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Multiattribute expected utility without the Archimedean axiom
JOURNAL
OF MATHEMATICAL
~u~tia~ribute
PSYCHOLOGY
573-591 (1992)
36,
Expected Utility without
the Arc~i~edean
Axiom
PETER C. FISHBURN AT&T
Bell Laboratories
AND IRVING H. LAVALLE Tulane
University
Two prominent themes in expected utility theory that were developed many years ago are non-Archimedean expected utility and independence conditions in multiattribute real-valued expected utility. This paper integrates these themes in an investigation of independence conditions for multiattribute vector-valued expected utility. The dimensionality of the utility space need bear no relationship to the number of attributes. The multiattribute independence conditions examined include value independence, utility independence, generaked utility inde~nden~, and two-factor mutual utility independence. Attribute de~om~sitions of components of utility vectors implied by some of the conditions are natural generalizations of forms that arise in the real-valued setting. 0 1992 Academic Press, Inc.
1. I~moDucT10N This paper develops implications of independence conditions for utility functions in multiattribute expected utility theory without the usual Archimedean axiom. Its independence conditions were first examined by Fishburn (1965, 1974), Pollak (1967), Keeney (1968), Raiffa (1969), and Fishburn and Keeney (1974) for von Neumann-Morgenstern expected utility follo~ng the axiomatizations of additive utility over Cartesian products by Debreu (19~), Lute and Tukey (1964), and others (see Fishburn, 1970; Krantz, Lute, Suppes, & Tversky, 1971; Keeney and Raiffa, 1976; Wakker, 1989). We depart from earlier work in expected utility by omitting or weakening its Archimedean axiom. This changes the setting for consideration of multiattribute independence conditions from real-valued to vectorvalued expected utility ordered lexicographically (Hausner, 1954; Chipman, 1960; Fishburn, 1971a, 1982). No particular relationship is presumed between the number of levels in the lexicographic hierarchy and the number of attributes in the outcome set. Two reasons motivate our generalization to lexicographic expected utility. The first is foundational. Archimedean axioms do not have the same normative status Address all reprint request to Peter C. Fishburn at AT&T Belt Laboratories, Murray Hill, NJ 07974.
573 0022-2496192 $5.00 Copyright ci 1992 by Academic Press. Inc. All rights of reproduction m any form res.xved.
574
FISHBURN
AND
LAVALLE
as axioms like transitivity and linearity. They serve the technical purpose of ensuring real-valued representations of preference, and hence it is important foundationally to consider their absence (Narens, 1985). The second reason is practical. Value hierarchies can induce defensible preference patterns that violate Archimedean conditions (Chipman, 1960; Georgescu-Roegen, 1966). One might associate such patterns with relative importance notions for attributes or enrich the lexicographic structure with threshold considerations (Tversky, 1969; Lute, 1978; Fishburn, 1980). An example of values-induced non-Archimedean preferences consists in observing that the ethical tenet “it is wrong to take an avoidable chance at a wrong outcome” makes every lottery that offers positive probability of a wrong outcome inferior to any undesirable but non-wrong alternative. The next section summarizes the lexicographic generalization of the linear utility model of von Neumann and Morgenstern (1944) which provides the setting for the rest of the paper. Its representation is
where > is a preference relation on a set P of probability distributions p, q, ... defined on a outcome set X, U(P) E II?, and >L denotes the usual lexicographic order on R”. In addition, U is a linear function whose uniqueness properties are central to some of our derivations of consequences of independence conditions. The lexicographic linear utility model does not presume any special structure for the outcome set X, but to develop the multiattribute theme we assume in Sections 3 through 5 that X=X,xX,x ... xX,. Section 3 notes that the value independence condition of p indifferent to q whenever p and q have the same marginal distribution on Xi for i= 1, .... n leads to the additive decomposition UP) = %(Pl) + %(P2) + ... + %l(P,h
where pi is the marginal of p on Xi and ui is a vector-valued linear function on the set of all such pi. A derivation of the same additive form with XEX,XX,X ... xX, in the context of a lexicographic generalization of Savage’s (1954) theory of decision under uncertainly appears in LaValle and Fishburn (1991). The additive form for real-valued utility on subsets of Cartesian products is discussed in Fishburn (1970, 1971b) and Wakker (1989). Section 4 considers utility independence conditions after attributes are partitioned into two sets with X= Y x Z. The simplest condition (Pollak, 1967; Keeney, 1968) says that (py, z)> (qy, z)-=- (py, z’)> (qy, z’), where (py, z) denotes the distribution in P that has marginal p y on Y and gives z E Z with probability 1. This condition of Y utility independent of Z allows U on P, x Z to be written as WPY, z) = A(z) VP,,
ZCJ + b(z),
where z0 is fixed in Z, b E 08”, and A(z) is a real m x m matrix of a type described later. We then examine a stronger version of the basic conditions as well as a
MULTIATTRIBUTE
575
EXPECTED UTILITY
relaxation known as generalized utility independence (Fishburn, 1974; Fishburn and Keeney, 1974). These modifications imply the same form for U(p,, z) but have different implications for the ,4(z) matrices. Since the utility independence relation is not symmetric, Section 5 looks at the joint effect of Y utility independent of 2 and Z utility independent of Y. We illustrate mutual utility independence with examples and then consider the case in which the Y-conditioned-on-Z form and the Z-conditioned-on-Y form both have full dimensionality, the same as the dimensionality d of U. In this case, with U(y, z) = WY, z), *.., Ud(Y, z)), each Uk on Y x Z decomposes as Uk(Y,
z,
=fk(Y)
+ gktZ)
+
i ;=I
C$fi(Y)
i j=l
gj(Z),
where they, are real-valued functions on Y, the g, are real-valued functions on Z, and the ci. are constants. We also prove that d can exceed the sum of the effective dimensions of the Y-conditioned-on-Z and Z-conditioned-on-Y forms. The paper concludes with a summary and open problems. In many places throughout, vectors can be viewed either as row or column vectors. However, when matrix multiplication is used, as in VP) = A U(P) + b, the vectors are assumed to be column vectors. 2. LEXICOGRAPHIC
LINEAR UTILITY
This section develops the aspects of lexicographic expected utility needed for ensuing sections. Our only requirement on X in the present section is that it contain at least two outcomes. Throughout the paper, > is a binary is preferred to relation on the set P of all finite-support probability distributions on an outcome set X. The convex (0 < L < 1) combination Ap + (1 - L)q of p, q E P is written as plq, so (plq)(x) = lP(X) + (1 -A) q(x ) for all x E X. We assumethat > is not empty, extend it to X by defining x > y as p > q when p(x) = q(y) = 1, and define its indifference relation N on P by p-4
A functional
if not(p > 4)
and
u: P + R is linear if, Vp, q E P, VO< A< 1, u(plq) = AU(P) + (I-
A function
not(q > P).
U: P + R” is linear if it is linear
1) u(q).
in each coordinate.
That
is, if
576
FISHBURN
AND
LAVALLE
u= (U’, .... Urn) with Uj: P -+ aB for each j, then Uj(pnq) = nUj(p) + (1 -A) U’(q) for all j. We define >L between real vectors a = (a,, .... a,) and b = (b,, .... b,) by a>,b
and the min j for aj # bj has aj > bj.
ifa#b
Hence U(p) > L U(q) if Uj(p) # U’(q) for some j, and U’(p) > U’(q) for the smallest such j. We say that > on P is (i)
a weak order if it is irreflexive and negatively transitive
[p>
q*p>r
or r>ql;
(ii) linear if Vp, q, r E P, VO < A < 1, p > q =>pLr > qlr; (iii) Archimedean if Vp, q, r E P, p > q > r *par > q and q >p/?r a, P E (0, 1).
for some
In addition, N on PislinearifVp,q,r~P,VO<~ on P is an Archimedean linear weak order if and only if there is a linear U: P -+ [w that preserves >, i.e., Vp, q E P, p > q o u(p) > u(q). Moreover, such a u is unique up to positive affine transformations: 0: P + R is also linear and > preserving if and only if there are real numbers a > 0 and j3 such that v(p) = au(p) + B
for all p E P.
When u is extended to X by U(X) = u(p) when p(x)= 1, linearity and the linitesupport hypothesis give u(p) = Cp(x) U(X) along with the familiar expected utility representation P~4~x~xPw4x)~
c
&)4x)*
XCX
This representation and X = X1 x . . . x X, provide the context for the independence conditions in our opening references. Those conditions lead to widely discussed decompositions of u(xi, .... x,), including additive and multiplicative forms. Keeney and Raiffa (1976) and successors describe many applications of these forms in decision analysis. The generalization of expected utility used in ensuing sections is summarized by Assumption 1. There is a positive integer m and a linear function such that Vp, q E P, p>q*
U(P)‘L
U: P -+ W”
U(q).
This presumes that > on P is a linear weak order and that N on P is linear (Hausner, 1954; Fishburn, 1971a, 1982). If X is finite, these axioms imply Assumption 1 with 1
MULTIATTRIBUTE
EXPECTED UTILITY
577
are given in Chapter 4 of Fishburn (1982). A representation which allows an infinite type of hierarchy is described in Hausner (1954), and relationships to ordered vectors spaces are discussed there and in LaValle and Fishburn (1992). We assumehenceforth that U and V are finite-dimensional linear functions on P that satisfy the lexicographic representation of Assumption 1. Call U parsimonious if its m is as small as possible. The uniqueness theorem for parsimonious functions is LEMMA 1. Suppose U and V are parsimonious into Rd. Then there is a d x d lower-triangular real matrix A with positive diagonal elementsand a b E W”, such that
V(p)=AU(p)+b Proof
forallpEP.
See Theorem 4.4 in Fishburn (1982).
1
If U is parsimonious and A and b are as just specified, then it follows easily that AU + b is parsimonious and satisfies the representation of Assumption 1. When u = (U’, ...) U”), V= (V’, .... Vd), A = [aik], and b= (b,, .... bd), the equation of Lemma 1 says that V’(p)=
i
ajkUk(p)+bj,
j = 1, .... d.
k=l
When d = 1, this reduces to v = CIU+ p, a > 0, for real-valued linear utility. We also consider nonparsimonious functions U: P + R’ with I> d when d is the dimension of a parsimonious function. The transformation matrix A for mappings between possibly nonparsimonious functions requires a special definition. DEFINITION
1 < d < min{m, indices l
1. An m x 1 real matrix A is a L-matrix of dimension d if l} and there are row indices 1
for i= 1, .... d,
(ii)
aj,, = 0
and
ajiki > 0
for all k > ki;
for l
for all k > min{k,
: ji >j>.
We refer to each (ji, ki) in the definition as a quasi-diagonal cell, to each ji as an essential row, and to each ki as an essential column. A partial picture of a L-matrix is
ji
ki . ..+oo
ki+l 0
o-e.0
j
. .. . . . 0
o...o
. . . +
o...o.
ji+l...
578
FISHBURN
Condition says that diagonal conditions be zero. LEMMA
AND
LAVALLE
(i) specifies positive ajk quasi-diagonal cells and if row j is not essential and the first essential cell (j’, k’) then row j has zeros from column k’ dictate that everything above and/or to the right
2.
Suppose a parsimonious
zeros to their right; (ii) row after j has quasion. In combination, the of a quasi-diagonal cell
function for the representation
1 is d-dimensional and that U: P --+ R’ and V: P --f R”’ with dd min{m, is an m x I real L-matrix
A of dimension d and a b E R” such that V(p)=AU(p)+b
ProojY
See Theorem
of Assumption I}. Then there
forallpEP.
1 in LaValle and Fishburn (1991).
Suppose parsimonious Let
functions
are d-dimensional
1
and U = (U’, .... U’) with
l>d.
0( Uj) = {(p, q): U’(p) > U’(q), Uk(p) = Uk(q) for all k resolved by UJ for U(p) >L U(q). When @(Vi) # @, we say that j or Uj is essential. Otherwise, if Uj resolves no preference, j or Uj is inessential. It is proved in LaValle and Fishburn (1991) that exactly d of the Uj are essential. Moreover, as suggested by Lemma 2, if j, < .‘.
for i = 1, .... d.
That is, the ith essential Vj resolves exactly the same preferences as the ith essential Vi, and the quasi-diagonal cells of the L-matrix A for Lemma 2 will be the (ji, ki). The equation of Lemma 2 says that V’(p)=
i
ajkUk(p)+bj,
j = 1, .... m.
k=l
Taking differences, we have v(p)-
v(q)=
i k=l
ajk[uk(P)-
uk(q)lv
which along with the properties of A from Definition 1 and the presumed representation by U shows that AU + b also represents > lexicographically. To summarize, suppose U: P+ R’ is a linear function that represents > lexicographically and d < I is the dimension of a parsimonious linear function that represents > lexicographically. Then: (a)
exactly d dimensions of U are essential;
MULTIATTRIBUTE
579
EXPECTED UTILITY
(b) if d < m, A is an m x 1 L-matrix of dimension d whose essential columns correspond to the essential dimensions of U, and b E R”, then AU + b is a linear function that represents > lexicographically and has exactly d essential dimensions; (c) if d < m, then the only linear K P + W” that represent > lexicographitally are those that relate to U by I’= AU+ b for an A and b as just specified in (b). Conclusion (c) follows immediately case of Lemma 2 for m = 1= d.
from Lemma
2, and Lemma
1 is the special
3. VALUE INDEPENDENCE
We assume henceforth that X= X, x . . . x X, with n 2 2. Also, as noted earlier, > on P is nonempty, and U and V are linear functions on P that represent > lexicographically. In addition, d is the minimum dimension of all linear functions that satisfy the representation of Assumption 1. Let P, be the set of all linite-support probability distributions on Xi and for any r E P let ri denote the marginal distribution of r on Xi, i = 1, .... n. The Xi are value independent if Yp, q E P, (P 1, *.a,P”) = (41, ...>4J *p - 4. THEOREM 1. Suppose U: P + R’, 12 d. Then the Xi are value independent if and only if there exist linear ui: Pi + @for i = 1, .... n such that, for all p E P,
VP) = U,(P,) + ... + hl(P”). Supposethis decomposition holdsfor U along with, Yp EP, VP) = V,(P,) + ... + Vn(Pnh where V, v,, .... v,: P, P,, .... P, + R”, m 2 d. Then there is an m x I real L-matrix A of dimension d and b, , .... b, E BY’ such that, Vi E { 1, .... n}, Ypi E Pi, Vi(pi) = Aui(p,) + bi. ProojI See Theorems 2 and 3 in LaValle and Fishburn (1991). Lemma 2 in the preceding section is used in the second part to give cvi(Pi)=
V(p)=AU(p)+b=ACui(pi)+b,
so for any fixed POEP we get v,(p,) = Au,(p,) + bi, where bi= b + A~,,,u,(p~) Cj+ivj(P,“h
I
-
580
FISHBURN
AND
LAVALLE
When I= d = 1, the ui: Pi + Iware unique up to similar positive affine transformations clui + pi, c(> 0, as in Theorem 11.1 in Fishburn (1970). When I = m = d > 1, the ui: Pi + IV’ are unique up to similar alline transformations Aui + b, with A a d x d lower-triangular real matrix with positive diagonal elements and bi E Rd (see Lemma 1). When min{ I, m} 2 d 2 1, we get the set of admissible transformations specified in the second part of Theorem 1 with the essential columns for A (given U) the same as the essential dimensions of U. There could be anywhere from 0 to d effective dimensions for each ui in the additive decomposition of Theorem 1 so long as d dimensions are collectively essential. Two extreme cases illustrate the possibilities. In the first, U, has d essential dimensions and ui is constant for i > 1. Then preference depends only on Xi. In the second, n = d and ui vanishes on all dimensions other than dimension i, so Ul(Pl) = (Wl(PlL
0, ..*901, k(P2) = (0, W*(P*), 0, “‘9 01, *-*,
with wi: Pi -+ I& Then preference between p and q is resolved by the first attribute unless w,(p,) = w,(q,), in which case it is resolved by the second attribute unless W,(Pd = w*(qd, ..‘9 with p N q if and only if w,(p,) = wi(qi) for i= 1, .... n. 4.
UTILITY
INDEPENDENCE
Along with the assumptions in the first paragraph of the preceding section, we now take
x= YXZ, where Y is the Cartesian product of some of the Xi, and 2 is the product of the remaining Xi. P, and P, denote the sets of all finite-support probability distributions on Y and Z, respectively, (p y, z) is the distribution in P that has marginal pv on Y and gives z E Z with certainty, and (y, pz) is defined similarly. We assume that U is parsimonious with U= (U’, .... Ud) since this simplifies notation and our analyses. Remarks in the two preceding sections suggest how a nonparsimonious U could be handled when utility independence holds. Despite our use of parsimonious functions for the basic preference representation, Lemma 2 continues to play a role since utility independence involves convex subsets of P on which U can have fewer than d essential dimensions. EXAMPLE
1. Let Y= [0, 11, Z= { zO, zi} and take d=2 with WY, %I = (Y9 0) KY? Zl) = (0, Y)
for all y E Y. The Archimedean condition fails since (1,0) > (0, 1) > (0,O) and (pr, 0) > (0,l) when ~~(0) < 1. Hence utility must be at least two-dimensional, so
MULTIATTRIBUTE
EXPECTED
UTILITY
581
d = 2 by the definition of U on X. On the convex subset of P in which z is fixed at zO, which is a copy of P,, U is one-dimensional: dimension 1 is the only essential dimension. U is also one-dimensional on the convex subset of P in which z is fixed at zI, which is another copy of P,. But now dimension 2 is the essential dimension. Moreover, the definition of U implies that > on the first copy of P,(z=z,) is the same as > on the second copy of P,(z = zl). According to our ensuing definition, Y is utility independent of Z.
For each z E Z define the z-conditional
if(~~,z)>(9~~z).
PY>Z4Y
The y-conditional
preference order >, on P, by
order >y on P, is defined analogously: Pz >.” 4z
if (Y, pz) > (y, qz).
Example 1 has >=,, = >2,, but not all >y are the same. We say that Y is utility independent (UI) of Z if >= on P, is the same for all z: Y(UI)Z
if >, = >;,
for all z, z’ E Z.
This
is weaker than value independence between Y and Z, which implies [see Theorem 11, and in this case both Y(UI)Z and Z(U1) Y. As shown by Pollak (1967), Keeney (1968), and others, the conjunction of Y(UI)Z and Z(U1) Y does not imply that Y and Z are value independent. Our basic theorem for Y(UI)Z in the lexicographic setting is as follows. U(p)=u.(p.)+u,(p,)
THEOREM 2. Suppose U: P + Rd and z0E Z. Let C be the subset of e < d dimensions on which U restricted to {(p y, zo):pye Py} is essential. Then Y(UI)Z if and only if either
(i) (ii)
e=O and U(py,z)
is constant on P,for
each ZEZ, or
e > 0 and for every z EZ there is a d x d real L-matrix A(z) of dimension e and essential column set C, and a b(z) E R”, such that
ProoJ The sufficiency of (i) or (ii) for Y(UI)Z is easily verified. Suppose Y(UI)Z. If e =0 then >zO is empty, hence >= is empty for every ZE Z because Y(UI)Z, and U(p,, z) must be constant on P, to give (py, z) N (qy, z) in all cases. Suppose e> 1. Since >,= >zO on P,, application of Lemma 2 to P y gives the relationship between U(p,, z) and U(p,, z,,) stated in (ii). Each A(z) has the same set C of essential columns because z0 is fixed. 1
Theorem 2 or the remarks following Lemma 2 show that if p y >=,, q y is resolved by the jth essential dimension of U restricted to {(p y, zO): pye Py}, then p y>, qy is resolved by the jth essential dimension of U restricted to { (pu, z): py6 Py}.
582
FISHBURN AND LAVALLE
Example 1 shows that e essential dimensions for U at z0 need not be the same dimensions as those for U at z. By modifying the definition of UI, we obtain more restrictive and less restrictive versions of utility independence for the lexicographic setting. Theorem 2 applies to the modifications under different restrictions on the A(z) matrices. The more restrictive version adds a special Archimedean axiom whose effect is to resolve (py, z) > (qy, z) by the same Vi for all z E Z. We say that Y is strongly utility independentof Z, and write Y(SUI)Z, if Y(UI)Z and for all pr, qrE P, and all z, z’ E Z, (py, z) > (qy, z) * there are ~1,BE (0, 1) such that (PY, z) 4q.9 2’) > (q.9 z) ap.7 z’)
and
(qn z) B(PY> z’) t (PYT z) 8(4r, z’).
It is easily seen when Y(SUI)Z that if (py, z) > (qy, z) with Uj(p,,, z) > Uj(q,, z) and Uk(p,,‘z) = Uk(q y, z) for all k < j, then the same things hold for every z’ E Z in place of z. The effect of Y(SUI)Z on Theorem 2 is to require A(z) in part (ii) to have C as its set of e essential rows. That is, the essential row set and essential column set of every transformation matrix are identical. When e = d, this is automatic and UI = SUI, so the notion of strong utility independence does not add anything to UI when utilities are real valued with d = 1. Our less restrictive version was considered in the real-valued setting by Fishburn (1974) and Fishburn and Keeney (1974). Let >* be the inverse of > so that p >*q if q>p. The condition of Y generalized utility independent (GUI) of Z allows complete reversals in preference and complete indifference in the >, comparisons: Y(GUI)Z if for all z, z’ E Z, >;#0*>;s E{&, >:,$a}. Thus I(>-: ZE Z>( < 3 when Y(GUI)Z. The definition of Z(GU1) Y is similar with Y and Z interchanged. As already noted, Example 1 has Y(UI).Z and not [Z(UI) Y]. But Z(GU1) Y since U(Y,z,~z,)=~(Y,O)+(1-~)(O,Y) =y(A
so >y=fa
when y=O, and >“=
1 -A),
>.V,#/zr whenever y, y’>O.
THEOREM 3. Suppose the hypotheses of Theorem 2 hold with >zO # 0. Then Y( GUZ)Z if and only if e > 0 andfor every z E Z there is a real d x d matrix A(z) and a b(z) E Rd such that
WP,, z) = A(z) VP,, A(z) = 0
zo) + 0)
for every pry P,;
if>:=JZf;
A(z) [respectively, -A(z)] is a L-matrix of dimension e and essential column set C if >, = > =. [respectively, >= = >:I.
MULTIATTRIBUTE
EXPECTED UTILITY
583
Proo$ The proof of the theorem is similar to the proof of Theorem 2. The only substantial difference occurs when >, = > z. Then the ujk in the e quasi-diagonal cells of A(z) must be negative to effect the complete reversal of conditional preferences. 1 A strong version of GUI has C as the essential row set of A(z) in the final clause of Theorem 3. The special Archimedean axiom for this version is a straightforward generalization of the one for SUI.
5. MUTUAL
UTILITY
INDEPENDENCE
We now assume that Y(UI)Z and Z(U1) Y under the formulation and assumptions in the opening paragraph of the preceding section. We assume also as a scaling convenience that U(y,, zO) = (0, .... 0) in Rd for a fixed (y,, zO) E X. When d= 1, mutual utility independence implies that there is a constant c0 such that, for all (Y, Z)EX U(YYz) = U(YYzo) + U(YO?z) + COU(Y?zo) U(YO?z). The additive form is obtained if c,, = 0; otherwise we get the multiplicative form by resealing. The constant must satisfy 1 + c0u( yO, z) > 0 and 1 + cOu( y, zO) > 0 for all y and z. Assume henceforth that d > 2. According to Theorem 2 applied to Y(UI)Z and then Z(U1) Y, WY, z) =&I
VY, zo) + b(z)
WY, z) = B(Y) WY,, z) + C(Y)l
where A and B are maps into d x d matrices, and b and c are maps into Rd. Taking y = y, in the first equation and z = z0 in the second, we have V( y,, z) = b(z) and U(Y, zo) = C(Y), so U(Y, z) = A(z)
U(Y, zo) + WY,,
U(Y, z) = B(Y) WY,,
z)
z) + U(Y, zo)
(1)
(2)
for all (y, z) E X. The particular form for U( y, z) implied by (1) and (2) will depend on the nature of the A(z) and B(y). We consider some examples to illustrate possibilities and then derive a few more general results. Throughout, we follow the notation of Theorem 2: ey, Cy, e,, and C, are the dimensionality of {(p y, zO): P,,E Py}, the essential column (dimension) set for ey, the dimensionality of {(y,, pz): pz E Pz}, and the essential column set for e,, respectively. 480/36/4-9
584
FISHBURN AND LAVALLE
EXAMPLE 2. Suppose e y = 0. By Theorem 2(i), U( y, z) is constant on Y for each ZEZ, and we have
U(Y> z) = U(Ylb z) for all (y, z) E X In this case e, = d, U( y, z,,) = U( y,, zO) = (0, .... 0), and B(y) is the d x d identity matrix Id for all y. EXAMPLE
3. Let Y = Z = [ 1,2] with y, = z0 = 1 and u y, z) = (U’( y, z), U2(Y, z)) = (YZ - 1, Y/Z - 1)
for all (y, z). The “indifference curves” in [l, 212 for U’ go from upper left to lower right, and those for U2 go from lower left to upper right. d= 2 is verified by (2,2)>(2,1)>(1,2)alongwith(p.,2)~(2,1)wheneverp.((l,2])>O.Wehave U’(y, l)= U’(l,z)=z-1,
V(y,
l)=y-
1,
ey= 1,
P(l,z)=l/z-1,
c,=
e,=2,
An example to illustrate the latter case is pz( 1) =p,(2) W?P,)
(1); C,={l,2}.
= l/2 and q,(3/2) = 1. Then
= U’(L 42) = l/2,
- l/4 = U’( 1, pz) > U2( 1, qz) = - l/3. Mutual utility independence is easily verified from the definition of U. Let E, denote expectation with respect to probability distribution r. For r, s E P ,,, (r,z)>(S,z)OE,(y)>E,(y) so Y(UI)Z;
for r, SE P,,
(y, r) >
(Y, s) -
E,(z)
> E,(z)
or {E,(z)
= E,(z)
and E,(llz)
> E,(llz))
so Z(U1) Y. We also have
for k = 1, 2. Mutual utility independence can be established directly from this form since 1 + Uk( y, 1) > 0 and 1 + U”( 1, z) > 0 for all y, z, and k. If we did not have constant signs here then utility independence but not generalized utility independence would fail. EXAMPLES.
Let Y=X,~X,=[1,2]~andZ=[1,2]
withy,=(l,l),z,=land
MULTIATTRIBUTE
EXPECTED
585
UTILITY
This is an extension of the preceding example. Its third dimension comes into play when equality holds for the first two dimensions. For example, when x=(x,,x,,z)=(1,2, 1) and ~‘=(a,,/?, l), Uk(x) = Uk(X’)
fork=
1,2
U3(x)=3-2>2$-2=
U3(xt).
We leave it as an exercise to show that Y(UI)Z, C,= { 1,3}, Z(UI)Y, and c,= (1,2}. The following theorem gives the general form of U when mutual utility independence holds and conditional utilities have full dimensionality with C.=C,={l,...,d}. THEOREM
constants
4. Suppose e y = e, = d, Y( UZ)Z, and Z( UZ) Y. Then ci for k = 1, .... d and, given k, for i, j< k, such that
+?Y,
U”(Y, z) = Uk(Y,&I) + Uk(Y,, z) + i i,j=
4 WY,, z)
there
are
(3)
1
for all ( y, z) E X and k = 1, .... d, In addition, for each k from
1 to d,
k 1+
1+
1 j=l
i
c;juj(y,,
z)
&Ui(y,
z,)>O
>
0
for all z E Z,
(4)
for ally E Y.
(5)
i=l
The sign restrictions of (4) and (5) are needed to prevent preference reversals or nullities of the type that can arise under GUI but are forbidden by UI. We verify them first and then establish (3). Proof of the Last Part. Assume that (3) holds along with the theorem’s hypotheses. Since the proofs of (4) and (5) are similar, we prove only (4). Consider (4) at k = 1. By (3) and linearity, WP,,
z) > wq,,
z) o WP,,
&I) Cl + c:, WY,,
’ wq,,
%I) Cl + 4 WYCI~z)l.
Since ey= d, there are pr, qrE P, such that U’(P y9 zo) > wq
z)l
Y, 4
586
FISHBURN
AND
LAVALLE
with (py, z,)>(q,, z,,). Since Y(UI)Z, (py, z)>(q*,z) for all z. Full dimensionality then requires U’(p,, z) > U’(q,, z) for all z, and this is true if and only if 1 + c:l U’(y,,
for all z E Z.
z) >o
This verifies (4) at k= 1. Suppose k> 1. By (3),
Uk(p,, z)> Uk(q,, z)* Uk(PY>4
i
1 + i CfjWY,, z) j=l
1
k-l
+ 1 CWP,, zo)- mq,, %)I i c; WY,~ z) i=
>
1
j=l
Uk(q*,z,)
l+ [
i j=l
cij
uj(YO?
z,
1 .
Since ey= d, there are pr, qre P, such that u’(P.,
for all i < k;
4 = U’(q.9 &I)
Uk(Pn
zo) > Uk(qn
zo).
Full dimensionality and Y(UI)Z then require U’(p,, z)= u’(q,, z) for all i< k, and Uk(p =, z) > Uk(q ., z), for all z E Z, and this clearly holds if and only if (4) holds at k. Proof of (3).
Fix k
l
{ 1, .... d} and define k x k matrices M=
[I”ql
=
C”‘(Yj,
zO)l~j=
N=
CNijl
=
C”‘(YO,
zj)l:j=l
1
for y,, .... y, E Y and zi, .... zk E Z. We wish to choose the yj and zj so that M and N are nonsingular. This can be done by induction on k. We demonstrate the essentials for M. For k = 1, choose any y, E Y for which U’( y,, zO) # 0, as guaranteed by ey = d. Suppose nonsingularity holds at k - 1 when k > 1 for
As assured by full dimensionality, Uk(P.? %I) + Uk(qn
z,),
let py, que P, satisfy for i < k.
U’(P Y1zcl) = U’(q Y, %)
Form M,+ from M0 by adding a kth row and column. Entry j< k in row k is U”( yj, zO), and entry i in column k is U’(p y, zO) - Vi(qr, z,,). Expand the determinant of M,+ by the kth column to get I%
= flM,I
CUk(PY, %I)-
Uk(q,v
%)I
zo.
MULTIATTRIBUTE
EXPECTED
Also expand the Uk by linearity
to obtain
and let m, be the determinant WY;, z,), ***, Uk( y;, zO). Then
of M,f
587
UTILITY
after its final column
is replaced by
so at least one m, is not zero. Let y, denote a yj for which m,#O. Then [ vi( yj, zo)]i,j.& k is nonsingular. Assume henceforth for fixed k that M and N are nonsingular k x k matrices as defined earlier. For all z and y, define k x k matrices M(z)=CU’(~j,z)-U’(~,,z)li,j~k
N(Y)=
C~‘(Y,
Zj)-
U’(Y,
ZO)li,j
We assume that (1) and (2) hold under mutual utility independence and, since eY= e, = d, we can presume that A(z) and B(y) are lower triangular (see Lemma 1). Because of lower triangularity and our focus on Uk, no generality is lost by truncating (1) and (2) to their first k rows and first k columns. Let (1) and (2) be understood in this way. Then M(z)=A(z)M by (1) and N(y)=B(y)N by (2). Because M and N are nonsingular, right multiplication by inverses gives A(z) = M(z)M-’ and B(y) = N(y) N- ‘, and substitution in (1) and (2) yields U(Y, z) = M(z)M-’ U(Y, z) = NY)N-’
VY, &I) + WY,, z)
(6)
WY,, z) + UY, 4
(7)
for all (y, z) E X. By (7),
so the jth column of M(z) equals the right Nj = N( yj) N- ’ - zk. Then, by column listing, CN,U(Y,,z)+U(Y,,z,)...NkU(Yo,z)+
M(Z)= =
Substitute
CN,U(y,,
z)...Nku(yo,z)]
side of this expression.
U(Yk,%)I +kf.
this into (6) to obtain
u(Yv Z) = u(Y, &I) + u(Y,,
Z) + CN, u(Y,v Z) .--N/c u(Y,, z)] M-‘U(Y,
zo).
Let
588
FISHBURN
AND
LAVALLE
Therefore @( y, z) equals Uk(y, zO) + Uk(y,,, z) plus the kth term of the final expression. That term is k rk c j=l
1
c (Nj)kiUi(yO, i= I
=
5 i=l
i /=I
;I]
U’(YOYZ)
[
i (Mpl)jlU’(y, /= 1
u’(Y,
ZO)
i j=l
zCl)]
(Nj)ki(M-‘)j,.
When the final sum over j is denoted by cf, we get (3).
1
Theorem 4 and our earlier examples illustrate the type of decomposition that arises under mutual utility independence. We do not have a general result for all possible forms of (1) and (2) under mutual utility independence, but conjecture that the general case uses only the real-valued functions U’(y, z,), .... V’(y, z,,), wYcl, z), .... Ud( y,, z) and multiplying constants in a bilinear form for the vi( y, zO) and Uj( yO, z), much as in (3). The examples preceding Theorem 4 illustrate some combinations for ey, e,, C,, and C,. An interesting question that we leave open asks for a description of all combinations that can arise when U has dimension d. We do have answers to two more limited but still interesting questions: 1. Is C, u C, = { 1, .... d}? 2. Is e,+e,>d? Surprisingly,
the answers are negative as shown by our concluding
example.
EXAMPLE 5. We use the simplest possible example to show a case of Y(UI)Z, Z(UI)Y, and e,+e,
U(y,z)=(y-l,z-l,yz-l),
y(J=z,=
1.
Then, with E denoting expectation,
VP,, z) = (E(y) - 1, z - 1, MY) - 1) ub, PA = (Y - 1, E(z) - 1, YE(Z) - 1). The first equation shows that only dimension 1 is essential for U on {(p y, zO): P,,E Py}, the second shows that only dimension 2 is essential for U on { (yO, pz): pz E Pz}, and clearly Y(UI)Z and Z(U1) Y. By the definition of U, d < 3. The fact that d = 3 is suggested by letting Xl = (2,2),
x2=(2, Ax1
l), 1 =PbJ
x3
= =
(1921,
l/Z
x4=
(41);
MULTIATTRIBUTE
and noting by the representation
EXPECTED
of Assumption
589
UTILITY
1 that
x,>x,>p>x,>x, X,>X,JX‘$>Xj
for all 0 < A < 1
P+x2%
for every 0 < 1< 1.
Let cli = CL(X~)for a E P, so c(~+ CQ+ c(~+ ~1~= 1. Then, for all c(,/I E P,
The three equations here imply a = /I, so CIN /I o CI= p. Now suppose that d= 2. We show that this is impossible. Let uk(xj) be the utility assigned to xi by the kth component of the two-dimensional utility function. We then require
aNBe i Ul(Xj)(aj-Bj)=O, jc,
u*(xj)(aj
-
Pi)
=
O.
j=l
Regardless of the values of the uk(xj), the two equations on the right side are insufficient to imply a =/I, which can only be guaranteed by three equations in the unknowns CI~-pi, CI~-p2, a3 -/I3 (with ad-/I4 given from these and 1 a, = C pi = 1). In other words, d = 2 forces a N fi for some a # j$ a contradiction. Hence d > 2, and since d < 3 by the definition of U, we conclude that d = 3.
6. DISCUSSION
This paper brings together two themes in expected utility theory that emerged many years ago. The first, non-Archimedean expected utility (Hausner, 1954; Chipman, 1960), was motivated by an interest in omitting the technically convenient Archimedean axiom. The second, independence conditions among attributes in multiattribute real-valued expected utility (Fishburn, 1965; Pollak, 1967; Keeney, 1968), was motivated by a desire to simplify the burden of utility assessment in multiattribute decision analysis. Our integrating framework considers an outcome set X= X, x X, x . . . x X, in the context of an m-dimensional linear function U = (U’, .... Urn) defined on the set P of linite-support probability distributions on X. Preference between members of P is assumed to coincide with the natural lexicographic order between their utility vectors. We noted that the marginal independence condition of value independence produces an additive decomposition of each Uj over the attributes (Theorem l), as in the real-valued case. Under a partition of the Xi that gives X = Y x 2, we then considered the conditional independence condition known as utility independence.
FISHBURN AND LAVALLE
590
This leads to a partial decomposition of U which confines one of Y and Z to onepoint marginals (Theorem 2). After commenting on a strengthening (SUI) and relaxation (GUI: Theorem 3) of utility independence, we examined the effects of mutual utility independence between Y and Z. Under full dimensionality of both conditional forms, mutual utility independence leads to a decomposition involving functions of Y and functions of Z (Theorem 4) that is a natural generalization of the real-valued mutual utility independence decomposition. But less-than-full dimensionalities are possible for the conditional forms, and there are even cases in which the sum of the numbers of essential dimensions for the two is less than the dimensionality of U (Example 5). Questions left open for mutual utility independence include the possible combinations of essential dimension sets for the Y-conditioned-on-Z form and the Z-conditioned-on-y form, and a general expression for U in terms of single-variable functions regardless of the dimensions of the conditional forms. The effects of other combinations of utility independence in the X= X, x X, x . . . x X, setting, such as Xi utility independent of the other attributes for i= 1, .... n, await study. Chapter 6 in Keeney and Raiffa (1976) describes results in this area for real-valued expected utility.
REFERENCES CHIPMAN, J. S. (1960). The foundations of utility. Econometrica, 28, 193-224. DEBREU, G. (1960). Topological methods in cardinal utility theory. In K. J. Arrow, S. Karlin, & P. Suppes (Eds.), Mathematical methods in the social sciences, 1959 (pp. 1626). Stanford: Stanford Univ. Press. FISHBURN,P. C. (1965). Independence in utility theory with whole product sets. Operations Research, 13, 2845.
FISHBURN,P. C. (1970). Utility theory for decision making. New York: Wiley. FISHBURN,P. C. (1971a). A study of lexicographic expected utility. Management Science, 17, 672478. FISHBURN,P. C. (1971b). Additive representations of real-valued functions on subsets of product sets. Journal
of Mathematical
Psychology,
8, 382-388.
FISHBURN, P. C. (1974). von Neumann-Morgenstern Research,
utility functions on two attributes. Operations
22, 35-45.
FISHBURN,P. C. (1979). On the nature of expected utility. In M. Allais & 0. Hagen (Eds.), Expected utility hypotheses and the Allais paradox (pp. 243-257). Dordrecht: Reidel. FISHBURN, P. C. (1980). Lexicographic additive differences. Journal of Mathematical Psychology, 21, 191-218. FISHBURN,P. C. (1982). The foundations of expected utility. Dordrecht: Reidel. FISHBURN,P. C., & KEENEY, R. L. (1974). Seven independence concepts and continuous multiattribute utility functions. Journal of Mathematical Psychology, 11, 294-327. GEQRGEXU-ROEGEN, N. (1966). Analytical economics: Issues and problems. Cambridge, MA: Harvard Univ. Press. HAUSNER,M. (1954). Multidimensional utilities. In R. M. Thrall, C. H. Coombs, & R. L. Davis (Eds.), Decision processes (pp. 167-180). New York: Wiley. JENSEN,N. E. (1967). An introduction to Bernoullian utility theory. I. Utility functions. Swedish Journal of Economics, 69, 163-183. KEENEY, R. L. (1968). Quasi-separable utility functions. Naval Research Logistics Quarterly, 15, 551-565.
MULTIATTRIBUTE
EXPECTED
UTILITY
591
KEENEY, R. L., & RAIFFA, H. (1976). Decisions with multiple objectives: Preferences and value tradeoffs. New York: Wiley. KRANTZ, D. H., LUCE, R. D., SUPPES, P., & TVERSKY, A. (1971). Foundations of measurement. I. New York: Academic Press. LAVALLE, I. H., & FISHBURN,P. C. (1991). Lexicographic state-dependent subjective expected utility. Journal
of Risk and Uncertainty,
4, 251-259.
LAVALLE, I. H., & FISHBURN,P. C. (1992). State-independent subjective expected lexicographic utility. Journal of Risk and Uncertainty, in press. LUCE, R. D. (1978). Lexicographic tradeoff structures. Theory and Decision, 9, 187-193. LUCE, R. D., & TUKEY, J. W. (1964). Simultaneous conjoint measurement: A new type of fundamental measurement. Journal of Mathematical Psychology, 1, l-27. NARENS, L. (1985). Abstract measurement theory. Cambridge, MA: MIT Press. PO~LAK, R. A. (1967). Additive von Neumann-Morgenstern utility functions. Econometrica, 35, 485-494. RAIFFA, H. (1969). Preferences for multiattributed alternatives. (Memorandum RM-5868-DOT RC) Santa Monica, CA: The Rand Corp. SAVAGE, L. J. (1954). The foundations of statistics. New York: Wiley. TVERSKY, A. (1969). Intransitivity of preferences. Psychological Review, 76, 3148. VON NEUMANN, J., & MORGENSTERN, 0. (1944). Theory of games and economic behavior. Princeton: Princeton Univ. Press. WAKKER, P. P. (1989). Additive representations of preferences. Dordrecht: Kluwer. RECEIVED:
January 1, 1991
OF MATHEMATICAL
~u~tia~ribute
PSYCHOLOGY
573-591 (1992)
36,
Expected Utility without
the Arc~i~edean
Axiom
PETER C. FISHBURN AT&T
Bell Laboratories
AND IRVING H. LAVALLE Tulane
University
Two prominent themes in expected utility theory that were developed many years ago are non-Archimedean expected utility and independence conditions in multiattribute real-valued expected utility. This paper integrates these themes in an investigation of independence conditions for multiattribute vector-valued expected utility. The dimensionality of the utility space need bear no relationship to the number of attributes. The multiattribute independence conditions examined include value independence, utility independence, generaked utility inde~nden~, and two-factor mutual utility independence. Attribute de~om~sitions of components of utility vectors implied by some of the conditions are natural generalizations of forms that arise in the real-valued setting. 0 1992 Academic Press, Inc.
1. I~moDucT10N This paper develops implications of independence conditions for utility functions in multiattribute expected utility theory without the usual Archimedean axiom. Its independence conditions were first examined by Fishburn (1965, 1974), Pollak (1967), Keeney (1968), Raiffa (1969), and Fishburn and Keeney (1974) for von Neumann-Morgenstern expected utility follo~ng the axiomatizations of additive utility over Cartesian products by Debreu (19~), Lute and Tukey (1964), and others (see Fishburn, 1970; Krantz, Lute, Suppes, & Tversky, 1971; Keeney and Raiffa, 1976; Wakker, 1989). We depart from earlier work in expected utility by omitting or weakening its Archimedean axiom. This changes the setting for consideration of multiattribute independence conditions from real-valued to vectorvalued expected utility ordered lexicographically (Hausner, 1954; Chipman, 1960; Fishburn, 1971a, 1982). No particular relationship is presumed between the number of levels in the lexicographic hierarchy and the number of attributes in the outcome set. Two reasons motivate our generalization to lexicographic expected utility. The first is foundational. Archimedean axioms do not have the same normative status Address all reprint request to Peter C. Fishburn at AT&T Belt Laboratories, Murray Hill, NJ 07974.
573 0022-2496192 $5.00 Copyright ci 1992 by Academic Press. Inc. All rights of reproduction m any form res.xved.
574
FISHBURN
AND
LAVALLE
as axioms like transitivity and linearity. They serve the technical purpose of ensuring real-valued representations of preference, and hence it is important foundationally to consider their absence (Narens, 1985). The second reason is practical. Value hierarchies can induce defensible preference patterns that violate Archimedean conditions (Chipman, 1960; Georgescu-Roegen, 1966). One might associate such patterns with relative importance notions for attributes or enrich the lexicographic structure with threshold considerations (Tversky, 1969; Lute, 1978; Fishburn, 1980). An example of values-induced non-Archimedean preferences consists in observing that the ethical tenet “it is wrong to take an avoidable chance at a wrong outcome” makes every lottery that offers positive probability of a wrong outcome inferior to any undesirable but non-wrong alternative. The next section summarizes the lexicographic generalization of the linear utility model of von Neumann and Morgenstern (1944) which provides the setting for the rest of the paper. Its representation is
where > is a preference relation on a set P of probability distributions p, q, ... defined on a outcome set X, U(P) E II?, and >L denotes the usual lexicographic order on R”. In addition, U is a linear function whose uniqueness properties are central to some of our derivations of consequences of independence conditions. The lexicographic linear utility model does not presume any special structure for the outcome set X, but to develop the multiattribute theme we assume in Sections 3 through 5 that X=X,xX,x ... xX,. Section 3 notes that the value independence condition of p indifferent to q whenever p and q have the same marginal distribution on Xi for i= 1, .... n leads to the additive decomposition UP) = %(Pl) + %(P2) + ... + %l(P,h
where pi is the marginal of p on Xi and ui is a vector-valued linear function on the set of all such pi. A derivation of the same additive form with XEX,XX,X ... xX, in the context of a lexicographic generalization of Savage’s (1954) theory of decision under uncertainly appears in LaValle and Fishburn (1991). The additive form for real-valued utility on subsets of Cartesian products is discussed in Fishburn (1970, 1971b) and Wakker (1989). Section 4 considers utility independence conditions after attributes are partitioned into two sets with X= Y x Z. The simplest condition (Pollak, 1967; Keeney, 1968) says that (py, z)> (qy, z)-=- (py, z’)> (qy, z’), where (py, z) denotes the distribution in P that has marginal p y on Y and gives z E Z with probability 1. This condition of Y utility independent of Z allows U on P, x Z to be written as WPY, z) = A(z) VP,,
ZCJ + b(z),
where z0 is fixed in Z, b E 08”, and A(z) is a real m x m matrix of a type described later. We then examine a stronger version of the basic conditions as well as a
MULTIATTRIBUTE
575
EXPECTED UTILITY
relaxation known as generalized utility independence (Fishburn, 1974; Fishburn and Keeney, 1974). These modifications imply the same form for U(p,, z) but have different implications for the ,4(z) matrices. Since the utility independence relation is not symmetric, Section 5 looks at the joint effect of Y utility independent of 2 and Z utility independent of Y. We illustrate mutual utility independence with examples and then consider the case in which the Y-conditioned-on-Z form and the Z-conditioned-on-Y form both have full dimensionality, the same as the dimensionality d of U. In this case, with U(y, z) = WY, z), *.., Ud(Y, z)), each Uk on Y x Z decomposes as Uk(Y,
z,
=fk(Y)
+ gktZ)
+
i ;=I
C$fi(Y)
i j=l
gj(Z),
where they, are real-valued functions on Y, the g, are real-valued functions on Z, and the ci. are constants. We also prove that d can exceed the sum of the effective dimensions of the Y-conditioned-on-Z and Z-conditioned-on-Y forms. The paper concludes with a summary and open problems. In many places throughout, vectors can be viewed either as row or column vectors. However, when matrix multiplication is used, as in VP) = A U(P) + b, the vectors are assumed to be column vectors. 2. LEXICOGRAPHIC
LINEAR UTILITY
This section develops the aspects of lexicographic expected utility needed for ensuing sections. Our only requirement on X in the present section is that it contain at least two outcomes. Throughout the paper, > is a binary is preferred to relation on the set P of all finite-support probability distributions on an outcome set X. The convex (0 < L < 1) combination Ap + (1 - L)q of p, q E P is written as plq, so (plq)(x) = lP(X) + (1 -A) q(x ) for all x E X. We assumethat > is not empty, extend it to X by defining x > y as p > q when p(x) = q(y) = 1, and define its indifference relation N on P by p-4
A functional
if not(p > 4)
and
u: P + R is linear if, Vp, q E P, VO< A< 1, u(plq) = AU(P) + (I-
A function
not(q > P).
U: P + R” is linear if it is linear
1) u(q).
in each coordinate.
That
is, if
576
FISHBURN
AND
LAVALLE
u= (U’, .... Urn) with Uj: P -+ aB for each j, then Uj(pnq) = nUj(p) + (1 -A) U’(q) for all j. We define >L between real vectors a = (a,, .... a,) and b = (b,, .... b,) by a>,b
and the min j for aj # bj has aj > bj.
ifa#b
Hence U(p) > L U(q) if Uj(p) # U’(q) for some j, and U’(p) > U’(q) for the smallest such j. We say that > on P is (i)
a weak order if it is irreflexive and negatively transitive
[p>
q*p>r
or r>ql;
(ii) linear if Vp, q, r E P, VO < A < 1, p > q =>pLr > qlr; (iii) Archimedean if Vp, q, r E P, p > q > r *par > q and q >p/?r a, P E (0, 1).
for some
In addition, N on PislinearifVp,q,r~P,VO<~
for all p E P.
When u is extended to X by U(X) = u(p) when p(x)= 1, linearity and the linitesupport hypothesis give u(p) = Cp(x) U(X) along with the familiar expected utility representation P~4~x~xPw4x)~
c
&)4x)*
XCX
This representation and X = X1 x . . . x X, provide the context for the independence conditions in our opening references. Those conditions lead to widely discussed decompositions of u(xi, .... x,), including additive and multiplicative forms. Keeney and Raiffa (1976) and successors describe many applications of these forms in decision analysis. The generalization of expected utility used in ensuing sections is summarized by Assumption 1. There is a positive integer m and a linear function such that Vp, q E P, p>q*
U(P)‘L
U: P -+ W”
U(q).
This presumes that > on P is a linear weak order and that N on P is linear (Hausner, 1954; Fishburn, 1971a, 1982). If X is finite, these axioms imply Assumption 1 with 1
MULTIATTRIBUTE
EXPECTED UTILITY
577
are given in Chapter 4 of Fishburn (1982). A representation which allows an infinite type of hierarchy is described in Hausner (1954), and relationships to ordered vectors spaces are discussed there and in LaValle and Fishburn (1992). We assumehenceforth that U and V are finite-dimensional linear functions on P that satisfy the lexicographic representation of Assumption 1. Call U parsimonious if its m is as small as possible. The uniqueness theorem for parsimonious functions is LEMMA 1. Suppose U and V are parsimonious into Rd. Then there is a d x d lower-triangular real matrix A with positive diagonal elementsand a b E W”, such that
V(p)=AU(p)+b Proof
forallpEP.
See Theorem 4.4 in Fishburn (1982).
1
If U is parsimonious and A and b are as just specified, then it follows easily that AU + b is parsimonious and satisfies the representation of Assumption 1. When u = (U’, ...) U”), V= (V’, .... Vd), A = [aik], and b= (b,, .... bd), the equation of Lemma 1 says that V’(p)=
i
ajkUk(p)+bj,
j = 1, .... d.
k=l
When d = 1, this reduces to v = CIU+ p, a > 0, for real-valued linear utility. We also consider nonparsimonious functions U: P + R’ with I> d when d is the dimension of a parsimonious function. The transformation matrix A for mappings between possibly nonparsimonious functions requires a special definition. DEFINITION
1 < d < min{m, indices l
1. An m x 1 real matrix A is a L-matrix of dimension d if l} and there are row indices 1
for i= 1, .... d,
(ii)
aj,, = 0
and
ajiki > 0
for all k > ki;
for l
for all k > min{k,
: ji >j>.
We refer to each (ji, ki) in the definition as a quasi-diagonal cell, to each ji as an essential row, and to each ki as an essential column. A partial picture of a L-matrix is
ji
ki . ..+oo
ki+l 0
o-e.0
j
. .. . . . 0
o...o
. . . +
o...o.
ji+l...
578
FISHBURN
Condition says that diagonal conditions be zero. LEMMA
AND
LAVALLE
(i) specifies positive ajk quasi-diagonal cells and if row j is not essential and the first essential cell (j’, k’) then row j has zeros from column k’ dictate that everything above and/or to the right
2.
Suppose a parsimonious
zeros to their right; (ii) row after j has quasion. In combination, the of a quasi-diagonal cell
function for the representation
1 is d-dimensional and that U: P --+ R’ and V: P --f R”’ with dd min{m, is an m x I real L-matrix
A of dimension d and a b E R” such that V(p)=AU(p)+b
ProojY
See Theorem
of Assumption I}. Then there
forallpEP.
1 in LaValle and Fishburn (1991).
Suppose parsimonious Let
functions
are d-dimensional
1
and U = (U’, .... U’) with
l>d.
0( Uj) = {(p, q): U’(p) > U’(q), Uk(p) = Uk(q) for all k
for i = 1, .... d.
That is, the ith essential Vj resolves exactly the same preferences as the ith essential Vi, and the quasi-diagonal cells of the L-matrix A for Lemma 2 will be the (ji, ki). The equation of Lemma 2 says that V’(p)=
i
ajkUk(p)+bj,
j = 1, .... m.
k=l
Taking differences, we have v(p)-
v(q)=
i k=l
ajk[uk(P)-
uk(q)lv
which along with the properties of A from Definition 1 and the presumed representation by U shows that AU + b also represents > lexicographically. To summarize, suppose U: P+ R’ is a linear function that represents > lexicographically and d < I is the dimension of a parsimonious linear function that represents > lexicographically. Then: (a)
exactly d dimensions of U are essential;
MULTIATTRIBUTE
579
EXPECTED UTILITY
(b) if d < m, A is an m x 1 L-matrix of dimension d whose essential columns correspond to the essential dimensions of U, and b E R”, then AU + b is a linear function that represents > lexicographically and has exactly d essential dimensions; (c) if d < m, then the only linear K P + W” that represent > lexicographitally are those that relate to U by I’= AU+ b for an A and b as just specified in (b). Conclusion (c) follows immediately case of Lemma 2 for m = 1= d.
from Lemma
2, and Lemma
1 is the special
3. VALUE INDEPENDENCE
We assume henceforth that X= X, x . . . x X, with n 2 2. Also, as noted earlier, > on P is nonempty, and U and V are linear functions on P that represent > lexicographically. In addition, d is the minimum dimension of all linear functions that satisfy the representation of Assumption 1. Let P, be the set of all linite-support probability distributions on Xi and for any r E P let ri denote the marginal distribution of r on Xi, i = 1, .... n. The Xi are value independent if Yp, q E P, (P 1, *.a,P”) = (41, ...>4J *p - 4. THEOREM 1. Suppose U: P + R’, 12 d. Then the Xi are value independent if and only if there exist linear ui: Pi + @for i = 1, .... n such that, for all p E P,
VP) = U,(P,) + ... + hl(P”). Supposethis decomposition holdsfor U along with, Yp EP, VP) = V,(P,) + ... + Vn(Pnh where V, v,, .... v,: P, P,, .... P, + R”, m 2 d. Then there is an m x I real L-matrix A of dimension d and b, , .... b, E BY’ such that, Vi E { 1, .... n}, Ypi E Pi, Vi(pi) = Aui(p,) + bi. ProojI See Theorems 2 and 3 in LaValle and Fishburn (1991). Lemma 2 in the preceding section is used in the second part to give cvi(Pi)=
V(p)=AU(p)+b=ACui(pi)+b,
so for any fixed POEP we get v,(p,) = Au,(p,) + bi, where bi= b + A~,,,u,(p~) Cj+ivj(P,“h
I
-
580
FISHBURN
AND
LAVALLE
When I= d = 1, the ui: Pi + Iware unique up to similar positive affine transformations clui + pi, c(> 0, as in Theorem 11.1 in Fishburn (1970). When I = m = d > 1, the ui: Pi + IV’ are unique up to similar alline transformations Aui + b, with A a d x d lower-triangular real matrix with positive diagonal elements and bi E Rd (see Lemma 1). When min{ I, m} 2 d 2 1, we get the set of admissible transformations specified in the second part of Theorem 1 with the essential columns for A (given U) the same as the essential dimensions of U. There could be anywhere from 0 to d effective dimensions for each ui in the additive decomposition of Theorem 1 so long as d dimensions are collectively essential. Two extreme cases illustrate the possibilities. In the first, U, has d essential dimensions and ui is constant for i > 1. Then preference depends only on Xi. In the second, n = d and ui vanishes on all dimensions other than dimension i, so Ul(Pl) = (Wl(PlL
0, ..*901, k(P2) = (0, W*(P*), 0, “‘9 01, *-*,
with wi: Pi -+ I& Then preference between p and q is resolved by the first attribute unless w,(p,) = w,(q,), in which case it is resolved by the second attribute unless W,(Pd = w*(qd, ..‘9 with p N q if and only if w,(p,) = wi(qi) for i= 1, .... n. 4.
UTILITY
INDEPENDENCE
Along with the assumptions in the first paragraph of the preceding section, we now take
x= YXZ, where Y is the Cartesian product of some of the Xi, and 2 is the product of the remaining Xi. P, and P, denote the sets of all finite-support probability distributions on Y and Z, respectively, (p y, z) is the distribution in P that has marginal pv on Y and gives z E Z with certainty, and (y, pz) is defined similarly. We assume that U is parsimonious with U= (U’, .... Ud) since this simplifies notation and our analyses. Remarks in the two preceding sections suggest how a nonparsimonious U could be handled when utility independence holds. Despite our use of parsimonious functions for the basic preference representation, Lemma 2 continues to play a role since utility independence involves convex subsets of P on which U can have fewer than d essential dimensions. EXAMPLE
1. Let Y= [0, 11, Z= { zO, zi} and take d=2 with WY, %I = (Y9 0) KY? Zl) = (0, Y)
for all y E Y. The Archimedean condition fails since (1,0) > (0, 1) > (0,O) and (pr, 0) > (0,l) when ~~(0) < 1. Hence utility must be at least two-dimensional, so
MULTIATTRIBUTE
EXPECTED
UTILITY
581
d = 2 by the definition of U on X. On the convex subset of P in which z is fixed at zO, which is a copy of P,, U is one-dimensional: dimension 1 is the only essential dimension. U is also one-dimensional on the convex subset of P in which z is fixed at zI, which is another copy of P,. But now dimension 2 is the essential dimension. Moreover, the definition of U implies that > on the first copy of P,(z=z,) is the same as > on the second copy of P,(z = zl). According to our ensuing definition, Y is utility independent of Z.
For each z E Z define the z-conditional
if(~~,z)>(9~~z).
PY>Z4Y
The y-conditional
preference order >, on P, by
order >y on P, is defined analogously: Pz >.” 4z
if (Y, pz) > (y, qz).
Example 1 has >=,, = >2,, but not all >y are the same. We say that Y is utility independent (UI) of Z if >= on P, is the same for all z: Y(UI)Z
if >, = >;,
for all z, z’ E Z.
This
is weaker than value independence between Y and Z, which implies [see Theorem 11, and in this case both Y(UI)Z and Z(U1) Y. As shown by Pollak (1967), Keeney (1968), and others, the conjunction of Y(UI)Z and Z(U1) Y does not imply that Y and Z are value independent. Our basic theorem for Y(UI)Z in the lexicographic setting is as follows. U(p)=u.(p.)+u,(p,)
THEOREM 2. Suppose U: P + Rd and z0E Z. Let C be the subset of e < d dimensions on which U restricted to {(p y, zo):pye Py} is essential. Then Y(UI)Z if and only if either
(i) (ii)
e=O and U(py,z)
is constant on P,for
each ZEZ, or
e > 0 and for every z EZ there is a d x d real L-matrix A(z) of dimension e and essential column set C, and a b(z) E R”, such that
ProoJ The sufficiency of (i) or (ii) for Y(UI)Z is easily verified. Suppose Y(UI)Z. If e =0 then >zO is empty, hence >= is empty for every ZE Z because Y(UI)Z, and U(p,, z) must be constant on P, to give (py, z) N (qy, z) in all cases. Suppose e> 1. Since >,= >zO on P,, application of Lemma 2 to P y gives the relationship between U(p,, z) and U(p,, z,,) stated in (ii). Each A(z) has the same set C of essential columns because z0 is fixed. 1
Theorem 2 or the remarks following Lemma 2 show that if p y >=,, q y is resolved by the jth essential dimension of U restricted to {(p y, zO): pye Py}, then p y>, qy is resolved by the jth essential dimension of U restricted to { (pu, z): py6 Py}.
582
FISHBURN AND LAVALLE
Example 1 shows that e essential dimensions for U at z0 need not be the same dimensions as those for U at z. By modifying the definition of UI, we obtain more restrictive and less restrictive versions of utility independence for the lexicographic setting. Theorem 2 applies to the modifications under different restrictions on the A(z) matrices. The more restrictive version adds a special Archimedean axiom whose effect is to resolve (py, z) > (qy, z) by the same Vi for all z E Z. We say that Y is strongly utility independentof Z, and write Y(SUI)Z, if Y(UI)Z and for all pr, qrE P, and all z, z’ E Z, (py, z) > (qy, z) * there are ~1,BE (0, 1) such that (PY, z) 4q.9 2’) > (q.9 z) ap.7 z’)
and
(qn z) B(PY> z’) t (PYT z) 8(4r, z’).
It is easily seen when Y(SUI)Z that if (py, z) > (qy, z) with Uj(p,,, z) > Uj(q,, z) and Uk(p,,‘z) = Uk(q y, z) for all k < j, then the same things hold for every z’ E Z in place of z. The effect of Y(SUI)Z on Theorem 2 is to require A(z) in part (ii) to have C as its set of e essential rows. That is, the essential row set and essential column set of every transformation matrix are identical. When e = d, this is automatic and UI = SUI, so the notion of strong utility independence does not add anything to UI when utilities are real valued with d = 1. Our less restrictive version was considered in the real-valued setting by Fishburn (1974) and Fishburn and Keeney (1974). Let >* be the inverse of > so that p >*q if q>p. The condition of Y generalized utility independent (GUI) of Z allows complete reversals in preference and complete indifference in the >, comparisons: Y(GUI)Z if for all z, z’ E Z, >;#0*>;s E{&, >:,$a}. Thus I(>-: ZE Z>( < 3 when Y(GUI)Z. The definition of Z(GU1) Y is similar with Y and Z interchanged. As already noted, Example 1 has Y(UI).Z and not [Z(UI) Y]. But Z(GU1) Y since U(Y,z,~z,)=~(Y,O)+(1-~)(O,Y) =y(A
so >y=fa
when y=O, and >“=
1 -A),
>.V,#/zr whenever y, y’>O.
THEOREM 3. Suppose the hypotheses of Theorem 2 hold with >zO # 0. Then Y( GUZ)Z if and only if e > 0 andfor every z E Z there is a real d x d matrix A(z) and a b(z) E Rd such that
WP,, z) = A(z) VP,, A(z) = 0
zo) + 0)
for every pry P,;
if>:=JZf;
A(z) [respectively, -A(z)] is a L-matrix of dimension e and essential column set C if >, = > =. [respectively, >= = >:I.
MULTIATTRIBUTE
EXPECTED UTILITY
583
Proo$ The proof of the theorem is similar to the proof of Theorem 2. The only substantial difference occurs when >, = > z. Then the ujk in the e quasi-diagonal cells of A(z) must be negative to effect the complete reversal of conditional preferences. 1 A strong version of GUI has C as the essential row set of A(z) in the final clause of Theorem 3. The special Archimedean axiom for this version is a straightforward generalization of the one for SUI.
5. MUTUAL
UTILITY
INDEPENDENCE
We now assume that Y(UI)Z and Z(U1) Y under the formulation and assumptions in the opening paragraph of the preceding section. We assume also as a scaling convenience that U(y,, zO) = (0, .... 0) in Rd for a fixed (y,, zO) E X. When d= 1, mutual utility independence implies that there is a constant c0 such that, for all (Y, Z)EX U(YYz) = U(YYzo) + U(YO?z) + COU(Y?zo) U(YO?z). The additive form is obtained if c,, = 0; otherwise we get the multiplicative form by resealing. The constant must satisfy 1 + c0u( yO, z) > 0 and 1 + cOu( y, zO) > 0 for all y and z. Assume henceforth that d > 2. According to Theorem 2 applied to Y(UI)Z and then Z(U1) Y, WY, z) =&I
VY, zo) + b(z)
WY, z) = B(Y) WY,, z) + C(Y)l
where A and B are maps into d x d matrices, and b and c are maps into Rd. Taking y = y, in the first equation and z = z0 in the second, we have V( y,, z) = b(z) and U(Y, zo) = C(Y), so U(Y, z) = A(z)
U(Y, zo) + WY,,
U(Y, z) = B(Y) WY,,
z)
z) + U(Y, zo)
(1)
(2)
for all (y, z) E X. The particular form for U( y, z) implied by (1) and (2) will depend on the nature of the A(z) and B(y). We consider some examples to illustrate possibilities and then derive a few more general results. Throughout, we follow the notation of Theorem 2: ey, Cy, e,, and C, are the dimensionality of {(p y, zO): P,,E Py}, the essential column (dimension) set for ey, the dimensionality of {(y,, pz): pz E Pz}, and the essential column set for e,, respectively. 480/36/4-9
584
FISHBURN AND LAVALLE
EXAMPLE 2. Suppose e y = 0. By Theorem 2(i), U( y, z) is constant on Y for each ZEZ, and we have
U(Y> z) = U(Ylb z) for all (y, z) E X In this case e, = d, U( y, z,,) = U( y,, zO) = (0, .... 0), and B(y) is the d x d identity matrix Id for all y. EXAMPLE
3. Let Y = Z = [ 1,2] with y, = z0 = 1 and u y, z) = (U’( y, z), U2(Y, z)) = (YZ - 1, Y/Z - 1)
for all (y, z). The “indifference curves” in [l, 212 for U’ go from upper left to lower right, and those for U2 go from lower left to upper right. d= 2 is verified by (2,2)>(2,1)>(1,2)alongwith(p.,2)~(2,1)wheneverp.((l,2])>O.Wehave U’(y, l)= U’(l,z)=z-1,
V(y,
l)=y-
1,
ey= 1,
P(l,z)=l/z-1,
c,=
e,=2,
An example to illustrate the latter case is pz( 1) =p,(2) W?P,)
(1); C,={l,2}.
= l/2 and q,(3/2) = 1. Then
= U’(L 42) = l/2,
- l/4 = U’( 1, pz) > U2( 1, qz) = - l/3. Mutual utility independence is easily verified from the definition of U. Let E, denote expectation with respect to probability distribution r. For r, s E P ,,, (r,z)>(S,z)OE,(y)>E,(y) so Y(UI)Z;
for r, SE P,,
(y, r) >
(Y, s) -
E,(z)
> E,(z)
or {E,(z)
= E,(z)
and E,(llz)
> E,(llz))
so Z(U1) Y. We also have
for k = 1, 2. Mutual utility independence can be established directly from this form since 1 + Uk( y, 1) > 0 and 1 + U”( 1, z) > 0 for all y, z, and k. If we did not have constant signs here then utility independence but not generalized utility independence would fail. EXAMPLES.
Let Y=X,~X,=[1,2]~andZ=[1,2]
withy,=(l,l),z,=land
MULTIATTRIBUTE
EXPECTED
585
UTILITY
This is an extension of the preceding example. Its third dimension comes into play when equality holds for the first two dimensions. For example, when x=(x,,x,,z)=(1,2, 1) and ~‘=(a,,/?, l), Uk(x) = Uk(X’)
fork=
1,2
U3(x)=3-2>2$-2=
U3(xt).
We leave it as an exercise to show that Y(UI)Z, C,= { 1,3}, Z(UI)Y, and c,= (1,2}. The following theorem gives the general form of U when mutual utility independence holds and conditional utilities have full dimensionality with C.=C,={l,...,d}. THEOREM
constants
4. Suppose e y = e, = d, Y( UZ)Z, and Z( UZ) Y. Then ci for k = 1, .... d and, given k, for i, j< k, such that
+?Y,
U”(Y, z) = Uk(Y,&I) + Uk(Y,, z) + i i,j=
4 WY,, z)
there
are
(3)
1
for all ( y, z) E X and k = 1, .... d, In addition, for each k from
1 to d,
k 1+
1+
1 j=l
i
c;juj(y,,
z)
&Ui(y,
z,)>O
>
0
for all z E Z,
(4)
for ally E Y.
(5)
i=l
The sign restrictions of (4) and (5) are needed to prevent preference reversals or nullities of the type that can arise under GUI but are forbidden by UI. We verify them first and then establish (3). Proof of the Last Part. Assume that (3) holds along with the theorem’s hypotheses. Since the proofs of (4) and (5) are similar, we prove only (4). Consider (4) at k = 1. By (3) and linearity, WP,,
z) > wq,,
z) o WP,,
&I) Cl + c:, WY,,
’ wq,,
%I) Cl + 4 WYCI~z)l.
Since ey= d, there are pr, qrE P, such that U’(P y9 zo) > wq
z)l
Y, 4
586
FISHBURN
AND
LAVALLE
with (py, z,)>(q,, z,,). Since Y(UI)Z, (py, z)>(q*,z) for all z. Full dimensionality then requires U’(p,, z) > U’(q,, z) for all z, and this is true if and only if 1 + c:l U’(y,,
for all z E Z.
z) >o
This verifies (4) at k= 1. Suppose k> 1. By (3),
Uk(p,, z)> Uk(q,, z)* Uk(PY>4
i
1 + i CfjWY,, z) j=l
1
k-l
+ 1 CWP,, zo)- mq,, %)I i c; WY,~ z) i=
>
1
j=l
Uk(q*,z,)
l+ [
i j=l
cij
uj(YO?
z,
1 .
Since ey= d, there are pr, qre P, such that u’(P.,
for all i < k;
4 = U’(q.9 &I)
Uk(Pn
zo) > Uk(qn
zo).
Full dimensionality and Y(UI)Z then require U’(p,, z)= u’(q,, z) for all i< k, and Uk(p =, z) > Uk(q ., z), for all z E Z, and this clearly holds if and only if (4) holds at k. Proof of (3).
Fix k
l
{ 1, .... d} and define k x k matrices M=
[I”ql
=
C”‘(Yj,
zO)l~j=
N=
CNijl
=
C”‘(YO,
zj)l:j=l
1
for y,, .... y, E Y and zi, .... zk E Z. We wish to choose the yj and zj so that M and N are nonsingular. This can be done by induction on k. We demonstrate the essentials for M. For k = 1, choose any y, E Y for which U’( y,, zO) # 0, as guaranteed by ey = d. Suppose nonsingularity holds at k - 1 when k > 1 for
As assured by full dimensionality, Uk(P.? %I) + Uk(qn
z,),
let py, que P, satisfy for i < k.
U’(P Y1zcl) = U’(q Y, %)
Form M,+ from M0 by adding a kth row and column. Entry j< k in row k is U”( yj, zO), and entry i in column k is U’(p y, zO) - Vi(qr, z,,). Expand the determinant of M,+ by the kth column to get I%
= flM,I
CUk(PY, %I)-
Uk(q,v
%)I
zo.
MULTIATTRIBUTE
EXPECTED
Also expand the Uk by linearity
to obtain
and let m, be the determinant WY;, z,), ***, Uk( y;, zO). Then
of M,f
587
UTILITY
after its final column
is replaced by
so at least one m, is not zero. Let y, denote a yj for which m,#O. Then [ vi( yj, zo)]i,j.& k is nonsingular. Assume henceforth for fixed k that M and N are nonsingular k x k matrices as defined earlier. For all z and y, define k x k matrices M(z)=CU’(~j,z)-U’(~,,z)li,j~k
N(Y)=
C~‘(Y,
Zj)-
U’(Y,
ZO)li,j
We assume that (1) and (2) hold under mutual utility independence and, since eY= e, = d, we can presume that A(z) and B(y) are lower triangular (see Lemma 1). Because of lower triangularity and our focus on Uk, no generality is lost by truncating (1) and (2) to their first k rows and first k columns. Let (1) and (2) be understood in this way. Then M(z)=A(z)M by (1) and N(y)=B(y)N by (2). Because M and N are nonsingular, right multiplication by inverses gives A(z) = M(z)M-’ and B(y) = N(y) N- ‘, and substitution in (1) and (2) yields U(Y, z) = M(z)M-’ U(Y, z) = NY)N-’
VY, &I) + WY,, z)
(6)
WY,, z) + UY, 4
(7)
for all (y, z) E X. By (7),
so the jth column of M(z) equals the right Nj = N( yj) N- ’ - zk. Then, by column listing, CN,U(Y,,z)+U(Y,,z,)...NkU(Yo,z)+
M(Z)= =
Substitute
CN,U(y,,
z)...Nku(yo,z)]
side of this expression.
U(Yk,%)I +kf.
this into (6) to obtain
u(Yv Z) = u(Y, &I) + u(Y,,
Z) + CN, u(Y,v Z) .--N/c u(Y,, z)] M-‘U(Y,
zo).
Let
588
FISHBURN
AND
LAVALLE
Therefore @( y, z) equals Uk(y, zO) + Uk(y,,, z) plus the kth term of the final expression. That term is k rk c j=l
1
c (Nj)kiUi(yO, i= I
=
5 i=l
i /=I
;I]
U’(YOYZ)
[
i (Mpl)jlU’(y, /= 1
u’(Y,
ZO)
i j=l
zCl)]
(Nj)ki(M-‘)j,.
When the final sum over j is denoted by cf, we get (3).
1
Theorem 4 and our earlier examples illustrate the type of decomposition that arises under mutual utility independence. We do not have a general result for all possible forms of (1) and (2) under mutual utility independence, but conjecture that the general case uses only the real-valued functions U’(y, z,), .... V’(y, z,,), wYcl, z), .... Ud( y,, z) and multiplying constants in a bilinear form for the vi( y, zO) and Uj( yO, z), much as in (3). The examples preceding Theorem 4 illustrate some combinations for ey, e,, C,, and C,. An interesting question that we leave open asks for a description of all combinations that can arise when U has dimension d. We do have answers to two more limited but still interesting questions: 1. Is C, u C, = { 1, .... d}? 2. Is e,+e,>d? Surprisingly,
the answers are negative as shown by our concluding
example.
EXAMPLE 5. We use the simplest possible example to show a case of Y(UI)Z, Z(UI)Y, and e,+e,
U(y,z)=(y-l,z-l,yz-l),
y(J=z,=
1.
Then, with E denoting expectation,
VP,, z) = (E(y) - 1, z - 1, MY) - 1) ub, PA = (Y - 1, E(z) - 1, YE(Z) - 1). The first equation shows that only dimension 1 is essential for U on {(p y, zO): P,,E Py}, the second shows that only dimension 2 is essential for U on { (yO, pz): pz E Pz}, and clearly Y(UI)Z and Z(U1) Y. By the definition of U, d < 3. The fact that d = 3 is suggested by letting Xl = (2,2),
x2=(2, Ax1
l), 1 =PbJ
x3
= =
(1921,
l/Z
x4=
(41);
MULTIATTRIBUTE
and noting by the representation
EXPECTED
of Assumption
589
UTILITY
1 that
x,>x,>p>x,>x, X,>X,JX‘$>Xj
for all 0 < A < 1
P+x2%
for every 0 < 1< 1.
Let cli = CL(X~)for a E P, so c(~+ CQ+ c(~+ ~1~= 1. Then, for all c(,/I E P,
The three equations here imply a = /I, so CIN /I o CI= p. Now suppose that d= 2. We show that this is impossible. Let uk(xj) be the utility assigned to xi by the kth component of the two-dimensional utility function. We then require
aNBe i Ul(Xj)(aj-Bj)=O, jc,
u*(xj)(aj
-
Pi)
=
O.
j=l
Regardless of the values of the uk(xj), the two equations on the right side are insufficient to imply a =/I, which can only be guaranteed by three equations in the unknowns CI~-pi, CI~-p2, a3 -/I3 (with ad-/I4 given from these and 1 a, = C pi = 1). In other words, d = 2 forces a N fi for some a # j$ a contradiction. Hence d > 2, and since d < 3 by the definition of U, we conclude that d = 3.
6. DISCUSSION
This paper brings together two themes in expected utility theory that emerged many years ago. The first, non-Archimedean expected utility (Hausner, 1954; Chipman, 1960), was motivated by an interest in omitting the technically convenient Archimedean axiom. The second, independence conditions among attributes in multiattribute real-valued expected utility (Fishburn, 1965; Pollak, 1967; Keeney, 1968), was motivated by a desire to simplify the burden of utility assessment in multiattribute decision analysis. Our integrating framework considers an outcome set X= X, x X, x . . . x X, in the context of an m-dimensional linear function U = (U’, .... Urn) defined on the set P of linite-support probability distributions on X. Preference between members of P is assumed to coincide with the natural lexicographic order between their utility vectors. We noted that the marginal independence condition of value independence produces an additive decomposition of each Uj over the attributes (Theorem l), as in the real-valued case. Under a partition of the Xi that gives X = Y x 2, we then considered the conditional independence condition known as utility independence.
FISHBURN AND LAVALLE
590
This leads to a partial decomposition of U which confines one of Y and Z to onepoint marginals (Theorem 2). After commenting on a strengthening (SUI) and relaxation (GUI: Theorem 3) of utility independence, we examined the effects of mutual utility independence between Y and Z. Under full dimensionality of both conditional forms, mutual utility independence leads to a decomposition involving functions of Y and functions of Z (Theorem 4) that is a natural generalization of the real-valued mutual utility independence decomposition. But less-than-full dimensionalities are possible for the conditional forms, and there are even cases in which the sum of the numbers of essential dimensions for the two is less than the dimensionality of U (Example 5). Questions left open for mutual utility independence include the possible combinations of essential dimension sets for the Y-conditioned-on-Z form and the Z-conditioned-on-y form, and a general expression for U in terms of single-variable functions regardless of the dimensions of the conditional forms. The effects of other combinations of utility independence in the X= X, x X, x . . . x X, setting, such as Xi utility independent of the other attributes for i= 1, .... n, await study. Chapter 6 in Keeney and Raiffa (1976) describes results in this area for real-valued expected utility.
REFERENCES CHIPMAN, J. S. (1960). The foundations of utility. Econometrica, 28, 193-224. DEBREU, G. (1960). Topological methods in cardinal utility theory. In K. J. Arrow, S. Karlin, & P. Suppes (Eds.), Mathematical methods in the social sciences, 1959 (pp. 1626). Stanford: Stanford Univ. Press. FISHBURN,P. C. (1965). Independence in utility theory with whole product sets. Operations Research, 13, 2845.
FISHBURN,P. C. (1970). Utility theory for decision making. New York: Wiley. FISHBURN,P. C. (1971a). A study of lexicographic expected utility. Management Science, 17, 672478. FISHBURN,P. C. (1971b). Additive representations of real-valued functions on subsets of product sets. Journal
of Mathematical
Psychology,
8, 382-388.
FISHBURN, P. C. (1974). von Neumann-Morgenstern Research,
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