decomposition of utility functions on subsets of product sets

Decomposition of Utility Functions on Subsets of Product SetsAuthor(s): François Sainfort and Jean M. DeichtmannSource: Operations Research, Vol. 44, No. 4 (Jul. - Aug., 1996), pp. 609-616Published by: INFORMSStable URL: http://www.jstor.org/stable/172003 .

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DECOMPOSITION OF UTILITY FUNCTIONS ON SUBSETS OF PRODUCT SETS

FRANPOIS SAINFORT

JEAN M. DEICHTMANN University of Wisconsin, Madison, Wisconsin

(Received December 1989; revision received February 1995, June 1995; accepted October 1995)

The standard decomposition theorem for additive and multiplicative utility functions (Pollak 1967, Keeney 1974) assumes that the outcome set is a whole product set. In this paper this assumption is relaxed, and the question of whether or not a natural revision of this theorem necessarily holds is investigated. This paper proves that two additional conditions are needed for the decomposition theorem to hold in the context where the outcome set is a subset of a Cartesian product. It is argued that these two new conditions are satisfied by a large family of subsets corresponding to significant real-world problems. Further research avenues are suggested including a generalization of this new decomposition result to nonexpected utility theories.

Preference elicitation and modeling is central to the theory of choice. Let X be a set of outcomes and P

the set of simple probability distributions (or gambles) on X. When outcomes are multi-attributed, the outcome set is usually represented by a Cartesian product X1 X X2 X ... X X, where each Xi corresponds to one of n "attributes" of a multidimensional decision problem. In the context where X is the whole product set, i.e., X = X1 X X2 x ...x XXn, axioms that imply the existence of a utility function u that can be written in a special simple form (additive, multiplicative, multilinear, diagonal, quasi- pyramidal, or semicube) have been proposed (Pollak 1967; Keeney 1974; Farquhar 1975, 1976).

The thrust of this paper is to investigate whether or not a natural revision of the main result obtained for additive and multiplicative utility functions in the context where the outcome set is a whole product set holds when X is only a subset of such a product set. The specific result we are referring to is Pollak's decomposition theorem for additive and multiplicative utility functions (Pollak). Hereafter, this result will be referred to as Theorem P. In this paper, two results will be proved. First, it will be proved that a natural revision of Theorem P does not hold. Second, it will be proved that with two additional conditions, a new decomposition result holds in the subset context. Then, it will be shown that this situation corresponds to a family of significant problems for practitioners.

The structure of this paper is as follows. Section 1 re- views Pollak's relevant work, states Theorem P and identi- fies the notions that need to be revised in order to propose an extension to the subset context. These notions-utility independence and component utility function-are dis- cussed in Sections 2 and 3, respectively. Section 4 shows that a simple revision of Theorem P does not necessarily hold. Therefore, additional axioms are proposed in Section

5, and a new decomposition theorem is proved. Section 6 discusses practical examples where the outcome set is only a subset of a product set and the preference structure satisfies the new set of axioms. These examples serve to illustrate the significance of the theoretical results obtained in Section 5. Section 7 is a concluding section and suggests further research avenues. All notations and definitions are summarized in Appendix 1.

1. POLLAK'S DECOMPOSITION THEOREM

Let us assume that > is a preference relation on P that satisfies von Neumann and Morgenstern's axioms (1944). There exists a utility function u on P and, with xo an arbitrary element of X; without loss of generality, we can set u(xO) = 0. In this section we assume that X is the whole product set, thus X = X1 x * . x X,. With I a nonempty subset of {1, . . ., n} and I its complement, let XI = li Xi and X= Ili Xi. Let P(X1) be the set of gambles on XI, and for.x, E X, and p E P(X1), let (x', p) be the gamble in P defined by prob (x,, xI) = p(x,), Vx1 E XI.

Definition 1. XI is utility independent of XI if for all x, y E XI, and all p, q E P(X1), (x', p) > (X', q) < (y', p) >

(YI, q).

Definition 2. The ith component utility function ui with respect to xo is defined by ui(xi) = u(xi, X?), Vxi E Xi.

With these definitions, Pollak's decomposition theorem (1967) for a whole product set can be stated as follows.

Theorem P. If the following two conditions hold:

> satisfies von Neumann and Morgenstern's axioms (1)

for all i E {1, . , n}, Xi is utility independent of Xi, (2)

Subject classifications: Utility/preference, multiattribute: utility theory on subsets of product sets. Utility/preference, theory: utility theory on subsets of product sets. Area of review: DECISION ANALYsIs, BARGAINING, AND NEGOTIATION.

Operations Research 0030-364X/96/4404-0609 $01.25 Vol. 44, No. 4, July-August 1996 609 ? 1996 INFORMS



610 / SAINFORT AND DEICHTMANN

yJX

Figure 1. Utility independence on a subset of a product set.

then there exists a constant k C 9l (the set of all real num- bers) such that:

n

u(x) = , ui(x1) + k E ui(x1)uj(xj) i=:i i>j

+ k2 E ui(xi)uj(Xj)ue(xf)

+ * + k n - u lxl . .. Un(Xn).

If k = 0, the decomposition is said to be additive; if k 0 0, the decomposition is said to be multiplicative and u(x) can be rewritten as

n

(1 + ku(x)) = fH (1 + kui(xi)). i=l

When X is no longer the whole product set, the notion of utility independence needs to be redefined since the gamble (x,, p) is no longer well defined because (x,, x,) is not necessarily an element of X. Similarly, the notion of component utility function needs to be reconsidered since (xi, xi) is not necessarily in X for all xi.

2. UTILITY INDEPENDENCE ON SUBSETS OF PRODUCT SETS

A natural revision of the notion of utility independence in the context where X is only a subset of a product set is the following. (This definition was proposed by Fishburn 1976 for the two-dimensional case.) For any x, in XI, let X,(x,) - {x, C XI: (x,, x,) E X}. Similarly, for any x, in XI, let X,(x,) = {, E XI: (x,, x,) C X}. X, is said to be utility independent of X, if, for all x,, y- in X,, the preference order over gambles defined on [X,(,1) n XI(YI)] x {XI} is the same as the preference order over gambles defined on [xj(_f,) n x1(y1)] x {5y}. A graphical illustration of utility independence is given in Figure 1.

In the subset context, for any vX C XI and p C P(X1(i(X)), the set of gambles on X1(.XI), let us redefine (x,, p) as the

gamble in P defined by prob (x1, Xc-I) = p(xI), Vxj E XI(XI), and prob (xI, YI) = 0, Vx, {i XI(YI), YI E XI, YI * X-, (Similarly, for any xI E XI and p C P(XI(xI)), the set of gambles on X1(x1), let (xI, p) be the gamble in P defined by prob (x,, XI) = p(), VXI E XI(xI), and prob (yI, XI) = 0, VX EC XI(yI), y, C XI, yI * xI.) Then, utility independence in the subset context can be defined as follows.

Definition 3. XI is utility independent of XI if for all XI, YI E XI, and all p, q E P(XI(XcI)) n P(xI(yI))q

(X I, P) > (Xc-I, q) <>(Y I, P) > (Y I, q) -

(Note that P(XI(XI)) n P(XI(YI)) = P(XI(XI) n xq(yI)), the set of gambles on XI(XI) n XI(YI).)

3. COMPONENT UTILITY FUNCTIONS ON SUBSETS OF PRODUCT SETS

To be able to define component utility functions in the subset context, we shall assume that there exists an element xo in X such that ui(xj) = u(xi, xi?) is well defined for any possible value of xi when x = (xi, xi) varies over X. Thus, we shall assume:

There exists x0 C X such that for all i E { 1, . . . , n}

and allx i C Xi, Xi(xG) D Xi(-i) * (3)

(Note: if X is a subset of nV, a geometric interpretation of (3) is that for any direction i, there exists a line Di parallel to the direction i such that for all x in X, the orthogonal projection of x on Di is also in X. The Dis are meant to be concurrent, and their intersection is xo.)

With such an xo, as in section 1, we can set u(x?) = 0. With Xl = {xi C Xi: Xi(xi) f 0}, an appropriate revision of the notion of component utility function for the subset context is:

Definition 4. The ith component utility function ui with respect to xo is defined by ui(xi) = u(xi, xo), Vxi E X.

4. THE NATURAL REVISION OF THEOREM P DOES NOT NECESSARILY HOLD

With these new definitions of the key notions of utility independence and component utility functions in the subset context, it is sensible to examine if a natural revision of Pollak's decomposition theorem hold, i.e., whether (1), (2) (based on Definition 3), and (3) imply the additive- multiplicative decomposition. The answer, however, is neg- ative. In effect, consider the following example, illustrated in Figure 2.

Let

X = {(XO, XO, XO), (X1, X2, X3), (X1, X2, X3), (X1, X2, X3),

with utility values u(x?, X2, X3) = u(xI, X2, X3) = 0 and u(xl, XO, XO) = U(XO, X1, X3) = U(XO~, XO, X3) = 1. If an individual's



SAINFORT AND DEICHTMANN / 611

x3

X31

X1~~~~~~~~~~~~~~~~~

xl Figure 2. Example of preference structure which verifies

Al, A2 and A3 but for which there is no additive or multiplicative utility decomposition.

xi

xo 0

Fr E p fs tc rs A

Figure 3. Example of a subset which verifies A4.

preferences on P are guided by the expected utility principle, then (1) is trivially verified. Furthermore, we have

xi (xi) n xi (xl)

- {(XO, xo), (x2, xo), (xO, X)} fl {(xo, xo), (x2, X3)}

- {(x2, X3)}v

Similarly, X2(xo) n X2(xl) = {(x?, x?)} and X3(xo) n X3(x3) = {(xO, xO)}. Therefore, (2) is automatically verified. Finally, it is very easy to see that (3) is verified with XO = (x0, xo, xo). Thus, (1), (2), and (3) hold; however, u cannot be decomposed additively or multiplicatively. In effect, suppose that there exists k such that

U(X) = Ul(Xl) + U2(X2) + U3(X3) + kul(Xl)U2(X2)

+ kul(x )u3(x3) + kU2(X2)U3(X3)

+ k2u (xl)u2(x2)u3(x3).

For x = (xl, xi, xl), we obtain the equation 0 = 3 + 3k + k2, which has no solution in 9R. Thus, the supposed existence of a real number k is contradicted. Therefore additional conditions are needed so that an additive or a multiplicative decomposition can be derived in the subset context.

5. TWO ADDITIONAL CONDITIONS

In this section, we introduce two additional conditions which, in combination with (1), (2), and (3), are sufficient to derive an additive or a multiplicative decomposition. A crucial step in proving Theorem P is to prove that for any i, u(x) can be written as ai(xi) + bj(xj)cj({c), for some functions ai, bi and ci. In the context of Theorem P, this is

a direct consequence of Xi utility independent of Xi. In the subset context, if we assume the following Condition (4), then it is fairly easy to see that the existence of the ai, bi and ci can be derived.

There exists xo E X such that for all i E {1, . , n}

and all xi E Xi,9 Xi (x?I) D X-i (xi ) * (4)

If X is a subset of gnf, a geometric interpretation of Con- dition (4) is that, for any direction i, there exists an orthogonal section Si such that for all x in X, the orthogonal projection of x on Si is also in X. The intersection of the Si's is xo. This is illustrated in Figure 3.

Lemma 0. Condition (4) implies Condition (3).

Proof. Let (xi, xi) be in X and let us prove that (xi, x?) is in X. This is immediate by applying repetitively (4):

(xi , XciJ X= (vX; Xi, XijJ E X=> Xko, Xj,g Xi, 9 TijkJ

EX= . *>(X9, x1p, . *, Xi) = (Xi, Xi) E X.

Therefore, xo is such that for all i E {1,..., n} and all xTi E Xi, Xi(x 5?) D Xi(xTi). Q.E.D.

Lemma 1. If Conditions (1), (2), and (4) hold, then for any i, there exists a function bi: Xi' - such that Vx E X, u(x) = ui(xi) + bi(xi)u(x?, 5ci)

Intuitively, with a slight abuse of notation, the reason for Lemma 1 to hold is that now there exists a reference set Xi(xio) which, for any xi, induces the preference ordering among all gambles on Xi(xi).

Proof. With i E {1, . . ., n} and xi E Xi', let f(.i) = u(xi, xi) defined on Xi(xi) and g(Xi) = u(xi?, xi) defined on Xi(x?) D Xi(xi). Let E(f, p) denote the expected value of f with respect to p, where p E P(Xi(xi)). For all xi let >x on P(Xi(xi)) be defined by p >xq if and only if (xi, p) > (xi, q). That is, Vxi E Xi', Vp, q E P(Xi(xi)), p >sxq < (xi, p) >




(xi, q) X E(f, p) > E(f, q). Similarly, Vp, q E _ki(x9)), p >x?q X (x4, p) > (x?, q) < E(g, p) > E(g, q). Then, X1 utility independent of Xi implies that Vp, q E P(X(xi)), p >xq < p >jxq. Thus, Vxi E Xi', Vp, q E P(Xi(xi)), E(f, p) > E(f, q) X* E(g, p) > E(g, q). Thus, f and g are two agreeing utility functions on Xi(xi). Since utility functions are unique up to a positive linear transformation, Vxi E X/, there exist two real constants a and b, b > 0, such that f = a + bg on Xi(xi). Let ai(xi) = a and bi(xi) = b. Then, Vxi E Xi', u(xi, xi) = ai(xi) + bi(xi)u(x??, xi). Thus, we have been able to construct ai: Xi'-- 9} and bi: Xi' -- 9I+, i = 1 ... n, such that for all x E X, u(x) = ai(xi) + bi(x1)u(x?, xi). For x = (xi, xi) E X, since (4) implies that (xi, X') E X (Lemma 0), then:

ui(xi) = u(xi, x?) = ai(xi) + bj(xi)u(xf?, x?) = ai(xi), which completes the proof. Q.E.D.

Although (4) is a step toward an additive or a multiplicative decomposition, it is not sufficient, in combination with (1) and (2) to derive the desired decomposition. To illustrate this fact, let us consider the following example.

Example1. LetX1 = {a,b,c},X2= {x,y,z},XCX1 x

X2. The elements of X with their utility values are shown in the following matrix:

z 2 * 8 y 1 2 * x 0 1 2

a b c

It is not difficult to verify that (1), (2), and (4) are verified. ((1) is verified by definition if preferences over P are defined according to the expected utility principle with the above utility values.) However, with xo = (a, x), there exists no k in 9J such that u(x) = u1(xO) + u2(x2) +

ku1(x1)u2(x2), since u(b, y) = 2 would imply k = 0 and u(c, z) = 8 would imply k = 1.

We shall now introduce a last condition, (5), which is sufficient (in combination with (1), (2), and (4)) to derive the desired decomposition. Condition (5) is useful in proving that the bis of Lemma 1 can be written bi = 1 + kui where k is a real constant. Let ki(a) = (bi(a) - 1)!ui(a) for all a C

Xi' such that ui(a) # 0. Our goal is to show that the functions ki(.) are constant and equal. Let - denote the indifference relation on P (defined byp - q X [p > q] A --[q > p])

and let + be defined byp + q X- [p - q]. (Note: with the usual definition of a "degenerate gamble," we will write x > y, x - y, or x -/- y with x and y elements of X.)

Two additional notions are necessary to formulate (5): the notion of isolated points in X, and the notion of k-comparability between points.

Definition 5. (xi, xc?) C X is isolated if X(x) = {x?}. Isolated points are important because they do not cause

any problem when trying to prove that bi = 1 + kui and because Condition (5) will not apply to them.

Definition 6. For i 0 j, (xi, x??) and (xj, Tcj) are directly k-comparable (denoted by xiKxj) if (xi, .c?) + x?, (xj, x?j + x?, and (xi, xj, j?y) E X.

(Note: (xi, o) -I- xo < ui(xi) t 0 and (xj, x?) + x? X0

uj(xi) 0 0)

Definition 7. (xi, .i?) and (x', .Tc) are directly k-comparable if there exist j, i ] j, and xj E Xj, such that xi Kxj and xi'Kxj.

Definition 8. (xi, xc?) and (xj, .jo)(i = j or i # j) are k-comparable if there exists a finite sequence a l, a, .

a JP (p - 2) with a il = xi, a = Xi, aJq E XjA, q = 2, p - 1 and ajiqKajq+l, q = 1, * * p - 1.

An important consequence of the notion of k-comparability is the following lemma.

Lemma 2. If (xi, Tco) and (yj, ?9)(i = j or i 0 j) are k-comparable, then ki(xi) = kj(yj).

Proof.

STEP 1. If (xi, x?) and (yj, xcj)(i 0 j) are directly k-comparable, then (xi, yj, 5?y) E X, ui(xi) t 0 and uj(yj) t 0. Thus, Lemma 1 applied to (xi, yj, .jX) E X implies:

u(xi, yj,' x) = ui(xi) + bi(xi)u(yj, 5jo) = uj(yj)

+ bj (yj)u (xi, x??).

Since by definition of the component utility functions, u(xi, xc?) = ui(xi) and u(yj, cj9) = uj(yj),

ui(xi) + bi(xi)uj(yj) = uj(yj) + bj(yj)ui(xi).

Thus, since ui(xi) # 0 and uj(yj) # 0:

bi(xi) - 1 bj(yj) - 1 ui(xi) u u(yj)

that is ki(xi) = kj(yj).

STEP 2: If (xi, Tc9) and (yi, TE9)(i = j) are directly k-comparable, then xiKze and yiKze for some ze E XI. Then Step 1 implies ki(xi) = k(ze) = ki(yi).

STEP 3: If (xi, xi?) and (yj, x>j)(i = j or i # j) are k-comparable, then Steps 1 and 2 imply ki(xi) = k (a j2

= = kj1 (a]P1) - ki(yi) with aJq as in Definition 8. Q.E.D.

We can now formulate Condition (5):

Foralli, j E {1, . . ., n}(i = j or i j), Vxi EXi,

Vxj E Xj, if (xi, xp?) and (xj, xp) are not isolated,

(xi, xp) - x0 and (xj, .jP) x x, then (xi, xp?) and

(yj', .X) are k-comparable. (5)

Lemma 3. If Conditions (1), (2), (4), and (5) are satisfied then there exists k E R such that:

V iCE{1, * **,n}, V iEX,b x =1+k x




Proof. To prove the existence of k, it is necessary to prove that (a) bi(xi) = 1 is appropriate for all xi such that ui(xj) = 0; and (b) ki(xi) = kj(xj) Vxi E Xi', Vxj E Xj such that ui(xi) f 0 and uj(xj) : 0 (i = j or i 4 j). To prove (a), let xi be such that ui(xi) = 0.

Case 1. Suppose u(xi, xi) = 0 for all xi E Xi(xi). Then, u(xi, xi) and u(x9, xi) are two utility functions on X(x-). Thus, u(xi, xi) = constant implies that u(x%, .i) = constant. Then, since u(xi, .i) = 0 and u(xj, -) = 0, these constants are the same and equal to zero. This shows that the value of bi(xi) does not matter at all in this case since 0 = u(xi, xi) = ui(xi) + bi(xi)u(xi?, xi) = 0 + bi(xi)O. Thus, bi(xi) can be set to any value, 1 for example.

Case 2. If we are not in Case 1, then there exists xi E Xi(xi), such that u(xi, xi) f 0. Repeated use of Lemma 1 yields

u(xi, 5Xi) = u 1 (xi) + bi (X1)U(X2, * . , Xn)

u(xi, Xi) = u1 (x1) + b (XU)(u2(X2)

+ b2(x2)u(x3, . xn))

u (xi, Xci) =

u(xi, xci) = u 1 (x1) + bi (xl)u2 (X2)

+ b1(xl)b2(x2)u3(X3) + * * -

+ bi (Xi) . .. bn, - 1 (xn, - 1)un, (xn,) .

Since u(xi, .i) 4 0, there exists j f i such that uj(xj) f 0. (xi, x, xi) E X => (xi, xj, x?-J) E X (from (4)) and Lemma 1 implies

u(xi, xj, x?)= ui (xi) + bi (xi)u(xj, 5) = uj (xj)

+ bj (xj)u (xi, .5b.

Hence, since ui(xi) 0, bi(xi)uj(xj) uj(xj), that is bi(xi) = 1.

To prove (b), consider xi E Xi', xj E Xj such that ui(xi)

f 0 and uj(xj) f 0. If (xi, Tc9) is isolated, then X(x-) =

{x ?}. Thus, u(xi, x??) = ui(xi) + bi(xi)u(xi?, x??) = ui(xi), so that the value of bi(xi) does not matter and therefore, ki(xi) can be set to any value. The same is true if (xj, 5VO) is isolated. If neither xi nor xj is isolated, then (5) implies that x,Kxj and Lemma 2 that k(xi) = k(xj). Q.E.D.

Theorem 2. Let P be a nonempty convex set of simple probability distributions on a set X If Conditions (1), (2), (4), and (5) hold, then there exists a constant k such that:

n

u(x) = Z ui(xi) + k E ui(xi)uj(xj) i=l i>j

+ k2 E ui(Xi)uj(xj)ue(xe)

+ k.. + kn - 1u(X1)

Proof. Once Lemmas 1 and 3 have been proved, the proof of Theorem 2 no longer departs from any stan-

dard proof of Theorem P (e.g., Keeney and Raiffa 1976). Using Lemma 1 iteratively, i = 1, . . ., n - 1, we obtain:

u(x) = ul(xl) + b (xl)u(x2, . . . , xn) I

u(x) = u1(x1) + b1 (xl)(u2(X2)

+ b2 (X2)U(X3, * ,Xn))

U (X) = * * * .

u(x) = ul(xl) + b, (Xl)U2(X2) + b1 (xl)b2(X2)U3(X3)

+ * - - + b l(X1 ) . .. bn - 1 (Xn - 1 )Un (Xn ) -

Using Lemma 3, i = 1, . . ., n - 1, yields:

u(x) = ul(xl) + [1 + kul(xl)]u2(x2) + [1 + kui(xl)]

[1 + kU2(X2)]U3(X3)

+ * [ - + [1 + kul(x1)]

***[1 + kun - 1 (Xn - 1 )]Un (Xn )

n

u (x) = E ui(xi) + k E ui(xi)uj(xj) i=1 i>j

+ k2 E ui(Xi)uj(xj)ue(xe) i>j>e

+ * * * + k n - lU 1 (Xl1) ..

Un (Xn )

which completes the proof. Q.E.D. Thus, with a proper redefinition of the notions of utility

independence and component utility functions and the ad- dition of Conditions (4) and (5), the additive-multiplicative decomposition theorem holds for subsets of product sets. Both conditions are satisfied when the outcome set is a Cartesian product so that the new decomposition theorem is a generalization of Pollak's theorem. It should be emphasized that the two new conditions are technical as opposed to behavioral, in the sense that they relate only to the structure of the outcome set and not to the decision maker's preference structure itself. While Condition (4) is fairly easy to visualize since it requires the existence of maximal sections, the interpretation of the notion of k-comparability and therefore of Condition (5) is not as easy. Fortunately however, as suggested by one of the anonymous reviewers, an important special case of an outcome set structure that verifies (5) is given in the following lemma. This is an important case since it covers a wide range of realistic subsets corresponding to many practical situations (see Section 6).

Lemma 4. If the outcome set X is a convex subset of gijn and if each component utility function is nonzero in the neighborhood of xo then Condition (5) is satisfied.

Proof. For i, j E {1, * * *, n} (i = j or i # j), consider xi E Xi', xj E Xj, such that (xi, 50) and (xj, xj?) are not isolated, (Xi, Xi0) +6 xo and (xj, .c-) + x?.

Case 1. Suppose that i = j and use xi instead of xj. With- out loss of generality, we may assume that xi > xi'. Further- more, since (xi, xi) is not isolated, we know that there




exists xi E Xi such that (xi, xi) E X and xci x,. Similarly, there exists .x Ee Xi such that (x', .x) E X and.V xl 5c?.

(a) For some j, i A j, xj and xj are different from xo. First, assume that xj and xj are on the same side of xjo.

Without loss of generality, we can assume that xj and xj are both greater than x. Since uj is nonzero in the neighborhood of xo, there exists Xf'. x9 < x min {xj, xj} and u1(x!) = 0. Because X is convex, (xi, x; X5c) E X and (x', x; x5?-) E X. Thus xi and x; are directly k-comparable as well as xi and x; Hence, xiKx<. Alternatively, assume that xj and xj are not on the same side of x90. Here without loss of generality, we can assume xj < x < xj. Then, because u; is nonzero in the neighborhood of xo and because X is convex, we can find x'" xi' < x" < xi, such that uj(xj) A 0, (xi, x;', X07) E X and (xi', x; X07) E X. Thus xjKxi'and x'Kx7; and therefore xi Kxi.

(b) There is no j, i A j, such that both xj and xj are different from xj9. Then necessarily xci is of the form (xj, .-)0, xi 0 x9, and xi is of the form (xj,, 5V,), xj, 4 xO, with j 0 j'. Here without loss of generality, we can assume xi > x and xj > x. Then there exists a1 E Xj with x9 < a' < x;, and a 2 E Xj, with x12 < a 2 < xj,, such that uj(al) 0 0 and uj, (a2) 0 0. Because X is convex, (xi, a1, X?J) E X, (a', a 2, ZJ0,) E X, and (xi', a 2 X.05j) E X. Then, xiKal, alKa2,

and a2Kx'; and thus (xi, 5i?) and (x', .T'?) are k-comparable.

Case 2. Suppose that i ] j. Without loss of generality, we can assume that xi > x?? and that xi > x9. Then there exists (xi', xj) such that x? < x' < xi, 9 < xj < xj, x'(xj - 9) +

xj(x- - x?) < xix - x?xy (i.e., (xi', xj) is within the triangle (xO?, xi); (x?, Xj0); (xi, xj?)), ui(x,') # 0 and uj(xj) A 0. Clearly then, xiKx1' (refer to Case 1), x1'Kxj, and xjKxj. So (xi, x?) and (xj, 50) are k-comparable. Q.E.D.

6. SIGNIFICANCE OF THE PROBLEM: UTILITY REPRESENTATIONS FOR REAL-WORLD DECISION MAKING PROBLEMS

In many practical situations, the outcome set X is not a whole product set, but is restricted to a non-rectangular subset of the whole product set (Krantz et al. 1971, p. 311). Such restrictions do exist, for example, because of the nonexistence of some combinations of attributes or because of logical/def- initional constraints on some attributes. Furthermore, even when the outcome set considered is the whole product set, it might be desirable to develop separate models on subsets of product sets (Keeney and Raiffa 1976, p. 255). Two examples forming a large class of problems are given below.

Example 2. This example concerns the development of an index to measure the severity of a burn (Fryback and Keeney 1983). The overall burn severity is a function of the percentages of the body area covered by burns of different degrees of severity. Fryback and Keeney (1983) consider three severity degrees, and define accordingly three attributes x1, x2, and X3, where x1 is the percentage of body area burnt at the first degree, x2 at the second degree, and X3 at the third and worst degree of severity. They

also consider an attribute x,, which indicates whether or not there is inhalation injury. Given mutual utility independence, the four attributes are combined in a multiplicative way as:

( 1 + ku (x)) = fl ( 1 + kki u i (Xi )), i= 1,2,3, I

where u(x) is the burn severity index scaled between 0 (no burn) and 1 (worst burns), the uis are univariate utility functions scaled between 0 (best level) and 1 (worst level), and the ks are scaling constants with the following values: ki = 0.40, k2 = 0.54, k3 = 0.90, k, = 0.65, and k = -0.99. However, such a multiplicative decomposition is not fully justified. In particular, Keeney and Raiffa's (1976) work cannot provide sufficient justification since it applies only to the whole product set case. Clearly, the outcome set for the burn severity index is a subset of a whole product set since x1 + x2 + X3 S 100. In using the usual decomposition results for utility functions on whole product sets, practical difficulties arise in the development of this particular model. For example, the usual relation:

1+k-= H (1+kki), i= 1,2,3, I

used to calculate the constant k does not seem appropriate since it corresponds to an "artificial" point defined by x1 = 100%, x2 - 100%, X3 = 100%, and x, = "inhalation injury" which does not exist. Although this is the "corner point" that Fryback and Keeney used, instead, an appropriate relation that could have been considered here is:

1 + k = (1 + kk3)(1 + kk),

since the "worst" burn severity level that actually exists is defined by: x1 = 0%, x2 = 0%, X3 = 100%, and x1 = "inhalation injury." By developing the model on the whole product set, the constants above verify the first equation and not the second equation.

Example 3. The previous example is not an isolated case. A similar example concerns the evaluation of pavement condition for stretches of highway (see for example Golabi et al. 1982). State departments of transportation routinely measure pavement distress indicators (for example, slab breakup, cracking, or rutting) in terms of the percentages of the area of a one-mile segment affected by the distress at different severity levels. This situation is structurally identical to the previous situation (the burn severity model) and is in fact illustrative of a whole class of similar problems where outcomes are described by the distribu- tion of percentages of a quantity corresponding to different categories. The development of an additive or multiplicative utility model to measure the overall condition of such a one-mile segment of highway requires new decomposition results for utility functions on subsets of a whole product set. In developing a multiplicative utility model, Sainfort et al. (1994) encountered problems similar to the ones described in Example 2.




Examples 2 and 3 correspond to a class of subsets of the whole product set X1 x X2 x ... x X,, defined by a con- straint on m attributes of the type ljm 1 Xj - C, where C is a known constant. While all subsets of potential interest do not have this particular form, this suggests that the subset problem is significant in practice.

7. CONCLUSION

Little research on utility theory has been devoted to utility functions on subsets of product sets (Fishburn 1967, 1971, 1976, 1993; Chateauneuf and Wakker 1993). This paper proves that useful results of the standard theory on the whole product set can be generalized to subsets provided that two additional technical conditions-the existence of maximal sections and the property of k-comparability-are verified. This paper proves the existence of an additive or multiplicative decomposition of a multiattribute utility function on subsets of Cartesian products. The proof applies to finite as well as infinite outcome sets and is based on a natural revision of the notion of utility independence. The family of subsets that satisfy the technical conditions of this decomposition theorem is quite large and includes situations of significance for the practitioners wishing to develop multiattribute utility models. The approach taken in this paper is different from Fishburn's (1976) approach. Fishburn proposed a stronger revision of the notion of utility independence and successfully treated the case of a finite outcome set. However, his revision of utility independence seems quite complex to test in practice, and further, the case of an infinite outcome set was left unre- solved. This paper focused on outcome sets (finite or infinite) that verify two technical conditions. It corresponds to a more applied orientation since, in practice, model devel- opers would presumably know how the whole product set is restricted and could verify that the subset of interest verifies these two conditions.

It is worth noting that these two conditions are purely technical, relating only to the "geometric" structure of the subset under consideration. Therefore, it is very likely that these properties could generalize additive-multiplicative decomposition results for nonexpected utility theories. In fact, when looking at the proof, as one anonymous re- viewer writes "the crucial question in such generalization [is] whether the utility function of the generalized utility theory is an interval scale, and whether the utility of a gamble is an additive function of the utility of the outcomes." Thus if an additive-multiplicative decomposition result exists for such a utility function, then the decomposition result presented in this paper could be generalized. For example, this would be the case for multiattribute measurable value functions (Dyer and Sarin 1979) and generic utility theory (Miyamoto 1988). Further research, however, is needed to fully generalize these results.

APPENDIX 1

Notation X Outcome set P Set of simple probability distributions on X

p, q Probability distributions > Preference relation on P

p - q _ [p > q] A -_ [q >p]

,,,p -/ q [p - q]

u Utility function Xi ith attribute set

I Nonempty subset of {1, ... , n} I Complement of I

Xi fi i Xi x,, y, Elements of XI

XI fi l Xi x,, y, Elements of XI

x = (x1, ,x,) Element of X x? Arbitrary element of X for which u(x?) = 0

ui(xi) u(xj, .xi), the ith component utility function X,(x,) { x,I E XI: (x,I, x,- ) E EX}

X,(x,) {x,I E XI: (x,I, x,I) E EX} P(X,) Set of gambles on XI (xc', p) Gamble in P induced by x, E XI and p E

P(X,(x,)) defined by: prob (x,, x,') = p(x,), Vx1 E X,(x,)

(x,, p) Gamble in P induced by x, E XI and p E P(X,(x,)) defined by: prob (x,, x,) = p(x,), Vx, E X,(x,)

Xi' {xi EXi:Xi (Xi) *0}

f(i) f(i) = U(Xi, Xi) gXli) gXli) = u (xi9, Xi)

E(f, p) Expected value of f with respect to p, where p E P(Xi(xi))

> x Preference relation on P(Xk(xi)), defined by p xqif and only if (xi, p) > (xi, q)

xiKxj (xi, Xi) -/ xo, (xj, Xj) -/- xO, and (xi, xj, XO) E x

ACKNOWLEDGMENTS

We thank the editor and the anonymous referees for their constructive comments and suggestions which significantly improved the paper.

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decomposition of utility functions on subsets of product sets

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