992), Jones and Sugden (1982), Klemisch-Ahlert (1993), Pattanaik and Xu (1990), Puppe (1996), Sen (1985, 1991, 1993), Steiner (1983) and Suppes (1987); see also Bossert, Pattanaik and Xu (1994) for a discussion of a related problem). One of the issues that has come up in this context is the role that preferences over the various alternatives should play in assessing an agent’s freedom. While Pattanaik and Xu (1990) introduced a set of axioms that did not incorporate preferences at all, and which characterized the ranking of opportunity sets on the basis of their cardinalities, Sen (1991, 1993), and subsequently Foster (1992) and Puppe (1996), have strongly argued in favor of introducing preference as a basic ingredient in the

Theory and Decision 44: 173–198, 1998. c 1998 Kluwer Academic Publishers. Printed in the Netherlands.

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evaluation of opportunity sets in terms of freedom. In this paper we undertake some further investigation into this problem. One of the basic conceptual issues that one faces in seeking to incorporate preferences in the assessment of freedom involves the interpretation of the preferences that are considered relevant for this purpose. One can think of several different types of preferences in this context: the present preferences of the agent whose freedom is under consideration, the preferences that the agent may have in the future, the preferences of all the individuals in the society, the preferences that a reasonable person may possibly have, and so on (see Section 2 below for a more detailed discussion of these). The formal analysis in this paper is compatible with more than one of these intuitive interpretations. However, in assessing the ‘intrinsic value’ of freedom offered by an opportunity set (this is the problem we address in this paper), our inclination is to take, as the point of reference, the preferences that a reasonable person may have over the options under consideration. It is clear that, in general, there can be several possible preference orderings each of which may be compatible with the notion of being ‘reasonable’. Thus, the starting point of our analysis is a reference set of orderings, interpreted as the set of all possible preference orderings that a reasonable person may have (of course, in special cases, this set may be a singleton1). We introduce a series of axioms that postulate restrictions on the ranking of opportunity sets in terms of freedom (all excepting one of these axioms make use of the notion of the reference set of orderings), and we explore the implications of several subsets of these axioms. In the process, we characterize certain rules for ranking opportunity sets in terms of freedom. We also show that certain apparently plausible axioms are logically incompatible. The plan of the paper is as follows. In Section 2 we discuss several intuitive interpretations of preferences that may be considered relevant in our context, and we clarify the basic intuition underlying our approach. In Section 3, after laying down some notation and definitions, we introduce a number of axioms in Section 4, each of which constitutes a restriction on the ranking of opportunity sets in terms of freedom. These axioms, which can be viewed as variants of axioms introduced in Pattanaik and Xu (1990), fall into three broad groups. First, we have axioms that are concerned with comparisons of sin-

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gleton opportunity sets (see Definition 4.1): these involve relatively simple cases where our intuition is likely to be somewhat clearer than the more complex cases considered in other axioms. Second, we have axioms which, given a certain ranking of two sets A and B and given a suitably specified alternative x, impose restrictions on the ranking of A [ fxg and B (see Definition 4.2). Our third group consists of two axioms (see Definition 4.3). Given the ranking over each of two pairs of opportunity sets, (A; B ) and (C; D), under certain conditions these two axioms impose restrictions on the ranking over the pair of opportunity sets (A [ B; C [ D). In Section 5 we explore the implications of several combinations of axioms introduced in the previous section. In this section we axiomatically characterize two distinct rules for ranking opportunity sets in terms of freedom. The first rule that we characterize (see Proposition 5.1) ranks opportunity sets on the basis of the cardinalities of their ‘maximal sets’ (the maximal set of a set A is defined as the set of all alternatives a in A such that a is best in A in terms of some ordering that a reasonable person may have). The second rule that we characterize (see Proposition 5.9) ranks two sets A and B on the basis of the cardinalities of: (i) the set derived by eliminating from the maximal set of A all those elements a such that a can never be at least as good as all the alternatives in B for any preference ordering of a reasonable person; and (ii) the set derived by eliminating from the maximal set of B all those alternatives b such that b can never be at least as good as all the alternatives in A for any preference ordering of a reasonable person. In Section 5, we also prove two ‘impossibility’ results each of which shows that certain plausible axioms regarding the ranking of opportunity sets are logically inconsistent (see Propositions 5.7 and 5.14). We conclude our paper by briefly outlining, in Section 6, the relationship between our analysis and the earlier contributions of some of the authors cited above.

2. PREFERENCE AND FREEDOM

Sen (1991, 1992, 1993) has forcefully argued that judgements about the degrees of freedom offered to an agent by different opportunity sets (‘menus’) must take into account the agent’s preferences over alternatives. Sen (1993) writes,

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The evaluation of freedom I enjoy from a certain menu must depend to a crucial extent on how I value the elements included in that menu.

He proceeds to give an example of how preferences over the elements included in opportunity sets (‘menus’) may influence the assessment of freedom. ... while it is certainly easy to maintain that ... an additional opportunity cannot reduce a person’s freedom ..., it may be too much to claim that there is necessarily an actual expansion of freedom. For example, the new alternative may be very uninteresting (e.g., the option of being beheaded at dawn), or thoroughly dominated by one of the existing ones (e.g., having another car much like the one already on offer except for a defective gear box).

The issue – namely the link between preferences and freedom – that Sen raises is important. In earlier papers, Steiner (1983) and Pattanaik and Xu (1990) have explored conceptual frameworks that rule out any role of preferences in judgements about freedom. The vigorous criticism of that position by Sen and several other writers raises important analytical issues that deserve further careful study. How do considerations of preferences enter into assessments of an individual’s freedom as reflected in his opportunity set? An attempt to answer this question inevitably leads us to the issue of why we value freedom in the sense of opportunity for choice. In this context, it may be helpful to consider a relatively simple case that involves a comparison between two opportunity sets A and B such that B is a proper subset of A. The bigger opportunity set A clearly offers more choices than the smaller opportunity set B . What can be the reasons for the agent to value such an expansion of his opportunities? First, consider the traditional reason based on indirect utility. In the traditional analysis, the agent is assumed to have a well defined preference ordering at the time he is confronted with an opportunity set. In fact, the preference ordering is one of the features that defines the decision-making agent (see Debreu’s (1959) characterization of a consumer). Given this preference ordering, and, given an opportunity set, the agent chooses the best alternative, defined in terms of his given preference ordering, in the opportunity set. It can then be claimed that the opportunity set A is more highly valued than opportunity set B if and only if the agent’s best option in A is better for him than his best option in B . This, of course, is a purely consequentialist or instrumentalist view of the value of freedom of choice as embodied in opportunity sets. Further, the only consequences that

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are taken into account here are the utilities: the opportunity sets are valued solely for the sake of the levels of ‘indirect utility’ that they enable an agent to attain, the indirect utility being (ordinally) measured in terms of a given preference ordering. The literature contains an important extension of the indirect utility approach (see, for example, Arrow, 1995; see also Kreps, 1979). In this extended framework, the individual is assumed to be uncertain about the preference ordering that he would have when he would actually have to choose an option from the opportunity set. Given this uncertainty, which may be either probabilistic uncertainty, as in Arrow (1995), or non-probabilistic, the individual has to value the different opportunity sets. While the introduction of uncertainty regarding preferences is a major extension of the indirect utility approach to freedom, note that, in the extended approach, the value of freedom is still perceived from a consequentialist (more specifically, utilitarian) perspective: freedom is now an instrument for achieving a better uncertain prospect, or, as in the expected utility framework of Arrow (1995), a higher level of expected utility. While not many people would like to deny the importance of indirect utility or ‘expected indirect utility’ in assessing the value of freedom of choice, to judge the value of freedom of choice exclusively in terms of such utility-based criteria would amount to ignoring a long standing libertarian tradition that believes that freedom is valuable in itself apart from whatever higher level of utility or expected utility it may enable us to achieve. Nor is such belief confined to a handful of ‘extreme’ libertarians. Consider an agent, assumed to be a woman, who, currently, has no intention of joining the army and who attaches probability zero to the possibility that she would like to join the army. A government ban on women’s joining the army would not reduce her freedom either in the standard indirect utility approach or in the probabilistic approach of Arrow (1995). However, we believe that a large number of people would feel that such a ban would reduce the woman’s freedom and we suspect that the reason for their feeling that way lies in their basically libertarian belief in the intrinsic value of freedom. While we do not want to belittle the importance of the utilitarian consequences of having one opportunity set or another, in this paper, we shall focus on the

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‘intrinsic’ value of freedom and how consideration of preferences can enter our assessment of this intrinsic value. The Libertarian belief in the intrinsic value of freedom of choice is often based on the view that, for a person to live a meaningful life, he must shape his life by making choices for himself: A person’s shaping his life in accordance with some overall plan is his way of giving meaning to his life; only a being with the capacity so to shape his life can have or strive for meaningful life. (Nozick, 1974, p. 50)

If, as Nozick claims, choosing for oneself and thereby shaping one’s own life is essential for meaningful human life, then such choice can hardly be visualized as the choice of a best feasible alternative, where the best is specified on the basis of a given preference ordering that comes as a defining feature of the identity of the individual. If the preference ordering is already given in this fashion, then the specification of a best alternative needs little volition on the part of the individual. Nor can shaping of one’s life be visualized in terms of choice of the expected utility maximizing feasible alternative where expected utility is calculated on the basis of a probability distribution over possible future preferences – a probability distribution that is somehow or other given to the individual. Rather, as Jones and Sugden (1982, p. 59) rightly summarize the view of choice underlying much of libertarian thought: To suppose that the act of choice requires the exercise of mental powers is to suppose that the chooser is in some considerable measure an autonomous agent; whatever he chooses, he might have chosen something else. There is a tension between this and the idea, implied in the economic theory of choice, that preferences are given. What makes significant choice possible is that preferences are not just part of a person’s physiology or psychology like the colour of his eyes or a tendency to depression. Given a person’s psychology, and given the situation in which he finds himself, there remain different preferences that he could hold, and he has to decide which is right for him.

The notion of choice that the libertarians value so much is not ‘passive’ choice in terms of a historically given preference ordering, or choice based on a probability distribution over preferences, the probability distribution being given independently of the agent’s volition. Rather, the act of choice is visualized as the end product of a conscious decision regarding what alternative in the opportunity set he should prefer to the rest, or, equivalently, what alternative he should choose from the opportunity set. This is so even when an agent starts with a historically given preference ordering. Even in this case, the

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agent is regarded as having ‘autonomy’ with respect to his preferences. Given his circumstances, he could still reasonably hold other preferences and choose on the basis of these other preferences. If he chooses to go by his historically inherited preferences, then that is his deliberate decision as an autonomous agent. Consider the woman in our earlier example. If we feel that the government’s ban on women’s joining the army reduces her freedom, it is because, given the woman’s situation, she could have reasonably chosen to join the army (or, equivalently, she could have reasonably preferred a career in the army to other available options), even though she actually does not do so and even though she attaches zero probability to her wanting to do so. Now consider Sen’s example where the opportunity set for an agent is expanded by adding the option of being beheaded at dawn. We believe that even people who share the libertarian belief in the intrinsic value of freedom will agree that ‘normally’ this would not constitute an increase in the agent’s freedom. The reason lies not so much in the fact that the agent himself actually prefers some already existing option(s) to the option of being beheaded at dawn. Rather, we believe, the reason lies in our presumption that, given the circumstances of the agent, no reasonable person would prefer the option of being beheaded at dawn over the other options. If further details are added to the example to remove this presumption, then there may be a change in our opinion that the addition of the option of being beheaded at dawn does not constitute an increase in freedom. Suppose that the sole existing alternative for the agent is to spend the rest of his life (assume that he is expected to live another 50 years) in a solitary cell. In this case, it is no longer clear to us that we would feel that his freedom is not increased when he is given the additional option of being beheaded at dawn. This is because now we can visualize a reasonable person in the agent’s situation preferring the option of being beheaded at dawn to confinement in a solitary cell for the rest of his life. We believe that consideration of preferences is important in assessing the intrinsic value of freedom of choice, as reflected in an agent’s opportunity set. However, it seems to us that, in many ways, the preferences that are crucial in such assessment are not the preferences that the agent actually has, nor the preference orderings that have positive probabilities of emerging as his future preference

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ordering, but the preference orderings that a reasonable person in the agent’s situation can possibly have. Suppose an agent now prefers vegetarian food and suppose the probability that he will like to eat non-vegetarian food in the future is zero. However, suppose, in the (objective) circumstances of the agent, a person can, without being unreasonable, prefer non-vegetarian food over vegetarian food in the future. This, of course, raises the question as to what constitute the criteria for judging the reasonableness of a person. This is not an issue we take up in this paper. Instead, we start with the primitive notion of a set } of preference orderings over the universal set of options. The specific interpretation of } that we adopt is that it is the set of all possible orderings that a reasonable person can have in the agent’s situation. At the risk of emphasizing the obvious, it may be worth clarifying further our interpretation of the set }. As we have said, we interpret } as the set of all preference orderings that a reasonable person in the agent’s situation can possibly have. Under this interpretation, if a preference ordering R over the relevant options does not belong to }, then it will mean that we cannot conceive of any reasonable person who, when placed in the agent’s circumstances, will have the ordering R. Thus, in this case, having R as one’s preference ordering in the agent’s situation is intuitively inconsistent with our prior notion of the reasonableness of a person. On the other hand, given our interpretation of }, if an ordering R0 belongs to }, then we can conceive of some reasonable person who, in the agent’s circumstances, will have the ordering R0 . Essentially, this means that having R0 as one’s preference ordering in the agent’s situation is intuitively consistent with our prior notion of the reasonableness of a person. In general, given the situation of the agent, there can be more than one preference ordering that is intuitively consistent with our prior notion of the reasonableness of a person. Therefore, in general, the set } may contain more than one ordering. We seek to explore the relationship between the intrinsic value of freedom and the set } interpreted as above. This relationship seems to be of considerable analytical interest. But we do recognize that, given the complexity of the notion of freedom, there may be conceptual links between preferences interpreted in other ways and one’s intuition about the intrinsic value of freedom. For example, alterna-

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tive interpretations of } can be as the set of all preference orderings R such that some individual in the society has the ordering R, or as the set of all preference orderings R0 such that there is a positive probability of R0 emerging as the ordering of some individual in the society in the future. We have chosen not to pursue these and several other interpretations in this paper. However, we believe that much of the formal analysis in the subsequent sections is compatible with many of these other interpretations.

3. THE BASIC NOTATION AND DEFINITIONS

Let X be the universal set of alternatives, assumed to be finite. One can think of several possible interpretations of the alternatives in X . For example, these alternatives can be thought of as ordinary commodity bundles. They may be interpreted as bundles of relevant characteristics of the commodities, in the sense of Lancaster (1966) and Gorman (1959), or as bundles of functionings in the sense of Sen (1985, 1987). Finally, the alternatives may have dimensions which refer to non-economic aspects of the individual’s life – his religion, expression of a specific political belief etc. At any given time, the set of all alternatives available to the individual will be a non-empty subset of X , and he has to choose exactly one alternative from this set of available alternatives. The individual’s freedom is seen as his freedom to choose from this set. Let Z be the set of all non-empty subsets of X . The elements of Z are the alternative feasible sets with which the agent may be faced. Let  be a binary relation defined over Z . For all A; B 2 Z; [A  B ] will be interpreted as ‘A offers at least as much freedom as B ’. For all A; B 2 Z , [A  B iff A  B and not (B  A)] and [A  B iff A  B and B  A]. Let } = fR1 ; . . . ; Rn g denote the reference set of preference orderings over X (a preference ordering over X is a reflexive, complete and transitive binary weak preference relation (‘at least as good as’) over X ). } will be interpreted as the set of all possible preference orderings over X , that a reasonable person may have. For all R 2 }, let P and I be, respectively, the asymmetric factor and the symmetric factor of R. For all A 2 Z , let max(A) denote the set of all a 2 A such that a is an R-greatest element in A for some R 2 }.

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Thus, max(A) is the set of all alternatives x in A such that x is a best alternative in A for some ordering in }. For all x 2 X and all A 2 Z,

x[I ]A iff max(A [ fxg) = A [ fxg; x[P ]A iff fxg = max(A [ fxg); A[P ]x iff x 62 max(A [ fxg): Thus, x[I ]A if and only if every alternative in A [ fxg is a best alternative in A [ fxg in terms of some ordering in }. Given our interpretation of }, this amounts to saying that x[I ]A if and only if every alternative in A [ fxg is a best alternative in A [ fxg in

terms of a preference ordering of a reasonable person in the agent’s situation. x[P ]A means that x is ranked strictly above all alternatives in A fxg in terms of every ordering that a reasonable person in the agent’s situation may possibly have. Finally, A[P ]x if and only if, for every ordering that a reasonable person can have in the agent’s situation, some alternative in A comes strictly higher than x. Remark 3.1. Note that, for all A; A0 2 max(A), then max(A0 ) = A0 ]; and [if a 2 max(A); a[I ]A0 ].

Z , [max(A) 6= ;]; [if A0  A0  max(A), then, for all

4. SOME PROPERTIES OF



In this section we consider a number of properties of the binary relation  over Z . These properties are defined in Definitions 4.1, 4.2 and 4.3 below. The first of these definitions, Definition 4.1, introduces three properties each of which deals with the ranking of singleton opportunity sets. DEFINITION 4.1. A binary relation  over Z satisfies: (4.1.1) indifference of no-choice situations (INS) iff, for all X; fxg  fyg;

x; y 2

(4.1.2) simple non-dominance (SND) iff, for all x; y 2 X , if x[I ]fy g, then fxg  fy g;

(4.1.3) simple dominance (SD) iff, for all x; y fxg  fyg.

2 X , if x[P ]fyg, then

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INS was discussed first by Jones and Sugden (1982) and subsequently by Pattanaik and Xu (1990).2 The intuition advanced for INS was that, since a singleton opportunity set did not offer any choice at all, if one was interested exclusively in the intrinsic value of freedom, then there was little to distinguish between one singleton opportunity set and another, though, in terms of indirect utility, one singleton opportunity set might be vastly superior to another singleton opportunity set. INS has been the subject of considerable criticism in the literature (see Sen (1991, 1993) and Foster (1992)). Sen, for example, argued that, if x is a much superior option than y , then fxg offers more freedom than fy g. SND and SD are formulated in response to Sen’s criticism of INS. Unlike INS, which stimulates that all singleton sets offer the same degree of freedom, SND stipulates that, if a reasonable person may consider x to be at least as good as y , and, further, if a reasonable person may also consider y to be at least as good as x, then fxg offers the same degree of freedom as fy g. INS implies but is not implied by SND. However, INS is incompatible with SD, which stipulates that, if a reasonable person will necessarily consider x to be better than y , then fxg offers more freedom than fy g. Therefore, the question naturally arises as to which of these two axioms correspond to our intuition about freedom. Here the intuitions of different people seem to differ sharply. In contrast to Sen (1991, 1993), Jones and Sugden (1982) feel that INS is a very reasonable condition, and so do we in the context of the intrinsic value of freedom. Given the complexity of the notion of freedom, such difference is to be expected at this early stage of the development of the formal analysis of freedom. Perhaps, at this stage, the best course of action open to the theorist is to explore formally the implications of different axioms or sets of axioms with the hope that this will lead to further sharpening of our intuition. Therefore, we shall consider INS as well as SND and SD, SND and SD being the product of our attempt to accommodate, within our general framework, Sen’s criticism of INS. Our next definition introduces four properties. Under each one of these, given a certain ranking of two opportunity sets G and H and given some suitably specified alternative g , a restriction is imposed on the ranking of G [ fg g and H . DEFINITION 4.2. A binary relation  over Z satisfies:

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(4.2.1) [I ]-monotonicity (IM) iff, for all A, B 2 Z and all X A, (x[I ]A and A  B ) implies [A [ fxg  B ];

x2

(4.2.2) weak [I ]-monotonicity (WIM) iff, for all A, B 2 Z and all x 2 X A, (x[I ]A and not (B [P ]x) and A  B ) implies [A [ fxg  B ]; (4.2.3) type 1 irrelevance of dominated alternatives (IDA-1) iff, for all A, B 2 Z and all x 2 X , if A[P ]x, then [A  B iff A [ fxg  B ] and [B  A iff B  A [ fxg]; (4.2.4) type 2 irrelevance of dominated alternatives (IDA-2) iff, for all A, B 2 Z and all x 2 X , if A[P ]x, then [A  B iff A  B [ fxg] and [B  A iff B [ fxg  A]. The two monotonicity properties in Definition 4.2 are related, in spirit, to Pattanaik and Xu’s (1990) axiom of ‘strict monotonicity’ (SM) which requires that, for all x; y 2 X , fx; y g offers strictly more freedom than each of the singleton sets fxg and fy g. As we noted in Section 2, SM has been questioned in the literature: Sen’s (1993) example, where the opportunity set is expanded by adding the option of being beheaded at dawn, was directed against the intuition underlying SM. However, as we also argued in Section 2, Sen’s example seems convincing because of the presumption that a reasonable person will never consider the new option (‘being beheaded at dawn’) to be at least as good as the options that existed to start with. If we remove this presumption, then it seems plausible to claim that the expanded opportunity set will be generally regarded as offering more freedom than the original opportunity set. Our IM is closely related to this intuitive claim, though the formal structure of IM is somewhat different. Under IM, if A offers at least as much freedom as B and if we add to A an alternative x 2 X A such that, for every alternative a in A [ fxg, a can be a best alternative in A [ fxg for a reasonable person, then A [ fxg offers strictly more freedom than B . Note that, intuitively, IM is a very weak property in so far as its hypothesis includes the stringent stipulation that every alternative in A [fxg is best in A [fxg in terms of some ordering in }. The intuition underlying IM can be further clarified by considering the simple case where the opportunity set of an individual contains only the option of spending the rest of his life in a solitary cell. Now suppose we add to this set the option of being beheaded at dawn.

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Does this increase the individual’s freedom? IM’s answer to this question is a conditional ‘yes’: if a reasonable person can consider being beheaded at dawn to be at least as good as spending the rest of his life in a solitary cell, and, further, if a reasonable person can consider spending the rest of his life in a solitary cell to be at least as good as being beheaded at dawn, then IM will declare that the expanded opportunity set offers strictly more freedom than the original opportunity set. WIM weakens IM by introducing in the hypothesis of IM the additional requirement that the alternative x that is added to A must be at least as good as all alternatives in B , in terms of some ordering in } (that is, in terms of the ordering of some reasonable person). Definitions 4.2.3 and 4.2.4 give two different versions of ‘irrelevance of dominated alternatives’. IDA-1 requires that, if x is such that, for every possible preference ordering of a reasonable person, at least one alternative in A will be ranked strictly above x, then the ranking of A [fxg and B must be exactly similar to the ranking of A and B . The intuition of IDA-1 is clear: if, in terms of every possible preference ordering of a reasonable person, x is strictly worse than at least one alternative in A, then adding x to A does not add to the agent’s freedom. Intuitively, IDA-2 requires the following: if x is ‘dominated’ by A in the sense that, for every possible preference ordering of a reasonable person, x is ranked strictly below at least one alternative in A, then the relative status of B [ fxg vis-`a-vis A should not be better than the relative status of B vis-`a-vis A; nor should it be worse than the relative status of B vis-`a-vis A. The intuition of IDA-2 is less transparent than that of IDA-1, though IDA-2 has some degree of plausibility. Our next definition, Definition 4.3, involves two properties which are intuitively related to the ‘independence’ axiom of Pattanaik and Xu (1990) and also to a property first introduced by Sen (1991). DEFINITION 4.3. A binary relation  over Z satisfies:

(4.3.1) composition (COM) iff, for all A; B; C; D 2 Z , such that (A \ C = B \ D = ; and max(A [ C ) = A [ C and max(B [ D) = B [ D), [A  B and C

 D] implies [A [ C  B [ D]; [A  B and C  D ] implies [A [ C  B [ D ];

and

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(4.3.2) weak composition (WCOM) iff, for all A; B; C; D 2 Z , such that (A \ C = B \ D = ; and max(A [ C ) = A [ C and max(A [ D ) = A [ D and max(B [ C ) = B [ C and max(B [ D) = B [ D), [A  B and C  D ] implies [A [ C  B [ D ]; and [A  B and C  D ] implies [A [ C  B [ D ]: COM is a weaker version of an axiom originally introduced by Sen (1991). To see the significance of COM, it may be helpful to start with a crude version of this property and show how successive refinements of this crude version lead us first to Sen’s axiom and then to COM. Consider four sets A; B; C and D . Prima facie, it seems plausible to say that, if A offers at least as much freedom as B , and C offers at least as much freedom as D , then A [ C offers more freedom than B [ D. However, a difficulty immediately arises in connection with this. Suppose A and C have a lot of elements in common while B and D have no common element. Then, adding the elements of C to A may not increase the agent’s freedom much, while adding the elements of D to B may increase his freedom considerably, in which case we may have B [ D  A [ C even though A  B and C  D . Sen’s axiom removes this difficulty by postulating that A and C have no common element and that B and D also have no common element. Thus, Sen’s axiom requires that, given A \ C = B \ D = ;, if [A  B and C  D], then [A [ C  B [ D], and, if [A  B and C  D], then [A [ C  B \ D]. However, there seems to be an intuitive difficulty even with Sen’s axiom. Suppose A = fx; yg, B = fz; wg, C = fx0 ; y0g and D = fz0 ; w0g and suppose the elements of these four sets are all distinct. Let } contain exactly two orderings R and R0 such that xPx0 Py 0PyPzPwPz 0Pw 0 and w 0 P 0 z 0 P 0 wP 0zP 0 yP 0 xP 0 y 0 P 0 x0 . This, of course means that a reasonable person will never consider either x0 or y 0 to be at least as good as x. On the other hand, if we consider x0 ; y 0 ; z 0 and w 0 , every one of these four alternatives can be considered by a reasonable person to be at least as good as the other three. In view of this, we can justifiably feel that adding x0 and y 0 to the set fx; y g does not result in any significant increase in freedom while adding z 0 and w 0 to the set fz; wg significantly increases the degree of freedom. In that case, we may be unwilling to accept that [A [ C  B [ D ] even if [A  B and C  D ]. Our axiom COM handles this problem of Sen’s

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axiom by restricting the applicability of the axiom to the case where every alternative in A [ C can be considered by a reasonable person to be at least as good as all alternatives in B [ D. Pursuing this line of thinking a step further, WCOM weakens COM by postulating further restrictions regarding A; B; C and D: the applicability of WCOM is limited only to those cases where, besides the restrictions imposed by COM on A; B; C and D, we have the further restriction that every alternative in A [ D can be considered by a reasonable person to be at least as good as all alternatives in A [ D, and every alternative in B [ C can be considered by a reasonable person to be at least as good as all alternatives in B [ C . 5. THE RESULTS

We first explore the implications of INS, IM, IDA-1 and COM: we show that these four conditions characterize a freedom ranking rule under which opportunity sets are ranked on the basis of the number of alternatives in an opportunity set, each of which is considered best in that opportunity set by a reasonable person. PROPOSITION 5.1. only if for all A; B

 satisfies INS, IM, IDA-1 and COM if and

2 Z; [A  B iff # max(A)  # max(B )]:

(5.1)

Proof. The necessity part of the proposition is obvious; we prove only the sufficiency part. Let  satisfy INS, IM, IDA-1 and COM. First, we show:

2 Z; if # max(A) = # max(B ); thenA  B: (5.2) Suppose A; B 2 Z and # max(A) = # max(B ) = g . Let max(A) = fa1; . . . ; ag g and max(B ) = fb1; . . . ; bg g. By INS, fa1g  fb1 g (5.3) for all A; B

and

fa2g  fb2 g:

(5.4)

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fa1g \ fa2 g = fb1 g \ fb2 g = ; and, further, max(fa1; a2 g) = fa1; a2g and max(fb1; b2 g) = fb1; b2 g (since a1; a2 2 max(A) and b1 ; b2 2 max(B )). Hence, by (5.3), (5.4) and COM, we have fa1; a2 g  fb1 ; b2g: (5.5) By INS, again,

fa3g  fb3 g:

(5.6)

By (5.5), (5.6) and COM,

fa1; a2 ; a3g  fb1 ; b2 ; b3g:

(5.7)

A  max(B ):

(5.8)

Proceeding in this fashion, finally, we have fa1 ; . . . ; ag g  fb1 ; . . ., bg g, that is, max(A)  max(B ). If A = max(A), then A  max(B ). Suppose A max(A) 6= ;. Let A max(A) = fa1 ; . . . ; am g 6= ;. It is clear that max(A)[P ] a1, max(A) [ fa1 g[P ]a2; . . . ; A famg[P ]am . Hence, by the repeated use of IDA-1, we have A  max(B ) in this case. Thus, in all cases,

Similarly, by the repeated use of IDA-1, from (5.8), we can derive A  B , which proves (5.2). Next, we show:

2 Z; if # max(A) > # max(B ); then A  B: (5.9) Suppose A; B 2 Z and # max(A) > # max(B ). Let # max(B ) = for all A; B

g and # max(A) = g + t (where t > 0). Further, let max(B ) = fb1; . . . ; bg g and max(A) = fa1; . . . ; ag ; . . . ; ag+t g. Note that, max(A) = fa1 ; . . . ; ag ; . . . ; ag+t g, max(B ) = fb1 ; . . . ; bg g, and max(fa1 ; . . . ; ag g) = fa1 ; . . . ; ag g. Hence, by (5.2), fa1; . . . ; ag g  B: (5.10) Since max(A) = fa1 ; . . . ; ag ; ag+1 ; . . . ; ag+t g, it is clear that ag+1 [I ] fa1; . . . ; ag g. By IM and (5.10), it follows that fa1; . . . ; ag+1g  B: (5.11) Taking (5.11), adding ag+2 ; . . . ; ag+t (one at a time) on the left hand side, and using IM repeatedly, we have

fa1; . . . ; ag+tg  B:

(5.12)

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Taking(5.12) and using reasoning similar to the reasoning used to prove (5.8), by IDA-1, we have A  B , which proves (5.9). (5.2) and (5.9) complete the proof of the sufficiency part of the proposition.

2

Remark 5.2. After we derived Proposition 5.1, we came to know of an elegant, unpublished result, proved independently by Kunal Sengupta, which is in some ways related to our Proposition 5.1. Using a notion of an equivalence class of alternatives, Sengupta provides a characterization of the following two-element class of freedom ranking rules: either [A  B for all A; B 2 Z ] or [for all A; B 2 Z; A  B iff the number of equivalence classes in max(A) is not smaller than the number of equivalence classes in max(B )]. Remark 5.3. In Proposition 5.1, we do not assume  to be complete or transitive, through completeness and transitivity of  follow as consequences of our assumptions (note that the ranking rule given by (5.1) yields an ordering regardless of how } is specified). It can be shown that, if  is assumed to be transitive, then INS and COM and the properties given by (5.13) and (5.14) below characterize the ranking rule in (5.1):

2 X; if x[I ]fyg; then fx; yg  fyg: (5.13) (5.14) for all x; y 2 X; if fxg[P ]y; then fx; y g  fxg: Note that, in the presence of the reflexivity of  (which is implied for all distinct x; y

by INS), (5.13) is much weaker than IM and (5.14) is much weaker than IDA-1. (It should be noted that (5.13) and (5.14) are the two most direct extensions of SM.)

Remark 5.4. If } happens to be the set of all possible preference orderings over X (that is, if every possible preference ordering could be held by a reasonable person), then the ranking rule defined by (5.1) coincides with the following ranking rule: for all A; B

2 Z; A  B iff #A  #B:

(5.15)

(5.15) is the rule that was originally characterized by Pattanaik and Xu (1990) using a framework where preferences did not play any

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rule in judgements about freedom. On the other hand, if } happens to contain only one ordering and that ordering is linear (that is, antisymmetric), then the binary relation  defined by (5.1) is such that: for all A; B 2 Z; A  B . Proposition 5.1 uses INS which, as we have mentioned earlier, has drawn considerable criticism from several writers. Now we consider SD and SND which, we believe, are likely to be more acceptable to many people who object to INS. We show that SD, SND and COM together yield a contradiction if } satisfies Assumption 5.5 below.

ASSUMPTION 5.5. There exist x; y; z 2 X such that, [for all R 2 }; xPy], and [for some R0; R00 2 }; xP 0yP 0z and zP 00 xP 00y].

Remark 5.6. Suppose Z  Rn , where the first two dimensions of each vector in X refer to attributes or commodities that every reasonable person in the agent’s situation will find desirable (that is, given the agent’s situation, not finding these attributes or commodities desirable is inconsistent with our notion of the reasonableness of a person). Thus, the first dimension may be some index of better health and the second dimension may be the extent of harmony in one’s family. Suppose x1 ; x2 ; x3 2 X are such that for all k 2 f3; . . . ; ng; x1k = x2k = x3k ; x31 > x11 > x21 ; and x12 > x22 > x32 . Then, while a reasonable person will necessarily consider x1 better than x2 , each of the possibilities, [x3 Px1 ] and [x2 Px3 ], may be consistent with our notion of a reasonable person. Assumption 5.5 will then be satisfied. PROPOSITION 5.7. Suppose Assumption 5.5 is satisfied. Then no reflexive binary relation  over Z exists that satisfies SD, SND and COM. Proof. Suppose  is a reflexive binary relation over Z satisfying SD, SND and COM. Let x; y; z 2 X be such that, for all R 2 }; xPy; and for some R0 ; R00 2 }; xP 0 yP 0 z and zP 00 xP 00 y . Then, by SD,

fxg  fyg:

(5.16)

By SND,

fxg  fzg

(5.17)

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and

fzg  fyg:

(5.18)

fzg  fzg:

(5.19)

fx; zg  fy; zg:

(5.20)

By the reflexivity of , Note that [max(fxg[fz g) = fx; z g and max(fy g[fz g) = fy; z g]. Hence, from (5.16) and (5.19), by COM, it follows that

Similarly, from (5.17) and (5.18), by COM, we have

fx; zg  fy; zg

(5.21)

(5.20) and (5.21) contradict each other, which completes the proof.

2

Remark 5.8. Given Assumption 5.5, not only do SD and SND together conflict with COM, but they are also inconsistent with the transitivity of . This is clear from the proof of Proposition 5.7, where, using Assumption 5.5, SD and SND we derived (5.16), (5.17) and (5.18), which together contradict the transitivity of . However, transitivity of  may be too demanding a property in our context: while we do not have any general result to confirm this, it is our conjecture that most rules for ranking opportunity sets that take into account a non-singleton reference set of preference orderings will violate transitivity of . While Proposition 5.7 shows that SD, SND and COM cannot be satisfied if the reference set of orderings, }, satisfies Assumption 5.5, SD and SND are compatible with WCOM even when no restriction is placed on }. In fact, as Proposition 5.9 below shows, SD, SND, WIM, IDA-1, IDA-2 and WCOM characterize a ranking rule of some interest. In ranking two opportunity sets A and B under this rule, one proceeds in the following way. First, one eliminates from A every alternative x in A such that either a reasonable person never considers x to be best in A [ fxg or a reasonable person never considers x to be best in B [ fxg. Similarly, one eliminates from B every alternative y in B such that either a reasonable person never

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considers y to be best in B [ fy g or a reasonable person never considers y to be best in A [ fy g. Next, one ranks A and B on the basis of the cardinalities of the reduced sets that one is left with after such elimination of alternatives from A and B , respectively. To present our next result, for all A; B 2 Z , let AB be the set of all a 2 A such that B [P ]a. Intuitively, AB refers to all those elements a of A such that a reasonable person may never consider a to be at least as good as all the elements of B . Note that AB is not necessarily identical with B A . PROPOSITION 5.9. A binary relation  over Z satisfies SD, SND, WIM, IDA-1, IDA-2 and WCOM iff for all A; B 2 Z; A  B iff #[max(A)  #[max(B ) B A]:

AB ]

(5:22)

Proof. Necessity can be checked easily. We prove only sufficiency. Let  satisfy SD, SND, WIM, IDA-1, IDA-2 and WCOM. First, we show that for all A; B 2 Z; if #[max(A) AB ] = #[max(B ) B A ]; then A  B: (5:23)

Suppose #[max(A) AB ] = #[max(B ) B A] = t. It can be checked that t 6= 0. Let (max(A) AB ) = fa1 ; . . . ; at g and (max(B ) B A ) = fb1 ; . . . ; bt g. By the definition of (max(A) AB ) and (max(B ) B A ) it follows that for all a 2 (max(A) AB ); all b 2 (max(B ) B A ); all A0  [max(A) AB ] and all B 0  [max(B ) B A ]; [max(A0 [ fag) = A0 [ fag and max(B 0 [ fag) = B 0 [ fag and max(A0 [ fbg) = A0 [ fbg and max(B 0 [ fbg) = B 0 [ fbg]:

(5:24)

By (5.24) and SND,

faig  fbig

for all

i 2 f1; . . . ; tg:

(5.25)

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Since, by (5.25), fa1 g  fb1 g and fa2 g  fb2 g, by (5.24) and WCOM, fa1 ; a2 g  fb1 ; b2 g. Again, since, by (5.24), fa3 g  fb3 g, given fa1 ; a2 g  fb1 ; b2 g, we have, by (5.24) and WCOM, fa1 ; a2 , a3g  fb1 ; b2; b3 g. Thus, by repeated application of (5.25), (5.24) and WCOM, we would have fa1 ; . . . ; at g  fb1 ; . . . ; bt g, that is: (max(A)

AB )  (max(B ) B A):

(5.26)

Then, by repeated application of IDA-2 followed by repeated application of IDA-1, from (5.26), we have [max(A)

AB ]  B:

(5.27)

Again, by repeated application of IDA-2 followed by repeated application of IDA-1, from (5.27), we have A  B . This completes the proof of (5.23). Now we prove that, for all A; B 2 Z; if #[max(A) AB ] > #[max(B ) B A ]; then A  B:

(5:28)

Suppose #[max(A) AB ] > #[max(B ) B A ]. First, consider the case where #[max(B ) B A ] = 0. It can be easily checked that, in this case, for all b 2 B and all a 2 max(A), max(A)[P ]b and not (fbg[P ]a). Let B = fb1 ; . . . ; bm g. Then, by SND and SD, for all a 2 max(A); fag  fb1 g. Note that for all a 2 max(A), a[I ] max(A) and not (fb1 g[P ]a). Then, by the repeated use of WIM, from fag  fb1 g, we have max(A)  fb1 g: From (5.29), noting [max(A)[P ]b for all b cation of IDA-2,

(5.29)

2 B ], by repeated appli-

max(A)  B:

(5.30)

fa1; . . . ; ag g  fb1 ; . . . ; bg g:

(5.31)

Again, by repeated application of IDA-1, from (5.30), we have A  B . Now consider the case where #[max(A) AB ] > #[max(B ) B A] > 0. Suppose max(B ) B A = fb1 ; . . . ; bg g and max(A) AB = fa1; . . . ; ag ; ag+1; . . . ; ag+hg. Then, following the method used to prove (5.23), we can show that

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By (5.31), (5.24) and the repeated application of WIM, we have fa1; . . . ; ag ; ag+1; . . . ; ag+hgfb1 ; . . . ; bg g, that is, (max(A) AB )  (max(B ) B A ). Then, by repeated application of IDA-1 and IDA-2, from (max(A) AB )  (max(B ) B A ), we have A  B . This completes the proof of (5.28). Given that we have proved (5.23) and (5.28), the proof of sufficiency is complete. 2

Remark 5.10. Suppose } contains only one ordering R and R is linear (that is, antisymmetric). It can be checked that, in this special case, the binary relation  defined by (5.22) is such that for all A; B

2 Z; A  B iff [max(A)R max(B )]:

(5.32)

(5.32) is, of course, the traditional rule based on the comparison of indirect utilities. Now consider another extreme case where } is the set of all possible orderings over X . In this special case, it can be checked that the binary relation  defined by (5.22) takes the simple form given by (5.15), which was characterized by Pattanaik and Xu (1990).

Remark 5.11. In Proposition 5.9, neither completeness nor transitivity of  is assumed. Completeness follows as a consequence of our axioms:  defined by (5.22) is complete. But (5.22) need not necessarily yield a transitive ranking. Indeed, in general, depending on },  defined by (5.22) may violate even acyclicity of . This is shown by Example 5.12 below. However, it is easy to see that, in the two special cases of (5.22), mentioned in Remark 5.10, where  takes a form given by either (5.32) or (5.15),  is transitive in addition to being reflexive and complete. EXAMPLE 5.12. To see that  defined by (5.22) may violate acyclicity of , consider the following example. Let A = fa1 ; a2 ; a3 g,

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B = fb1 ; b2g, C = fc1; c2; c3g, and } = (R1; . . . ; R5 ) be such that R1 R2 R3 R4 R5 c3 c3 c3 b1 b2 a1 a2 a3 c1 c2 a2 a3 a1 c3 c3 a3 a1 a2 a1 a1 b1 b2 b1 a2 a2 b2 b1 b2 a3 a3 c1 c2 c1 b2 b1 c2 c1 c2 c2 c1 (in each ranking shown above, the elements are arranged in the strictly descending order of preference). Then max(A) = fa1 ; a2 ; a3 g, max(B ) = fb1 ; b2 g, max(C ) = fc1 ; c2 ; c3 g, AB = B A = B C = C A = ;, AC = fa1; a2 ; a3g, and C B = fc1; c2g. Therefore, max(A) AB = fa1; a2; a3g, max(B ) B A = fb1 ; b2g, max(C ) C A = fc1; c2; c3g, max(A) AC = ;, max(B ) B C = fb1; b2 g, max(C ) C B = fc3g. Consequently, we have the following ranking: A  B , B  C , and C  A. Therefore,  violates acyclicity. Remark 5.13. As may be recalled, our Proposition 5.7 shows that, in the presence of Assumption 5.5, SD, SND and COM are incompatible. At the same time, Proposition 5.9 shows that, SD and SND are compatible with WCOM, WIM, IDA-1 and IDA-2. In view of this, it is tempting (especially if one finds the intuition underlying SD and SND persuasive) to think that COM is too strong and should be relaxed. However, while WCOM (which is compatible with SND and SD) is formally weaker than COM (which is incompatible with SD and SND, given Assumption 5.5), intuitively, we do not find COM any less appealing than WCOM. Also, as our following proposition shows, we continue to get an ‘impossibility result’, if, in Proposition 5.7, we replace SD and COM by IM, IDA-1 and IDA-2. PROPOSITION 5.14. Suppose Assumption 5.5 is satisfied. Then no binary relation  over Z satisfies SND, IM, IDA-1 and IDA-2. Proof. Suppose  is a binary relation over Z satisfying SND, IM, IDA-1 and IDA-2. Let x; y; z 2 X be such that, for all R 2 },

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xPy, and for some R0, R00 2 }, xP 0yP 0z and zP 00 xP 00y. Then, by SND,

fxg  fzg: (5.33) Note that fxg[P ]y . Hence, from (5.33), by IDA-1, it follows that fx; yg  fzg: (5.34) Since y [I ]fz g, from (5.34), by IM, it then follows that fz; yg  fx; yg: (5.35) From (5.33), noting that fxg[P ]y , by IDA-2, it follows that fxg  fy; zg: (5.36) Then by IDA-1, recalling that fxg[P ]y , from (5.36), we have fx; yg  fy; zg: (5.37) (5.35) and (5.37) contradict each other, which completes the proof.

2

6. CONCLUDING REMARKS

We conclude this paper by briefly commenting on the works of some other authors, which are related to our approach to the problem of freedom and preferences. In an earlier paper (Pattanaik and Xu, 1990), we introduced the axioms of INS and SM and a third axiom called ‘Independence’. We showed that these three axioms together characterized the ranking rule under which opportunity sets were ranked on the basis of their cardinalities. Since this is a rather naive rule, the result had the flavor of an impossibility result. INS and SM have been criticized (see, for example, Sen, 1993) for ignoring the role of preferences in assessing the freedom of an individual. Sen (1991, 1992, 1993) has argued vigourously for taking account of preferences in arriving at judgements about freedom. Though Sen is not quite explicit about the issue as to whose preference should count in this context, he seems to be mainly concerned with the case where the relevant set of preference

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orderings is a singleton; presumably, the sole element of this set is the present preference ordering of the agent whose freedom is under consideration. Puppe (1996) also uses this framework where the reference set of preference orderings contains exactly one ordering. Our approach in this paper, where the reference set of orderings is not constrained to be a singleton, is close to the approach of Jones and Sugden (1982) and Foster (1992). However, our focus is somewhat different. Jones and Sugden’s (1982) incisive analysis concentrates on philosophical and conceptual issues relating to freedom, while our main concern has been to explore the formal implications of alternative sets of axioms. Foster (1992) concentrates on the properties and representability of the ‘unanimity ordering’  defined as follows: for all A; B 2 Z; A  B iff, for all R 2 }, aRb where a and b are R-greatest elements in A and B , respectively (recall that } is the reference set of orderings). ACKNOWLEDGMENTS

We are grateful to James Foster, Wulf Gaertner, Marlies KlemischAhlert, Amartya Sen, Kunal Sengupta and Robert Sugden for many helpful comments. Several improvements are due to the suggestions of a referee and Antonio Medina. NOTES 1. The notion of a reference set of preference orderings originates in Jones and Sugden (1982) and Foster (1992). 2. At the time of writing our earlier paper, Pattanaik and Xu (1990), we were not aware of the very important paper of Jones and Sugden (1982) and their formulation of INS. We are, of course, happy to acknowledge their priority in formulating this property. REFERENCES Arrow, K. (1995), A note on freedom and flexibility, in K. Basu, P.K. Pattanaik and K. Suzumura (eds.), Choice, Welfare and Development, A Festschrift in Honour of Amartya K. Sen. Oxford: Oxford University Press. Bossert, W., Pattanaik, P.K. and Xu, Y. (1994), Ranking opportunity set: An axiomatic approach, Journal of Economic Theory 63, 326–345.

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Debreu, G. (1959), Theory of Value. New York: Wiley. Dutta, B. (1990), Interest and advantage (mimeo). Foster, J. (1992), Notes on effective freedom (mimeo). Gorman, W.M. (1959), Separable utility and aggregation, Econometrica 27, 469– 481. Jones, P. and Sugden, R. (1982), Evaluating choice, International Review of Law and Economics 2, 47–65. Klemisch-Ahlert, M. (1993), A comparison of different rankings of opportunity sets, Social Choice and Welfare 10, 189–207. Kreps, D.M. (1979), A representation theorem for ‘Preference for Flexibility’, Econometrica 47, 565–577. Lancaster, K.J. (1966), A new approach to consumer theory, Journal of Political Economy 74, 132–157. Nozick, R. (1974), Anarchy, State and Utopia. Oxford: Blackwell. Pattanaik, P.K. and Xu, Y. (1990), On ranking opportunity sets in terms of freedom of choice, Recherches Economiques do Louvain 56, 383–390. Puppe, C. (1996), An axiomatic approach to ‘Preference for Freedom of Choice’, Journal of Economic Theory 68, 174–199. Sen, A.K. (1985), Commodities and Capabilities. Cambridge: Cambridge University Press. Sen, A.K. (1987), The standard of living (Lectures I and II), in G. Hawthorn (ed.), The Standard of Living. Cambridge: Cambridge University Press. Sen, A.K. (1991), Welfare, preference and freedom, Journal of Econometrics 50, 15–29. Sen, A.K. (1992), Inequality Reexamined. Cambridge, Mass.: Harvard University Press. Sen, A.K. (1993), Markets and freedoms, Oxford Economic Papers 45, 519–541. Steiner, H. (1983), How free: Computing personal liberty, in A. Phillips-Griffiths (ed.), Of Liberty. Cambridge: Cambridge University Press. Suppes, P. (1987), Maximizing freedom of decision: An axiomatic Analysis, in G.R. Feiwel (ed.), Arrow and the Foundations of Economic Policy. New York: New York University Press. Addresses for correspondence: Prasanta K. Pattanaik, Department of Economics, University of California, Riverside, CA 92521, USA Phone: (909) 787-5037; E-mail: [email protected] Yongsheng Xu, Department of Economics, University of Nottingham, University Park, Nottingham, NG7 2RD, UK Phone: +44 115 951-5481; E-mail: [email protected]

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