Ambiguous Beliefs Research Articles

Vigilant Measures of Risk and the Demand for Contingent Claims

I examine a class of utility maximization problems with a not necessarily law-invariant utility, and with a not necessarily law-invariant risk measure constraint. The objective function is a Lebesgue integral of some function U with respect to some probability measure P, and the constraint set contains some risk measure constraint which is not necessarily P-law-invariant. This introduces some heterogeneity in the perception of uncertainty. The primitive U is a function of some given underlying random variable X and of a contingent claim Y on X. Many problems in economic theory and financial theory can be formulated in this manner, when a heterogeneity in the perception of uncertainty is introduced. Under a consistency requirement on the risk measure that I call Vigilance, supermodularity of the primitive U is sufficient for the existence of optimal contingent claims, and for these optimal claims to be comonotonic with the underlying random variable X. Vigilance is satisfied by a large class of risk measures, including all distortion risk measures. In the latter case, I give a quantile characterization of optimal solutions, which can be useful in many applications. As an illustration, I consider a problem of demand for insurance indemnity schedules, where the insurer has ambiguous beliefs about the realizations of the insurable loss and is ambiguity-averse in the sense of Schmeidler. In this case, I give an explicit characterization of the optimal indemnity schedule for the insured and I show how the result naturally extends the classical result of Arrow on the optimality of the deductible indemnity schedule. Arrow’s result is then obtained as a special case.

Ambiguous beliefs and mechanism design

This paper develops a payoff equivalence theorem for mechanisms with ambiguity averse participants with preferences of the Maxmin Expected Utility (MEU) form (Gilboa and Schmeidler, 1989). We use our payoff equivalence result to explicitly characterize the revenue maximizing private value auction mechanism for agents with arbitrary forms of ambiguous beliefs. We also show that the revenue ranking between first and second price auctions is sensitive to the form of ambiguity aversion. Our payoff equivalence techniques allow us to study the constrained efficient, budget balanced bilateral trade mechanism and show that increased ambiguity improves the efficiency of the mechanism. In addition, we characterize the revenue maximizing, efficient bilateral trade mechanism and show that heightened ambiguity lowers ex ante budget deficits.

Ambiguity aversion, higher-order risk attitude and optimal effort

In this paper, we examine whether a more ambiguity-averse individual will invest in more effort to shift her initial starting wealth distribution toward a better target distribution. We assume that the individual has ambiguous beliefs regarding two target (starting) distributions and that one distribution is preferred to the other. We find that an increase in ambiguity aversion will decrease (increase) the optimal effort when the cost of effort is non-monetary. When the cost of effort is monetary, the effect depends on whether the individual would make more effort when the target (starting) distribution is the preferred distribution than the target (starting) distributions, the inferior one. We further characterize the individual’s higher-order risk preferences to examine the sufficient conditions.

Beliefs correspondences and equilibria in ambiguous games

The Nash equilibrium concept combines two fundamental ideas. First, rational players choose the most preferred strategy given their beliefs about what other players will do. Second, it imposes the consistency condition that all players' beliefs are correct. This consistency condition has often been considered too strong and different solution concepts have been introduced in the literature in order to take into account ambiguous beliefs. In this paper, we show, by means of examples, that in some situation beliefs might be dependent on the strategy profile and that this kind of contingent ambiguity affects equilibrium behavior differently with respect to the existing models of ambiguous games. Hence we consider a multiple prior approach and subjective beliefs correspondences which depend on the strategy profile; we investigate existence of the equilibrium concepts corresponding to different attitudes towards ambiguity (namely optimism and pessimism). Finally we analyze particular beliefs correspondences: beliefs given by correlated equilibria and by ambiguity levels on events.

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About the Authors

Second-order ambiguous beliefs

This paper axiomatizes models of second-order ambiguous beliefs in the original domain of preferences of Anscombe and Aumann (Ann Math Stat 34:199–205, 1963) by weakening the first-stage independence postulate. The models we propose include the second-order subjective expected utility (SOSEU) of Seo (Econometrica 77:1575–1605, 2009) as a particular case. We characterize the intersection of our models of second-order ambiguity with the canonical models of (first-order) ambiguity aversion and provide a further generalization of SOSEU by relaxing the completeness axiom.

Dynamically consistent updating of multiple prior beliefs – An algorithmic approach

This paper develops algorithms for dynamically consistent updating of ambiguous beliefs in the maxmin expected utility model of decision making under ambiguity. Dynamic consistency is the requirement that ex-ante contingent choices are respected by updated preferences. Such updating, in this context, implies dependence on the feasible set of payoff vectors available in the problem and/or on an ex-ante optimal act for the problem. Despite this complication, the algorithms are formulated concisely and are easy to implement, thus making dynamically consistent updating operational in the presence of ambiguity.

Ambiguity attitude, R&D investments and economic growth

The process aimed at discovering new ideas is an economic activity the returns from which are intrinsically uncertain. We extend the neo-Schumpeterian growth framework to investigate the role of strong uncertainty in the innovative process. In particular, we postulate that, when deciding upon R&D efforts, investors hold ‘ambiguous beliefs’ about the exact probability of arrival of the next vertical innovation, and that they face ambiguity via the α −MEU decision rule (Ghirardato et al., J Econ Theory 118:133–173, 2004). Along the balanced growth path, the higher the agent’s ambiguity aversion (α), the lower the R&D efforts and the economic performance. Consistent with cross-country empirical evidence, this causal mechanism suggests that, together with the profitability conditions of the economy, different ‘cultural’ attitudes towards ambiguity may help explain the different R&D efforts observed across countries.

A class of incomplete and ambiguity averse preferences

Abstract This paper characterizes models of ambiguous beliefs in the absence of the completeness axiom. We axiomatize multiple-selves versions of some of the most important examples of complete and ambiguity averse preferences, and characterize when those incomplete preferences are ambiguity averse.

On robust asymmetric equilibria in asymmetric R&D-driven growth economies

In an R&D-driven growth model with asymmetric fundamentals the steady state equilibrium R&D investments are industry-speci…c and they are such that R&D returns are equalized across industries. Return equalization, however, makes investors indierent as to where to target research and, hence, the prob- lem of allocation of R&D investments across industries is indeterminate. Agents' indierence creates an investment scenario. We assume that agents hold ambiguous beliefs on the per-industry pro…tability of their R&D invest- ments. Investors'aversion towards ambiguity (in the sense of Gilboa-Schmeidler, 1989) eliminates the indeterminacy of the R&D investment problem. In particu- lar, we prove that the asymmetric return-equalizing equilibrium is robust against a however small degree of investors'aversion to ambiguity.

Knightian games and robustness to ambiguity

Abstract This paper introduces a notion of robustness to ambiguous beliefs for Bayesian Nash equilibria. An equilibrium is robust if the corresponding strategies remain approximately optimal for a class of games with ambiguous beliefs that results from an appropriately defined perturbation of the belief structure of the original non-ambiguous belief game. The robustness definition is based on a novel definition of equilibrium for games with ambiguous beliefs that requires equilibrium strategies to be approximate best responses for all measures that define a player's belief. Conditions are derived under which robustness is characterized by a newly defined strategic continuity property, which can be verified without reference to perturbations and corresponding ambiguous belief games.

Jaffray’s ideas on ambiguity

This paper discusses Jean-Yves Jaffray’s ideas on ambiguity and the views underlying his ideas. His models, developed 20 years ago, provide the most tractable separation of risk attitudes, ambiguity attitudes, and ambiguity beliefs available in the literature today.

Why women of lower educational attainment struggle to make healthier food choices: The importance of psychological and social factors

Women of lower educational attainment are more likely to eat unhealthy diets than women of higher educational attainment. To identify influences on the food choices of women with lower educational attainment, 11 focus groups (eight with women of lower, and three with women of higher educational attainment) were held. Using a semi-structured discussion guide, environmental, social, historical and psychological factors known to be associated with food choice were explored. Audio recordings were transcribed verbatim and thematically analysed. Compared to women of higher educational attainment, women of lower educational attainment had less control over their families’ food choices, less support for attempts to eat healthily, fewer opportunities to observe and learn good food-related practices, more negative affect, more perceived environmental constraints and more ambiguous beliefs about the consequences of eating a nutritious diet. These findings provide a starting point for taking forward the design of an intervention to improve the diets of young women.

On attitude polarization under Bayesian learning with non-additive beliefs

Ample psychological evidence suggests that people’s learning behavior is often prone to a “myside bias” or “irrational belief persistence” in contrast to learning behavior exclusively based on objective data. In the context of Bayesian learning such a bias may result in diverging posterior beliefs and attitude polarization even if agents receive identical information. Such patterns cannot be explained by the standard model of rational Bayesian learning that implies convergent beliefs. Based on Choquet expected utility theory, we therefore develop formal models of Bayesian learning with psychological bias as alternatives to rational Bayesian learning. We derive conditions under which beliefs may diverge in the learning process despite the fact that all agents observe the same sample drawn from an i.i.d. process. Key to our approach is the description of ambiguous beliefs as neo-additive capacities (Chateauneuf et al., J Econ Theory 137:538–567, 2007), which allows for a flexible and parsimonious parametrization of departures from additive probability measures.

Half empty, half full and why we can agree to disagree forever

Aumann [Aumann R., 1976. Agreeing to disagree. Annals of Statisitics 4, 1236–1239] derives his famous we cannot agree to disagree result under the assumption that people are expected utility ( = EU ) decision makers. Motivated by empirical evidence against EU theory, we study the possibility of agreeing to disagree within the framework of Choquet expected utility ( = CEU ) theory which generalizes EU theory by allowing for ambiguous beliefs. As our first main contribution, we show that people may well agree to disagree if their Bayesian updating of ambiguous beliefs is psychologically biased in our sense. Remarkably, this finding holds regardless of whether people with identical priors apply the same psychologically biased Bayesian update rule or not. As our second main contribution, we develop a formal model of Bayesian learning under ambiguity. As a key feature of our approach the posterior subjective beliefs do, in general, not converge to “true” probabilities which is in line with psychological evidence against converging learning behavior. This finding thus formally establishes that CEU decision makers may even agree to disagree in the long-run despite the fact that they always received the same information.

Hierarchies of ambiguous beliefs

We present a theory of interactive beliefs analogous to Mertens and Zamir [Formulation of Bayesian analysis for games with incomplete information, Int. J. Game Theory 14 (1985) 1–29] and Brandenburger and Dekel [Hierarchies of beliefs and common knowledge, J. Econ. Theory 59 (1993) 189–198] that allows for hierarchies of ambiguity. Each agent is allowed a compact set of beliefs at each level, rather than just a single belief as in the standard model. We propose appropriate definitions of coherency and common knowledge for our types. Common knowledge of coherency closes the model, in the sense that each type homeomorphically encodes a compact set of beliefs over the others’ types. This space universally embeds every implicit type space of ambiguous beliefs in a beliefs-preserving manner. An extension to ambiguous conditional probability systems [P. Battigalli, M. Siniscalchi, Hierarchies of conditional beliefs and interactive epistemology in dynamic games, J. Econ. Theory 88 (1999) 188–230] is presented. The standard universal type space and the universal space of compact continuous possibility structures are epistemically identified as subsets.

An experimental study of updating ambiguous beliefs

‘Ambiguous beliefs’ are beliefs which are inconsistent with a unique, additive prior. The problem of their update in face of new information has been dealt with in the theoretical literature, and received several contradictory answers. In particular, the ‘maximum likelihood update’ and the ‘full Bayesian update’ have been axiomatized. This experimental study attempts to test the descriptive validity of these two theories by using the Ellsberg experiment framework.

Families of update rules for non-additive measures: Applications in pricing risks

Wang et al. [Axiomatic characterization of insurance prices, Insurance: Mathematics and Economics 21 (1997) 173–183] propose axioms for prices in an insurance market. Chateauneuf et al. [Choquet pricing for financial markets with frictions, Mathematical Finance 6 (1996) 323–330] propose similar axioms for prices in a financial market with frictions. As a result of these axioms, market prices can be represented by the Choquet integral with respect to a non-additive measure. In both insurance and financial pricing, it is important to update prices in light of newly available information. This updating can be achieved by conditioning the underlying non-additive measure. Denneberg [Conditioning (updating) non-additive measures, Annals of Operations Research 52 (1994) 21–42] studies three conditioning rules for updating non-additive measures. Two of these update rules, the Bayes' and the Dempster-Shafer, are extreme cases of a family of update rules, [Gilboa, Schmeidler, Updating ambiguous beliefs, Journal of Economic Theory 59 (1993) 33–49]. In this paper, we introduce a family of update rules more general than the one of Gilboa and Schmeidler. We also show how to embed the general and Dempster-Shafer update formulas in another family of update rules. We examine the properties of these two families of update rules and the resulting conditional prices.

A content, correlational and factor analytic study of four tolerance of ambiguity questionnaires

A review of the scattered and diffuse tolerance of ambiguity (AT) literature from 1946 revealed 4 fairly frequently used self-report measures of this concept (AT). This study set out to compare and contrast these different measures by content, correlational and factor analysis. Nearly 250 people completed all four dimensions. Each scale was then subject to content and factor analyses which agreed with each other, as did correlations with the various factors. First-and higher-order factor analysis yielded a number of interpretable factors. The results are discussed in terms of the multi-dimensional nature of AT beliefs, the psychometric equivalence of these various scales, and the difficulty of measuring the concept in the first place.

Updating Ambiguous Beliefs

We present and axiomatize several update rules for probabilities (and preferences) where there is no unique additive prior. In the context of non-additive probabilities we define and axiomatize Bayesian update rules; in the context of multiple (additive) priors we define maximum likelihood rules. It turns out that for decision makers which can be described by both theories, the two approaches coincide. Thus, we suggest an axiomatically based ambiguous beliefs update rule, which is needed for applications in many economic theory models. Journal of Economic Literature classification numbers: D80, D81, C11, C71.