Revealed preference and linear utility

We study a Linear Axiom of Revealed Preference (LARP) that characterizes the consistency of a choice function with respect to a preference order satisfying the independence axiom. In addition, LARP characterizes lexicographic linear utility rationality when the choice space is a convex subset of a finite-dimensional real vector space, and LARP characterizes linear utility rationality when the choice space corresponds to a finite choice experiment.     

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.     

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.     

Axiomatization of a Preference for Most Probable Winner

In binary choice between discrete outcome lotteries, an individual may prefer lottery L₁ to lottery L₂ when the probability that L₁ delivers a better outcome than L₂ is higher than the probability that L₂ delivers a better outcome than L₁. Such a preference can be rationalized by three standard axioms (solvability, convexity and symmetry) and one less standard axiom (a fanning-in). A preference for the most probable winner can be represented by a skew-symmetric bilinear utility function. Such a utility function has the structure of a regret theory when lottery outcomes are perceived as ordinal and the assumption of regret aversion is replaced with a preference for a win. The empirical evidence supporting the proposed system of axioms is discussed.     

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.     

Reference Wealth Effects in Sequential Choice

It is argued that in order to accommodate experimentally-observed choice patterns, it is not enough to model the utility function as being dependent on changes from a reference wealth point. Instead, individuals should be modeled as treating decisions as part of an identifiable sequence of decisions, and utility should be a function of reference wealth, income so far from the sequence, and payoffs from the current decision. The three-argument utility function allows for risk aversion over gains and risk seeking over losses for the first choice in the sequence, and for the house money and break-even effects in later decisions.     

Evaluating Time Streams of Income: Discounting What?

For decisions whose consequences accrue over time, there are several possible techniques to compute total utility. One is to discount utilities of future consequences at some appropriate rate. The second is to discount per-period certainty equivalents. And the third is to compute net present values (NPVs) of various possible streams and to then apply the utility function to these net present values. We find that the best approach is to first compute NPVs of various possible income streams and then take the utility of such NPVs. We show the drawbacks of other alternative models of evaluating income streams. The article discusses the advantages of the power and logarithmic forms in the modeling of time preference. These are the only forms for which utility of income and utility of consumption are strategically equivalent. Further, these forms permit the flexibility in the choice of a time period (e.g., monthly or quarterly) without modifying the utility function, thus simplifying analysis.     

Ambiguity aversion in the small and in the large for weighted linear utility

The widely observed preference for lotteries involving precise rather than vague of ambiguous probabilities is called ambiguity aversion. Ambiguity aversion cannot be predicted or explained by conventional expected utility models. For the subjectively weighted linear utility (SWLU) model, we define both probability and payoff premiums for ambiguity, and introduce alocal ambiguity aversion function a(u) that is proportional to these ambiguity premiums for small uncertainties. We show that one individual's ambiguity premiums areglobally larger than another's if and only if hisa(u) function is everywhere larger. Ambiguity aversion has been observed to increase 1) when the mean probability of gain increases and 2) when the mean probability of loss decreases. We show that such behavior is equivalent toa(u) increasing in both the gain and loss domains. Increasing ambiguity aversion also explains the observed excess of sellers' over buyers' prices for insurance against an ambiguous probability of loss.     

Elicitation of Subjective Probabilities when the Initial Endowment is Unobservable

In the framework of subjective expected utility theory we develop two distinct procedures for the elicitation of a person's subjective probabilities when the initial endowment is random and unobservable. Procedures of the first kind rely on the boundedness of the utility function to elicit the person's subjective probabilities directly. Procedures of the second kind infer the person's initial endowment from his choice behavior prior to the application of probability elicitation.     

A Method for Eliciting Utilities and its Application to Collective Choice

Designing a mechanism that provides a direct incentive for an individual to report her utility function over several alternatives is a difficult task. A framework for such mechanism design is the following: an individual (a decision maker) is faced with an optimization problem (e.g., maximization of expected utility), and a mechanism designer observes the decision maker’s action. The mechanism does reveal the individual’s utility truthfully if the mechanism designer, having observed the decision maker’s action, infers the decision maker’s utilities over several alternatives. This paper studies an example of such a mechanism and discusses its application to the problem of optimal social choice. Under certain simplifying assumptions about individuals’ utility functions and about how voters choose their voting strategies, this mechanism selects the alternative that maximizes Harsanyi’s social utility function and is Pareto-efficient.     

Revealed Preference and Expected Utility

This essay gives necessary and sufficient conditions for recovering expected utility from choice behavior in several popular models of uncertainty. In particular, these techniques handle a finite state model; a model for which the choice space consists of probability densities and the expected utility representation requires bounded, measurable utility; and a model for which the choice space consists of Borel probability measures and the expected utility representation requires bounded, continuous utility. The key result is the identification of the continuity condition necessary for the revelation of linear utility.     

Testing the Effects of Similarity on Risky Choice: Implications for Violations of Expected Utility

Our aim in this paper was to establish an empirical evaluation for similarity effects modeled by Rubinstein; Azipurua et al.; Leland; and Sileo. These tests are conducted through a sensitivity analysis of two well-known examples of expected utility (EU) independence violations. We found that subjective similarity reported by respondents was explained very well by objective measures suggested in the similarity literature. The empirical results of this analysis also show that: (1) the likelihood of selection for the riskier choice increases as the pair becomes more similar, (2) these choice patterns are consistent with well-known independence violations of expected utility, and (3) a significant proportion of individuals exhibit intransitive choice patterns predicted under similarity effects, but not allowed under generalized expected utility models for risky choice.     

A Note on the Portfolio Selection Problem

In this note we provide new results of interest in the portfolio choice problem when the risky opportunities are correlated: for a general vector (X₁, X₂,..., X_n) of risky opportunities we give new conditions for stochastic comparison among different portfolios choices and new necessary and sufficient conditions to characterize the portfolio which gives the maximal expected utility.     

Subjective Probability Weighting and the Discovered Preference Hypothesis

Numerous studies have convincingly shown that prospect theory can better describe risky choice behavior than the classical expected utility model because it makes the plausible assumption that risk aversion is driven not only by the degree of sensitivity toward outcomes, but also by the degree of sensitivity toward probabilities. This article presents the results of an experiment aimed at testing whether agents become more sensitive toward probabilities over time when they repeatedly face similar decisions, receive feedback on the consequences of their decisions, and are given ample incentives to reflect on their decisions, as predicted by Plott’s Discovered Preference Hypothesis (DPH). The results of a laboratory experiment with N = 62 participants support this hypothesis. The elicited subjective probability weighting function converges significantly toward linearity when respondents are asked to make repeated choices and are given direct feedback after each choice. Such convergence to linearity is absent in an experimental treatment where respondents are asked to make repeated choices but do not experience the resolution of risk directly after each choice, as predicted by the DPH. I thank Peter P. Wakker for useful comments and suggestions.     

The value of information in anticipated utility theory

A well-known property of expected utility theory is that the value of information is nonnegative. Given the widespread dissatisfaction with the expected utility hypothesis, a natural question to ask is whether competing theories of choice preserve this property. This article considers one widely discussed alternative to expected utility, anticipated utility theory. We show that, like expected utility, the anticipated value of perfect information is always nonnegative. The value of imperfect information, however, may be negative, though the precise valuation of information depends upon whether the reduction of compound lotteries axiom is used to derive the anticipated utility functional.I am indebted to Edi Karni, Peter Wakker, and an anonymous referee for helpful comments on earlier versions of this article. They are of course in no way responsible for errors or obscurities in the present version.     

The impact of risk sharing on efficient decision

A group of risk-averse members must choose among monetary risks and payoff-sharing rules. Departure from the status quo requires unanimous consent. Such groups drill for oil, bail out nations, and make hostile takeover bids. Assume agreement on probabilities. As is well known, if all members have identically shaped HARA utility functions, efficient group act-choices follow another such function independently of payoff sharing. We show that all other groups inevitably have complex efficient behavior, accepting gambles among individually unacceptable lotteries in almost every status quo position. We also develop proper risk aversion for groups, and treat disagreement on probabilities.Support of the Associates of the Harvard Business School (Pratt) and the Business and Government and Energy and Environmental Policy Centers (Zeckhauser) is gratefully acknowledged. We received helpful comments from Scott Johnson, Mark Machina, and a referee. Jay Patel provided assistance beyond the call of collegiality.     

Eliciting beliefs

We develop an algorithm that can be used to approximate a decisionmaker’s beliefs for a class of preference structures that includes, among others, α-maximin expected utility preferences, Choquet expected utility preferences, and, more generally, constant additive preferences. For both exact and statistical approximation, we demonstrate convergence in an appropriate sense to the true belief structure.     

Expected Utility Consistent Extensions of Preferences

We consider the problem of extending a (complete) order over a set to its power set. The extension axioms we consider generate orderings over sets according to their expected utilities induced by some assignment of utilities over alternatives and probability distributions over sets. The model we propose gives a general and unified exposition of expected utility consistent extensions whilst it allows to emphasize various subtleties, the effects of which seem to be underestimated – particularly in the literature on strategy-proof social choice correspondences.