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.     

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.     

Generalized similarity judgments: An alternative explanation for choice anomalies

Rubinstein's (1988) procedure for choosing between risky prospects, based, in part, upon similarities between prizes and probabilities across lotteries, is modified and extended to apply to a more general class of binary choices. This modified procedure is shown to imply behaviors following from Loomes and Sugden's (1982) Regret Theory, although under more general conditions, and provides an alternative explanation for much of the data which led to the specification of Prospect Theory's value and decision weighing functions. The procedure also explains observed violations of stochastic dominance, transitivity, and invariance not accounted for in available alternatives to expected utility.     

The inferred referendum — A rule for committee decisions

We present a new method of social choice. The result of our method coincides with that of majority voting when it does not produce an intransitivity among the alternatives under consideration. When majority voting would produce an intransitivity, our method orders the alternatives in the same way as the transitive constituency would whom the committee members are most likely to represent. Analysis of the application of our method to three alternatives shows that (a) the resulting order depends only on the committee members' votes between pairs of alternatives (b) the resulting order is less in conflict with Sen's Property than the orders provided by other schemes (c) when majority voting provides an intransitivity, the hypothesis that, in fact, the committee's constituency is as we assume it to be is almost as likely as the hypothesis that it precisely mirrors the committee.     

Testing transitivity in choice under risk

Recently proposed models of risky choice imply systematic violations of transitivity of preference. This study explored whether people show the predicted intransitivity of the two models proposed to account for the certainty effect in Allais paradoxes. In order to distinguish “true” violations from those produced by “error,” a model was fit in which each choice can have a different error rate and each person can have a different pattern of preferences that need not be transitive. Error rate for a choice is estimated from preference reversals between repeated presentations of the same choice. Results showed that few people repeated intransitive patterns. We can retain the hypothesis that all participants were transitive.     

Stochastic expected utility theory

This paper proposes a new decision theory of how individuals make random errors when they compute the expected utility of risky lotteries. When distorted by errors, the expected utility of a lottery never exceeds (falls below) the utility of the highest (lowest) outcome. This assumption implies that errors are likely to overvalue (undervalue) lotteries with expected utility close to the utility of the lowest (highest) outcome. Proposed theory explains many stylized empirical facts such as the fourfold pattern of risk attitudes, common consequence effect (Allais paradox), common ratio effect and violations of betweenness. Theory fits the data from ten well-known experimental studies at least as well as cumulative prospect theory.

Reconsidering the common ratio effect: the roles of compound independence,reduction, and coalescing

Common ratio effects should be ruled out if subjects’ preferences satisfy compound independence, reduction of compound lotteries, and coalescing. In other words, at least one of these axioms should be violated in order to generate a common ratio effect. Relying on a simple experiment, we investigate which failure of these axioms is concomitant with the empirical observation of common ratio effects. We observe that compound independence and reduction of compound lotteries hold, whereas coalescing is systematically violated. This result provides support for theories which explain the common ratio effect by violations of coalescing (i.e., configural weight theory) instead of violations of compound independence (i.e., rank-dependent utility or cumulative prospect theory).     

Ambiguity framed

In his exposition of subjective expected utility theory, Savage (1954) proposed that the Allais paradox could be reduced if it were recast into a format which made the appeal of the independence axiom of expected utility theory more transparent. Recent studies consistently find support for this prediction. We consider a salience-based choice model which explains this frame-dependence of the Allais paradox. We then derive the novel prediction that the presentation format responsible for reductions in Allais-style violations of expected utility theory will also reduce Ellsberg-style violations of subjective expected utility theory. This format makes the appeal of Savage’s “sure thing principle” more transparent. We design an experiment to test this prediction and find strong support for such frame-dependence of ambiguity aversion in Ellsberg-style choices. In particular, we observe markedly less ambiguity-averse behavior in Savage’s matrix format than in a more standard “prospect” format. This finding poses a new challenge for the leading models of ambiguity aversion.     

Neglecting Disaster: Why Don't People Insure Against Large Losses?

This paper provides a theoretical explanation for the common observation that people often fail to purchase insurance against low-probability high-loss events even when it is offered at favorable premiums. We hypothesize that individuals maximize expected utility but face an explicit or implicit cost to discovering the true probability of rare events. This cost constitutes a threshold that may inhibit purchase but may be offset in several ways by suppliers of insurers and state regulators.     

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.     

Does Learning Diminish Violations of Independence, Coalescing and Monotonicity?

Violations of expected utility theory are sometimes attributed to imprecise preferences interacting with a lack of learning opportunity in the experimental laboratory. This paper reports an experimental test of whether a learning opportunity which engenders accurate probability assessments, by enhancing understanding of the meaning of stated probability information, causes anomalous behaviour to diminish. The data show that whilst in some cases expected utility maximising behaviour increases with the learning opportunity, so too do systematic violations. Therefore, there should be no presumption that anomalous behaviour under risk is transient and that discovered preferences will be appropriately described by expected utility theory.     

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.     

Convex stochastic dominance with finite consequence sets

Stochastic dominance is a notion in expected-utility decision theory which has been developed to facilitate the analysis of risky or uncertain decision alternatives when the full form of the decision maker's von Neumann-Morgenstern utility function on the consequence space X is not completely specified. For example, if f and g are probability functions on X which correspond to two risky alternatives, then f first-degree stochastically dominates g if, for every consequence x in X, the chance of getting a consequence that is preferred to x is as great under f as under g. When this is true, the expected utility of f must be as great as the expected utility of g.Most work in stochastic dominance has been based on increasing utility functions on X with X an interval on the real line. The present paper, following [1], formulates appropriate notions of first-degree and second-degree stochastic dominance when X is an arbitrary finite set. The only structure imposed on X arises from the decision maker's preferences. It is shown how typical analyses with stochastic dominance can be enriched by applying the notion to convex combinations of probability functions. The potential applications of convex stochastic dominance include analyses of simple-majority voting on risky alternatives when voters have similar preference orders on the consequences.     

Testing independence conditions in the presence of errors and splitting effects

This paper presents experimental tests of several independence conditions implied by expected utility and alternative models. We perform repeated choice experiments and fit an error model that allows us to discriminate between true violations of independence and those that can be attributed to errors. In order to investigate the role of event splitting effects, we present each choice problem not only in coalesced form (as in many previous studies) but also in split forms. It turns out previously reported violations of independence and splitting effects remain significant even when controlling for errors. However, splitting effects have a substantial influence on tests of independence conditions. When choices are presented in canonical split form, in which probabilities on corresponding probability-consequence ranked branches are equal, violations of the properties tested could be reduced to insignificance or even reversed.     

Common Consequence Conditions in Decision Making under Risk

We generalize the Allais common consequence effect by describing three common consequence effect conditions and characterizing their implications for the probability weighting function in rank-dependent expected utility. The three conditions—horizontal, vertical, and diagonal shifts within the probability triangle—are necessary and sufficient for different curvature properties of the probability weighting function. The first two conditions, shifts in probability mass from the lowest to middle outcomes and middle to highest outcomes respectively, are alternative conditions for concavity and convexity of the weighting function. The third condition, decreasing Pratt-Arrow absolute concavity, is consistent with recently proposed weighting functions. The three conditions collectively characterize where indifference curves fan out and where they fan in. The common consequence conditions indicate that for nonlinear weighting functions in the context of rank-dependent expected utility, there must exist a region where indifference curves fan out in one direction and fan in the other direction.     

被访者驱动抽样：基于多种方法的估计诊断

本文利用RDS样本数据，使用RDS估计器、收敛图、瓶颈图、经纬度信息等，对违反“随机招募假设”的RDS估计进行综合诊断。诊断结果表明，适度违反“随机招募假设”并不会导致严重的RDS偏差，RDS估计仍然是有效的。因此，本文较为系统地介绍了多种诊断方法的实际操作及判断假设违反适度的可能标准，为国内研究者理解RDS方法，推动和发展RDS抽样和统计估计提供了思路。     

Estimating individual and group preference functionals using experimental data

In this paper, the empirical performance of several preference functionals is assessed using individual and group experimental data. We investigate if there is a risky choice theory that fits group decisions better than alternative theories, and if there are significant differences between individual and group choices. Experimental findings reported in this paper provide answers to both of those questions showing that expected utility gains a “winning” position over higher-level functionals (we considered disappoint aversion and two variants of rank-dependent utility) when risky choices are undertaken by individuals as well as by small groups. However, in the group experiment, alternatives (and, most notably, disappoint aversion) improve their relative performance, a fact that hints at the existence of differences between individual and group choices. We interpreted this result as evidence that feelings-like disappointment aversion become stronger in group decision.     

Explaining robust additive utility models by sequences of preference swaps

As decision-aiding tools become more popular everyday—but at the same time more sophisticated—it is of utmost importance to develop their explanatory capabilities. Some decisions require careful explanations, which can be challenging to provide when the underlying mathematical model is complex. This is the case when recommendations are based on incomplete expression of preferences, as the decision-aiding tool has to infer despite this scarcity of information. This step is key in the process but hardly intelligible for the user. The robust additive utility model is a necessary preference relation which makes minimal assumptions, at the price of handling a collection of compatible utility functions, virtually impossible to exhibit to the user. This strength for the model is a challenge for the explanation. In this paper, we come up with an explanation engine based on sequences of preference swaps, that is, pairwise comparison of alternatives. The intuition is to confront the decision maker with “elementary” comparisons, thus building incremental explanations. Elementary here means that alternatives compared may only differ on two criteria. Technically, our explanation engine exploits some properties of the necessary preference relation that we unveil in the paper. Equipped with this, we explore the issues of the existence and length of the resulting sequences. We show in particular that in the general case, no bound can be given on the length of explanations, but that in binary domains, the sequences remain short.