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Article Excerpt
1. Introduction
Many decision situations involve outcomes that consist of several attributes. In applied decision analyses, it is useful to decompose the utility function over these multiattribute outcomes into separate utility functions over the different attributes to reduce the number of preference elicitations. This is only justified if the decision-maker's preferences satisfy particular assumptions. Several authors have identified the preference conditions that allow decomposing multiattribute utility functions into additive, multiplicative, and related decompositions (e.g., Farquhar 1975, Fishburn 1965, Keeney and Raiffa 1976).
Most of these decomposition results have been derived under expected utility. Abundant evidence exists, however, that (subjective) expected utility is not valid as a descriptive theory of decision under (uncertainty) risk. The descriptive deficiencies of expected utility complicate the empirical assessment of the preference conditions underlying decompositions: it cannot be excluded that observed violations of preference conditions are due to violations of expected utility rather than to violations of a decomposition. To obtain robust tests of the appropriateness of decompositions, it is desirable to derive conditions that are valid even when expected utility is violated.
In this paper, we study multiattribute utility theory under prospect theory (Kahneman and Tversky 1979, Tversky and Kahneman 1992). Prospect theory is currently the most influential theory of decision making under uncertainty. It models two major deviations from expected utility: nonlinear decision weighting and loss aversion, i.e., the tendency that people treat outcomes as deviations from a reference point and are more sensitive to losses than to gains of the same magnitude. Both nonlinear decision weighting and loss aversion are widely documented in the empirical literature. Despite its popularity, some evidence has been accumulated recently revealing limitations of the theory (see the summaries of Marley and Luce 2005 and Birnbaum 2008).
Fishburn (1984), Miyamoto (1988), Dyckerhoff (1994), and Miyamoto and Wakker (1996) also studied multiattribute utility under nonexpected utility, but only considered outcomes of the same sign. Like us, Zank (2001) and Bleichrodt and Miyamoto (2003) studied multiattribute utility theory under prospect theory but their approach was different than the approach of this paper, as we explain next.
A central issue in multiattribute prospect theory is to determine when an attribute yields a gain or a loss. Consider, for example, a research associate (RA) who contemplates changing jobs. In evaluating different jobs, the RA has to consider several aspects, e.g., salary, commuting time, cost of living, amount of research time, etc. How does the RA determine whether a particular job offer is an improvement (a gain) compared with her reference point (presumably her current job)? One possibility is that she first determines whether the job offer, as a whole, is a gain or a loss compared to her reference point, and then applies the decomposition to determine how attractive the job offer is compared with other offers. This holistic evaluation was used by Zank (2001) and by Bleichrodt and Miyamoto (2003).
Another approach, the focus of this paper, is that the RA determines a reference point for each attribute and evaluates job offers as gains and losses on each attribute. This attribute-specific evaluation seems plausible when the number of attributes is large and the choice is complex. A decision context where the attribute-specific evaluation is particularly intuitive is welfare theory: there we are interested in whether each individual's welfare is above some reference level. The attribute-specific evaluation is commonly assumed in empirical studies on loss aversion for trade-offs under certainty and was found to be descriptively accurate by some studies (Bateman et al. 1997, Bleichrodt and Pinto 2002, Tversky and Kahneman 1991). Also empirical studies for decision under risk relied on the attribute-specific evaluation. See, for example, Payne et al. (1984) who study managers' choices among capital budget proposals involving cash flows at two points in time, and Fischer et al. (1986) who consider both risky, multiperiod cash flows and risky job alternatives. Both studies use attribute-specific reference points. No preference foundation of the attribute-specific evaluation existed until now. Providing such a foundation is the topic of this paper.
The difference between the holistic and the attribute-specific evaluation is that in the former, loss aversion and decision weighting are attribute-independent, whereas in the latter they depend on the attributes. As we show in [section]5, the holistic and the attribute-specific evaluation are in general equivalent only when people behave according to expected utility, i.e., when loss aversion does not affect people's preferences and there is no decision weighting. An example to further clarify the difference between the holistic and the attribute-specific evaluation is in [section]3.
This paper gives preference foundations for additive utility under prospect theory and the attribute-specific evaluation. We restrict our attention to the additive decomposition for two reasons: First, it is commonly applied in many areas of economics and decision analysis. Second, other decompositions, such as multiplicative and multilinear utility, raise special problems under the attribute-specific evaluation. Solving these problems is beyond the scope of this paper.
The remainder of this paper is organized as follows. Section 2 gives notation and explains prospect theory for single-attribute outcomes. In [section]3, we move to multiattribute utility where we first assume, for ease of exposition, that there are just two attributes, both numerical. Section 4 gives preference foundations for prospect theory with additive utility under the attribute-specific evaluation. As mentioned, weighting functions are defined per attribute and they may differ across attributes in the attribute-specific evaluation. To force them to be equal across attributes requires additional conditions. We will characterize this special case in [section]5. We extend our results to the case where there are more than two attributes in [section]6 and to the case of nonnumeric outcomes in [section]7. Section 8 concludes the paper with some observations on the empirical measurement of additive utility in prospect theory under the attribute-specific evaluation. All proofs are in the appendix.
2. Prospect Theory for Single-Attribute Outcomes
We consider a decision maker in a situation where there is a finite number, n [greater than or equal to] 2 of distinct states of nature, exactly one of which obtains. S = {1, ..., n} denotes the finite set of states of nature. Subsets of S are called events. In a medical decision problem, the states of nature can, for example, be mutually exclusive diseases, and the decision maker has to choose between different treatments before knowing what the actual disease is. We consider decision under uncertainty where the probabilities for the states of nature may, but need not, be given. The assumption of a finite number of states of nature is made for expositional purposes. The results of this paper can be extended to an infinite state space using tools from Wakker (1993). The extension to decision under risk, i.e., the case where probabilities are objectively given, is as in Kobberling and Wakker (2003, [section]5.3).
The decision-maker's problem is to choose between prospects. Each prospect is an n-tuple of outcomes, one for each state of nature. Formally, a prospect is a function from the set of states of nature to the set of outcomes C. We denote the set of prospects as P = [C.sup.n] We shall write ([[Florin].sub.1], ..., [[Florin].sub.n]) for the prospect [Florin] that gives [[Florin].sub.j]. if state j occurs. A constant prospect gives the same outcome for each state of nature. For ease of exposition, we assume in this section that outcomes are one-dimensional. The set of outcomes C is a nondegenerate convex subset of R. Outcomes are defined with respect to a reference point. The reference point is a constant prospect, that we will denote as r. We assume that the reference point is fixed, i.e., we restrict attention to preferences with respect to one reference point. Variations in the reference point are analyzed by Schmidt (2003).
Let [greater than or equal to] denote a preference relation on the set of prospects. As usual, > denotes the asymmetric part of [greater than or equal to] (strict preference) and ~ denotes the symmetric part of [greater than or equal to] (indifference), and [less than or equal to] and r is a gain and an outcome x < r is a loss.
A prospect [Florin] is rank-ordered if [[Florin].sub.1] [greater than or equal to] ... [greater than or equal to] [[Florin].sub.n]. For each prospect, there exists a permutation [rho], such that [[Florin].sub.[rho](1)] [greater than or equal to] ... [greater than or equal to] [[Florin].sub.[rho](n)]. For each permutation [rho], let [[P.sub.[rho]] = {[Florin] [member of] P: [[Florin].sub.[rho](1)] [greater than or equal to] ... [greater than or equal to] [[Florin].sub.[rho](1)] [greater than or...
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