The concepts of conditional risk aversion, the conditional risk premium, and risk independence pertaining to multiattributed utility functions are defined. The latter notion is then generalized to what is called utility independence. A number of theorems useful for simplifying the assessment of multiattributed utility functions given certain risk independence and utility independence assumptions are stated.U. S. Army Research Office (Durham) under Contract No. DAHCO4-70-C-0058

Keeney, Ralph L.

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ArmyResearch Office (Durham) under Contract No. DAHCO4-70-C-0058.Risk Independence and Multiattributed Utility Functions *-Ralph L. Keeney-'Massachusetts Institute of TechnologyAbstractThe concepts of conditional risk aversion, the conditional risk premium,and risk independence pertaining to multiattributed utility functions are defined.The latter notion is then generalized to what is called utility independence. Anumber of theorems useful for simplifying the assessment of multiattributedutility functions given certain risk independence and utility independence assump-tions are stated.*Adapted from a thesis, supervised by Professor Howard Raiffa (Harvard),presented to the M.I. T. Operations Research Center in partial fulfillment ofthe requirements for the degree of Doctor of Philosophy, June, 1969. Addi-tional supervision and other expenditures associated with the work reportedherein were supported (in part) by the U.S. Army Research Office (Durham),under Contract Number DA-31-124-ARO-D-209 and (in part) by the Office ofNaval Research under Contract Nonr-3963 (06), NR 276-004, at the M.I. T.Operations Research Center. During his doctoral program the author wassupported by the Bell Telephone Laboratories' Doctoral Study Program.**Currently Assistant Professor of Civil Engineering and Staff Member of theOperations Research Center.2.1. Introduction and SummaryIn assessing cardinal utility functions for assets or-any other singleattribute, it has proven to be useful to begin by specifying certain qualitativecharacteristics to which the decision-maker subscribes. Aside frommonotonicity the important characteristics are those concerning the decision-maker's attitude toward risk, in particular, risk aversion and decreasingrisk aversion. Given one's risk characteristics, his utility function can oftenbe restricted to one or a few functional forms. The problem is then reducedto finding a member of these families of utility functions appropriate to theparticular decision-maker. This is usually done by assessing certaintyequivalents for a few simple lotteries and using this information to fix theparameters of the family of utility functions.In this paper, we attempt to extend this idea. More specifically, ameasure of risk relevant to multiattribute cardinal utility functions is de-fined. Restrictions on the functional form of the utility functions are indicatedprovided this measure satisfies certain conditions. In related work,Fishburn [1, 2, 3 and Pollak [7] have looked at the functional forms of multi-attribute utility functions implied by assumptions about the decision maker'spreferences for various lotteries. Stiglitz [9] recently investigated restric-tions on the indifference map implied by assumptions about the multiattributecardinal utility function, and restrictions on that utility function implied by as-sumptions about the indifference map.The concepts of a conditional utility function and risk independence andthe notation to be used are defined in the next section, followed by a proof ofour main result for two-attribute utility functions in section 3. In section 4,3.the conditional risk premium is introduced. The notion of risk independenceis then considered in a different context which permits extensions and general-izations of the results. These are presented in the final section as represen-tation theorems which simplify the assessment of multiattribute utility functions.2. Definition of Risk IndependencePratt [8] defines the local risk aversion r(x) byr. (x) = - u(x) (1)-i) u'(x) 'where u(x) is a utility functionI for the continuous scalar attribute X andu'(x) and u"(x) are respectively the first and second derivatives of u(x). Byintegrating (1), exponentiating, and integrating again, he showedu(x) = kl e (x)dxdx + k 2 , (2)where kl and k 2 are constants of integration. One can observe that the riskaversion function r(x) contains all the essential information about u(x) whileeliminating the arbitrariness introduced by positive linear transformations.For multiattributed utility functions, it seems natural to develop con-ditional risk aversion functions on the same basis as r(x). More specifically,consider the utility function u(x 1,xZ . .,xn ) for attributes X 1, i = 1, 2,..., nand for notational convenience, let us designate X 1 x X 2 x... x X as X and1 2 n4.X x ... x X x X x ... x X as X-. Then the conditional risk aversion for1 i-i i+1 n 1X., which we denote by ri(x), will be defined by1 1u'(x)r(x) = - (3)1 -) u' (x)1-where ul(x) and u!' (x) are the first and second partial derivatives of u(x)1- 1 - -with respect to x.We will say that X. is risk independent of X- if r.(x) does not depend on1 1 1-x-. In other words, X. is risk independent of the other attributes if the1 1riskiness (as measured by r.) of lotteries involving only uncertain amounts of1X. does not depend on the fixed amounts of the other attributes. This may be1a reasonable assumption in many situations. That is, if all of the risk isassociated with only one attribute and the other attribtes are all fixed, thedecision-maker's attitudes toward risk will depend only on that attribute in-volving the risk.oLet x- represent any amount of X- and let x- be a specific amount.oThen we can define the conditional utility function for X. given x- x- to1 1 10mean any positive linear transformation of u(x, x-).1 1Given this notation, we can efficiently prove an importantLEMMA. If X. is risk independent of X-, then1 1ou(x.,x-) = f(x-) u(x., x-) + g(x-) , (4)1 1 1 1 1 1where f(x-)> 0.15.Proof. Given r.(x.,x-.) = r.(x., x-), it follows from (3) that1 1 1 1 1 1a a hog 'ax [log u(x.,x )= [log (xi, X)]X. 1 1 i ax. I 'i 1so by partial integration and exponentiation,a (x-) o bui(X x-) e =u (x.,x-) e11 1 1 1 1where a(x-) and b are integration constants. Integrating again,a (x-) o bu(x.,x-) e i +c(x-) u(x.,x-) e + db-a(x-)which becomes (4) when rearranged and f(x-) e i and g(x-) =1 1[d- c(xT-)] [e-a(xi )].The lemma becomes almost obvious when we consider that r.(x)1 mspecifies the conditional utility function for X. uniquely up to positive trans-formations and that r.(x) does not depend on x- .This result is useful in a number of situations. For example, supposeX. is risk independent of X- and suppose ri(xi,xT) = c. > 0. That is, the de-cision maker is constantly risk averse over X.' for some x- in X-, Given these1 1 i- , -f 6.conditions, it follows directly from Pratt[8] that u(x., x:) must be a positive1 1-c.x.. Then from the lemma, we knowlinear transformation of -e 1 1u(xi, x-) for all x- in X- can be expressed by-Cx.u(x.,x-) -f(x-) e 1ii + g(x-), f(x-) > 0.Stiglitz[9] considers a utility function of this form in the context of consumerbehavior under uncertainty.3. Utility Functions and Risk IndependenceIn this section, we derive the functional form of a utility function withtwo attributes given each attribute is risk independent of the other. As before,if u(x, y) represents the utility function for attributes X and Y, it must beassumed u is increasing and twice continuously differentiable in each attribute.In section 5, the concept of risk independence is viewed in a slightly differentcontext which allows us to eliminate these restrictions.An important result isTHEOREM 1. Given X is risk independent of Y and Y is risk independent ofX, then u(x, y) can be expressed byu(x, y) = U(X (x, y ) + u(  y) + k u(x, y ) u(x, y), (5)where k is an empirically evaluated constant and u(x, yo) andu(x , y) are consistently scaled conditional utility functions.Proof. For reference, let us define the origin of u(x, y) byu(x ,yo ) = 0. (6)7.Since X is risk independent of Y, from (4) we knowu(x, y) = f() U(x y) + gl(y)' (7)Similarly, Y is risk independent of X sou(x, Y) = f2 (X) u(xo, Y) + g( (8)Then by evaluating (7) at x = x , we haveou(x, y) = fl(y) U(X, Yo) + gl( Y ) =gl(y) (9)and likewise, evaluating (8) at y = y yieldsU(X, o) = g2() . (10)Now substituting (9) into (7) and (10) into (8) and equating the resultingequations,fl(y) u(x, yo) = f() U(X') + (x y) y) (11)which, after rearranging, isf 1 (Y) - 1 f 2 (X) - 1u(x ,y) ) u(x,Xyo' YY (12)In (12), a function of x is equal to a function of y, therefore, they both mustequal a constant. Call this constant k, and we havef2(x) - 1= k, xx, (13)u(x,yo) oor(14)f2(x) = k u(x, y) + 1 .The restriction x x of (13) is not necessary in (14) since f(Xo) = 1 as can0 Zo'be verified by evaluating (11) at x = x . Substituting (10) and (14) into (8) weconcludeu(x, y) = [k u(x, Yo ) + 1] u(xO, y) + u(X, Y)0u(x, yo ) +u(x, y) + k u(x, y) u(x, y).00 0 0By evaluating u(xl, Y1) for arbitrary xl and Y1, we find k can be evaluatedfromu(xl Y1) - U(Xo' Y1) - U(X 1 ' Y)u(Xo, Y1) U(x1, Yo)The converse of theorem 1 is also true and easily proven workingdirectly with the definition of conditional risk aversion in (3). That is,given a utility function for two continuous scalar attributes is of the form(5), then X is risk independent of Y and Y is risk independent of X.The usefulness of theorem 1 is that it simplifies the assessment ofu(x, y) provided the requisite risk independent assumptions hold. Theassessment of the two-attribute utility function is reduced to assessing twoone-attribute conditional utility functions and the utilities of two additionalconsequences. The latter are necessary to consistently scale the conditionalutility functions and to evaluate k.4. Conditional Risk PremiumsOne of the important results of Pratt's work was his relating the localrisk aversion to the intuitively appealing concept of a risk premium. Theseideas are also valid in the current context concerning conditional utilityfunctions.9.Consider the lottery represented by (x.,x-), where the "-" repre-1 1sents a random outcome, and let Pi(Xi ) represent the probability densityfunction describing this outcome. Then the conditional certainty equivalentAfor x. given x- is defined as the amount of X., call it x., such that the de-1 1 1 1Acision maker is indifferent between (x., x-) and (x., x-). The conditionalrisk premium for this lottery 7ri is defined as the amount such that the de-cision maker is indifferent between (x.-.,x-. ) and (x.,x-), where x. is the- Aexpected value of x.. It should be clear that ir = x. - x.1 1 1 1In general, there is no reason why the conditional certainty equivalent andconditional risk premium for x. would not depend on x- . However, it follows1 1from (4) that when X. is risk independent of X-, the conditional risk premium1 1and conditional certainty equivalent for x. will not in fact depend on the con-1dition x- . This is useful in that it allows us in some situations to assess theexpected utility of lotteries in terms of conditional certainty equivalents.To be specific, consider the lottery (x, y), where we have verifiedX and Y are risk independent of each other. We can now calculate the ex-pected utility of (x, y) using (5) to find10.E[u(x Y)] = S S(x, YO+ ux o, y) + k u(x, y,) u(xO, y)1 p(x, y) dxdyx yE[u(x, yo)] + E[u(x[u(x, yo u(x ,7)], (15)where E denotes expectation and p(x, y) is the joint probability density func.-tion for (x, y). When X and Y are probabilistically independent, (15) becomesE[u(x, y)] = E[u(x, yo)] + E[u(x, y)] + k E[u(x, y)] E[u(xo , y)1 (16)But since X and Y are risk independent, we can reduce (16) to[u(x, y) = u(x, y) + u(x, y) + k u(x, yo) u(x o , y), (17)A Awhere x and y are respectively the conditional certainty equivalents for xand y . It immediately follows from (17) thatE[u(x, y)] =u(, ) = u(x , -T),x Ywhere 7T and r7 are the conditional risk premiums.x y5. Generalization of Risk IndependenceResults analogous to theorem 1 could be derived for n-attribute utilityfunctions provided each attribute X. was risk idependent of all the otherattributes. However, there would be some unnecessary limitations of suchresults. First of all, u(x) would have to be increasing and twice continuouslydifferentiable in each attribute for each ri(x) to be defined. Also, since therisk aversion function is only defined for scalar attributes, there is nocorresponding risk aversion function for vector attributes. 3Thus, an11.expression like (5) would not be valid if for instance, y = ( Y2). Riskindependence can be considered in a slightly different context such that theresults are not limited by such restrictions.In the notation of section 2, recall that the lemma states providedX. is risk independent of X-, the conditional utility function for X. given1 1 1any fixed x_ is a positive linear transformation of the conditional utility0function for X. given x- = x-. The converse is easily shown to be true, so1 i 1this condition is equivalent to risk independence for increasing and twicecontinuously differentiable scalar attributes. However, one can see from(4) that such a condition by itself does not require restrictions on theattributes, and that it may hold for non-increasing, non-continuous, vectorattributes. Thus, by stating our assumptions in terms of conditional utilityfunctions rather than in terms of the conditional risk aversions, the resultscan be extended to include many additional situations.6. Additional ResultsRather than repeat derivations that are found elsewhere [4], we will stateonly one important result which simplifies the assessment of a multidimensionalutility function provided the requisite assumptions hold.Let u(x 1, x 2 , .. ., x ) be a utility function over consequence spacenX X1 x X x ... x X . Then define vector attributes Y and Z such thathere y = ( nX = Y x Z, where y = (x 1, x 2, ... ) and z = (x m+ 1 x m + 2 ... ,x ) represent12.specific amounts of Y and Z respectively. Given u(y, z), Y is said to beutility independent of Z if the decision-maker's relative preferences overlotteries on Y, when Z is held fixed at z , are the same regardless of the0amount z . Given this condition, since utility functions are unique up to posi-0tive linear transformations, the conditional utility function for Y, given anyamount of Z, is a positive linear transformation of the conditional utilityfunction for Y given any other amount of Z. Mathematically stated, if Y isutility independent of Z, then for any z.u(y, z) = f(z) + g(z) u(y, z ), all z.0If Y and Z are utility independent of each other, they are said to be mutuallyutility independent. Similarily, if each of the Xi's is utility independent of all1the others, they are mutually utility independent.In the n-dimensional case, if the conditional utility function for X. is1denoted by u.(x., x-), where x- represents a fixed amount of all the otheri1 1 1attributes, and if xi* and *x. are arbitrarily chosen such that u. (xY, x.) > u.i(*., x -),1 i 11 1 1 1 1then one can proveTHEOREM 2. Given X X X X X X X and the X. are mutually utilityTHOEM2 1 X  n 1independent, u(x 1 , x 2 , . . x ) is completely specified by(a) u.(xi, x-) for arbitrary x-, for each X., and1 1 1 1(b) u(x', x' .. .,x ) for all x! = x' or *x .1' 2 n 1 1 113.When X is partitioned into Y x Z,there is the generalization of Theorem 1 whichwe state as aCOROLLARY. If Y and Z are mutually utility independent, u(y, z) can beevaluated fromu(y (yz ) = + u(y o, z ) + k u(y,z ) u(y o z )where y and z are arbitrarily chosen and k is an empirically evaluatedconstant.Other related results using the concept of utility independence are foundin Meyer [6] and Keeney [5].7. ConclusionsWe have used the concept of risk aversion developed by Pratt [81 whichhas proven to be important in assessing single-attribute utility functions indefining a conditional risk aversion function relevant to multiattribute utilityfunctions. The functional form of the two-attribute utility function satisfyingcertain reasonable assumptions concerning one's conditional risk aversionattitudes was derived. The notion of risk independence was then generalizedand renamed utility independence. Finally, two representation theorems werestated which simplify the assessment of a multidimensional utility function pro-vided specified utility independence assumptions hold.8. AcknowledgmentI thank Professor Howard'Raiffa of Harvard University who supervisedthe thesis from which this paper was adapted.14.References1. Fishburn, Peter C., "Independence in Utility Theory with Whole ProductSets," Operations Research, Vol. 13, 1965, pp. 28-45.2. , "Additivity in Utility Theory with Denumerable ProductSets," Econometrica, Vol. 34, 1966, pp. 500-503.3. , "Interdependence and Additivity in Multivariate,Uni4imensional Expected Utility Theory," International Economic Review,Vol. 8, 1967, pp. 335-342.4. Keeney, Ralph L., "Utility Functions for Multiattributed Consequences,"Management Science, Vol. 18, 1972, pp. 276-287.5. , "Utility Independence and Preferences for MultiattributedConsequences," Operations Research, Vol. 19, 1971, pp. 875-893.6. Meyer, Richard F., "On the Relationship Among the Utility of Assets,the Utility of Consumption, and Investment Strateg,- in an Uncertain, butTime Invariant World," to be published in the Proceedings of the FourthIFORS Conference, Venice, Italy, 1969.7. Pollak, Robert A., "Additive Von Neumann-Morgenstern Utility Functions,"Econometrica, Vol. 35, 1967, pp. 485-494.8. Pratt, John W., "Risk Aversion in the Small and in the Large,"Econometrica, Vol. 32, 1964, pp. 122-136.9. Stiglitz, Joseph E., "Behavior Towards Risk with Many Commodities,"Econometrica, Vol. 37, 1969, pp. 660-667.15.Note s1. It is assumed u(x) is increasing and twice continuously differentiable.2. It is assumed u(x) is increasing in each X. and the first and secondpartial derivatives exist and are continuous.3. We define a scalar attribute as a single-valued attribute and a vectorattribute as a many-valued attribute. For instance, if the set ofconsequences X X x X2 x X3, where the Xi are scalar attributes, thenY X X X 2 would be a vector attribute.

Risk Independence and Multiattributed Utility Functions

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