Ellsberg Paradox Research Articles

Decision Making in Phantom Spaces

The phantom space, with its phantom probability measure, allows the framing of ambiguous probabilities of events, as well as their vague consequences. This paper introduces a decision-making model under uncertainty, established on a phantom generalization of the von Neumann-Morgenstern axiomatization. Beliefs in our model are subjective variations of objective probabilities, recorded in a framework comprising not only risk but also phantom effects. Uncertainty measures are carried over naturally into this setting such that many of the familiar attributes of objective probabilities are preserved. The degree of uncertainty, which is determined by the available information and subjective beliefs of the decision maker, is distinguished from the attitude toward uncertainty, which is drawn from her preferences. Decision making under ambiguity is a special case of our model in which probabilities are vague but outcomes of events are clearly forecasted. The Ellsberg paradox and an insurance dilemma are the main examples we present.

Betting on Machina’s reflection example: an experiment on ambiguity

In a recent article, Machina (Am Econ Rev forthcoming, 2008) suggested choice problems in the spirit of Ellsberg (Q J Econ 75:643-669, 1961), which challenge tail-separability, an implication of Choquet expected utility (CEU), to a similar extent as the Ellsberg paradox challenged the sure-thing principle implied by subjective expected utility (SEU). We have tested choice behavior for bets on one of Machina's choice problems, the reflection example. Our results indicate that tail-separability is violated by a large majority of subjects (over 70% of the sample). These empirical findings complement the theoretical analysis of Machina (Am Econ Rev forthcoming, 2008) and, together, they confirm the need for new approaches in the analysis of ambiguity for decision making.

A theory of subjective compound lotteries

We develop a Savage-type model of choice under uncertainty in which agents identify uncertain prospects with subjective compound lotteries. Our theory permits issue preference; that is, agents may not be indifferent among gambles that yield the same probability distribution if they depend on different issues. Hence, we establish subjective foundations for the Anscombe–Aumann framework and other models with two different types of probabilities. We define second-order risk as risk that resolves in the first stage of the compound lottery and show that uncertainty aversion implies aversion to second-order risk which implies issue preference and behavior consistent with the Ellsberg paradox.

Risk, uncertainty and discrete choice models

This paper examines the cross-fertilizations of random utility models with the study of decision making under risk and uncertainty. We start with a description of the expected utility (EU) theory and then consider deviations from the standard EU frameworks, involving the Allais paradox and the Ellsberg paradox, inter alia. We then discuss how the resulting non-EU framework can be modeled and estimated within the framework of discrete choices in static and dynamic contexts. Our objectives in addressing risk and ambiguity in individual choice contexts are to understand the decision choice process and to use behavioral information for prediction, prescription, and policy analysis.

Cognition and Wealth: The Importance of Probabilistic Thinking

This paper utilizes a large set of subjective probability questions from the Health and Retirement Survey to construct an index measuring the precision of probabilistic beliefs (PPB) and relates this index to household choices about the riskiness of their portfolios and the rate of growth of their net worth. A theory of uncertainty aversion based on repeated sampling is proposed that resolves the Ellsberg Paradox within a conventional expected utility model. In this theory, uncertainty aversion is implied by risk aversion. This theory is then used to propose a link between an individual's degree of uncertainty and his propensity to give focal answers of 0, 50_50 or 100 or exact answers to survey questions and the validity of this interpretation is tested empirically. Finally, an index of the precision of probabilistic thinking is constructed by calculating the fraction of probability questions to which each HRS respondent gives a non-focal answer. This index is shown to have a statistically and economically significant positive effect on the fraction of risky assets in household portfolios and on the rate of growth of these assets longitudinally. These results suggest that there is systematic variation in the competence of individuals to manage investment accounts that should be considered in designing policies to create individual retirement accounts in the Social Security system.

Affective Decision Making and the Ellsberg Paradox

Affective decision-making is a strategic model of choice under risk and uncertainty where we posit two cognitive processes -- the rational and the emoitonal process. Observed choice is the result of equilibirum in this intrapersonal game. As an example, we present applications of affective decision-making in insurance markets, where the risk perceptions of consumers are endogenous. We derive the axiomatic foundation of affective decision making, and show that affective decision making is a model of ambiguity-seeking behavior consistent with the Ellsberg paradox.

Option Pricing Model with Fuzzy Measures under Knightian Uncertainty

Abstract Conventionally, the risk is described with a unique probability measure. However, Ellsberg paradox indicates that the existence of Knightian uncertainty would have an effect on both decision-makers' behavior and asset pricing. In this article, an option-pricing model is proposed under Knightian uncertainty using the λ-fuzzy measure and the Choquet integral, and the equilibrium price of European option on a non-dividend-paying stock is deduced. The equilibrium price is found to be an interval instead of a determinate number, which is in accordance with the conclusion of Epstein conclusion. Subsequent experimental research and the outcome indicate that the parameter λ which can describe human subjective sentimental will change with volatility of personal mood. Moreover, this study will pave a novel way to cope with other derivatives pricing under Knightian uncertainty.

Utility of gambling II: risk, paradoxes, and data

We specialize our results on entropy-modified representations of event-based gambles to representations of probability-based gambles by assuming an implicit event structure underlying the probabilities, and adding assumptions linking the qualitative properties of the former and the latter. Under segregation and under duplex decomposition, we obtain numerical representations consisting of a linear weighted utility term plus a term corresponding to information-theoretical entropies. These representations accommodate the Allais paradox and most of the data due to Birnbaum and associates. A representation of mixed event-and probability-based gambles accommodates the Ellsberg paradox. We suggest possible extensions to handle the data not accommodated.

On the relationship between Keynes’s conception of evidential weight and the Ellsberg paradox

A number of scholars have noted that Ellsberg’s seminal 1961QJE critique of the subjective expected utility model bears certain resemblances to ideas expressed in J. M. Keynes’s 1921 A Treatise on Probability. Ellsberg did not mention Keynes’s work in his article and referred instead to F. Knight’s distinction between ‘risk’ and ‘uncertainty’, thus inspiring a literature on various aspects of ‘Knightian uncertainty’. Nevertheless, the recent publication of Ellsberg’s PhD dissertation [Ellsberg, D. (2001). Risk, ambiguity and decision. New York: Garland Publishing], submitted to the University of Harvard in 1962, reveals that Ellsberg was actually aware of Keynes’s work. This gives rise to a number of interesting questions concerning the relation between the two authors’ works. The present paper, drawing in part on a conversation with Ellsberg, attempts to answer these questions. It turns out that the ‘mystery’ of why Ellsberg did not mention Keynes in his QJE article has a simple solution, namely that his dissertation was only completed after he had written the QJE article and that he had only come across Keynes after writing the QJE article. However, it is argued that although Ellsberg recognised the link between his notion of ambiguity and Keynes’s conception of the weight of argument in his PhD dissertation, he did not fully appreciate the fact that Keynes was more concerned with ‘practical’ rather than ‘conventionalised’ choice situations. To this extent, therefore, it is fair to say that ‘Knightian uncertainty’ is in many ways closer to the ideas expressed by Keynes than by Knight, and Keynes’s actual contribution to modern decision theory has been underestimated.

Cominimum additive operators

This paper proposes a class of weak additivity concepts for an operator on the set of real valued functions on a finite state space Ω , which include additivity and comonotonic additivity as extreme cases. Let E ⊆ 2 Ω be a collection of subsets of Ω . Two functions x and y on Ω are E -cominimum if, for each E ∈ E , the set of minimizers of x restricted on E and that of y have a common element. An operator I on the set of functions on Ω is E -cominimum additive if I ( x + y ) = I ( x ) + I ( y ) whenever x and y are E -cominimum. The main result characterizes homogeneous E -cominimum additive operators in terms of the Choquet integrals and the corresponding non-additive signed measures. As applications, this paper gives an alternative proof for the characterization of the E-capacity expected utility model of Eichberger and Kelsey [Eichberger, J., Kelsey, D., 1999. E-capacities and the Ellsberg paradox. Theory and Decision 46, 107–140] and that of the multiperiod decision model of Gilboa [Gilboa, I., 1989. Expectation and variation in multiperiod decisions. Econometrica 57, 1153–1169].

Rationality crossovers

Herein we further explore whether the power of arbitrage to induce people to exhibit more rational behavior extends to diverse decision-making tasks and stated valuation over preferences for gambles. We examine how arbitrage in a preference reversal setting affects behavior for the valuation of low probability food safety risks, the Allais Paradox, and the Ellsberg paradox. We design a three-stage experiment that elicits choices and values over gambles, with and without the experience of arbitrage. Our results suggest that a rationality crossover can exist – arbitrage in one setting can crossover to affect the choices in unrelated tasks. Stated values for safer food dropped by 20–50%, and the frequency of the Allais paradox is cut in half. People acted more rationally by reducing their stated value for a lottery, or if monetary adjustments are impossible they adjust their choice away from the lottery. Rationality crossovers have their predicted limits in that the frequency of the Ellsberg paradox, the most distinct decision environment, remained the same. We also found that the form of arbitrage as captured by stricter real marketlike experience or a weaker version of cheap-talk (i.e., hypothetical) arbitrage did not affect the results. This paper shows that arbitrage-induced rationality can transfer across contexts.

Distinguishing indeterminate belief from “risk-averse” preferences

I focus my discussion on the well-known Ellsberg paradox. I find good normative reasons for incorporating non-precise belief, as represented by sets of probabilities, in an Ellsberg decision model. This amounts to forgoing the completeness axiom of expected utility theory. Provided that probability sets are interpreted as genuinely indeterminate belief (as opposed to “imprecise” belief), such a model can moreover make the “Ellsberg choices” rationally permissible. Without some further element to the story, however, the model does not explain how an agent may come to have unique preferences for each of the Ellsberg options. Levi (1986, Hard choices: Decision making under unresolved conflict. Cambridge, New York: Cambridge University Press) holds that the extra element amounts to innocuous secondary “risk” or security considerations that are used to break ties when more than one option is rationally permissible. While I think a lexical choice rule of this kind is very plausible, I argue that it involves a greater break with xpected utility theory than mere violation of the ordering axiom.

Subjective probabilities on “small” domains

The Savagian choice-theoretic construction of subjective probability does not apply to preferences, like those in the Ellsberg Paradox, that reflect a distinction between risk and ambiguity. We formulate two representation results—one for expected utility, the other for probabilistic sophistication—that derive subjective probabilities but only on a “small” domain of risky events. Risky events can be either specified exogenously or in terms of choice behavior; in the latter case, both the values and the domain of probability are subjective. The analysis identifies a mathematical structure—called a mosaic—that is intuitive for both exogenous and behavioral specifications of risky events. This structure is weaker than an algebra or even a λ -system.

Updating Preferences with Multiple Priors

We propose and axiomatically characterize update rules for the preferences with multiple priors of Gilboa and Schmeidler (J. of Math. Econ., 18, pp. 141-153, 1989) for decision making under ambiguity. These rules are the first, for any model of preferences over acts, to be able to reconcile typical behavior in the face of ambiguity (as exemplified by Ellsberg's paradox) with dynamic consistency for all appropriately non-null events. Updating takes the form of applying Bayes' rule to subsets of the set of priors, where the specific subset depends on the preferences, the conditioning event, and the choice problem (i.e., a feasible set of acts together with an act chosen from that set).

Partially-Specified Probabilities: Decisions and Games

In Ellsberg paradox, decision makers that are partially informed about the actual probability distribution violate the expected utility paradigm. This paper develops a theory of decision making with a partially specified probability. The paper takes an axiomatic approach using Anscombe-Aumann's (1963) setting, and is based on a concave integral for capacities (see Lehrer, 2005). The partially-specified decision making is then carried on to games in order to introduce partially-specified equilibrium.

Do Trade Union Leaders Violate Subjective Expected Utility? Some Insights From Experimental Data

This paper presents the results of two experiments designed to test violations of Subjective Expected Utility Theory (SEUT) within a sample of Italian trade union delegates and leaders. Subjects priced risky and ambiguous prospects in the domain of gains. Risky prospects were based on games of chance, while ambiguous prospects were built on the standard Ellsberg paradox and on event lotteries whose outcomes were based either on the results of a fictional election or on the future results of the 1999 European Parliamentary election in Italy and the U.K. The experiments show that, although risky prospects were priced at their expected values on average, trade union delegates and leaders did violate SEUT when assessing ambiguous prospects. Moreover, their behaviour depended on the source of uncertainty (Ellsberg paradox vs. electoral results; fictional vs. real election; Italy vs. U.K. election outcomes). We discuss the implications of these results for the economic theory of the trade union.

A Theory of Belief for Scientific Refutations

A probability function \(\mathbb{P}\) on an algebra of events is assumed. Some of the events are scientific refutations in the sense that the assumption of their occurrence leads to a contradiction. It is shown that the scientific refutations form a a boolean sublattice in terms of the subset ordering. In general, the restriction of \(\mathbb{P}\) to the sublattice is not a probability function on the sublattice. It does, however, have many interesting properties. In particular, (i) it captures probabilistic ideas inherent in some legal procedures; and (ii) it is used to argue against the commonly held view that behavioral violations of certain basic conditions for qualitative probability are indicative of irrationality. Also discussed are (iii) the relationship between the formal development of scientific refutations presented here and intuitionistic logic, and (iv) an interpretation of a belief function used in the behavioral sciences to explain empirical results about subjective, probabilistic estimation, including the Ellsberg paradox.

A Bayesian Approach to Uncertainty Aversion

The Ellsberg paradox demonstrates that peoples belief over uncertain events might not be representable by subjective probability. We relate this paradox to other commonly observed anomalies, such as a rejection of the backward induction prediction in the one-shot Ultimatum Game. We argue that the pattern common to these observations is that the behavior is governed by rational rules. These rules have evolved and are optimal within the repeated and concurrent environments that people usually encounter. When an individual relies on these rules to analyze one-shot or single circumstances, paradoxes emerge. We show that when a risk averse individual has a Bayesian prior and uses a rule which is optimal for simultaneous and positively correlated ambiguous risks to evaluate a single vague circumstance, his behavior will exhibit uncertainty aversion. Thus, the behavior predicted by Ellsberg may be explained within the Bayesian expected utility paradigm.

Diminishing Impatience and Non-Expected Utility: A Unified Framework

This work suggests that diminishing impatience originates in the distinction between the certain present and the risky future. It models this difference using the metaphor of uncertain lifetime. This metaphor captures not only the real risk of mortality, which can never be assumed away, but also the implicit risk that a promise of future benefit may not be kept. Uncertain lifetime transforms any planned consumption path into a lottery. A simple functional representation of preferences over stochastic consumption paths is derived. It explicitly allows for time inconsistency when the future is risky, and can be used to study present bias, commitment devices and sophisticated behavior. This representation allows to relate diminishing impatience to well known behavioral regularities in the field of choice under risk and uncertainty. A simple parametric condition can account for the modal choice pattern observed in diminishing impatience, the Common Ratio Effect and the Ellsberg Paradox. Furthermore, it is shown that existing experimental evidence supports this work's hypothesis that diminishing impatience may be traced to the certainty of the present contrasted with the risk associated with any future path. The representation developed is applied to analyze the optimal rate of resource depletion when there exists uninsurable uncertainty that the unconsumed stock may disappear.

Probabilistic versus Non-probabilistic Decision Making: Savage, Shackle and Beyond

This paper discusses the evolution of decision theory after Savage's Foundations. Two developments are examined. First, it is presented the rationale of Shackle's proposal to abandon probabilistic decision making. Second, it is discussed the axiomatisation provided by the non-additive probability approach to account for the experimental evidence originated by the Ellsberg Paradox. An attempt is made to establish a connection between Shackle's non-probabilistic instances and non-additive probabilistic decision making. The main outgrowth of the paper is that the similarities between the non-additive approach and Shackle's theory are not limited to a number of methodological statements. In fact, it is also the formal measures used in the two contexts for representing individual preferences in uncertain environments that resemble each other.