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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. REFERENCES DEBKEU. G. (1960). Topological methods in cardinal utility theory. In K. J. Arrow, S. Karlin and P. Suppcs (Eds.). Mathematical methods in rhe social sciences 1959 (pp. 1626). Stanford, CA: Stanford Univ. Press. GOKMAN. W. H. (1968). The structure of utility functions. Review of Economic Srudies, 35, 367-390. LEHNEK. P. E. (1981). Folded additive structures: A nonpolynomial model of some factorial interactions. Journal qf Mathematical P.ycho1og.v. 23. 99-l 14. LEONTIEF. W. (1947a). A note on the interrelation of subsets of independent variables of a continuous function with continuous first derivatives. BuNetin qf the American Mathematical Society. 58, 343-350. LEONTIEF, W. (1947b). Introduction to a theory of the internal structure of functional relationships. Econometrica, 15, 361-373. LUCE. R. D. & TUKEY, J. W. (1964). Simultaneous conjoint measurement: A new type of fundamental measurement. Journal o/’ Mathemarical Psvcholog.v, I, l-27. NAHAS. K. H. ( 1977). Pwfrrence modelling 0f‘ufilit.v surfaces. Ph.D. thesis, Department of EngineeringEconomic Systems, Stanford University. PIAGGIO. H. T. H. (1960). Diflerenrial equations. London: Bell. TING. H. M. ( 1971). Aggregation of artrihute.s,for mulriatrributed uti1if.v nssessment Technical Report 66. Cambridge, MA: Operations Research Center, MIT. RECEIVED: January 13, 1986.