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Choice Functions and Revealed Preference" ,2

AMARTYA K. SEN University of Delhi

1. MOTIVATION The object of this paper is to provide a systematic treatment of the axiomatic structure

of the theory of revealed preference. In particular it is addressed to the following problems in revealed preference theory.

(1) Much of revealed preference theory has been concerned with choices restricted to certain distinguished subsets of alternatives, in particular to a class of convex polyhedra (e.g., " budget triangles " in the two commodity case). This restriction may have some rationale for analyzing the preferences of competitive consumers, but it makes the results unusable for other types of choices, e.g., of government bureaucracies, of voters, of con- sumers in an imperfect market. If the restriction is removed, the axiomatic structure of revealed preference theory changes radically. This axiomatic structure is studied in Sections 2-5. In Section 6 the rationale of restricting the domain of choice functions and that of rationality conditions is critically examined.

(2) While some revealed preference theories are concerned with element-valued choice functions (i.e., with choice functions the range of which is restricted to unit sets), others assume set-valued choice functions. It is interesting to analyze the problem generally in terms of set-valued choice functions and then study the consequence of an additional restriction that all choice sets be unit sets. Section 7 is devoted to this additional restriction.

(3) While revealed preference theory has been obsessed with transitivity, certain weaker requirements have come to prominence in other branches of choice theory. It is interesting to investigate the conditions that guarantee that a choice function is representable by a binary relation of preference whether or not that relation is transitive. Also the conditions that ensure transitivity of strict preference though not necessarily of indifference are worth studying because of the relevance of this case to demand theory (see Armstrong [1], Majumdar [14], Luce [13], Georgescu-Roegen [7]) and to the theory of collective choice (see Pattanaik [15], Sen [18], Inada [11], Fishburn [4]). The axiomatic structure of these requirements is studied in Sections 8-10. This also helps to achieve a factorization of the conditions for full transitivity.

2. CHOICE FUNCTIONS AND BINARY RELATIONS Let X be the set of all alternatives. For any subset S of X, a " choice set " C(S)

represents the chosen elements of S. A " choice function " is a functional relation that specifies a choice set C(S) for any S in a particular domain K of non-empty subsets of X. We can represent a choice function as C(.), or more loosely as C(S) taking S as a variable within K.

1 First version received March 1970; final version received Nov. 1970 (Eds). 2 For comments and criticisms I am most grateful to Hans Herzberger and Roy Radner.

307

308 REVIEW OF ECONOMIC STUDIES Definition 1. A function C(.) that specifies a non-empty choice set for every non-

empty set S in K is called a choice function defined over K. It will be assumed for the moment that K includes all finite subsets of X. This assump-

tion will be examined further in Section 6. There are many alternative ways of generating binary relations of preference from

any choice function. Three different ones will now be introduced. The first is that x is " at least as good as " y if x is chosen when y is available. This is to be denoted xRy. Strict preference (P) and indifference (I) are defined correspondingly. For all x, y in X:

Definition 2. xRy if and only if for some S in K, x E C(S) and y E S. Definition 3. xPy if and only if xRy and not yRx. Definition 4. xIy if and only if both xRy and yRx. A second interpretation corresponds to Uzawa's [19] and Arrow's [2] definition of a

" relation generated " by the choice function in terms of choice over pairs, and we shall say xRy if x is chosen (not necessarily uniquely) in a choice over the pair [x, y]. For all x, y in X:

Definition 5. xRy if and only if x E C([x, y]). Definition 6. xPy if and only if xRy and not yRx. Definition 7. xly if and only if both xRy and yRx. A third interpretation corresponds to what Arrow [2] calls " revealed preference" P.

We say xPy if x is chosen when y is available and rejected. We define R and I correspond- ingly. For all x, y in X:

Definition 8. xPy if and only if there is some S in K such that x E C(S) and y E [S- C(S)].

Definition 9. xRy if and only if not yPx. Definition 10. xIy if and only if xRy and yRix. The contrast between the definitions is illustrated in terms of an example. Example 1. x = C([x, y]), y = C([y, z]),

x = C([x, z]), y = C([x, y, z]). It follows from the definitions that:

(1) xIy, yPz, and xPz;

(2) xPy, yPz, and xPz; (3) xPy, yPz, xPz, and yPx. All the interpretations have some problem. R has the problem that x and y are treated

as indifferent even though x is chosen and y rejected over the pair [x, y]. R has, on the other hand, the problem that x is declared as strictly preferred to y, even though y is chosen and x rejected in the choice over the triple [x, y, z]. However, R involves the problem that P is not asymmetric, and x is declared preferred to y and y to x. Another way of viewing the problem in (3) is that R is not " complete " over [x, y}. On the other hand, R and R are always complete if K includes all finite subsets of X, or even if it only includes all pairs in X.

It can be established that for any C(S), (R = R) if and only if (P = P & I = I), (K = R) if and only if (PP = = I), and (R = R) if and only if (P = P & I = I). The proofs are straightforward.

SEN CHOICE FUNCTIONS 309

3. IMAGE AND NORMALITY Corresponding to each choice function C(S) we may define its " image " C(S) as the

choice function generated by the binary relation R revealed by C(S). Definition 11. For any S in K, (S) = [x I x E S and for all y in S, xRy]. That is C(S) consists of the " best " elements of S in terms of the relation R. Clearly,

C(S) c(S), since x E C(S) implies that for all y in S, xRy. But the converse may not hold as is clear from Example 1.

Definition 12. A choice function C(S) is normal if and only if C(S) = (S) for all S in K.

It is clear that a choice function being normal is equivalent to its being essentially binary in composition. Further, for a normal choice function, R = R. But R = R does not imply that the choice function is normal. The last is clear from the following example.

Example 2. [x] = C([x, y]), [x, z] = C([x, z]), [z] = ([y, z]),

[x] = C([x, y, z]). This implies xPy, xIz, zPy, and xPy, xlz, zPy. But [x, z] = ([x, y, z]). Note also that a normal choice function does not guarantee that R = R, thought the converse will be shown to be true (see T.3).

4. AXIOMS OF REVEALED PREFERENCE AND CONGRUENCE The concept of indirect revealed preference discussed by Houthakker [9] makes use

of the finite closure of P. Indirect revealed preference in the " wide " sense (see Richter [16]) is based on the finite closure of R.

Definition 13. For any pair x, y in X, x is indirectly revealed preferred to y (denoted xP*y) if and only if there exists in X a sequence xi, i = 0, ..., n, such that x0 = x, xn = y, and for all i = 1, ..., n, xi- 1Pxi.

Definition 14. For any pair x, y in X, x is indirectly revealed preferred to y in the wide sense (denoted xWy) if and only if there exists in X a sequence xi, i = 0, ..., n, such that x? = x, x' = y, and for all i = 1, ... n, x- 1Rx'.

The following conditions of rationality have been much discussed in the literature. For all x, y in X: WEAK AXIOM OF REVEALED PREFERENCE (WARP):

If xPy, then not yRx. STRONG AXIOM OF REVEALED PREFERENCE (SARP):

If xP*y, then not yRx. STRONG CONGRUENCE AXIOM (SCA):

If xWy, then for any S in K such that y E C(S) and x E S, x must also belong to C(S). WARP and SARP were respectively proposed by Samuelson [17] and by Houthakker

[9], Ville [20] and von Neumann and Morgenstern [21], adapted here as in Arrow [2] to correspond to set-valued choice functions. SCA is Richter's [16] " Congruence Axiom ", renamed to permit a weak version of it to be proposed, which is done below. WEAK CONGRUENCE AXIOM (WCA):

If xRy, then for any S in K such that y E C(S) and x E S, x must also belong to C(S). (T. 1). The Weak Congruence Axiom implies that the revealed preference R is an

ordering, and if the choice function is normal then the converse is also true.

310 REVIEW OF ECONOMIC STUDIES

Proof Since the domain of the choice function includes all finite sets, R must be complete and reflexive. To prove that WCA implies the transitivity of R take any triple T = [x, y, z] such that xRy and yRz. In view of WCA if z E C(T), then y E C(T), and if y E C(T) then x E C(T). But at least one of x, y and z must be in C(T). Hence x E C(T) and thus xRz. R is, therefore, transitive.

To show the converse, for any S in K, let y E C(S), x E S and xRy. Clearly, yRz for all z in S, and in view of xRy and the transitivity of R, xRz for all z in S. Thus x E C(S). And since the choice function is normal, x E C(S). Hence WCA holds.

While WCA seems necessary and sufficient for an ordinal preference structure, the same result is guaranteed by the convergence of R and .

(T.2). The Weak Congruence Axiom holds if and only if R = R.

Proof. Let WCA be violated. Evidently for some S in K, for some x, y E S, while y E C(S) and xRy, x does not belong to C(S). Obviously yPx, so that not xRy. Thus R # R, and therefore R = R implies WCA.

To show the converse it is noted that xRy implies not yFx, which guarantees that x E C([x, y]) and therefore xRy. On the other hand, not xAy implies y.Px so that for some S in K, we have y E C(S) and x E [S- C(S)]. If it is now assumed that xRy, then WCA will be violated, and therefore not xRy. Thus WCA implies R = R.

5. EQUIVALENCE OF AXIOMS This equivalence can be extended to cover all the " rationality" conditions proposed

so far. (T.3). The following conditions are equivalent:

(i) R is an ordering and C(S) is normal; (ii) R is an ordering and C(S) is normal; (iii) Weak Congruence Axiom; (iv) Strong Congruence Axiom; (v) Weak Axiom of Revealed Preference; (vi) Strong Axiom of Revealed Preference;

(vii) R = R; and (viii) R = R and C(S) is normal.

Proof. The equivalence of (i) and (ii) is immediate since C(S) being normal implies R = R.

From (T. 1), (i) implies (iii) and (iii) implies that R is an ordering. The equivalence of (i) and (iii) is completed by showing that WCA guarantees the normality of the choice function. Since C(S)c= (S), it is sufficient to show that WCA implies that C(S)(c C(S). Let y E C(S) while x E (S). Obviously xRy, and hence by WCA, x E C(S).

By definition (iv) implies (iii). To prove the converse assume that WCA holds and the antecedent of SCA holds, i.e., y E C(S), x E S, and xWy. Since y E C(S), yRz for all z in S. Using the notation of Definition 14, if x" 'Py, then xn-l 'Rz for all z in S since R is transitive given WCA. Thus xn-1 E Q(s'), where S' is the union of S and the unit set [xn-1]. Hence xn 'Py implies that xn-1 E C(S'). If, on the other hand, yIxn 1, then y e Q(S') since y eC(-S). Thus by normality y c C(S), and by WCA, xn' -e C(S1). Similarly xn-2 E C(S2) where S2 is the union of S' and the unit set [xn-2]. Proceeding this way x = x is in the choice set of s'. But if x is in C(S'), then x E C(S), since SC Sn. Hence x E C(S) and SCA must hold.

SEN CHOICE FUNCTIONS 311

Next the equivalence of (iii) and (v). If WARP is violated, then for some x, y E X, we have xPy and yRx. Since xPy implies that there is an S in K such that x E C(S), y E S, and y is not in C(S), WCA must be false in view of yRx. Thus WCA implies WARP. Conversely, let WCA be violated. Then for some x, y e S we have xRy, when y is in C(S) but x is not in C(S). Evidently xRy and yPx. This is a violation of WARP.

Next, (vi) is taken up. Obviously, (vi) implies (v), since SARP subsumes WARP. On the other hand, by (T. 1), WCA implies that R is an ordering. Hence WARP, which is equivalent to WCA, implies that R is an ordering. But by (T.2), R = R in view of WCA. Hence WARP guarantees that R is an ordering and that xP*y implies xPy for all x, y in X. But then WARP implies SARP.

Equivalence of (vii) and (iii) is given by (T.2) and that of (viii) and (vii) follows from the fact that R = R whenever C(S) is normal. This establishes (T.3).

This demonstration of the complete equivalence of all the rationality conditions pro- posed so far would seem to complete a line of enquiry initiated by Arrow [2] who proved the equivalence of (ii), (v) and (vi).' Some of the results contained in (T.3) have apparently been denied and it is worth commenting on a few of the corners. For example, Richter [16], who has proved the equivalence of SCA and (i), has argued that " it can be shown that the Congruence Axiom does not imply the Weak Axiom [of revealed preference] and hence not the Strong" (p. 639). But the difference seems to arise from the fact that Richter applies Samuelson's and Houthakker's definition of the revealed preference axioms, which are made with the assumption of element-valued (as opposed to set-valued) choice functions, to a case where there are a number of best elements in the set (Richter [16], Figure 2, p. 639). Similarly, Houthakker's [9] argument for the necessity of bringing in the Strong Axiom of Revealed Preference, rather than making do with the Weak Axiom, arises from the fact that Houthakker considers a choice function that is defined over certain distinguished sets only, viz., a class of convex polyhedra representing budget sets, and which is undefined over finite subsets of elements. The same is true of Gale's [6] demonstration that WARP does not imply SARP in a rejoinder to Arrow's paper [2].

Finally, it may be noted that rationality may be identified with the systematic converg- ence of different interpretations of the preference revealed by a choice function. As is clear from (T.3), R = R implies complete rationality in the sense of transitivity and normal- ity, and so does R = R if the choice function is normal. That a primitive concept like the coincidence of different interpretations of revealed preference can be taken to be a complete criterion of rationality of choice is of some interest in understanding this problematic concept.

6. THE DOMAIN OF THE CHOICE FUNCTION AND THE RATIONALITY AXIOMS

It is conventional to assume in revealed preference theory that the domain of the choice function includes only the class of convex polyhedras that represent " budget sets " in some real (commodity) space, e.g. " budget triangles " in the two-commodity case. In contrast it has been assumed here that the domain includes all finite subsets of X whether or not it includes any other subset. The difference is significant for Arrow's [2] result on the equivalence of the " weak " and the " strong " axioms of revealed preference and more generally for the set of equivalences established here in (T.3).

Evidently if there is some argument for confining the domain to the budget polyhedras it applies to the study of the competitive consumer and not to choices of other agents, e.g., a non-competitive consumer, a voter or a government bureaucracy. But does it make sense even for the competitive consumer? The question deserves a close examination.

1 Some weaker rationality conditions will be discussed in the following sections of the paper. See also Herzberger [8] for a vast collection of results involving some rationality axioms not covered here.

312 REVIEW OF ECONOMIC STUDIES

It is certainly the case that the observed behaviour of the competitive consumer will include choices only over budget sets. What can then be the operational significance, it might be asked, of including other subsets in the domain of the choice function? The significance lies in the fact that the interpretation of observed choices will depend on whether it is assumed that the rationality axioms would hold over other potential choice situations as well even though these choices cannot be observed in the competitive market. For example if it is assumed that the Weak Congruence Axiom would hold over finite subsets as well, then the observation that x is chosen when y is available in the budget set and y is chosen when z is in the budget set can be used as a basis for deducing that x is regarded as at least as good as z even without observing that x is chosen when z is available. This is because by (T.2) WCA defined over all finite subsets implies that R is transitive. Thus the interpretation of observed choices will vary with what is assumed about the applic- ability of the rationality axioms to unobserved choices.

Thus a real difference with operational significance is involved. But it may be argued that if some choices are never observed how can we postulate rationality axioms for these choices since we cannot check whether the rationality axioms assumed would hold? This problem raises important questions about the methodological basis of revealed preference theory. In particular the following two questions are relevant.

(1) Are the rationality axioms to be used only after establishing them to be true? (2) Are there reasons to expect that some of the rationality axioms will tend to be

satisfied in choices over " budget sets " but not for other choices? Suppose a certain axiom is shown to guarantee some rationality results if the axiom

holds over every element in a certain class of subsets of X, i.e., over some domain K* c K. In order to test this axiom before using, we have to observe choices over every element of K*. Consider K* as the class of budget sets. We know that any observed choice will be from K* in the competitive market, but this is not the same thing as saying that choice from every element of K* will, in fact, be observed. There are an infinite (and uncountable) number of budget sets even for the two-commodity case and choices only over a few will be observed. What is then the status of an axiom that is used in an exercise having been seen to be not violated over a certain proper subset of K* ? Clearly it is still an assumption rather than an established fact. There is, of course, nothing profound in this recognition, and it is in the nature of the theory of revealed preference that the exercise consists of taking axioms in the strictly logical sense and then deriving analytical results assuming these axioms to be true. But then the question arises: why assume the axioms to be true only for " budget sets " and not for others? Such results as the non-equivalence of the " weak" and the " strong " axioms relate to this issue.

This takes us to question (2). Are there reasons to expect the fulfilment of these axioms over budget sets but not over other subsets of K? No plausible reasoning seems to have been put forward to answer the question in the affirmative. The difference lies in the ability to observe violation of axioms and not in any inherent reason to expect violations in one case and not in the other. But even for K* not all choices will be observed Why then restrict the domain of an axiom to K* only and not to entire K when (a) verification is possible in fact neither for K nor for K*, and (b) there are no a priori reasons to expect the axiom to hold over K* but not over (K- K*)?

The validity of the theorems obtained does not, of course, depend on whether we find the above line of argument to be convincing, but the importance of the results clearly does, especially for demand theory.

Two final remarks may be made. First, while it is not required that the domain includes all infinite sets as well, nothing would of course be affected in the results and the proofs even if all infinite sets are included in the domain. Second, it is not really necessary that even all finite sets be included in the domain. All the results and proofs would continue to hold even if the domain includes all pairs and triples but not all finite sets.

SEN CHOICE FUNCTIONS 313

7. ELEMENT-VALUED CHOICE FUNCTIONS Much of revealed preference theory has been concerned with element-valued choice

functions. Certain special results are true for such functions. (T.4). A choice function the range of which is restricted to the class of unit sets is

normal if and only if R is an ordering. Proof Let C(S) be normal. For some x, y, z in X let xRy & yRz. Hence xRy & yRz

since C(S) is normal, and xPy & yPz as the choice sets are unit sets. Thus xPy & yPz due to normality. If we now take [z] = C([x, z]), then zPx, but that will make C([x, y, z]) empty. Since this is impossible, [x] C([x, z]), and hence xPz. Thus R is transitive.

It is obvious that choice sets being unit sets would imply P = R = P. And if R is an ordering, R = R, and this implies that the choice function is normal by (T.3).

Consider now a weak property of rationality originally introduced by Chernoff [3]. PROPERTY ae. For any pair of sets S and T in K and for any x E S, if x E C(T) and

Sc T, then x e C(S). That is, if x is " best " in a set it is best in all subsets of it to which x belongs. (T.5). Any normal choice function satisfies Property ae. Proof. Let x E C(T). Then xRy for all y in T. If Sc T, then xRy for all y in S.

Thus, x e 0(S). By normality x e C(S). (T.6). A choice function the range of which is restricted to the class of unit sets satisfies

Property oc if and only if it satisfies WCA. Proof WCA implies normality and by (T.5) this implies Property oc. Regarding the

converse it is sufficient to show that a implies that the antecedent of WCA must always be false. If it were not false, then for some S, x e S, y E C(S) and xRy. Since y e C(S) and x e S, by Property ae, y e C([x, y]). On the other hand, since xRy, obviously for some H such that y E H, x belongs to C(H). Then by ac, x E C([x, y]). But this is impossible since C([x, y]) must be a unit set.

As a corollary to (T.6) it is noted that for element-valued choice functions Property a implies all the conditions covered in (T.3).'

8. TRANSITIVITY AND FACTORIZATION For element-valued choice functions Property a implies complete rationality, but what

does cc imply in general? Not even normality as is clear from Example 2 in Section 3, but it does guarantee that R = R.

(T.7). For any choice function Property a implies R R. Proof. xRy implies xRy by definition. With xRy we know that for some S in K,

x E C(S) and y E S. By oc, x E C([x, y]). Hence xRy. How can a be supplemented in the general case to get complete rationality? Property

,B was introduced for this purpose (see Sen [18]). PROPERTY P3. For all pairs of sets S and T in K and for all pairs of elements x and y

belonging to C(S), if Scz T, then x E C(T) if and only if y E C(T). That is, if x and y are both best in S, a subset of T, then x is best in T if and only if

y is best in T. 1 For a direct proof that for element-valued choice functions x is equivalent to condition (v) in (T. 3),

see Houthakker [10].

314 REVIEW OF ECONOMIC STUDIES

(T.8). A choice function satisfies the Weak Congruence Axiom if and only if it satisfies Properties a and P.

Proof By (T.3), WCA implies normality, which in turn implies a, by (T.5). Let P be violated with x but not y belonging to C(T). But since x e S and y e C(S), we have yRx. This is a violation of WCA. Thus WCA implies a and ,B both.

Now the converse. Let the antecedent of WCA hold, and for some S, x e S, y e C(S) and xRy. By oc, y e C([x, y]). Since xRy, for some H, x E C(H) and y E H. By a, x E C([x, y]). Thus [x, y] = C([x, y]). But then by ,B, y E C(S) if and only if x e C(S). Since y is, in fact, in C(S), so must be x. Thus WCA must hold and it is shown that a and ,B together imply WCA.

A corollary to (T.8) is that a and ,B together imply all the conditions listed in (T.3) and thus amount to complete rationality in the usual sense.

9. AXIOMS FOR NORMALITY AND BINARINESS Properties a and ,B imply normality and transitivity of R. What implies normality

alone? PROPERTY y. Let M be any class of sets chosen from K and let V be the union of all

sets in M. Then any x that belongs to C(S) for all S in M must belong to C(V). That is, if x is best in each set in a class of sets such that their union is V, then x must

be best in V. (T.9). A choice function is normal if and only if it satisfies Properties a and y. Proof. Let C(S) be normal. By (T.5) it satisfies Property oc. Let the antecedent of

Property y be fulfilled. Thus xRy for all y in V. Hence x E C(V). But since C(S) is normal, x E C(V). Hence Property y is also satisfied.

To prove the converse let x belong to C(V). Thus xRy for all y in V. By (T.7), xRy for all y in V. Therefore, x E C([x, y]) for all y in V. Therefore by Property y, x E C(V). Hence C(S) is normal.'

10. AXIOMS FOR QUASI-TRANSITIVITY An intermediate property between normality alone and that with full transitivity is

normality coupled with " quasi-transitivity " of revealed preference (see Sen [18]). Armstrong [1] had shown the plausibility of intransitive indifference combined with transitive strict preference. This possibility has been discussed further by Georgescu-Roegen [7], Majumdar [14] and others in the context of demand theory, and by Pattanaik [15], Sen [18], Inada [11], Fishburn [4] and others in the context of the theory of collective choice.

Luce [12], [13] has studied extensively the case of " semi-orders " which is a special case of quasi-transitivity.2 Our concentration here will be on the more general condition. What guarantees quasi-transitivity as such?

1 For any finite class K properties a nd y can be redefined in terms of pairs of sets in the following way.

For all X and Y in K: Property oc*: C(X U Y)' [C(X) U C(Y)]; Property y*: [C(X) n c(Y)] C c(x u Y).

It is clear that a* amounts to cx, and y is the finite closure of y*. This formulation brings out the com- plementary nature of the two properties in an illuminating way and I am grateful to Hans Herzberger for drawing my attention to this.

2 The axiomatic structure of semiorders has recently been subjected to a searching examination by Dean Jamison and Lawrence Lau in " Semiorders, Revealed Preference, and the Theory of the Consumer Demand ", Technical Report No. 31, Institute for Mathematical studies in the Social Sciences, Stanford University, July 1970, presented at the World Econometric Congress in Cambridge, September 1970.

SEN CHOICE FUNCTIONS 315

PROPERTY 6. For any pair of finite sets S, T in K, for any pair of elements x, y e C(S), if Sc T, then [x] # C(T).

That is, if x and y are both best in S, a subset of T, then neither of them can be uniquely best in T. However, unlike in the case of Property f, it is not required that if one of x and y is best in T, then so should be the other.

______________________________________ _i _ - quasi-transitivity

transitivity

normal C(S)

R ~ ~ ~ RR7

WCA 4 -- SCA 4 - SARP * WARP

Diagram

(T.10). For a normal choice function, R is quasi-transitive (i.e. strict P is transitive) if and only if Property 6 is satisfied.

Proof. Let xPy and yPz hold. If not xRz, then C([x, y, z]) will be empty, which is impossible since C(S) is normal. Hence xRz. If xIz, then [x, z] = C([x, z]). Property a will now imply that x should not be uniquely best in [x, y, z], but we know that

[x] = ([x, y, z]). This is a contradiction since C(S) is normal. Hence xPz. So R is quasi-transitive.

To prove the converse let 6 be violated, and in spite of the antecedent holding, let [x] = C(T). Since y e C(S), yRx holds, so that y can fail to belong to (T) only if some z' in T exists such that z'Py. But z1 does not belong to C(T) = (T), so that there exists

316 REVIEW OF ECONOMIC STUDIES

z2 z2pzl, and by quasi-transitivity z2py. Obviously, z2 9 x, since yRx, and there exists z3 : z3pz2. Proceeding this way we get a sequence z1, z2, ... zn, such that zi # x and zi + lpzi, for i = 1, .. ., n-I, and such that all elements of T other than x and y are exhausted.

If xPZ, then by quasi-transitivity xPy. This is impossible and hence zn E C(T). But this is a contradiction since [x] = C(T).'

Finally, the main results presented in the paper are represented in an Implication Diagram with the direction of the arrow representing that of implication.

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SEN CHOICE FUNCTIONS 317

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