AN AXIOM B.

OF

MATHEMATICAL

UTILITY

THEORY

G. M i r k i n

UDC 518.9

Lately, in the mathematical theory of utility, an approach has developed involved with the renunciation of r e q u i r i n g t r a n s i t i v i t y of the relationship of subjective indistinguishability (for example, in [115. This r e q u i r e m e n t ("if a is indistinguishable f r o m b, and b is indistinguishable f r o m c, t h e n a is indistinguishable f r o m c"5 provides for the existence of nonintersecting sets (surfaces5 of s o - c a l l e d indifference, consisting of objects with identical "utility," i.e., indistinguishable f r o m the point of view of their utility ([2], w8.6). Renunciation of this postulate of transitivity is connected with the acceptance of a threshold of distinguishability as c o n c e r n s subjective choice. Indeed, it may turn out that, although for certain objects a and b the difference in t h e i r preferabilities is less than the distinguishability threshold, so that a and b a r e subjectively r e c o g n i z e d as indistinguishable, and the analogous situation holds for b and c, still, the total difference in the p r e f e r a b i l i t i e s of objects a and e might exceed the threshold of distinguishability, so that one of them is p r e f e r a b l e to the other. The violation of transitivity of the indistinguishability r e l a t i o n ship related t o this effect leads to a logical difficulty of the well-known type of the "sophism of the pile of beans" [3], to lack of clarity and indistinctness in the meaning of many concepts used by people. The p r o b l e m a r i s e s of an a b s t r a c t description of the relationship of p r e f e r e n c e for which the r e l a tionship of indistinguishability is not n e c e s s a r i l y transitive. The f i r s t step in the solution of this p r o b l e m was made in [4, 5] (cL also [6], p. 465. The authors considered the case when c o m p a r i s o n of objects is made in a c c o r d a n c e with one indicator, with the threshold of distinguishability being identical for all obj eels. A p r e c i s e formulation of their r e s u l t is as follows. We denote by J5 the set of all intervals of the real axis of identical length 5 >0. Consider the following binary relation ~ on set J5 : x ~ g if and only if interval x lies completely to the right of interval y; i.e., its left end is g r e a t e r than the right end of interval y. Let A be an a r b i t r a r y finite set, and let P C A • A be an antireflexive binary relation on A (i.e, (a, a)q~P for each ac-A). The fact that s y s t e m (A, P) is h o m o m o r p h i c a l l y embedded in s y s t e m (J~, 2), means that t h e r e exists a singlevalued mapping g : A ~ J5 such that (a, b) E P ~. g (b) ~ g (a)

(15

In turns out [6] that for s y s t e m (A, P) to be h o m o m o r p h i c a l l y embedded in s y s t e m (Js' ~'), it is n e c e s s a r y and sufficient that P be antireflexive and satisfy the conditions (a, b) E Pand (c, d) E P -+ (a, d) E P or

(c, b) E P,

(2)

(a,b) EPand~b,c)EP = . ( a , d ) E P or

(d,c)EP

(3)

Conditions (2) and (3) provide a description of the p r e f e r e n c e relationship P C A x A arising in the case when c o m p a r i s o n is by one indicator with indistinguishability threshold 5; with this, the corresponding indistinguishability relation I = PUP 1 is, in general, intransitive. In the p r e s e n t note we consider the case when c o m p a r i s o n is by one indicator, but when the threshold of indistinguishability may be different for different objects. This basic result is the p r o o f that axiom (2) provides the description of the corresponding antireflexive p r e f e r e n c e relation. This explains, in p a r t i c u l a r , the c i r c u m s t a n c e that, in the existing works on utility theory, only this condition is utilized, while (35, in fact, is not used (ef., for example, [1]). T r a n s l a t e d f r o m Kibernetika, No. 6, pp. 60-62, N o v e m b e r - D e c e m b e r , 1970. Original article submitted July 28, 1969. 9 1973 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

776

Thus, let A be an finite set, and P C A x A an antireflexive binary relation on A. THEOREM 1. Relation P satisfies condition (2) if and only if it satisfies the condition

P(a>~P(b)

or P ( b ) ~ P ( a ) ,

(4)

where P ~P(b) and P ( b ) ~ P lust as it is on Js"

being defined on them

THEOREM 2. F o r s y s t e m (A, P) to be i s o m o r p h i c a l l y embedded in s y s t e m (I > ) , it is n e c e s s a r y and sufficient that antireflexive relation P C A x A satisfy condition (4). P r o o f . It is obvious that condition (4) holds for ~>, T h e r e f o r e , if s y s t e m (A, P) is i s o m o r p h i c a l l y embedded in (I, > ) , then P also s a t i s f i e s (4). Now, let P c A x A be antireflexive and Satisfy condition (4). We shall p r o v e that s y s t e m (A, P) is i s o m o r p h i c a l l y embedded in (I ~>). We note f i r s t of all that condition (4) means in fact that the entire set A is partitioned into nonintersecting nonvacuous c l a s s e s A1, A 2. . . . . Am in such fashion that, first, for each i (i= 1 . . . . . m), P ~ ~ P (am) , where the 1 ., 9 9 9 inclusions a r e always s t r i c t . We now fix element bi~P (ai~ Ai; i = 1 . . . . . --I

--I

--1

m - 1 ) , and we con-

--1

s i d e r the sets P(bL) . . . . . P ( b _ ~ ) , where P(b) = {al(a,b)EP} . The fact that aEP(bl)(i= 1 . . . . . m-l) means that biEP
of the sets P i), so that a E L~ Ai have proven the f o r m u l a

-, P(~,)

r = UA~

(5)

(i= I . . . . . r n - - 1) -1

We now p r o v e that each nonvacuous set of the f o r m

P (b>(bEA) can be p r e s e n t e d by f o r m u l a (5).

Indeed, the fact that a E Pi b) me ans b ~ P< a >. If f o r some i(i = 1 . . . . . m) a EA i, then, by the definition of sets A 1. . . . .

Am, f o r a n y

a'EUAi, bEP(a'>,i.e.,a'-CP
UAi~P(b)

.

/=1 --1

It is obvious that this inclusion b e c o m e s an identity for the maximal i (over all a E P
Pl(b) = 6 A; =

A, so that (b, b)~P, which would contradict the anti-

i=l

reflexivity of P .

The a s s e r t i o n is proven.

Its c o r o l l a r y is the fact that set A is partitioned into the nonvacuous nonintersecting sets B l , . o., Bm_l, B 0 which satisfy the following conditions: (a) b l a B 1. . . . .

bm_lC-Bm_l; _I

(b) for each i(i = 1 . . . . . 1

(c) q ) = ~ l < b 0 ) C P

I

I

m - 1 ) P = P for any b~EB i and ~ = g) (for b0-cB0); _l



Thus, the sets B0, B 1. . . . . Bm_l, play, with r e s p e c t to P l = { ( b , a)l (a, b)EP}, the s a m e role as do sets A1, . . . , A m with r e s p e c t to P. The quantity m of sets A i for relation P is called its rank. We have obtained the following a s s e r t i o n which is of independent interest.

777

-1 THEOREM 3. If antireflexive binary r e l a t i o n P s a t i s f i e s condition (4), then relation P also s a t i s f i e s (4), with the r a n k s of P and ~1 coinciding. To given a ~ A we m a k e c o r r e s p o n d the following n u m b e r s , l a and ra:

l~-]+~,

l

if

"~EBi(]-~.O,I . . . . . m - - l ) ,

(6) r --=i, if

a E A ( i = l . . . . ,m).

We shall show that for each aEA I a < ra, i.e., the p a i r (l a, ra) defines an i n t e r v a l of the r e a l axis. i

Indeed, since aEBj, then ~-I < a ) = ~ l
m-l),

and, in view of (5), P t
T h e r e f o r e , if aEA i then, by virtue of the antireflexivity of P, it is i m p o s s i b l e that i- j and, since i a n d j a r e natural n u m b e r s , i > j + 1/2~ Thus, we can define the mapping g : A ~ I by the f o r m u l a

(7)

g (a) = (l~, r,)

Obviously, g(a) is univocally defined for all aEA, since the s y s t e m s {A1. . . . . p a r t i t i o n s of set A.

Am} and {B0. . . . .

Bm_l) a r e

We now show that g is a h o m o m o r p h i s m of s y s t e m (A, P) in s y s t e m (J, > ) -

Indeed, l e t (a, b ) E P .

We shall show that g ( b ) ~ g ( a ) , i.e., l b > ra~ L e t / b = j + 1/2. Then

--I

i

P ( b ) = U Ak,

]

which m e a n s that

aE[_J Ak, so that r a -
C o n v e r s e l y , let g(b) ~-g(a} . We now show that (a, b ) E P . ]

Indeed, in this c a s e l b > r a. Let l b = j +

--!

1/2, r a = i. Then i -< j, s o that a E U Ak = P (b), i.eo, (a, b)EP, q~ We so redefine mapping g that it becomes o n e - t o - o n e , while still r e m a i n i n g a h o m o m o r p h i s m . This is not difficult to do in view of the finitude of set A, by the following path, for example. Let e l e m e n t s al . . . . .

a n be such that g ( a l ) = g ( a 2 ) = . . . = g(an). We define mapping g* as follows: (a ~ al . . . . . an), i--1 ! g * ( a i ) = ( l a i + - ~i ---- , r1~ i + --2-n- ) (i=1 ..... n). Ig* (a)

:

g (a)

Since the shift of all the i n t e r v a l s g*(a i) with r e s p e c t to the i n t e r v a l g(al) is l e s s than 1/4, the p r o p e r t y of h o m o m o r p h i s m of the mapping is retained: it is p o s s i b l e to violate the mutual positioning of the i n t e r v a l s g(a), by the definition of g, only by a right shift in e x c e s s of 1/2. P r o c e e d i n g analogously each t i m e t h e r e a r e s e v e r a l e l e m e n t s having identical i m a g e s , we a r r i v e at a o n e - t o - o n e h o m o m o r p h i s m , i.e., the i s o m o r p h i s m g*o T h e o r e m 2 is p r o v e n . We note in conclusion that all the reasoning, with insignificant modifications, r e m a i n s valid in the c a s e when s e t A is of d e n u m e r a b l e cardinality~ Another r e m a r k r e l a t e s to the fact that a binary r e l a t i o n satisfying condition (4) is well known in the theory of b i n a r y r e l a t i o n s as the F e r r e t s relation (cf. [7], T h e o r e m 11.4.1.). T h e o r e m 2 can be treated as a new c h a r a c t e r i s t i c of antireflexive F e r r e r s r e l a t i o n s on finite and d e n u m e r a b l e s e t s . LITERATURE

CITED

11 2.

P. Fishburn, " S e m i - o r d e r s and r i s k y choices, ~ J. Mathem. Psychology, 5, pp. 315-361 (1968). S. Karlin, M a t h e m a t i c a l Methods and Theory in G a m e s , P r o g r a m m i n g and Economics, Addison-Wesley

3.

(1959). E. Borel, Probabilities and Life, Dover (1962)o

778

4.

5. 6. 7.

R. D. Luce, " Sem i - or de r s and a theory of utility discrimination," Econometrica, 24,178-191

(1956). D. Scott and P. Suppes, "Foundational aspects of theories of measurement," J. Symb. Logic, 23, 113-128 (1958). P. Suppes and J. Zines, "Fundamentals of measurement theory," in: Psychological Measurement [Russian translation], Moscow (1967). O. Ore, Theory of Graphs, AMS Colloquium Publ., Vol. 38 (1962).

779