Ellsberg Paradox Research Articles

Is ambiguity aversion a preference? Ambiguity aversion without asymmetric information

Ambiguity aversion is the interpretation of the experimental finding (the Ellsberg paradox) that most subjects prefer betting on events whose probabilities are known (objective) to betting on events whose probabilities are unknown (subjective). However in typical experiments these unknown probabilities are known by others. Thus the typical Ellsberg experiment is a situation of asymmetric information. People may try to avoid situations where they are the less informed party, which is normatively appropriate. We find that eliminating asymmetric information in the Ellsberg experiment while leaving ambiguity in place, makes subjects prefer the ambiguous bet over the objective one, reversing the prior results.

Contingent Thinking and the Sure-Thing Principle: Revisiting Classic Anomalies in the Laboratory

Abstract We present an experimental framework to study the extent to which failures of contingent thinking explain classic anomalies in a broad class of environments, including overbidding in auctions and the Ellsberg paradox. We study environments in which the subject’s choices affect payoffs only in some states but not in others. We find that anomalies are in large part driven by incongruences between choices in the standard presentation of each problem and a “contingent” presentation, which focuses the subject on the set of states where her actions matter. Additional evidence suggests that this phenomenon is in large part driven by people’s failure to put themselves in states that have not yet happened even though they are made aware that their actions only matter in those states.

Eliciting second-order beliefs

We study elicitation of subjective beliefs of an agent facing ambiguity (model uncertainty): the agent has a non-singleton set of (first order) priors on an event and a second-order distribution on these priors. Such a two-stage decomposition of uncertainty and non-reduction of subjective compound lotteries resulting from non-neutrality to the second-order distribution plays an important role in resolving the Ellsberg Paradox. However, a key unanswered question is whether we can actually observe and separate the sets of first and second order subjective probabilities. We answer this question and show that it is indeed possible to pin the two sets down uniquely and therefore separate them meaningfully. We introduce prize variations and ensure that the tangent plane at any point on the surface of the certainty-equivalent function is reported truthfully. Any basis for this plane consists of derivatives in two different directions, and these combine first and second order beliefs in different ways. We show that this is enough to ensure that reporting both sets of beliefs truthfully is uniquely optimal. The technique requires knowledge of the changes in certainty equivalent of a mixture of acts as a result of variations in prizes, which we also elicit.

The transport problem for non-additive measures

Non-additive measures, also known as fuzzy measures, capacities, and monotonic games, are increasingly used in different fields. Applications have been built within computer science and artificial intelligence related to e.g., decision making, image processing, machine learning for both classification, and regression. Tools for measure identification have been built. In short, as non-additive measures are more general than additive ones (i.e., than probabilities), they have better modeling capabilities allowing to model situations and problems that cannot be modeled by the latter. See e.g., the application of non-additive measures and the Choquet integral to model both Ellsberg paradox and Allais paradox. Because of that, there is an increasing need to analyze non-additive measures.The optimal transport problem is a useful tool for probabilities. It is used to define the Wasserstein distance, and also to solve problems as data matching. In this work with tackle the problem of defining the optimal transport problem for non-additive measures. We consider that it is necessary to provide appropriate definitions with a flavour similar to the one for probabilities, but for non-additive measures. The solutions should generalize the classical ones.We provide definitions based on the Möbius transform, but also based on the (max,+)-transform that we consider that has some advantages. We will discuss in this paper the problems that arise to define the transport problem for non-additive measures, and provide ways to solve them. We include the definitions of the optimal transport problem, and prove some properties. We also conclude providing a definition of the Wasserstein distance for non-additive measures.

Counterfactual Priors: A Bayesian Response to Ellsberg's Paradox

Counterfactual Priors: A Bayesian Response to Ellsberg's Paradox

A Two-Ball Ellsberg Paradox

A Two-Ball Ellsberg Paradox

Development of ambiguity aversion from early adolescence to adulthood: New insights from the Ellsberg paradox

This research aimed to clarify whether middle adolescents’ risk-taking is driven by reduced ambiguity aversion. In Study 1, we explored the development of ambiguity aversion using an adaptation of the classic Ellsberg paradox with early adolescents (10–11 years old), middle adolescents (14–16 years old), and young adults (20–25 years old). Study 2 examined the development of ambiguity aversion depending on the ambiguity level in middle adolescence compared with adults. These two studies revealed that only early adolescents did not demonstrate ambiguity aversion. In contrast, middle adolescents and adults showed strong ambiguity aversion irrespective of ambiguity level. These findings support the idea that the period of young adolescence could be the start of ambiguity aversion development, although this tendency to avoid ambiguous options is already developed in middle adolescence. This finding might have important public health implications and suggests that prevention campaigns should consider early adolescence to be a particularly vulnerable age group for risky behaviors.

More Causes Less Effect: Destructive Interference in Decision Making.

We present a new experiment demonstrating destructive interference in customers’ estimates of conditional probabilities of product failure. We take the perspective of a manufacturer of consumer products and consider two situations of cause and effect. Whereas, individually, the effect of the causes is similar, it is observed that when combined, the two causes produce the opposite effect. Such negative interference of two or more product features may be exploited for better modeling of the cognitive processes taking place in customers’ minds. Doing so can enhance the likelihood that a manufacturer will be able to design a better product, or a feature within it. Quantum probability has been used to explain some commonly observed “non-classical” effects, such as the disjunction effect, question order effect, violation of the sure-thing principle, and the Machina and Ellsberg paradoxes. In this work, we present results from a survey on the impact of multiple observed symptoms on the drivability of a vehicle. The symptoms are assumed to be conditionally independent. We demonstrate that the response statistics cannot be directly explained using classical probability, but quantum formulation easily models it, as it allows for both positive and negative “interference” between events. Since quantum formalism also accounts for classical probability’s predictions, it serves as a richer paradigm for modeling decision making behavior in engineering design and behavioral economics.

Quantum Explanations for Interference Effects in Engineering Decision Making

<div class="section abstract"><div class="htmlview paragraph">Engineering practice routinely involves decision making under uncertainty. Much of this decision making entails reconciling multiple pieces of information to form a suitable model of uncertainty. As more information is collected, one expectedly makes better and better decisions. However, conditional probability assessments made by human decision makers, as new information arrives does not always follow expected trends and instead exhibits inconsistencies. Understanding them is necessary for a better modeling of the cognitive processes taking place in their mind, whether it be the designer or the end-user. Doing so can result in better products and product features. Quantum probability has been used in the literature to explain many commonly observed deviations from the classical probability such as: question order effect, response replicability effect, Machina and Ellsberg paradoxes and the effect of positive and negative interference between events. In this work, we present results from a survey demonstrating responses that while difficult to explain using classical probability, can be explained using a quantum formulation - highlighting its potential in engineering applications. Since quantum formulism is more general and can also match the predictions of classical probability, it serves as a richer paradigm for modeling decision making behavior in engineering practice.</div></div>

Eliciting Second-Order Beliefs

We study elicitation of subjective beliefs of an agent facing ambiguity (model uncertainty): the agent has a non-singleton set of (first-order) priors on an event and a second-order prior on these first-order belief-states. Such a two-stage decomposition of uncertainty and non-reduction of compound lotteries resulting from nonneutrality to the second-order distribution plays an important role in resolving the Ellsberg Paradox. The problem of eliciting beliefs on unobservable belief-states with ambiguity-sensitive agents is novel, and we introduce new elicitation techniques using report-dependent prize variations. We construct a direct revelation mechanism that induces truthful reporting of the first-order belief states as well as the secondorder distribution on the belief-states as the unique best response. The technique requires knowledge of the sensitivity function to second-order distribution (capturing ambiguity attitude) and the vN-M utility function, which we also elicit.

Can redescriptions of outcomes salvage the axioms of decision theory?

The basic axioms or formal conditions of decision theory, especially the ordering condition put on preferences and the axioms underlying the expected utility (EU) formula, are subject to a number of counter-examples, some of which can be endowed with normative value and thus fall within the ambit of a philosophical reflection on practical rationality. Against such counter-examples, a defensive strategy has been developed which consists in redescribing the outcomes of the available options in such a way that the threatened axioms or conditions continue to hold. We examine how this strategy performs in three major cases: Sen's counterexamples to the binariness property of preferences, the Allais paradox of EU theory under risk, and the Ellsberg paradox of EU theory under uncertainty. We find that the strategy typically proves to be lacking in several major respects, suffering from logical triviality, incompleteness, and theoretical insularity (i.e., being cut off from the methods and results of decision theory). To give the strategy more structure, philosophers have developed “principles of individuation”; but we observe that these do not address the aforementioned defects. Instead, we propose the method of checking whether the strategy can overcome its typical defects once it is given a proper theoretical expansion (i.e., it is duly developed using the available tools of decision theory). We find that the strategy passes the test imperfectly in Sen's case and not at all in Allais's. In Ellsberg's case, however, it comes close to meeting our requirement. But even the analysis of this more promising application suggests that the strategy ought to address the decision problem as a whole, rather than just the outcomes, and that it should extend its revision process to the very statements it is meant to protect. Thus, by and large, the same cautionary tale against redescription practices runs through the analysis of all three cases. A more general lesson, simply put, is that there is no easy way out from the paradoxes of decision theory.

LEONARD SAVAGE, THE ELLSBERG PARADOX, AND THE DEBATE ON SUBJECTIVE PROBABILITIES: EVIDENCE FROM THE ARCHIVES

This paper explores archival material concerning the reception of Leonard J. Savage’s foundational work of rational choice theory in its subjective-Bayesian form. The focus is on the criticism raised in the early 1960s by Daniel Ellsberg, William Fellner, and Cedric Smith, who were supporters of the newly developed subjective approach but could not understand Savage’s insistence on the strict version he shared with Bruno de Finetti. The episode is well known, thanks to the so-called Ellsberg Paradox and the extensive reference made to it in current decision theory. But Savage’s reaction to his critics has never been examined. Although Savage never really engaged with the issue in his published writings, the private exchange with Ellsberg and Fellner, and with de Finetti about how to deal with Smith, shows that Savage’s attention to the generalization advocated by his correspondents was substantive. In particular, Savage’s defense of the normative value of rational choice theory against counterexamples such as Ellsberg’s did not prevent him from admitting that he would give careful consideration to a more realistic axiomatic system, should the critics be able to provide one.

If It Is Surely Better, Do It More? Implications for Preferences Under Ambiguity

Consider a canonical problem in choice under uncertainty: choosing from a convex feasible set consisting of all (Anscombe–Aumann) mixtures of two acts f and g, [Formula: see text]. We propose a preference condition, monotonicity in optimal mixtures, which says that surely improving the act f (in the sense of weak dominance) makes the optimal weight(s) on f weakly higher. We use a stylized model of a sales agent reacting to incentives to illustrate the tight connection between monotonicity in optimal mixtures and a monotone comparative static of interest in applications. We then explore more generally the relation between this condition and preferences exhibiting ambiguity-sensitive behavior as in the classic Ellsberg paradoxes. We find that monotonicity in optimal mixtures and ambiguity aversion (even only local to an event) are incompatible for a large and popular class of ambiguity-sensitive preferences (the c-linearly biseparable class. This implies, for example, that maxmin expected utility preferences are consistent with monotonicity in optimal mixtures if and only if they are subjective expected utility preferences. This incompatibility is not between monotonicity in optimal mixtures and ambiguity aversion per se. For example, we show that smooth ambiguity preferences can satisfy both properties as long as they are not too ambiguity averse. Our most general result, applying to an extremely broad universe of preferences, shows a sense in which monotonicity in optimal mixtures places upper bounds on the intensity of ambiguity-averse behavior. This paper was accepted by Manel Baucells, decision analysis.

Representing Attitudes Towards Ambiguity in Hilbert Space: Foundations and Applications

We provide here a general mathematical framework to model attitudes towards ambiguity which uses the formalism of quantum theory as a “purely mathematical formalism, detached from any physical interpretation”. We show that the quantum-theoretic framework enables modelling of the Ellsberg paradox, but it also successfully applies to more concrete human decision-making tests involving financial, managerial and medical decisions. In particular, we elaborate a mathematical representation of various empirical studies which reveal that attitudes of managers towards uncertainty shift from ambiguity seeking to ambiguity aversion, and viceversa, thus exhibiting hope effects and fear effects. The present framework provides a promising direction towards the development of a unified theory of decisions in the presence of uncertainty.

Expected utility theory, Jeffrey’s decision theory, and the paradoxes

In Richard Bradley’s book, Decision Theory with a Human Face (2017), we have selected two themes for discussion. The first is the Bolker-Jeffrey (BJ) theory of decision, which the book uses throughout as a tool to reorganize the whole field of decision theory, and in particular to evaluate the extent to which expected utility (EU) theories may be normatively too demanding. The second theme is the redefinition strategy that can be used to defend EU theories against the Allais and Ellsberg paradoxes, a strategy that the book by and large endorses, and even develops in an original way concerning the Ellsberg paradox. We argue that the BJ theory is too specific to fulfil Bradley’s foundational project and that the redefinition strategy fails in both the Allais and Ellsberg cases. Although we share Bradley’s conclusion that EU theories do not state universal rationality requirements, we reach it not by a comparison with BJ theory, but by a comparison with the non-EU theories that the paradoxes have heuristically suggested.

Learning about the Ellsberg Paradox reduces, but does not abolish, ambiguity aversion.

Ambiguity aversion-the tendency to avoid options whose outcome probabilities are unknown-is a ubiquitous phenomenon. While in some cases ambiguity aversion is an adaptive strategy, in many situations it leads to suboptimal decisions, as illustrated by the famous Ellsberg Paradox. Behavioral interventions for reducing ambiguity aversion should therefore be of substantial practical value. Here we test a simple intervention, aimed at reducing ambiguity aversion in an experimental design, where aversion to ambiguity leads to reduced earnings. Participants made a series of choices between a reference lottery with a 50% chance of winning $5, and another lottery, which offered more money, but whose outcome probability was either lower than 50% (risky lottery) or not fully known (ambiguous lottery). Similar to previous studies, participants exhibited both risk and ambiguity aversion in their choices. They then went through one of three interventions. Two groups of participants learned about the Ellsberg Paradox and their own suboptimal choices, either by actively calculating the objective winning probability of the ambiguous lotteries, or by observing these calculations. A control group learned about base-rate neglect, which was irrelevant to the task. Following the intervention, participants again made a series of choices under risk and ambiguity. Participants who learned about the Ellsberg Paradox were more tolerant of ambiguity, yet ambiguity aversion was not completely abolished. At the same time, these participants also exhibited reduced aversion to risk, suggesting inappropriate generalization of learning to an irrelevant decision domain. Our results highlight the challenge for behavioral interventions: generating a strong, yet specific, behavioral change.

Can Redescriptions of Outcomes Salvage the Axioms of Decision Theory?

The basic axioms or formal conditions of decision theory, especially the ordering condition put on preferences and the axioms underlying the expected utility (EU) formula, are subject to a number of counter-examples, some of which can be endowed with normative value and thus fall within the ambit of a philosophical reflection on practical rationality. Against such counter-examples, a defensive strategy has been developed which consists in redescribing the outcomes of the available options in such a way that the threatened axioms or conditions continue to hold. We examine how this strategy performs in three major cases: Sen's counterexamples to the binariness property of preferences, the Allais paradox of EU theory under risk, and the Ellsberg paradox of EU theory under uncertainty. We find that the strategy typically proves to be lacking in several major respects, suffering from logical triviality, incompleteness, and theoretical insularity (i.e., being cut off from the methods and results of decision theory). To give the strategy more structure, philosophers have developed principles of individuation; but we observe that these do not address the aforementioned defects. Instead, we propose the method of checking whether the strategy can overcome its typical defects once it is given a proper theoretical expansion (i.e., it is duly developed using the available tools of decision theory). We find that the strategy passes the test imperfectly in Sen's case and not at all in Allais's. In Ellsberg's case, however, it comes close to meeting our requirement. But even the analysis of this more promising application suggests that the strategy ought to address the decision problem as a whole, rather than just the outcomes, and that it should extend its revision process to the very statements it is meant to protect. Thus, by and large, the same cautionary tale against redescription practices runs through the analysis of all three cases. A more general lesson, simply put, is that there is no easy way out from the paradoxes of decision theory.

Migration decisions under ambiguity: Migration distance effects revisited

AbstractThere is ample evidence that distance has a strong negative effect on migration. Despite its significance in migration decisions, scholars rarely explore the migration distance effect deeper than the first level of interpretation derived from the simple neoclassical theory of migration. This study revisits the migration distance effect in the spirit of Knightian distinction between risk and uncertainty. To this end, this study attempts to construct estimates of the risk premium migrants are willing to pay to avoid the risk associated with migration distance. The results show that the magnitude of the distance effect is not rationalized by risk aversion in the Arrow–Pratt sense (Arrow, 1965, The theory of risk aversion, Yrjo Jahnssonin Saatio, Helsinki, Finland, 90; Pratt, 1964, Econometrica, 32, 122). In particular, the risk premium demanded by distant rural residents is unjustifiably higher compared with that of those living closer to urban areas in their migration decisions. The results suggest that the migration distance effect is analogous to the equity home bias puzzle and the Ellsberg paradox in that the assumption of subjective expected utility can lead to seemingly irrational human behaviors. Some of the findings in this study shed light on the role of subjective aspects such as perception, confidence, and pessimism in migration decisions, which have been neglected in the literature.

Resolution of Behavioral Paradoxes in Dual Decision Theory

A novel approach to dual decision making is suggested. The basic notion is the behavioral probability of an event, which consists of two terms. One of the terms defines the probability of the event based on the rational estimation of utility. While the other term takes into account irrational processes in decision making. The sum of these terms is the behavioral probability taking into account the rational-irrational duality of decision making. Despite that the term characterizing irrational corrections is random, its non-informative prior can be estimated, which makes it possible to predict, without any fitting parameters, the average ratio of decision makers taking the corresponding decisions. It is shown that the well known paradoxes in behavioral decision making, such as Allais paradox, Ellsberg paradox, Kahneman-Tversky paradox, conjunction fallacy, disjunction effect, prisoner dilemma, and Ariely paradox, find natural explanations in the frame of the suggested dual decision theory.

Ellsberg's Hidden Paradox

Ellsberg (1961) proposes two alternative frames to elicit individuals' preferences for ambiguity. Through an experiment, we find that Ellsberg's three-color one-urn frame induces very different revealed preferences than the two-color two-urn frame. In both frames, we document ambiguity aversion for likely gains and (weak) ambiguity seeking for unlikely gains. The intensity of both these attitudes, however, is strongly curbed in the three-color frame. These findings may imply that perceived probabilities are biased toward uniformly distributed states and consequences, introducing a tension between them in the three-color frame. Surprisingly, we find that subjects who are simply asked to fill urns with colored bingo chips for a fixed payment also reveal a similar uniformity bias. We argue that both urn fillers and incentivized decision makers are influenced by similar perceptually salient features of probability intervals. Our findings have important implications for decision theory, and, by extension, for economic, financial, political, and medical decisions.