A Note on an Application of the Trade-off Method in Evaluating a Utility Function

ABRAHAM MEHREZ

and AMIRAM GAFNI

Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer Sheva, Israel

Faculty of Management, The Leon Recanati Graduate School of Business Administration, Tel Aviv University, Tel Aviv, Israel

In this note we suggest a possible use for the trade-off technique to exclude possible forms of the utility function that are common to the same set of assumptions. Our discussion is focused on the specific case of two attributes. This case is becoming more common in coping with practical problems in the health care field, where the two attributes are life years and health status.

A diffcult task in theory and even more so in practice is the decomposition of a multi-attribute utility func- tion. There are three basic stages in deriving the specific form of a utility function:

(1) Validating the assumptions leading to a specific form which is not necessarily unique (e.g. Fishburn, 1965; Farquhar, 1975; Keeney and Raiffa, 1976; Hiroyuki and Yutaka, 1983);

(2) Evaluating some coefficients of the utility model; (3) Determining the exact form of the utility function.

This task can also be difficult for the case of two attributes ( X , , X,), where the utility function is additive or multiplicative. In other words, the utility function satisfies the condition that X, is utility-independent of X, and X, is utility-independent of X, (see Keeney, 1972); and each attribute is relatively independent (see Keeney, 1981; Richard, 1975). Note that the last condition guarantees that the utility function over X is either exponential or linear.

In addition to the above procedure the analyst (who is, in fact, an external observer) may also like to

(1) Increase hisher understanding of the decision- maker and thus improve hisher ability to predict the decision-maker’s future behavior;

(2) Increase the decision-maker’s insight into hisher own preferences which will help himher in making subsequent choices.

(3) Exclude possible forms of the utility function that are common to the same set of assumptions.

Such information can be obtained by using trade-off

techniques. Detailed discussion and illustrations of the use of such techniques for providing information on (1) and (2) can be found, for example, in MacCrimmon and Toda (1969); MacCrimmon and Siu (1974) and Keeney and Raiffa (1976). The purpose of this note is to illustrate how the use of trade-off techniques can be useful in obtaining information on (3), or more specifically, how the set of possible forms of utility models can be narrowed, or alternatively how some possibilities can be excluded. Our analysis, presented below, will focus on the specific case of two attributes.

THE ANALYSIS

Let X = X, x X , represent a rectangular subset of a two-dimensional Euclidean space. Then, a specific consequence can be designated by x or (x,, x,), where x i is a particular amount of X i . Assume that these assumptions above hold (Keeney, 1972; Keeney, 1981; Richard, 1975) such that the utility function is either

= k,u,(x,) + k,u,(x,) (1)

(2) where U and Ui are utility functions scaled from zero to one; the ki are scaling constants with 0 < ki < 1; and k > - 1 is a non-zero scaling constant. ui(xi) must be either one of the exponential utility forms

(3) where ci is a positive or negative constant or the linear

or 1 + ku(x)=(l + kk,u,(xJ)(l + kkzuz(x,))

uAxi) = cieciXtVi = 1,2

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MANAGERIAL AND DECISION ECONOMICS, VOL. 6. NO. 3. 1985 191

utility function

ui(xi) = xi V i = 1,2 (4)

Following Keeney (1974), x t is designated the most desirable, and x: the least desirable consequence.

It can be shown that the trade-off ratio

(2lXlJ for the additive case is

(5 )

where: u:, is the first-order partial derivative of u with respect to Xi, and for the multiplicative case is

where qx, is the elasticity of ui. Using the relationship described in Eqns ( 5 ) and (6)

it is easy to show that when the u;s are linear utility functions and the form is additive or when the uI)s are exponential utility functions and the form is multi- plicative then the trade-off ratio is constant; otherwise it is not. Such a property can easily be tested empiri- cally by direct questioning of the decision-maker; and if the trade-off ratio found is not constant then the decompositions mentioned can be excluded. In fact, such an investigation can raise the model’s reliability in the decision-maker’s eyes. On the other hand, as

mentioned before, it can also help us explore the consistency of the decision-maker.

The observation made in this note calls for further research that relates the trade-off concept and different utility decomposition. For example, with respect to the exponential case and the multiplicative form, the trade-off ratio can be interpreted in terms of measures of risk (see Pratt, 1964). This might help us in some cases to formulate trade-off questions in a different manner.

Finally, it should be mentioned that applications for such research can be found in many areas. For example, in the health care field, multi-attribute utility theory is used to suggest a form for a utility function over life years and health status; the choice of treatment may depend on the patient’s trade-off between long- evity and quality of life (health status). In other words, some treatment alternatives result in longer survival but with a low quality of life compared with other alter- natives where the quality of life is higher but with a lower survival (fewer years to live). Examples for such cases are laryngeal cancer, coronary artery disease and chronic kidney disease. Illustrations of the use and measurement of a utility function for such cases can befoundin McNeiletal.(1981)and PliskinetaL(1981).

Acknowledgement

Part of this work was done while Dr Gafni was a visitor to McMaster University, Hamilton, Ontario, Canada.

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