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Many of the complex problems faced by decision makers involve uncertainty as well as multiple conflicting objectives. This book provides a complete understanding of the types of objective functions that should be used in multiattribute decision making. By using tools such as preference, value, and utility functions, readers will learn state-of-the-art methods to analyze prospects to guide decision making and will develop a process that guarantees a defensible analysis to rationalize choices. Summarizing and distilling classical techniques and providing extensive coverage of recent advances in the field, the author offers practical guidance on how to make good decisions in the face of uncertainty. This text will appeal to graduate students and practitioners alike in systems engineering, operations research, business, management, government, climate change, energy, and healthcare

This paper introduces the notion of a multiattribute utility tree. This graphical representation decomposes the von Neumann–Morgenstern utility of a multiattribute consequence into a sum of products of indifference probability assessments of binary gambles. The utility tree displays the sequence of gambles required to elicit the utility value of a consequence. In addition, it enables the analyst to conduct consistency checks on the indifference assessments provided by the decision maker and to change the order of the assessments based on her comfort level. Once the indifference assessments are provided, the utility value of a consequence can be obtained by direct rollback analysis. On a continuous domain, the utility tree decomposes the functional form of a multiattribute utility function into a sum of products of normalized conditional utility functions. Each attribute in the expansion is conditioned on the boundary values of the attributes expanded before it. This formulation provides a general method for deriving the functional form of a multiattribute utility function under a wide variety of conditions. It also leads to several new independence concepts such as "boundary independence," which is a weaker condition than utility independence, and "corner independence," which makes higher-order independence assertions. Reversing the order of the nodes in the tree relates several widely used notions of utility independence found in the literature.

The cross derivatives of a multiattribute utility function play an important role in the choice between multivariate lotteries and in multiattribute Taylor expansions of the utility function. This paper decomposes the cross derivatives into two components: the derivatives of a single-attribute utility function over value and the cross derivatives of the value function. This approach provides a simple method for reasoning about the signs of the cross derivatives of a multiattribute utility function using derivatives of a univariate utility function and the properties of the value function. To illustrate the approach, we relate the multivariate risk aversion concept, which involves the mixed partial derivative of the utility function, to the Arrow–Pratt risk aversion function. We show that for additive value functions, a decision maker is multivariate risk averse if and only if he is risk averse over value in the Arrow–Pratt sense. For other value functions, however, a decision maker can be risk averse or risk seeking over value and still exhibit multivariate risk aversion. The approach also derives the conditions on the value function that relate two important classes of utility functions: single attribute utility functions whose derivatives alternate in sign and multiattribute utility functions whose cross derivatives alternate in sign. These two classes are widely used in practice and form the basis of univariate and multivariate stochastic dominance. Several examples illustrate the approach.

AbstractThe construction of a multiattribute utility function is significantly simplified if it is possible to decompose the function into lower‐order utility assessments. When every attribute is utility independent of its complement, we have a powerful property that reduces the functional form into a multilinear combination of single‐attribute functions. We propose a natural extension to the multilinear form requiring the milder set of independence conditions: every attribute is utility independent of a subset of the attributes in its complement. We introduce a graph, which we call the bidirectional utility diagram, to facilitate the elicitation of these utility independence conditions. We also define a class of bidirectional diagrams, which we refer to as the canonical form. We show how this canonical representation leads to tractable functional forms of utility functions that may appeal to the decision analyst and can be used in practice. We also discuss situations where a canonical form does not exist. We then present an iterative approach to determine the lower‐order utility assessments that are required for a given set of utility independence assertions. We conclude with some extensions of this graph‐based approach to independence relations of the form: a subset of the attributes is utility independent of another subset. Copyright © 2010 John Wiley & Sons, Ltd.

Linear and log-linear pools are widely used methods for aggregating expert belief. This paper frames the expert aggregation problem as a decision problem with scoring rules. We propose a scoring function that uses the Kullback-Leibler (KL) divergence measure between the aggregate distribution and each of the expert distributions. The asymmetric nature of the KL measure allows for a convenient scoring system for which the linear and log-linear pools provide the optimal assignment. We also propose a "goodness-of-fit" measure that determines how well each opinion pool characterizes its expert distributions, and also determines the performance of each pool under this scoring function. We work through several examples to illustrate the approach.

This paper defines invariant utility functions to continuous monotonic transformations. We also define transformation invariance as the condition in which the certain equivalent of a lottery follows a continuous monotonic transformation that is applied to its outcomes. We show that invariant utility functions uniquely satisfy transformation invariance, and we illustrate how knowledge of an invariance criterion determines the functional form of the utility function. This formulation extends the widely used notions of invariance to shift and scale transformations on the outcomes of a lottery to more general monotonic transformations. Moreover, we interpret any continuous and strictly monotonic utility function as an invariant utility function to a composite monotonic transformation. Furthermore, we show how this composite transformation uniquely characterizes the utility function up to a linear transformation. We derive the invariance formulations that lead to the assignment of hyperbolic absolute risk-averse (HARA) utility functions, linear plus exponential utility functions, and a two-parameter power-logarithmic utility function that generalizes the logarithmic utility function. We work through several examples to illustrate the approach.

This paper presents the results of four lottery-type experiments that investigate the effects of incentive structures on decision-making under uncertainty. We compare choices made with and without incentives, with fixed targets, with binary targets, and with four-outcome targets that are discretized from a logistic distribution. The results of the behavioral experiments (i) validate theoretical findings of utility functions induced by fixed and uncertain targets. Further, the behavioral results show that (ii) individuals' choices are indeed affected by incentive structures, which we quantify by several deviation measures. (iii) Defined consistency measures show that choices under uncertain targets become less consistent as the number of uncertain target outcomes increases. The results of these experiments provide insights into the effects of setting incentive structures on decision-making behavior.

Archimedean utility copulas comprise the general class of multiattribute utility functions that have additive ordinal preferences and are strictly increasing with each argument for at least one reference value of the complementary attributes. The construction of an Archimedean utility copula requires an assessment of an individual utility function for each attribute as well as a single generating function. The assessment of individual utility functions for the attributes of a decision has had a large share of literature coverage, but there has been much less literature on the construction of the generating function for the Archimedean functional form. This paper focuses on the assessment of Archimedean utility copulas with polynomial generating functions. We provide methods to assess these generating functions and derive bounds on the types of utility surfaces that they provide. We demonstrate that linear generating functions correspond to the multiplicative form of mutual utility independence, and then we show how higher-order polynomial generating functions allow more flexibility in the types of multiattribute utility functions and corner values that can be modeled. The results of this paper provide a new method to help the analyst construct multiattribute utility functions in a simple way when utility independence conditions do not hold.

This paper presents some functional equations that have played an essential role in the characterization of utility and probability functions in decision analysis. We survey some previous results with improvements and derive several new results. We also discuss some simple methods for the solution of these equations and highlight some subtle points about their use. We show that one functional equation can determine several unknown functions within it, and that relaxing differentiability and requiring only continuity of the functions leads to generalizations of many well-known results in utility and Bayesian probability theory.

We present an analogy between joint cumulative probability distributions and a class of multiattribute utility functions, which we call attribute dominance utility functions. Attribute dominance utility functions permit assessing multiattribute utility functions using common techniques of joint probability assessment such as marginal-conditional assessments and the method of copulas. By itself, this class of utility functions appears in many cases of decision analysis practice. Furthermore, we show that many functional forms of multiattribute utility function can be decomposed into attribute dominance utility functions that are easier to elicit. We introduce the notion of utility inference analogous to Bayes' rule for probability inference and provide a graphic representation of attribute dominance utility functions, which we call utility diagrams.

AbstractThis article presents a public value measure that can be used to aid executives in the public sector to better assess policy decisions and maximize value to the American people. Using Transportation Security Administration (TSA) programs as an example, we first identify the basic components of public value. We then propose a public value account to quantify the outcomes of various risk scenarios, and we determine the certain equivalent of several important TSA programs. We illustrate how this proposed measure can quantify the effects of two main challenges that government organizations face when conducting enterprise risk management: (1) short‐term versus long‐term incentives and (2) avoiding potential negative consequences even if they occur with low probability. Finally, we illustrate how this measure enables the use of various tools from decision analysis to be applied in government settings, such as stochastic dominance arguments and certain equivalent calculations. Regarding the TSA case study, our analysis demonstrates the value of continued expansion of the TSA trusted traveler initiative and increasing the background vetting for passengers who are afforded expedited security screening.

This paper measures the risk aversion coefficients exhibited by online customers at a name-your-own-price (NYOP) intermediary that allowed the submission of subsequent offers. We present a decision analytic model to determine the optimal sequence of offers for a decision maker with a constant risk aversion coefficient. We show how the optimal sequence of offers can be determined by simple sensitivity analysis. We then discuss the inverse problem of estimating the risk aversion coefficient from a sequence of offers placed by a customer. We provide a direct expression for the risk aversion coefficient in terms of the increments between each two successive offers. This expression separates the estimation of the risk aversion from the remaining model parameters and enables its calculation directly by observing a sequence of offers from a customer. We then use field data from an NYOP firm to estimate the risk aversion that is exhibited throughout the online process. We find that customers exhibit relatively high risk aversion coefficients on Internet sites. Based on three different product groups, we also find that risk aversion is a significant variable in the offer strategy of the customers. To our knowledge, this is the first work that empirically examines risk aversion in a nonexperimental Internet-based NYOP setting.