Probability Weighting in Choice under Risk: An Empirical Test

This paper reports a violation of rank-dependent utility with inverse S-shaped probability weighting for binary gambles. The paper starts with a violation of expected utility theory: one-stage gambles elicit systematically different utilities than theoretically equivalent two-stage gambles. This systematic disparity does not disappear, but becomes more pronounced after correction for inverse S-shaped probability weighting. The data are also inconsistent with configural weight theory and Machina's fanning out hypothesis. Possible explanations for the data are loss aversion and anchoring and insufficient adjustment.     

Expectations, Disappointment, and Rank-Dependent Probability Weighting

We develop a model of Disappointment in which disappointment and elation arise from comparing the outcome received, not with an expected value as in previous models, but rather with the other individual outcomes of the lottery. This approach may better reflect the way individuals are liable to experience disappointment. The model obtained accounts for classic behavioral deviations from the normative theory, offers a richer structure than previous disappointment models, and leads to a Rank-Dependent Utility formulation in a transparent way. Thus, our disappointment model may provide a clear psychological rationale for the subjective transformation of probabilities.     

Error Propagation in the Elicitation of Utility and Probability Weighting Functions

Elicitation methods in decision-making under risk allow us to infer the utilities of outcomes as well as the probability weights from the observed preferences of an individual. An optimally efficient elicitation method is proposed, which takes the inevitable distortion of preferences by random errors into account and minimizes the effect of such errors on the inferred utility and probability weighting functions. Under mild assumptions, the optimally efficient method for eliciting utilities and probability weights is the following three-stage procedure. First, a probability is elicited whose subjective weight is one half. Second, the utility function is elicited through the midpoint chaining certainty equivalent method using the probability elicited at the first stage. Finally, the probability weighting function is elicited through the probability equivalent method.     

Lottery-Dependent Utility via Stochastic Benchmarking

The possibility to interpret expected and nonexpected utility theories in purely probabilistic terms has been recently investigated. Such interpretation proposes as guideline for the Decision Maker the comparison of random variables through their probability to outperform a stochastic benchmark. We apply this type of analysis to the model of Becker and Sarin, showing that their utility functional may be seen as the probability that an opportune random variable, depending on the one to be evaluated, does not outperform a non-random benchmark. Further, the consequent choice criterion is equivalent to a sort of probability of ruin. Possible interpretations and financial examples are discussed.     

Lottery Decisions and Probability Weighting Function

This article analyzes the implications of basic lottery tests for the probability weighting function w(p). We first show that the w(p) function with one argument cannot accommodate three basic tests of lottery choice. We also discuss in detail the links between the w(p) function with one argument and the preference reversal paradox. In the last section we propose an evaluation function that accommodates the restrictions imposed by the tests and we provide an example of such function.     

Dynamic Choice and NonExpected Utility

This paper explores how some widely studied classes of nonexpected utility models could be used in dynamic choice situations. A new "sequential consistency" condition is introduced for single-stage and multi-stage decision problems. Sequential consistency requires that if a decision maker has committed to a family of models (e.g., the multiple priors family, the rank-dependent family, or the betweenness family) then he use the same family throughout. Conditions are presented under which dynamic consistency, consequentialism, and sequential consistency can be simultaneously preserved for a nonexpected utility maximizer. An important class of applications concerns cases where the exact sequence of decisions and events, and thus the dynamic structure of the decision problem, is relevant to the decision maker. It is shown that for the multiple priors model, dynamic consistency, consequentialism, and sequential consistency can all be preserved. The result removes the argument that nonexpected utility models cannot be consistently used in dynamic choice situations. Rank-dependent and betweenness models can only be used in a restrictive manner, where deviation from expected utility is allowed in at most one stage.     

Is Probability Weighting Sensitive to the Magnitude of Consequences? An Experimental Investigation on Losses

Experimental investigations of the probability weighting function over losses are scarce and all involve small payoffs. The paper aims to give new insight into the probability weighting function for losses, by eliciting it through a simple two-stage semi-parametric procedure over more realistic losses, and by investigating its sensitivity to the magnitude of the payoffs. Current data confirm previous evidence of convex utility functions and inverse-S-shaped weighting functions. Still, at least for small probabilities, probability weighting appears to be affected by the size of consequences: the larger the losses, the more aversive the gambles and the more pessimistic the subjects are.     

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     

Local Utility Functions and Local Probability Transformations

This note shows that, under appropriate conditions, preferences may be locally approximated by the linear utility or risk-neutral preference functional associated with a local probability transformation.     

Lifetime Uncertainty and Time Preference

Despite Fisher's (1930) psychological intuitions of and the formal treatment given by Yaari (1965, Review of Economic Studies 32, 137), the intertemporal model of choice is mainly a model with certain lifetime. The purpose of this paper is to reconsider this assumption, starting from a very simple two-period model of choice with lifetime uncertainty. We examine the comparative statics of the model at the first two orders and replace the concept of `pure time preference' by taking into account the subjective treatment of the probability of survival. This revised version was published online in June 2006 with corrections to the Cover Date.     

The Becker-DeGroot-Marschak mechanism and generalized utility theories: Theoretical predictions and empirical observations

Karni and Safra [8] prove that the Becker-DeGroot-Marschak mechanism reveals a decision maker's true certainty equivalent of a lottery if and only if he satisfies the independence axiom. Segal [17] claims that this mechanism may reveal a violation of the reduction of compound lotteries axiom. This paper empirically tests these two interpretations. Our results show that the second interpretation fits better with the collected data. Moreover, we show by means of some nonexpected utility examples that these results are consistent with a wide range of functionals.     

The Becker-DeGroot-Marschak mechanism and nonexpected utility: A testable approach

The Becker-DeGroot-Marschak mechanism is widely used to elicit decision makers' selling prices of lotteries. This mechanism leads, however, to the preference reversal phenomenon, which seemed to indicate nontransitive preferences. To solve this puzzle, Karni and Safra (1987) introduced a new interpretation of this mechanism based on two-stage lotteries without the independence axiom. In this article, we suggest a set of empirically testable hypotheses based on their interpretation of the mechanism. One of these tests can be used to find the utility and the probability transformation functions of an anticipated utility maximizer.This article consists of an earlier paper with a similar title (University of Toronto WP #8809) and the paper Elicitation of Certainty Equivalents and the Becker-DeGroot-Marschak Mechanism.     

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.     

Empirical evidence of two-attribute utility dependence on probability

We investigate utility dependence on probability using a new methodology that examines how indifference statements vary with the probability of obtaining times and costs of individual trips. Of 127 subjects, 8 supplied 3 (out of 3) sets of indifference statements consistent with probability independence. Those subjects with 2 or more sets of indifference statements violating probability independence exhibited a systematic dependence, in that knowing the direction of a subject's violation in one set of indifference statements would increase the likelihood of his or her violating other sets of indifference statements in the same direction. Data show that this systematic violation of dependence should not be attributed to artifacts of the experiment.     

Recent developments in modeling preferences: Uncertainty and ambiguity

In subjective expected utility (SEU), the decision weights people attach to events are their beliefs about the likelihood of events. Much empirical evidence, inspired by Ellsberg (1961) and others, shows that people prefer to bet on events they know more about, even when their beliefs are held constant. (They are averse to ambiguity, or uncertainty about probability.) We review evidence, recent theoretical explanations, and applications of research on ambiguity and SEU.Thanks to Jonathan Baron, James Dow, Peter Fishburn, Itzhak Gilboa, Gordon Hazen, Howard Kunreuther, Tomas Phillipson, David Schmeidler, Amos Tversky, the editor, and several anonymous referees for corrections and helpful comments. Camerer's contribution to this work was supported by the National Science Foundation, grant no. SES 88-09299. Weber's contribution was supported by the Deutsche Forschungsge-meinschaft, grant no. WE 993/5-1.     

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.     

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.     

An Analysis of Welfare Participation: Rational-Choice Perspective and Group-Threat Hypothesis

ABSTRACT

This secondary data analysis examined the impacts of human capital, child care, fringe benefits, child support, ethnicity, county economy, and county minority-population size on participation in Temporary Assistance for Needy Families (TANF) by parents living at or below the federal poverty threshold. Longitudinal records of 1,789 such parents were extracted from the National Longitudinal Survey of Youth (1996–2008). Multilevel mixed-effects linear regression showed that having personal/family problems, being African American, and residing in a county with widespread unemployment were associated positively with TANF receipt. TANF receipt was associated negatively with employment offering a retirement plan, full-time employment, part-time employment, and residence in a county with a large Hispanic population. Future research might explore the relationship between subpopulations’ sizes and states’ TANF policies.     

Testing Prospect Theories Using Probability Tradeoff Consistency

The two versions of prospect theory, original prospect theory (OPT; Kahneman and Tversky, 1979) and cumulative prospect theory (CPT; Tversky and Kahneman, 1992), use different composition rules to combine the value function and the probability weighting function and hence value gambles with two or more non-zero outcomes differently. Previous tests of OPT and CPT have yielded mixed results, with some investigations supporting OPT and some supporting CPT. We extend the probability tradeoff consistency axiom used in Abdellaoui (2002) for CPT to OPT, and develop a critical test of the two prospect theories based on their respective probability tradeoff consistency conditions. An empirical investigation of the critical test shows that choices are consistent with OPT, but not CPT, for gambles that do not involve a certainty effect, and consistent with both CPT and OPT for gambles that do involve a certainty effect, provided that an editing operation is invoked for OPT.     

Utility theory with probability dependent outcome valuation: Extensions and applications

This article extends Karni's (1992) Utility Theory with Probability Dependent Outcome Valuation (UTP-DOV) to accommodate a wider set of preferences, and applies the new representation to a variety of decision problems under risk. First, we present a new, more general, axiomatization of UTPDOV that renders the underlying intuition more transparent. Second, we develop sufficient conditions under which UTPDOV satisfies the betweenness and the ordinal independence axioms. Finally, we examine functional forms of the UTPDOV model that are consistent with several well known departures from the independence axiom.