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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 article is based on the executive summary of a book in the Arabic language, The State of the Arab Nation 2014–2015, edited by Ali E.Hillal Dessouki and published by the Center for Arab Unity Studies. The book analyzes events in the Arab region from 2014 to the first part of 2015. The chapters examine the international order, the Arab regional system, and domestic conditions in the Arab states and neighbouring countries, such as Turkey and Iran. There is also particular focus on the countries of the Arab Spring and the remaining Arab countries, as well as the outlook for the youth in Arab countries and their role in future. Other chapters consider economic developments and their link to political developments and issues relating to science, technology and digital technologies. The final chapters cover the major political hotspots in the region, namely Palestine, Syria, Iraq, Libya and Yemen. The conclusion points to the main challenges facing the Arab nation in 2015.