Ellsberg paradox: Ambiguity and complexity aversions compared

We present a simple model where preferences with complexity aversion, rather than ambiguity aversion, resolve the Ellsberg paradox. We test our theory using laboratory experiments where subjects choose among lotteries that “range” from a simple risky lottery, through risky but more complex lotteries, to one similar to Ellsberg’s ambiguity urn. Our model ranks lotteries according to their complexity and makes different—at times contrasting—predictions than most models of ambiguity in response to manipulations of prizes. The results support that complexity aversion preferences play an important and separate role from beliefs with ambiguity aversion in explaining behavior under uncertainty.     

An experimental test of prospect theory for predicting choice under ambiguity

Prospect theory is the most popular theory for predicting decisions under risk. This paper investigates its predictive power for decisions under ambiguity, using its specification through the source method. We find that it outperforms its most popular alternatives, including subjective expected utility, Choquet expected utility, and three multiple priors theories: maxmin expected utility, maxmax expected utility, and a-maxmin expected utility.     

Learning in the Allais paradox

Whereas both the Allais paradox, the first empirical challenge of the classical rationality assumptions, and learning have been the focus of many experimental investigations, no experimental study exists today into learning in the pure context of the Allais paradox. This paper presents such a study. We find that choices converge to expected utility maximization if subjects are given the opportunity to learn by both thought and experience, but less so when they learn by thought only. To the extent that genuine preferences should be measured with proper learning and incentives, our study gives the first pure demonstration that irrationalities such as in the Allais paradox are less pronounced than often thought. Paul Anand and Stefan Trautmann made helpful comments.     

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.

A simplified axiomatic approach to ambiguity aversion

This paper takes the Anscombe–Aumann framework with horse and roulette lotteries, and applies the Savage axioms to the horse lotteries and the von Neumann–Morgenstern axioms to the roulette lotteries. The resulting representation of preferences yields a subjective probability measure over states and two utility functions, one governing risk attitudes and one governing ambiguity attitudes. The model is able to accommodate the Ellsberg paradox and preferences for reductions in ambiguity.     

Reverse common ratio effect

The results of a new experimental study reveal highly systematic violations of expected utility theory. The pattern of these violations is exactly the opposite of the classical common ratio effect discovered by Allais (1953). Two recent decision theories—stochastic expected utility theory (Blavatskyy 2007) and perceived relative argument model (Loomes 2008)—predicted the existence of a reverse common ratio effect. However, these theories can rationalize only one part of the new experimental data reported in this paper. The other part appears to be neither predicted by existing theories nor documented in the existing empirical studies.     

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.     

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.     

Ambiguity aversion in the small and in the large for weighted linear utility

The widely observed preference for lotteries involving precise rather than vague of ambiguous probabilities is called ambiguity aversion. Ambiguity aversion cannot be predicted or explained by conventional expected utility models. For the subjectively weighted linear utility (SWLU) model, we define both probability and payoff premiums for ambiguity, and introduce alocal ambiguity aversion function a(u) that is proportional to these ambiguity premiums for small uncertainties. We show that one individual's ambiguity premiums areglobally larger than another's if and only if hisa(u) function is everywhere larger. Ambiguity aversion has been observed to increase 1) when the mean probability of gain increases and 2) when the mean probability of loss decreases. We show that such behavior is equivalent toa(u) increasing in both the gain and loss domains. Increasing ambiguity aversion also explains the observed excess of sellers' over buyers' prices for insurance against an ambiguous probability of loss.     

Subjectively weighted linear utility

An axiomatized theory of nonlinear utility and subjective probability is presented in which assessed probabilities are allowed to depend on the consequences associated with events. The representation includes the expected utility model as a special case, but can accommodate the Ellsberg paradox and other types of ambiguity sensitive behavior, while retaining familiar properties of subjective probability, such as additivity for disjoint events and multiplication of conditional probabilities. It is an extension, to the states model of decision making under uncertainty, of Chew's weighted linear utility representation for decision making under risk.     

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.     

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.     

Resolving Rabin’s paradox

We present a theoretical model of Rabin’s famous calibration paradox that resolves confusions in the literature and that makes it possible to identify the causes of the paradox. Using suitable experimental stimuli, we show that the paradox truly violates expected utility and that it is caused by reference dependence. Rabin already showed that utility curvature alone cannot explain his paradox. We, more strongly, do not find any contribution of utility curvature to the explanation of the paradox. We find no contribution of probability weighting either. We conclude that Rabin’s paradox underscores the importance of reference dependence.

     

The preservation of multivariate comparative statics in nonexpected utility theory

This article investigates the preservation of multivariate expected utility comparative statics for “smooth” nonexpected utility representations. Specifically, we answer the following question: if an expected utility comparative statics property depends only on preferences over sure prospects, then when will a nonexpected utility maximizer with identical sure preferences also satisfy that property? We demonstrate that the effects of increased risk aversion are preserved under the “Almost Degenerate Independence” axiom, but that those of distribution changes of exogenous risks are not preserved under stringent assumptions. Hence, nonexpected utility comparative statics may diverge from expected utility, even for “first-order” properties—those whose effect is determinable from restrictions on “local” utility functions.     

Aggregating imprecise or conflicting beliefs: An experimental investigation using modern ambiguity theories

Two experiments show that violations of expected utility due to ambiguity, found in general decision experiments, also affect belief aggregation. Hence we use modern ambiguity theories to analyze belief aggregation, thus obtaining more refined and empirically more valid results than traditional theories can provide. We can now confirm more reliably that conflicting (heterogeneous) beliefs where some agents express certainty are processed differently than informationally equivalent imprecise homogeneous beliefs. We can also investigate new phenomena related to ambiguity. For instance, agents who express certainty receive extra weight (a cognitive effect related to ambiguity-generated insensitivity) and generate extra preference value (source preference; a motivational effect related to ambiguity aversion). Hence, incentive compatible belief elicitations that prevent manipulation are especially warranted when agents express certainty. For multiple prior theories of ambiguity, our findings imply that the same prior probabilities can be treated differently in different contexts, suggesting an interest of corresponding generalizations.     

Capacity Updating Rules and Rational Belief Change

Choquet expected utility substitutes capacities for subjective probabilities to explain uncertainty aversion and related phenomena. This paper studies capacities as models of belief. The notions of inner and outer acceptance context are defined. These are shown to be the natural acceptance contexts when belief expansion is described by naïve Bayesian and Dempster–Shafer updating of capacities respectively. We also show that Eichberger and Kelsey's (1999b) use of Dempster–Shafer updating as a model of belief revision may lead to violations of the AGM axioms for rational belief change.     

The two faces of independence: betweenness and homotheticity

This paper shows that expected utility belongs to the intersection of models satisfying betweenness and a homotheticity condition for risky choice. Betweenness models can accommodate variable risk attitudes, originally highlighted by the Allais paradox, by restricting indifference curves to be linear while allowing non-parallelism. Homotheticity, in contrast, restricts indifference curves to be parallel while permitting non-linearities, such as those highlighted by inverse-S probability weighting. Data from an experiment indicate that approximately 2/3s of participants satisfied homotheticity. Of this group, about half satisfied the expected utility hypothesis. The other half violated expected utility and therefore betweenness. These results indicate that homothetic preferences, and in particular those violating betweenness, can be empirically relevant for decision making under risk.

     

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.