Testing between alternative models of choice under uncertainty—Comment

Battaglio, Kagel, and Jiranyakul use experimental tests to compare rank-dependent expected utility (RDEU), regret theory, prospect theory, and Machina's generalized smooth preferences model. They conclude that none of these models consistently organizes the data. The purpose of this note is to point out that RDEU theory was tested in combination with a hypothesis on the choice of functional form that has been explicitly rejected by the original author of the model (Quiggin, 1982, 1987). When the original form of RDEU theory is tested, it performs quite well.     

An empirical test of ordinal independence

In this article, we test Green and Jullien's (1988) Ordinal Independence (OI) Axiom, an axiom necessary for any rank-dependent expected utility (RDEU) model, including Cumulative Prospect Theory (Tversky and Kahneman, 1992). We observe systematic violations of OI (some within-subject violation rates of over 50%). These patterns of choice cannot be explained by any RDEU theory alone. We suggest that subjects are employing an editing operation prior to evaluation: if an outcome-probability pair is common to both gambles, it is cancelled when the commonality is transparent; otherwise, it is not cancelled. We interpret the results with respect to both original and cumulative prospect theory and the known empirical properties of the weighting function.     

Violations of the betweenness axiom and nonlinearity in probability

Betweenness is a weakened form of the independence axiom, stating that a probability mixture of two gambles should lie between them in preference. Betweenness is used in many generalizations of expected utility and in applications to game theory and macroeconomics. Experimental violations of betweenness are widespread. We rule out intransitivity as a source of violations and find that violations are less systematic when mixtures are presented in compound form (because the compound lottery reduction axiom fails empirically). We also fit data from nine studies using Gul's disappointment-aversion theory and two variants of EU, which weight separate or cumulative probabilities nonlinearly. The three theories add only one parameter to EU and fit much better.     

Decision analysis using lottery-dependent utility

In this article we show how the lottery-dependent expected utility (LDEU) model can be used in decision analysis. The LDEU model is an extension of the classical expected utility (EU) model and yet permits preference patterns that are infeasible in the EU model. We propose a framework for constructing decision trees in a particular way that permits us to use the principle of optimality and thus the divide and conquer strategy for analyzing complex problems using the LDEU model. Our approach may be applicable to some other nonlinear utility models as well. The result is that, if desired, decision analysis can be conducted without assuming the restrictive substitution principle/independence axiom.     

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.     

The Firm Under Uncertainty with General Risk-Averse Preferences: A State-Contingent Approach

This paper summarizes and synthesizes recent developments in the state-contingent theory of production under uncertainty presented by Chambers and Quiggin (2000) with a particular focus on the case of generalized expected utility preferences. The problem of the risk-averse firm under price and production uncertainty is analyzed using a state-contingent production technology and general risk-averse preferences. The concept of an efficient frontier, which identifies all potentially optimal production plans for weakly risk-averse decisionmakers, is introduced and used to develop comparative static results. For constant absolute risky technologies, the efficient frontier is shown to correspond to a unique isocost contour.     

Expected utility: An anniversary and a new era

During the past generation, expected utility theory has been widely accepted as the normative standard for decision making under risk and under uncertainty. However, it is now known that reasonable people often violate its assumptions, and a number of generalizations of the theory have been developed to accommodate some of the more common violations. This essay recalls the origins of expected utility in the early 1700s, notes its axiomatizations on the basis of preference comparisons in the mid-1900s, describes violations of those axioms uncovered since then, outlines new theories stimulated by the violations, and suggests where the field might be headed in the next few decades.     

Risk seeking with diminishing marginal utility in a non-expected utility model

The present work takes place in the framework of a non-expected utility model under risk: the RDEU theory (Rank Dependent Expected Utility, first initiated by Quiggin under the denomination of Anticipated Utility), where the decision maker's behavior is characterized by two functionsu andf. Our first result gives a condition under which the functionu characterizes the decision maker's attitude towards wealth. Then, defining a decision maker as risk averter (respectively risk seeker) when he always prefers to any random variable its expected value (weak definition of risk aversion), the second result states that a decision maker who has an increasing marginal utility of wealth (a convex functionu) can be risk averse, if his functionf issufficiently below his functionu, hence if he is sufficientlypessimistic. Obviously, he can also be risk seeking with a diminishing marginal utility of wealth. This result is noteworthy because with a stronger definition of risk aversion/risk seeking, based on mean-preserving spreads, Chew, Karni, and Safra have shown that the only way to be risk averse (in their sense) in RDEU theory is to have, simultaneously, a concave functionu and a convex functionf.     

Economic choice in generalized expected utility theory

Generalized expected utility models have enjoyed considerable success in explaining observed choices under uncertainty. However, there has been only limited progress in deriving comparative static results. This paper presents a general framework which permits the incorporation of a wide range of generalized expected utility models, but is sufficiently powerful to permit the derivation of comparative static results. The central idea is to represent preferences by the expected utility of a transformed probability distribution.     

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 note on deriving rank-dependent utility using additive joint receipts

Luce and Fishburn (1991) derived a general rank-dependent utility model using an operation ⊕ of joint receipt. Their argument rested on an empirically supported property (now) calledsegregation and on the assumption that utility is additive over ⊕. This note generalizes that conclusion to the case where utility need not be additive over ⊕, but rather is of a more general form, which they derived but did not use in their article. Tversky and Kahneman (1992), conjecturing that the joint receipt of two sums of money is simply their sum, criticized that original model because ⊕=+ together with additive utility implies the unacceptable conclusion that the utility of money is proportional to money. In the present generalized theory, if ⊕=+, utility is a negative exponential function of money rather than proportional. Similar results hold for losses. The case of mixed gains and losses is less well understood.     

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.     

Interval scalability of rank-dependent utility

Luce and Narens (Journal of Mathematical Psychology, 29:1–72, 1985) showed that rank-dependent utility (RDU) is the most general interval scale utility model for binary lotteries. It can be easily established that this result cannot be generalized to lotteries with more than two outcomes. This article suggests several additional conditions to ensure RDU as the only utility model with the desired property of interval scalability in the general case. The related axiomatizations of some special cases of RDU of independent interest (the quantile utility, expected utility, and Yaari’s dual expected utility) are also given.     

Moment characterization of higher-order risk preferences

This article presents a characterization of higher-order risk preferences such as prudence or temperance in terms of statistical moments. Our results, which are generalizations of Roger (Theory Decis, 70(1):27–44, 2011) and Ekern (Econ Lett, 6(4), 329–333, 1980), give a better understanding of how higher-order risk preferences relate to skewness preference and kurtosis aversion. While they are not based on expected utility theory, an implication within that theory is that all commonly used utility functions exhibit skewness preference and kurtosis aversion.     

A new test of convexity–concavity of discount function

Discounted utility theory and its generalizations (e.g., quasihyperbolic discounting, generalized hyperbolic discounting) use discount functions for weighting utilities of outcomes received in different time periods. We propose a new simple test of convexity–concavity of discount function. This test can be used with any utility function (which can be linear or not) and any preferences over risky lotteries (expected utility theory or not). The data from a controlled laboratory experiment show that about one third of experimental subjects reveal a concave discount function and another one third of subjects reveal a convex discount function (for delays up to two month).

     

Elementary proof that mean–variance implies quadratic utility

An extensive literature overlapping economics, statistical decision theory and finance, contrasts expected utility [EU] with the more recent framework of mean–variance (MV). A basic proposition is that MV follows from EU under the assumption of quadratic utility. A less recognized proposition, first raised by Markowitz, is that MV is fully justified under EU, if and only if utility is quadratic. The existing proof of this proposition relies on an assumption from EU, described here as “Buridan’s axiom” after the French philosopher’s fable of the ass that starved out of indifference between two bales of hay. To satisfy this axiom, MV must represent not only “pure” strategies, but also their probability mixtures, as points in the (σ, μ) plane. Markowitz and others have argued that probability mixtures are represented sufficiently by (σ, μ) only under quadratic utility, and hence that MV, interpreted as a mathematical re-expression of EU, implies quadratic utility. We prove a stronger form of this theorem, not involving or contradicting Buridan’s axiom, nor any more fundamental axiom of utility theory.