Download Multilinear Representations for Ordinal Utility

JOURNAL
OF MATHEMATICAL
PSYCHOLOGY
Multilinear
31, 44-59 (1987)
Representations
Utility Functions
DAVID
E. BELL
Harvard
Universiry
for Ordinal
An ordinal utility function u over two attributes A’, , A’, is additive if there exists a strictly
monotonic function cpsuch that q(u) = v,(.Y,) + u2(x2) for some functions v,, v2. Here we consider the class of ordinal utility functions over n attributes for which each pair of attributes is
additive, but not necessarily separable. for any fixed levels of the remaining attributes. We
show that while this class is more general than those that are ordinally additive, the
assessment task is of the same order of difftculty, and involves a hierarchy of multilinear
1 19X7 Academic Press. Inc.
rather than additive decompositions.
1. INTR~DLJCTI~N
When making comparisons a person will rely on certain descriptors or attributes
of the alternatives. Let (X, ,..., X,) be the dimensions on which the decision maker
evaluates alternatives. An ordinal utility function 4x1,..., x,) is any function with
the property that if alternative A = (a, ,..., a,) is preferred to B= (b,,..., 6,) then
u(a, )...) a,) > u(b, )...) b,) and vice versa. Clearly if u is an ordinal utility function
then so is p(u) so long as cp is strictly monotonically
increasing. More generally, u
may be thought of as a model of any factorial interactions (Lehner, 1981).
A utility function is said to be ordinally additive if there exist such a cp and
functions u, ,..., u, with the property
444x1 ,...>x,)1 = f Ui(Xi).
i=
I
This property is especially useful. It aids assessment, because each attribute has its
own scale, ui, that is independent of the levels of the other attributes. For the same
reason evaluation of alternatives is especially simple.
Even if the whole function is not additive, it could be that additivity could hold
within a subset of attributes. For example, the simple function u(x, y, z) = x + yz is
not ordinally additive, but when rewritten as
x + exp( log y + log z)
or
u*(x) + dUz(Y) + h(Z))
44
0022-2496/87$3.00
CopyrIght (3 1987 by Academic Press, Inc.
All rights of reproduction
in any form reserved.
MULTILINEAR
REPRESENTATIONS
45
we see that the subset of attributes { Y, Z} is additive. Moreover, once the function
o,(y) + v3(z) has been assessed, call this a new attribute T, then we see that
u(x, y, z) may be written as
u,(x) + v(t).
Thus, although u(x, y, z) is not additive, there is a way to structure the assessment
process so that complicated interactions are avoided. Ting (1971) gives a highly
readable account of the literature on this subject. Nahas (1977) relates and adapts
this work to cardinal utility functions.
Some very simple functions are not amenable to decomposition
into additive
hierarchies. For example, u(x, y, z) = xy + yz + xz is not additive nor is any subset
conditionally additive. However, if we fix the attribute Z at some specific level, say
3, then u(x,y, 3)=xy+3y+3x=exp(log(x+3)+log(y+3))-9=cp(u,(x)+u,(y))
is additive. The problem is that the functions u, and u2 depend on the level of z.
Though cp does not change in this example, any utility function of the form
u(x, y, z) = (p(u,(x, z) + u,(y, z), z) would still have the property that for any fixed
value of 1 the subset {X, Y} was conditionally additive.
It is important to distinguish between the following properties.
A subset of attributes (X,, X2,..., X,) is strongly conditionally
additive with respect to {X,, ,,..., X,,} if there exist functions ui, i= l,..., k, and q
such that
DEFINITION
1.
4x1, x2 ,..., .u,) = cp i Ui(Xi), Xk+ I’..., x, .
( i= 1
)
DEFINITION
2. A subset of attributes (X,, X2,..., X, > is weakly conditionally
additive with respect to (Xk+ , ,..., X,} if there exist functions ui, i= l,..., k, and cp
such that
24(x,)...,
X~,Xk+,,-Yk+Z,...,xn)r-~k+,,...,~~,
Evidently a subset which is weakly conditionally additive is strongly conditionally
additive if and only if it is also separable (Leontief, 1947a; Gorman, 1968).
All decomposition results for ordinal utility functions that I am aware of use
Definition 1 as the fundamental independence assumption. In this paper I will use
Definition 2 and thus not rely on separability.
The implications of this weakening may be described as follows. If each pair of
attributes is weakly conditionally
additive (of the remaining attributes) then the
utility function may be decomposed into a hierarchy of subsets such that at each
level a subset has either an additive or a multilinear representation.
46
DAVID E. BELL
For example, when II = 3, the utility
following four forms:
function
u(x, y, z) must have one of the
0) $Cdu + w) + tl,
(ii) $Cdu + t) + ~1,
(iii)
(iv)
$[q(w+t)+u],
~[a,v+a~w+a,t+a,,uw+a,,ut+a,,wt+a,,,uwt],
where u, w, and t are functions of x, y, and Z, respectively, and + and cp are
arbitrary functions.
For general n the results consist of repeated application of additive or multilinear
forms (type (iv) above) to a hierarchy of subsets of attributes. This provides a set of
decompositions
strictly broader than the purely additive hierarchy, yet which
retains the key simplicity of having one scale for each attribute.
The next section provides some background theorems and elementary lemmas
necessary for the main proofs. Section 3 contains a proof of the main theorem
providing the result for three attributes described in (i) to (iv) above. Section 4
provides the necessary generalizations for larger n. The proofs are by no means
elegant. A simple proof might be constructed if the following corollary of the main
result could be demonstrated directly. If each pair of attributes is weakly conditionally additive with respect to the remainder then either there exists a subset of
attributes (of size greater than one but less than n) that is separable or the utility
function is multilinear (i.e., of the form (iv)).
2. PRELIMINARY RESULTS
An important result, due to Debreu (1960) and strengthened by Gorman (1968),
that helps to detect the presence of additivity in three or more attributes is as
follows.
THEOREM 1. If the conditional utility function over any pair of attributes
independent of the fixed levels of the other attributes then the utility function
additive.
is
is
This result allows us to test for ordinal additivity either by asking the decision
maker to introspect about his preferences or by testing a utility function for this
property. Another result, due to Leontief (1947b) and Debreu (1960), is useful for
all n but only when a utility function has already been assessed. As stated here the
result requires differentiability
of U. Though this is not a restriction (Lute and
Tukey, 1964) we will assume differentiability
throughout and so use the more
specific statement.
REPRESENTATIONS
47
u(x,, x2) is additive
if and only if, for some
MULTILINEAR
2. A utility
functions v I and v2,
function
THEOREM
logs-log&=v,(x,)-v,(x,).
I
2
In generalizing to n attributes we will use the notation x, to mean all attributes
other than X,, and x, for all attributes other than Xi and X,.
DEFINITION 3. A set of attributes is pairwise weakly conditionally
(PWCA) if each pair of attributes is weak1.y conditionally additive (WCA).
We may adapt Theorem 2 to provide
additive
LEMMA 1. Two attributes X,, X, are WCA if and only if there exist functions vf
and vt such that
log $ - log & = IIf,
I
J
Note that for PWCA there are potentially
following result reduces this number to n.
- I+;).
n(n - 1) functions of the form II:.. The
LEMMA 2. Attributes (X, ,..., X,) are PWCA zf and only if there exist functions
vi, i = l,..., n, such that, for each pair of attributes Xi, Xi
log~~-log&(n’;)-vj(.~,),
I
J
Proof.
To prove n = 3 we begin with
log&10g~=v~*(x,,x,)+v~,(x,,x,)
I
2
log~-log~=v~~(~~,,x2)+v:,(x,,~~~)
I2
3
log~-log~=v:,(x2,x3)+~:,(x,,r,).
I
3
Adding the three equations together we find
vi2 + VT2+ vi3 + v& + vi1 + ?I;, = 0.
Setting x3 at any value, we see that
ui3(x19
480/31/l-4
x2)+C(xl,
x2)=al(x1)+a2(x2)
48
DAVID E. BELL
for some functions a,, a*. Using this equation to replace o$, and repeating.for vf2
and u& we see that only three two-dimensional functions are required, and that,
with some redefinition, the equations may be written as in the statement of the
lemma. The case n = 4 is more tedious, but is also found by repeated substitution,
This case is also equivalent to the general induction procedure. The details are
omitted since this paper only makes use of the three-attribute result.
LEMMA 3. Functions u and v are related as u = q(v) ,for some cp if and only ij
log(du/a.u,) - log(dv/ds,) is independent of i.
Proof: If u = c~(u) then a@.~, = (8v/ax,) $(v) SO that log(&@x,) - log(&@x,)
= log q’(u) for all i. If log(au/ax,) - log(&/&,)
= k(xr,..., x,) for all i then
&.&3x, = ek( &$.I~,). Since du = L(&@xj) dx, and dv = .Z(&$7x,) dx, we have
du = ek dv. Hence u is constant when u is constant and vice versa. Hence u = q(v)
for some cp.
COROLLARY 1. Functions u and v are each PWCA
i= 11..., n, if and only if u = cp(v) for some cp.
Proof.
and have the same ui(?c,),
Since
log&log~,=log~-1ogg
1
.I
I
’ I
for all i and j, then
au
au
l%~--og~=l%~px;
au
a0
Vi, j.
LEMMA 4. Zf u,(x, ,..., xk) and u2(xI, + , ,..., x,) are, separately, PWCA,
u = au, + hu2 + cu, u2for arbitrary constants a, 6, c.
Proof:
&=(b+cu,)g
I
=(a+cu2)~
I
if
i>k.
if
i<k.
I
If Xi and Xj are both in (XI,..., xk) then
log g
I
- log $log$log~
I
I
I
then so is
MULTILINEAR
and, similarly,
if both are in (X,,
log;
I
49
REPRESENTATIONS
, ,..., X,). If i < k and j > k then
- log g = log 2
+ log(a + CUJ
I
I
- log 2
- log(b + cu,).
I
The first and fourth terms of the right-hand
and third, of xi.
3. DIFFERENTIAL
side are independent
EQUATIONSFOR
of xj; the second
THREE ATTRIBUTES
We will concentrate for the time being on the case n = 3, and rewrite the
equations in Lemma 2 as
dz= W, Y)k(x,Y,~1.
Note that these three equations
PWCA.
(3)
are necessary and sufficient for (X, Y, Z) to be
LEMMA 5. The partial differential equations au/ax = jk, &lay = gk, aujaz = hk
are integrable if and only if
h($$+g(g-;)+f($-$)=O.
Proqf:
(4)
We may derive two equations for a2u/ax dy and equate them:
ak ag
fk+f-=-k+gz.
ay ax
ay
Similarly
and
~k+g!%~hk+h!!t~
z
aZ ay
ak
50
DAVID
E. BELL
Weighting these equations in the ratio h: g: -f and adding, then clearing the term
in k leaves the equation of the lemma. The converse is shown by an argument in
Piaggio (1960, p. 141). 1
We may rewrite (4) as a partial differential
equation in h(x, y):
Choose three values of Z, say z’, ?, and z3, and substitute each in (5) and then
solve the resulting equations for h, ah/as, and &/SJ’. For this procedure to work,
the ,-I must generate independent equations, and functions for h, ah/ax, and ah/+
that are internally consistent.
In fact we will show that it is not possible to have three linearly independent
versions of (5), but that in the case where there are two, the corresponding utility
function is of the multilinear form, i.e.,
where u = u(x), M’= M’(Y), and t = t(z).
We will also show that if there is only one linearly independent
11must have one of the forms
equation (5) then
II = cp(v+ w) + t
u=cp(u+t)+u
or
u = (P(M’ + 1) + u.
If (5) is, up to a factor, the same equation for all z, thenf(y, z) = /I(z) f(y, z”) = flf,
and g(x, 2) =/3(z) g(x, z ‘) =/?g, for some p(z) and z”. The function h is thus a
solution of
This may be rewritten as
Now g,(aL/ax) - fo(cX/dy) = 0 has the solution L = @(F, + Go), where F. = l/f,
and G& = l/g,, so h = $(Fo + G,)/I$G& for some general $.
MULTILJNEAR
LEMMA
6.
There exist ,functions (p,, (p2, and ‘p3 such that
au
-=
2X
rpl(w+t)k
W’f
’
fh
G=
Proof.
au
z=
cpz(u+ Qk
3
u’f’
tf and only if u has one of the following
cp,ICI,4-x), 4.vL t(z):
(i)
(ii)
(iii)
(iv)
51
REPRESENTATIONS
forms
v3(“+w)k
ujwl
for
arbitrary
functions
$(cp(u+w)+t),
Ncp(o + t) + w),
$(q~(w+t)+v),
~(a,2,+a,M’+a3t+a12uU’+aI~ut+a23wt+a123uUlt).
Substituting
the forms off, g, and h into (4) we find
or
An obvious solution is 50, = (p2 = constant (which corresponds to the solution of (6)
above). By inspection u = cp(v + w) + t is a solution in this case where cp’ = l/cp, (set
k = v’w’t’p;). By Corollary 1 the general solution is $(cp(v + W) + t). This reasoning
gives us cases (i), (ii), (iii).
Since cp,( MI+ t) E (p2(u + t) only if both are constant we need only consider the
case rp, # cpz. In this case
Transform the variables to 0, = w + t, 0, = v + t, 0, = v + w and view this last
equation as one in 0, giving the solution cp3= a + be’@] for some constants a, b, and
c, where c=(cp’,-&)/(cp,-cp,)
so that cp’,-c’p,=cp;--c’p2=c0,
say. Hence
p, = -c,/c+ b,e”“‘+“,
cpz= -co/c+ b2e“D+r’r
and 9, = -co/c+ b3e”“‘+““.
A
solut;on using these cp, is
U=b,e~‘.“+b2e~‘.“‘+b3e-‘.‘-COe-‘.’”+”.+’).
P
(7)
52
DAVID
E. BELL
For example,
au
CO
h, --,-c(w+r)
C
--cv'e-""
z=
[
=(-c,/c+h,e”“+“)
cv’w’t’e -c(tr+w+f)
w’t’
=cP,(w+t)
w’t’
I
k.
Equation (7) is equivalent to a multilinear form (e.g., e -“’ = a,,/,/&
+&
u2 if
a,*3 # 0). It is straightforward to show that the forms specified in the lemma satisfy
the differential equations. 1
Now suppose that there are two linearly independent versions of (5), one when
z=z ‘, one when z = z2. For all other values of z, Eq. (5) is a linear combination of
the cases when z=z’ and z= z2. Then, for some ,X,(Z), a,(z),
and
f(y, =)= a,f(y,=‘I+a2f(y,z2)- a,f, +a2f2.
These expressions are sufficient to give
but not sufficient to give
f$-gz=a,
Imposing
(aI
g,
a2(f2(ll
this
+
latter
a2g2)(aifl
gl
+
I2 g2)
+
-
(f,P--g,!$)+aZ(f2%--g2$).
z
g2(llfl
condition
4f2)
=
+
12f2)),
requires
al(fl(kl
that
gl
+
k2g2)
where k, = ai(
(a,
f,
+
-
a2f2)(a’,
g,(k,f,
k, = ai(
l2 = a;(z’).
This expression may be rearranged to give
(f1g~-f~~~)(a~a~-a~a~-~,a,+k~a~)=0.
By choice of the fi, gj we know f, g, #f2 g, for some x, y. Hence
a,a;-a,a;--l,a,+k,a,=O
g,
+
+k,f,))
a;
g2)
+
I, = a;(z2),
MULTILINEAR
53
REPRESENTATIONS
a; - 1,
a;-k2
-=-z=‘(z),
a1
a2
say. Hence
1, je-
a’, - a’~, = k2 so (d/dz)(e-“a,) = k2 j eea dz and (d/dz)(e-“a,)
=
dz. Hence a, = (k,A + a,)/A’ and a, = (I, A + a,)/A’ for some constants CI~
and a,. We may write
and
g(su
,-
_)=(a,g,fa,g,)+(k,gl+l,g,)A
A’
Now we appeal to symmetry. At least one offi/fi and g,/g, must be non-constant.
Suppose the former. Consider (4) as a differential equation in g. As y varies does
this equation have one, two, or three linearly independent forms? We know it has
at least two, becausef(y, z) =f, aI(z) +f2a2(z). Could it have three? That is, could
we have h(x, y) =x7=, /Ii(y) hi(x)? If so, then also f(y, z) = C:= r pi(y) q,(z) for
some ‘p,. But then CT= r pi(y) v;(z) =f,(y) a,(z) +fz(y) a*(z). Now take any three
values of z to show that the /?; are linearly dependent. Thus we may conclude thatf,
g, and h all have at most two generators on each dimension.
We have already shown that
f(y
7) =
PI
+
r2b)
A(z)
9’
A’(z)
so by symmetry we can argue that
f(y
+m+~2ww
1’
B’(Y)
for some B. Hence for some constants ai,
.f(Y, z)=
a,+a,A+a,B+a,AB
A’B’
which we will write, more suggestively, as
f(Y, z)=
a,+a,w+a,t+a,wt
W’i’
9
54
DAVID
E. BELL
Thus we may write
all a,+a,w+a,t+a,wtk
a,y=
W’f’
du
$i=
b,+b,v+b,t+b,vtk
v’t’
au C,+C2v+c3W+~4~~lk
z=
V’W’
Substituting these equations in (4) shows that the constants a,, hi, ci must be consistent with a multilinear form.
Now consider the case when there are more than two linearly independent versions of (5) as a function of z. Since there can be at most three such equations, at
-,’ ,* -2 3 and z3, say, we have
f(y, =) = a,(=)f(y, z’) + a2(=).m, z2)+ ~&).f(Jh z3)
=~,f,+az.f2+Qf3
and
Proceeding as before,
(~I.6 + a2f2+ ~,f,)(a;g,
=
',[f,ck,g,
+ @,cfi(l,g,
+
a;g,
+
a;g31
+k2fZ+k3f3)]
+k2g2+k3g3)-g,(k,f,
+
+~3Cf3(m,g,
lzg,
+ 13g,) - g2(1,f, + 1,f* +
+~2g,+~~,g,)-g,(m,f,
l3f3)]
+m2f*+m3f3)]
or
(f,g2-fZg,)(~,~;-~;~,--,k2+~2~‘)
+(f,g3-f3g,)(~3~;--cr;~,+~lk,--a,~,)
+
(f2g3
-f3g2)(t12c1;
-
a;cr3
-
@,l,
+
@3m2)
=
O.
All the OLexpressions must be zero; otherwise there would be a contradiction with
the assumption that (4), viewed as an equation in h, ah/ax, and ah/@, has three
linearly independent cases. Hence
MULTILINEAR
so, as before, ~1,= (/,A + a,)/A’
a,k,-m,a,=O,
we have
55
REPRESENTATIONS
and a2 = (k,A + az)/A’. Since also a3a’, -a;~,
I,-m,-(“A~~~‘A~]+(~)(k,-a3)=0
A’
or
a;-a3
(I,-m,)A’
I,A+a,
-- A”
A’
Let
d=
(I, -ml) A’ -- A”
1,A+a,
A’
Now
e -“ai - n1a3e-s = k3epn
so that
or
a3=k3~e-rdz+kq
e -7?
Now
so
-ko
e-“dzze m,(I,A+a,)““‘l+k,.
s
We have
a3=Ck
3epk0(f, A + a,)“‘l”l + kb]
(I,A+u,)‘~“““.
m, edkOA’
1
=k
3’
+
56
DAVID E.BELL
Now we cannot have k, = 0 or m, = I, ; otherwise c(, and c(~ would be linearly
dependent. But by using the third x expression we can also determine that
x3 =
[lie-‘“(krA
+aJ”‘:“‘+k,][k~
FN?c?
+a,],-““jk?
-‘o/i
For both of these equations for x3 to hold, we require that
Fyi, [/,(k2.d
+ a?)
= m,[k,(l,A
+
k,(k,A
+ a,)’
+a,)+k,(l,A
‘We’ e”]
+.,)‘-‘f’%k”].
This is impossible unless kz/a, = 1,/a, and mJk, = m,ll, , i.e., kJ1, = 0,/a, = m2/m,.
But this means IX?= (kJf,) CI,, which is a contradiction.
We summarize our results as follows.
THEOREM 3. The attributes X, Y, Z
the following four forms:
are
PWCA if and only ifu(.u, y, z) has one of
(i) (c/Cdu+ lrl)+ fl
(ii)
(iii)
$[q(u + t) + 12.1
$[~(M.+~)+L~]
where 21,II‘, and t are,functions ef.~, y, and z, respectively, and cpand $ are arbitrary
transformations.
Thus the class of PWCA functions (for n = 3) is relatively straightforward to
assess. First one tests if any pair of attributes is separable, that is, if trade-offs
between X and Y, say, are independent of Z. If so, then the utility function must
have form (i). Having assessed u(x, y, ;O) = cp’(v(x) + w(y)) in the usual way we
may rename the quantity q”(u + w) as a new attribute X’. Now the original utility
function has the form $[q(x’) + t(z)], so again traditional techniques for ordinally
additive functions may be used.
If no pair is separable, then form (iv) must apply. Only two functions of the form
u(x, y, z”) and u(x,, y, :) are needed to determine u, u’, t, and the constants. (For a
multilinear function with n attributes one need only assess n/2 functions of the form
24(x,, XIX;).)
4. MORE THAN THREE ATTRIBUTES
It is a simple matter to generate examples of n-attribute utility functions that are
PWCA, yet complex. Using Lemma 4 and Corollary 1 we see that the following
procedure produces PWCA functions.
MULTILINEAR
Step 1.
Step 2.
Step 3.
that subset.
Step 4.
REPRESENTATIONS
57
Make arbitrary transformations of the attributes.
Make an arbitrary partition of the attributes into subsets.
For the ith subset, let (pi be a multilinear function of the attributes in
Consider the ‘pi functions as new attributes
and return to Step 1.
This cycle of steps is repeated until there is only one element of the partition
Step 1. For example, for n = 2 we have
in
Step 1. x + u(x) y -+ w(y).
Step 2.
Partition
= (X, Y}.
Step3.
cp,(x, y)=a,u+a,w+a,zw.
Hence u( x, y) = cp(a, r + a, ~1-I- u3 VW), which of course is additive. For n = 3 we may
have, at Step 2, either
(i)
(ii)
Partition=
{X, Yj, {Z}, or
Partition = (X, Y, Z}.
The first leads to form (i) of Theorem 3, the second to form (iv) of Theorem
II = 4, we may have at the first step 2 either
(i)
(ii)
(iii)
(iv)
(X
IX,
(X,
{X
Y), (-5 W),
Y, Z}, { W},
Y, Z, W}, or
Y}, {Z), (W}.
To prove that this procedure provides all PWCA
from the three attribute case.
COROLLARY 2. If
X,, X,, X, are
u= $[u,(x,,
.x3) + uz(xz, x3), x3],
where
av,lii.u,=o.
3. For
PWCA
either
functions we need a corollary
then
u muy
be
Mritten
(ajay) t,b(x, y) =0
or av,/?.u, +
Prooj
Since we know that u has only four possible forms, the assertion may be
checked directly. 1
With four attributes X,, X,, X3, X, we cannot assume that u(x,, x2, x3 1x4) will
be the same one of the four forms of the theorem for each value of x4. But let S be
an open subset of X, on which this conditional utility function does have the same
form. We may write, as an example,
24= \c/Cd~l(Xl?x4) + ~z(Xz,x4), x4)+ m,,
x4), d.
(8)
58
DAVID
E. BELL
Since this is true for each X, , and since X2, X3, and X4 are PWCA,
Corollary 2, either
then by
Thus u may be written either as
l)[cp(U,(.K,1x4) + U?(.KZ,
x4), .KJ)+ Oj(.Kj,-%,I
or as
~[q(o,(.u, I+ Lqx,)) i- UJ(.Y3),.Y41.
In the second case, since for fixed levels of X, , .Y?we know that, for some n, rc, , nz,
n(u) = n,(-Yj) + 7f,(-Y‘t),
we know that
for some 7~:.
In the first case, since X3 and X4 are WCA we can see that either
(i)
(ii)
(iii)
u3(-x3, -b) = 4V-u3) + u:(.h),
u= $[q$u,(.u,) + u2(x1)) + cp”(u$s3) + u:(.K~))], or
~=~C(cp(u,(-~,)+u~(.~~))+u~(.~~))
u~(-K~)].
By similar reasoning regarding the other possible variations of (8) we may show
that, for .y4 E S, u must be one of the forms generated by the stepwise procedure. It
remains to argue that the form of u will be constant over the whole of X4.
Since X,, X4 are WCA then for any fixed levels of X, and X, we must have
II = rr(a(.u,) + h(~~)). If we had
for .x4 ES, but
U(X,,
.Kz,
X3,
-Yq)
=
U’,(.K,)
+
&:Vz(.K2)
+
W3(X3))
+
for .Y~E T, then from (9) we know
or a constant linear transformation
thereof, and from (10) that
a(x*) = M’,(X,).
W4(Xq)
(10)
MULTILINEAR
REPRESENTATIONS
59
Since the choice of .Y! can vary, we deduce that cp, is linear and uI = We. It is clear
that two forms of u for different values of x4 are untenable. The proof for n
attributes proceeds by induction.
5. SUMMARY
A common practice by those faced with the task of assessing a two-dimensional
function is to assume ordinal additivity. This reduces the assessment task to one of
finding two one-dimensional scales. The natural generalization of this idea to n attributes has been to assume unconditional
ordinal additivity, which requires the
assessment of n one-dimensional scales, or a hierarchy of additive subsets, requiring
as many as 2n - 2 scales.
It is clear that if, for each subset, we permit a multilinear representation rather
than simply an additive representation then the resulting set of decompositions is
broader and thus more flexible, yet without requiring the assessment of any more
scales. In this paper we have demonstrated that such a class of decompositions is
equivalent to an assumption of pairwise weak conditional additivity among the
attributes.
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RECEIVED:
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