Revealed preference and intransitive indifference

Revealed preference and intransitive indifference

JOURNAL OF ECONOMIC Revealed THEORY 98-105 54, Preference (199 1) and Intransitive indifference SUSAN H. GENSEMER Department q/Economics, Sy...

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JOURNAL

OF ECONOMIC

Revealed

THEORY

98-105

54,

Preference

(199 1)

and Intransitive

indifference

SUSAN H. GENSEMER Department q/Economics, Syracuse University, Syracuse, New York 13244-1090 Received

October

18, 1988; revised

August

15, 1990

This paper approaches revealed preference from an assumption of a type of bounded rationality on the part of the consumer. We assume that the individual may have discrimination problems with regard to alternative choices (specifically, we assume that the underlying preferences of the consumer are described by a semiorder). The main result of this paper is the description of a revealed preference condition which is satisfied by a set of partial choice data if and only if the partial choice function can be rationalized by semiorder preferences. Journal of Economic ‘? 1991 Academic Press, Inc. Literature Classification Number: 022.

1. INTRODUCTION Revealed preference has a long history in economics. The underlying preferences of concern here are semiorders and interval orders. The semiorder, introduced by Lute [lo], is a preference order which allows for imperfect discrimination between alternatives on the part of the individual. The semiorder is different from a weak order (complete and transitive order) in that indifference under a semiorder might be intransitive. The interval order includes the semiorder as a special case and is presented by Fishburn [2,4]. Of the works which discuss relationships between observed choices and preferences, three seem to be particularly related to this investigation. Specifically, Jamison and Lau [7] (also, see Jamison and Lau [S] and Fishburn [3]) consider an individual’s choices of alternatives from a set of alternatives. They discussconditions on the choices of an individual which are necessary and sufficient to guarantee that the individual’s underlying preference order is a semiorder or an interval order. They assumethat the observer’s information on the individual’s choices is complete in the sense that observations are available on the individual’s choices on each subset of his alternative set. Richter [ 111 dispenseswith the latter assumption and presents a condition, the Congruence Axiom, which is necessary and sufficient to guarantee that an individual’s choices are explicable by an 98 0022-0531/91

$3.00

Copyright 0 1991 by Acadermc Press. Inc All rights of reproduction in any form reserved.

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underlying weak order. Finally, Kim [9] allows for the possibility of incompleteness of observations and allows for the possibility of semiorder or interval order preferences; however, he assumes that the consumer always chooses exactly one item from the available items. The latter is a very restrictive assumption from a realistic point of view and is a considerably simplifying assumption from a technical point of view. The following development represents a synthesis of the investigations by Jamison and Lau and Richter in that a condition is presented which is necessary and sufficient to guarantee that an individual’s choices can be explained by underlying semiorder preferences when the (possibly incomplete) set of data on the individual’s choices is finite. In addition, a similar condition is discussed for interval orders. In the following section, notation and definitions are presented. Then the relationships between this study and the studies of Jamison and Lau and Richter are explicitly described. Section 2 ends with the presentation of some preliminary lemmas. In the third section, the main theorem is presented. 2. NOTATION, DEFINITIONS, AND PRELIMINARY RESULTS

Let S denote a known set of alternatives faced by the individual and let Y’= 2’, the set of all subsets of S. We assume that observations are available on the individual’s choices when he or she is faced with sets in a nonempty set 9 where Y E Y’ - { @zr>.A function h: Y + Y’ such that h(B) c B for every B E Y is a partial choice function. If h is a partial choice function, the set {(B, h(B)): BE Y} is the set of partial choice data. Most of the results in this paper assume that Y and BE 9 are finite and in this case, the set of partial choice data is said to be finite. In the following definition and elsewhere in this study, P can be interpreted as strict preference and Z can be interpreted as indifference. Lower case italic letters are used to denote alternatives of S. DEFINITION 1.

Al. A2. A3. A4

(S, P, Z) is a semiorder if the structure satisfies:

If x, y E S, exactly one of the following holds: xPy, yPx, or xZv. For all x E S, SZX. If w, x, y, z E S and wPxZyPz, then wPz. If w, x, y, z E S and xP7yPz, then not both xZw and wlz.

The order (P, Z) rationalizes the set of choice data if h(B) = {x E B : xPy or XZJ?for every y E B}

for every BE 9’.

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SUSAN H.GENSEMER

The main result of this paper is the description of a condition which is satisfied by a set of partial choice data if and only if the partial choice function can be rationalized by semiorder preferences, where it is assumed that the set of partial choice data is finite. This study is similar to Richter’s in that Richter considers a partial choice function as in this paper; on the other hand, this study is distinguished from Richter’s in that he considers the rationalizability of a partial choice function of an individual with an underlying weak preference order. While Jamison and Lau consider conditions on the set of choice data which are satisfied if and only if the individual’s preference order is a semiorder, they assume that Y = 9” - {a}; that is, they assume that information is available on the individual’s entire choice function. In contrast, in this study we simply assume that Y E Y’(a>, where 9’ is nonempty. Finally, in contrast to the latter two studies, this study assumes that the set of partial choice data is finite. This is a simplifying assumption which is further discussed in Section 3. Given that our observations might be incomplete (that is, we might not we need to consider the concept of an incomplete have Y=Y’{a}), semiorder. The concept of an incomplete semiorder is introduced in Gensemer [S] and related material is developed by Doignon et al. [ 1 ] and Roy [ 121. In the following definition and throughout the paper, lower case Greek letters, usually subscripted, stand for symbols such as P or I. We consider sequences such as x, c11 . . ~1,~ 1x,, where xi E S (i = 1, .... n), n b 2, and crj=PorZ(i=l ,..., n-l). Wedenote #(cc,:a,=P, i=l,..., n-l} by N(P) and denote # {ui: a;=Z, i= 1, .... n- l> by N(Z). DEFINITION 2. satisfies : Bl. B2. x1 zx,.

(S, P, Z) is an incomplete

semiorder

if the structure

If x, y E S and xZy, then ylx. Let X;ES (i= 1, .... n) and .~,a, . ..c(.-,x,.

If N(P)>,N(Z),

then

The following lemma relates an incomplete semiorder to a semiorder. We say that an order (P, Z) on the set S is embeddable in an order (P’, Z’) on the set S if xPy implies that xP’y and xZy implies that xZ’y. We also say that (P’, Z) is an extension of (P, Z). The following result is shown by Gensemer [S]. See also the results by Doignon et al. [ 1 ] and Roy [ 121. LEMMA 1. An incomplete semiorder is embeddable in a semiorder. The following are definitions of the revealed preference notions used in this presentation. In the following, the set B - h(B) contains exactly those alternatives which appear in B, but not in h(B).

REVEALED

101

PREFERENCE

DEFINITION 3. If BEG, h(B)= {x}, and y E B - h(B), we say that x is revealed strongly preferred to y and write xP’y. DEFINITION

indifferent

4. If BE Y and x, y E h(B), to y and write xl’y.

we say that x is revealed

DEFINITION 5. If BE 9’, x E h(B), y E B - h(B), not xP’y, and not xl’y, we say that x is revealed weakly preferred to y and write xR’y.

Much of the following lemma relates the observed revealed preference relations to an assumed underlying semiorder of the individual. LEMMA 2. If a finite set of partial choice data can be rationalized by a semiorder (P, I>, then:

(i) (ii)

h(B) # @ for every BE Y.

(iii) (iv)

(P’, I’)

I~BEY and B-h(B)#@, for every y E B - h(B).

there exists x’~h(B)

such that x’Py

is embeddablein (P, I).

if xR’y, there exists z E S such that xIzPy.

Proof: Let BE 9, x E B, and define the dominated set in B with respect to x as D(x, B) = (y E B : xPy) where x E B. We first show that given D(x, B) and D(x’, B) where x, x’ E B, either D(x, B) G D(x’, B) or D(x’, B) cD(.u, B). If neither holds, then there exist y, y’ E B such that y E D(x, B), y’ $ D(x, B), y’ E D(x’, B), and y $ D(x’, B). Therefore, xPy, y’crx where o!= P or Z, x’Py’, and yfix’ where j3 = P or Z. Then xPyBx’Py’ where /I = P or I. By the properties of a semiorder, this implies that xPy’. However, since y’ctx where c(= P or Z, we have a contradiction by Al. Therefore, if X, x’ E B, either D(x, B) G D(x’, B) or D(x’, B) c D(x, B). Now it is possible to number the elements of B = (x, , .... x,} such that D(x,, B) s . . . cD(x,, B). Note that xi .$0(x,, B) for any i. Otherwise, X, ED(x,, B) and x,Px,, impossible by Al and A2. Therefore, by using Al, we find that for every i= 1, .... it, .~i Px, or x,Zx,. It follows that x1 E/Z(B) and (i) holds. If y E B- h(B) is arbitrary, then y~D(x~, B) for some i = 1, .... n. It follows that y E D(x,, B) and X, Py. Therefore, (ii) holds. If xP’y, then it is clear from (ii) that xPy. If xl’y, then x, YE h(B) for some BE 9, and it is clear that xly. Therefore, (iii) holds. Finally, if xR’y, then x E h(B) and y E B - h(B) for some BE Y. By (ii), there exists z E h(B) such that zPy. By (iii), xii. Therefore, xZzPy and (iv) is established. 1

Condition (iv) illustrates one possible difference between choice results under semiorder and weak order preferences. Under semiorder preferences, it is possible to have x, y E h(B), z E B - h(B), and zZxZyPz. In contrast,

102

SUSANH.GENSEMER

given the latter observations on choices, under weak order preferences we would have xly, XPZ, and yPz. As previously stated, the results in this paper assume that the set of partial choice data is finite. In this case, Y = {B,, B,, .... B,}. Assume that these sets are numbered such that for i= 1, .... s where 0 6 s < t, Bi - h(Bi) # @ and there does not exist x( E h(B;) such that x:P’v for every y E Bi - h(B,). The set of choice data is assumed to satisfy or will be shown to satisfy the following condition. In the condition, yi represents a nonnegative integer. While it is not explicitly indicated, yi (i = 1, .... S) can vary from sequence to sequence. Condition a. For each i= 1, .... s, there exists a distinguished element x7 E h(B,) such that x)P’y or xi* R’y for every y E B, - h(Bi). These XT’S are such that if y,cx, . ..a.+ iym is any sequence in which x,*R’y (for any y E Bj- h(B;)) appears yi times (i = 1, .... s), N(P’) > N(I’) - C;=, yi, and N(R’) 3 0, it follows that y, # y,.

Note that if ri = 0 (i= 1, .... S) in Condition CL,then the condition is analogous, but not identical to, a revealed preference condition for weak orders. The complicated nature of Condition a arises because the structure of the choice data might not yield information on which element(s) in the chosen set h(B) strictly dominates those that are not chosen (elements in B-h(B)). Condition LYrecognizes that we might have to search for an element of h(B) which can be used to strictly dominate the elements in B - h(B). In contrast, in a weak order context, any element of h(B) strictly dominates any element of B - h(B). Given a finite set of partial choice data, assume that we have derived the revealed preference relations (P’. R’, Z). We derive an order (P”, Z”) from (P’, R’, I’). Then in Lemma 3, we show that the derived order is an incomplete semiorder, assuming that the data satisfies Condition LX.Define: xP”y if

(i)

xP’y or

(ii)

x = x7 and y E Bj - h( BJ for some i = 1, .... s,

(1)

and xZ”y LEMMA

if

xl’y.

(2)

3. Zf a finite set of partial choice data satisfies Condition a, then:

(i) (ii)

(S, PI’, Y) is an incomplete semiorder. Zf BE Y and B-h(B) # 0, there exists x* Eh(B) such that x*P”y for every y E B - h(B). Proof

To see that (i) holds, first note that Bl of the definition

of an

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incomplete semiorder follows immediately by the way in which I” is defined. Now assume that there is a sequence XlMl

. ..%-l

X n,

where

N(P”) 2 N(Z”).

By the way in which (P”, Z”) is defined, (3) implies sequence XlPl . ..llkXH.

(3)

that there is a (4)

where x: R’y, y E Bj - h(B,), appears yi > 0 times (i = 1, .... s), N(P) > N(Z)-Cj=, ri, and N(R’)20. If x1 =x,, then we obtain an immediate violation of the way in which, by Condition CI,the x7 E h(B;) (i = 1, .... S) were to be chosen. Therefore, B2 of the definition of incomplete semiorder holds and the proof of (i) is complete. To see that (ii) holds, note that if i = 1, .... S, then x*P”y where XT E h(B,) for every y E Bi - h(B’) by (1 )(ii). Otherwise, if i = s + 1, .... t, by the way in which the Bi (i = 1, .... t) were numbered, there exists x( E h(B,) such that xjP’y for every y E Bi- h(Bi). By (l)(i), it follows that x;P”y for every y E Bi - h(Bj). Therefore, (ii) holds. i

3. THE MAIN

RESULT

Now we present the main theorem which establishes that Condition c( is a necessary and suffkient condition for a finite set of partial choice data to be rationalizable by a semiorder. THEOREM. A finite set of partial choice data can be rationalized by a semiorder if and only if the set of data satisfies Condition x.

Proof First, the forward implication of the theorem is demonstrated. Assume that a finite set of partial choice data can be rationalized by a semiorder. We show that Condition CI is satisfied by the set of partial choice data. By Lemma 2(ii), for each i = 1, .... s, there exists x( E h(Bi) such that XI Py for every y E Bi - h( Bi). Consider any sequence y 1c(, . LX,_ 1 y, , where x;R’y, y E Bi - h(Bi), appears ri > 0 times (i = 1, .... s), N(P’) > N(Z’) - CT=, ri, and N(E) > 0. By Lemma 2(ii)-(iv), we have a sequence y,fi,z, ...zkPljk y,, where x:Py, YE B,- h(Bi), appears ri times (i= 1, .... s), and N(P) 2 N(Z). By the properties of a semiorder, this implies that y1 Z-v,,,. The preceding arguments establish that Condition a is satisfied. Now the reverse implication of the theorem is established. Suppose that

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H. GENSEMER

a finite set of partial choice data satisfies Condition u. As previously demonstrated, we can define (P”, Z”) from (P’, R’, I’) which satisfies the conditions of Lemma 3. Since (P”, Z”) is an incomplete semiorder, it is embeddable in a semiorder (P, I), by Lemma 1. By Lemma 3(ii) and since xPy if xP”y, (P, Z) has the property that if y E B - h(B), there exists x* E/Z(B) such that x*Py. We show that (P, Z) rationalizes h; that is, defining g(B) = {x E B : xPy or xly for every y E B}, we establish that g(B) = h(B). First, we show that h(B) c g(B). If x E h(B), then xr’y for every y E h(B). Therefore, xZ”y and xly for every y E h(B). Let z E B - h(B). As previously stated, there exists x* E h(B) such that x*Pz. Since x E h(B), we have xZ’x*, xI”X*, and xZx*. Therefore, xZx*Pz. By the properties of a semiorder, we cannot have zPx. By Al, either XPZ or xl,-. Therefore, if XE h(B) and z E B - h(B), then xPz or xlz and as shown previously, if y E h(B), xly. It follows that x E g(B) and h(B) G g(B). Now we show that g(B) s h(B). Let x E g(B) and assume that x 4 h(B). Then x E B - h(B) and, as previously stated, there exists x* E h(B) such that .u*P.x. However, since x E g(B), we also have that xPx* or xZx*, a violation of Al. Therefore, x E h(B) and g(B) = h(B). The preceding arguments establish that the semiorder (P, Z) rationalizes the set of partial choice data. m Without the finiteness assumption, we can develop a condition similar to (but more complicated than) Condition ~1.The assumption of a finite set of data simplifies the analysis in that, under semiorder preferences, when B is finite and B - h(B) # 0, we can find one x’ E h(B) such that x’Py for every y E B- h(B), by Lemma 2(ii). Without the assumption, we would seek x,. E h(B) (that is, an x which depends on y) such that x,,Py; that is, there is no guarantee that Lemma 2(ii) holds without the assumption. Therefore, the statement of a condition like c( is simply more complicated without the finiteness assumption. As shown by Gensemer [6], a similar (but seemingly more complicated) result can be derived for interval orders. The same paper contains an extension of the result in this paper which can be used to simplify the testing of a finite set of partial choice data for consistency with semiorder preferences and an illustration of the method here. Preliminary work has been done on computerizing these results.

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of two relations

by

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PREFERENCE

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2. P. FISHBURN, Intransitive indifference with unequal indifference intervals, J. Math. Psych. 7 (1970), 144-149. 3. P. FISHBURN, Semiorders and choice functions, Economefrica 43 (1975), 975-977. 4. P. FISHBURN, “Interval Orders and Interval Graphs,” Wiley, New York, 1985. 5. S. GENSEMER, Intransitive indifference and incomparability, Econ. Letters 26 (1988). 311-314. 6. S. GENSEMER, “Revealed Preference and Intransitive Indifference,” Working Paper No. 22, Syracuse University. 1988. 7. D. JAMISON AND L. LAU, Semiorders and the theory of choice, Econometrica 41 (1973). 901-912. 8. D. JAMBON AND L. LAU, Semiorders and the theory of choice: A correction, Econorne~%~~ 43 (1975), 979-980. 9. T. KIM. Intransitive indifference and revealed preference. Econometrica 55 (1987), 163-167. 10. R. LUCE, Semiorders and a theory of utility discrimination, Econometrica 24 (1956), 178-191. 11. M. RICHTER, Revealed preference theory, Econometrica 34 (1966), 6355645. 12. B. ROY, “Preference, indifference, incomparabiliti,” Documents du LAMSADE, No. 9, Universite de Paris-Dauphine, 1980.