The value of perfect information in nonlinear utility theory

Wakker (1988) has recently shown that, in contrast to an expected utility maximizer, the value of information will sometimes be negative for an agent who violates the independence axiom of expected utility theory. We demonstrate, however, that the value ofperfect information will always be nonnegative if the agent satisfies a weak dominance axiom. This result thus mitigates to some degree the normative objection to nonlinear utility theory implicit in Wakker's finding.     

Under stochastic dominance Choquet-expected utility and anticipated utility are identical

The aim of this paper is to convince the reader that Choquet-expected utility, as initiated by Schmeidler (1982, 1989) for decision making under uncertainty, when formulated for decision making under risk naturally leads to anticipated utility, as initiated by Quiggin/Yaari. Thus the two generalizations of expected utility in fact are one.     

On the nonexistence of Blackwell's theorem-type results with general preference relations

A well-known theorem of Blackwell states that, when quantity of information is properly defined, every expected utility decision maker prefers more information to less; for more general preferences, however, the theorem is no longer true. In this article, we investigate the extent to which Blackwell's Theorem does not hold and describe conditions, and situations, under which information is still valuable. We also show that, for many types of additions of information, there exists a decision maker who will reject this information.We thank Niv Ahituv, Larry Epstein, Uzi Segal, and an anonymous referee for their helpful comments. This article was partially financed by the Israel Institute of Business Research.     

Subjective expected utility theory revisited: A reductio ad absurdum paradox

A paradox is posed and analyzed in which people reverse their preferences for information on probabilities versus prizes once the range of the unknown probabilities is sufficiently narrowed. This reversal is shown to be incompatible with both objective expected utility (EU) as well as subjective versions in which the same probability transformation applies to all random variables. Experimental data are presented showing that the reversals occur with small, medium and large payoffs.The present paradox is compared with those of Allais and Ellsberg, and found to differ in substantive ways. It raises further questions about the normative status of expected utility theory, especially its treatment of probability and value. The paradox specifically calls into question EU's substitution and compound probability axioms.     

Prospective reference theory: Toward an explanation of the paradoxes

This article develops a variant of the expected utility model termed prospective reference theory. Although the standard model occurs as a limiting case, the general approach is that individuals treat stated experimental probabilities as imperfect information. This model is applied to a wide variety of aberrant phenomena, including the Allais paradox, the overweighting of low-probability events, the existence of premiums for certain elimination of risks, and the representativeness heuristic. The prospective reference theory model predicts most of the observed behavioral patterns rather than being potentially reconcilable with such phenomena.Kenneth Arrow and Robert Viscusi provided helpful comments. A preliminary version of this article was presented at the 1987 AEA meetings.     

Expected utility without utility

This paper advances an interpretation of Von Neumann-Morgenstern's expected utility model for preferences over lotteries which does not require the notion of a cardinal utility over prizes and can be phrased entirely in the language of probability. According to it, the expected utility of a lottery can be read as the probability that this lottery outperforms another given independent lottery. The implications of this interpretation for some topics and models in decision theory are considered.     

Preferences for information on probabilities versus prizes: The role of risk-taking attitudes

Many real-world decisions entail choices between information on either probabilities or payoffs (i.e., prizes). Simplified versions of such decisions are examined to gain insight into preferences for different types of information as a function of risk-attitudes. General and simple decision rules are derived for cases where the utility function is concave (or convex) over the relevant payoff interval.The article further describes several experiments to test business students' intuitions concerning these optimal decision rules. In general, risk-taking attitudes did not correlate significantly with subjects' preferences for information, in violation of theorems regarding mean-preserving spreads of risk. Other tests, e.g., narrowing certain probability ranges, also resulted in preferences contrary to expected utility (EU) theory.     

A Process Approach to the Utility for Gambling

This paper argues that any specific utility or disutility for gambling must be excluded from expected utility because such a theory is consequential while a pleasure or displeasure for gambling is a matter of process, not of consequences. A (dis)utility for gambling is modeled as a process utility which monotonically combines with expected utility restricted to consequences. This allows for a process (dis)utility for gambling to be revealed. As an illustration, the model shows how empirical observations in the Allais paradox can reveal a process disutility of gambling. A more general model of rational behavior combining processes and consequences is then proposed and discussed.     

Testing between alternative models of choice under uncertainty: Some initial results

Experiments have identified a number of well-known violations of expected utility theory, giving rise to alternative models of choice under uncertainty, all of which are able to explain these violations. In this article, predictions of several prominent rival formulations are examined. No single alternative consistently organizes choices. Among the more important inconsistencies, we identify conditions generating systematic fanning in of indifference curves in the unit probability triangle, and find risk-loving over a number of gambles with all positive payoffs, in cases where prospect theory predicts risk aversion.     

The comonotonic sure-thing principle

This article identifies the common characterizing property, the comonotonic sure-thing principle, that underlies the rank-dependent direction in non-expected utility. This property restricts Savage's sure-thing principle to comonotonic acts, and is characterized in full generality by means of a new functional form—cumulative utility—that generalizes the Choquet integral. Thus, a common generalization of all existing rank-dependent forms is obtained, including rank-dependent expected utility, Choquet expected utility, and cumulative prospect theory.     

An experimental evaluation of the descriptive validity of lottery-dependent utility theory

This article compares the performance of the expected utility (EU) and lottery-dependent expected utility (LDEU) models in predicting the actual choices of experimental subjects among risky options. In the process, we present two approaches for calibrating the LDEU model for an individual decision maker. The results indicate that while LDEU exhibits a higher potential for correctly predicting choice, the version of the model calibrated by indifference judgments does not outperform EU. We suggest a functional form for the parametric functions that defines the LDEU model, and discuss ways in which this function can be incorporated into choice-based assessment approaches to improve predictions.This research was supported in part by the Business Associates Fund at the Fuqua School of Business, Duke University.     

Local utility functions

In Machina's approach to generalised expected utility theory, decision makers maximise a choice functional which is smooth but not linear in the probabilities. When evaluating small changes, the choice functional can be approximated by the expectation of a local utility function. This local utility function is not however invariant under large changes in risk. This paper gives a simple explicit formula which can be used to write down the local utility functions of some common decision rules.     

Expected utility with lower probabilities

An uncertain and not just risky situation may be modeled using so-called belief functions assigning lower probabilities to subsets of outcomes. In this article we extend the von Neumann-Morgenstern expected utility theory from probability measures to belief functions. We use this theory to characterize uncertainty neutrality and different degrees of uncertainty aversion.We are grateful to Birgit Grodal, Salvatore Modica, David Schmeidler, and an anonymous referee for comments, help, and encouragement. Financial support from the Danish Social Sciences Research Council is acknowledged.     

Lottery Dependent Utility: a Reexamination

In order to accommodate empirically observed violations of the independence axiom of expected utility theory Becker and Sarin (1987) proposed their model of lottery dependent utility in which the utility of an outcome may depend on the lottery being evaluated. Although this dependence is intuitively very appealing and provides a simple functional form of the resulting decision criterion, lottery dependent utility has been nearly completely neglected in the recent literature on decision making under risk. The goal of this paper is to revive the lottery dependent utility model. Therefore, we derive first a sound axiomatic foundation of lottery dependent utility. Secondly, we develop a discontinuous variant of the model which can accommodate boundary effects and may lead to a lexicographic non-expected utility model. Both analyses are accompanied by considering some functional specifications which are in accordance with recent experimental results and may have significant applications in business and management science.     

Estimating the demand for risky assets via the indirect expected utility function

This article obtains demand functions for risky assets without making a priori assumptions about the form of the utility function. In a simple portfolio model, the envelope theorem is applied to the indirect expected utility function to derive estimating equations. Tests for the existence of constant absolute or constant relative risk aversion are also developed. Empirical estimation of the demand for financial assets held by U.S. households for the period 1946–1985 indicates that aggregate household behavior is consistent with the existence of constant relative risk aversion, with the coefficient of risk aversion having a value of approximately 1.3.The authors gratefully acknowledge helpful comments from the editor and an anonymous referee.     

An experimental test of several generalized utility theories

There is much evidence that people willingly violate expected utility theory when making choices. Several axiomatic theories have been proposed to explain some of this evidence, but there are few data that discriminate between the theories. To gather such data, an experiment was conducted using pairs of gambles with three levels of outcomes and many combinations of probabilities. Most typical findings were replicated, including the common consequence effect and different risk attitudes for gains and losses. There is evidence of both fanning out and fanning in of indifference curves, and both quasiconcavity and quasiconvexity of preferences. No theory can explain all the data, but prospect theory and the hypothesis that indifference curves fan out can explain most of them.The Wharton School,University of Pennsylvania     

Valuing Others’ Information under Imperfect Expectations

Sometimes we believe that others receive harmful information. However, Marschak’s value of information framework always assigns non-negative value under expected utility: it starts from the decision maker’s beliefs – and one can never anticipate information’s harmfulness for oneself. The impact of decision makers’ capabilities to process information and of their expectations remains hidden behind the individual and subjective perspective Marschak’s framework assumes. By introducing a second decision maker as a point of reference, this paper introduces a way for evaluating others’ information from a cross-individual, imperfect expectations perspective for agents maximising expected utility. We define the cross-value of information that can become negative – then the information is “harmful” from a cross-individual perspective – and we define (mutual) cost of limited information processing capabilities and imperfect expectations as an opportunity cost from this same point of reference. The simple relationship between these two expected utility-based concepts and Marschak’s framework is shown, and we discuss evaluating short-term reactions of stock market prices to new information as an important domain of valuing others’ information.      

Expected utility from additive utility on semigroups

In the present paper we study the framework of additive utility theory, obtaining new results derived from a concurrence of algebraic and topological techniques. Such techniques lean on the concept of a connected topological totally ordered semigroup. We achieve a general result concerning the existence of continuous and additive utility functions on completely preordered sets endowed with a binary operation ``+', not necessarily being commutative or associative. In the final part of the paper we get some applications to expected utility theory, and a representation theorem for a class of complete preorders on a quite general family of real mixture spaces.     

Some quantitative concepts of value and utility from a utilitarian point of view

This paper clarifies and interprets some basic quantitative concepts of value, utility and utility function from a utilitarian point of view. First, I discuss the question as to whether value is objective or subjective. I hold that value is subjective and statistical in nature (although from the various subjective values of a certain object a norm can usually be obtained). Second, I emphasize the distinction between use value and exchange value in relation to utility. Third, I propose a law of diminishing incremental interest, which refers to the incremental (marginal) utility of money. Fourth, I identify the utility of money with the von Neumann-Morgenstern utility. Fifth, I question the necessity of the usual normalization of utility functions and the restricted linear transformation (and the consequent concept of strategic equivalence). Sixth, I discuss in detail the terminal values and utilities of a utility function from a philosophical rather than mathematical point of view, particularly the boundedness of a utility function and the magnitudes of V ₀ and U ₀. Finally, I conclude that, in order to be able to have interpersonal comparisons of utility, utility should have the same dimension as value rather than no dimension, and the normalization problem should be reconsidered in the light of terminal values and utilities.