Dyer Sarin or 1979

Measurable Multiattribute Value FunctionsAuthor(s): James S. Dyer and Rakesh K. SarinReviewed work(s):Source: Operations Research, Vol. 27, No. 4 (Jul. - Aug., 1979), pp. 810-822Published by: INFORMSStable URL: http://www.jstor.org/stable/170296 .Accessed: 04/01/2012 20:15

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

INFORMS is collaborating with JSTOR to digitize, preserve and extend access to Operations Research.

http://www.jstor.org

Measurable Multiattribute Value Functions JAMES S. DYER

University of California, Los Angeles, California

RAKESH K. SARIN Purdue University, West Lafayette, Indiana (Received October 1977; accepted April 1978)

This paper presents a theory of measurable multiattribute value functions. Measurable value functions are based on the concept of a ''preference difference'' between alternatives and provide an interval scale of measurement for preferences under certainty. We present conditions for additive, multiplicative, and more complex forms of the measurable multiattribute value function. This development provides a link between the additive value function and multiattribute utility theory.

T HIS PAPER presents a theory of measurable multiattribute value functions that extends the familiar theory regarding additive value

functions due to Debreu [1] and to Luce and Tukey [14]. Following Keeney and Raiffa [12], we refer to a preference representation function under certainty as a value function, and to a preference representation function under uncertainty as a utility function. We use the term measurable value function for a value function that may be used to order the differences in the strength of preference between pairs of alternatives or, more simply, the "preference differences" between the alternatives.

Let X denote the set of all possible consequences in a decision situation, with w, xI y, z, w', x', y' E X; define X* as a nonempty subset of X x X, and > * as a binary relation on X*. We shall interpret wx > * yz to mean that the strength of preference for w over x is greater than or equal to the strength of preference for y over z. The notation wx -* yz means both wx >* yz andyz >* wx, and wx >* yz means notyz >* wx.

There are several alternative axiom systems for measurable value functions, including the topological results of Debreu [1] and the algebraic development by Scott and Suppes [16]. Some of these systems allow both "positive" and "negative" preference differences and are called algebraic difference structures. For example, the "degree of preference" for x over w would be "negative" if w is preferred to x. Our development is based on an axiom system presented by Krantz et al. [13, Definition 4.1] that does not allow negative differences; hence it is called a positive difference structure.

810 Operations Research 0030-364X/78/2704-0810 $01.25 Vol. 27, No. 4, July-August 1979 ? 1979 Operations Research Society of America

Multiattribute Value Functions 811

The axioms of [13] imply that there exists a real-valued function v on X such that, for all w, x, y, z E X, if w is preferred to x and y to z, then wx > * yz if and only if

v(w) - v(x) _ v(y) - v(z). (1) Further, since v is unique up to a positive linear transformation, it is a cardinal function (i.e., v provides an interval scale of measurement). That is, if v' also satisfies (1), then there are real numbers a > 0 and b such that v'(x) = av(x) + b for all x E X (Theorem 4.1, [13]).

We define the binary preference relation >_ on X from the binary relation >_ * on X* in the natural way by requiring wx > * yx if and only if w >_ y for all w, x, y E X. Then from (1) it is clear that w > y if and only if v(w) _ v(y). Thus, v is a value function on X and, by virtue of (1), it is a measurable value function.

A von Neumann-Morgenstern utility function [17] also faithfully re- produces a preference order on X and is unique up to a positive linear transformation; hence it is also a cardinal function. However, there is no reason why a utility function should satisfy the interpretation of (1) regarding preference differences. In general, a measurable value function and a utility function defined on the same set of alternatives will not bear any particular relation to each other (see Ellsberg [4] and Fishburn [7]).

We recently presented an axiomatization of a measurable additive value function for multiple attributes in [3]. This theory is reviewed in Section 1. In Section 2 we demonstrate how this idea of a measurable value function may be extended to recover the multiplicative and multi- linear forms as well, providing a link between the additive value function and multiattribute utility functions. This link is investigated in further detail in Section 3. Some appropriate methods of assessment for these measurable value functions are surveyed in Section 4, and the implications for practical applications are discussed. Finally, Section 5 gives a brief summary and the conclusions.

1. A REVIEW OF MEASURABLE ADDITIVE VALUE THEORY Again let X denote the set of all possible consequences in a particular

decision problem. In the multiattribute problem Xin l Xi where Xi is the set of possible consequences for the ith attribute. We use the letters w, x, y, and z to indicate distinct elements of X. For example, w E X is represented by (w1, ..., wA), where w- is a level in the nonempty attribute set X, for i = 1, ... , n. Given I C (1, . .. , n}, we occasionally partition the attributes into two sets X, and X,, where X, represents those attribute sets in {Xl, X2, ... , Xn} with indices that are elements of I, and X, represents those attribute sets with indices that are elements of the complement of I. By defining wI, xi, yi, z, E X, and wij, xi, y5i, z- E XI, we may write w = (w,, WI-) or use the notation (we, wi) and (xi, i) to denote

812 Dyer and Sarin

two elements of X that differ only in the level of the ith attribute. Finally, we also assume that the preference relation > on X is a weak order.

One of the most important concepts in the theory of additive value functions is that of preferential independence.

DEFINITION 1. Xi is preferentially independent of XI if (w,, w,I) > (x1, wI) for any w,, x, E XI and w,I E XI implies (w,, x,) > (x,, xj) for all x-I

DEFINITION 2. The attributes X1, ..., Xn are mutually preferentially independent if for every subset I of { 1, . .. , n} the set XI of these attributes is preferentially independent of XI.

When coupled with a solvability condition and some technical assumptions, mutual preferential independence implies the existence of an additive value function for n ' 3 attributes [13]. DEFINITION 3. The set of attributes X1, ... , Xn is bounded from above (below) if there exists xi*(x-o) E Xi such that (xi*, xih) > (xi, xih) ((xi, xi.) > (x-0, xi-)) for all x- fE Xi, all x-i E Xi, and all i E t1, ..., n}. The attributes are bounded if they are bounded from above and below.

Next we introduce the notation necessary to define preferences among positive differences between vector-valued outcomes.

DEFINITION 4. X* = {wx I w, x E X and w > x} is a nonempty subset of X x X, and > * denotes a weak order on X*.

We now wish to specify a relationship between > on X and > * on X*.

DEFINITION 5. The set of mutually preferentially independent attributes XI . . . , Xn is difference consistent if, for all wi, xi E Xi, (wi, Wi) > (xi, wi) if and only if (wi, wP9)(xi', wi) >* (xi, w-i)(xi?, wi) for some wPi E Xi, and for any i E t1, ...,n}, and if w -'x then wy -*xy oryw -*yx or both for any y E X.

The second condition is much more powerful. Loosely speaking, this relationship says that the preference difference between (wi, wi-) and (xi, wi) is not affected by w-9. DEFINITION 6. The attribute Xi is difference independent of Xi if, for all w1, xi E Xi such that (wi, - -) > (xi, wi) for some wi E Xi, (wi, Wi) (xi, wPi)

* (w1, Yi)(x , x-i) for any x-i E Xi. We now state the basic representation theorem of the measurable

additive value theory. In addition to the conditions explicitly stated in


the following theorem, some additional structural and technical conditions are required.1

THEOREM 1. Assume n _ 3, X1, ..., Xn are mutually preferentially independent, difference consistent, and X1 is difference independent of X1. Then there exist functions vi: Xi -* Re, i = 1, ... , n, such that for all wi, xi, yi, zi E Xi,

(i) if wx, yz E X*, then wx >Z * yz if and only if I Vi(wi) - = Vi(xi) I

_= Vi(yi) - I=i vi(z1); (ii) x < y if and only if J'=i vO(xi) E?'=i vi(yi);

(iii) if vi', i = 1, ..., n are n other functions with the same properties, then there exist constants a > O, /3i, ..., /3n such that vi' = av, + p3i, i = 1,..., n.

Results (ii) and (iii) are well-known and follow immediately from the assumption that the attributes are mutually preferentially independent [1]. The new result is (i), which means that v = vi also provides difference measurement on X.

Interested readers are referred to [3] for a detailed proof of Theorem 1. Here we briefly summarize its logic. The strategy is to combine results in difference measurement theory that provide the basis for measurement on an interval scale under conditions of certainty with those of additive conjoint measurement theory. Using difference measurement theory (e.g., [13], Ch. 4), we first establish the existence of a measurable value function v':X -* Re. To complete the proof, we establish a relationship between v' and the functions vi:Xi -* Re. Specifically, we show that there must exist constants a > 0 and b such that v' = a (E>=I vi + b). This establishes the new result.

Results similar to Theorem 1 have been obtained by others using slightly stronger sets of assumptions, but their implications for multiattribute utility theory have not been exploited. Notice that we do not explicitly assume the existence of a measurable value function defined on X. Krantz et al. [13] assume a measurable function on X rather than difference consistency and derive essentially the same results for the case of n = 2. Fishburn [private correspondence] notes that the difference independence condition is similar to the notion of "persistence," which means that similar preferences over a single attribute hold in different time periods. Again assuming the existence of a measurable function on X, his result [6, Theorem 7.2] for homogeneous product sets can be given an interpretation similar to Theorem 1.

'Specifically, we assume restricted solvability from below, an Archimedian property, at least three attributes are essential, and that the attributes are bounded from below. If n = 2, we assume that the two attributes are preferentially independent of one another and that the Thomsen condition is satisfied (see [3]).

814 Dyer and Sarin

2. MULTIPLICATIVE AND OTHER NONADDITIVE FORMS OF MEASURABLE VALUE FUNCTIONS

In this section we identify a weaker condition than difference independence, which we refer to as weak difference independence. This condition plays a role similar to the utility independence condition in multiattribute utility theory. We show how this condition can be exploited to obtain multiplicative and other nonadditive forms of the measurable multiattribute value function.

In this development we explicitly assume the existence of a measurable value function v on X that is induced by the preference relation > * on X* and is related to the binary preference relation >_ on X in the natural way, as described earlier.2 We also assume the set of attributes X1, . . X, is bounded.

DEFINITION 7. XI is weak difference independent of XI if, given any wI, Xj, yi, ZI E XI and some 17v, C Xl such that

(WI, dI)(Xi, W1) > (yI' lI)(zI, WuiI), (WI, if)(Xi, XI) (y, 5XcI)(zi, Yi) for any =-I E XI. (That is, the ordering of preference differences depends only on the values of the attributes XI and not on the fixed values of XI.)

Finally, we introduce the notion of strategic equivalence for measurable value functions.

DEFINITION 8. Two measurable value functions v and v' are strategically equivalent, written v v', if they imply the same ranking of preference differences for any two pairs of alternatives. That is, for any wx, yz E X*, v - v' is equivalent to v(w) - v(x) _ v(y) - v(z) if and only if v'(w) - v'(x) _ v'(y) - v'(z). (Given v -v', then there exist constants a > 0 and b such that v'(x) = av(x) + b for all x E X. This follows immediately from the uniqueness result for measurable value functions [13, Theorem 4.1].) THEOREM 2. X, is weak difference independent of XI if and only if

v(xI, Xi) -g(XI) + h(x1)v(x1, wii) (2) for all xj, I, and iii4, where g(.i) and h(k1) > 0 depend only on x,.

Proof If X, is weak difference independent of XI, then the ordering of preference differences depends only on the values of the attributes XI.

2 For ease of exposition, we continue to use the notation introduced in the context of a positive difference structure. However, the results in this section could be obtained by assuming either the existence of a positive difference structure or an algebraic difference structure.


Thus, V(XI, I) must be strategically equivalent to V(XI, WIi) for any xi, kh and wib. Therefore, v(xj, 1w,i) must be a positive linear transformation of v(x,, Xl) that depends only on I.

Notice that this result is identical to the implication of utility independence for multiattribute utility functions, as discussed in [12]. We shall refer to (2) as conditional cardinality. THEOREM 3. If there exists a measurable value function v on X, if xi.., X, are mutually preferentially independent, and if X1 is weak difference independent of X1, then either

1 + Xv(x) = JJn-= [1 + xxivi(xi)] if>i=i X= # 1 (3a) or v(x) = X7-v A1v-(x-) if=1 AY, 1 (3b) where v(x*) = 1, v(x?) = 0, vi(x-*) = 1, vi(xi0) = 0, Ai = v(xi*, xi?), and A is a scaling constant such that A > -1, A #A 0, and A solves 1 + A = JjJ=i (1 + AXi).

Proof The proof of Theorem 3 can be done along the lines of the development in Keeney [11] for multiplicative utility functions. His arguments are valid because utility independence produces a conditionally cardinal utility function for each subset of attributes. Since weak difference independence produces a conditionally cardinal measurable value function in precisely the same way, the proof of Theorem 3 is identical, given the obvious substitution of language.

If the conditions in Theorem 3 are satisfied, the following result identifies whether v is additive or multiplicative.

COROLLARY 1. If, in addition to the conditions of Theorem 3, (wi, wj, ivij) (xi, wj, ivij) -* (wi, xj, i7ij) (xe, xj, ivi) for the following:

1. some wi, xi E Xi such that (wi, 17i) > (xi, 1w,) for some %i E X, 2. some wj, xj E Xj such that not (wj, Wj) (xi, w1;) for some 1; E X1,

and 3. some tij E Xi>,

then (3b) holds. Otherwise, (3a) holds. We can now extend the idea of Theorem 3 to obtain other nonadditive

forms of the measurable value function. The following result is an immediate consequence of the observation that conditional cardinality is the key feature in the corresponding proofs in multiattribute utility theory.

COROLLARY 2. The following theorems in Keeney and Raiffa [12] are valid if the expressions "utility function" and "utility independence" are replaced by the expressions "measurable value function" and "weak

816 Dyer and Sarin

difference independence," respectively, each time they appear: 5.2, 5.3, 5.6-5.9, 6.1-6.3, and 6.6-6.10.

3. RELATIONSHIPS BETWEEN THE NONMEASURABLE VALUE FUNCTION, THE MEASURABLE VALUE FUNCTION, AND THE

UTILITY FUNCTION The necessary conditions for the additive and multiplicative measura-

ble value functions and risky utility functions-notably, mutual preferential independence-are also necessary and sufficient for the additive value function that does not provide difference measurement. Therefore, it is natural to investigate the relationships among them. In order to distinguish between the two value functions, we shall use the notation vY for the nonmeasurable function. The following choice of scaling will also be imposed. For f= Uv, v, or u, f is normalized by f(xi*, . . ., xn*) = 1 and f(Xi?,... , xno) = 0 and fi(xi) is a conditional function on Xi scaled by fi(xi*) 1 and fi(xi0) = 0. Finally, X, X, and k will be used as scaling constants for the nonmeasurable and measurable value functions and the utility function, respectively.

The Additive Functions The relationships among the alternative developments of the additive

forms of real-valued functions on X follow immediately from their re- spective uniqueness properties. This may be summarized as follows:

THEOREM 4. Assume n _ 3 and X1, . .. , Xn are mutually preferentially independent. Then

(A) if Xi, ... , Xn are difference consistent and Xi is difference independent of X1, then vY = v;

(B) if there exists a utility function u on X and if preferences over lotteries on X1, . .. , Xn depend only on their marginal probability distributions and not on theirjoint probability distributions, then 0

v= u;

(C) if both A and B are satisfied, v- = v u. Proof The proof of A follows immediately from the uniqueness result

(iii) in Theorem 1. Result B is well-known [15] and C obviously follows from A and B.

Part B of Theorem 4 has been noted by others. For example, Dyer et al. [2] use this observation to justify using assessment techniques based on the concept of preference differences to determine an additive value function that was subsequently treated as a utility function in an analysis involving risk. They erred, however, since their assessment strategy implicitly assumed result C rather than simply B.


The Multiplicative Functions

Throughout this section we assume that the following conditions are satisfied:

1. n ' 3, and X1, ... , X, are mutually preferentially independent and bounded;

2. There exists a measurable value function v on X and X1 is weak difference independent of X1, and

3. There exists a utility function u on X and X1 is utility independent of X1.

Suppose we have assessed the additive value function vY and wish to obtain either v or u. We can use the results of the following theorem and its corollary.

THEOREM 5. (A) Either

1. v(x) = v(x) and V'Y(x-) = vi(xi), i = 1, ..., n, or 2. v3(x) ln(1 + X) = In[t + Xv(x)] and

X,i 0(x_) In(l + A) = in[l + ?AAv.(xi)], i = 1, ..., n. (B) Either

1. v?(x) = u(x) and v?i(x.) = ui(x-), i = 1, ..., n, or 2. v(x) ln(l + k) = In[l + ku(x)] and

Xiv?i(xi) ln(l + k) = In[l + kk-u-(x-)], i-1, ..., n. Proof We present the proof for part A only since the logic of the proof

for part B is identical. By Theorem 3, v is either additive or multiplicative. When v is additive, the uniqueness properties ensure that v(x) = vY(x) and vi(x) = vi3(x). When v is multiplicative, we obtain ln(l + Xv(x)) = >=i ln(l + XXAv-(xi)) and, since A > -1, this transformation preserves the ordering of v(x).

Because of the uniqueness of the additive representation, we know that ln(l + Xv(x)) is a positive linear transformation of v(x). With our choice of scaling, the range of ln(1 + Av(x)) is [O, ln(l + A)j if A > 0 and [ln(l + A), 0] if A < 0. Therefore, vi(x) ln(l + A) = ln(l + Av(x)) so that Xiii(xi) ln(1 + A) = ln(1 + AA1v (x )) for i = 1, ... , n, establishing Part A. DEFINITION 9. For each attribute Xi, if there exists xi' E Xi such that (xi', t)(xi', x-i) -* (xi*, Yi)(x,', x-) for any x-i E Xi, then xi' is the equal difference point for Xi.

Notice that vi(xi') = 1/2 because of our choice of scaling. Given v, Corollary 3 shows that the assessment of xi' for any attribute Xi is enough to completely specify v.

818 Dyer and Sarin

COROLLARY 3. If the conditions for Part A of Theorem 5 hold and 'i( xi') = 1/2 for some i E {1, 9.. , n}, then v = vi. Otherwise,

1 + (1 + A)A = 2(1 + A)XiV1(x1')

Remark. If Vi(x.') > 1/2, then A > 0; and if V3i(xi') < 1/2, then A < 0. Given A, v is obtained from Part A2 of Theorem 5.

Proof. From Part A2 of Theorem 5, ln(1 + A) = ln[1 + AAivi(xj)]/ Nivi (xi). Evaluating this equation at x = x* and at x = (xi', Xi o), we obtain ln(1 + A) = ln(1 + AAi)/X) = ln(1 to(1/2)AAi)/X01iv(xi'), respectively. Simpli- fying yields (1 + (1/2)AAi) = (1 + \)X (x) and AN = (1 + A) 1. Combining these expressions, we obtain (1/2 + (1/2)(1 + A)XL) = (1 + \);i"i(x') which gives our result.

Finally, to derive u from vi, find xi" for some attribute Xi such that the decision maker is indifferent between xi" and an equal chance lottery between xi* and xi0, with the other criteria held fixed. The following result is stated without proof (see also Theorem 6.11 of [12]). COROLLARY 4. If the conditions for Part B of Theorem 5 hold and vi(xi") =/2 or some i E {1, . n. , n}, then u = Vi. Otherwise, 1 + (1 + k)

= 2(1 + k)Aitvi(*ri ) Theorem 5 and Corollaries 3 and 4 can also be used to derive v after u

has been assessed, or vice versa. For example, suppose u has been assessed using appropriate procedures. To obtain v, we find the equal difference point xi' for some criterion Xi. Part B of Theorem 5 is used to obtain Xi and vi3 for each criterion, and then Corollary 3 gives v. In a similar manner, u can be obtained from v after assessing xi" for some criterion Xi.

4. ASSESSMENT OF MEASURABLE MULTIATTRIBUTE VALUE FUNCTIONS

Verification of the Independence Conditions

The first issue to be considered is the verification of the independence conditions. Since methods for verifying mutual preferential independence are discussed in [12], we focus on the independence conditions involving preference differences.

Difference consistency is so intuitively appealing that it could simply be assumed to hold in most practical applications. The following procedure could be used to verify difference independence. We determine wi, xi E X1 such that (wi, W1i) > (xi, W1) for some w-, E X1. We then ask the decision maker to imagine that he is in situation 1: He already has (xi, &i), and he can exchange it for (w1, W1). Next, we arbitrarily choose xl


E X1 and ask him to imagine situation 2: He already has (x1, x-,), and he can exchange it for (wl, xl). Would he prefer to make the exchange in situation 1 or in situation 2, or is he indifferent between the two exchanges? If he is indifferent between the two exchanges for several different values of wi, xi E Xi and wi, xi E X1, then we can conclude that X1 is difference independent of X1.

Before using this procedure, we must ensure that the decision maker understands that we are asking him to focus on the e- X' ,finge rather than the final outcomes. For example, if he states that he -fers an exchange of $1,000,000 for $1,000,001 to an exchange of $5 for $500, then he undoubtedly is not focusing on the substitution of one outcome for another. Thus, some training may be required before this approach to verification is attempted.

Alternatively, we might display the pair of alternatives (wi, W-1) and (xi, ti3v) and the pair (wi, xl) and (x1, x-i). We then ask which pair is more "similar" in terms of the decision maker's preferences. If the decision maker considers the pairs to be equally similar, we interpret this to mean that the preference differences between the pairs are equal and so difference independence is satisfied. Similarity judgments are well accepted in multidimensional scaling applications (e.g., see Green and Rao [10]), and considerable empirical evidence is available to support their use.

The results of Section 3 involving weak difference independence also require the explicit assumption of a measurable value function defined on the set X. In principle each of the axioms stated by Krantz et al. [13, Definition 4.1] could be verified through a series of tests, but in practice the existence of the measurable function could simply be assumed unless contrary behavior on the part of the decision maker is noted.

To verify weak difference independence, partition X into X, and XI, and choose wI, Xi, YI, zI E X, and &i E XI so that (WI, WiI) > (XI, WI), (yI, WI) > (ZI, WIi), and the exchange of (XI, &I) for (wI, 1WvI) is preferred to the exchange of (zi, WIv) for (yI, WhI). Then pick another value XI of X, and ask if the decision maker still prefers the exchange of (XI, XI) for (WI, JXI) to the exchange of (zI, XI) for (yI, XI). This must be true if the subset X, is weakly difference independent of XI. If the decision maker's response is affirmative, we repeat the question for other quadruples of consequences from X, with the values of the criteria in XI fixed at different levels. Continuing in this manner and asking the decision maker to verbally rationalize his responses, the analyst can either verify that X, is weakly difference independent of X, or discover that the condition does not hold. Note that for the multiplicative measurable value function, it would only be necessary to verify weak difference independence for the special case of I= (1}.

820 Dyer and Sarin

Assessment of the Measurable Value Functions

If difference independence or weak difference independence holds, each conditional measurable value function vi can be assessed while holding xi constant at any arbitrary value (generally at x-i?). With the additive value function that does not provide difference measurement, this strategy cannot be used.

Fishburn [5] reviews three approaches for assessing each vi that require strength of preference judgments. He identifies these methods as direct rating, direct midpoint, and direct ordered metric. Since he gives a simple example of each method as well as additional references, we will omit any further discussion of them here.

If the measurable value function is additive, the scaling constants may be assessed using the same trade-off approach suggested for estimating the scaling constants for the additive ordinal value function [12, Ch. 3]. As an alternative, the following direct rating approach could be used. Suppose An E Ai for all i E {1, ..., n - 1}, and assign AAn an arbitrary value, say 10. Then, directly assign each Ai some value from 0 to 10 that reflects the relative importance of the difference from (xi*, x-iO) to (xi', XOi) versus the difference from (x,*, ixo) to (xno, X-n). For example, if the decision maker considers the preference difference between (xi*, Oi) and (xI0, x-) to be half of the difference between (xn*, X-n) and (xno, x- ?) then A, = 5.

In the multiplicative model we must estimate A as well as each Xi. If An >-Xi for all i E { 1, ... , n}, then any of the scaling constant estimation procedures for the additive model can be used to establish equations of the form Xi = Xnvn(xun') for i = 1, ... , n - 1. However, we cannot arbitrarily choose An, to obtain each of the other constants as in the additive model.

One alternative is to assess An directly. We have chosen a scaling such that v(x*) = 1 and v(x?) = 0. The difference between (xn*, i-0) and xo could be compared directly to the difference between x* and xo. If, for example, the decision maker considers the preference difference between (xn*, *-n) and xo to be one-tenth of the difference between x* and xo, then An = 1Ao, and the values- of A1, i = 1, . .. , n - 1, can be obtained using the equations Xi = Anvn(xnt). The value of A is then calculated by solving 1 + A = fi7i=i (1 + AAX) for A.

As an alternative, we can multiply each side of the n - 1 equations Ai = Avnn(xn') by A. One additional trade-off judgment establishing (xi*, x-1)

(xi', xn', x-ino) provides

Ai = (1 + AAivi( xi')) (1 + AAXvn(xn') )-1. (3) Treating each Ak as a single variable, we can solve the n equations simultaneously to obtain XAi, i = 1,..., n. The value of A is simply fl7= (1 + Aki) - 1.


5. CONCLUSIONS

This paper has presented a theory for measurable multiattribute value functions. In the case of certainty, it provides an alternative to the cumbersome assessment procedures associated with the additive value function that does not provide difference measurement. In addition, the results give an interval scale of measurement that may be given a "strength of preference" interpretation. This theory also provides alternative assessment techniques for multiattribute utility functions since the additive and multiplicative forms of the utility and value functions have simple relationships that can be exploited.

No doubt there are other theoretical contributions that can be made in this area. For example, Fishburn and Keeney [8, 9] have generalized the concepts of preference independence and utility independence to allow for reversals of preference. A similar extension to the concept of weak difference independence should be straightforward. A more critical issue, we feel, is an examination of this theory from an empirical standpoint. The concept of the measurable value function has been criticized as being non-operational, at least in the context of a single attribute. Whether these objections can be overcome is an issue that needs to be explored.

ACKNOWLEDGMENTS

It is a pleasure to acknowledge the helpful comments of D. Erlenkotter, R. Keeney, and the anonymous referees.

REFERENCES 1. G. DEBREU, "Topological Methods in Cardinal Utility Theory," in Mathe-

matical Methods in the Social Sciences, 1959, pp. 16-26, K. J. Arrow, S. Karlin, and P. Suppes (eds.), Stanford University Press, Stanford, 1960.

2. J. DYER, W. FARRELL, AND P. BRADLEY, Utility Functions for Test Perform- ance," Management Sci. 20, 507-519 (1973).

3. J. DYER, AND R. SARIN, "An Axiomatization of Cardinal Additive Conjoint Measurement Theory," Working Paper No. 265, Western Management Science Institute, UCLA, February 1977.

4. D. ELLSBERG, "Classic and Current Notions of 'Measurable Utility,"' Eco- nomic J. 64, 528-556 (1954).

5. P. FISHBURN, "Methods of Estimating Additive Utilities," Management Sci. 13, 435-453 (1967).

6. P. FISHBURN, Utility Theory for Decision Making, John Wiley & Sons, New York, 1970.

7. P. FISHBURN, "Cardinal Utility: An Interpretive Essay," Rivista Internazion- ale di Scienze Economiche e Commerciali 13, 1102-1114 (1976).

8. P. FISHBURN, AND R. KEENEY, Seven Independence Concepts and Contin- uous Multiattribute Utility Functions," J. Math. Psychol. 11, 294-327 (1974).

822 Dyer and Sarin

9. P. FISHBURN, AND R. KEENEY, Generalized Utility Independence and Im- plications," Opns. Res. 23, 928-940 (1975).

10. P. GREEN, AND V. RAO, "Conjoint Measurement for Quantifying Judgmental Data," J. Marketing Res. 8, 355-363 (1971).

11. R. KEENEY, "Multiplicative Utility Functions," Opns. Res. 22, 22-34 (1974). 12. R. KEENEY, AND H. RAIFFA, Decisions with Multiple Objectives, John Wiley

& Sons, New York, 1976. 13. D. KRANTZ, R. LuCE, P. SUPPES, AND A. TVERSKY, Foundations of Mea-

surement, Vol. I, Academic Press, New York, 1971. 14. R. LuCE, AND J. TUKEY, "Simultaneous Conjoint Measurement: A New Type

of Fundamental Measurement," J. Math. Psychol. 1, 1-27 (1964). 15. H. RAIFFA, "Preferences for Multi-attributed Alternatives," RM-5868-DOT/

RC, The Rand Corporation, Santa Monica, Calif., 1969. 16. D. SCOTT, AND P. SUPPES, "Foundational Aspects of Theories of Measure-

ment," J. Symbolic Logic 23, 113-128 (1958). 17. J. VON NEUMANN, AND 0. MORGENSTERN, Theory of Games and Economic

Behavior, Princeton University Press, 1947.

Article Contentsp. 810p. 811p. 812p. 813p. 814p. 815p. 816p. 817p. 818p. 819p. 820p. 821p. 822

Issue Table of ContentsOperations Research, Vol. 27, No. 4 (Jul. - Aug., 1979), pp. i-vi+629-862Front Matter [pp. i-vi]Feature ArticleAggregate Advertising Models: The State of the Art [pp. 629-667]Markov Models of Advertising and Pricing Decisions [pp. 668-681]Optimizing Advertising Expenditures in a Dynamic Duopoly [pp. 682-692]A Queueing Theory, Bayesian Model for the Circulation of Books in a Library [pp. 693-716]On the Stationary Analysis of Continuous Review (s, S) Inventory Systems with Constant Lead Times [pp. 717-729]Infinite-Horizon Dynamic Programming Models-A Planning-Horizon Formulation [pp. 730-742]Bounding Global Minima with Interval Arithmetic [pp. 743-754]The Sum of Serial Correlations of Waiting and System Times in GI/G/1 Queues [pp. 755-766]Queues Solvable without Rouch's Theorem [pp. 767-781]The Two-Machine Maximum Flow Time Problem with Series Parallel Precedence Relations [pp. 782-791]The Two-Machine Maximum Flow Time Problem with Series-Parallel Precedence Constraints: An Algorithm and Extensions [pp. 792-798]An Analysis of Approximations for Finding a Maximum Weight Hamiltonian Circuit [pp. 799-809]Measurable Multiattribute Value Functions [pp. 810-822]Computing Network Reliability [pp. 823-838]

Technical NotesA Note on the Nerlove-Arrow Model under Uncertainty [pp. 839-842]Constraint Qualifications for Inexact Linear Programs [pp. 843-847]

Letters to the EditorComments on the Possibility of Cycling with the Simplex Method [pp. 848-852]To the Editor [p. 852]To the Editor [pp. 852-854]Errata: Direct Solutions of M/G/1 Processor-Sharing Models [p. 854]

Back Matter [pp. 855-862]

Dyer Sarin or 1979

Documents

Transcript of Dyer Sarin or 1979