Ali E. Abbas

Maximum Entropy Utility

This paper presents a method to assign utility values when only partial information is available about the decision makers preferences. We introduce the notion of a utility density function and a maximum entropy principle for utility assignment. The maximum entropy utility solution embeds a large family of utility functions that includes the most commonly used functional forms. We discuss the implications of maximum entropy utility on the preference behavior of the decision maker and present an application to competitive bidding situations where only previous decisions are observed by each party. We also present minimum cross entropy utility, which incorporates additional knowledge about the shape of the utility function into the maximum entropy formulation, and work through several examples to illustrate the approach.

Entropy methods for joint distributions in decision analysis

A fundamental step in decision analysis is the elicitation of the decision makers information about the uncertainties of the decision situation in the form of a joint probability distribution. This paper presents a method based on the maximum entropy principle to obtain a joint probability distribution using lower order joint probability assessments. The approach reduces the number of assessments significantly and also reduces the number of conditioning variables in these assessments. We discuss the order of the approximation provided by the maximum entropy distribution with each lower order assessment using a Monte Carlo simulation and discuss the implications of using the maximum entropy distribution in Bayesian inference. We present an application to a practical decision situation faced by a semiconductor testing company in the Silicon Valley.

Attribute Dominance Utility

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.

A Comparison of Two Probability Encoding Methods: Fixed Probability vs. Fixed Variable Values

We present the results of an experiment comparing two popular methods for encoding probability distributions of continuous variables in decision analysis: eliciting values of a variable, X , through comparisons with a fixed probability wheel and eliciting the percentiles of the cumulative distribution, F ( X ), through comparisons with fixed values of the variable. We show slight but consistent superiority for the fixed variable method along several dimensions such as monotonicity, accuracy, and precision of the estimated fractiles. The fixed variable elicitation method was also slightly faster and preferred by most participants. We discuss the reasons for its superiority and conclude with several recommendations for the practice of probability assessment.

Multiattribute Utility Copulas

We introduce the notion of a multiattribute utility copula that expresses any (i) continuous; (ii) bounded multiattribute utility function that is (iii) nondecreasing with each of its arguments, and (iv) strictly increasing with each argument for at least one reference value of the complement attributes, in terms of single-attribute utility assessments. This formulation provides a wealth of new functional forms that can be used to model preferences over utility-dependent attributes and enables sensitivity analyses to some of the widely used functional forms of utility independence. We introduce a class of utility copulas, called Archimedean utility copulas, and discuss the conditions under which it yields the additive and multiplicative forms. We also discuss linear and composite transformations of utility copulas that construct utility functions with partial utility independence. We conclude with the risk aversion functions that are induced by utility copula formulations and work through several examples to illustrate the approach.

One-Switch Independence for Multiattribute Utility Functions

Assessment of multiattribute utility functions is significantly simplified if it is possible to decompose the function into more manageable pieces. Utility independence is a powerful property that serves well for this purpose, but if it is not appropriate in a given situation, what options does the analyst have? We review some possibilities and propose a new independence assumption based on the one-switch property. We argue that it is a natural generalization of utility independence and show how it leads to tractable multiattribute utility functions.

A Kullback-Leibler View of Linear and Log-Linear Pools

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.

Invariant Utility Functions and Certain Equivalent Transformations

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.

Entropy methods for adaptive utility elicitation

This paper presents an optimal question-selection algorithm to elicit von Neumann and Morgenstern utility values for a set of ordered prospects of a decision situation. The approach uses information theory and entropy-coding principles to select the minimum expected number of questions needed for utility elicitation. At each stage of the questionnaire, we use the question that will provide the largest reduction in the entropy of the joint distribution of the utility values. The algorithm uses questions that require binary responses, which are easier to provide than numeric values, and uses an adaptive question-selection scheme where each new question depends on the previous response obtained from the decision maker. We present a geometric interpretation for utility elicitation and work through a full example to illustrate the approach.

An Entropy Approach for Utility Assignment in Decision Analysis

A fundamental step in decision analysis is the elicitation of the decision‐maker’s preferences about the prospects of a decision situation in the form of utility values. However, this can be a difficult task to perform in practice as the number of prospects may be large, and eliciting a utility value for each prospect may be a time consuming and stressful task for the decision maker. To relieve some of the burden of this task, this paper presents a normative method to assign unbiased utility values when only incomplete preference information is available about the decision maker. We introduce the notion of a utility density function and propose a maximum entropy utility principle for utility assignment.