AD-ALIS OP" CALIFORNIA WNiY DAVIS IADATE SCHOOL OF AD14INISTRATION F/6 12/1DECOPOSITIONS OF MULTIATTRIBJTE UTILITY FUNCTIONS BASED ON CON--ETC(U)

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DECOMPOSITIONS OF MULTIATTRIBUTE

UTILITY FUNCTIONS BASED ON CONVEX DEPENDENCE

Hiroyuki Tamura* and Yutaka Nakamura**

*Department of Precision Engineering **Graduate School of Administration

Osaka University, Suita University of California, Davis

Osaka 56r, Japan Davis, California 95616

DTIG

Working Paper 82-1 JUNO 21982

March 1982

This research was supported in part by the Office of Naval Research underContract #N00014-80-C-0897, Task #NR-277-258 to the University of California,Davis (Peter H. Farquhar, Principal Investigator).

Aproved for public release; distribution unlimited.

All rights reserved. Reproduction in whole or in part is not permittedwithout the written consent of the authors, except for any purpose of theUnited States Government.

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Decompositions of Multiattribute Utility Technical ReportFunctions Based on Convex Dependence

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Hiroyuki Tamura and Yutaka Nakamura N00014-80-C-0897

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Multiattribute utility theory Preference modelsDecision analysis ApproximationsConvex dependence

20. 0.STRACT (Continue an reverse d.* It necose.y m idt .ty by block mber)o'we- describesa method of assessing multiattribute utility functions. Firt-

w e--Atroduce- the concept of convex dependence, whe e consider the change of ,shapes of conditional utility functions. Then, v.-.ata t1sh theorems which showhow to decompose multiattribute utility functions using convex dependence. Theconvex decomposition includes as special cases Keeney's additive/multiplicativedecompositions, Fishburn's bilateral, decomposition, and Bell's decompositionunder the interpolation independence. Moreover, the convex decomposition is anexact grid model which was axiomatized by Fishburn and Farquhar.

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2

DECOMPOSITIONS OF MULTIATTRIBUTE UTILITY FUNCTIONS

BASED ON CONVEX DEPENDENCE

Hiroyuki Tamura and Yutaka Nakamurat

Department of Precision EngineeringOsaka University, Suita, Osaka 565, Japan

ABSTRACT

We describe a method of assessing von Neumann-Morgenstern utility func-

tions on a two-attribute space and its extension to n-attribute spaces. First,

we introduce the concept of convex dependence between two attributes, where we

consider the change of shapes of conditional utility functions. Then, we esta-

blish theorems which show how to decompose a two-attribute utility function

using the concept of convex dependence. This concept covers a wide range of

situations involving trade-offs. The convex decomposition Includes as special

cases Keeney's additive/multiplicative decompositions, Fishburn's bilateral

decomposition, and Bell's decomposition under the interpolation independence.

Moreover, the convex decomposition is an exact grid model which was axiomatized

by Fishburn and Farquhar. Finally, we extend the convex decomposition theorem

from two attributes to an arbitrary number of attributes.Acoession For

NTIS GRA&IDTIC TABUnannouncedJustification

Distribution/tPresently at the University of California, Davis. DistriitodeAvail____ability CodesAvail ad/or

Dist Special

wIpc"

3

This paper deals with individual decision making where the decision

alternatives are characterized by multiple attributes. The problem Is to pro-

vide conditions describing how a decision maker trades off conflicting attri-

butes In evaluating decision alternatives. These conditions then restrict the

form of a multiattribute utility function in a decomposition theorem. In many

situations, it is practically impossible to directly assess a multiattribute

utility function, so it is necessary to develop conditions that reduce the

dimensionality of the functions that are required in the decomposition.

Much of the research in utility theory deals with additive decompositions

[5, 16]. Pollak [16], Keeney [11, 12, 13, 14], and others, however, develop a

"utility independence" condition that implies non-additive utility decomposi-

tions. Although these decompositions have been applied to many real-world

decision problems, there are situations, such as conflict resolution between

pollution and consumption 117], where the utility independence condition does

not hold. Fishburn [6] and Farquhar [3, 4] have investigated more general

independence conditions that imply various non-additive utility decomposi-

tions. For example, Farquhar's fractional decompositions include nonseparable

attribute Interactions.

In this paper, we introduce the concept of convex dependence as an exten-

sion of utility independence. In our methodology, normalized conditional

utility functions play an Important role. Utility independence implies that

the normalized conditional utility functions do not depend on different condi-

tional levels. On the other hand, convex dependence implies that each nor-

malized conditional utility functions can be represented as a convex combina-

tion of some specified normalized conditional utility functions. Keeney 1121

described interpolation in motivating utility independence. If we find that

4

the utility independence condition does not hold in the process of assessing

normalized conditional utility functions, we can repeat the procedure 1171 to

test the convex dependence condition to derive the utility representations as

approximations, the concept of the convex dependence covers a wide range of

situations involving trade-offs. The convex decomposition Includes as special

cases Keeney's 112, 131 multilinear and multiplicative decompositions,

Fishburn's [6] bilateral decomposition, and Bell's (1] decomposition under

Interpolation independence, which is the same as first-order convex dependence

in this paper. Bell [21 has developed ways to reduce the number of constants

to be assessed and has provided a generalization of additive and multiplica-

tive forms in the multiattribute case. Moreover, the convex decomposition is

an exact grid model as defined by Fishburn [7]. Our approach gives an

approximation of utility functions but recently Fishburn and Farquhar 181

derived a preference axiom which provides a general exact grid model, and

provided a procedure for selecting the normalized conditional utility

functions.

1. PRELIMINARIES

Let X X1 x ... x X. denote the consequence space which, for simplicity,

is a rectagular subset of a finite-dimensional Euclidean space. A specific

consequence xcX is represented by (xi, ..., x.), where xi is a particular

level in the attribute set Xi. We consider Y x Z as two-attribute space,

where Y - Xil x x Xir, Z - Xir+1 x ... x Xin and {ti, ... , i

(1, ... , nj. Throughout the paper, we assume that appropriate conditions

are satisfied for the existence of von Neumann-Morgenstern utility function

u(yz) on Y x Z 118). Moreover, we assume that there exist distinct y*, y0 €¥

which satisfy u(y ,z) * u(yO ,z) for all zcZ. Similarly, we assume that there

exist distinct z *,z cZ, which satisfy u(y,z ) * u(y,zO ) for all ycY.

DEFINITION 1. Given an arbitrary zcZ, a normaZixed conditionaZ utiZity

function vz(y) on Y is defined as

v2 (y) - u(yz) - u(y 0,z)u(y ,z) - u(y ,z)"

From Definition I it is obvious that vz(y ) 0 and vz(y ) - 1. Moreover,

if a decision maker prefers y *to yO then v(y) represents his utility, and

if a decision maker prefers yO to y , then v,(y).represents his disutilty.

To represent the decomposition forms and proofs simply, we need to

introduce some notation. First, we define three functions f(y,z), G(y,z) and

H(y,z) which will be used to represent the decompostion forms. We assume

0 0u(yO,zO ) i 0 without loss of generality.

0 0f(y,z) i u(y,z) - u(y ,z) - u(y,zO), (1)

*0 0 *G(y,z) i u(y ,z )f(yz) - u(y,z )f(y*,z), (2)

00*H(y,z) E u(yO,z )f(y,z) - u(y ,Z)f(y,z*). (3)

The two functions G(y,z) and H(y,z) are related to each other as follows.

0* 0 * 0 0 *u(y ,z )G(y,z) - u(y ,z)G(y,z ) * u(y ,zO)H(y,z) - u(y,z )H(y*,z). (4)

We define F(y,z) as

F(yz) 1 u(y , *)G(yz) - u(y ,z)G(y,z ). (5)

To represent the constants simply in our decomposition forms, three matrices

Gn Hn and Fn are defined for yl ..- , yncY and zI . ... , zncZ. Let the (i,J)

6

element of the matrix G n be denoted by (Gn),j, which is defined as G(yi,zJ),df ne IHn=*

where Zn = * Similarly, define H) H(yJ,z), where y" yz) , and define

(Fn) f(yj,zi), where yn a y and z - z *. Let G be the (n-1) x (n-1)

matrix obtained from Gn by deleting the i-th row and the J-th column, and let

"det" denote the determinant on square matrices. Define

IGnI . det(G0), - (-1)i+JIGn 1, i,j - 1, ... , no

Let iHni, i' IFn, and ' be defined similarly. Moreover, for n 1 1, we define

Gij - Hij - FIj - 1. We define an n x n matrix Cn for distinct y,, yney,

and distinct z0,z, ..., znCZ as (Gn)ij Vzj(yi) - Vz0 (yi).

2. CONVEX DEPENDENCE AND ITS PROPERTIES

In this section, we define the concept of convex dependence and discuss

some of its properties. In the following, let 6ij be the Kronecker delta

function.

DEFINITION 2. Y is n-th order convex dependent on Z, denoted Y(CDn)Z, if there

exist distinct zO, zj, o.., znCZ and real functions gl, ... , gn on Z with gi(zj)

6ij for icji, ..., n} and Je{O, 1, ... , n} such that the normalized

conditional utility function vz(y) can be written as

n nv Z(y) - - gi(z)l vZO(y) + gi(z) vzL(y) (6)

for all ycY and zeZ, where n is the smallest non-negative integer for which (6)

holds.

7

For n - 1, relation (6) implies "Y is interpolation independent of Z in Bell's

11, 21 terminology. When Y and Z are scalar attributes, a geometric illustration

of Definition 2 is in Figure 1. Suppose three arbitrary normalized conditional

utility functions v z(y), vz (y), and v z(y) are assessed on Y. If Y(CD0 )Z, all

the normalized conditional utility functions are identical as shown in Figure 1(a).

If Y(CDI)Z, an arbitrary normalized conditional utility function vz(y) can be

obtained as a convex combination of v z(y) and v (y) as shown in Figure 1(b).

Moreover, Figure 1(b) shows that the preferential independence condition 19)

need not hold (Note that v z(y) is monotonic and v z(y) is not.).

Figure I goes here

We now establish several properties of convex dependence. Let Y(GUI)Z

denote Y is generalized utility independent of Z: see Fishburn and Keeney [101

for a definition.

PROPERTY 1. Y(CD0)Z, if and only if Y(GUI)Z.

Proof. If Y(GUI)Z, the following equation holds

u(y,z) - M(z)u(y,z0 ) + O(z) (7)

for some z0 cZ. Setting y - y0 and y - y * in (7) where u(y0 ,z) * u(y*,z) for all

zcZ by the assumption in section 1, we obtain

u(y0 ,z) - o(z)u(y0 ,zO) + O(z), ($a)

u(y*,z) - a(z)u(y*,Zo) + B(z). (8b)

8

Therefore,

u(y,z) - u(yOz) a(z)[u(yz 0 ) - u(y0 'zo)] u(YzO) - u(y0 'zo)

u(y*,z) - u(y0 ,z) a(z)[u(y* ,z0 ) - u(yO,z0 ) u(y*,zO) - u(yOz 0 ).

From the Definition 1, (9) implies that vz(y) - vz 0(y) which shows that Y(CD0 )Z.

If Y(CD0 )Z, (9) holds. Rearranging (9), we obtain

u(y,z) - u(y* ,z) - u(y0 z) u(y,z0) + u(yO'z)u(y*zo) - u(yOpz 0)u(y*,z) (10)

u(y*,z0) - u(yO,z0 ) u(y *,z0) - u(y0 ,z0)

which shows that Y(GUI)Z. U

This property shows that the convex dependence is a natural extension of general-

ized utility independence except for null zones.

PROPERTY 2. If Y(CDn)Z, then there exist distinct yl, ".., YncY, and distinct

z0 ,zl, .. , ZnCZ which satisfy rank Gn - n.

Proof. On the contrary, suppose rank Gn * n for all distinct yl, "'., YnCY

and z0 ,z1, -9o, zneZ. Then there exist real numbers hi (i - 1, ... , n) such that

for all ycY, we have

n-i(Y) -VZ0(y) I hi|Vzi(Y) - Vz(y)]

which implies Y(CDn-I)Z. U

Using Property 2, we can assess the order of convex dependence 117).

For n - 1, 2, ... sequentially we test the rank condition of Gn for arbitrary

distinct Yl, "'', yncY. Then if rank Gn - n and rank Gn+l - n for arbitrary

distinct yl "", yn+cY, we can conclude Y(CDn)Z.

9

nIt is obvious that relation between Gn and C is as follows

rank Gn = rank Gn

I n 01 n-Ifor distinct y ... , yy and distinct z ,z , ..., z , z cZ, because G(y,z)

u(y*,zO)[u(y ,z) - u(y0,z)][vz(y) - vz0 (y)] from (1) and (2). Thus we immedi-

ately get the following property.

1 nPROPERTY 3. If Y(CDn)Z, then there exist distinct yl ... , y cY and distinct1n-i n.

z I ... , z cZ which satisfy rank C n n.

Obviously the same property of rank condition for Hn holds. Property 3 guarantees

that the following property holds, which shows the relation of the order of

convex dependence between two attributes.

PROPERTY 4. For n - 0, 1, ... , if Y(CDn)Z, then Z is at most (n + I)-th order

convex dependent on Y.

Proof. See appendix.

A few aspects of these Properties deserve brief comment. If Y is utility

independent of Z which is denoted Y(UI)Z, then Y is obviously convex dependent on

Z; the converse is not true. The concept of convex dependence asserts that

when Y is utility independent of Z, Z must be utility independent or first-order

convex dependent on Y. Moreover, if Y is n-th order convex dependent on Z, then Z

satisfies one of the three properties, Z(CDn-1)y, Z(CDn)Y, or Z(CDn+I)Y, because

if Z(CD)Y for m < n - 1, then Y(CD,+I)Z at most and a + I < n.

10

PROPERTY 5. If rank G n n for distinct y y .. , eY and distinct z ..

z n- Z, then rank F n n.

Proof. By using (2), we obtain the following relation between Gn and F n

n, y*0)Jn { u ~ "i- O TGiu- n1 I I z F If(yn',z ) u~y n,z 0)IF nj},

where summation i - 1 to n means I - 1, 2, .. ,n-1, ~FIn frdsic n-i 1 n-I

on the contrary, if rank F * .frditnc, y c-Y and z Z .. , Z,

then,

IF~ 0 and F~ i f(yT ZJ) - 0 for £ - 1, 2, .. ,n

1 2 n-i n nI Tbecause even if we transform one of y , y , .,y and y into y In F

rank Fn* n by the assumption. N

3. CONVEX DECOMPOSITION THEOREMS ON TWO-ATTRIBUTE SPACE

This section uees convex dependence to establish two decomposition theorems

and a corollary for two-attribute utility functions. We further discuss the

relation of these results with the previous researches.

* I *THEOREM 1. For n 1, 2, ... , Y(CDn)Z, if and only if

u(y'Z) -u(y0,Z) + u(y,Z) + V(y)f(y*,z) + n,.~ G ~G(y,z )G(y,!), (11)

IGn ii j I

where v(y) uyz) cy 1* 0' 00u(y'z) U(Y'z)

Proof. See appendix.

11

THEOREM 2. For n 1 1, 2, ... , Y(CDn)Z and Z(CDn)Y, if and only if

u(y,z) u(y0,z) + u(y,z0 ) +L I -n(J,z)1 -I 11I

+ C - G(yz )H(y 0z), (12)

I l -i nj

I In* n*where c - CyCz [f(yn,z n) - IF j f(yn,zi)f(yj zn

I I 1andc C ,

y u(y u(y z)

Proof. See appendix.

We have obtained two main decomposition theorems which can represent a wide

range of utility functions. Moreover, when the utility on the arbitrary point

(yn,zn) has a particular value, that is, c - 0 in (12), we can obtain one more

decomposition of utility functions which does not depend on the point (yn,zn).

This decomposition still satisfies Y(CDn)Z and Z(CDn)Y, so we will call this new

property reduced n-th order convex dependence and denote it by Y(RCDn)Z. It is

obvious that Z(RCDn)Y when Y(RCDn)Z.

COROLLARY 1. For n 1, 2, ... , Y(RCDn)Z, if and only if

u(yz) - u(y0 ,z) + u(y,z0) + I n n F f(yz)f(yJz). (13)IF l i-jai

We note that when n - 1, (13) reduces to Fishburn's 16J bilateral decomposition,

u(y,z) - u(yO,z) + u(y,z0 ) + f(y,z*)f(y*,z). (14)f(y*,z )

12

In Figure 2, we show on two scalar attributes the difference between the

conditional utility functions necessary to construct the previous decomposition

models and our decomposition models. By assessing utilities on the heavy shaded

lines and points, we can completely specify the utility function in the cases

indicated in Figure 2. As seen from Figure 2, an advantage of the convex

decomposition is that only conditional utility functions with one varying

attribute need be assessed even for high-order convex dependent cases.

Figure 2 goes here

4. CONVEX DECOMPOSITION THEOREM ON N-ATTRIBUTE SPACE

There are many ways to extend the two-attribute convex decomposition

theorems in Section 3 to n-attribute decompositions. In this paper, we extend

Theorem I to n attributes in a way which might be useful in the practical

situations discussed later.

We partition X Into Xi and X, where X1 2 X, x ... x X,_, K X,+ .x Xn.

When we consider Y - Xi and Z - Xi in Theorem 1, all notation and definitions

in the previous section are suffixed with I. The representation and its proof

of n-attribute convex decomposition theorem requires some additional terminology

and notation as shown in Farquhar 13]. First, we define the following function

for i 1 1, ... , n,

Gm ( X ki,ki ki i(ki,k)Gl(xi , ),

wi kwhere i(j,k)s (j,k)-cofactor of G and x eX,, xik Xi . The delta operator

A is defined as follows. Suppose X - X for some 1 I r n and

IrCei, ... , }. Let ye X1r and a - {(i: "Id} Gle1, ... , , *, blank).

13

Then delta operator A is defined aa

u60 1 (_,)b u--,O Ii iu( , y) a I se*, ... , 1, y): aj -1 if jJ.

r J C I r r

aj - 0 and mj 0 if J J J1, (16)

rwhere b r + I aj.

J=1

We shall often omit attributes that are at the level x0, when it will not

0~be confusing. For instance, u(x) u(xl xy0). The utility function is always

scaled so that u(x~, ... , xn) - 0. From the definition of the delta operator

aj Au

and (1), fj(x , x , y) for all JcIr , J=Ir -J are equal each other. Using the

Gi Aurelation of fi(x , x) -f(x, , x) for i = 1, ..., n, we can get the following

notation

u AQfir(Y) fi(Xjr , y) for all iCIr . (17)

The coefficient function A(ir,O)(Y) for Ir flI, n}, 0 1 {Bi: Iclrl and

Oic{I, ... , *1 is defined as

A )b u(x ) f (y): j and c -0

(IrB)" -JCIr iIr Ir ' , d j

if jJ, aj - * and cj I 1 if j J1, (18)

rwhere b r + I c and y X

JT1 j h er l

The coefficient function has the relation with (2) as follows.

14

PROPERTY 6.

(i) A(i- G(x xi) for i = 1, ... , n.

(ii) A (ir.)(y) - u(xI) A(j.0)(xi Y) - u(x1i ) 6(j,0)(xi . y)

for icIr and J - Ir - W.

(iii) s(JIB)(Y)= ( )b Gi(XA , y) A U(X ): a j W O ; , 0 andKCJ J E'r

aj n 0 if JcK, j = and aj m I if J k KI,

rwhere b s r + aiJ Ir + {ii Ir and y-X.

Proof. (i), (ii), and (iii) are easily obtained from (2) and (18). 1

THEOREM 3. Suppose that for ieN - {1, ... , ni, mi are nonnegative integers.

For i 1 1, ..., n, X I(CD m)XI if and only if

u~x , xn) " I {c1 n vI (iX)I C N icI

Di mi+ I H di I Gi (xi)[A(I + v (x~), (19)I CN icI ji(10 1

A*

where VI(Xj) Z A(IB)(xJ ) v7:] JCNl-I JeJ ;

iC =I "-u(xI )

di = i for i - 1, ..., n,

" as 10: ic~l and Bic{l, ... , si,

Proof. See appendix.

15

Decomposition form in Theorem 3 gives a wide range of utility functions

on n-attribute space because it is possible to allow for the various orders

of convex dependence among attributes. The order of convex dependence is the

number of normalized conditional utility functions which must be evaluated to

construct a multiattribute utility function. Therefore, Theorem 3 provides the

general decomposition form which has mi conditional utility functions on each

Xi to be evaluated. Nahas [151 discussed the order of conditional utility

function on each Xi when utility independence holds among attributes. In this

paper, we show the relation among orders of convex dependence on each Xj, which

is one extension of Nahas' discussion. As Property 4 holds with respect to the

order of convex dependence between attributes, the following property holds

with respect to the order of convex dependence in Theorem 3.

PROPERTY 7. When XI(CDNi)XI for I - 1, ... , n, if m2, ..., m. are arbitrary

orders of convex dependence, the order mI must satisfy the following two

Inequalities.

n(i) A (ai+2) ml + I

1-2

(ii) ml + 2-> max {a2 , ... , an),

where ai = (mi + ) /n

11 (mj + 2), i - 2, ..., no~J -2

Proof: (i) When m2, ... , mn are arbitrarily given, we can obtain the

upperbound of mI by the following term in (19).

n mi al

din Gil J(x I)A N,O(0n-

16

The upperbound of mI is determined by the number of normalized conditional

utility function on X1 included in (20). Then, it is sufficient to take into

account the following term In (20).

G1 (xl)a (21)1Joel 1' ( ,)

By Property 6 it is obvious that (21) is constructed by the linear

combination of the following terms.

mli J B2 B),

di m G ml (x )G (x J 2 .. 0 xn (22)I I .' 1 1 1 x 2 n

where ictO, 1, ... , mi, *}, i m 2, ... , n.

Substituting (15) into (22), we have

ml l( X 1 1 k B2 n2d 1 kl- G j(k'j) Cl(xlI x2, x . (23)

0 /2 on

Setting x- (x x ) In (23), we have1 2 n

1 02 OnG I - (x1, x2 , 0 i ). (24)U(x1)

Then, the decomposition (19) includes Gl(xl, 2 , ,, 1, .. , mi, ,

._ iI - 1, ... , n, that is, 11 (mj + 2) normalized conditional utility functions

V. 1-2at most.

(11) When XI(CDmi)XI, i - 1, a.., n, the orders m, **, mn mut satisfy the

folloving inequalities by (1).

I (mj +2) mi+ 1, i-I, ... , nj-i

j *

17

Then for uI we have

, + 2 > max {a2,.., an),

where ai - (mi + 1)/ nfl (mj + 2), i m 2, ... , n. U

j-2J1*i

In some decision problems, utility independence may not hold in one or

more attributes. In such cases the convex decomposition theorem may give a

representation of the utility function. We illustrate how the convex

decomposition theorem decomposes the utility function when n - 3.

When ml " m2 - m3 - 0 in (19), we have obviously

u(x= x2, x3) . c I vi(xi).I C {1,2,3} c,

This decomposition is a multilinear utility function [11].

When 22 and 33 are arbitrary orders of convex dependence, we obtain the

following inequalities from Property 7.

(22 + 2)(m3 + 2) aI + 1, (25)

.E2 + I a3 + I.m1 + 2 > max 3+1 33+-1 (26)tM+ 2' W2 + 21

When *2 - 33 - 0 in (25), that is, X2 (CD0 )XIX 3 and X3(CD0 )XIX 2 , X1 is at most

third-order convex dependent on X2X3. In this case the decomposition form in

Theorem 3 is reduced to

u(x 1 3x 2 ,x 3 ) = c a vi(xi)SIC 1,2,31 I I

31, m' * o+d II(xl,x2 ,x3)v2(x2)

10 i ()21 XlX 2 ,x 3 )V3 (x 3 ) + G(XlX2) v3(X (27)

1 1 2 3 112

18

Therefore, we can construct (27) by evaluating one conditional utility

function on X2 and X3 , m1 conditional utility functions on X1 , where ml - 1,

2, or 3, and constants. When ml - 3, that is, XI(CD3 )X2X3 , (27) is reduced to

A* 0u(x 1 ,X2,x 3 ) C I vl(x I ) + u(xIx 2 ,x 3 )v 2 (X 2 )

0 A*+ u(xIx 2 ,X 3 )v3(x3) + u(xl,x 2,x3 )v2 (x 2 )v 3 (x 3 ). (28)

This decomposition form is the same as the one which Keeney showed in [14) and

Nahas discussed in [15] when X2 (UI)XlX 3 and X3(UI)XIX 2 . Keeney said nothing

about what property holds between X1 and X2X3 in this case. Convex dependence

asserts that (28) holds if and only if XI(CD3)X2X3 as shown above. Moreover,

from Property 7 (11) convex dependence allows for XI(CD2 )X2X3 or XI(CDI)X 2X3

which are stronger conditions than XI(CD3)X2X3. In these cases, we could

obtain decomposition forms easily as shown in (27) where aI - I and 2 are

corresponding to XI(CDI)X 2X3 and XI(CD2)X2X3, respectively.

5. SUMMARY

The concept of convex dependence is introduced for decomposing

multiattribute utility functions. Convex dependence is based on normalized

conditional utility functions. Since the order of convex dependence can be an

arbitrary finite number, many different forms can be produced from the convex

decomposition theorems. We have shown that the convex decompositions include

the additive, multiplicative, multilinear and bilateral decompositions as

special cases. A major advantage of the convex decompositions is that only

single-attribute utility functions are used in the utility representations

even for high-order convex dependent cases. Therefore, it is relatively easy

19

to assess the utility functions. Moreover, In the multiattribute case the

orders of convex dependence among the attributes have much freedom even If the

restrictions in Property 7 are taken into account. So even in the practical

situations where utility independence, which is the O-th order convex depen-

dence, holds for all but one or two the attributes, the convex decompositions

produce an appropriate representation.

Our approach is an approximation method based upon the exact grid model

defined by Fishburn [7). We note that Fishburn and Farquhar [8) recently

established an jaiomatic approach for a general exact grid model and provided

a procedure for selecting a basis of normalized conditional utility functions.

*

V

20

ACKNOWLEDGEMENT

The authors wish to express their sincere thanks to Professor Peter H.

Farquhar for his extensive comments on earlier drafts of this paper. This

research was supported in part by the Ministry of Education, Japan, for the

science research program of "Environmental Science" under Grant No. 303066,

403066, and 503066; by the Toyota Foundation, Tokyo, under Grant No. 77-1-203,

and by the Office of Naval Research, Contract #N00014-80-C-0897, Task

#NR-277-258, with the University of California, Davis.

a

21

APPENDIX

To represent simply an arbitrary linear combination of normalized

conditional utility functions, we define the following notation

mC[v (y), ... , v (y)] 6 1 v zi(y),ziZn I=

m

where e- i 1.

By using this notation, the following equations hold.

f(y,z) - f(y*,z)C[Vzo(y), vz(y)], (29a)

- f(y,z*)C[V y0(Z), v y(z)], (29b)

G(y,z) - G(y,z*)C[v yO(z), v y(z), v y*(z)], (29c)

H(y,z) = H(y ,z)C[vz0 (y), vz(y), vz*(y)I. (29d)

Proof of Property 4: When n - 0, if Y(CD0)Z, then vZ(y) Vz0 (y).

Using (1), we have

f(y,z) - vz0(y)f(y*,z). (30)

Substituting (29b) into (30), we have

C[vy(z), VyO(Z)] - C[vy*(Z), Vyo(Z)] .

This concludes Z(CDI)Y at most.

When n 1 1, if Y(CDn)Z, then for distinct zO , zI, . n1, Z CZ

nt

n

V(y) 1- I gj(z)]vzO(y) + I g1 (z)v i(y)i-l i-i

n

- [vzi(y)- Vz0(y)]gi(z) + VzO(y). (31)i-i

N -

22

By Property 2 we can select distinct yl, ... , yney and zI, ... , zn-cZ which

make Cn a nonsingular matrix. Then, substituting these y1, ... , yncy into

(31), we have the following matrix equation,

Gn a v, (32)

where & and v are column vectors and these i-th elements are gj(z) and vz(yi)

v zo(YI), respectively.

Using G(y,z), (32) is transformed into

G - u, (33)

where u(y*z) $ u(yO,z) for all zcZ from the previous assumption, and (-G)ij -

G(yi,zJ)/[u(y*,zi) - u(yO,zJ)], where zn - z*, and u is a column vector and

Its i-th element is G(yi,z)/[u(y*,z) - u(yO,z)].

Solving (33) for gj(z) (I - 1, ..., n) and substituting these g1 (z) into

(31), we obtain *!1 n G(y,zi) n

G(y,z) Fn i-i L.) (34)ii

where Gn is nonsingular by Property 3.

* iBy (29c) we have

G(y,z* )C[v yO(Z), v y(z), vy(z)

n n. (y,z1) G iGI(yj, a*)C[vyO(z), vyj(z), v,*(z)). (35)

IGnl 1-1 j-i y

Suming up all the coefficients of C[vyO(z), vyj(z), v y*(z)] for j - 1, 2, n...

in the right hand side of (35) yields

I i G(yzi" J G' G(Y' z) -io-- ii J-l

23

which implies

Vy(Z) - C[vyO(Z),y ..., Vyn(z), vye(Z)].

This concludes Z(CDn+I)Y at most. I

Proof of Theorem 1: Suppose Y(CDn)Z, and (34) holds. Substituting (2) into

the left hand side of (34) and solving it with respect to u(y,z), then we have (11).

Conversely, suppose that (11) holds. By definition (2), it is obvious

that Y(CDn)Z. *

Proof of Theorem 2: Suppose Y(CDn)Z and Z(CDn)y. Using Theorem 1, we

get two equations,u(y)y 0 ,), 0 n(36a)

u(yz) u(yOz) + u(yz 0 ) + v(y)f(y ,z) + c I (y)G(yiZ),

and

u(y,z) - u(yO'z) + u(yzO) + v(z)f(yZ ) + cz I I

wheren In

(y) n )E G(yz ) and H_(Z) _ k(y z).IG nI k-i ' IH nI kL A

Substituting (36b) into f(ya,z) for act{, 2, ... , n, *1, we have

f(ya,z) = v(z)f(y, z*) + CzH(yoz), (37)

where we use v(z0) = 0, H(y,z0) -0 and H(y,z) H Hn(z)H(y,z).i-i

Substituting (36b) into G(y a z), we have

G(y z) - v(z)G(y * c H(z)F(yz (38)

Substituting (37) and (38) into (36a), and using (2) and (3), we have

u(y,z) = u(y 0,z) + u(y,Z ) + v(y)f(y ,z) + v(z)f(y,z )

v(y)v(z)f(y*,z) + C c i G, (y)H (z)F(y ,z). (39)Y z

24

We can assume that F n is a nonsingular matrix because Property 5 holds.

Considering next equation and transforming it, we obtain

V(y)f(y ,Z) + v(z)f(y~z )-v(y)v(z)f(y ,z)

- j~nji- I I F n f[()f*)f(y M ,z)

i-I j=-ij~

+ v(z)f(y j z* )f(y,zi) -v(y)v(Z)f(y z *)f(y*,zi)] (40)

By definition (2) and (3), the following relation holds.

V(y)f(y A.,' )f(y ,z) + V(z)f(yJ, z* )f~y,z ) - v(y)V(Z)f(y *,z )f(yz)

- f(y,z i )f(yi ,z) - c Vc G(y,z i )H(yjz) (41)

Substituting (40) and (41) into (39), we obtain

f(y,z) - 11 j(F(Y [ )f(Y.z Myyz(Yz)(Y,7FI i- i ij

ccz G ()n (42a)

i-i J-1

f(y,z) I IF~ F, j-1z~fyj +

IFn _1 j 'Ii

cyc n n knl rn ki , F(y k,z r n- )~ ~'Hy~) (42b)

In (42a), setting y - yP, z - zq for p, qcfl, ... , n}, and solving it with

respect to cyczF(yp,zq), and then substituting it into the following

FnCccZ n n -n k r Fj

n G kij F(y z 1nICYCz

- i nj[IF nIf(Yn,z ) I I n f(yn, 2 q)f(yp,3n)], (43)

25

where we use the following relations

nf n* nFn f(y kzq)f(yPzr)= IFnl n 6qf(yk zq),

p=1 q=1 qlr

n

k i y " and r=1 rjyz)= 6jpIHn,

where 6tj denotes the Kronecker's delta.

Substituting (43) into (42b), then we have (12). Therefore, sufficient

condition is proved.

n n nConversely, suppose that (12) holds, then we assume C , Hn , and F are

nonsingular matrices. Substituting (3) and (29a) into (12), we have

f(y,z*)C[v 0(Z), V (z)]y y

n 1 n y, u)f(y,z)ctv O(z ) -yj 'z ) ]

n * -n-- v yJ(Z) - v x (Z)

i-c j-i H jGyz u(y0 z * )[u(y i I*) - j' 'O(4

Summing up the coefficients of C[vy0(z). vyj(z)], vyj(z) for j = 0, 1, 2,

* and v 0 (z) of the right hand side of (44), we have f(y,z*). Then, wey

conclude Z(CDn)Y, and the same procedure for Y concludes Y(CDn)Z. I

Proof of Theorem 3: We can prove this theorem in the same way as Farquhar

[3]. If Xi(CD mi)Xi for i - 1, ..., n, then by Theorem 1, (15) and (18) the

following equation holds.

U(Xl ..., xn) u(xi + U(Xl, ..., x i-. X+ 1, *me, x)

+ Vi N )f (x1' *got X A*, "", )

ni Ciiji)i.i)(

+ d( ..., xi, x I, ..., x ) (45)IiJ Ij1(,01 1 11 l1 n

26

If i I in (45), then we have

u(x I ... , x n) = u(x1) + u(x2, ..., x) + vl(xl)fl(x1 , x 2 , --., x)

Cln(X2' l, x 2, I..., 2 xn).(

j I

If 1 - 2 In (45), then we have

U(X P ..., xn) u(x2) + u(x1, x3 , ..., xn ) + v2 (x2 )f2(x1 , x2, x3 , .., x.)

+ 2G m2 ,2(xl, 0 x(7+ d2 G2 ,j(x 2 )A( 2 , 3 , ... , xn) (47)

i '2

We consider to substitute (46) into (47). First, we substitute (46) into

the following

f xsxAc x3 x u(xl ' x Acxu(x2

Ac Accf2(X1, x2, x, -.., xn ) = ~ 1 2 13, ... , xn) -ux)

f 0 Ac xn + vl(xl)f( xn)= f2 x, x2, x3, ...,n Kx3, ...

+ d U1 G G (x )A ( x39 .". Xn) (48)

where celo, 1, .*, a2' * K - {1,21, a - [a, ,12}, a I a 2 m c and we use

the relation (17).

Secondly, we substitute (48) into the following

A 2, ) (xX3' , w n)'2A A(,2 (x 03 09, xn +N)A NA

(2,02)(x 1x) + V(x1 )( 2 , 2)(, 3n)

p"d 1 (x1)AaK,1 N(3' xn)' (49)whed 1 K"'1,21, and 0 (x, 021

wher K, a- 4.21. and B - {B;. 021. a

27

From (46) we obtain the following

A* 0U(Xl' x3 "''' xn) - U(xl) + u(x3 1 ""', x) + V l(xl)fl(X1, xO, x 3 , .. , xn)

J 1 0+ dI GI1 j(Xl)A(I ,8l)(X2 , 23, ... , Xn). (50)

Substituting (48), (49), and (50) into (47), we have

u(x1 , ..., xn) - u(x1) + u(x2) + u(x3, ..,* Xn )

Vl(Xl)fl(X1, x2, x3, ... ,x )

+ v2(x2 )f2 (x1 , x2, x3, ..., x)

+ Vl(xl)v 2 (x 2 )fa(x 3 , .. o, x)

+ dI J G1,j (x1 AIO's)(x 2, x3 " X' n) + v 2 (X2)A (l,0)(x2 9 x3' .... X n)

Sm2 ( 0 ) + (X )A A*+ d2 G2,j(x 2) {A(2B2 x3, ... , (8 2) V 1 )(Xl , x, ni- , 1'

d2 m2 al m2+ dld 2 G ( ) (x A xx)

1 2 Jol k-l I l) 2,k(X2)A(Kf)(x31 . )

where K - 11,21, a - 1a1, a2 } , 81 -f, and a2 - C.

This procedure is repeated for steps i - 1, ..., n. Hence, we have (19) by

using Property 6 and the following relation

, - u(X I) and u(xi) - u(x I) vi(x) Ifor I - 1, ..., n,r r

where Ir , ..., i C N, a (, and ai * for all I.

Conversely, if (19) holds, It is evidenly that XI(CD a)Xi for I -I, ... , U

by (29) and the property of convex combination. U

28

REFERENCES

1. Bell, D.E. (1979). Consistent assessment procedure using conditionalutility functions. Operations Reseach, 27, 1054-1066.

2. Bell, D.E. (1979). Multlattribute utility functions: Decompositionsusing interpolation. Management Science, 25, 744-753.

3. Farquhar, P.R. (1975). A fractional hypercube decomposition theorem formultiattribute utility functions. operations Research, 23, 941-967.

4. Farquhar, P.H. (1976). Pyramid and semicube decompositions of multi-attribute utility functions. Operatione Research, 24, 256-271.

5. Fishburn, P.C. (1965). Independence in utility theory with whole productsets. Operations Research, 13, 28-45.

6. Fishburn, P.C. (1974). von Neumann-Morgenstern utility functions on twoattributes. Operations Research, 22, 35-45.

7. Fishburn, P.C. (1977). Approximations of two-attribute utility functions.Mathematice of Operations Research, 2, 30-44.

8. Fishburn, P.C. and P.R. Farquhar (1981). Finite-degree utility independ-ence. Working Paper 81-3, Graduate School of Administration, Universityof California, Davis, California. To appear in Mathematics of OperationsResearch.

9. Fishburn, P.C. and R.L. Keeney (1974). Seven independence concepts andcontinuous multiattribute utility functions. JournaZ of MathematicaZPsychoZogy, 11, 294-327.

10. Fishburn, P.C. and R.L. Keeney (1975). Generalized utility independenceand some implications. Operations Research, 23, 928-940.

11. Keeney, R.L. (1971). Utility independence and preferences for multi-attributed consequences. Operations Research, 19, 875-893.

12. Keeney, R.L. (1972). Utility functions for multiattributed consequences.Management Science, 18, 276-287.

13. Keeney, R.L. (1974). Multiplicative utility functions. OperationsResearch, 22, 22-34.

14. Keeney. R.L. and H. Raiffa (1976). Decisions with Multiple Objectives:Preferences and Value Tradeoffs. John Wiley and Sons, New York.

15. Nahas, K.H. (1977). Preference modeling of utility surfaces.Unpublished doctoral dissertation, Department of Engineering-EconomicSystems, Stanford University, Stanford, California.

29

16. Pollak, R.A. (1967). Additive von Neumann-Horgenstern utility functions.Eeonometrica, 35, 485-494.

17. Tamura, H. and Y. Nakamura (1978). Constructing a two-attribute utilityfunction for pollution and consumption based on a new concept of convexdependence. In H. Myoken (Ed.), Information, Decision and Control inDynamic Socdo-Economics, pp. 381-412. Bunshindo, Tokyo, Japan.

18. Von Neumann, J. and 0. Morgenstern (1944). Theory of Cames and EconomicBehavior. 2nd Ed., Princeton University Press, Princeton, New Jersey,1947; 3rd Ed., John Wiley and Sons, New York, 1953.

I

I

30

4- -

4-)

V(y= (y).V (y)z0z z

00

y y Y(a) Y(CD 0)z

V =) v(Y) for all zcZ

P- V (y)

0 0y y y Y

(b) Y(CD )z

"V ZY = (l-g(z))v z0(Y) + g(z)v zI(Y) for all zcZ

Figure I. The relations among normalized conditional utility functions Whenthe convex dependence holds.

31

z I z zZ -- - - - -, --- ZI .. . IZ

,-I I,I I

I I"I I

0I I 0

0 z- 0 0 zy y Y y YY Y Y Y(a) additive model (b) multiplicative model (c) Y(CD0 )zZ Y(CD0 )Z, Z(CD0 )z z z

oz 0 1 ----

0 z 0 1 0 1 *Y Y Y y Y yY y y Y Y

(d) Fishburn's model (e) Y(CD )Z, Z(CD )y (f) Y(CDI)ZY(RCD )z

z z z

z 2, I

z z z

0 0 01-.-20 1 *z 2y y y Y 0 1 2 f 0 1 2.,,Y Y Y Y Y Y Y Y Y(g) Y( CD2)z (h) Y(CD2)Z, Z(CD2 )Y (i) Y(CD2)Z

z z z

z - -- -n-i -n-l

- An.-* z z j 2l

oo ... zo .* . Iz0 0n- -- O z

y y1... y y yO yl...y n-y y Y yO y'...y y Y* Y() Y (RCD n)Z (k) Y(C n )Z, Z(CD )y (1) Y (CD in)Z

Figure 2. Assigning utilities for heavy shaded consequences completelyspecifles the utility function In the cases Indicated.

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Professor Thomas S. Wallsten Professor Mustafa R. YilmazL.L. Thurstone Psychometric Lab. Management Science Dept.Department of Psychology College of Business Admin.University of North Carolina Northeastern UniversityChapel Hill, NC 27514 360 Huntington Avenue

Boston, MA 02115Professor S.R. WatsonEngineering Department Professor Po-Lung YuControl & Mgmt Systems Div. School of BusinessUniversity of Cambridge Summerfield HallMill Lane University of KansasCambridge CB2 1RX Lawrence, KS 66045ENGLAND

Professor Milan ZelenyDr. Martin 0. Weber Graduate School of Business Admin.Institut fur Wirtschaftswissenschaften Fordham Univ., Lincoln CenterTemplergraben 64 New York, NY 10023D-5100 AachenWEST GERMANY Professor Stanley Zionts

Dept. of Mgmt Sci. & SystemsProfessor Donald A. Wehrung School of ManagementFaculty of Commerce & Bus. Adm. State University of New YorkUniversity of British Columbia Buffalo, NY 14214Vancouver, B.C. V6T IW5CANADA

Professor Chelsea C. WhiteDept. of Engrg. Sci. & SystemsThornton HallUniversity of VirginiaCharlottesville, VA 22901

* Dr. Andrzej WierzbickiInternational Institute for

Applied Systems AnalysisSchloss LaxenburgLaxenburg A-2361AUSTRIA

Professor Robert B. WilsonDept. of Decision SciencesGrad. School of Business Admin.Stanford UniversityStanford, CA 94305