- Email: [email protected]
Sub-semiorder: A model of multidimensional choice with preference intransitivity
fOURNAt
OF MATHEMATICAL
PSYCHOLOGY
Sub-semiorder:
16,
51-59
(1977)
A Model of Multidimensional with Preference Intransitivity” YEW-KWANG
Monash
University,
Clayton,
Choice
NG Victoria
3168,
Australia
The paper develops a model of choice called a sub-semiorder which is a generalization of Lute’s semiorder to multidimensional choice. The same reason (imperfect discrimination) that gives rise to the intransitivity of indifference in a semiorder gives rise to the intransitivity of preference in a sub-semiorder. This provides a rational explanation of intransitivity of preference without resorting to the lexicographic semiorder of Tversky. It is shown that the “apparent underlying preference” of a sub-semiorder is transitive but unfortunately it is not complete. However, with a mild condition, there exists a maximal element.
The traditional algebraic theories of choice assume transitivity of both preference and indifference. Transitivity of indifference involves perfect discriminatory power clearly contrary to common sense and experimental studies. Lute’s definition of a semiorder (Lute, 1956, p. 181) overcomes this shortcoming of the traditional theory of an ordering. However, a semiorder still involves transitivity of preference which has been seen to be violated in many experiments. This violation has not usually been observed if transitivity is defined stochastically. In a remarkable paper, Tversky (1969) shows that consistent and predictable violation of stochastic transitivity also can be demonstrated. The theory advanced by Tversky to explain this intransitivity is that of a lexicographic semiorder (LS) where a semiorder is imposed on a lexicographic ordering. In this paper, I advance the hypothesis that choices are characterized by a sub-semiorder, which is a generalization of a semiorder to multidimensional choice. It can be seen that rational choice without lexicographic ordering may give rise to preference intransitivity (Section 1). Section 2 gives a forma1 definition of a sub-semiorder and Section 3 shows that the “apparent underlying preference” of a sub-semiorder is noncyclic, transitive, but not, in general, complete.
1.
MULTIDIMENSIONAL
CHOICE
AND
PREFERENCE
INTRANSITIVITY
The definition of a semiorder was prompted by the recognition of imperfect discrimination which gives rise to intransitivity of indifference. Transitivity of preference is nevertheless preserved if we confine our study to variation over one dimension only. Intuitively, * I am grateful
to the editor
and a referee
for very
helpful
comments.
51 Copyright AS1 rights
Q 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
0022-1496
52
YEW-KWANG
NG
a dimension refers to any aspect over which the elements of the set may differ from each other. But obviously, we will be interested only in those aspects that affect the preference of the individual concerned. The number of relevant dimensions can be any positive integer. If we represent the set in an n-dimensional Euclidean space (really, just the ordered n-tuple property of the space is needed) or some subset thereof, 12is the number of dimensions. If we examine choices involving more than one dimension, it can be shown that imperfect discrimination alone may give rise to intransitive preference. For example, suppose that I prefer thick white paper. In Fig. 1 the dashed lines mark off the level of my discrimination for whiteness and thickness. Along any vertical or horizontal line, my preference is a semiorder. However, if we let both dimensions vary, my preference is not a semiorder. For example, I prefer x to y as I can observe that x is thicker than y but cannot observe any difference in whiteness. Similarly, I prefer y to z. But I am indifferent to the choice between x and z.
In the above two-dimensional case with equal weight on each dimension, we do not have cyclicity of preference. However, if we have unequal weights, or if we have three or more dimensions, cyclicity may arise. For example, suppose a subject is able to discriminate a difference of any dimension involving more than one unit. He prefers high values to low and gives equal weight to each of the three dimensions. His preference over variation in any one dimension is a semiorder. But (1.2, 0, 0) P (0, 0.5, 0.5) P (0.5, 1.2, -0.6) P (1.2,0,0), which is cyclic. To show cyclic&y for the case of two dimensions with unequal weights, suppose he regards the second dimension as three times more important than the first, but his discrimination ability is the same with respect to both dimensions, being able to tell a difference of one unit. Then (1.2, 0) P (0, 0.6) P (- 1.2, 1.2) P (1.2, 0).
MULTIDIMENSIONAL
2.
CHOICE
SUB-SEMIORDER
A binary relation R over a set X is called an ordering of X if it is complete, reflexive, and transitive. For most purposes, xRy represents “x is preferred to or indifferent from Y,” or “x is perceived to possess at least as high a value (m certain attributes) as y,” or some other similar meaning. If both xRy and yRx, we write xly; if xRy and not yRx, we write xPy. A semiorder was first defined rigorously by Lute (1956, p. 181). In the following definition, I use a slightly different formulation which proves to be useful in the generalization to a sub-semiorder. DEFINITION
following (Al) (A2) (A3)
1. A binary relation R over the set X is called a semiorder of X if the are satisfied (throughout the paper, Vx stands for Vx E X, etc.): Rejexivity: Completeness: Weak Transitivity:
(‘c’x) ww (Vx, y: x # y) (X&J v yRx) (Vr, x, y, z) (rPxIyPz v rIxPyPx
G- rPz)
It may be noted that the requirement rPxPyIz =r rPz is not needed as it is implied by Al-A3. It may also be noted that Al and A3 imply xPyPx 3 xPz. In other words, transitivity of preference is implied. For the definition of a sub-semiorder, let us adopt the following notations. For all x, xe’ (note the superscript; subscripts are used later to denote different x’s) denotes the value of x in dimension i. If xi = y” for all but (possibly) one i E {I, 2,..., n}, we write xDy, where D reads “does not differ in more than one dimension.” DEFINITION 2. A binary relation R on the set X is a sub-semiorder reflexive and complete and satisfies:
(A3’)
Mild
Transitivity:
(xPrRyPs
of X iff it is
(Vx, y: xDy)
v xRrPyPs
* xPs; sPxRrPy
v sPxPrRy
G- spy).
It is clear that (A3’) is a weakening of (A3) since an additional requirement regarding dimensionality is imposed. Hence, a sub-semiorder is a generalization of a semiorder. In other words, a semiorder is a sub-semiorder, but a sub-semiorder is not necessarily a semiorder.
3. THE
NATURE
OF A SUB-SEMIORDER
AND
ITS
UNDERLYING
PREFERENCE
If a binary preference relation R is a semiorder, we may adopt the following notations to represent its “underlying preference structure”: (i) (ii) (iii)
xBy 0 (jr: xPrIy v xIrPy); xAy 0 (not xBy and not yBx); xLy 0 (xBy V XAy)
YEW-KWANG
54
NG
where “B,” ” A,” and “L” may be interpreted as “better than,” “as good as,” and “at least as good as,” respectively. One can show that, for a semiorder, the underlying preference relation L is an ordering, i.e., it is complete, reflexive, and transitive. With some additional requirement, it can then be represented by a real-valued utility function.1 For a sub-semiorder, since the preference relation is necessarily a semiorder only over variation involving one dimension, we adopt the following notations: (iv)
xBy o
(v)
xJy o not xBy and not yBx and
xDy
and 3:
xPrIy
v XIYPY
(vi)
xLy t> xBy v x2y
(vii)
x2y cz 3x,, x2 ,..., x, : x2x12x,
xDy
b.a 2xmAy
where 7n may be any nonnegative integer, including zero (i.e., xJy j xwy) (viii)
xBy 0 3x, , X2 ,..., x, : XLX&va **. ExJy,
where at least one of the z is B (ix)
xEy 0 xJy
v xBy
It may be noted that, if a sub-semiorderis not also a semiorder, it is not appropriate to use L as its underlying preference, and B cannot be appropriately used as “better than,” as it is not transitive (similarly with A). It is for this reasonthat the barred and double-barred notations are designedinstead. The binary relation E may be regarded as the “apparent underlying preference” of a sub-semiorder. The reasonfor adding the adjective “apparent” is because the “intrinsic” underlying preference may not be completely revealed in the definition of J?. This will becomeclearer later, when we have proved a few propositions. PROPOSITION one and
only
1.
For
of the following
a sub-semiorder, is true:
for xAy,
xBy,
any yBx
pair
of alternatives (ok xAy,
xBy,
x, y
such
that
xDy,
yBx).
Proof. Since the definitions of “A” and “B” are exhaustive, it is tautologically true that at least one of xAy, xBy, yBx is true. So we only have to show that only one is true. If this is not so, then there exist somex and y such that, either (i) xAy and xBy; or (ii) xBy and yBx. The former cannot be true from the definition of “A”. If (ii) is true, then 3: xPrRy v xRrPy and 3s: yPsRx v yRsPx. Hence at least one of the following
1 This representation and related problems have been tackled by a number of writers. In particular, Scott and Suppes (1968) prove that a semiorder over a finite set is closed interval representable with constant intervals. Fishbum (1973, Theorem 6) provides necessary and sufficient conditions for a semiorder to be closed-interval representable. These conditions involve some density requirement as well as countability whose meaning is not intuitively clear. In Ng (I 975), representation is proved by adopting standard assumptions on explicit preference and then showing that the underlying preference must also satisfy certain conditions well known to ensure the existence of utility function. The definition of “underlying preference” here is slightly different from the one used in Ng (1975), making the proof slightly more tedious but involving similar arguments.
MULTIDIMENSIONAL
CHOICE
55
has to be true: (a) xPrRyPsRx, (b) xPrRyRsPx, (c) xRrPyPsRx, (d) xRrPyRsPx. It is easy to check that each of these violates (A3’) since xDy. Q.E.D.
For a sub-semiorder, x&x, .*. Lx, if none(at least one)of the E’s is B.
PROPOSITION 2.
xIAx,(xIBx,)
3 (i) not x,Bxl
; (ii) x&x,
, with
Proof. First take the case where none of thee is B, so x,Ax, ... 2x,, . If the proposition is not true for this case, we have x,Bx, or x,Bx, . Then 3r: x,,,PrRx, v x,RrPx, v xlPrRx, v x,RrPx,,, . Suppose x,PrRx, is the case. (Other cases are similarly proved and will not be repeated.) Compare x,-r with r. From A2, either x,,-,Pr, x,-$r, or rPx,-, . The last two cannot be true because otherwise x,PrRx,-, * x,Bx,-, , violating PI (i.e., Proposition l), since x,Dx,-, . So x,-~ PrRx, * x,-,Bx, . Proceed to compare x,,+* with r in a similar fashion, and we can establish that x,-sBxI . Repeating the process, we get x,Bx, , violating the given fact that x,&(Pl). So x,4x, and not x,Bxl establishing the proposition for this case. Now take the case where at least one of the e is B, and take this to be xiBxi+r . So 3r: xiPrRx,+, v x,RrPxi,, . Again suppose xiPrRxi+, . (The other case is similarly proved.) Compare r with xi+a . It cannot be true that x,+sPr since Xi+rExi+a . So, rRx,+%, giving xiBx,+, . Repeating the process, we have xiBx, . Then compare xi-r with r, establishing xi-,Bx, . Repeating the process, we get xlBxm . To show that x,Bxl cannot be true, we do the similar comparison as we did in proving the first part of the proposition. Q.E.D. COROLLARY. The apparent underlying preferenceof a sub-semiorder is noncyclic, i.e., XEXJ -*aLx, +- not x,8x. PROPOSITION 3. The apparent underlying preference9, of a sub-semiorder is a partial ordering (satisfpng rejexivity and transitivity).
Proof. Reflexivity is easily established, since it is easy to see that x,4x and xAx. Either one of these is sufficient for xAx. Transitivity is also not difficult to establish. If xEyrz, taking y to be one of the connecting links between x and z, we have XJ%. Q.E.D. PROPOSITION 4. In general, the apparent underlying preferenceof a sub-semiorder is not an ordering.
Proof. Consider an explicit preference relation R” on E2 (Euclidean plane) defined by: yPOx u (yl > x1 + k & y2 3 x2 - k) v (y2 > x”+k&yl>xl-k)v(y1+y2> xl + xa + k) where k is a constant. yPx o not yPOx and not xPOy. This is illustrated in Fig. 2. The idea is that the individual can observe a difference of larger than k in each dimension. If there is an observable difference in each of the two dimensions, the difference in the differences has to be larger than K for the individual to prefer one to another. Incidentally, the set {y: yPoxj is closed at the boundaries indicated by the dotted lines. Any point (excluding the lower extreme points) on these dotted lines is a member of {y: yPox}, but any arbitrary neighborhood of it contains a point not in {y: ypOx}. This is
56
YEW-KWANG
NG
clearly incompatible with the usual continuity requirement (e.g., see Debreu, 1959, p. 56). Hence, an interesting technical problem is raised which, however, need not concern us here.2
{y
: xR”y)
FIGURE
2
From the definition of RO, it is clear that it satisfies reflexivity and completeness. To check that mild transitivity is also satisfied, note that x differs from y in no more than one dimension for this property (i.e., A3’). Since the definition of RO is symmetrical about the two dimensions, we may take x1 = y1 without loss of generality. From the definition of PO, xP”r implies (x’ >
y1
+ k & x2 >
y2
- k) v (x” >
~2
+ k&xl
v (x’ + x2 >
y1
>
11
- k)
+ r2 + k).
(1)
And rROy implies (yl <
y1
+ k v y2 < r2 - k) & (y2 <
y2
+ k v y’ < r1 - k)
& (yl + y2 G Since x1 = yl, it is easy that xDy, xP”rRoy Z- xi > this result, it can easily xPOrROyP% rj xP%. Since This demonstrates that R” 2 If we make
the model
y1
+
r2
+ k).
(2)
to deduce from (1) and (2) that x2 > y2. Hence, for x, y such y”. Similarly, it can be shown that xROrPOy s- xi > yi. With be seen that mild transitivity is satisfied. For example, xi > yi, yP% must imply xP% from the definition of PO. is a sub-semiorder on E2.
stochastic,
I think
the discontinuity
will
be eliminated.
MULTIDIMENSIONAL
CHOICE
To show that the apparent underlying preferenceslO to show that it is not complete. First note that &Oy or due south of x, i.e., if x1 = yl, x’3 3 y2. Thus, X&J (including due south and due west) of x, i.e., xi > yi consider a point x which is northwest or southwest violating completeness.
57
is not an ordering on E2, it suffices implies that y = x or y is due west implies that y = x ory is southwest i = 1, 2. Now for any given point x, of x. Then, not xE”z and not zz”x, Q.E.D.
The fact that the apparent underlying preference of a sub-semiorder is not, in general, an ordering does not mean that its “intrinsic” underlying preference cannot be an ordering. Thus in the example used in the proof of Proposition 4, it is reasonable to regard the intrinsic underlying preference i; as defined by xLy 0 x1 + x2 > y1 + y2, which is an ordering on @. However, the ease with which we discover the intrinsic underlying preference 20 from R” is deceptive. This is due to the highly simple nature of the case in question. First the relevant set E2 is only two-dimensional and convex. Second, R” is symmetrical about the two dimensions (equal weights) and monotonic. Third, the discrimination threshold k is constant throughout. In the general case where the set of alternatives may involve many dimensions and may not be convex, where the discrimination threshold may vary, and where different dimensions may have disimilar effects on preference, it is very difficult, if not impossible, to discover the intrinsic underlying preference. Moreover, even in the above simple case, the definition of the intrinsic underlying preference is in terms of the objective measures (the numerical values of x1 and x2) and not in terms of the explicit binary preference relation P and I. I must admit that I have not been able to construct a satisfactory definition of the intrinsic underlying preferences of a sub-semiorder in terms of the explicit preference, for the general case anyway. It can, however, be shown that there is a B-maximal element x in X (i.e., yBx for no y in X) for a sub-semiorder that satisfies a rather mild condition. DEFINITION 3. A binary relation ) on a set X is said to satisfy the Well-Ordering Condition if, for each subset of X on which ) is a linear order (as defined in Fishburn, 1972, p. 29), there is an element in X which is ) maximal on that subset. In the example used in the proof of Proposition 4, each straight line is linearly ordered by B. But there does not exist a maximal element if X is unbounded. If we are confined to compact (bounded and closed) sets, then the Well-Ordering Condition is very likely to be satisfied. For unbounded sets, the condition may also be satisfied if we have satiation.
5. If R on X is a sub-semiorder whose apparent underlying preference B satisfies the Well-Ordering Condition, then there is an x in X that is B-maximal
PROPOSITION
relation on X.
Proof. The proof is a straightforward application of Zorn’s lemma (for a discussion of which, see Fishburn, 1972, Chap. 6), since, from Proposition 2, B on X is a strict partial order if A on X is a sub-semiorder. Q.E.D.
58
YEW-KWANG
4.
CONCLUDING
NC F&MARKS
Multidimensional choices may, but need not, necessarily violate the requirement of a semiorder. If a number of dimensions combines to exert their influence on preference in a related way, the preference relation may be a semiorder over these dimensions. For example, my preference for a dish of sweet and sour fish may be a function of both sweetness and sourness. But the two dimensions may combine to assert their influence such that my preference is a semiorder. This is likely to be so because both sweetness and sourness are tasted at the same instance. For other cases, such as the location and the appearance of a house, the different dimensions are likely to be more independent of each other and hence preferences are more likely to violate the requirement of a semiorder. Many problems associated with the weakening of preference from an order into a semiorder can be resolved relatively easily since the underlying preference of a semiorder is an order. The weakening of preference into a sub-semiorder seems to raise more formidable problems since its apparent underlying preference is not generally an order and its intrinsic underlying preference is difficult to define in an operationally meaningful way. However, it is reasonable to regard the intrinsic underlying preference as that pattern of preference if the individua1 has perfect discriminating ability. The explicit binary preference is a reflection of this intrinsic preference subject to imperfect discriminatory ability. Hence, the explicit preference, while not a perfect representation of the intrinsic preference, is nevertheless “close” to it in some sense. Hence, if we are careful enough in interpreting the explicit preference, and if we adopt some helping devices, such as confming direct comparisons to variation over one dimension only and using indirect comparison (Ng. 1975, Sect. 9) for variations over many dimensions, it would not be impossible to get a close enough representation of the intrinsic preference. The imperfection in preference discrimination may be due to two different reinforcing causes. First, it may be due to imperfect knowledge or inadequate anticipatory ability. For example, I may be unable to decide whether I prefer house A to house B or the other way round and hence register an indifference between them because I am not sure which house will suit my requirements better at the moment of choice. However, if I am given a chance to live in both houses for a sufficient period of time, I may be able to be quite sure that I prefer A to B. This is similar to what Georgescu-Roegen (1936, 1958) calls the threshold in choice. This threshold is, as a rule, a decreasing function of the amount of time allowed for the choice. Georgescu-Roegen believes that, at “the limit when the time of experimenting is infinite, the threshold is zero” (1966, p. 152). Preference threshold need not be zero if we recognize a second cause of imperfect discrimination. This is what I call “finite sensibility,” which refers to the fundamental limitation in human capability of feeling. If the amounts of sugar in two cups of coffee are close enough, one cannot tell a difference no matter how carefully one tastes it. Thus, even if we rule out by assumption imperfect knowledge, uncertainties, etc., we may still have imperfect discrimination due to finite sensitivity. The distinction of imperfect discrimination as due to imperfect knowledge and as due to finite sensibility is of some importance. For some problems we may be more
MULTIDIMENSIONAL
CHOICE
59
interested in individual preference representing his actual feeling without “distortion” by imperfect knowledgeand the like. Now it canbe arguedthat the feeling of an individual at any moment of time is either affected by one factor (dimension) or affected by a number of factors which combine to exert their influences such asthe caseof sweetand sour fish discussedabove, since, by definition, an individual hasonly one mind. Hence, individual feeling at any moment of time is a semiorder. A fuller development of this argument as well asthe consideration of the complication introduced by the element of time can only be dealt with in another paper. Apart from this extension, the concept of sub-semiorder seemsto provide a few other challengesfor further development such as making the model stochastic (Lute, 1958), analyzing the intrinsic underlying preference, conducting experiments to ascertain the usefulnessof the model, and comparing the model with other psychological theories of choice such as the elimination-by-aspects theory (Tversky, 1972), the similarity-between-stimuli theory (Rumelhart and Greeno, 1971), etc.
REFERENCES
G. Theory of value. New Haven, Conn.: Yale Univ. Press, 1959. P. C. Muthenatics of decision theory. Atlantic Highlands, N. J.: Mouton. 1972. P. C. Interval representations for interval orders and semiorders. Journal of Mathematical Psychology, 1973, 10, 91-105. GEORGESCU-ROEGEN, N. The pure theory of consumer behaviour. Quarterly Journal of Economics, 1936, 50, 545-593. [Reprinted in Analytical economics. Cambridge, Mass.: Harvard Univ. Press, 1966).] GEORGESCU-ROEGEN, N. Threshold in choice and the theory of demand. Econometrica, 1958, 26, 157-l 68. [Reprinted as above.] LUCE, R. D. Semiorders and a theory of utility discrimination. Econometrica, 1956, 24, 178-191. LUCS, R. D. A probabilistic theory of utility. Econometrica, 1958, 26, 193-224. Nc, Y.-K. Bentham or Bergson I Finite sensibility, utility functions, and social welfare functions. Review of Economic Studies, 1975.42, 545-570. RUMELHART, D. L., and GRBBNO, J. G. Similarity between stimuli: An experimental text of the Lute and Restle choice models. ]ournaZ of Muthemudcal Psychology, 1971, 8, 370-381. SCOTT, D., AND SUPPBS, P. Fundamental aspects of theories of measurement. Journal of Symbolic Logic, 1958. 23, 113-128. T~ERSKY, A. Intransitivity of preferences, Psychological Review, 1969, 76, 31-48. TVERSKY, A. Elimination by aspects: A theory of choice, Psychological Review, 1972, 79, 281-299. DEBREU,
FISHBURN, FISHBURN,
RECEIVED:
February 24, 1976
OF MATHEMATICAL
PSYCHOLOGY
Sub-semiorder:
16,
51-59
(1977)
A Model of Multidimensional with Preference Intransitivity” YEW-KWANG
Monash
University,
Clayton,
Choice
NG Victoria
3168,
Australia
The paper develops a model of choice called a sub-semiorder which is a generalization of Lute’s semiorder to multidimensional choice. The same reason (imperfect discrimination) that gives rise to the intransitivity of indifference in a semiorder gives rise to the intransitivity of preference in a sub-semiorder. This provides a rational explanation of intransitivity of preference without resorting to the lexicographic semiorder of Tversky. It is shown that the “apparent underlying preference” of a sub-semiorder is transitive but unfortunately it is not complete. However, with a mild condition, there exists a maximal element.
The traditional algebraic theories of choice assume transitivity of both preference and indifference. Transitivity of indifference involves perfect discriminatory power clearly contrary to common sense and experimental studies. Lute’s definition of a semiorder (Lute, 1956, p. 181) overcomes this shortcoming of the traditional theory of an ordering. However, a semiorder still involves transitivity of preference which has been seen to be violated in many experiments. This violation has not usually been observed if transitivity is defined stochastically. In a remarkable paper, Tversky (1969) shows that consistent and predictable violation of stochastic transitivity also can be demonstrated. The theory advanced by Tversky to explain this intransitivity is that of a lexicographic semiorder (LS) where a semiorder is imposed on a lexicographic ordering. In this paper, I advance the hypothesis that choices are characterized by a sub-semiorder, which is a generalization of a semiorder to multidimensional choice. It can be seen that rational choice without lexicographic ordering may give rise to preference intransitivity (Section 1). Section 2 gives a forma1 definition of a sub-semiorder and Section 3 shows that the “apparent underlying preference” of a sub-semiorder is noncyclic, transitive, but not, in general, complete.
1.
MULTIDIMENSIONAL
CHOICE
AND
PREFERENCE
INTRANSITIVITY
The definition of a semiorder was prompted by the recognition of imperfect discrimination which gives rise to intransitivity of indifference. Transitivity of preference is nevertheless preserved if we confine our study to variation over one dimension only. Intuitively, * I am grateful
to the editor
and a referee
for very
helpful
comments.
51 Copyright AS1 rights
Q 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
0022-1496
52
YEW-KWANG
NG
a dimension refers to any aspect over which the elements of the set may differ from each other. But obviously, we will be interested only in those aspects that affect the preference of the individual concerned. The number of relevant dimensions can be any positive integer. If we represent the set in an n-dimensional Euclidean space (really, just the ordered n-tuple property of the space is needed) or some subset thereof, 12is the number of dimensions. If we examine choices involving more than one dimension, it can be shown that imperfect discrimination alone may give rise to intransitive preference. For example, suppose that I prefer thick white paper. In Fig. 1 the dashed lines mark off the level of my discrimination for whiteness and thickness. Along any vertical or horizontal line, my preference is a semiorder. However, if we let both dimensions vary, my preference is not a semiorder. For example, I prefer x to y as I can observe that x is thicker than y but cannot observe any difference in whiteness. Similarly, I prefer y to z. But I am indifferent to the choice between x and z.
In the above two-dimensional case with equal weight on each dimension, we do not have cyclicity of preference. However, if we have unequal weights, or if we have three or more dimensions, cyclicity may arise. For example, suppose a subject is able to discriminate a difference of any dimension involving more than one unit. He prefers high values to low and gives equal weight to each of the three dimensions. His preference over variation in any one dimension is a semiorder. But (1.2, 0, 0) P (0, 0.5, 0.5) P (0.5, 1.2, -0.6) P (1.2,0,0), which is cyclic. To show cyclic&y for the case of two dimensions with unequal weights, suppose he regards the second dimension as three times more important than the first, but his discrimination ability is the same with respect to both dimensions, being able to tell a difference of one unit. Then (1.2, 0) P (0, 0.6) P (- 1.2, 1.2) P (1.2, 0).
MULTIDIMENSIONAL
2.
CHOICE
SUB-SEMIORDER
A binary relation R over a set X is called an ordering of X if it is complete, reflexive, and transitive. For most purposes, xRy represents “x is preferred to or indifferent from Y,” or “x is perceived to possess at least as high a value (m certain attributes) as y,” or some other similar meaning. If both xRy and yRx, we write xly; if xRy and not yRx, we write xPy. A semiorder was first defined rigorously by Lute (1956, p. 181). In the following definition, I use a slightly different formulation which proves to be useful in the generalization to a sub-semiorder. DEFINITION
following (Al) (A2) (A3)
1. A binary relation R over the set X is called a semiorder of X if the are satisfied (throughout the paper, Vx stands for Vx E X, etc.): Rejexivity: Completeness: Weak Transitivity:
(‘c’x) ww (Vx, y: x # y) (X&J v yRx) (Vr, x, y, z) (rPxIyPz v rIxPyPx
G- rPz)
It may be noted that the requirement rPxPyIz =r rPz is not needed as it is implied by Al-A3. It may also be noted that Al and A3 imply xPyPx 3 xPz. In other words, transitivity of preference is implied. For the definition of a sub-semiorder, let us adopt the following notations. For all x, xe’ (note the superscript; subscripts are used later to denote different x’s) denotes the value of x in dimension i. If xi = y” for all but (possibly) one i E {I, 2,..., n}, we write xDy, where D reads “does not differ in more than one dimension.” DEFINITION 2. A binary relation R on the set X is a sub-semiorder reflexive and complete and satisfies:
(A3’)
Mild
Transitivity:
(xPrRyPs
of X iff it is
(Vx, y: xDy)
v xRrPyPs
* xPs; sPxRrPy
v sPxPrRy
G- spy).
It is clear that (A3’) is a weakening of (A3) since an additional requirement regarding dimensionality is imposed. Hence, a sub-semiorder is a generalization of a semiorder. In other words, a semiorder is a sub-semiorder, but a sub-semiorder is not necessarily a semiorder.
3. THE
NATURE
OF A SUB-SEMIORDER
AND
ITS
UNDERLYING
PREFERENCE
If a binary preference relation R is a semiorder, we may adopt the following notations to represent its “underlying preference structure”: (i) (ii) (iii)
xBy 0 (jr: xPrIy v xIrPy); xAy 0 (not xBy and not yBx); xLy 0 (xBy V XAy)
YEW-KWANG
54
NG
where “B,” ” A,” and “L” may be interpreted as “better than,” “as good as,” and “at least as good as,” respectively. One can show that, for a semiorder, the underlying preference relation L is an ordering, i.e., it is complete, reflexive, and transitive. With some additional requirement, it can then be represented by a real-valued utility function.1 For a sub-semiorder, since the preference relation is necessarily a semiorder only over variation involving one dimension, we adopt the following notations: (iv)
xBy o
(v)
xJy o not xBy and not yBx and
xDy
and 3:
xPrIy
v XIYPY
(vi)
xLy t> xBy v x2y
(vii)
x2y cz 3x,, x2 ,..., x, : x2x12x,
xDy
b.a 2xmAy
where 7n may be any nonnegative integer, including zero (i.e., xJy j xwy) (viii)
xBy 0 3x, , X2 ,..., x, : XLX&va **. ExJy,
where at least one of the z is B (ix)
xEy 0 xJy
v xBy
It may be noted that, if a sub-semiorderis not also a semiorder, it is not appropriate to use L as its underlying preference, and B cannot be appropriately used as “better than,” as it is not transitive (similarly with A). It is for this reasonthat the barred and double-barred notations are designedinstead. The binary relation E may be regarded as the “apparent underlying preference” of a sub-semiorder. The reasonfor adding the adjective “apparent” is because the “intrinsic” underlying preference may not be completely revealed in the definition of J?. This will becomeclearer later, when we have proved a few propositions. PROPOSITION one and
only
1.
For
of the following
a sub-semiorder, is true:
for xAy,
xBy,
any yBx
pair
of alternatives (ok xAy,
xBy,
x, y
such
that
xDy,
yBx).
Proof. Since the definitions of “A” and “B” are exhaustive, it is tautologically true that at least one of xAy, xBy, yBx is true. So we only have to show that only one is true. If this is not so, then there exist somex and y such that, either (i) xAy and xBy; or (ii) xBy and yBx. The former cannot be true from the definition of “A”. If (ii) is true, then 3: xPrRy v xRrPy and 3s: yPsRx v yRsPx. Hence at least one of the following
1 This representation and related problems have been tackled by a number of writers. In particular, Scott and Suppes (1968) prove that a semiorder over a finite set is closed interval representable with constant intervals. Fishbum (1973, Theorem 6) provides necessary and sufficient conditions for a semiorder to be closed-interval representable. These conditions involve some density requirement as well as countability whose meaning is not intuitively clear. In Ng (I 975), representation is proved by adopting standard assumptions on explicit preference and then showing that the underlying preference must also satisfy certain conditions well known to ensure the existence of utility function. The definition of “underlying preference” here is slightly different from the one used in Ng (1975), making the proof slightly more tedious but involving similar arguments.
MULTIDIMENSIONAL
CHOICE
55
has to be true: (a) xPrRyPsRx, (b) xPrRyRsPx, (c) xRrPyPsRx, (d) xRrPyRsPx. It is easy to check that each of these violates (A3’) since xDy. Q.E.D.
For a sub-semiorder, x&x, .*. Lx, if none(at least one)of the E’s is B.
PROPOSITION 2.
xIAx,(xIBx,)
3 (i) not x,Bxl
; (ii) x&x,
, with
Proof. First take the case where none of thee is B, so x,Ax, ... 2x,, . If the proposition is not true for this case, we have x,Bx, or x,Bx, . Then 3r: x,,,PrRx, v x,RrPx, v xlPrRx, v x,RrPx,,, . Suppose x,PrRx, is the case. (Other cases are similarly proved and will not be repeated.) Compare x,-r with r. From A2, either x,,-,Pr, x,-$r, or rPx,-, . The last two cannot be true because otherwise x,PrRx,-, * x,Bx,-, , violating PI (i.e., Proposition l), since x,Dx,-, . So x,-~ PrRx, * x,-,Bx, . Proceed to compare x,,+* with r in a similar fashion, and we can establish that x,-sBxI . Repeating the process, we get x,Bx, , violating the given fact that x,&(Pl). So x,4x, and not x,Bxl establishing the proposition for this case. Now take the case where at least one of the e is B, and take this to be xiBxi+r . So 3r: xiPrRx,+, v x,RrPxi,, . Again suppose xiPrRxi+, . (The other case is similarly proved.) Compare r with xi+a . It cannot be true that x,+sPr since Xi+rExi+a . So, rRx,+%, giving xiBx,+, . Repeating the process, we have xiBx, . Then compare xi-r with r, establishing xi-,Bx, . Repeating the process, we get xlBxm . To show that x,Bxl cannot be true, we do the similar comparison as we did in proving the first part of the proposition. Q.E.D. COROLLARY. The apparent underlying preferenceof a sub-semiorder is noncyclic, i.e., XEXJ -*aLx, +- not x,8x. PROPOSITION 3. The apparent underlying preference9, of a sub-semiorder is a partial ordering (satisfpng rejexivity and transitivity).
Proof. Reflexivity is easily established, since it is easy to see that x,4x and xAx. Either one of these is sufficient for xAx. Transitivity is also not difficult to establish. If xEyrz, taking y to be one of the connecting links between x and z, we have XJ%. Q.E.D. PROPOSITION 4. In general, the apparent underlying preferenceof a sub-semiorder is not an ordering.
Proof. Consider an explicit preference relation R” on E2 (Euclidean plane) defined by: yPOx u (yl > x1 + k & y2 3 x2 - k) v (y2 > x”+k&yl>xl-k)v(y1+y2> xl + xa + k) where k is a constant. yPx o not yPOx and not xPOy. This is illustrated in Fig. 2. The idea is that the individual can observe a difference of larger than k in each dimension. If there is an observable difference in each of the two dimensions, the difference in the differences has to be larger than K for the individual to prefer one to another. Incidentally, the set {y: yPoxj is closed at the boundaries indicated by the dotted lines. Any point (excluding the lower extreme points) on these dotted lines is a member of {y: yPox}, but any arbitrary neighborhood of it contains a point not in {y: ypOx}. This is
56
YEW-KWANG
NG
clearly incompatible with the usual continuity requirement (e.g., see Debreu, 1959, p. 56). Hence, an interesting technical problem is raised which, however, need not concern us here.2
{y
: xR”y)
FIGURE
2
From the definition of RO, it is clear that it satisfies reflexivity and completeness. To check that mild transitivity is also satisfied, note that x differs from y in no more than one dimension for this property (i.e., A3’). Since the definition of RO is symmetrical about the two dimensions, we may take x1 = y1 without loss of generality. From the definition of PO, xP”r implies (x’ >
y1
+ k & x2 >
y2
- k) v (x” >
~2
+ k&xl
v (x’ + x2 >
y1
>
11
- k)
+ r2 + k).
(1)
And rROy implies (yl <
y1
+ k v y2 < r2 - k) & (y2 <
y2
+ k v y’ < r1 - k)
& (yl + y2 G Since x1 = yl, it is easy that xDy, xP”rRoy Z- xi > this result, it can easily xPOrROyP% rj xP%. Since This demonstrates that R” 2 If we make
the model
y1
+
r2
+ k).
(2)
to deduce from (1) and (2) that x2 > y2. Hence, for x, y such y”. Similarly, it can be shown that xROrPOy s- xi > yi. With be seen that mild transitivity is satisfied. For example, xi > yi, yP% must imply xP% from the definition of PO. is a sub-semiorder on E2.
stochastic,
I think
the discontinuity
will
be eliminated.
MULTIDIMENSIONAL
CHOICE
To show that the apparent underlying preferenceslO to show that it is not complete. First note that &Oy or due south of x, i.e., if x1 = yl, x’3 3 y2. Thus, X&J (including due south and due west) of x, i.e., xi > yi consider a point x which is northwest or southwest violating completeness.
57
is not an ordering on E2, it suffices implies that y = x or y is due west implies that y = x ory is southwest i = 1, 2. Now for any given point x, of x. Then, not xE”z and not zz”x, Q.E.D.
The fact that the apparent underlying preference of a sub-semiorder is not, in general, an ordering does not mean that its “intrinsic” underlying preference cannot be an ordering. Thus in the example used in the proof of Proposition 4, it is reasonable to regard the intrinsic underlying preference i; as defined by xLy 0 x1 + x2 > y1 + y2, which is an ordering on @. However, the ease with which we discover the intrinsic underlying preference 20 from R” is deceptive. This is due to the highly simple nature of the case in question. First the relevant set E2 is only two-dimensional and convex. Second, R” is symmetrical about the two dimensions (equal weights) and monotonic. Third, the discrimination threshold k is constant throughout. In the general case where the set of alternatives may involve many dimensions and may not be convex, where the discrimination threshold may vary, and where different dimensions may have disimilar effects on preference, it is very difficult, if not impossible, to discover the intrinsic underlying preference. Moreover, even in the above simple case, the definition of the intrinsic underlying preference is in terms of the objective measures (the numerical values of x1 and x2) and not in terms of the explicit binary preference relation P and I. I must admit that I have not been able to construct a satisfactory definition of the intrinsic underlying preferences of a sub-semiorder in terms of the explicit preference, for the general case anyway. It can, however, be shown that there is a B-maximal element x in X (i.e., yBx for no y in X) for a sub-semiorder that satisfies a rather mild condition. DEFINITION 3. A binary relation ) on a set X is said to satisfy the Well-Ordering Condition if, for each subset of X on which ) is a linear order (as defined in Fishburn, 1972, p. 29), there is an element in X which is ) maximal on that subset. In the example used in the proof of Proposition 4, each straight line is linearly ordered by B. But there does not exist a maximal element if X is unbounded. If we are confined to compact (bounded and closed) sets, then the Well-Ordering Condition is very likely to be satisfied. For unbounded sets, the condition may also be satisfied if we have satiation.
5. If R on X is a sub-semiorder whose apparent underlying preference B satisfies the Well-Ordering Condition, then there is an x in X that is B-maximal
PROPOSITION
relation on X.
Proof. The proof is a straightforward application of Zorn’s lemma (for a discussion of which, see Fishburn, 1972, Chap. 6), since, from Proposition 2, B on X is a strict partial order if A on X is a sub-semiorder. Q.E.D.
58
YEW-KWANG
4.
CONCLUDING
NC F&MARKS
Multidimensional choices may, but need not, necessarily violate the requirement of a semiorder. If a number of dimensions combines to exert their influence on preference in a related way, the preference relation may be a semiorder over these dimensions. For example, my preference for a dish of sweet and sour fish may be a function of both sweetness and sourness. But the two dimensions may combine to assert their influence such that my preference is a semiorder. This is likely to be so because both sweetness and sourness are tasted at the same instance. For other cases, such as the location and the appearance of a house, the different dimensions are likely to be more independent of each other and hence preferences are more likely to violate the requirement of a semiorder. Many problems associated with the weakening of preference from an order into a semiorder can be resolved relatively easily since the underlying preference of a semiorder is an order. The weakening of preference into a sub-semiorder seems to raise more formidable problems since its apparent underlying preference is not generally an order and its intrinsic underlying preference is difficult to define in an operationally meaningful way. However, it is reasonable to regard the intrinsic underlying preference as that pattern of preference if the individua1 has perfect discriminating ability. The explicit binary preference is a reflection of this intrinsic preference subject to imperfect discriminatory ability. Hence, the explicit preference, while not a perfect representation of the intrinsic preference, is nevertheless “close” to it in some sense. Hence, if we are careful enough in interpreting the explicit preference, and if we adopt some helping devices, such as confming direct comparisons to variation over one dimension only and using indirect comparison (Ng. 1975, Sect. 9) for variations over many dimensions, it would not be impossible to get a close enough representation of the intrinsic preference. The imperfection in preference discrimination may be due to two different reinforcing causes. First, it may be due to imperfect knowledge or inadequate anticipatory ability. For example, I may be unable to decide whether I prefer house A to house B or the other way round and hence register an indifference between them because I am not sure which house will suit my requirements better at the moment of choice. However, if I am given a chance to live in both houses for a sufficient period of time, I may be able to be quite sure that I prefer A to B. This is similar to what Georgescu-Roegen (1936, 1958) calls the threshold in choice. This threshold is, as a rule, a decreasing function of the amount of time allowed for the choice. Georgescu-Roegen believes that, at “the limit when the time of experimenting is infinite, the threshold is zero” (1966, p. 152). Preference threshold need not be zero if we recognize a second cause of imperfect discrimination. This is what I call “finite sensibility,” which refers to the fundamental limitation in human capability of feeling. If the amounts of sugar in two cups of coffee are close enough, one cannot tell a difference no matter how carefully one tastes it. Thus, even if we rule out by assumption imperfect knowledge, uncertainties, etc., we may still have imperfect discrimination due to finite sensitivity. The distinction of imperfect discrimination as due to imperfect knowledge and as due to finite sensibility is of some importance. For some problems we may be more
MULTIDIMENSIONAL
CHOICE
59
interested in individual preference representing his actual feeling without “distortion” by imperfect knowledgeand the like. Now it canbe arguedthat the feeling of an individual at any moment of time is either affected by one factor (dimension) or affected by a number of factors which combine to exert their influences such asthe caseof sweetand sour fish discussedabove, since, by definition, an individual hasonly one mind. Hence, individual feeling at any moment of time is a semiorder. A fuller development of this argument as well asthe consideration of the complication introduced by the element of time can only be dealt with in another paper. Apart from this extension, the concept of sub-semiorder seemsto provide a few other challengesfor further development such as making the model stochastic (Lute, 1958), analyzing the intrinsic underlying preference, conducting experiments to ascertain the usefulnessof the model, and comparing the model with other psychological theories of choice such as the elimination-by-aspects theory (Tversky, 1972), the similarity-between-stimuli theory (Rumelhart and Greeno, 1971), etc.
REFERENCES
G. Theory of value. New Haven, Conn.: Yale Univ. Press, 1959. P. C. Muthenatics of decision theory. Atlantic Highlands, N. J.: Mouton. 1972. P. C. Interval representations for interval orders and semiorders. Journal of Mathematical Psychology, 1973, 10, 91-105. GEORGESCU-ROEGEN, N. The pure theory of consumer behaviour. Quarterly Journal of Economics, 1936, 50, 545-593. [Reprinted in Analytical economics. Cambridge, Mass.: Harvard Univ. Press, 1966).] GEORGESCU-ROEGEN, N. Threshold in choice and the theory of demand. Econometrica, 1958, 26, 157-l 68. [Reprinted as above.] LUCE, R. D. Semiorders and a theory of utility discrimination. Econometrica, 1956, 24, 178-191. LUCS, R. D. A probabilistic theory of utility. Econometrica, 1958, 26, 193-224. Nc, Y.-K. Bentham or Bergson I Finite sensibility, utility functions, and social welfare functions. Review of Economic Studies, 1975.42, 545-570. RUMELHART, D. L., and GRBBNO, J. G. Similarity between stimuli: An experimental text of the Lute and Restle choice models. ]ournaZ of Muthemudcal Psychology, 1971, 8, 370-381. SCOTT, D., AND SUPPBS, P. Fundamental aspects of theories of measurement. Journal of Symbolic Logic, 1958. 23, 113-128. T~ERSKY, A. Intransitivity of preferences, Psychological Review, 1969, 76, 31-48. TVERSKY, A. Elimination by aspects: A theory of choice, Psychological Review, 1972, 79, 281-299. DEBREU,
FISHBURN, FISHBURN,
RECEIVED:
February 24, 1976