Causes of Allais common consequence paradoxes: An experimental dissection

The common consequence paradox of Allais can be decomposed into three simpler principles: transitivity, coalescing, and restricted branch independence. Different theories attribute such paradoxes to violations of restricted branch independence only, to coalescing only, or to both. This study separates tests of these two properties in order to compare these theories. Although rank-dependent utility (RDU) theories, including cumulative prospect theory (CPT), violate branch independence, the empirical pattern of violations is opposite that required by RDU theories to account for Allais paradoxes. Data also show systematic violations of coalescing, which refute RDU theories. The findings contradict both original and CPTs with or without their editing principles of combination and cancellation. Modal choices were well predicted by Birnbaum's RAM and TAX models with parameters estimated from previous data. The effects of event framing on these tests were also assessed and found to be negligible.     

Evidence against Rank-Dependent Utility Theories: Tests of Cumulative Independence,Interval Independence,Stochastic Dominance,and Transitivity

This study tests between two modern theories of decision making. Rank- and sign-dependent utility (RSDU) models, including cumulative prospect theory (CPT), imply stochastic dominance and two cumulative independence conditions. Configural weight models, with parameters estimated in previous research, predict systematic violations of these properties for certain choices. Experimental data systematically violate all three properties, contrary to RSDU but consistent with configural weight models. This study also tests whether violations of stochastic dominance can be explained by violations of transitivity. Violations of transitivity may be evidence of a dominance detecting mechanism. Although some transitivity violations were observed, most choice triads violated stochastic dominance without violating transitivity. Judged differences between gambles were not consistent with the CPT model. Data were not consistent with the editing principles of cancellation and combination. The main findings are interpreted in terms of coalescing, the principle that equal outcomes can be combined in a gamble by adding their probabilities. RSDU models imply coalescing but configural weight models violate it, allowing configural weighting to explain violations of stochastic dominance and cumulative independence.     

VIOLATIONS OF BRANCH INDEPENDENCE IN JUDGMENTS OF THE VALUE OF GAMBLES

Abstract —Branch independence is weaker than Savage's "sure thing" principle. It requires that judgments of gambles with a common outcome produced by the same probability-event must not reverse order when that common outcome is changed. Subjects judged 168 gambles from viewpoints of both buyer (highest buying price) and in both viewpoints. Violations also changed systematically between viewpoints, consistent with the theory that viewpoint affects configural weighting but not the utility function. Violations of branch independence were opposite those predicted by the model of cumulative prospect theory. The middle of three equally likely outcomes received the most weight in the seller's viewpoint. In the buyer's, lower outcomes received greater weights. In both viewpoints, the ratio of weights of the middle outcome to the highest outcome exceeded the ratio of weights of the lowest outcome to the middle outcome.     

Do people make decisions under risk based on ignorance? An empirical test of the priority heuristic against cumulative prospect theory

Brandstätter, Gigerenzer, and Hertwig [Brandstätter, E., Gigerenzer, G., & Hertwig, R. (2006). The Priority Heuristic: Making choices without trade-offs. Psychological Review, 113(2), 409–432] put forward the priority heuristic (PH) as a fast and frugal heuristic for decisions under risk. According to the PH, individuals do not make trade-offs between gains and probabilities, as proposed by expected utility models such as cumulative prospect theory (CPT), but use information in a non-compensatory manner and ignore information. We conducted three studies to test the PH empirically by analyzing individual choice patterns, decision times and information search parameters in diagnostic decision tasks. Results on all three dependent variables conflict with the predictions of the PH and can be better explained by the CPT. The predictive accuracy of the PH was high for decision tasks in which the predictions align with the predictions of the CPT but very low for decision tasks in which this was not the case. The findings indicate that earlier results supporting the PH might have been caused by the selection of decision tasks that were not diagnostic for the PH as compared to CPT.     

The exchangeable multinomial model as an approach to testing deterministic axioms of choice and measurement

The multinomial (Dirichlet) model, derived from de Finetti's concept of exchangeability, is proposed as a general Bayesian framework to test axioms on data, in particular, deterministic axioms characterizing theories of choice or measurement. For testing, the proposed framework does not require a deterministic axiom to be cast in a probabilistic form (e.g., casting deterministic transitivity as weak stochastic transitivity). The generality of this framework is demonstrated through empirical tests of 16 different axioms, including transitivity, consequence monotonicity, segregation, additivity of joint receipt, stochastic dominance, coalescing, restricted branch independence, double cancellation, triple cancellation, and the Thomsen condition. The model generalizes many previously proposed methods of axiom testing under measurement error, is analytically tractable, and provides a Bayesian framework for the random relation approach to probabilistic measurement (J. Math. Psychol. 40 (1996) 219). A hierarchical and nonparametric generalization of the model is discussed.     

Method effects due to social desirability as a parsimonious explanation of the deviation from unidimensionality in LOT-R scores

Dispositional optimism measured by the Life Orientation Test (Revised; LOT-R, Scheier et al., 1994) was originally proposed as a unidimensional construct pertaining to individual differences in positive outcome expectancies. However, factor analytic research has suggested two separate “optimism” and “pessimism” factors associated with positively and negatively worded items respectively. Using confirmatory factor analysis with data from N = 658 students it is shown that a general optimism model with method effects of the positively worded items fits even better than the “optimism–pessimism” model in terms of the AIC statistic, which accounts for both model parsimony and model fit. Furthermore, an attempt is made to explain the method effects by social desirability: impression management, one of two components of socially desirable responding, correlates significantly with the method factor whereas the other component, self-deceptive enhancement, is related to the general optimism factor. Thus it is shown that the deviation from unidimensionality of observed scores does not imply deviation from unidimensionality of optimism when method effects are incorporated in the model.     

Revisiting the Gain–Loss Separability Assumption in Prospect Theory

A fundamental assumption of prospect theory is gain–loss separability (GLS)—the assertion that the overall utility of a prospect can be expressed as a function of the utilities of its positive and negative components. Violations of GLS may potentially limit the generalization of results from studies of single‐domain prospects to mixed prospects and systematically distort the predictions of the theory. Violations also have implications for how choices with positive and negative components should be presented to decision makers. Previous studies, using different elicitation methods, have documented different rates, and types, of systematic violations of GLS. We discuss the differences between two specific elicitation methods—binary choice and certainty equivalents—and report results of a new study of GLS using both methods and randomly generated prospects. We compare the extent and nature of GLS violations under the two elicitation methods using between‐subject and within‐subject analyses. We find (i) systematic violations of GLS under both methods, (ii) higher rates of violations under choice, (iii) higher sensitivity to the outcomes for the certainty equivalents, which is consistent with the predictions of the scale‐compatibility hypothesis, and (iv) different patterns of violations under the two methods, which are explained by method‐specific preferences. We discuss the psychological mechanisms underlying the findings and the implications for presenting information with gain and loss components. Copyright © 2012 John Wiley & Sons, Ltd.     

Violations of Branch Independence in Choices between Gambles

Branch Independence is weaker than Savage's independence axiom; it holds that if two gambles have a common outcome for an event of known probability, the value of that common outcome should have no effect on the preference order induced by other probability-outcome branches. Systematic violations of branch independence were obtained in two experiments with choices between gambles composed of three equally likely, positive outcomes. Most people prefer ($2, $40, $44) over ($2, $10, $98); however, most people prefer ($10, $98, $136) over ($40, $44, $136). These results refute Expected Utility theories. They also refute the theory that people edit and cancel common components in choice. The pattern is opposite that predicted by the weighting function of cumulative prospect theory. Results are consistent with rank dependent, configural weight theory, with w_L> w_M> w_H, where w_L, w_M, and w_Hare the weights of the lowest, medium, and highest outcomes, respectively. In this theory, violations of branch independence depend on relations among weights: results indicate that w_L/w_M< w_M/w_H.     

Additivity and expected utility in risky multiattribute preferences

This experiment analyzed human preferences among even chance gambles for commodity bundles. The purpose of the experiment was to test several independence assumptions that distinguish between models for risky multiattribute preferences. In particular, the additivity and the expected utility part of the additive expected utility model were tested symmetrically. The degree and form of model violations were established, and the effects of instructions and of response modes were examined. All independence assumptions were violated by a bias to prefer a gamble or a commodity bundle that was previously matched against a standard. Systematic and strong violations that superseded this bias questioned the validity of the additive expected utility model. In violation of the additivity part of that model subjects consistently preferred the gamble with more balanced outcomes when comparing gambles with identical marginal probability distributions. This trend, called multiattribute risk aversion, was independent of subjects' single attribute risk attitude. Instructions and response modes had no noticeable impacts on these response patterns.     

Inspection time and everyday functioning: A longitudinal study

We tested whether level and/or changes in inspection time (IT) in an elderly sample (70–85 years at baseline) predicted future independence in everyday functioning, independent from chronological age. Sixty-eight participants were initially assessed on IT and cognitive abilities in 2003, reassessed on IT in 2005, and then completed the Everyday Problems Test, a test of independence in everyday living, in 2008. Controlling for age, lower fluid reasoning at baseline (Raven’s-subset) and slowing IT over 18 months independently predicted poorer everyday functioning in 2008, together accounting for one-third variance. This supports the proposition that IT has utility as a biomarker for less favourable ageing.     

Experimental test of the prospect theory value function: A stochastic dominance approach

According to prospect theory, individuals are risk averse regarding gains but risk seeking regarding losses, implying an S-shaped value function. The S-shaped value function hypothesis is based on experiments in which subjects are asked to choose separately between alternatives with either only positive or only negative outcomes, alternatives which rarely exist in the capital market. In addition, the S-shaped findings may be biased by the “certainty effect” and by probability distortion. In this paper we employ the recently developed prospect stochastic dominance criterion to test the prospect theory S-shaped value function hypothesis with mixed outcomes and with no “certainty effect.” Assuming that subjects do not distort moderate probabilities, we strongly reject the prospect theory S-shaped value function, with at least 76–86% of the choices being inconsistent with such preferences. When possible subjective probability distortions are taken into account, we find that at least 50–66% of the choices are inconsistent with an S-shaped value function.     

Cognitive models of risky choice: parameter stability and predictive accuracy of prospect theory

In the behavioral sciences, a popular approach to describe and predict behavior is cognitive modeling with adjustable parameters (i.e., which can be fitted to data). Modeling with adjustable parameters allows, among other things, measuring differences between people. At the same time, parameter estimation also bears the risk of overfitting. Are individual differences as measured by model parameters stable enough to improve the ability to predict behavior as compared to modeling without adjustable parameters? We examined this issue in cumulative prospect theory (CPT), arguably the most widely used framework to model decisions under risk. Specifically, we examined (a) the temporal stability of CPT’s parameters; and (b) how well different implementations of CPT, varying in the number of adjustable parameters, predict individual choice relative to models with no adjustable parameters (such as CPT with fixed parameters, expected value theory, and various heuristics). We presented participants with risky choice problems and fitted CPT to each individual’s choices in two separate sessions (which were 1 week apart). All parameters were correlated across time, in particular when using a simple implementation of CPT. CPT allowing for individual variability in parameter values predicted individual choice better than CPT with fixed parameters, expected value theory, and the heuristics. CPT’s parameters thus seem to pick up stable individual differences that need to be considered when predicting risky choice.     

Field articulation and working memory

This experiment assessed the proposal that performance differences associated with field independence/field articulation reflect differential efficiency in the use of a limited capacity information-processing system termed working memory. Extreme scores on a group-administered version of the embedded figures test were used to identify subjects “high” and “low” in degree of field articulation. The task required that a subject retain a series of digits for subsequent report while performing a semantic modification of a target phrase. Concurrent information load on working memory was varied in two ways: by increasing memory load from 3 to 6 digits, and by increasing the complexity of the semantic processing required. In low information load conditions no differences were found between high FA and low FA groups. In high information load conditions low FA subjects made more errors in digit recall and took longer to perform the difficult semantic modification.     

Subjective utility,anticipated utility,and the Allais paradox

One of the most notable counterexamples to expected utility theory is the “Allais paradox” (M. Allais, 1953, Econometrica, 31, 503–546). A number of alternative theories have been proposed in an attempt to resolve this paradox, notably including Karmarkar, 1978, Karmarkar, 1979, 24, 67–72). It is shown that SWU theory necessarily involves violations of dominance, but that the theory can be modified to avoid these violations. The result is a special case of J. Quiggin's anticipated utility theory (1982, Journal of Economic Behaviour and Organisation, 3, 323–343).