# A Bayesian approach to the Machina paradox

Mateus Joffily, Thijs van de Laar

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# A Bayesian approach to the Machina paradox

Mateus Joffily, Thijs van de Laar

**Abstract:**

Variants of the Ellsberg urn experiments introduced by Machina (Am. Econ. Rev., 99(1), 385-392, 2009) have challenged several prominent models of ambiguity aversion. We show that our Bayesian hierarchical model - originally developed to explain Ellsberg-type preferences - also captures the ambiguity preferences observed in Machina's reflection example. Our findings indicate that ambiguity aversion in both the Ellsberg and Machina paradoxes can be attributed to pessimistic prior beliefs about unobserved outcomes. Moreover, the model predicts an asymmetric pattern of preferences across intermediate payoff levels in the reflection example: ambiguity aversion is stronger when the intermediate payoff lies closer to the worst outcome, while the opposite holds for ambiguity-seeking preferences.

**Keywords:**

Machina Paradox; Ambiguity Aversion; Bayesian Modeling

**JEL codes:**

C63; D81; D91
# A Bayesian approach to the Machina paradox

Mateus Joffily* Thijs van de Laar†

November 20, 2025

## Abstract

Variants of the Ellsberg urn experiments introduced by Machina (Am. Econ. Rev., 99(1), 385-392, 2009) have challenged several prominent models of ambiguity aversion. We show that our Bayesian hierarchical model - originally developed to explain Ellsberg-type preferences - also captures the ambiguity preferences observed in Machina's reflection example. Our findings indicate that ambiguity aversion in both the Ellsberg and Machina paradoxes can be attributed to pessimistic prior beliefs about unobserved outcomes. Moreover, the model predicts an asymmetric pattern of preferences across intermediate payoff levels in the reflection example: ambiguity aversion is stronger when the intermediate payoff lies closer to the worst outcome, while the opposite holds for ambiguity-seeking preferences.

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

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

## Introduction

In his seminal work, Ellsberg (1961) proposed a series of urn experiments that revealed the limitations of Savage (1954)'s subjective expected utility (SEU) theory in accounting for ambiguity-averse behavior. To address these limitations, Schmeidler (1989) developed the Choquet expected utility (CEU) theory, which incorporated nonadditive probabilities to capture ambiguity preferences in contexts where SEU failed. Machina (2009) subsequently extended Ellsberg's urn experiments by considering gambles that combined elements of both risk and ambiguity, and demonstrated that CEU also fails to accommodate the observed preferences in these scenarios.

Further research has shown that Machina's examples challenge not only CEU but also several other prominent generalizations of SEU (see Baillon et al. (2011), Blavatskyy (2013) and Machina (2014)). To date, only a limited number of models have proven capable of withstanding Machina's sieve. Notable examples include Blavatskyy (2025)'s modified Choquet expected utility model, Sharpe (2023)'s two-dimensional vector space model, He (2021)'s two-stage evaluation model, Aerts et al. (2018)'s quantum theoretical

<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>
<table><caption>Table 1: Machina (2009)'s reflection example with payoffs parameterized by $0 < \pi < 1$. The events $E_1$, $E_2$, $E_3$ and $E_4$ involve drawing balls marked with numbers from 1 to 4 from a single urn. In each decision problem, the decision maker places a bet on H or L, which assigns different payoffs to the events.</caption><thead><tr><th rowspan="2">problem</th><th rowspan="2">bet</th><th colspan="2">50 balls</th><th colspan="2">50 balls</th></tr><tr><th>E<sub>1</sub></th><th>E<sub>2</sub></th><th>E<sub>3</sub></th><th>E<sub>4</sub></th></tr></thead><tbody><tr><td rowspan="2">1</td><td>H</td><td>100π</td><td>100</td><td>100π</td><td>0</td></tr><tr><td>L</td><td>100π</td><td>100π</td><td>100</td><td>0</td></tr><tr><td rowspan="2">2</td><td>L</td><td>0</td><td>100</td><td>100π</td><td>100π</td></tr><tr><td>H</td><td>0</td><td>100π</td><td>100</td><td>100π</td></tr></tbody></table>

framework, Dillenberger and Segal (2015)'s two-stage recursive ambiguity model and Gul and Pesendorfer (2014)'s expected uncertain utility theory.

In this paper, we show that our Bayesian hierarchical framework, based on the Bolker-Jeffrey theory of conditional subjective expected utility (Jeffrey, 1990) and originally developed to explain Ellsberg-type preferences (Joffily and van de Laar, 2025), can also account for the prevailing preferences observed in Machina's reflection example. The remainder of the paper is organized as follows. Section 1 reviews the reflection example introduced by Machina (2009). Section 2 presents our Bayesian hierarchical model of the reflection example. Section 3 describes the approximate Bayesian method used to estimate the conditional distributions required for the expected utility computation. Section 4 presents the model's predictions for preferences in the reflection example. Finally, Section 5 summarizes our findings and concludes.

# 1 The reflection example

The reflection example proposed by Machina (2009) involves a single urn containing one hundred balls, fifty of which are marked with the numbers 1 or 2, and the rest with 3 or 4. Drawing a ball marked with the number $i$ is denoted by the event $E_i$. Consequently, the probabilities of the composite events $E_1 \cup E_2$ and $E_3 \cup E_4$ are both equal to 0.5, but the individual probabilities of $E_i$ remain unknown. As shown in Table 1, decision makers are asked to choose between two bets, labeled H and L<fnref n="1" />, across two decision problems with distinct payoff schemes parameterised by $0 < \pi < 1$ (Baillon et al., 2011). In contrast to the original Ellsberg urn experiments, in which each bet can be classified as either risky (known probabilities) or ambiguous (unknown probabilities), each bet in the reflection example involves a mixture of known and unknown probabilities.

Two aspects of the reflection example are of particular interest. First, since bets H and L in problem 1 are an informationally symmetric left-right reflection of bets H and L in problem 2, it has been hypothesized that a decision maker who prefers H to L (or L to H) in problem 1 should also prefer H to L (or L to H) in problem 2 (Machina, 2009). By informationally symmetric, it is meant that the evidence for $E_1$ is identical to that for $E_2$, and the evidence for $E_3$ is identical to that for $E_4$, in addition to the fact that $E_1 \cup E_2$ and $E_3 \cup E_4$ are equally likely (Baillon et al., 2011). Second, since the probability

<footnote n="1">¹The labels H and L refer to high and low uncertainty bets, respectively, as will be discussed later.</footnote>
of obtaining the intermediate payoff $(100\pi)$ is objectively known for bet L but not for bet H, and given empirical evidence that preferences in scenarios of uncertainty are typically ambiguity-averse, it has been further hypothesized that bet L should be preferred to bet H in both decision problems (Machina, 2009; L'Haridon and Placido, 2010; Baillon et al., 2011).

L'Haridon and Placido (2010) provided empirical support for these two hypotheses in their study of the reflection example (but see Aerts et al. (2018) for conflicting results). They found that more than 90% of subjects exhibited strict preferences for either L or H, even when indifference was allowed. Of these, around 50% preferred L to H and 30% preferred H to L in both decision problems. Furthermore, the majority of subjects who were ambiguity-averse in the Ellsberg paradox also preferred L to H in the reflection example, suggesting a consistent pattern of ambiguity-averse behavior across related choice contexts.

These findings challenged many prominent models of choice under uncertainty that were specifically designed to address the Ellsberg paradox. In particular, Machina (2009) demonstrated that a CEU maximizer should behave contrary to the left-right reflection hypothesis, meaning that if H was preferred to L in problem 1, then L should be preferred to H in problem 2, and vice-versa. This is due to a property that rank-dependent utility models retain from SEU, namely tail-separability (Machina, 2009). Tail-separability rests on the premise that if two acts share a common payoff over an event $E_i$, then changes in that payoff should not affect the preference ordering between the acts (Machina, 2009; L'Haridon and Placido, 2010).

Baillon et al. (2011) further demonstrated that these preferences also pose challenges for other generalizations of SEU, including Gilboa and Schmeidler (1989)'s maxmin expected utility model, Ghirardato et al. (2004)'s $\alpha$-maxmin expected utility model, Klibanoff et al. (2005)'s smooth model of ambiguity aversion and Maccheroni et al. (2006)'s variational-preference model, but primarily due to the overly restrictive nature of the uncertainty aversion axiom. However, Dominiak and Lefort (2021) argued that these models can indeed accommodate Ellsberg-type preferences when both ambiguity aversion and pessimism about known probabilities are taken into account.

Finally, Blavatskyy (2013) proposed a variant of the reflection example and demonstrated that Siniscalchi (2009)'s vector expected utility model and Nau (2006)'s model of second-order utilities and probabilities cannot explain ambiguity aversion in both the Ellsberg and Machina paradoxes at the same time.

## 2 Theory

In our Bayesian framework, decision problems such as the Ellsberg urn experiments and Machina's examples are modeled on the basis of a generative model and a decision rule (Joffily and van de Laar, 2025). The generative model is a probabilistic representation of how states of the world and actions produce observable outcomes. The decision rule is a function that maps pairs of beliefs and desires into actions. In this section, we describe the generative model and the decision rule that comprise our Bayesian model of the reflection example. 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.
<figure><img src="image_2.png" /><figcaption>Figure 1: Forney-style factor graphs used in the approximate Bayesian computation for the reflection example in (a) the framing stage and (b) the valuation stage. In an FFG, edges represent variables and nodes represent factors. The solid nodes indicate model parameters ($\alpha$, $\alpha'$, $d$, $d'$, $\gamma$, $c$) and a specific realization of action $a$. Dashed boxes summarize the indicated distributions.</figcaption></figure>

## 2.1 The generative model

We formally define the generative model as a joint probability distribution over outcomes $y$, hidden states of the world $x$, actions $a$, and the parameters of the likelihood matrix $\mathbf{A}$, conditional on the set of model assumptions $\mathcal{M}_g$. Accordingly, the generative model is expressed as

$$p(y, x, a, \mathbf{A} | \mathcal{M}_g) = p(y, x, \mathbf{A} | a, \mathcal{M}_g) p(a),$$

where the conditional distribution given $a$ factorizes as

$$p(y, x, \mathbf{A} | a, \mathcal{M}_g) \propto p(y | x, \mathbf{A}) p(x) p(\mathbf{A}) p(y | a, \mathcal{M}_g). \quad (1)$$

Since the right-hand side of (1) defines a double constraint on $y$, the symbol $\propto$ indicates proportionality up to an implicit normalizing constant, which renders the left-hand side a proper probability distribution. The model assumptions $\mathcal{M}_g$ include the payoff vector $\mathbf{c}$, specific to decision problem $g \in \{1, 2\}$, and the optimistic/pessimistic quality $\gamma$ of prior beliefs about unobserved outcomes, which will be explained in more detail below. The outcome $y \in \mathcal{O}$ denotes the drawing of a ball marked with the number $i$ from the urn, where the set $\mathcal{O} = \{E_1, E_2, E_3, E_4\}$. We assume that the events $E_1 \cup E_2$ and $E_3 \cup E_4$ depend on the hidden state of the world $x \in \mathcal{X}$, which can take one of two possible values in the set $\mathcal{X} = \{\text{"E}_1 \cup \text{"E}_2\text{"}, \text{"E}_3 \cup \text{"E}_4\text{"}\}$ with equal probability. Given the true state of the world, one of the two outcomes congruent with that state is realized with a probability unknown to the decision maker. The action $a \in \mathcal{A}$ represents a specific bet from the set of possible actions $\mathcal{A} = \{\text{L}, \text{H}\}$.
We consider that outcomes are governed by the categorical distribution given by

$$p(y = i | x = j, \mathbf{A}) = A_{ij}, \qquad (2)$$

where the likelihood matrix $\mathbf{A}$ in $\mathcal{O} \times \mathcal{X}$ encodes the state-dependent probabilities of outcomes. Prior beliefs about the state-dependent probabilities in $\mathbf{A}$ are represented by independent Dirichlet distributions for the rows of $\mathbf{A}$, and parameterized by the concentration hyperparameter matrix $\alpha$, as

$$p(\mathbf{A}) = \mathrm{Dir}(\mathbf{A}|\boldsymbol{\alpha}), \quad \boldsymbol{\alpha} = \begin{pmatrix} 1.0 & 0.1 \\ 1.0 & 0.1 \\ 0.1 & 1.0 \\ 0.1 & 1.0 \end{pmatrix}.$$

Accordingly, the outcome $E_i$ is represented in the $i$-th row, and the hidden states “$E_1 \cup E_2$” and “$E_3 \cup E_4$” are encoded in the first and second columns of $\mathbf{A}$, respectively. The concentration hyperparameters specify weak (vague) prior beliefs about the probabilities of individual outcomes, yet assume strong (precise) beliefs about the relative likelihood of state-congruent versus state-incongruent outcomes. Specifically, high probabilities are assigned to outcomes congruent with the current state (i.e., events $E_1$ and $E_2$ under state “$E_1 \cup E_2$”, and events $E_3$ and $E_4$ under state “$E_3 \cup E_4$”), while low probabilities are assigned to incongruent outcomes (i.e., events $E_3$ and $E_4$ under state “$E_1 \cup E_2$”, and events $E_1$ and $E_2$ under state “$E_3 \cup E_4$”). Put simply, decision makers are uncertain about the exact probabilities of individual state-congruent outcomes but hold strong beliefs that one of these congruent outcomes will be observed.

Prior beliefs about hidden states are represented by a categorical distribution, as

$$p(x) = \mathrm{Cat}(x|\mathbf{d}), \qquad (3)$$

where the two-elements vector $\mathbf{d} = (0.5, 0.5)$ specifies equal prior probabilities for the two hidden states “$E_1 \cup E_2$” and “$E_3 \cup E_4$”.

The conditional distribution $p(y|a, \mathcal{M}_g)$ represents motivated prior beliefs about unobserved outcomes under the supposition that action $a$ will be performed. Motivated priors are represented by the categorical distribution, as

$$p(y|a, \mathcal{M}_g) = \mathrm{Cat}(y|\sigma(\gamma \mathbf{c}(a))). \qquad (4)$$

We refer to them as motivated because they are a function of the desirability of outcomes, instantiated by the action-dependent payoff vector $\mathbf{c}(a)$ in $\mathcal{O}$. The $i$-th element in $\mathbf{c}(a)$ corresponds to the outcome $E_i$. The softmax (normalized exponential) function $\sigma$ converts the vector $\mathbf{c}(a)$ into a probability distribution. Therefore, for a specific element of $\mathbf{c}(a) = (c_i)$, the corresponding entry in $\sigma(\gamma \mathbf{c}(a)) = (p_i)$ is evaluated as

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

The parameter $\gamma \in \mathbb{R}$ specifies the nature of motivated priors: optimistic ($\gamma > 0$), neutral ($\gamma = 0$), or pessimistic ($\gamma < 0$). Optimistic priors assign higher probabilities to high-desirability outcomes and lower probabilities to low-desirability outcomes; neutral priors assign equal probabilities to all outcomes regardless of their desirability; and
pessimistic priors assign higher probabilities to low-desirability outcomes and lower probabilities to high-desirability outcomes.

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

$$p(a) = \delta[a - a^*], \qquad (5)$$

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

## 2.2 The decision rule

We adopt a decision rule based on the Bolker-Jeffrey conditional expected utility equation (Jeffrey, 1990). The conditional expected utility (EU) of action $a$ is, thus, the sum of the utilities of outcomes weighted by their probabilities under the supposition that $a$ will be performed. In our framework, the utilities of outcomes are defined as the logarithm of the softmax-normalized payoff vector, given by

$$\mathbf{u}(a) = \log \sigma(\beta \mathbf{c}(a)),$$

where $\beta > 0$ is a scale factor and $\sigma$ is the softmax function. Thus, for a specific element of $\mathbf{c}(a) = (c_i)$, the corresponding entry in $\mathbf{u}(a) = (u_i)$ evaluates to

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

Because the term inside the logarithm, known as the partition function, is the same for all outcomes, the utility of the outcome $E_i$ is simply the $\beta$-scaled payoff $c_i$ added to a constant term. 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 (Ortega and Braun, 2009; Friston et al., 2015; Vries et al., 2025). This definition of the utility function further emphasizes the close relationship between utility and motivated priors in (4).

The conditional EU of action $a$ in decision problem $g$ is thus expressed as

$$\mathrm{EU}(a, \mathcal{M}_g) = \sum_i u_i p(y = E_i | a, \mathcal{M}_g), \qquad (6)$$

where the conditional probability distribution over outcomes is given by

$$p(y|a, \mathcal{M}_g) = \sum_x \int_{\mathbf{A}} p(y|x, \mathbf{A}) p(x, \mathbf{A}|a, \mathcal{M}_g) d\mathbf{A}, \qquad (7)$$

and the updated prior beliefs about $x$ and $\mathbf{A}$ is obtained by marginalizing out the outcomes in (1), as

$$p(x, \mathbf{A}|a, \mathcal{M}_g) = \sum_y p(y, x, \mathbf{A}|a, \mathcal{M}_g). \qquad (8)$$

Since $p(x, \mathbf{A}|a, \mathcal{M}_g)$ is conditioned on motivated priors $p(y|a, \mathcal{M}_g)$, as opposed to actual observations, we refer to them as “updated prior beliefs” rather than “posterior beliefs” (Vries et al., 2025). Finally, the optimal action $a^*$ is the one that maximizes EU, as

$$a^* = \arg\max_a \mathrm{EU}(a, \mathcal{M}_g).$$
## 3 Approximate Bayesian computation

In order to calculate the EU of each action, we used an approximate Bayesian computation method 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 a FFG, edges represent variables and nodes represent the functional relationships between variables.

We employed a two-stage procedure to compute the approximate conditional probability of outcomes<fnref n="2" />. The corresponding FFGs used in each stage are depicted in Figures 1.a and 1.b. Conceptually, this two-stage scheme recalls the two phases of the decision-making process devised by Tversky and Kahneman (1992) in cumulative prospect theory, namely the framing and valuation phases<fnref n="3" />.

In the framing stage, a variational message passing procedure is performed on the FFG depicted in Figure 1.a to approximate the updated prior distribution in (8) (Dauwels, 2007). This yields approximate (updated) marginal priors $q(x)q(A) \approx p(x, A|a, M_g)$, where $q(x) = \text{Cat}(x|\mathbf{d}')$ and $q(A) = \text{Dir}(\mathbf{A}|\alpha')$. In the valuation stage, $q(x)$ and $q(A)$ serve as priors for message passing on the second FFG depicted in Figure 1.b. This FFG reuses the likelihood model in (2) to compute the conditional probability of outcomes in (7), which is then used to estimate the value of each action within the decision rule.

## 4 Results

The adequacy of our model in explaining the preferences observed in the reflection example was evaluated for $\gamma$ values in the interval $[-8.0, 8.0]$, with $\beta$ fixed at 1.0. For convenience, the payoffs indicated in table 1 were divided by 100, so that the payoff vector $\mathbf{c}$ can be written as

$$ \mathbf{c}(a) = \begin{cases} (\pi, \ 1.0, \ \pi, \ 0.0) & \text{if a bet on H in problem 1,} \\ (\pi, \ \pi, \ 1.0, \ 0.0) & \text{if a bet on L in problem 1,} \\ (0.0, \ 1.0, \ \pi, \ \pi) & \text{if a bet on L in problem 2,} \\ (0.0, \ \pi, \ 1.0, \ \pi) & \text{if a bet on H in problem 2.} \end{cases} $$

The model's predicted preferences for each problem were evaluated using the difference between the expected conditional utilities of bets H and L for intermediate payoff values within the range $0 \le \pi \le 1$. A positive difference indicated ambiguity seeking, a negative difference indicated ambiguity aversion, and a value of zero implied indifference. The absolute value of this difference was interpreted as the strength of preference (Alós-Ferrer and Garagnani, 2022).

<footnote n="2">²We 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>

<footnote n="3">³"In the framing phase, the decision maker constructs a representation of the acts, contingencies, and outcomes that are relevant to the decision. In the valuation phase, the decision maker assesses the value of each prospect and chooses accordingly." (Tversky and Kahneman, 1992).</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 2: Difference between the conditional expected utilities of bets H and L as a function of the intermediate payoff ($\pi$) and optimism/pessimism ($\gamma$) in problems 1 and 2.</figcaption></figure>

As shown in Figure 2, preferences depend on both the magnitude of $\gamma$ and the intermediate payoff. Across both problems, the model predicts ambiguity-seeking, indifference, and ambiguity-averse preferences when motivated priors are optimistic ($\gamma > 0$), neutral ($\gamma = 0$), and pessimistic ($\gamma < 0$), respectively. When the intermediate payoff equals either the worst (0) or the best (1) payoff, the model predicts indifference for all values of $\gamma$. These results indicate that the prevalent ambiguity aversion observed in the reflection example can be attributed to pessimistic prior beliefs in our model (L’Haridon and Placido, 2010).

Notably, ambiguity aversion becomes stronger as the intermediate payoff moves from the midpoint ($\pi = 0.5$) toward the worst outcome. Conversely, the strength of ambiguity seeking increases as the intermediate payoff moves from the midpoint toward the best outcome. This asymmetry becomes more pronounced as the magnitude of $\gamma$ increases. Given empirical evidence that choice frequencies in experimental settings are proportional to the strength of preferences (Alós-Ferrer and Garagnani, 2022), these results suggest that ambiguity-averse behavior should occur more frequently when the intermediate payoff approaches the worst outcome, while ambiguity-seeking behavior should occur more frequently when the intermediate payoff approaches the best outcome.

To elucidate the beliefs underlying these preferences, the updated prior beliefs about hidden states, $q(x)$, and the conditional probability of outcomes given a bet, $p(y|a, \mathcal{M}_g)$, are presented in tables 2 and 3, respectively, for the cases of $\gamma \in \{-8, 0, 8\}$ and $\pi \in \{0.25, 5.0, 0.75\}$. As shown in table 2, the probabilities of the hidden states “$E_1 \cup E_2$” and “$E_3 \cup E_4$” deviate from 0.5 when motivated priors are pessimistic ($\gamma = -8$) or optimistic ($\gamma = 8$). Although this may appear to contradict the initial prior, in which both states are assumed to be equally likely, such asymmetry naturally arises from Bayesian updating when the motivated priors are pessimistic or optimistic ($\gamma \neq 0$). To see this, notice that in our model the observation of a specific event $E_i$ resolves uncertainty about the hidden state. For instance, after observing $E_1$, the decision maker can infer that the true state is “$E_1 \cup E_2$”. Similarly, pessimistic and optimistic priors are informative themselves and thus influence the updated prior beliefs about hidden states. As a result, $q(x)$ is biased toward the state containing the best outcome ($1$) under optimistic priors, and toward the state containing the worst outcome ($0$) under pessimistic priors.

Finally, as shown in table 3, the conditional probabilities of individual outcomes depend both on their own desirability and on that of the paired outcome. In general, less
<table><caption>Table 2: Updated prior beliefs about hidden states, $q(x)$, under the assumption that a bet will be placed on H and L for $\pi \in \{0.25, 0.50, 0.75\}$ and $\gamma \in \{-8, 0, 8\}$ in problems 1 and 2. The pair of payoffs associated with each state is indicated between parentheses above the probabilities.</caption><thead><tr><th rowspan="2">problem</th><th rowspan="2">bet</th><th rowspan="2">π</th><th colspan="2">γ = -8</th><th colspan="2">γ = 0</th><th colspan="2">γ = 8</th></tr><tr><th>"E<sub>1</sub> ∪ E<sub>2</sub>"</th><th>"E<sub>3</sub> ∪ E<sub>4</sub>"</th><th>"E<sub>1</sub> ∪ E<sub>2</sub>"</th><th>"E<sub>3</sub> ∪ E<sub>4</sub>"</th><th>"E<sub>1</sub> ∪ E<sub>2</sub>"</th><th>"E<sub>3</sub> ∪ E<sub>4</sub>"</th></tr></thead><tbody><tr><td rowspan="12">1</td><td rowspan="3">H</td><td>0.25</td><td>(π, 1)</td><td>(π, 0)</td><td>(π, 1)</td><td>(π, 0)</td><td>(π, 1)</td><td>(π, 0)</td></tr><tr><td>0.50</td><td>0.078</td><td>0.922</td><td>0.500</td><td>0.500</td><td>0.998</td><td>0.002</td></tr><tr><td>0.75</td><td>0.011</td><td>0.989</td><td>0.500</td><td>0.500</td><td>0.989</td><td>0.011</td></tr><tr><td rowspan="3">L</td><td>0.25</td><td>0.002</td><td>0.998</td><td>0.500</td><td>0.500</td><td>0.922</td><td>0.078</td></tr><tr><td>0.50</td><td>(π, π)</td><td>(1, 0)</td><td>(π, π)</td><td>(1, 0)</td><td>(π, π)</td><td>(1, 0)</td></tr><tr><td>0.75</td><td>0.145</td><td>0.855</td><td>0.500</td><td>0.500</td><td>0.003</td><td>0.997</td></tr><tr><td rowspan="3">2</td><td>0.25</td><td>0.021</td><td>0.979</td><td>0.500</td><td>0.500</td><td>0.021</td><td>0.979</td></tr><tr><td>0.50</td><td>0.003</td><td>0.997</td><td>0.500</td><td>0.500</td><td>0.145</td><td>0.855</td></tr><tr><td>0.75</td><td>(0, 1)</td><td>(π, π)</td><td>(0, 1)</td><td>(π, π)</td><td>(0, 1)</td><td>(π, π)</td></tr><tr><td rowspan="6">H</td><td rowspan="3">L</td><td>0.25</td><td>0.855</td><td>0.145</td><td>0.500</td><td>0.500</td><td>0.997</td><td>0.003</td></tr><tr><td>0.50</td><td>0.979</td><td>0.021</td><td>0.500</td><td>0.500</td><td>0.979</td><td>0.021</td></tr><tr><td>0.75</td><td>0.997</td><td>0.003</td><td>0.500</td><td>0.500</td><td>0.855</td><td>0.145</td></tr><tr><td rowspan="3">L</td><td>0.25</td><td>(0, π)</td><td>(1, π)</td><td>(0, π)</td><td>(1, π)</td><td>(0, π)</td><td>(1, π)</td></tr><tr><td>0.50</td><td>0.922</td><td>0.078</td><td>0.500</td><td>0.500</td><td>0.002</td><td>0.998</td></tr><tr><td>0.75</td><td>0.989</td><td>0.011</td><td>0.500</td><td>0.500</td><td>0.011</td><td>0.989</td></tr><tr><td rowspan="3">L</td><td>0.25</td><td>0.001</td><td>0.269</td><td>0.729</td><td>0.250</td><td>0.250</td><td>0.001</td><td>0.269</td></tr><tr><td>0.50</td><td>0.009</td><td>0.267</td><td>0.715</td><td>0.250</td><td>0.250</td><td>0.009</td><td>0.267</td></tr><tr><td>0.75</td><td>0.001</td><td>0.269</td><td>0.729</td><td>0.250</td><td>0.250</td><td>0.066</td><td>0.251</td></tr><tr><td rowspan="6">H</td><td rowspan="3">L</td><td>0.25</td><td>(0)</td><td>(1)</td><td>(π)</td><td>(π)</td><td>(0)</td><td>(1)</td><td>(π)</td></tr><tr><td>0.50</td><td>0.618</td><td>0.251</td><td>0.066</td><td>0.066</td><td>0.250</td><td>0.250</td><td>0.001</td><td>0.269</td></tr><tr><td>0.75</td><td>0.715</td><td>0.267</td><td>0.009</td><td>0.009</td><td>0.250</td><td>0.250</td><td>0.267</td><td>0.009</td></tr><tr><td rowspan="3">L</td><td>0.25</td><td>0.729</td><td>0.269</td><td>0.001</td><td>0.001</td><td>0.250</td><td>0.250</td><td>0.251</td><td>0.066</td></tr><tr><td>0.50</td><td>(0)</td><td>(π)</td><td>(1)</td><td>(π)</td><td>(0)</td><td>(π)</td><td>(1)</td><td>(π)</td></tr><tr><td>0.75</td><td>0.649</td><td>0.282</td><td>0.032</td><td>0.037</td><td>0.250</td><td>0.250</td><td>0.001</td><td>0.269</td></tr></tbody></table>

<table><caption>Table 3: Conditional probability of outcomes, $p(y|a, M_g)$, under the assumption that a bet will be placed on H and L for $\pi \in \{0.25, 0.50, 0.75\}$ and $\gamma \in \{-8, 0, 8\}$ in problems 1 and 2. The payoff in each event is indicated between parentheses above the probabilities.</caption><thead><tr><th rowspan="2">problem</th><th rowspan="2">bet</th><th rowspan="2">π</th><th colspan="4">γ = -8</th><th colspan="4">γ = 0</th><th colspan="4">γ = 8</th></tr><tr><th>E<sub>1</sub></th><th>E<sub>2</sub></th><th>E<sub>3</sub></th><th>E<sub>4</sub></th><th>E<sub>1</sub></th><th>E<sub>2</sub></th><th>E<sub>3</sub></th><th>E<sub>4</sub></th><th>E<sub>1</sub></th><th>E<sub>2</sub></th><th>E<sub>3</sub></th><th>E<sub>4</sub></th></tr></thead><tbody><tr><td rowspan="12">1</td><td rowspan="3">H</td><td>0.25</td><td>(π)</td><td>(1)</td><td>(π)</td><td>(0)</td><td>(π)</td><td>(1)</td><td>(π)</td><td>(0)</td><td>(π)</td><td>(1)</td><td>(π)</td><td>(0)</td></tr><tr><td>0.50</td><td>0.037</td><td>0.032</td><td>0.282</td><td>0.649</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.269</td><td>0.729</td><td>0.001</td><td>0.001</td></tr><tr><td>0.75</td><td>0.005</td><td>0.005</td><td>0.271</td><td>0.720</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.271</td><td>0.720</td><td>0.005</td><td>0.005</td></tr><tr><td rowspan="3">L</td><td>0.25</td><td>0.001</td><td>0.001</td><td>0.269</td><td>0.729</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.282</td><td>0.649</td><td>0.037</td><td>0.032</td></tr><tr><td>0.50</td><td>(π)</td><td>(π)</td><td>(1)</td><td>(0)</td><td>(π)</td><td>(π)</td><td>(1)</td><td>(0)</td><td>(π)</td><td>(π)</td><td>(1)</td><td>(0)</td></tr><tr><td>0.75</td><td>0.066</td><td>0.066</td><td>0.251</td><td>0.618</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.001</td><td>0.001</td><td>0.729</td><td>0.269</td></tr><tr><td rowspan="3">L</td><td>0.25</td><td>0.009</td><td>0.009</td><td>0.267</td><td>0.715</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.009</td><td>0.009</td><td>0.715</td><td>0.267</td></tr><tr><td>0.50</td><td>0.001</td><td>0.001</td><td>0.269</td><td>0.729</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.066</td><td>0.066</td><td>0.618</td><td>0.251</td></tr><tr><td>0.75</td><td>(0)</td><td>(1)</td><td>(π)</td><td>(π)</td><td>(0)</td><td>(1)</td><td>(π)</td><td>(π)</td><td>(0)</td><td>(1)</td><td>(π)</td><td>(π)</td></tr><tr><td rowspan="6">H</td><td rowspan="3">L</td><td>0.25</td><td>(0)</td><td>(1)</td><td>(π)</td><td>(π)</td><td>(0)</td><td>(1)</td><td>(π)</td><td>(π)</td><td>(0)</td><td>(1)</td><td>(π)</td><td>(π)</td></tr><tr><td>0.50</td><td>0.618</td><td>0.251</td><td>0.066</td><td>0.066</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.269</td><td>0.729</td><td>0.001</td><td>0.001</td></tr><tr><td>0.75</td><td>0.715</td><td>0.267</td><td>0.009</td><td>0.009</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.267</td><td>0.715</td><td>0.009</td><td>0.009</td></tr><tr><td rowspan="3">L</td><td>0.25</td><td>0.729</td><td>0.269</td><td>0.001</td><td>0.001</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.251</td><td>0.618</td><td>0.066</td><td>0.066</td></tr><tr><td>0.50</td><td>(0)</td><td>(π)</td><td>(1)</td><td>(π)</td><td>(0)</td><td>(π)</td><td>(1)</td><td>(π)</td><td>(0)</td><td>(π)</td><td>(1)</td><td>(π)</td></tr><tr><td>0.75</td><td>0.649</td><td>0.282</td><td>0.032</td><td>0.037</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.250</td><td>0.001</td><td>0.001</td><td>0.729</td><td>0.269</td></tr></tbody></table>

(more) desirable outcomes are assigned higher probabilities under pessimistic (optimistic) priors. However, because the worst and best outcomes are always paired in bet L but not in bet H, the probability mass is allocated differently across paired outcomes in the two bets. Consequently, the expected utility of bet L is higher (lower) than that of bet H when motivated priors are pessimistic (optimistic).
# 5 Conclusion

Our Bayesian framework has successfully accounted for the prevailing preferences observed in two classical paradoxes that have long inspired the development of decision models under uncertainty: the Ellsberg (1961) and Machina (2009) paradoxes (see also Joffily and van de Laar (2025)). In both cases, pessimistic prior beliefs about unobserved outcomes emerged as a common determinant of ambiguity aversion.

Importantly, our results revealed an asymmetric pattern of preferences across intermediate payoff levels. Among pessimistic decision makers, the strength of ambiguity aversion increases as the intermediate payoff approaches the worst outcome, whereas among optimistic decision makers, the strength of ambiguity seeking increases as the intermediate payoff approaches the best outcome. Considering the monotonic relation between choice frequency and strength of preference reported in experimental studies (Alós-Ferrer and Garagnani, 2022), we predict that ambiguity-averse behavior should occur more frequently when the intermediate payoff is closer to the worst outcome, while ambiguity-seeking behavior should occur more frequently when the intermediate payoff is closer to the best outcome. Empirical evaluation of these predictions represent a promising avenue for future research.

Furthermore, stochastic choice behavior can be incorporated into our framework by replacing the marginal distribution over actions in (5) with a Boltzmann distribution, using the conditional expected utility in (6) as its argument, and assuming that the selected bet is sampled from this distribution (see Ortega and Braun (2013) for a discussion of Boltzmann distributions in decision-making). This formulation yields a softmax decision rule, in which the probability of selecting action *a* is proportional to its relative conditional expected utility (Sutton and Barto, 1998; Friston et al., 2015).

Finally, due to its inherent flexibility, our Bayesian framework is particularly well-suited for modeling decision problems that extend beyond the classical urn paradigm. It can effectively accommodate more complex, naturalistic scenarios, such as those involving sequential choices, belief updating, and larger outcome spaces.

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