Symmetry axioms and perceived ambiguity
 894 Downloads
 1 Citations
Abstract
Since at least de Finetti (Annales de l’Institut Henri Poincare 7:1–68, 1937), preference symmetry assumptions have played an important role in models of decision making under uncertainty. In the current paper, we explore (1) the relationship between the symmetry assumption of Klibanoff et al. (KMS) (Econometrica 82:1945–1978, 2014) and alternative symmetry assumptions in the literature, and (2) assuming symmetry, the relationship between the set of relevant measures, shown by KMS (2014) to reflect only perceived ambiguity, and the set of measures (which we will refer to as the Bewley set) developed by Ghirardato et al. (J Econ Theory 118:133–173, 2004), Nehring (Ambiguity in the context of probabilistic beliefs, working paper, 2001, Bernoulli without Bayes: a theory of utilitysophisticated preference, working paper, 2007) and Ghirardato and Siniscalchi (A more robust definition of multiple priors, working paper, 2007, Econometrica 80:2827–2847, 2012). This Bewley set is the main alternative offered in the literature as possibly representing perceived ambiguity. Regarding symmetry assumptions, we show that, under relatively mild conditions, a variety of preference symmetry conditions from the literature [including that in KMS (2014)] are equivalent. In KMS (2014), we showed that, under symmetry, the Bewley set and the set of relevant measures are not always the same. Here, we establish a preference condition, No Half Measures, that is necessary and sufficient for the two to be the same under symmetry. This condition is rather stringent. Only when it is satisfied may the Bewley set be interpreted as reflecting only perceived ambiguity and not also taste aspects such as ambiguity aversion.
Keywords
Symmetry Beliefs Ambiguity Ambiguity aversion Model uncertainty EllsbergJEL Classifications
D01 D80 D81 D831 Introduction
Recent literature on ambiguity (meaning subjective uncertainty about probabilities over states) or model uncertainty (see e.g., surveys by Gilboa and Marinacci [17] and Marinacci [22]), makes heavy use of models of decision making under uncertainty. In this literature, a concern relevant to applying these models is how to connect the representation of preferences to perceived ambiguity or the models over which you are uncertain. In a recent contribution, Klibanoff, Mukerji and Seo (henceforth KMS) [21], we proposed an answer to this question for a broad class of symmetric preferences. In particular, we defined a notion of relevant measures and showed that these measures reflect only perceived ambiguity and not ambiguity aversion or risk aversion.
Since at least de Finetti [7], preference symmetry assumptions have played an important role in models of decision making under uncertainty. In the current paper we explore (1) the relationship between the symmetry assumption of KMS [21] and alternative symmetry assumptions in the literature, and (2) assuming symmetry, the relationship between the set of relevant measures developed by KMS [21] and the set of measures (which we will refer to as the Bewley set) developed by Ghirardato et al. [14], Nehring [24, 25] and Ghirardato and Siniscalchi [15, 16]. This latter relationship is of particular interest because this Bewley set is the main alternative offered in the literature as possibly representing perceived ambiguity. Regarding symmetry assumptions, we show that, under relatively mild conditions, a variety of preference symmetry conditions from the literature (including that in KMS [21]) are equivalent.
In KMS [21], we showed that, under symmetry, the Bewley set and the set of relevant measures are not always the same. This is important because, when they differ, the Bewley set is affected by changes in ambiguity aversion, and thus cannot be interpreted as reflecting only perceived ambiguity. In the present paper, we establish a preference condition, No Half Measures, that is necessary and sufficient for the two to be same for symmetric preferences. In contrast to the mild conditions used for the symmetry equivalence result, No Half Measures is rather stringent. Yet our results, combined with those in KMS [21], imply that it is only under these stringent conditions that the Bewley set may be interpreted as reflecting only perceived ambiguity and not also taste aspects such as ambiguity aversion.
When incorporating considerations of ambiguity or model uncertainty in finance models it is natural for the modeler to want to impose constraints on the preferences of the decision maker(s) in the model so that they reflect some type of calibration of perceived ambiguity to external data. For example, in an asset pricing model, the modeler may want to impose the restriction that an investor seeks to make her portfolio robust against only a certain limited set of stochastic processes or probabilitistic forecasts or different forecasting models (when incorporating model uncertainty) because these processes or forecasts pass certain tests of inference on past data. The theory developed in KMS [21] gives reasons why, if such constraints were to be imposed, one should do so through constraints on the relevant measures. KMS [21] shows where the relevant measures appear in representation functionals for symmetric versions of a variety of popular ambiguity models. For instance, in the \(\alpha \)MEU model (see e.g., Ghirardato et al. [14]) and the smooth ambiguity model (see e.g., Klibanoff et al. [20], Nau [23], Seo [26]) and the extended MEU with contraction model (see e.g., Gajdos et al. [13]), the set of probabilistic forecast models entertained by the decision maker should be the set of measures that appear in the functional representations. Thus, reassuringly, the theory of relevant measures provides foundations to the common and intuitive practice in the way these ambiguity models are used.
However, KMS’s theory [21] is predicated on a specification of symmetry. It is common to invoke some notion of symmetry/exchangeability, at least implicitly, to underpin statistical inference (e.g., of the kind used to justify restrictions on the set of forecasting models considered when incorporating model uncertainty). Our results in Sect. 3 demonstrate that KMS’s specification of symmetry is not special and is in fact equivalent to notions invoked elsewhere in the literature under conditions that would be met in most settings.
Our study of the No Half Measures condition, in Sect. 4, shows that the possibilities of establishing foundations for practices concerning the way contraints on model uncertainty are incorporated in ambiguity models via theories invoking the Bewley set are limited. This is particularly true in the case of functional forms which allow for nonextreme attitudes to ambiguity that may be varied parametrically in the functional representation (such as in the \(\alpha \)MEU model, the extended MEU with contraction model, and the smooth ambiguity model). For instance, in the \(\alpha \)MEU model, the Bewley set can be identified with the set of measures appearing in the model only if \(\alpha \) is 0 or 1.
2 Setting and preferences
In this section, we describe aspects of the formal setting, notation and preferences from KMS [21] that will be useful here.
Let S be a compact metric space and \(\Omega =S^{\infty }\) the state space with generic element \(\omega =\left( \omega _{1},\omega _{2},\ldots \right) \). The state space \(\Omega \) is also compact metric (Aliprantis and Border [1, Theorems 2.61 and 3.36]). Denote by \(\Sigma _{i}\) the Borel \(\sigma \)algebra on the ith copy of S, and by \(\Sigma \) the product \(\sigma \)algebra on \(S^{\infty }\). An act is a simple Anscombe–Aumann act, a measurable \(f:S^{\infty }\rightarrow X\) having finite range (i.e., \(f\left( S^{\infty }\right) \) is finite) where X is the set of lotteries (i.e., finite support probability measures on an outcome space Z). The set of acts is denoted by \({\mathcal {F}}\), and \(\succsim \) is a binary relation on \({\mathcal {F}}\times {\mathcal {F}}\). As usual, we identify a constant act (an act yielding the same element of X on all of \(S^{\infty }\)) with the element of X it yields.
Denote by \(\Pi \) the set of all finite permutations on \(\left\{ 1,2,\ldots \right\} \) i.e., all onetoone and onto functions \(\pi :\left\{ 1,2,\ldots \right\} \rightarrow \left\{ 1,2,\ldots \right\} \) such that \(\pi (i)=i\) for all but finitely many \(i\in \left\{ 1,2,\ldots \right\} \). For \(\pi \in \Pi \), let \(\pi \omega =\left( \omega _{\pi \left( 1\right) },\omega _{\pi \left( 2\right) },\ldots \right) \) and \(\left( \pi f\right) \left( \omega \right) =f\left( \pi \omega \right) \).
For any topological space Y, \(\Delta \left( Y\right) \) denotes the set of (countably additive) Borel probability measures on Y. Unless stated otherwise, a measure is understood as a countably additive Borel measure. For later use, \(ba\left( Y\right) \) is the set of finitely additive bounded realvalued set functions on Y, and \(ba_{+}^{1}\left( Y\right) \) the set of nonnegative probability charges in \(ba\left( Y\right) \). A measure \(p\in \Delta \left( S^{\infty }\right) \) is called symmetric if the order doesn’t matter, i.e., \(p\left( A\right) =p\left( \pi A\right) \) for all \(\pi \in \Pi \), where \(\pi A=\left\{ \pi \omega :\omega \in A\right\} \). Denote by \(\ell ^{\infty }\) the i.i.d. measure with the marginal \(\ell \in \Delta \left( S\right) \). Define \(\int _{S^{\infty }}fdp\in X\) by \(\left( \int _{S^{\infty }}fdp\right) \left( B\right) =\left( \int _{S^{\infty }}f\left( \omega \right) \left( B\right) dp\left( \omega \right) \right) \). (Since f is simple, this is welldefined.)
Fix \(x_{*},x^{*}\in X\) such that \(x^{*}\succ x_{*}\). For any event \(A\in \Sigma \), \(1_{A}\) denotes the act giving \(x^{*}\) on A and \(x_{*}\) otherwise. Informally, this is a bet on A. More generally, for \(x,y\in X\), xAy denotes the act giving x on A and y otherwise. A finite cylinder event \(A\in \Sigma \) is any event of the form \(\left\{ \omega :\omega _{i}\in A_{i}\text { for }i=1,\ldots ,n\right\} \) for \(A_{i}\in \Sigma _{i}\) and some finite n.
Endow \(\Delta \left( S\right) \), \(\Delta \left( \Delta \left( S\right) \right) \) and \(\Delta \left( S^{\infty }\right) \) with the relative weak* topology. To see what this is, consider, for example, \(\Delta \left( S\right) \). The relative weak* topology on \(\Delta \left( S\right) \) is the collection of sets \(V\cap \Delta \left( S\right) \) for weak* open \(V\subseteq ba\left( S\right) \), where the weak* topology on ba(S) is the weakest topology for which all functions \(\ell \longmapsto \int \psi d\ell \) are continuous for all bounded measurable \(\psi \) on S. Also note that a net \(\ell _{\alpha }\in ba\left( S\right) \) converges to \(\ell \in ba\left( S\right) \) under the weak* topology if and only if \(\int \psi d\ell _{\alpha }\rightarrow \int \psi d\ell \) for all bounded measurable \(\psi \) on S. For a set \(D\subseteq \Delta (S)\), denote the closure of D in the relative weak* topology by \(\overline{D}\).
The support of a probability measure \(m\in \Delta \left( \Delta \left( S\right) \right) \), denoted \({\text {supp}}m\), is a relative weak* closed set such that \(m\left( \left( {\text {supp}}m\right) ^{c}\right) =0\) and if \(L\cap {\text {supp}}m\ne \emptyset \) for relative weak* open L, \(m\left( L\cap {\text {supp}}m\right) >0\). (See e.g., Aliprantis and Border [1, p.441].)
Let \(\Psi _{n}\left( \omega \right) \in \Delta \left( S\right) \) denote the empirical frequency operator \(\Psi _{n}\left( \omega \right) \left( A\right) =\frac{1}{n}\sum _{t=1}^{n}I\left( \omega _{t}\in A\right) \) for each event A in S. Define the limiting frequency operator \(\Psi \) by \(\Psi \left( \omega \right) \left( A\right) =\lim _{n}\Psi _{n}\left( \omega \right) \left( A\right) \) if the limit exists and 0 otherwise. Also, to map given limiting frequencies or sets of limiting frequencies to events in \(S^{\infty }\), we consider the natural inverses \(\Psi ^{1}\left( \ell \right) =\left\{ \omega :\Psi \left( \omega \right) =\ell \right\} \) and \(\Psi ^{1}\left( L\right) =\left\{ \omega :\Psi \left( \omega \right) \in L\right\} \) for \(\ell \in \Delta \left( S\right) \) and \(L\subseteq \Delta \left( S\right) \).
The following axioms are used in describing assumptions on preferences \(\succsim \) at various points in this paper. For remarks and interpretation of these axioms see KMS [21].
Axiom 1
(Ccomplete Preorder) \(\succsim \) is reflexive, transitive and the restriction of \(\succsim \) to X is complete.
Axiom 2
(Monotonicity) If \(f\left( \omega \right) \succsim g\left( \omega \right) \) for all \(\omega \in S^{\infty }\), \(f\succsim g\).
Axiom 3
(Risk Independence) For all \(x,x^{\prime },x^{\prime \prime }\in X\) and \(\alpha \in \left( 0,1\right) \), \(x\succsim x^{\prime }\) if and only if \(\alpha x+\left( 1\alpha \right) x^{\prime \prime }\succsim \alpha x^{\prime }+\left( 1\alpha \right) x^{\prime \prime }\).
Axiom 4
(Nontriviality) There exist \(x,y\in X\) such that \(x\succ y\).
The key axiom is Event Symmetry which implies that the ordinates of \(S^{\infty }\) are viewed as interchangeable.
Axiom 5
We now state three forms of continuity that are used in the paper.
Axiom 6
(Mixture Continuity) For all \(f,g,h\in {\mathcal {F}}\), the sets \(\{\lambda \in \left[ 0,1\right] :\lambda f+\left( 1\lambda \right) g\succsim h\}\) and \(\{\lambda \in \left[ 0,1\right] :h\succsim \lambda f+\left( 1\lambda \right) g\}\) are closed in \(\left[ 0,1\right] \).
Axiom 7
(Mixture Continuity of \(\succsim ^{*}\)) For all \(f,g,h\in {\mathcal {F}}\), the sets \(\{\lambda \in \left[ 0,1\right] :\lambda f+\left( 1\lambda \right) g\succsim ^{*}h\}\) and \(\{\lambda \in \left[ 0,1\right] :h\succsim ^{*}\lambda f+\left( 1\lambda \right) g\}\) are closed in \(\left[ 0,1\right] \).
Finally, borrowed from Ghirardato et al. [14], we have the following adaptation of Arrow’s [3] monotone continuity:
Axiom 8
(Monotone Continuity of \(\succsim ^{*}\)) For all \(x,x^{\prime },x^{\prime \prime }\in X\), if \(A_{n}\searrow \emptyset \) and \(x^{\prime }\succ x^{\prime \prime }\), then \(x^{\prime }\succsim ^{*}xA_{n}x^{\prime \prime }\) for some n.
Definition 2.1
\(\succsim \) satisfies Weak Continuous Symmetry if it satisfies all of the above axioms except possibly Mixture Continuity.
When we say that \(\succsim \) is Weak Continuous Symmetric, we mean that it satisfies Weak Continuous Symmetry.
3 Relating Event Symmetry to the literature
A key axiom in our approach is Event Symmetry. We now show that this condition relates quite closely to a variety of other conditions from the literature, including strengthenings of de Finetti [7]’s Exchangeability, of Hewitt and Savage [19]’s Symmetry, of Seo [26]’s Dominance and of Klibanoff et al. [20]’s Consistency. One of those conditions (condition (viii) below) requires some additional definitions which we state next.
Definition 3.1
Note this definition associates with each act f an act \(f^{\Psi }\) that, for each event \(\left\{ \omega :\Psi \left( \omega \right) =\ell \right\} \) corresponding to the limiting frequencies generated by \(\ell \), yields the lottery generated by f under the assumption that the i.i.d. process \(\ell ^{\infty }\) governs the realization of the state.
Since \(f^{\Psi }\) need not be simple, but is an element of the space \({\mathcal {\hat{F}}}\) of all bounded and measurable functions from \(\Omega \) to X, it is necessary to consider extending \(\succsim \) to \({\mathcal {\hat{F}}}\). In particular, we consider extensions continuous in the following sense: \({\hat{\succsim }}\) on \({\mathcal {\hat{F}}}\) satisfies Norm Continuity if \(f{\hat{\succsim }}g\) whenever \(f_{k}{\hat{\succsim }}g_{k}\) for all \(k=1,2,\ldots \) and \(f_{k}\) and \(g_{k}\) normconverge to f and g respectively.
We now show that, under mild conditions, Event Symmetry is in fact equivalent to a variety of other conditions appearing in the literature. A first contribution, then, is to show that despite the seeming variety of preference formalizations of symmetry in the ambiguity literature, reassuringly their differences will not matter in most settings. Secondly, this also shows that results from KMS [21] derived assuming Event Symmetry are robust in the sense that they would continue to hold under alternative symmetry formulations.
Theorem 3.1
 (i)
for every \(f\in {\mathcal {F}}\) and \(\pi \in \Pi \), \(f\sim \frac{1}{2}f+\frac{1}{2}\pi f\),
 (ii)
for every \(f\in {\mathcal {F}}\), \(\pi \in \Pi \) and \(\alpha \in \left[ 0,1\right] \), \(f\sim \alpha \pi f+\left( 1\alpha \right) f\),
 (iii)
for every \(f\in {\mathcal {F}}\) and \(\pi _{i}\in \Pi \), \(f\sim \frac{1}{n}\sum _{i=1}^{n}\pi _{i}f\),
 (iv)
for every \(f\in {\mathcal {F}}\), \(\pi _{i}\in \Pi \) and \(\alpha _{i}\in \left[ 0,1\right] \) with \(\sum _{i=1}^{n}\alpha _{i}=1\), \(f\sim \sum _{i=1}^{n}\alpha _{i}\pi _{i}f\), and
 (v)
for every \(f\in {\mathcal {F}}\) and \(\pi \in \Pi \), \(f\sim ^{*}\pi f\). Moreover, the above are equivalent to each of the following under the assumption that \(\succsim \) satisfies Ccomplete Preorder, Mixture Continuity, Monotonicity, Risk Independence, Nontriviality and Monotone Continuity of \(\succsim ^{*}\):
 (vi)
Event Symmetry,
 (vii)
for every \(f,g\in {\mathcal {F}}\), if \(\int fdp\succsim \int gdp\) for all symmetric \(p\in \Delta \left( S^{\infty }\right) \), then \(f\succsim g\). Finally, if, in addition, there exists an extension of \(\succsim \) to \({\mathcal {\hat{F}}}\) that is reflexive, transitive and satisfies Norm Continuity, then the following is equivalent to all of the above:
 (viii)
for \(f,g\in {\mathcal {F}}\), \(f\succsim g\) if and only if \(f^{\Psi }{\hat{\succsim }}g^{\Psi }\), if \({\hat{\succsim }}\) is any such extension.
All of these conditions are strengthenings of Hewitt and Savage [19]’s symmetry: given \(p\in \Delta \left( S^{\infty }\right) \), \(p\left( A\right) =p\left( \pi A\right) \) for all finite cylinder events \(A\in \Sigma \). In terms of preference, this translates into \(1_{A}\sim 1_{\pi A}\) for all such events and all permutations \(\pi \in \Pi \).^{1} Event Symmetry strengthens this by requiring the indifference to be preserved under mixture with any common third act. Under reflexivity, transitivity and Mixture Continuity of \(\succsim \), conditions (i)–(v) each imply Event Symmetry.
Condition (ii) is closely related to Epstein and Seo [10]’s Strong Exchangeability, the first behavioral axiom in the literature that captures the idea that the agent views all experiments as identical, i.e., i.i.d. (See Epstein and Seo [10] for a behavioral interpretation of condition (ii). A similar interpretation applies to (i), (iii) and (iv).) Their axiom states that condition (ii) holds when f depends only on a finite number of experiments. However, under their regularity axiom, their Strong Exchangeability extends to every act f and hence is equivalent to condition (ii).
Epstein and Seo [10], under MEU, characterize two models of preferences where the decision maker is indifferent to permutations—one that restricts to identical experiments (no ambiguity on idiosyncratic factors) and permits ambiguity about parameters, and the other that allows ambiguity on idiosyncratic factors but rules out ambiguity about parameters. Only the former of the two models satisfies (ii). Epstein and Seo [12] go further and characterize a single model within MEU that reflects ambiguity on both at the same time. The unifying framework permits a behavioral distinction between idiosyncratic factors and parameters as two sources of ambiguity.
Condition (i) is a special case of condition (ii) when \(\alpha =\frac{1}{2}\). Since Epstein and Seo [10] consider MEU models, they could restrict to the case \(\alpha =\frac{1}{2}\). The above theorem shows that \(\alpha =\frac{1}{2}\) is sufficient to capture the same idea in general as long as reflexivity, transitivity and Mixture Continuity of \(\succsim \) hold.
De Castro and AlNajjar [2, 5] use condition (iii) and its generalization to collections of transformations \(\Gamma \) other than the finite permutations \(\Pi \). They provide conditions on \(\Gamma \) under which complete, transitive, monotonic, continuous and risk independent preferences satisfying (iii) with respect to \(\Gamma \) are such that each act is indifferent to an associated act based on limiting frequencies (where the notion of limiting frequencies uses the given \(\Gamma \)) where the associated act is constructed much like \(f^{\Psi }\) above with \(\Gamma \)ergodic measures replacing the i.i.d. measures as parameters. They also prove a representation theorem for a utility function similar to (4.2) using an expected utility assumption on parameterbased acts added to the conditions on preferences and on \(\Gamma \) mentioned in the previous sentence.
Condition (iv) strengthens condition (iii).
Condition (v) is stronger than Event Symmetry in that f is not necessarily a binary act.
CerreiaVioglio et al. [6]’s Consistency is similar to condition (vii), but instead of considering transformations (e.g. permutations) and/or all symmetric measures, they assume a family of objectively rational probability measures on a general state space and require \(\int fdp\succsim \int gdp\) for all measures p in that family. Taking the family to be all symmetric measures makes CerreiaVioglio et al.’s Consistency exactly condition (vii). With their Consistency they provide representation theorems for Bewley preference, Choquet expected utility, variational preferences, uncertainty averse preferences and the smooth ambiguity model. Also, they prove that the generalization of condition (v) to other collections of transformations \(\Gamma \) together with some (mild) axioms imply their Consistency when the family of objectively rational measures is the \(\Gamma \)invariant measures.
The content of condition (viii) is that given the “knowledge” that everything is driven by some (as yet unknown) i.i.d. process, it seems reasonable that when evaluating an act, the individual would ultimately care only about the induced mapping from the space of i.i.d. processes to the lotteries generated under each process. Klibanoff et al. [20]’s Consistency assumption says that when evaluating an act, an individual cares only about the induced mapping from probability measures on the state space to lotteries. Their assumption was stated in terms of acts and “second order acts” (maps from probability measures on the state space to outcomes). The latter objects do not appear as such in the present paper, but their role is played by the subset of acts measurable with respect to limiting frequency events, the acts \(f^{\Psi }\in {\mathcal {\hat{F}}}\). The identification (in terms of preference) of f with \(f^{\Psi }\) stated in condition (viii) is analogous to the identification of f with an “associated second order act” in Klibanoff et al. [20]’s Consistency with the qualification that \(f^{\Psi }\) only induces the same mapping from probability measures to lotteries as f for i.i.d. probability measures. Thus, condition (viii) strengthens Klibanoff et al. [20]’s Consistency to incorporate the known symmetry of the ordinates in the same way that condition (vii) strengthened Seo [26]’s Dominance. Theorem 3.1 says that, under our other axioms, each of these strengthenings is equivalent to Event Symmetry. One can view the equivalence of condition (iii) and condition (viii) as following from the i.i.d. case of the sufficient statistic result in AlNajjar and de Castro [2].
4 When do relevant measures and the Bewley set agree?
We begin by recalling definitions of these sets and some key results from KMS [21] that we build upon.
KMS [21] defined what it means for a marginal \(\ell \in \Delta \left( S\right) \) to be a relevant measure. For notational convenience, let \({\mathcal {O}}_{\ell }\) be the collection of open subsets of \(\Delta \left( S\right) \) that contains \(\ell \). That is, for \(\ell \in \Delta \left( S\right) \), \({\mathcal {O}}_{\ell }=\{L\subseteq \Delta \left( S\right) :L\) is open, \(\ell \in L\}\).
Definition 4.1
A measure \(\ell \in \Delta \left( S\right) \) is relevant (according to preferences \(\succsim \)) if, for any \(L\in {\mathcal {O}}_{\ell }\), there are \(f,g\in {\mathcal {F}}\) such that \(f\not \sim g\) and \(\int fd\hat{\ell }^{\infty }=\int gd\hat{\ell }^{\infty }\) for all \({\widehat{\ell }}\in \Delta \left( S\right) \backslash L\).
In words, \(\ell \) is relevant if it satisfies the following property: For each open set containing \(\ell \), there are acts that are not indifferent despite generating identical induced distributions over outcomes when any measure outside this set governs the independent realization of each ordinate S. From KMS [21], we know that the set of relevant measures, R, is closed and that, a measure is not relevant if and only if the limiting frequency event generated by some open neighborhood of that measure is null according to the preferences. Thus, behavior is as if relevant measures were the possible resolutions of ambiguity. In this sense, the relevant measures reflect perceived ambiguity. Furthermore KMS [21, Theorem 3.4] showed that, under mild conditions, the set of relevant measures reflects only perceived ambiguity. Notably, the set is not affected by increases or decreases in ambiguity aversion.
The Bewley set, C, is defined using a Bewleystyle (Bewley [4]) representation of the induced relation \(\succsim ^{*}\) (defined by (2.2)). For Weak Continuous Symmetric preferences violating Anscombe–Aumann Independence, \(\succsim ^{*}\) is incomplete relative to \(\succsim \). In the context of Weak Continuous Symmetric preferences, KMS [21, Theorem 4.5] provided a Bewleystyle (Bewley [4]) representation result for \(\succsim ^{*}\), identifying both the Bewley set C and the set of relevant measures R. It is of interest to note that this result is a generalization of de Finetti’s theorem. We restate it here, and will make use of it in proving several of the results of this section:
Theorem 4.1
The main contribution of this section is to show that preference condition No Half Measures, defined below, is equivalent to the set of relevant measures fully determining the Bewley set, and thus necessary and sufficient for the Bewley set to inherit the interpretation of reflecting only perceived ambiguity from the set of relevant measures.
Condition
(No Half Measures) For \(A\in \Sigma ^{\Psi }\), at least one of the following holds:
 (i)

If \(1_{S^{\infty }}\succ x\in X\), then \(\alpha x+(1\alpha )f\not \succsim \alpha 1_{A}+(1\alpha )f\) for some \(f\in {\mathcal {F}}\) and \(\alpha \in (0,1]\);
 (ii)

For all \(f\in {\mathcal {F}}\) and \(\alpha \in (0,1]\), \(\alpha 1_{\emptyset }+(1\alpha )f\thicksim \alpha 1_{A}+(1\alpha )f\).
We now show the connection between No Half Measures and the relation between the Bewley set and the set of relevant measures.
Theorem 4.2
Suppose \(\succsim \) is Weak Continuous Symmetric. Then \(\succsim \) satisfies No Half Measures if and only if \(C=\left\{ \int \ell ^{\infty }dm\left( \ell \right) :m\in \Delta (R)\right\} \) in (4.1).
Recall that, without No Half Measures, \(C=\left\{ \int \ell ^{\infty }dm\left( \ell \right) :m\in M\right\} \) where \(M\subseteq \Delta (R)\), so that knowing R need not pin down C. Our theorem shows that the No Half Measures condition is identifying those cases in which C reflects only the set R of processes considered relevant. In this sense, No Half Measures is required for the approach in Ghirardato et al. [14], Nehring [24] and Ghirardato and Siniscalchi [15, 16] based on C to agree with the relevant measures and thus be interpretable as reflecting only perceived ambiguity and not perceived ambiguity intertwined with ambiguity aversion.
How restrictive is the No Half Measures condition?
One restriction is that No Half Measures implies there is only a finite set of relevant measures, and so only a finite number of possible resolutions of ambiguity.
Theorem 4.3
If a Weak Continuous Symmetric \(\succsim \) satisfies No Half Measures then R is finite.
What are the further implications of No Half Measures in the context of specific decision models? We begin with the Maxmin expected utility (MEU) model (Gilboa and Schmeidler [18]). For this model, No Half Measures implies the set of measures in the representation are the i.i.d. products generated by the measures in R. We then examine the \(\alpha \)MEU model and show there, when R is nonsingleton No Half Measures is only compatible with extreme values of \(\alpha \) (\(\alpha \) equals 0 or 1). Similarly, we conclude this subsection with an examination of the smooth ambiguity model—when R is nonsingleton we show No Half Measures places nontrivial restrictions on the function \(\phi \).^{3}
4.1 MEU model
Theorem 4.4
The contribution of Theorem 4.4 is that in this symmetric MEU context, No Half Measures is equivalent to limiting the set of measures to the i.i.d. products generated by a finite set of relevant measures, R. The proof uses Ghirardato et al. [14, Theorem 14] to show that F is the Bewley set, and then applies Theorems 4.1, 4.2 and 4.3 to obtain the result. Note that the supposition in the first sentence of the theorem could equivalently be replaced by assuming \(\succsim \) satisfies Weak Continuous Symmetry and the axioms in Gilboa and Schmeidler [18].
We can apply Theorem 4.4 to the Extended MEU with contraction model to get the following:
Corollary 4.1
To see that this is a strong condition, observe that the restriction (\(\beta =1\) or (\(q\in \left\{ \ell ^{\infty }:\ell \in D\right\} \) and (\(D=\{q\}\) or \(\beta =0\)))) is equivalent to MEU without contraction (\(\beta =1\)) or expected utility with an i.i.d. prior, ruling out all intermediate, nonextreme, cases of ambiguity attitude.
4.2 \(\alpha \)MEU model
According to KMS [21, footnote 9] (and the example below), the Bewley set, C, generally will depend on \(\alpha \) as well as the set of measures appearing in the representation. Since No Half Measures ensures C is fully determined by R, it must eliminate the dependence of C on \(\alpha \). When does this dependence occur? Our next example (see also Eichberger et al. [8, Section 4] and Eichberger et al. [9, section 3.2.2]), suggests that \(\alpha \in (0,1)\) creates dependence.
Example 4.1
Our next result shows that the troublesome case in the example (i.e., \(\alpha \in (0,1)\) and R nonsingleton) is exactly what is ruled out by No Half Measures. In this sense, No Half Measures turns out to be extremely restrictive in the context of \(\alpha \)MEU:
Theorem 4.5
Eichberger et al. [9, Theorem 2] prove a related result – for any finite state space, when C is nonsingleton, any \(\succsim \) having an \(\alpha \)MEU representation with Bewley set C as the set of measures has \(\alpha \) equal to 0 or 1. Since S itself, let alone \(S^{\infty }\), need not be finite our result requires different arguments. In fact, Eichberger et al. [9] provide an example with an infinite compact metric state space showing that their result may fail in such settings. In light of this, our theorem shows that Event Symmetry and our continuity together with the product structure of the state space is sufficient to extend their finite state space conclusion.
4.3 Smooth ambiguity model
Theorem 4.6
Some examples violating the condition \(\sup {\textstyle \bigcup _{r,t\in u\left( X\right) }}\frac{\phi ^{\prime }\left( t\right) }{\phi ^{\prime }\left( r\right) }=+\infty \) include \(\phi (x)=x\) or \(\phi (x)=2x\cos (x)\) with \(u\left( X\right) ={\mathbb {R}}_{++}\) or \({\mathbb {R}}_{+}\), and \(\phi (x)=x+\ln (1+x)\) with \(u\left( X\right) ={\mathbb {R}}_{++}\) or \({\mathbb {R}}_{+}\). When utility is bounded both above and below (e.g., there is a best and a worst outcome), it is easy to find examples violating the condition (see Section 4.3 in KMS [21]). Examples satisfying the condition in the theorem include, \(\phi \left( u\right) =\exp \left( u/\theta \right) \) or \(\phi (x)=x\exp \left( x/\theta \right) \) with \(u\left( X\right) ={\mathbb {R}}\) and \(\phi \left( x\right) =\frac{x^{1\gamma }}{1\gamma }\) or \(\phi (x)=x+\frac{x^{1\gamma }}{1\gamma }\) for \(\gamma \ne 1\) with \(u\left( X\right) ={\mathbb {R}}_{++}\) or \({\mathbb {R}}_{+}\).
Following the statement of the No Half Measures condition, we explained that this condition fails when those measures that sometimes get positive weight differ from those that sometimes get arbitrarily close to full weight (where “weight” is measured, as explained earlier, by utility tradeoffs between the limiting frequency event corresponding to the measure and its complementary event). By characterizing No Half Measures in the context of the Extended MEU with contraction, \(\alpha \)MEU and the smooth ambiguity models, we have seen three concrete illustrations of when the Bewley set, based on (magnitudes of) these positive weights or marginal rates of substitution, will involve parameters usually thought of as tastes (e.g., \(\beta \), \(\alpha \) and \(\phi \)) and when it will not. The cases where it will involve these parameters are extensive: For Extended MEU with contraction, No Half Measures fails except for MEU with no contraction or expected utility with an i.i.d. prior. For \(\alpha \)MEU, No Half Measures fails in essentially all the cases beyond MEU and max–max EU. For the smooth ambiguity model, No Half Measures fails whenever the variation in the slope of \(\phi \) is bounded and \(\mu \) is nondegenerate.
5 Appendix: Proofs
Denote by \(B\left( S\right) \) the set of bounded measurable functions on S. Similarly for \(B\left( \Delta \left( S\right) \right) \) and \(B\left( S^{\infty }\right) \). When B is replaced by \(B_{0}\) this denotes the restriction to simple measurable functions. Similarly, when \(B_{0}\) is replaced by \(B_{0}^{l}\) (resp. \(B_{0}^{u}\)) this denotes the restriction to lower (resp. upper) semicontinuous simple functions. For \(b\in B\left( S^{\infty }\right) \), we write \(\left\ b\right\ \) for the supnorm of b (i.e., \(\sup _{\omega }\left b(\omega )\right \)).
5.1 Proof of Theorem 3.1
Step 1 \(g\sim g^{*}\).
Step 2 \(\alpha f+\left( 1\alpha \right) h\sim \alpha \pi f+\left( 1\alpha \right) h\).
Clearly, (iv) implies (ii) and (iii). Moreover, (ii) implies (i) and so does (iii). Thus, we have shown that (i)–(v) are equivalent.
Assume the additional axioms to show equivalence of (i)–(vii).
(v)\(\Leftrightarrow \)(vi): Clearly, (v) implies (vi). By Theorem 4.1, the converse holds.
Our argument proceeds by steps along the following lines: In steps 1–3, given an arbitrary \(h^{\prime }\in {\mathcal {\hat{F}}}\), we construct sequences of simple acts that normconverge to \(h^{\prime }\) from above and from below. In steps 4–7, we show that \({\hat{\succsim }}^{*}\) is the unique extension of \(\succsim ^{*}\) satisfying Ccomplete Preorder, Norm Continuity, Monotonicity, Independence and Nontriviality. In step 8, we use the representation of \(\succsim ^{*}\) in Theorem 4.1 to construct a representation for \({\hat{\succsim }}^{*}\), and use that representation to show (vi) implies (viii).
Step 3 For any \(h^{\prime }\in {\mathcal {\hat{F}}}\), there is \(h_{k}\in {\mathcal {F}}\) such that \(h_{k}\) normconverges to \(h^{\prime }\) and \(h^{\prime }\left( \omega \right) \succsim h_{k}\left( \omega \right) \) for all \(\omega \in \Omega \): Slightly change Steps 1 and 2.
Step 4 \({\hat{\succsim }}\) satisfies Monotonicity: Take \(f,g\in {\mathcal {\hat{F}}}\) such that \(f\left( \omega \right) {\hat{\succsim }}g\left( \omega \right) \) for all \(\omega \in \Omega \). By Steps 2 and 3, there are \(f_{k},g_{k}\in {\mathcal {F}}\) such that \(f_{k},g_{k}\) normconverge to f, g respectively, and \(f_{k}\left( \omega \right) {\hat{\succsim }}f\left( \omega \right) \) and \(g\left( \omega \right) {\hat{\succsim }}g_{k}\left( \omega \right) \) for all \(\omega \in \Omega \). Then, \(f_{k}\left( \omega \right) \succsim g_{k}\left( \omega \right) \) for all \(\omega \in \Omega \), and Monotonicity of \(\succsim \) implies \(f_{k}\succsim g_{k}\), hence \(f_{k}{\hat{\succsim }}g_{k}\). Norm Continuity guarantees \(f{\hat{\succsim }}g\).
Step 5 \({\hat{\succsim }}^{*}\) satisfies Ccomplete Preorder, Norm Continuity, Monotonicity, Independence and Nontriviality: \({\hat{\succsim }}^{*}\) inherits Ccomplete Preorder, Norm Continuity, Monotonicity and Nontriviality from the corresponding properties of \({\hat{\succsim }}\) and satisfies Independence by (5.3).
Step 6 \({\hat{\succsim }}^{*}\) extends \(\succsim ^{*}\): Take \(f,g\in {\mathcal {F}}\) such that \(f\succsim ^{*}g\), that is, \(\alpha f+\left( 1\alpha \right) h\succsim \alpha g+\left( 1\alpha \right) h\) for all \(\alpha \in \left[ 0,1\right] \) and \(h\in {\mathcal {F}}\) (and thus \(\alpha f+\left( 1\alpha \right) h{\hat{\succsim }}\alpha g+\left( 1\alpha \right) h\) for all \(\alpha \in \left[ 0,1\right] \) and \(h\in {\mathcal {F}}\) since \({\hat{\succsim }}\) is an extension of \(\succsim \)). Now fix \(h^{\prime }\in {\mathcal {\hat{F}}}\). By Step 2, there is \(h_{k}\in {\mathcal {F}}\) normconverging to \(h^{\prime }\). Moreover, by the mixture linearity of u, \(\alpha f+\left( 1\alpha \right) h_{k}\) and \(\alpha g+\left( 1\alpha \right) h_{k}\) normconverge to \(\alpha f+\left( 1\alpha \right) h^{\prime }\) and \(\alpha g+\left( 1\alpha \right) h^{\prime }\) respectively. Since \(\alpha f+\left( 1\alpha \right) h_{k}{\hat{\succsim }}\alpha g+\left( 1\alpha \right) h_{k}\) for all \(\alpha \in \left[ 0,1\right] \) and \(k=1,2,\ldots \), Norm Continuity implies \(\alpha f+\left( 1\alpha \right) h^{\prime }{\hat{\succsim }}\alpha g+\left( 1\alpha \right) h^{\prime }\) for all \(\alpha \in \left[ 0,1\right] \). Since \(h^{\prime }\) is arbitrary, \(f{\hat{\succsim }}^{*}g\).
Step 7 All extensions of \(\succsim ^{*}\) satisfying the axioms in Step 5 are the same: Assume \(\succsim _{1}^{*},\succsim _{2}^{*}\) on \({\mathcal {\hat{F}}}\) are two such extensions. It is enough to show that \(f\succsim _{1}^{*}g\) for \(f,g\in {\mathcal {\hat{F}}}\) implies \(f\succsim _{2}^{*}g\) since the labeling of the extensions is arbitrary. By Steps 2 and 3, there are \(f_{k},g_{k}\in {\mathcal {F}}\) such that \(f_{k}\) and \(g_{k}\) normconverge to f and g respectively and \(f_{k}\left( \omega \right) \succsim _{1}^{*}f\left( \omega \right) \), \(g\left( \omega \right) \succsim _{1}^{*}g_{k}\left( \omega \right) \) for all \(\omega \in \Omega \). Thus, \(f_{k}\succsim _{1}^{*}f\succsim _{1}^{*}g\succsim _{1}^{*}g_{k}\). Since \(\succsim _{1}^{*},\succsim _{2}^{*}\) coincide on \({\mathcal {F}}\), \(f_{k}\succsim _{2}^{*}g_{k}\). Norm Continuity of \(\succsim _{2}^{*}\) implies \(f\succsim _{2}^{*}g\).
5.2 Proofs for Sect. 4
It is sometimes convenient in the proofs to use an alternative formulation of No Half Measures. The following lemma gives the alternative formulation and shows that it is equivalent to No Half Measures. We may rely on this lemma without reference to it.
Lemma 5.1
 (a)
No Half Measures;
 (b)
For \(A\in \Sigma ^{\Psi }\), if there exists an \(x\in X\) such that \(1_{S^{\infty }}\succ x\succsim ^{*}1_{A}\), then \(1_{A}\sim ^{*}1_{\emptyset }\).
Proof of Lemma 5.1
(a)\(\Rightarrow \)(b): Fix \(A\in \Sigma ^{\Psi }\). Suppose for each \(x\in X\) with \(1_{S^{\infty }}\succ x\), there exists \(f\in {\mathcal {F}}\) and \(\alpha \in (0,1]\) such that \(\alpha x+(1\alpha )f\not \succsim \alpha 1_{A}+(1\alpha )f\). Then \(x\not \succsim ^{*}1_{A}\) and (b) is vacuously satisfied. Suppose instead that for all \(f\in {\mathcal {F}}\) and \(\alpha \in (0,1]\), \(\alpha 1_{\emptyset }+(1\alpha )f\thicksim \) \(\alpha 1_{A}+(1\alpha )f\). Then \(1_{A}\sim ^{*}1_{\emptyset }\) and (b) is satisfied.
(b)\(\Rightarrow \)(a): Fix \(A\in \Sigma ^{\Psi }\). Suppose (b) holds. Can it be that neither of the possibilities in (a) hold? If neither holds, \(1_{A}\not \sim ^{*}1_{\emptyset }\) and for some \(x\in X\) with \(1_{S^{\infty }}\succ x\), for all \(f\in {\mathcal {F}}\) and \(\alpha \in (0,1]\), \(\alpha x+(1\alpha )f\succsim \alpha 1_{A}+(1\alpha )f\). But then, \(1_{S^{\infty }}\succ x\succsim ^{*}1_{A}\) and \(1_{A}\not \sim ^{*}1_{\emptyset }\) contradicting (b). \(\square \)
5.2.1 Proof of Theorem 4.2
Normalize u so that \(u\left( x^{*}\right) =1\) and \(u\left( x_{*}\right) =0\).
(if): We show No Half Measures. Fix any \(A\in \Sigma ^{\Psi }\) and assume \(1_{S^{\infty }}\succ x\succsim ^{*}1_{A}\). Then, \(u\left( x\right) \ge \int u\left( 1_{A}\right) d\ell ^{\infty }=\ell ^{\infty }\left( A\right) \) for all \(\ell \in R\). Note that \(\ell ^{\infty }\left( A\right) \in \left\{ 0,1\right\} \) for every \(A\in \Sigma ^{\Psi }\). Since \(1_{S^{\infty }}\succ x\), \(1>u\left( x\right) \) and thus \(\ell ^{\infty }\left( A\right) =0\) for all \(\ell \in R\). Then, \(1_{A}\sim ^{*}0\). By Lemma 5.1, No Half Measures holds.
(only if): Since \(\succsim \) satisfies Weak Continuous Symmetry, Theorem 4.1 delivers sets C, R and M. By definition, \(C\subseteq \left\{ \int \ell ^{\infty }dm\left( \ell \right) :m\in \Delta (R)\right\} \). For the other inclusion, follow the steps.
Step 1 For \({\widehat{\ell }}\in \Delta (S)\), if \(\delta _{{\widehat{\ell }}}\notin M\), then \(\max _{m\in M}m\left( L\right) <1\) for some \(L\in {\mathcal {O}}_{{\widehat{\ell }}}\). We show this by contradiction. Suppose that, for every \(L\in {\mathcal {O}}_{{\widehat{\ell }}}\), there is \(m_{L}\in M\) such that \(m_{L}\left( L\right) =1\). We can view \({\mathcal {O}}_{{\widehat{\ell }}}\) as a directed set with the inverse inclusion. Then, \(\left\{ m_{L}\right\} \) is a net and it has a limit point in M because M is compact. The limit point has to be \(\delta _{{\widehat{\ell }}}\), a contradiction.
5.2.2 Proof of Theorem 4.3
Suppose R is not finite. Then, we can take distinct \(\ell _{n}\in R\) for each n. Let \(A_{n}=\bigcup _{k\ge n}\Psi ^{1}\left( \ell _{k}\right) \). Then, \(A_{n}\searrow \emptyset \). To try to verify Monotone Continuity of \(\succsim ^{*}\), take three constant acts \(x\succ x^{\prime }\succ x^{\prime \prime }\). Let \(p_{n}=\ell _{n}^{\infty }\). Then, \(p_{n}\left( A_{n}\right) =1\) and \(u\left( x^{\prime }\right) <p_{n}\left( A_{n}\right) u\left( x\right) +\left( 1p_{n}\left( A_{n}\right) \right) u\left( x^{\prime \prime }\right) \). By Theorem 4.2, \(p_{n}\in C\) and Theorem 4.1 implies that it is not possible that \(x^{\prime }\succsim ^{*}xA_{n}x^{\prime \prime }\) for any n. This contradicts Monotone Continuity of \(\succsim ^{*}\).
5.2.3 Proof of Theorem 4.4
Fix the representation V.
(if) If \(\succsim \) is represented by \(\min _{\ell \in R}\int _{S^{\infty }}u(f)d\ell ^{\infty }\) then the fact that \(\succsim \) satisfies No Half Measures follows from one direction of Theorem 4.5.
5.2.4 Proof of Theorem 4.5
We next show the other direction—that No Half Measures implies (\(\alpha =0\) or \(\alpha =1\) or D singleton). To see this, we assume D nonsingleton and \(\alpha \in (0,1)\) and show that No Half Measures is violated.
5.2.5 Proof of Theorem 4.6
Normalize \(u(x^{*})=1\) and \(u(x_{*})=0\).
Footnotes
 1.
Epstein and Seo [11] assume \(f\sim \pi f\) for all permutations \(\pi \) and acts f. They show that there is a modeling tradeoff between this symmetry axiom, a type of dynamic consistency and ambiguity.
 2.
A measure that always gets less than full belief will not be relevant, which motivates the name, No Half Measures.
 3.
Similarly, though we do not fully investigate it, No Half Measures imposed on the Vector expected utility model (Siniscalchi [28]) will imply nontrivial restrictions on the adjustment function A and the adjustment factors \(\zeta _{i}\) appearing in that model.
References
 1.Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis, 3rd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
 2.AlNajjar, N.I., de Castro, L.: Parametric representation of preferences. J. Econ. Theory 115, 118–133 (2014)MathSciNetzbMATHGoogle Scholar
 3.Arrow, K.: Essays in the Theory of RiskBearing, NorthHolland (1970)Google Scholar
 4.Bewley, T.F.: Knightian decision theory, Part I, Decisions in Economics and Finance 25 (2002), 79–110, first version 1986Google Scholar
 5.de Castro, L., AlNajjar, N.I.: A subjective foundation of objective probability, working paper (2009)Google Scholar
 6.CerreiaVioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L.: Ambiguity and robust statistics. J. Econ. Theory 148, 974–1049 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
 7.de Finetti, B.: La prevision: ses lois logiques, ses sources subjectives. Annales de l’Institut Henri Poincare 7, 1–68 (1937)MathSciNetzbMATHGoogle Scholar
 8.Eichberger, J., Grant, S., Kelsey, D.: Differentiating ambiguity: an expository note. Econ. Theory 36, 327–336 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
 9.Eichberger, J., Grant, S., Kelsey, D., Koshevoy, G.: The \(\alpha \)MEU model: a comment. J. Econ. Theory 146, 1684–1698 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
 10.Epstein, L.G., Seo, K.: Symmetry of evidence without evidence of symmetry. Theor. Econ. 5, 313–368 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
 11.Epstein, L.G., Seo, K.: Symmetry or dynamic consistency?. B.E. J. Theor. Econ. 11, Article 11 (2011)Google Scholar
 12.Epstein, L.G., Seo, K.: De Finetti meets Ellsberg. Res. Econ. 68, 11–26 (2014)CrossRefGoogle Scholar
 13.Gajdos, T., Hayashi, T., Tallon, J.M., Vergnaud, J.C.: Attitude toward imprecise information. J. Econ. Theory 140, 23–56 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
 14.Ghirardato, P., Maccheroni, F., Marinacci, M.: Differentiating ambiguity and ambiguity attitude. J. Econ. Theory 118, 133–173 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
 15.Ghirardato, P., Siniscalchi, M.: A more robust definition of multiple priors, working paper (2010)Google Scholar
 16.Ghirardato, P., Siniscalchi, M.: Ambiguity in the small and in the large. Econometrica 80, 2827–2847 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 17.Gilboa, I., Marinacci, M.: Ambiguity and the Bayesian paradigm, advances in economics and econometrics: theory and applications. In: Acemoglu D., Arellano M., Dekel E. (eds.) Tenth World Congress of the Econometric Society. New York: Cambridge University Press (2013)Google Scholar
 18.Gilboa, I., Schmeidler, D.: Maxmin expected utility with nonunique prior. J. Math. Econ. 18, 141–153 (1989)CrossRefzbMATHGoogle Scholar
 19.Hewitt, E., Savage, L.J.: Symmetric measures on Cartesian products. Trans. Am. Math. Soc. 80, 470–501 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
 20.Klibanoff, P., Marinacci, M., Mukerji, S.: A smooth model of decision making under ambiguity. Econometrica 73, 1849–1892 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
 21.Klibanoff, P., Mukerji, S., Seo, K.: Perceived ambiguity and relevant measures. Econometrica 82, 1945–1978 (2014)Google Scholar
 22.Marinacci, M.: Model Uncertainty. J. Euro. Econ. Assoc. 13, 998–1076 (2015)CrossRefGoogle Scholar
 23.Nau, R.: Uncertainty aversion with secondorder utilities and probabilities. Manag. Sci. 52, 136–145 (2006)CrossRefzbMATHGoogle Scholar
 24.Nehring, K.: Ambiguity in the context of probabilistic beliefs, working paper (2001)Google Scholar
 25.Nehring, K.: Bernoulli without Bayes: a theory of utilitysophisticated preference, working paper (2007)Google Scholar
 26.Seo, K.: Ambiguity and secondorder belief. Econometrica 77, 1575–1605 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
 27.Siniscalchi, M.: A behavioral characterization of plausible priors. J. Econ. Theory 128, 91–135 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
 28.Siniscalchi, M.: Vector expected utility and attitudes toward variation. Econometrica 77, 801–855 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.