Yutaka Nakamura

Subjective expected utility with non-additive probabilities on finite state spaces

Abstract Schmeidler and Gilboas representation generalizes subjective expected utility to cope with the Ellsberg paradox, so that the probability measure over states of the world need not be additive. This paper examines a similar generalization in Savages framework when the set of states is finite, while Savages states are continuously divisible. Our axiomatization requires that the set X of consequences be infinite in contrast to Savages arbitrary X . Three representational forms are axiomatized to give non-additivity, complementary additivity, and additivity of probability measures, respectively.

Decompositions of Multiattribute Utility Functions Based on Convex Dependence

We describe a method of assessing von Neumann-Morgenstern utility functions on a two-attribute space and its extension to n-attribute spaces. First, we introduce the concept of convex dependence between two attributes, where we consider the change of shapes of conditional utility functions. Then, we establish theorems which show how to decompose a two-attribute utility function using the concept of convex dependence. This concept covers a wide range of situations involving trade-offs. The convex decomposition includes as special cases Keeneys additive/multiplicative decompositions, Fishburns bilateral decomposition, and Bells decomposition under the interpolation independence. Moreover, the convex decomposition is an exact grid model which was axiomatized by Fishburn and Farquhar. Finally, we extend the convex decomposition theorem from two attributes to an arbitrary number of attributes.

Constant exchange risk properties

This paper develops a methodology using risk properties to characterize the functional form of a utility measure for decision making under uncertainty. The constant absolute risk property, for example, is known to be necessary and sufficient, with appropriate regularity conditions, for the utility function to have either a linear or an exponential form. A new generalization of this property, called the constant exchange risk property, gives a characterization of six utility functions: the linear function, the exponential function, the quadratic function, the sum of two exponential functions, the sum of a linear and an exponential function, and the product of a linear and an exponential function. Since all of these functional forms have been used previously as approximations, this methodology allows analysts to distinguish beforehand between alternative forms and thus properly specify the utility function in applications.

Rank dependent utility for arbitrary consequence spaces

Abstract Quiggins anticipated utility, sometimes called rank dependent utility, generalizes von Neumann and Morgensterns expected utility to accommodate Allais type violations of preference judgments. His theory and the subsequent axiomatic refinements presume that the underlying consequence spaces are rich, so that certainty equivalents of gambles exist. This paper develops an axiomatic characterization of rank dependent utility for arbitrary consequence spaces, so that certainty equivalents of gambles do not necessarily exist.

Multisymmetric structures and non-expected utility

Abstract This paper develops multisymmetric structures and their numerical representations, which generalize the bisymmetric structure in additive conjoint measurement. Then we discuss applications of those structures to axiomatic characterizations of non-expected utility theories.

Expected utility with an interval ordered structure

Abstract This paper examines an interval ordered structure under risky conditions and proves that it has an expected utility representation with a threshold function. In addition to the assumption of an interval order, two independence axioms and a strong Archimedean axiom are necessary and sufficient for the representation. The threshold is given by a nonnegative linear functional. We also explore a special structure which gives a nonnegative constant threshold function.

Sumex utility functions

Abstract This paper develops necessary and sufficient axioms for a decreasingly absolute risk averse von Neumann-Morgenstern utility function on the real line to be representable by a sumex functional form, i.e. the sum of a finite number of exponential or linear functions. Furthermore, we discuss constructive procedures that identify parameters in that form.

Bilinear utility and a threshold structure for nontransitive preferences

Abstract This paper develops a generalization of Fishburns SSB utility theory. Let P be a set of probability measures. A new representational form yields a bilinear functional u on P × P for which u ( p , q )> 0 if and only if p is preferred to q , where u ( p , p )≤0, and u ( q , p ) u ( p , q )> 0. Since u is not necessarily skew-symmetric, i.e., u ( p , q )≠− u ( q ,), it provides imprecise preference discriminability. It is also discussed that u is related to SSB utility with a threshold structure.

LEXICOGRAPHIC ADDITIVITY FOR MULTI-ATTRIBUTE PREFERENCES ON FINITE SETS

This paper explores lexicographically additive representations of multi-attribute preferences on finite sets. Lexicographic additivity combines a lexicographic feature with local value tradeoffs. Tradeoff structures are governed by either transitive or nontransitive additive conjoint measurement. Alternatives are locally traded off when they are close enough within threshold associated with a dominant subset of attributes.

Utility assessment procedures for polynomial-exponential functions

This article develops a methodology for testing constant exchange risk properties and identifying an appropriate form for a decision makers utility function. These risk properties characterize six different utility functions which are sums of products of polynomials and exponential functions. Such functional forms are commonly used in decision analysis applications. The practical advantage of this methodology is that these constant exchange risk properties eliminate the usual arbitrariness in the selection of a parametric utility function and often reduce the data requirements for subsequent estimation. The procedure is straightforward to apply. The decision maker need only provide certainty equivalents for two‐outcome gambles and determine the more‐preferred gamble in paired comparisons. The technical details of the procedure can be handled by interactive computer software.