# A Bayesian hierarchical model of Ellsberg-type preferences

Mateus Joffily, Thijs van de Laar

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# A Bayesian hierarchical model of Ellsberg-type preferences

Mateus Joffily, Thijs van de Laar

**Abstract:**

We consider a Bayesian hierarchical model of Ellsberg-type preferences based on conditional subjective expected utility. In our model, optimism and pessimism emanate from motivated prior beliefs about unobserved outcomes. This implies that optimism and pessimism can only propagate along the structure of the hierarchical model when the probability of outcomes is unknown (ambiguity) rather than known (risk). We show that the prevailing preferences observed in Ellsberg (1961)'s two-urn experiment and Jabarian and Lazarus (2023)'s two-ball experiment are reproduced by our model when motivated priors are pessimistic.

**Keywords:**

Ellsberg Paradox; Ambiguity Aversion; Bayesian Modeling

**JEL codes:**

C63; D81; D91
# A Bayesian hierarchical model of Ellsberg-type preferences

Mateus Joffily* Thijs van de Laar†

June 12, 2025

## Abstract

We consider a Bayesian hierarchical model of Ellsberg-type preferences based on conditional subjective expected utility. In our model, optimism and pessimism emanate from motivated prior beliefs about unobserved outcomes. This implies that optimism and pessimism can only propagate along the structure of the hierarchical model when the probability of outcomes is unknown (ambiguity) rather than known (risk). We show that the prevailing preferences observed in Ellsberg (1961)'s two-urn experiment and Jabarian and Lazarus (2023)'s two-ball experiment are reproduced by our model when motivated priors are pessimistic.

**Keywords:** Ellsberg Paradox; Ambiguity Aversion; Bayesian Modeling

**JEL Classification:** C63; D81; D91

## Introduction

How people perceive and deal with uncertainty when making decisions remains an open question in behavioral economics and related fields. A common approach is to associate uncertainty with probabilistic beliefs about variables of interest, such as states of the world and outcomes. Uncertainty about the probability of outcomes has traditionally been classified into two types: risk and ambiguity (Camerer and Weber, 1992). In the case of risk, the decision maker is fully informed about the probability of outcomes, so probabilities are said to be known and beliefs about them precise. In ambiguity, on the other hand, the decision maker is not at all, or only partially, informed about the probability of outcomes, so probabilities are considered unknown and beliefs about them vague.

In his seminal work, Leonard Savage (1954) laid the foundations of subjective expected utility (SEU) and showed that, given a set of rationality principles, choices under ambiguity can be reduced to choices under risk. Later, Daniel Ellsberg (1961) challenged SEU with a set of experiments that became known as the Ellsberg paradox. Ellsberg's two-urn

<footnote>*CNRS, Université Lumière Lyon 2, Université Jean Monnet Saint-Etienne, emlyon business school, GATE, 69007 Lyon, France. mateus.joffily@cnrs.fr.</footnote>

<footnote>†Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands. t.w.v.d.laar@tue.nl.</footnote>
experiment compared a risky urn with an ambiguous one, and showed that ambiguity-
averse preferences violate the SEU axioms or, more specifically, the sure-thing principle.
The sure-thing principle postulates that a decision maker who is determined to take a
certain action, either knowing that an event will happen or knowing that it will not hap-
pen, must also take that same action when knowing nothing about the event. Despite
its broad acceptance, the principle has been shown to be valid only in cases where states
of the world are independent of actions, which has direct implications for Ellsberg-type
preferences (Jeffrey, 1982). Since then, the Ellsberg paradox has been the subject of
much debate, leading to the development of models of decision under uncertainty that go
beyond the realms of Bayesian decision theory (Gilboa and Marinacci, 2016; Etner et al.,
2012).

In what follows, we present a Bayesian hierarchical model that can account for Ellsberg-
type preferences. Rather than proposing a new theory of subjective expected utility for
decisions under uncertainty, we draw on the Bolker-Jeffrey (1990) theory of conditional
subjective expected utility (CSEU) and show the existence of prior beliefs that are con-
sistent with Ellsberg-type preferences. The contribution of our model is threefold. First,
it dissolves the paradoxical nature of Ellsberg-type preferences from the perspective of
Bayesian decision theory. Second, it proposes a general framework that incorporates op-
timism and pessimism into Bayesian decision models, thereby highlighting a common
ground between Bayesian reasoning and motivated beliefs (Bénabou and Tirole, 2016).
Third, as an inherent feature of being Bayesian, our model can comfortably accommo-
date not only single choice problems, but also sequential choices that depend on updating
beliefs.

The paper is organized as follows. In section 1, we give a general description of our Bayesian framework based on the Bolker-Jeffrey (1990) theory of CSEU and describe the two-urn experiment of Ellsberg (1961) and the two-ball experiment of Jabarian and Lazarus (2023). In section 2, our decision model is specified in terms of a generative model and a decision rule. In section 3, we describe the approximate Bayesian method employed to compute the conditional distributions in our model. In section 4, we present the theoretical results for the two-urn and two-ball experiments based on the predictions of our model. In section 5, we discuss our approach in the light of alternative models and conclude.

# 1 Problem Statement

Bayesian models of decision making are generally specified on the basis of two compo-
nents. The first is a probabilistic model of how states of the world and actions generate
observable outcomes, called the generative model. The second is a function that maps
pairs of beliefs and desires to actions, known as the decision rule. In our Bayesian frame-
work, the generative model includes not only beliefs about the structure of the task, which
encodes the environmental causes of observations, but also motivated prior beliefs about
unobserved outcomes. Motivated priors can be optimistic, neutral or pessimistic, de-
pending on whether the probability they assign to the outcomes is directly proportional,
independent or inversely proportional to their desirability, respectively. However, as we
will see below, Bayes' rule ensures that motivated priors can only propagate along the
structure of the model when the probability of outcomes is unknown (ambiguity) rather
than known (risk). This feature leads to updated prior beliefs about the probability of
outcomes that differ between ambiguous and risky situations. Crucially, this difference
<table><caption>Table 1: The two-urn experiment. Subjects consider two gambles based on drawing balls from either the risky urn R or the ambiguous urn A. In gamble 1, a prize of $100 is offered if a red ball is drawn from the chosen urn and $0 otherwise. In gamble 2, a prize of $100 is offered if a blue ball is drawn from the chosen urn and $0 otherwise.</caption><thead><tr><th rowspan="2">gamble</th><th colspan="2">urn R</th><th colspan="2">urn A</th></tr><tr><th>red<br>(50)</th><th>blue<br>(50)</th><th>red<br>(100)</th><th>blue<br>(100)</th></tr></thead><tbody><tr><td>1</td><td>$100</td><td>$0</td><td>$100</td><td>$0</td></tr><tr><td>2</td><td>$0</td><td>$100</td><td>$0</td><td>$100</td></tr></tbody></table>

is such that a decision rule based on maximizing conditional expected utility will imply Ellsberg-type preferences when prior beliefs are pessimistic.

## 1.1 The Bolker-Jeffrey theory

The Bolker-Jeffrey (1990) theory of CSEU, as it has been known after Ethan Bolker (1967)'s representation theorem, was proposed as an alternative to Savage (1954)'s SEU, considered too restrictive and artificial to model real-world problems. While SEU treated actions as functions attaching a unique outcome to each state of the world, Bolker-Jeffrey's CSEU considers actions as propositions, just like outcomes and states, but which the decision maker can choose to make true. Consequently, beliefs about outcomes and their hidden causes can be expressed conditional upon the supposition that an action will be performed. It can be demonstrated that the Bolker-Jeffrey's equation of expected utility is partition invariant. That is to say, the desirability of a proposition can be expressed as a conditional expected utility over any countable set of mutually exclusive, jointly exhaustive propositions (Joyce, 2000). Interest in the Bolker-Jeffrey framework has long been limited due to the fact that pairs of beliefs and desirabilities cannot be uniquely determined on the basis of pairwise choices, as is the case with SEU (Bolker, 1967). However, alternative representation theorems for subjective probability and desirability have recently been proposed to address this non-uniqueness problem (Joyce, 1999; Bradley, 2017).

## 1.2 The two-urn experiment

The two-urn experiment proposed by Ellsberg (1961) involved drawing balls from either a risky urn R or an ambiguous urn A. The urn R contained a known number of fifty red balls and fifty blue balls, while the urn A contained an unknown mixture of one hundred red and blue balls, as described in Table 1. Subjects were confronted with two gambles. In gamble 1, they were offered to bet on one of the two urns for a prize of $100 if a red ball was drawn, and $0 otherwise. In gamble 2, a prize of $100 was offered if a blue ball was drawn, and $0 otherwise. According to SEU, if subjects believe that the number of red balls is less than the number of blue balls in urn A, they must choose urn R in gamble 1 but urn A in gamble 2. Conversely, if they believe the opposite to be true, they must choose urn A in gamble 1 but urn R in gamble 2. Both preferences are equally valid, but they depend on the subjects' prior beliefs about the composition of urn A. Subsequent experimental studies confirmed that most people strictly prefer urn R to urn
<table><caption>Table 2: The two-ball experiment. Subjects are offered to draw two balls with replacement from the (R)isky urn or the (A)mbiguous urn for a prize of $100 if the two balls match in color and $0 otherwise.</caption><thead><tr><th rowspan="2">gamble</th><th rowspan="2">draws</th><th colspan="2">urn R</th><th colspan="2">urn A</th></tr><tr><th>red<br>(50)</th><th>blue<br>(50)</th><th>red<br>(100)</th><th>blue<br>(100)</th></tr></thead><tbody><tr><td rowspan="2">3</td><td>red</td><td>$100</td><td>$0</td><td>$100</td><td>$0</td></tr><tr><td>blue</td><td>$0</td><td>$100</td><td>$0</td><td>$100</td></tr></tbody></table>

A in both gambles, indicating an ambiguity-averse preference that cannot be explained by SEU (Trautmann and van de Kuilen, 2015).

## 1.3 The two-ball experiment

Jabarian and Lazarus (2023) proposed a variant of the two-urn experiment, which relied on two successive draws from the same urn. The interest of the two-ball experiment is that it involves a gamble in which betting on urn A, although ambiguous, is objectively more advantageous than urn R. As described in Table 2, subjects were offered to draw two balls with replacement from one of the two urns for a prize of $100 if the two balls matched in color, and $0 otherwise. From the SEU perspective, the worst-case scenario is that urn A contains 50 red balls and 50 blue balls, just like urn R, implying a winning probability of 0.5. Any other composition of urn A implies a higher winning probability for a bet on urn A than urn R. Intriguingly, Jabarian and Lazarus (2023) observed that 55% of subjects preferred R to A, leading the authors to conclude that “subjects avoid ambiguity per se rather than avoiding ambiguity because it may yield a worse outcome”.

# 2 Theory

In this section, we formally introduce our Bayesian framework, which consists of two components, namely the generative model and the decision rule. As a notational convention, we will write random variables and their realizations in lowercase letters, matrices and collections in boldface, and sets in calligraphic letters. In addition, a collection of ordered variables will be written in bold with a ranged subscript, $z_{1:K} := (z_1, \dots, z_K)$.

## 2.1 The generative model

The generative model is defined as a joint probability distribution over a collection of outcomes $\mathbf{y}_{1:K}$, outcome probabilities $\boldsymbol{\theta}$ and action $a$ given by

$$p(\mathbf{y}_{1:K}, \boldsymbol{\theta}, a | \mathbf{m}_g) = p(\mathbf{y}_{1:K}, \boldsymbol{\theta} | a, \mathbf{m}_g) p(a), \quad (1)$$

where the conditional distribution given $a$ is further factorized as

$$p(\mathbf{y}_{1:K}, \boldsymbol{\theta} | a, \mathbf{m}_g) \propto p(\mathbf{y}_{1:K} | \boldsymbol{\theta}) p(\boldsymbol{\theta} | a) p(\mathbf{y}_{1:K} | \mathbf{m}_g). \quad (2)$$

Since the right-hand side of (2) defines a double constraint on $\mathbf{y}_{1:K}$, the symbol $\propto$ indicates proportionality up to an implicit normalizing constant, which renders the left-hand side a proper probability distribution. The model assumptions denoted by $\mathbf{m}_g$ refer
<figure><img src="image_3.png" /><figcaption>Figure 1: Forney-style factor graphs of the models used for the approximate Bayesian computation of (a) the updated priors, and (b) the conditional beliefs about unobserved outcomes in the two-urn experiment. In an FFG, edges represent variables and nodes represent factors. The solid nodes indicate a specific realization for action a, the deterministic values of the hyperparameters α and α', the payoff matrix C, the optimistic/neutral/pessimistic quality γ of prior beliefs about outcomes. Dashed boxes summarize the indicated distributions.</figcaption></figure>

to parameters specific to each gamble $g \in \{1, 2, 3\}$, including the number of ball draws K, the payoff matrix C and the optimistic/pessimistic quality γ of prior beliefs about unobserved outcomes, which will be explained in more detail below. The outcome $y_k \in \mathcal{O}$ denotes the color of the k-th ball drawn from the set of possible ball colors $\mathcal{O} = \{\text{red, blue}\}$, and the ordered collection $\mathbf{y}_{1:K} := (y_1, \dots, y_K)$ refers to K balls drawn with replacement from the same urn. In the case of the two-urn experiment, $K = 1$, whereas in the case of the two-ball experiment, $K = 2$. The action $a \in \mathcal{A}$ represents a bet on a specific urn from the set $\mathcal{A} = \{\text{R, A}\}$, and the vector of probabilities $\boldsymbol{\theta} = (\theta_1, \theta_2)$ encodes the ratio of red ($\theta_1$) and blue balls ($\theta_2$) in the selected urn.

We consider independent and identically distributed outcomes, so that the distribution over outcomes is given by

$$p(\mathbf{y}_{1:K}|\boldsymbol{\theta}) = \prod_{k=1}^{K} p(y_k|\boldsymbol{\theta}), \qquad (3)$$

and each outcome is governed by the categorical distribution, as

$$p(y_k|\boldsymbol{\theta}) = \text{Cat}(y_k|\boldsymbol{\theta}). \qquad (4)$$

Prior beliefs about the probability of outcomes are represented by a Dirichlet distribution.
<figure><img src="image_4.png" /><figcaption>Figure 2: Forney-style factor graphs of the models used for the approximate Bayesian computation of (a) the updated priors, and (b) the conditional beliefs about unobserved outcomes in the two-ball experiment. In an FFG, edges represent variables and nodes represent factors. The solid nodes indicate a specific realization for action a, the deterministic values of the hyperparameters α and α', the payoff matrix C, the optimistic/neutral/pessimistic quality γ of prior beliefs about outcomes. Dashed boxes summarize the indicated distributions.</figcaption></figure>

bution, which is the conjugate prior of the categorical distribution, as

$$p(\theta|a) = \text{Dir}(\theta|\alpha(a)), \qquad (5)$$

where α(a) is a two-element vector containing the concentration hyperparameters of the Dirichlet distribution controlled by action a. We conceptualise ambiguity as a continuum, with risk falling at the extreme where beliefs about the probability of outcomes are extremely precise. Accordingly, we assign large and small concentration values to α(R) and α(A), respectively, so that the precision of prior beliefs about θ is high for the risky urn but low for the ambiguous one.

We introduce motivated priors $p(y_{1:K}|\mathbf{m}_g)$ over unobserved outcomes that depend on the desirability of the outcomes. In the two-urn experiment, since a single ball is drawn from the selected urn, motivated priors are represented by a univariate categorical distribution, as

$$p(y_1|\mathbf{m}_g) = \text{Cat}(y_1|\sigma(\gamma C)), \qquad (6)$$

where C is a single row payoff matrix in $O$, $\gamma \in \mathbb{R}$ encodes the optimistic, neutral or pessimistic quality of motivated beliefs, and $\sigma$ denotes the softmax (normalized exponential)
function. The softmax function converts the vector of payoffs into a probability distribution, so that for a specific element of $\mathbf{C} = (c_i)$, the corresponding entry in $\sigma(\gamma \mathbf{C}) = (p_i)$ is evaluated as

$$p_i = \frac{\exp(\gamma c_i)}{\sum_j \exp(\gamma c_j)},$$

Therefore, motivated priors can be:

* optimistic, when $\gamma > 0$, with high probabilities assigned to high-desirability outcomes and low probabilities assigned to low-desirability outcomes;

* neutral, when $\gamma = 0$, with equal probability assigned to all outcomes regardless of desirability;

* pessimistic, when $\gamma < 0$, with high probabilities assigned to low-desirability outcomes and low probabilities assigned to high-desirability outcomes.

In the two-ball experiment, since two balls are drawn instead of one, $\mathbf{C}$ is a matrix in $\mathcal{O} \times \mathcal{O}$ whose rows and columns represent the payoff on each pair of colors. Therefore, motivated priors are captured by a contingency table that parameterizes a bivariate generalization of the categorical distribution (contingency distribution), as

$$p(\mathbf{y}_{1:2} | \mathbf{m}_g) = \mathrm{Con}(\mathbf{y}_{1:2} | \sigma(\gamma \mathbf{C})). \qquad (7)$$

Finally, beliefs about actions are represented by a delta distribution, expressed as

$$p(a) = \delta[a - a^*],$$

where $\delta$ is the Kronecker delta function and $a^*$ is the optimal action according to the decision rule described below.

## 2.2 The decision rule

We adopt a decision rule based on Bolker-Jeffrey (1990)'s equation of expected utility. Hence, the expected utility (EU) of action $a$ in gamble $g$ is the sum of the utilities of outcomes $\mathbf{y}_{1:K}$ weighted by their updated conditional probabilities, as

$$\mathrm{EU}(a, \mathbf{m}_g) = \sum_{\mathbf{y}_{1:K}} u(\mathbf{y}_{1:K}) p(\mathbf{y}_{1:K} | a, \mathbf{m}_g),$$

where the conditional probability distribution over outcomes under the supposition that $a$ will be performed is given by

$$p(\mathbf{y}_{1:K} | a, \mathbf{m}_g) = \int p(\mathbf{y}_{1:K} | \boldsymbol{\theta}) p(\boldsymbol{\theta} | a, \mathbf{m}_g) d\boldsymbol{\theta}, \qquad (8)$$

and the updated prior beliefs about the probability of outcomes is obtained by marginalizing out the outcomes in (2), as

$$p(\boldsymbol{\theta} | a, \mathbf{m}_g) = \sum_{\mathbf{y}_{1:K}} p(\mathbf{y}_{1:K}, \boldsymbol{\theta} | a, \mathbf{m}_g). \qquad (9)$$
This means that $p(\theta|a, \mathbf{m}_g)$ is conditioned on motivated priors $p(\mathbf{y}_{1:K}|\mathbf{m}_g)$, as opposed to actual observations. For this reason, we refer to them as “updated prior beliefs” instead of “posterior beliefs” (Vries et al., 2025). It is important to note that the updated prior $p(\theta|a, \mathbf{m}_g)$ will be very similar to the prior $p(\theta|a)$ in (5) for bets on the risky urn, but will often differ for bets on the ambiguous urn. This is because prior beliefs about the probability of outcomes are, by definition, precise for the risky urn and thus less susceptible to be modified by motivated priors. As we shall see, the different role played by motivated priors on updated priors in the context of risk and ambiguity will be key to explaining Ellsberg-type preferences.

We define the utility of outcomes as the logarithm of the softmax-normalized payoff matrix, as

$$u(\mathbf{y}_{1:K}) = \log \sigma(\beta \mathbf{C}),$$

where $\beta > 0$ is a scale factor. Thus, for a specific element of $\mathbf{C} = (c_i)$, the corresponding entry in the vector $u(\mathbf{y}_{1:K}) = (u_i)$ evaluates to

$$u_i = \beta c_i - \log \left( \sum_j \exp(\beta c_j) \right).$$

Since the term inside the logarithm, known as the partition function, is the same for all outcomes, the utility of outcome $i$ is simply the $\beta$-scaled payoff $c_i$ added to a constant term. Interestingly, this softmax-normalization of utility is common practice in information-theoretic models of bounded rationality, so that utility is given the same unit of measurement as Shannon information and entropy (Ortega and Braun, 2009; Friston et al., 2015; Vries et al., 2025). Furthermore, this emphasizes the close relationship between utility and motivated priors in (6) and (7).

Finally, the optimal action $a^*$ is the one that maximizes the EU, as

$$a^* = \arg\max_a \text{EU}(a, \mathbf{m}_g).$$

## 3 Approximate Bayesian computation

In order to calculate the EU of each bet, the conditional probability of outcomes in (8) was estimated using an approximate Bayesian computation based on factor graphs (Yedidia et al., 2003). A Forney-style factor graph (FFG) offers a convenient way to represent the factorization of probabilistic models and to estimate marginal probability distributions (Cox et al., 2019). In an FFG, edges represent variables and nodes represent the functional relationships between variables. The FFG of the conditional distribution in (2) is shown in Figure 1.a for the two-urn experiment and in Figure 2.a for the two-ball experiment.

We employed a two-step approach to compute updated priors and conditional beliefs about the probability of unobserved outcomes<fnref n="1" />. In the first step, a message passing procedure approximates (9) by computing the approximate (marginal) updated prior $q(\theta) \sim p(\theta|a, \mathbf{m}_g)$, where $q(\theta) = \text{Dir}(\theta|\alpha'(a))$. For acyclic factor graphs like the one of

<footnote n="1">¹We used the RxInfer.jl probabilistic programming package (version 4.0.1) as a tool for automated derivation of message passing algorithms (Bagaev et al., 2023). The codes written in the Julia programming language (Bezanson et al., 2017) that generate the results in this paper are available upon request.</footnote>
<figure><img src="image_5.png" /><figcaption>Figure 3: Difference between the EU of bets on urn A and urn B as a function of $\gamma \in [-8.0, 8.0]$ and $\beta \in \{0.5, 1.0, 2.0\}$ for (a) gamble 1 and (b) gamble 2.</figcaption></figure>

the two-urn experiment, the sum-product message passing algorithm provides the exact updated distribution (Yedidia et al., 2003). However, for factor graphs like the one of the two-ball experiment with loops, an approximate solution needs to be found based on variational message passing (Dauwels, 2007).

In the second step, $q(\theta)$ is used as a prior for message passing on another (reduced) FFG shown in Figures 1.b and 2.b for the two-urn and the two-ball experiments, respectively. This reduced model reuses the likelihood model (3) to compute the (approximate) conditional probability of outcomes (8) used in the decision rule. Note that in this second step both graphs are acyclic.

# 4 Results

To illustrate the suitability of our model, we estimated the preferences in the two-urn and the two-ball experiments for different values of $\gamma \in [-8.0, 8.0]$ and $\beta \in (0.5, 1.0, 2.0)$. We considered prior beliefs about $\theta$ that are centered at 0.5 for every gamble, but with greater precision for the urn $R$ than for the urn $A$, so that $\alpha(R)$ was set to (100,100) and $\alpha(A)$ to (1,1).

For convenience, we replaced $100 in tables 1 and 2 with 1.0 in the payoff matrices, so that the payoff matrices for the two-urn experiment in table 1 were

$$C = \begin{cases} (1.0 \quad 0.0) & \text{in gamble 1,} \\ (0.0 \quad 1.0) & \text{in gamble 2,} \end{cases}$$

where the first element corresponds to the payoff on a red ball and the second on a blue ball. For the two-ball experiment in table 2, we considered two alternative payoff schemes, named gamble 3a and gamble 3b, given by
$$ C = \begin{cases} \begin{pmatrix} 1.0 & 0.0 \\ 0.0 & 1.0 \end{pmatrix} & \text{in gamble 3a,} \\ \begin{pmatrix} 1.0 & 0.0 \\ 0.0 & 0.1 \end{pmatrix} & \text{in gamble 3b,} \end{cases} $$

where the first row of the payoff matrices refers to the outcome pairs (red,red) and (red,blue), and the second row to (blue,red) and (blue,blue).

Gamble 3a is the most straightforward implementation of the two-ball experiment, in which equally high payoffs are assigned to both winning outcome pairs (red,red) and (blue,blue), and low payoffs to the losing outcome pairs (red,blue) and (blue,red). Thus, the entries in the payoff matrix satisfy the condition $c_{1,1} = c_{2,2} > c_{1,2} = c_{2,1}$.

Gamble 3b introduces a slight modification to this payoff scheme and considers that one of the winning outcome pairs has a higher payoff than the other, so that $c_{1,1}$ is either higher or lower than $c_{2,2}$. This inequality is based on the speculative assumption that decision-makers evaluate each color in isolation and that being the color under consideration increases its value relative to the alternatives. It is important to note that the alternation of values between the two best outcomes, i.e. (red,red) and (blue,blue), does not alter the overall EU value of the bets. In other words, the value of each bet is invariant to alternating values between $c_{1,1}$ and $c_{2,2}$. For this reason, we only considered the case where $c_{1,1} > c_{2,2} > c_{1,2} = c_{2,1}$ in our analysis.

We calculated the difference between EU(A, $m_g$) and EU(R, $m_g$) for each gamble g in the two-urn and two-ball experiments. Ambiguity-seeking, -neutral, and -averse preference was considered whenever that difference was greater than, equal to or less than zero, respectively. To show how optimism ($\gamma > 0$), neutrality ($\gamma = 0$) and pessimism ($\gamma < 0$) affect beliefs about $\theta$, we plotted the latter against $\gamma$ based on the supposition that a bet would be made on the risky and the ambiguous urns separately.

## 4.1 The two-urn experiment

The difference between EU(A, $m_g$) and EU(R, $m_g$) for gambles 1 and 2 in the two-urn experiment are shown in figure 3. We can see that ambiguity-seeking, -neutral and -averse preferences are predicted whenever motivated priors are, respectively, optimistic, neutral and pessimistic. This indicates that the ambiguity-averse preference observed empirically in the two-urn experiment can be explained by pessimism in our model. Furthermore, the strength of ambiguity attitudes, measured as the magnitude of the difference between EU(A, $m_g$) and EU(R, $m_g$), is directly proportional to the scale of the utility of outcomes parameterized by $\beta$.

The relationship between motivated priors and updated priors is illustrated in figure 4 for three values of $\gamma$. In gamble 1 and gamble 2, in which the winning color is red and blue, respectively, we can see that the updated prior distributions over the probability of drawing a red ball, i.e., $\theta_1$, remain unchanged when a bet is made on the risky urn for all $\gamma$. However, when a bet is made on the ambiguous urn, the updated prior distributions are skewed towards low (high) probabilities of the winning color if motivated priors are pessimistic (optimistic). Nevertheless, they remain centered at 0.5 if motivated priors are neutral in gambles 1 and 2. This distortion of updated priors due to motivated priors
<figure><img src="image_6.png" /><figcaption>Figure 4: Updated prior beliefs about the probability of drawing a red ball ($\theta_1$) based on the supposition that a bet is made on (a) the risky urn and (b) the ambiguous urn in gamble 1, and on (c) the risky urn and (d) the ambiguous urn in gamble 2. Note that $q(\theta_1) \sim p(\theta_1|a, m_g)$. The plots in each row correspond to pessimistic ($\gamma = -8$), neutral ($\gamma = 0$) and optimistic ($\gamma = 8$) motivated priors, and the parameter $\beta$ was set to 1 for all analysis.</figcaption></figure>

when a bet is made on the ambiguous urn but not on the risky urn explains the ambiguity-seeking, neutral and ambiguity-averse preferences observed in the two-urn experiment.

## 4.2 The two-ball experiment

The difference between EU(A, $m_g$) and EU(R, $m_g$) for gambles 3a and 3b in the two-ball experiment are shown in figure 5. In gamble 3a, only ambiguity-seeking preferences are predicted for all $\gamma$, suggesting that pessimism is unable to reverse the preference for the ambiguous urn in this payoff scheme. However, in gamble 3b, we can see that ambiguity-averse preferences arise for large enough pessimism, suggesting that an unequal payoff between the two winning outcome pairs, i.e. (red,red) and (blue,blue), combined with pessimism can explain the ambiguity-averse preferences observed empirically in the two-ball experiment (Jabarian and Lazarus, 2023).

To clarify these results, we also plotted the updated priors for three values of $\gamma$ in figure 6. As in the two-urn experiment, motivated priors can only change prior beliefs about the probability of drawing a red ball, $\theta_1$, when a bet is made on the ambiguous urn, but not the risky one. In gamble 3a, due to the symmetry of the payoff matrix, the updated priors given a bet on the ambiguous urn remain centered at 0.5 regardless of optimism, neutrality or pessimism. However, when the payoff on one of the winning outcome pairs
<figure><img src="image_7.png" /><figcaption>Figure 5: Difference between the EU of bets on urn A and urn B as a function of $\gamma \in [-8.0, 8.0]$ and $\beta \in \{0.5, 1.0, 2.0\}$ for (a) gamble 3a and (b) gamble 3b.</figcaption></figure>

is larger than the other in gamble 3b, the updated prior distributions over $\theta_1$ are skewed toward low (high) probabilities when motivated priors are pessimistic (optimistic). Thus, preferences can also be reversed from ambiguity-seeking to ambiguity-avoidance when pessimism is sufficiently large and the utilities of the winning pairs are slightly different in the two-ball experiment.

# 5 Discussion and conclusion

Previous attempts to explain the Ellsberg paradox from the perspective of Bayesian decision theory have largely relied on Savage (1954)'s SEU framework and non-hierarchical generative models. In this paper, we presented a Bayesian hierarchical model based on Jeffrey (1990)'s theory of CSEU that proved able to explain the prevailing preferences observed in both the Ellsberg (1961)'s two-urn experiment and the Jabarian and Lazarus (2023)'s two-ball experiment. A distinctive feature of our model is that it includes motivated prior beliefs about unobserved outcomes that can only propagate along the structure of the hierarchical model when the prior beliefs about the probability of outcomes is unknown (ambiguity) rather than known (risk).

Other models have also attempted to offer a Bayesian explanation for Ellsberg-type preferences in the past. Dobbs (1991) supposed that decision makers calculate an *ex ante* expected utility of actions based on a concave/convex ambiguity function of their *ex post* expected utility, which implies an assumption similar to pessimism/optimism in our framework. Halevy and Feltkamp (2005) showed that, if the Ellsberg two-urn experiment is treated as a sequential choice problem, risk-averse decision makers with a well-defined Bayesian prior should also be ambiguity-averse. Bradley (2017) claimed that Ellsberg-type preferences are both rational and consistent with Savage's axioms, provided that decision makers do not value chances linearly. Ortega and Braun (2013) proposed a thermodynamically inspired model of bounded rationality, which shares similarities with ours in terms of optimistic and pessimistic distortions of beliefs in the context of ambiguity, but relies on variational (Bayes) free-energy minimization under an (inverse) temperature constraint. Vieider (2024) approached ambiguity-averse preferences as a consequence of noisy mental signals representing choice quantities. Our model adds to these by relying
<figure><img src="image_8.png" /><figcaption>Figure 6: Updated prior beliefs about the probability of drawing a red ball ($\theta_1$) based on the supposition that a bet is made on (a) the risky urn and (b) the ambiguous urn in gamble 3a, and on (c) the risky urn and (d) the ambiguous urn in gamble 3b. Note that $q(\theta_1) \sim p(\theta_1|a, m_g)$. The plots in each row correspond to pessimistic ($\gamma = -8$), neutral ($\gamma = 0$) and optimistic ($\gamma = 8$) motivated beliefs, and the parameter $\beta$ was set to 1 for all analysis.</figcaption></figure>

exclusively on motivated priors under the principle of expected utility maximization and the Baye's rule. Moreover, the way in which motivated priors are incorporated into our model guarantees that all updated beliefs throughout the structure of the hierarchical model are internally consistent. Our framework is flexible enough to accommodate a wide range of decision problems, as long as the generative model and the desirability of the outcomes are specified.

The assumption that pessimism and optimism play a role in attitudes toward ambiguity is not unique to our model. Other non-Bayesian models have also explored this hypothesis in the past (Gilboa and Marinacci, 2016). One of the most influential is the maxmin expected utility (MEU) model developed by Gilboa and Schmeidler (1989). MEU considers a non-unique prior over the probability of outcomes and the maximization of the (minimum) expected utility given the worst possible state of the world. Attitudes of ambiguity aversion predicted by MEU are similar to those of our Bayesian model when prior beliefs about the probability of outcomes are replaced by a delta distribution centered on the worst-case scenario, i.e. the lowest probability of the winning outcome. Other models extended MEU and considered not only ambiguity-averse, but also ambiguity-seeking attitudes, such as the α-MEU (Ghirardato et al., 2004) and the smooth model of ambiguity (Klibanoff et al., 2005). More recently, Chandrasekher et al. (2022) proposed the dual-self
expected utility (DSEU) model, in which optimism and pessimism coexist to characterize
degrees of ambiguity preference ranging from ambiguity aversion to ambiguity seeking. On
the other hand, although our model also considers degrees of attitudes towards ambiguity
ranging from aversion to seeking, it does not require conflicting motives to coexist.

Experimental studies have investigated the relationship between subjective beliefs, pessimism and ambiguity aversion (Bier and Connell, 1994; Pulford, 2009; Zhang, 2022). Of particular interest, Zhang (2022) assessed the entire subjective prior distribution of decision makers and showed that the shape of prior beliefs correlates with the degree of ambiguity aversion. More specifically, ambiguity aversion was positively associated with subjective probabilities on states representing undesired outcomes, i.e. negative stock price growth, and negatively associated with subjective probabilities on states representing desired outcomes, i.e. positive stock price growth. This finding corroborates the relationship between the shape of motivated priors and ambiguity-averse preferences predicted by our model. In a similar vein, source theory (Baillon et al., 2025) also relies on subjective probability distortions to explain ambiguity aversion, but these distortions are obtained by means of a probability weighting function.

Moreover, like previous works, our hierarchical model relies on second-order proba-
bilities (beliefs about probabilities) to explain Ellsberg-type preferences (Seo, 2009; Nau,
2006; Klibanoff et al., 2005; Segal, 1987, 1990). However, unlike most of these works, it
does not require the violation of the principle of reduction of compound lotteries. On the
contrary, it is based on motivated priors in compliance with Bayes' rule.

In his seminal paper, Ellsberg (1961) concluded that “[…], the Bayesian or Savage approach gives wrong predictions and, by their lights, bad advice.” Instead, we would say that this may be true for the Savage approach, but not for the Bayesian approach in general.

6 Acknowledgments

This study was supported by the LABEX CORTEX of the University of Lyon [ANR-11-LABX-0042] to MJ.

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