Decomposition,Interdependence and Precision in Multiattribute Utility Measurements

Traditionally, parameters of multiattribute utility models, representing a decision maker's preference judgements, are treated deterministically. This may be unrealistic, because assessment of such parameters is potentially fraught with imprecisions and errors. We thus treat such parameters as stochastic and investigate how their associated imprecision/errors are propagated in an additive multiattribute utility function in terms of the aggregate variance. Both a no information and a rank order case regarding the attribute weights are considered, assuming a uniform distribution over the feasible region of attribute weights constrained by the respective information assumption. In general, as the number of attributes increases, the variance of the aggregate utility in both cases decreases and approaches the same limit, which depends only on the variances as well as the correlations among the single-attribute utilities. However, the marginal change in aggregate utility variance decreases rather rapidly and hence decomposition as a variance reduction mechanism is generally useful but becomes relatively ineffective if the number of attributes exceed about 10. Moreover, it was found that utilities which are positively correlated increase the aggregate utility variance, hence every effort should be made to avoid positive correlations between the single-attribute utilities. We also provide guidelines for determining under what condition and to what extent a decision maker should decompose to obtain an aggregate utility variance that is smaller than that of holistic assessments. Extensions of the current model and empirical research to support some of our behavioural assumptions are discussed. © 1997 John Wiley & Sons, Ltd.     

A simple model for the utility of gambling

A model of the utility of gambling is presented in a modified von Neumann-Morgenstern format. Axioms imply a utility function that preserves preferences between sure things and between gambles. The addition of a utility of gambling term to the expected utility of a gamble preserves preference comparisons between gambles and sure things. Aspects of the utility of gambling are noted, and comparisons are made to standard concepts of risk attitudes.The author is indebted to Joseph Sani for valuable discussions on the topic of this paper.     

RATIONAL VERSUS PLAUSIBLE ACCOUNTING EQUIVALENCES IN PREFERENCE JUDGMENTS

Subjective expected utility (SEU) embodies four distinct principles of rational behavior. Although all have been called into some question empirically, the least plausible and least studied is the property that formally equivalent gambles are treated as indifferent in preference. The paper describes some results that arise when this property is sharply weakened and to some degree replaced by alternative rational and not-so-rational-assumptions. The resulting utility representations, like SEU, are weighted averages of the utilities of consequences, but with the weights dependent on more than the underlying chance event. In rank-dependent cases, which arise from a restricted assumption about formally equivalent gambles, the weights depend on the rank position of the corresponding consequence. In ank-and sign-dependent models, they depend both on the rank position of the consequence associated to the event and on whether it is a gain or a loss. The theory giving rise to the latter involves an additional primitive, namely, joint receipt of gambles, in terms of which new rational and irrational assumptions are invoked. The result generalizes prospect theory to gambles with more than a single gain and a single loss.     

Functional equations arising in a theory of rank dependence and homogeneous joint receipts

This paper focuses on a class of utility representations of uncertain alternatives with two possible consequences (binary gambles) when they are linked via a distributivity property called segregation to an operation of joint receipt, which may be non-commutative. The assumption that the gambling structure and the joint receipt operation both have homogeneous representations that are order preserving leads to a functional equation that has too many solutions to be useful for characterizing a reasonably specific utility representation. A plausible restriction on the form of the utility of gambles leads to the functional equation

     

Empirical comparisons of bilinear and nonbilinear utility theories

Several bilinear and nonbilinear utility theories are evaluated using individual data from 144 informants. The nonbilinear models are best for 67% of the informants. The nonrational property of duplex decomposition linking joint receipts and mixed gambles of gains and losses is more adequate than the rational link of general segregation for 73% of the informants. The correlations are very high and linear, even in the worst fitting case. The weighting functions are mostly inverse-S-shaped and for 84% of the informants are fit best by the class of functions proposed by D. Prelec. These fits were generally excellent, with a minimum correlation of.81, a maximum of.99, and an average of.97.     

Utility Functions: A Compromise Programming Approach to Specification and Optimization

Nowadays, utility theory and compromise programming (CP) are considered very different paradigms and methodologies to measure preferences as well as to determine decision maker's optima on an efficient frontier. In this paper, however, we show that a utility function with separate variables (presented in the form of a Taylor series around the ideal point) is reducible to a weighted sum of CP distances. This linkage between utility and compromise (based on a main assumption in which the usual utility functions hold) leads to (i) a method for specification and optimization of usual utility functions by operational technique and (ii) a reformulation of standard CP with the advantage of determining the best CP solution from a utility perspective. © 1997 John Wiley & Sons, Ltd.     

Random error in holistic evaluations and additive decompositions of multiattribute utility — an empirical comparison

This paper details the results of an empirical investigation of the random errors associated with decomposition estimates of multiattribute utility. In a riskless setting, two groups of subjects were asked to evaluate multiattribute alternatives both holistically and with the use of an additive decomposition. For one group, the alternatives were described in terms of three attributes, and for the other in terms of five. Estimates of random error associated with the various elicitations (holistic, single-attribute utility, scaling constants, or weights) were obtained using a test-retest format. It was found for both groups that the additive decomposition had significantly smaller levels of random error than the holistic evaluation. However, the number of attributes did not seem to make a significant difference to the amount of random error associated with the decomposition estimates. The levels of error found in the various elicitations were consistent with theoretical bounds that have recently been proposed in the literature. These results show that the structure imposed on the problem through decomposition results in measurable improvement in quality of the multiattribute utility judgements, and contribute to a greater understanding of the decomposition method in decision analysis.     

Probabilistic choice (models) as a result of balancing multiple goals

We conceptualize probabilistic choice as the result of the simultaneous pursuit of multiple goals in a vector optimization representation, which is reduced to a scalar optimization that implies goal balancing. The majority of prior theoretical and empirical work on such probabilistic choice is based on random utility models, the most basic of which assume that each choice option has a valuation that has a deterministic (systematic) component plus a random component determined by some specified distribution. An alternate approach to probabilistic choice has considered maximization of one quantity (e.g., utility), subject to constraints on one or more other quantities (e.g., cost). The multiple goal perspective integrates the results regarding the well-studied multinomial logit model of probabilistic choice that has been derived from each of the above approaches; extends the results to other models in the generalized extreme value (GEV) class; and relates them to recent axiomatic work on the utility of gambling.     

Decomposition of multivariate utility functions in non-additive expected utility theory

In expected utility many results have been derived that give necessary and/or sufficient conditions for a multivariate utility function to be decomposable into lower-dimensional functions. In particular, multilinear, multiplicative and additive decompositions have been widely discussed. These utility functions can be more easily assessed in practical situations. In this paper we present a theory of decomposition in the context of nonadditive expected utility such as anticipated utility or Choquet expected utility. We show that many of the results used in conventional expected utility carry over to these more general frameworks. If preferences over lotteries depend only on the marginal probability distributions, then in expected utility the utility function is additively decomposable. We show that in anticipated utility the marginality condition implies not only that the utility function is additively decomposable but also that the distortion function is the identity function. We further demonstrate that a decision maker who is bivariate risk neutral has a utility function that is additively decomposable and a distortion function q for which q(½) = ½.     

Luck and Decision

abstract Much recent work on moral responsibility and on distributive justice has addressed the concept of luck. Very little attention has been given to the relation of luck to rationality. How does luck bear on our choices? Can beliefs about luck lead to unwise decisions? These questions have particular relevance for understanding gambling behaviour, and for public policy on gambling. In this paper I argue that no one is reliably lucky, and that projecting luck can undermine rational decision‐making. I give various examples to show the conceptual connection between luck and unpredictability. I present an a posteriori conception of projectibility, and argue that because lucky events are rationally unexpected, regularity statements about luck fail to satisfy the conditions of projectibility. I reject the claim that ‘lucky’ is a dispositional term, and thus projectible, on the ground that a dispositional interpretation leads to contradiction. I then defend my claim that luck is not projectible against three objections. I conclude with some thoughts about rational responses to luck, using gambling as an illustration.     

Preference as fulfilment of desires

We consider preferences as fulfillment of conditional desires, which can be either positive or negative, or both. We go beyond the standard multi-attributive additive utility theory in the sense that we separate the data given by the preference relation over an unstructured space from the property structure representing (conditional) attributes or desires. The model accounts for the psychologically motivated and empirically confirmed asymmetry between desire fulfillment and disappointment (loss aversion). The only restriction on the set of desires is, loosely speaking, a kind of mutual logical independence. We formulate a representation theorem characterising when a weak order (i.e. complete and transitive) preference is compatible with the logical structure of desires and has an additive representation over it. It is unique in the sense that each utility function representing the preferences has at most one such additive decomposition.     

A Simple Axiomatization of Binary Rank-Dependent Utility of Gains (Losses)

For binary gambles composed only of gains (losses) relative to a status quo, the rank-dependent utility model with a representation that is dense in intervals is shown to be equivalent to ten elementary properties plus event commutativity and a gamble partition assumption. The proof reduces to a (difficult) functional equation that has been solved by Aczél, Maksa, and Páles (in press).     

A solution to a problem raised in Luce and Marley (2005)

Luce and Marley [2005. Ranked additive utility representations of gambles: Old and new axiomatizations. Journal of Risk and Uncertainty, 30, 21-62] examined various relations between mathematical forms for the utility of joint receipt ⊕ of gambles and for the utility of uncertain gambles. Their assumptions lead to a bisymmetry functional equation which, when the gambles are ranked, is defined on a restricted domain. Maksa [1999. Solution of generalized bisymmetry type equations without surjectivity. Aequationes Mathematicae, 57, 50-74] solved the general case and Kocsis [2007. A bisymmetry equation on restricted domain. Aequationes Mathematicae, 73, 280-284] presents the solution for the ranked case. The latter solution allows us to solve open problem 5 in Luce and Marley (2005) by showing that the assumptions of their Theorem 19 for an order-preserving ranked additive utility (RAU) representation U imply that U is a ranked weighted utility (RWU) representation that is additive over ⊕.     

An Interactive Approach for Multicriteria Decision Making Using a Tchebycheff Utility Function Approximation

Multicriteria decision making (MCDM) can provide an efficient mean for considering various and conflicting objectives to reveal the alternative that maximizes the decision maker's (DM) utility. In this paper, we propose a new interactive MCDM method for implicit alternatives to help a DM obtain a most preferred solution. We employ a Tchebycheff function to generate weights for objectives consistent with the DM's responses to pairwise comparisons between alternatives and present a mixed integer linear programming formulation to generate these weights. Thus, we approximate the DM's utility function by a Tchebycheff function and generate weights consistent with the DM's responses. We test our approach with different true utility functions on various sized multiple criteria linear programming problems. The computational results show that even with non‐Tchebycheff true utility functions, our method can generate alternatives very close to the optimal solution with few questions. The comparison of our results with other methods reveals its advantages. Copyright © 2013 John Wiley & Sons, Ltd.     

The use of messages in altering risky gambling behavior in experienced gamblers

The present study was an experimental analogue that examined the relationship between gambling-related irrational beliefs and risky gambling behavior. Eighty high-frequency gamblers were randomly assigned to four conditions and played a chance-based computer game in a laboratory setting. Depending on the condition, during the game a pop-up screen repeatedly displayed either accurate or inaccurate messages concerning the game, neutral messages, or no messages. Consistent with a cognitive-behavioral model of gambling, accurate messages that correctly described the random contingencies governing the game decreased risky gambling behavior. Contrary to predictions, inaccurate messages designed to mimic gamblers' irrational beliefs about their abilities to influence chance events did not lead to more risky gambling behavior than exposure to neutral or no messages. Participants in the latter three conditions did not differ significantly from one another and all showed riskier gambling behavior than participants in the accurate message condition. The results suggest that harm minimization strategies that help individuals maintain a rational perspective while gambling may protect them from unreasonable risk-taking.     

Duplex decomposition and general segregation of lotteries of a gain and a loss: An empirical evaluation

We investigate the empirical validity of two hypotheses, duplex decomposition (DD) and general segregation (GS), regarding decomposition of a binary gamble of a gain and a loss into two unitary gambles in which one consequence of each gamble is no change from the status quo. Four binary lotteries (money gambles with specified probabilities) and four decomposed lotteries designed to test GS and four decomposed lotteries to test DD were constructed, and certainty equivalents (CEs) were estimated for each lottery. Respondents’ indifference between a binary lottery and a decomposed lottery was determined by evaluating the equality between the CE of a mixed binary lottery and the CE of the corresponding decomposed lottery. Given the variability of estimates of CEs and the lack of a clear statistical definition for the equality between two CEs, we applied several criteria: we counted responses where the difference between two CEs was either ±2% (the strictest criterion), ±4%, ±6%, or ±8% (the most lenient criterion) of the range of lottery outcomes. The results showed that under the strictest criterion, GS held for 25% of the responses and DD for 22%. Under the most lenient criterion, GS held for 56% of the responses and DD for 52%. Depending upon the criterion used, between 39 and 75% of the responses were consistent with at least one of the hypotheses. Several methodological problems in determining the indifference between two lotteries are discussed.     

Dominance Measuring Method Performance under Incomplete Information about Weights

In multi‐attribute utility theory, it is often not easy to elicit precise values for the scaling weights representing the relative importance of criteria. A very widespread approach is to gather incomplete information. A recent approach for dealing with such situations is to use information about each alternative's intensity of dominance, known as dominance measuring methods. Different dominance measuring methods have been proposed, and simulation studies have been carried out to compare these methods with each other and with other approaches but only when ordinal information about weights is available. In this paper, we use Monte Carlo simulation techniques to analyse the performance of and adapt such methods to deal with weight intervals, weights fitting independent normal probability distributions or weights represented by fuzzy numbers. Moreover, dominance measuring method performance is also compared with a widely used methodology dealing with incomplete information on weights, the stochastic multicriteria acceptability analysis (SMAA). SMAA is based on exploring the weight space to describe the evaluations that would make each alternative the preferred one. Copyright © 2012 John Wiley & Sons, Ltd.     

Testing the indifference between a binary lottery and its edited components using observed estimates of variability

Past studies have shown inconsistent conclusions on the empirical validity of the duplex decomposition assumption that Luce proposed (1997). The duplex decomposition assumption states that a person would feel indifferent between playing a mixed (gain and loss outcomes) lottery and playing two lotteries simultaneously or successively where one lottery is the gain part and the other lottery is the loss part of the original mixed lottery. The current study tested this assumption by investigating the equality between the certainty equivalent of a binary lottery, CE (BL), and the certainty equivalent of the duplex decomposed lottery, CE (DDL), while controlling for the response variability in the estimated CEs. The results from two experiments showed that for 69% of responses, the difference between the CE (BL) and CE (DDL) was smaller than or equal to the response variability in estimating CEs, implying that duplex decomposition cannot be rejected for a majority of responses. The size of response variability in estimated CEs was discussed.     

Combining additive representations on subsets into an overall representation

Many traditional conjoint representations of binary preferences are additively decomposable, or additive for short. An important generalization arises under rank-dependence, when additivity is restricted to cones with a fixed ranking of components from best to worst (comonotonicity), leading to configural weighting, rank-dependent utility, and rank- and sign-dependent utility (prospect theory). This paper provides a general result showing how additive representations on an arbitrary collection of comonotonic cones can be combined into one overall representation that applies to the union of all cones considered. The result is applied to a new paradigm for decision under uncertainty developed by Duncan Luce and others, which allows for violations of basic rationality properties such as the coalescing of events and other framing conditions. Through our result, a complete preference foundation of a number of new models by Luce and others can be obtained. We also show how additive representations on different full product sets can be combined into a representation on the union of these different product sets.     

Lexicographic order preservation and stochastic dominance

Stochastic dominance concerns conditions on outcome probabilities that are necessary and sufficient for one act to be (strictly) preferred to another according to all preference relations that share certain properties, one of which customarily is an Archimedean property sufficient to entail existence of real-valued representations. We relax this assumption to permit linear lexicographic utility of finite and known dimensionality. In some situations, levels of the lexicographic hierarchy could correspond to explicit criteria or attributes. In our model, subjective probabilities emerge as matrix premultipliers of the outcome utility vectors. We thus obtain matrix probability generalizations of the familiar cumulative probability conditions for stochastic dominance.