2.1. Ellsberg paradox

Ellsberg (Reference Ellsberg1961) proposed two thought-experiment decision problems, which have been the primary examples motivating research on ambiguity. One of them, known as the two-colour problem, goes as follows.^{Footnote 2}

The four possible event-contingent payment schemes of the problem are presented in Table 1. They depend on the colour of the ball drawn from one of the urns.

Table 1.

Ellsberg two-colour problem

Typically, what is observed is that people are indifferent on the colour on which to bet: they express indifference between f and f’, and between g and g’ (which is noted $f \sim f'$ and $g \sim g'$ , respectively); but prefer betting on $R100$ rather than on $E100$ : they express a preference for g over f and for g’ over f’ (noted $g \succ f$ and $g' \succ f'$ ). Such preferences are, however, incompatible with the existence of subjective probabilities: the ranking $f \prec g$ would imply that the probability of drawing a red ball in urn $E100$ is considered to be less than 0.5 (i.e. less than the probability of drawing a red ball in urn $R100$ ), while at the same time the ranking $f' \prec g'$ would imply that the probability of drawing a black ball in urn $E100$ is considered to be less than 0.5 (i.e. less than the probability of drawing a black ball in $R100$ ). Overall, this would result in having the probabilities of complementary events summing up to less than 1, which is inconsistent. As bets f and f’ involve ambiguity, whereas bets g and g’ involve risk, these typically observed preferences indicate ambiguity aversion, that is, a preference for known-probability bets over unknown ones.

2.2. Decomposing ambiguity into layers of uncertainty

Following the works of Hansen (Reference Hansen2014), Marinacci (Reference Marinacci2015) and Hansen and Marinacci (Reference Hansen and Marinacci2016), themselves building on early insights of Arrow (Reference Arrow1951), ambiguity can generally be decomposed into different layers of uncertainty: (i) risk within a model, in which the uncertainty is about the outcomes of a given (probability) model that specifies fully the outcome probabilities; (ii) model uncertainty, in which the uncertainty is about which probability model (among a given set) should be used to assign these probabilities; and (iii) model misspecification, in which the uncertainty is induced by whether or not the correct model lies among the set under consideration (Aydogan et al. Reference Aydogan, Berger, Bosetti and Liu2020). While most uncertain situations in real-life usually encompass the three layers of uncertainty together, Ellsberg’s (Reference Ellsberg1961) examples are particularly remarkable because they generate ambiguity using only the layers of type (i) and (ii) (indeed, model misspecification is nonexistent in Ellsberg’s setup). For example, in the two-colour problem described above, the composition of the urn $E100$ is necessarily one of the 101 compositions physically possible.^{Footnote 3}

The main distinction between the layers of risk and model uncertainty is the nature of the uncertainty they feature. A risk represents an objectively uncertain prospect. It refers to the physical quantification of uncertainty by means of a probability model (and as such is sometimes referred to as an instance of physical or aleatory uncertainty). It is usually referred to as a lottery, represented by the gamble $L = \left( {{x_1},{p_1}; \ldots ;{x_n},{p_n}} \right)$ that yields outcome ${x_i}$ with a well-defined objective probability ${p_i}$ . This probability ${p_i}$ is a measure of randomness. For example, a risk may be characterized by a specific composition of an urn from which a ball is randomly drawn. If the balls in the urn can only have two colours (e.g. red and black), the risk may further be characterized by the binary lottery ${x_p}y$ , yielding x with probability p and y otherwise.

The uncertainty about which probability model (or risk) should be considered is not of the same nature.^{Footnote 4} In Ellsberg’s problem, this second layer of uncertainty has an epistemic nature, as no objective probabilities may be assigned to the possible compositions of the urn. The prospect determining the correct composition of the urn is thus subjectively uncertain. In theory, a DM can still form a probability measure over the possible urn compositions, but these probabilities are necessarily subjective, reflecting the DM’s degree of belief in each possible model. Under this decomposition of ambiguity into different layers, ambiguity aversion results from a preference for risk over model uncertainty (see Marinacci Reference Marinacci2015, for a theoretical discussion; and Berger and Bosetti Reference Berger and Bosetti2020, for experimental evidence).

3.1. Subjective expected utility

Under the benchmark approach of subjective expected utility (SEU), a preference satisfying Savage’s (Reference Savage1954) axioms is represented by the criterion

${U_{SEU}}({\rm{a}}) = \sum\limits_\theta (\sum\limits_s u({\rm{a}}(s))\,{m_\theta }(s))\,\mu (\theta ) = \sum\limits_s u({\rm{a}}(s))\,\overline m(s).$

In these expressions, u is a von Neumann–Morgernstern utility function. It translates economic consequences, measured in monetary terms, into utility levels, and is known to capture risk attitudes. As shown within the bigger parentheses, it is possible, for each model ${m_\theta }$ , to compute the expected payoff of a given act. This expected payoff takes an expected utility form: it is computed as the weighted sum of utilities obtained from the consequences in each state of the environment. To address model uncertainty, the DM has a subjective prior probability distribution $\mu $ that quantifies her belief about the true parameter $\theta $ . Thus, $\mu (\theta )$ is the DM’s subjective belief that ${m_\theta }$ is the true model. As the DM does not know which is the correct model, she considers the expected payoff of each possible model and aggregates them out by performing a weighted average according to the relative weights that she associates with each of them (i.e. her prior beliefs $\mu (\theta )$ ). The first representation hereabove is the two-layer version of SEU that has been axiomatized by Cerreia-Vioglio et al. (Reference Cerreia-Vioglio, Maccheroni, Marinacci and Montrucchio2013). Under this framework, the layers of risk and model uncertainty are implicitly treated in the same way. The second representation is the original reduced form of Savage (Reference Savage1954), in which $\overline m$ is the predictive probability induced by the prior $\mu $ : $\overline m(s) = \sum\nolimits_\theta {m_\theta }(s)\,\mu (\theta )$ for all states s. In Ellsberg’s experiment, for example, a bet on the ambiguous urn $E100$ , giving $x = 100$ if correct and $y = 0$ otherwise, yields $${U_{SEU}} = {1 \over {101}}\sum\nolimits_{\theta = 0}^{100} {{\theta \over {100}}} u(100) = {1 \over 2}u(100)$$ , which is exactly the same as a bet on the risky urn $R100$ .^{Footnote 6}

3.2. Smooth ambiguity

To accommodate the phenomenon of ambiguity aversion highlighted in Ellsberg (Reference Ellsberg1961), the theoretical economic literature has followed different perspectives. One of them, known as the smooth ambiguity approach (Klibanoff et al. Reference Klibanoff, Marinacci and Mukerji2005, KMM hereinafter), also adopts a Bayesian perspective by having the two layers of uncertainty quantified by a single probability measure. However, in contrast with SEU, a distinct treatment is allowed for the layers of risk and model uncertainty. By letting the function v represent the DM’s attitude toward model uncertainty, a natural generalization of the SEU criterion emerges as a version of the smooth ambiguity model of KMM. Under this framework, acts are ranked according to the criterion

${U_{KMM}}({\rm{a}}) = \sum\limits_\theta \phi (\sum\limits_s u({\rm{a}}(s))\,{m_\theta }(s))\,\mu (\theta ).$

A stronger aversion to the layer of model uncertainty than to that of risk leads to ambiguity aversion, modelled by a concave function $\phi = v \circ {u^{ - 1}}$ . Under this criterion, a bet on urn $E100$ yields $${U_{KMM}} = {1 \over {101}}\sum\nolimits_{\theta = 0}^{100} \phi \left( {{\theta \over {100}}u(100)} \right)$$ , and it is easy to show that ${U_{KMM}} \le \phi \left( {{U_{SEU}}} \right)$ under ambiguity aversion. This interpretation of a non-neutral attitude toward ambiguity stems from non-reduction of objectively and subjectively determined probabilistic judgements (i.e. different confidence in such judgements).

3.3. Other approaches

The smooth ambiguity approach allows for viewing ambiguity à la Ellsberg as a two-stage uncertain prospect, in which the uncertainty in each stage corresponds to a distinct layer. Other approaches also model ambiguity as multiple stages of uncertainty while not necessarily making a distinction between the layers of risk and model uncertainty (see Aydogan et al. Reference Aydogan, Berger, Bosetti and Liu2020, for more details). This is for example the case of the approaches proposed by Segal (Reference Segal1987) (recursive rank dependent utility approach, RRDU) and by Seo (Reference Seo2009). As these approaches do not distinguish between objectively and subjectively determined probabilistic judgements in the distinct layers, they made no distinction between compound risk (in which there are two stages of risk) and ambiguity à la Ellsberg (in which there are a layer of risk and a layer of model uncertainty). Hence, as non-neutral ambiguity attitudes may result from the violation of reduction of compound lotteries, an elementary rationality condition, these theories assign to ambiguity attitudes a purely descriptive status.

4.1. Numbers of potential models

The first, natural, way to interpret these situations derives from the number of possible compositions an ambiguous urn may entail. From Table 2, we can observe that:^{Footnote 7}

Observation 1. The number of potential models, noted $|M|$ , physically compatible with each form of partial ambiguity is increasing in n (strictly increasing in the cases of ${I_n}$ and ${D_n}$ , and weakly increasing – first increasing, then constant– in the case of ${T_n}$ ).

More specifically, the number of possible models lies between 1, when there is no ambiguity and 101, when there is full ambiguity. Under this interpretation, what makes a situation of partial ambiguity thus corresponds to lying between these boundaries: $|M| \in (1,101)$ .

4.2. Information and its structure

Amount of information. The partial ambiguity situations can also be analysed in terms of the amount of information they carry. For example, in the case of interval ambiguity, the situations lie between one in which you know the colour of each individual ball in the urn (in the no ambiguity situation, ${I_0}$ ) and a situation in which you do not know the colour of any of the balls (in the full ambiguity situation, ${I_{50}}$ ). More generally, the information available in the different forms of partial ambiguity can be summarized as follows:

• In situations ${I_n},$ it is known that the urn is composed of at least $\left( {50 - n} \right)$ red balls, and at least $\left( {50 - n} \right)$ black balls,

• In situations ${D_n},$ it is known that at least $\left( {100 - n} \right)$ balls in the urn have the same colour,

• In situations ${T_n},$ it is known that there are exactly either $\left( {50 - n} \right)$ or $\left( {50 + n} \right)$ red (or black) balls in the urn.

We can thus observe the following.

Observation 2. The amount of information available, within each form of partial ambiguity, is decreasing in n (strictly decreasing in the cases of ${I_n}$ and ${D_n}$ , and weakly decreasing – first decreasing, then constant– in the case of ${T_n}$ ).

Note that this interpretation, in terms of the amount of information, has sometimes been related to the one in terms of the numbers of potential models presented before. This is, for example, the case in the work of Einhorn and Hogarth (Reference Einhorn and Hogarth1985: 435), who wrote: ‘Thus, ambiguity results from the uncertainty associated with specifying which of a set of distributions is appropriate in a given situation. Moreover, the amount of ambiguity is an increasing function of the number of distributions that are not ruled out (or made implausible) by one’s knowledge of the situation.’

Quality of information. Alternatively, the partial ambiguity situations may also be presented in terms of the quality of information they carry, using their information structure. The DM’s information about the likelihoods of the different states (and thus the outcome of the bet) is a priori modelled by a set M. By letting $S = \{ r,b\} $ be the state space, in which the ball drawn from the urn is either red (r) or black (b), the set of models in each uncertain situation can be expressed as $M = \left\{ {m\,\,{\rm{such \,\,that}}\,\,P(r) \in I} \right\}.$ In this expression, $P(r)$ is the probability that the ball drawn is red, and $I \subseteq \left[ {0,1} \right]$ is a set-theoretic modelling of information. It represents the available information characterizing the chances to make a correct bet. In urn $E100$ for example, $$I = \left\{ {{0 \over {100}},{1 \over {100}}, \ldots ,{{100} \over {100}}} \right\}$$ . Assuming that the DM has information about M, her acts needs to be measurable with respect to M. The set of models can then be partially ordered by the ‘finer than’ and the ‘coarser than’ relations:

Definition. If M’ and M are such that I’ is a subset of I, then M’ is said to be finer than M, and M is said to be coarser than M’.

In words, this means that better information regarding the structure of ambiguity may be naturally modelled by a smaller set of potential models. As such, a situation is called full ambiguity if its associated set of models is maximal as in $E100$ , and is called partial ambiguity if its associated set of models is finer than in $E100$ . We can then observe the following.

Observation 3.

• In situations ${I_n}$ and ${D_n}$ , the set of models becomes coarser as n increases,

• The use of set theory remains silent on the ranking of situations under ${T_n}$ .

Regarding the second point, note that the use of set theory does not enable us to distinguish ${T_{25}}$ from ${T_{50}}$ , whose associated sets of models M can be ranked by neither the finer nor the coarser relation.