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.     

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).     

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.     

Testing the Intransitivity Explanation of the Allais Paradox

This paper uses a two-dimensional version of a standard common consequence experiment to test the intransitivity explanation of Allais-paradox-type violations of expected utility theory. We compare the common consequence effect of two choice problems differing only with respect to whether alternatives are statistically correlated or independent. We framed the experiment so that intransitive preferences could explain violating behavior when alternatives are independent, but not when they are correlated. We found the same pattern of violation in the two cases. This is evidence against intransitivity as an explanation of the Allais Paradox. The question whether violations of expected utility are mainly due to intransitivity or to violation of independence is important since it is exactly on this issue the main new decision theories differ.     

Violations of betweenness and choice shifts in groups

In decision theory, the betweenness axiom postulates that a decision maker who chooses an alternative A over another alternative B must also choose any probability mixture of A and B over B itself and can never choose a probability mixture of A and B over A itself. The betweenness axiom is a weaker version of the independence axiom of expected utility theory. Numerous empirical studies documented systematic violations of the betweenness axiom in revealed individual choice under uncertainty. This paper shows that these systematic violations can be linked to another behavioral regularity—choice shifts in a group decision making. Choice shifts are observed if an individual faces the same decision problem but makes a different choice when deciding alone and in a group.     

The independence condition in the theory of social choice

Arrow's theorem is really a theorem about the independence condition. In order to show the very crucial role that this condition plays, the theorem is proved in a refined version, where the use of the Pareto condition is almost avoided.A distinction is made between group preference functions and group decision functions, yielding respectively preference relations and optimal subsets as values. Arrow's theorem is about the first kind, but some ambiguities and mistakes in his book are explained if we assume that he was really thinking of decision functions. The trouble then is that it is not clear how to formulate the independence condition for decision functions. Therefore the next step is to analyse Arrow's argument for accepting the independence condition.The most frequent ambiguity depends on an interpretation of A as the set of all conceivable alternatives, while the variable subset B is the set of all feasible or available alternatives. He then argues that preferences between alternatives that are not feasible shall not influence the choice from the set of available alternatives. But even if this principle is accepted, it only forces us to require independence with respect to some specific set B and not to every B simultaneously. Therefore the independence condition cannot be accepted on these grounds.Another argument is about an election where one of the candidates dies. On one interpretation this argument can be taken to support an independence requirement which leads to a contradiction. On another interpretation it is a condition about connexions between choices from different sets.The so-called problem of binary choice is found to be different from the independence problem and it plays no essential role in Arrow's impossibility result. Other impossibility results by Sen, Batra and Pattanaik and by Schwartz are of a different character.In the last section, several weaker independence conditions are presented. Their relations to Arrow's condition are stated and the arguments supporting them are discussed.     

Hidden information acquisition and static choice

This note explores the consequence of hidden information acquisition for static choice theory. We show that any choice function in the observable problem can be consistent with some well-behaved choice function in a metaproblem with unobservable costly information acquisition. This illustrates how choices may not satisfy consistency conditions because a decision maker's decision process (in this case, information acquisition) depends on her feasible set. It also illustrates the importance of modeling the source of violations of consistency conditions, rather than simply weakening axioms on preferences.     

When Allais meets Ulysses: Dynamic axioms and the common ratio effect

We report experimental findings about subjects’ behavior in dynamic decision problems involving multistage lotteries with different timings of resolution of uncertainty. Our within-subject design allows us to study violations of the independence and dynamic axioms: Dynamic Consistency, Consequentialism and Reduction of Compound Lotteries. We investigate the effects of changes in probability and outcome levels on the pattern of choices observed in the Common Ratio Effect (CRE) and in the Reverse Common Ratio Effect (RCRE) and on their dynamic counterparts. We find that the probability level plays an important role in violations of Reduction of Compound Lottery and Dynamic Consistency and the outcomes levels in violations of Consequentialism. Moreover, more than one quarter of our subjects satisfy the Independence axiom but violate two dynamic axioms. We thus suggest that there is a greater dissociation that might have been expected between preferences captured by dynamic axioms and those observed over single-stage lotteries.     

An expected utility maximizer walks into a bar...

We conducted field experiments at a bar to test whether blood alcohol concentration (BAC) correlates with violations of the generalized axiom of revealed preference (GARP) and the independence axiom. We found that individuals with BACs well above the legal limit for driving adhere to GARP and independence at rates similar to those who are sober. This finding led to the fielding of a third experiment to explore how risk preferences might vary as a function of BAC. We found gender-specific effects: Men did not exhibit variations in risk preferences across BACs. In contrast, women were more risk averse than men at low BACs but exhibited increasing tolerance towards risks as BAC increased. Based on our estimates, men and women’s risk preferences are predicted to be identical at BACs nearly twice the legal limit for driving. We discuss the implications for policy-makers.     

Lottery Dependent Utility: a Reexamination

In order to accommodate empirically observed violations of the independence axiom of expected utility theory Becker and Sarin (1987) proposed their model of lottery dependent utility in which the utility of an outcome may depend on the lottery being evaluated. Although this dependence is intuitively very appealing and provides a simple functional form of the resulting decision criterion, lottery dependent utility has been nearly completely neglected in the recent literature on decision making under risk. The goal of this paper is to revive the lottery dependent utility model. Therefore, we derive first a sound axiomatic foundation of lottery dependent utility. Secondly, we develop a discontinuous variant of the model which can accommodate boundary effects and may lead to a lexicographic non-expected utility model. Both analyses are accompanied by considering some functional specifications which are in accordance with recent experimental results and may have significant applications in business and management science.     

Regret theory with general choice sets

The regret theory of choice under uncertainty proposed by Loomes and Sugden has performed well in explaining and predicting violations of Expected Utility theory. The original version of the model was confined to pairwise choices, which limited its usefulness as an economic theory of choice. Axioms for a more general form of regret theory have been proposed by Loomes and Sugden. In this article, it is shown that a simple nonmanipulability requirement is sufficient to characterize the functional form for regret theory with general choice sets. The stochastic dominance and comparative static properties of the model are outlined. A number of special cases are derived in which regret theory is equivalent to other well-known theories of choice under uncertainty.I would like to thank Graham Loomes, Robert Sugden, Peter Wakker, and Nancy Wallace for helpful comments and criticism.     

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.     

Rationality with preference discovery costs

Economic theory assumes that preferences are rational. However, experiments have found small violations of transitivity. This paper develops a model of rationality with preference discovery costs. Introspection is costly. Thus, agents may find it optimal to use less than full effort, even though this raises the risk of making a poor choice. This model could potentially explain the intransitivities observed in the data while retaining rationality and optimization.     

Testing between alternative models of choice under uncertainty: Some initial results

Experiments have identified a number of well-known violations of expected utility theory, giving rise to alternative models of choice under uncertainty, all of which are able to explain these violations. In this article, predictions of several prominent rival formulations are examined. No single alternative consistently organizes choices. Among the more important inconsistencies, we identify conditions generating systematic fanning in of indifference curves in the unit probability triangle, and find risk-loving over a number of gambles with all positive payoffs, in cases where prospect theory predicts risk aversion.     

Experimental payment protocols and the Bipolar Behaviorist

If someone claims that individuals behave as if they violate the independence axiom (IA) when making decisions over simple lotteries, it is invariably on the basis of experiments and theories that must assume the IA through the use of the random lottery incentive mechanism (RLIM). We refer to someone who holds this view as a Bipolar Behaviorist, exhibiting pessimism about the axiom when it comes to characterizing how individuals directly evaluate two lotteries in a binary choice task, but optimism about the axiom when it comes to characterizing how individuals evaluate multiple lotteries that make up the incentive structure for a multiple-task experiment. We reject the hypothesis about subject behavior underlying this stance: we find that preferences estimated with a model that assumes violations of the IA are significantly affected when one elicits choices with procedures that require the independence assumption, as compared to choices elicited with procedures that do not require the assumption. The upshot is that one cannot consistently estimate popular models that relax the IA using data from experiments that assume the validity of the RLIM.     

Security level,potential level,expected utility: A three-criteria decision model under risk

Among the violations of expected utility (E.U.) theory which have been observed by experimenters, the violations of its independence axiom is, by far, the most common. It seems that, in many cases, these inconsistencies can be ascribed to the desire for security - called the security factor by L. Lopes (1986) - which makes people attach special importance to the worst outcomes of risky decisions as well as to the sole outcomes of riskless decisions (certainty effect). J.-Y. Jaffray (1988) has proposed a model which generalizes E.U. theory by taking into account this factor and is then able to account for certain violations. However, especially in experiments on choice involving prospective losses, violations of the von Neumann-Morgenstern independence axiom cannot be explained by the security factor alone and have to be partially ascribed to the potential factor (L. Lopes, 1986) which reflects heightened attention to the best outcomes of decisions, especially when the best outcome is the status quo. In this paper, we construct an axiomatic model for subjects taking into account simultaneously or alternatively the security factor and the potential factor. For this, as in Jaffray's model, it has been necessary to weaken not only the standard independence axiom but also the continuity axiom and, in the same time, to reinforce the dominance axiom. In the resulting model, choices are partially determined by the mere comparison of the (security level, potential level) (i.e. the (worst outcome, best outcome)) pairs offered, and completed by the maximization of an affine function of the expected utility, the coefficients of which depend on both the security level and potential level.In this model, a decision maker who (i) has constant marginal utility for money, (ii) is sensitive to the security factor alone in the domain of gains, (iii) is sensitive to the potential factor alone in the domain of losses, behaves as a risk averter for gains and a risk seeker for losses.     

A “Pseudo-endowment” effect,and its implications for some recent nonexpected utility models

This article describes a modification of the Allais paradox that induces preferences inconsistent with two conditions weaker than the independence axiom, namely quasi-convexity (a special case of which is the betweenness axom), and Hypothesis II of Machina (also called fanning-out). These violations can be formally derived from prospect theory by invoking a nonliner transformation of probability into decision weight.I would like to thank David Bell, Vijay Krishna, John Pratt, and especially Colin Camerer for helpful comments and criticism.