# Wishful Thinking is Risky Thinking:
A Statistical-Distance Based Approach

Jarrod Burgh Emerson Melo

###### Abstract.

We develop a model of wishful thinking that incorporates the costs and
benefits of biased beliefs. We establish the connection between distorted
beliefs and risk, revealing how wishful thinking can be understood in terms of
risk measures. Our model accommodates extreme beliefs, allowing wishful-
thinking decision-makers to assign zero probability to undesirable states and
positive probability to otherwise impossible states. Furthermore, we establish
that wishful thinking behavior is equivalent to quantile-utility maximization
for the class of threshold beliefs distortion cost functions. Finally,
exploiting this equivalence, we derive conditions under which an optimistic
decision-maker prefers skewed and riskier choices.

Department of Economics, Indiana University, Bloomington, IN 47408, USA.
Email<EMAIL_ADDRESS>
JEL classification: D01, D80, D84
Keywords: wishful thinking, cognitive dissonance, risk, optimism, quantile
maximization, preference for skewness

##  1\. Introduction

Wishful thinking (WT) behavior refers to the inclination to overestimate the
probability of favorable events while underestimating the likelihood of
unfavorable events (Aue et al. (2012)). In economic terms, WT-biased beliefs
lead to overconfidence and optimism in decision-making (Malmendier and Taylor
(2015)).

Extensive and robust evidence supports the relevance of WT behavior in
economic decision-making. Studies have examined various domains to illustrate
this phenomenon. For instance, Puri and Robinson (2007) explore overconfidence
regarding life expectancy, Orhun et al. (2021) analyze beliefs about the risks
of returning to work during a pandemic, and Seaward and Kemp (2000)
investigate WT behavior concerning the repayment time for student loan debt.
In the context of markets, Grubb (2015) and Stone and Wood (2018) provide
theoretical and empirical evidence of consumer and firm overoptimism,
respectively. Malmendier and Tate (2015) offer insights into CEOs’
overconfidence, while Daniel and Hirshleifer (2015) discuss WT’s role in
explaining investors’ optimistic behavior in financial markets. Furthermore,
Lovallo and Kahneman (2003) delve into the optimistic tendencies and
overconfidence displayed by business executives and entrepreneurs in their
decision-making processes.

In this paper, we develop a model of WT that considers both the benefits and
costs associated with biased beliefs and optimistic behavior. Drawing on the
framework established by Caplin and Leahy (2019), we propose a two-stage model
where the decision maker (DM) confronts uncertainty about future events and
makes choices involving actions and belief structures. By actively selecting
their beliefs, DMs aim to maximize subjective utility for a given alternative,
considering the cost of deviating from objective beliefs. To quantify this
cost, we introduce a belief distortion function proportional to the
$\phi$-divergence (Csiszar (1967)) between subjective and objective beliefs.
This cost function penalizes deviations from objective beliefs by measuring
the statistical distance between subjective and objective beliefs. Our
approach accommodates various examples, including the Kullback-Leibler, Burg,
and modified $\chi^{2}$ distances.

We contribute to the literature on WT by making at least four key
contributions. First, we establish a direct link between WT behavior and the
concept of convex risk measures, as introduced by Artzner et al. (1999) and
Frittelli and Rosazza Gianin (2002). Using Lagrangian duality arguments, we
demonstrate that the problem of selecting an optimal belief vector can be
reformulated as a minimization problem, which determines the level of risk
associated with the optimal belief vector. Specifically, for the Kullback-
Leibler distance, our analysis reveals that the WT problem corresponds to the
DM analyzing the well-known entropic risk functional, as discussed in the
works of Föllmer and Schied (2002) and Föllmer and Schied (2016). This
behavioral perspective highlights that optimistic DMs tend to adopt beliefs
that lead them to behave as risk seekers.

Second, we provide a complete characterization of optimal beliefs in the
context of WT and offer new behavioral insights. We show that optimal beliefs
exhibit a distinct pattern of “twisting” the baseline probabilities toward
states with high utilities. Importantly, this finding extends beyond the
Kullback-Leibler case and applies to a broad class of $\phi$-divergence
functions, going beyond previous research by Caplin and Leahy (2019) and
Mayraz (2019). Moreover, our model captures situations where optimistic DMs
assign a subjective probability of zero to states with low utilities,
reflecting a phenomenon we refer to as “cognitive suppression.” We establish
the necessary conditions on the cost function to capture this kind of bias.
Similarly, we identify necessary conditions that allow WT to lead the DM to
assign positive subjective probabilities to states that the baseline deems
impossible or highly improbable, a behavior we term “cognitive emergence.”
Notably, cognitive emergence is only observed for the state with the highest
utility level. Our formalization of cognitive suppression and cognitive
emergence aligns with concepts found in the psychology literature, such as
“wishing” (Bury et al. (2016)) and “false hope” (Korner (1970)), and sheds new
light on these cognitive biases by emphasizing the significant role that
subjective beliefs play in reshaping the perception of what is possible. To
our knowledge, cognitive suppression and cognitive emergence represent novel
contributions to the WT literature.

Third, we establish a connection between WT behavior and quantile-utility
maximization. We uncover this relationship by introducing a new belief
distortion cost. In particular, we introduce the threshold beliefs distortion
cost function, which penalizes deviations from objective beliefs in a binary
fashion. Our analysis reveals that a WT agent focuses on the upper quantile
utility of different alternatives. Subsequently, the DM selects the option
with the highest average conditional expected utility, given that they are in
the upper quantile. As a result, an optimistic DM concentrates solely on the
favorable outcomes in the upper tail of the distribution, disregarding less
favorable outcomes in the rest of the distribution. This finding establishes a
formal connection between WT and models of quantile-utility maximization, such
as those found in Chambers (2009), Manski (1988), de Castro and Galvao (2019,
2022), and Rostek (2010). However, our approach differs from the literature on
quantile preferences in two key aspects. First, the connection between WT and
quantile-utility maximization arises as a consequence of optimism and not as a
consequence of some primitive (or axiom) of the model. Second, our analysis
demonstrates that an optimistic DM is not solely concerned with the utility
associated with a specific quantile but also with the conditional average
utility related to the upper tail defined by that quantile. In other words,
our analysis accounts for magnitudes of low-probability large-utility tail
events, while the quantile approach does not account for this information.

Finally, we leverage the connection between quantile-utility maximization and
WT behavior to shed light on DM’s preference for skewness. Specifically, when
the utility of different alternatives follows a Generalized Pareto
Distribution (GPD), we demonstrate that the optimal choice of a WT agent
depends on the degree of positive skewness. The shape parameters associated
with the GPD distributions are crucial in determining the DM’s optimal action.
This finding offers a straightforward explanation for an optimistic DM’s
observed preference for skewness. To illustrate the implications of our
result, we discuss its application to entry market decisions and discrete
choice models.

The rest of the paper is organized as follows: in §2, we introduce the model
and characterize the optimal beliefs. Furthermore, in §2 we discuss the
connection between risk measures and WT models. §3 discusses how our model can
generate cognitive emergence and suppression. §4 discusses the connection
between WT behavior and quantile-utility maximization. In addition, this
section provides a characterization of preference for skewness in our WT
model. §5 discusses some of the related literature. §6 concludes. Proofs and
technical discussions are gathered in Appendices A, B, and C respectively.

##  2\. The Model

In this section, we develop a WT model that incorporates the benefits and
costs of distorted beliefs. As the introduction section mentions, we build
upon Caplin and Leahy (2019)’s framework.111 In particular, we follow their
idea that the DM maximizes her current subjective expected utility, which
incorporates utility from current experience (assumed zero) and utility from
the DM’s anticipated future realization. This relies on the view that an
agent’s subjective utility depends on beliefs regarding future outcomes.

Formally, we consider an environment where the DM is confronted with the task
of selecting an action $a$ from a set $A=\\{a_{1},\ldots,a_{n}\\}$ in the
presence of uncertainty regarding a utility-relevant state $\omega\in\Omega$.
The DM’s utility function is defined as
$u:A\times\Omega\longrightarrow\mathbb{R}$, which assigns a real-valued number
to each pair consisting of an action and a state. Let
$U(a)=\left(u(a,\omega)\right)_{\omega\in\Omega}$ and $U=(U(a))_{a\in A}$
respectively. An _exogenous_ objective belief is present in the form of a
probability distribution $q$ over the state space $\Omega$. The objective
belief assigns a probability $q(\omega)$ to each state $\omega$, representing
the objective likelihood of that state occurring. The DM holds a subjective
belief represented by the probability distribution $p\in\Delta(\Omega)$, where
$p(\omega)$ denotes the subjective probability assigned to each state
$\omega\in\Omega$. The subjective expected utility (SEU) of alternative $a\in
A$ for the DM is given by:

(1) $\mathbb{E}_{p}(u(a,\omega))=\sum_{\omega\in\Omega}p(\omega)u(a,\omega)$

The expected payoff (1) makes explicit that the DM uses her subjective beliefs
$p$ to evaluate utility-maximizing actions.

To account for the impact of deviating from the objective beliefs $q$, we
introduce a cost of belief distortion in evaluating utility-maximizing
actions. This cost reflects the DM’s preference for accurate beliefs. We
assume belief distortion costs increase as the deviation from
$q\in\Delta(\Omega)$ increases. Formally, we model the cost of belief
distortion as the $\phi$-divergence (Csiszar (1967)) between the subjective
belief $p$ and the objective belief $q$, denoted as $C_{\phi}(p\|q)$. This
cost function captures the statistical distance between subjective and
objective beliefs. The details of $C_{\phi}(p\|q)$ will be discussed in the
subsequent section.

Accordingly, and given a cost function $C_{\phi}(p\|q)$, the WT agent chooses
an optimal pair $(a^{\star},p^{\star})$ that maximizes:

(2) $\max_{a\in
A}\max_{p\in\Delta(\Omega)}\left\\{\mathbb{E}_{p}(u(a,\omega))-\delta
C_{\phi}(p\|q)\right\\}$

where $\delta>0$ represents the marginal cost of deviations from $q$.

The problem (2) presents a framework in which the agent simultaneously chooses
both an action and a belief structure. Importantly, this framework considers
beliefs to be contingent on the chosen action. In other words, the DM assigns
a specific belief structure to each possible action, allowing for the
possibility of holding seemingly contradictory beliefs. This notion of beliefs
being action-contingent is reminiscent of the concept of cognitive dissonance,
as discussed in Akerlof and Dickens (1982) and relates to situations where
agents may hold contradictory beliefs. For an illustrative example clarifying
this concept of action contingency, please refer to Section 3.3.

Our approach to WT behavior is closely connected to motivated reasoning, as
discussed in studies such as Kunda (1990) and Bénabou and Tirole (2016).
According to this theory, when choosing their optimal beliefs, a motivated DM
is driven by the subjective expected utility $\mathbb{E}_{p}(u(a,\omega))$,
representing the anticipated rewards associated with different actions and
outcomes. However, in addition to the rewards, a motivated DM also considers
the importance of accuracy. In our framework, this is captured by the term
$\delta C_{\phi}(p\|q)$ in expression (2), where $\delta$ represents the
weight placed on the cost of belief distortion. Thus, our model incorporates
the motivational aspect of maximizing SEU and considering accuracy in belief
formation.

###  2.1. Belief distortion and $\phi$-divergences

Intuitively, the term $C_{\phi}(p\|q)$ captures the distance or divergence
between $p$ and $q$. We formalize this interpretation by employing the concept
of statistical divergence, which is a measure of dissimilarity between
probability distributions (Csiszar (1967); Liese and Vajda (1987); Pardo
(2005)). To do so, we utilize a specific class of functions, which is given
by:

######  Definition 1.

Consider the class of $\phi$-divergence functions $\Phi$. A function
$\phi\in\Phi$ must satisfy:

  1. (1)

$\phi:\mathbb{R}\rightarrow(-\infty,+\infty]$ is a proper closed convex
function

  2. (2)

$\phi$ is non-negative and attains its minimum at $1$, and furthermore
$\phi(1)=0$.

  3. (3)

Define undefined arguments as:
$0\phi(\frac{c}{0})=clim_{t\rightarrow\infty}\frac{\phi(t)}{t}$ $\forall c>0$
and $0\phi(\frac{0}{0})=0$.

We can now utilize our definition of the class $\Phi$ to formalize our cost
function:

######  Definition 2.

Let $\phi\in\Phi$. The $\phi$-divergence of the probability vector $p$ with
respect to the baseline belief $q$ is

(3)
$C_{\phi}(p\|q)=\sum_{\omega\in\Omega}q(\omega)\phi\left(\frac{p(\omega)}{q(\omega)}\right).$

where $p,q\in\Delta(\Omega)$.

In equation (3), the cost function explicitly depends on the choice of
$\phi\in\Phi$. Additionally, we observe that $C_{\phi}(p\|q)$ can be expressed
as the expected value under the objective belief $q$ of the function $\phi$
applied to the ratio of subjective probabilities $p(\omega)/q(\omega)$.

While $C_{\phi}(p\|q)$ is a statistical distance, it is worth mentioning that
it does not necessarily satisfy the triangle inequality. Moreover, for $p$ and
$q$ in the interior of the probability simplex, it is generally true that
$C_{\phi}(p\|q)\neq C_{\phi}(q\|p)$. In other words, the function
$C_{\phi}(\cdot\|\cdot)$ is generally asymmetric.

A key element in our framework will be the convex conjugate of the function
$\phi$, denoted as $\phi^{\ast}$. The conjugate function is defined as:

(4)
$\phi(s)=\sup_{t\in\mathbb{R}}{st-\phi(t)}=\sup_{t\in\operatorname{dom}\phi}{st-\phi(t)}=\sup_{t\in\operatorname{int}\operatorname{dom}\phi}{st-\phi(t)},$

where the last equality follows from (Rockafellar, 1970, Cor. 12.2.2). The
conjugate function $\phi^{\ast}$ is a closed proper convex function, with
$\operatorname{int}\operatorname{dom}\phi^{\ast}=(a,b)$, where

$a=\lim_{t\rightarrow-\infty}t^{-1}\phi^{\ast}(t)\in[-\infty,+\infty);b=\lim_{t\rightarrow+\infty}t^{-1}\phi^{\ast}(t)\in(-\infty,+\infty].$

Additionally, it is essential to note that for the convex and closed function
$\phi$, its bi-conjugate is given by $\phi^{**}=\phi$, as shown in Rockafellar
(1970).

A key observation is that since $1$ is the minimizer of $\phi$ and lies in the
interior of its domain, we have $\phi^{\prime}(1)=0$. Moreover, utilizing the
property of convex and closed functions, known as the Fenchel equality, we
have the equivalence $y=\phi^{\prime}(x)$ if and only if
$x=\phi^{*^{\prime}}(y)$. By applying this observation to $x=1$ and $y=0$, we
obtain $\phi^{\ast\prime}(0)=1$.

Throughout the paper, we make the following assumption.

######  Assumption 1.

$\phi^{*}(s)$ is strictly convex and differentiable with
$\phi^{*\prime}(s)\geq 0$ for all $s$.

Divergence | $\phi(t)$ | $C_{\phi}(p\|q)$ | $\phi^{*}(s)$
---|---|---|---
Kullback-Leibler | $\phi(t):=t\log t,\quad t>0$ | $\sum_{\omega\in\Omega}p(\omega)\log\frac{p(\omega)}{q(\omega)}$ | $e^{s}-1$
Hellinger | $(1-\sqrt{t})^{2},\quad t>0$ | $\sum_{\omega\in\Omega}\left(\sqrt{p(\omega)}-\sqrt{q(\omega}\right)^{2}$ | ${s\over 1-s}\quad s<1$
Modified $\chi^{2}$ distance | $\phi(t):=(t-1)^{2},\quad t>0$ | $\sum_{\omega\in\Omega}\frac{\left(p(\omega)-q(\omega)\right)^{2}}{q(\omega)}$ | $\phi^{\ast}(s)=\begin{cases}-1,&\text{if $s<-2$ }\\\ s+{s^{2}\over 4},&s\geq-2.\end{cases}$
Burg Entropy | $-logt+t-1,\quad t>0$ | $\sum_{\omega\in\Omega}q(\omega)\log\frac{p(\omega)}{q(\omega)}$ | $\phi^{*}(s)=-log(1-s),\quad s<1$
Table 1. Example of $\phi$-divergences and its conjugates.

Table 1 provides some popular examples of $\phi$-divergences and their
conjugates. As it is easy to see, the Kullback-Leibler distance is a
particular case of a wider class of tractable cost functions. Furthermore, the
conjugate $\phi^{\ast}$ has a very tractable form for these four cost
functions. As we shall see, this later property will be useful in
characterizing the optimal beliefs. In doing so, we make use of the following
technical lemma.

######  Lemma 1.

Let Assumption 1 hold. Let
$V_{\phi}(U(a))\triangleq\max_{p\in\Delta(\Omega)}\left\\{\mathbb{E}_{p}(u(a,\omega))-\delta
C_{\phi}(p\|q)\right\\}$ for all $a\in A$. Then, the following holds:

(5)
$V_{\phi}(U(a))=\min_{\lambda_{a}\in[\underline{u}_{a},\bar{u}_{a}]}\left\\{\lambda_{a}+\delta\mathbb{E}_{q}(\phi^{\ast}((u(a,\omega)-\lambda_{a})/\delta))\right\\},$

where $\underline{u}_{a}=\min_{\omega\in\Omega}{u(a,\omega)}$ and
$\bar{u}_{a}=\max_{\omega\in\Omega}u(a,\omega)$. Furthermore, the minimization
problem (5) has a unique optimal solution $\lambda_{a}^{\star}$.

Proof. All proofs are gathered in Appendix A. $\square$

The previous lemma establishes that finding the optimal beliefs is equivalent
to solving a one-dimensional minimization problem.

In what follows, let $\lambda_{a}^{\star}$ be the unique solution to the
problem (5). By using Lemma 1, we are able to characterize the DM’s optimal
belief choice vector.

######  Proposition 1.

Let Assumption 1 hold and define
$w_{a}(\omega)\triangleq\phi^{\ast\prime}((u(a,\omega)-\lambda^{\star}_{a})/\delta)$
for all $a\in A,\omega\in\Omega$. Then for each $a\in A$ the optimal belief
choice $p^{\star}(a)$ satisfies

(6) $\nabla V_{\phi}(U(a))=p^{\star}(a),$

where

(7) ${\partial V_{\phi}(U(a))\over\partial
u(a,\omega)}=w_{a}(\omega)q(\omega)=p^{\star}(\omega|a),\quad\forall\omega\in\Omega.$

Some remarks are in order. First, Proposition 1 characterizes the optimal
belief vector $p^{\star}(a)$ as the product of the weight vector
$w_{a}=(w_{a}(\omega))_{\omega\in\Omega}$ and the objective belief $q$.
Additionally, from the definition of $w_{a}(\omega)$, the distorted belief
vector $p^{\star}(a)$ depends on the particular choice of $\phi$. Each term
$w_{a}(\omega)$ captures how the DM “twists” the truth (Kovach (2020)) for
each state $\omega$. For instance, if $w_{a}(\omega)=1$ for some
$\omega\in\Omega$, then $p(\omega|a)=q(\omega)$. Similarly, if
$w_{a}(\omega)>1$ for some $\omega\in\Omega$, then $p(\omega|a)>q(\omega)$. In
the latter case, we say that the DM exhibits overprecision (Moore et al.
(2015)).222It is worth pointing out that we can weaken Assumption 1 by
considering the subgradient $\partial\phi^{\ast}(s)$ instead of the gradient
$\phi^{\ast\prime}$. Given the convexity of $\phi^{\ast}$, the subgradient
$\partial\phi^{\ast}(s)$ always exists. Then the weight $w_{a}(\omega)$ will
correspond to a selection of $\partial\phi^{\ast}(s)$. Additionally, given
that the subgradient of a convex function is a maximal monotone operator, the
monotonicity result in Corollary 1 will also hold.

Second, the result in Proposition 1 is related to Mayraz (2019). In his paper,
the weights $w_{a}(\omega)$ are interpreted as “desires” that capture the DM’s
overoptimism. Specifically, Mayraz (2019) models the weights $w_{a}(\omega)$
as proportional to $e^{u(a,\omega)/\delta}$. Therefore, our characterization
(7) generalizes the incorporation of desires in WT models. Proposition 1
generalizes the result in Caplin and Leahy (2019) non-trivially. They
characterize the optimal beliefs $p^{\star}(a)$ under the assumption that
$C_{\phi}(p\|q)$ is given by the Kullback-Leibler distance. Their
characterization is a particular case of the expression (7).

Third, the expression (7) helps us to understand how our model captures WT
behavior in a general way. To see this, let us consider the ratio of the
optimal beliefs for two states and assume that for
$\omega,\omega^{\prime}\in\Omega$ the associate utilities satisfy
$u(a,\omega)>u(a,\omega^{\prime})$. Then for $q(\omega)>q(\omega^{\prime})$,
the likelihood ratio

(8) ${p(\omega|a)\over p(\omega^{\prime}|a)}={w_{a}(\omega)q(\omega)\over
w_{a}(\omega^{\prime})q(\omega^{\prime})}$

implies that $p(\omega|a)>p(\omega^{\prime}|a).$

The likelihood ratio allows a more precise understanding of the behavioral
implication of WT subjective beliefs. Formally, the ratio in (8) formalizes
the fact that when comparing two states, $\omega$, and $\omega^{\prime}$, the
relative probability assigned to $p(\omega)$ is higher than
$p(\omega^{\prime})$. Thus, the DM assigns a higher probability to more
desirable outcomes in relative terms. In other words, the DM’s optimal belief
vector $p^{\star}(a)$ biases the objective belief $q$ toward states with
higher utilities. This optimistic biased behavior is generated by all
$\phi\in\Phi$. The following corollary formalizes the previous discussion.

######  Corollary 1.

Let Assumption 1 hold. Then $p^{\star}(\omega|a)$ is increasing in both
$u(a,\omega)$ and $q(\omega)$. Therefore, given states $\omega$ and
$\omega^{\prime}$ with $q(\omega)>q(\omega^{\prime})$ and
$u(a,\omega)>u(a,\omega^{\prime})$, we have
$p^{\star}(\omega|a)>p^{\star}(\omega^{\prime}|a)$.

######  Example 1.

To understand how Proposition 1 works, let us revisit the case where
$C_{\phi}(p\|q)$ represents the Kullback-Leibler distance. It is
straightforward to show that
$\lambda^{\star}_{a}=V_{\phi}(U(a))=\delta\log\mathbb{E}_{q}(e^{u(a,\omega)/\delta})$.
Therefore, according to Proposition 1, expression (7) tells us that subjective
beliefs are obtained by

$p^{\star}(\omega|a)=w_{a}(\omega)q(\omega)$

where

$w_{a}(\omega)={\frac{e^{u(a,\omega)/\delta}}{\sum_{\omega^{\prime}\in\Omega}q(\omega^{\prime})e^{u(a,\omega^{\prime})/\delta}}}$

Accordingly, the DM chooses the optimal action $a^{\star}\in A$ such that

$a^{\star}=\arg\max_{a\in
A}\delta\log(\mathbb{E}_{q}(e^{u(a,\omega)/\delta})).$

$\square$

######  Example 2.

Let the DM solve the problem (2), where the modified $\chi^{2}$ distance gives
the deviation cost from $q$. Assume
$\delta>(\mathbb{E}_{q}(u(a,\omega))-\underline{u}_{a})/2\delta$ for all $a\in
A$ and $\omega\in\Omega$. It is straightforward to show that

$\lambda_{a}^{\star}=\mathbb{E}_{q}(u(a,\omega)).$

For a given action $a$, let us define the following difference:

$g(a,\omega)\triangleq u(a,\omega)-\mathbb{E}_{q}(u(a,\omega)).$

Accordingly, we can then write

$V_{\phi}(U(a))=\mathbb{E}_{q}(u(a,\omega))+{1\over
4\delta}\sum_{\omega\in\Omega}q(\omega)g(a,\omega)^{2}.$

Defining $Var(U(a))\triangleq\mathbb{E}_{q}(g(a,\omega)^{2})$, we rewrite
$V_{\phi}(U(a))$ as

$V_{\phi}(U(a))=\mathbb{E}_{q}(u(a,\omega))+{1\over 4\delta}Var(U(a)).$

Thus the optimized valued associated with the optimal beliefs $p^{\star}(a)$
takes the form of a mean-variance model. To recover the subjective beliefs, we
use Proposition 1 and compute ${\partial V_{\phi}(U(a))\over\partial
u(a,\omega)}$:

${\partial V_{\phi}(U(a))\over\partial
u(a,\omega)}=q(\omega)+\frac{1}{2\delta}q(\omega)g(a,\omega)-\frac{1}{2\delta}q(\omega)\sum_{\omega^{\prime}\in\Omega}q(\omega^{\prime})g(a,\omega^{\prime})$

Using the fact that $\sum_{\omega\in\Omega}q(\omega)g(a,\omega)=0$, the
subjective beliefs are:

$p^{\star}(\omega|a)=q(\omega)\left(1+\frac{g(a,\omega)}{2\delta}\right)\quad\forall\omega\in\Omega.$

The previous expression shows that $p^{\star}(\omega|a)$ is increasing in
$g(a,\omega)$. Finally, the DM chooses the optimal action $a^{\star}\in A$
such that:

$a^{\star}=\arg\max_{a\in A}\left\\{\mathbb{E}_{q}(u(a,\omega))+{1\over
4\delta}Var(U(a))\right\\}.$

$\square$

In Example 2, the DM prefers states with both a high mean and a high variance.
This preference is reflected in the optimal belief vector $p^{\star}(a)$.

By examining the likelihood ratio, we can further explore the subjective
probabilities. In the case of the Kullback-Leibler distance, the likelihood
ratio is given by:

(9)
$\frac{p^{\star}(\omega|a^{\star})}{p^{\star}(\omega^{\prime}|a^{\star})}=\frac{q(\omega)}{q(\omega^{\prime})}e^{u(a^{\star},\omega)-u(a^{\star},\omega^{\prime})}.$

The likelihood ratio in the Kullback-Leibler case has been widely discussed in
the WT literature, including works by Caplin and Leahy (2019), Mayraz (2019),
and Kovach (2020). It captures the relative weighting of states $\omega$ and
$\omega^{\prime}$ based on the baseline probabilities $q$, as well as the
differences in utilities $u(a^{\star},\omega)-u(a^{\star},\omega^{\prime})$.

In the case of the modified $\chi^{2}$ distance, the likelihood ratio is given
by:

(10)
$\frac{p(\omega|a^{)}}{p(\omega^{\prime}|a^{)}}=\frac{q(\omega)(g(a^{,}\omega)+2\delta)}{q(\omega^{\prime})(g(a^{,}\omega^{\prime})+2\delta)}.$

In this case, the likelihood ratio depends on the extent to which the
utilities associated with states $\omega$ and $\omega^{\prime}$ are better or
worse than the average utility.

It is important to note that closed-form solutions for the likelihood ratios
are possible in the cases of KL and modified $\chi^{2}$ distances. However, in
general, obtaining closed-form solutions is not always feasible. Analyzing the
likelihood ratio provides valuable insights into how the objective beliefs and
utility differences between states influence the DM’s subjective
probabilities.

###  2.2. Wishful thinking and risk

In the context of portfolio allocation, Föllmer and Schied (2002) and
Frittelli and Rosazza Gianin (2002) have introduced the notion of _convex risk
measures_ in an attempt to quantify the riskiness of different financial
portfolio decisions.333In a fundamental paper, Artzner et al. (1999)
introduced the notion of coherent risk measures. Their approach is axiomatic
and relies heavily on the properties of subadditivity and homogeneity. Föllmer
and Schied (2002) and Frittelli and Rosazza Gianin (2002) replaced these two
conditions by focusing in convexity properties. It is worth pointing out that
the notion of coherent and convex risk measures applies far beyond the case of
portfolio allocation problems. In our WT model, the notion of convex risk
measures emerges naturally to quantify the risk associated with choosing a
pair $(a,p(a))$. Formally, $V_{\phi}$ possesses all the defining properties of
a convex risk measure. The following result formalizes this connection.

######  Proposition 2.

Let $\phi\in\Phi$. Then for all $a\in A$, the following properties hold:

  * (i)

$V_{\phi}(U(a)+c)=V_{\phi}(U(a))+c,\forall c\in\mathbb{R}$.

  * (ii)

$V_{\phi}(c)=c$, for any constant $c\in\mathbb{R}$ (considered as a degenerate
random variable).

  * (iii)

If $u(a,\omega)\leq\tilde{u}(a,\omega),\forall\omega\in\Omega$, then,
$V_{\phi}(U(a))\leq V_{\phi}(\tilde{U}(a))$.

  * (iv)

For any random variables $U_{1}(a),U_{2}(a)$ with finite moments and any
$\kappa\in(0,1)$, one has

$V_{\phi}\left(\kappa U_{1}(a)+(1-\kappa)U_{2}(a)\right)\leq\kappa
V_{\phi}\left(U_{1}(a)\right)+(1-\kappa)V_{\phi}\left(U_{2}(a)\right).$

The previous result is a simple adaptation of (Ben-Tal and Teboulle, 2007,
Thm. 2.1 ). Part (i) is known as _translation invariance_ and establishes that
adding a constant $c$ to $U(a)$ is equivalent to adding the same constant to
$V_{\phi}(U(a))$. Part (ii) is known as _consistency_ and states that when
$u(a,\omega)=c$ for all $\omega$, then the value of $V_{\phi}(U(a))$ is
constant and equal to $c$. Part (iii) is just a _monotonicity_ condition, in
the sense that $V_{\phi}(U(a))$ is monotone increasing on $U(a)$ (in a
stochastic sense). Finally, condition (iv) establishes that $V_{\phi}(U(a))$
is a _convex_ function.

As we said before, the properties in Proposition 2 establish that
$V_{\phi}(U(a))$ is a convex risk measure. The following corollary formalizes
this fact.

######  Corollary 2.

For all $a\in A$, $V_{\phi}(U(a))$ is a convex risk measure.

Some remarks are in order. First, to see why the interpretation of $V_{\phi}$
as a risk measure is useful, we revisit the Kullback-Leibler case. In this
case, we know that
$V_{\phi}(U(a))=\delta\log(\mathbb{E}_{q}(e^{u(a,\omega)/\delta}))$, which is
known as the entropic risk measure (Föllmer and Schied (2016)). Similarly, in
the case of the Example 2, we know that for each alternative $a\in A$, we get
$V_{\phi}(U(a))=\mathbb{E}_{q}(u(a,\omega))+{1\over 4\delta}Var(U(a))$. Noting
that $\delta>0$, the DM will choose the alternative with the highest expected
utility and variance combination. Corollary 2 establishes that this pattern
generalizes beyond the entropic and the mean-variance cases. However, it’s
important to highlight that in our WT model, the implementation of the notion
of convex risk measure differs from its traditional application in portfolio
allocation problems. The key distinction lies in the fact that in our model,
each alternative $a\in A$ is associated with an uncertain prospect $U(a)$, and
the associated risk is measured by $V_{\phi}(U(a))$. This framework allows us
to analyze risk and decision-making under uncertainty in a more general
context beyond traditional portfolio allocation settings.

Second, it is worth noting that Corollary 2 implies that a WT agent will
choose the riskiest alternative from the set $A$. This can be observed by
combining problem (2) with Lemma 1, which shows that the WT problem is
equivalent to $\max_{a\in A}V_{\phi}(U(a))$. Thus, the WT agent aims to
maximize the risk measure $V_{\phi}$ applied to the uncertain prospects $U(a)$
associated with each alternative $a\in A$. This behavior can be seen in
Examples 1 and 2, where the WT agent selects the alternative with the highest
risk according to the specified risk measure.

To our knowledge, the connection between risk measures and WT behavior has not
been explored in the existing literature. In Section 4, we take advantage of
this connection to investigate the relationship between WT behavior,
conditional value-at-risk (CVaR), and the concept of “preference for
skewness.”

###  2.3. Expected Utility Equivalence

This section establishes the behavioral equivalence between WT decision-making
and expected utility (EU) behavior. Recalling that $U=(U(a))_{a\in A}$ and
$U(a)=(u(a,\omega))_{\omega\in\Omega}$, we define

$A_{eu}(U)\triangleq\arg\max_{a\in A}\mathbb{E}_{q}(u(a,\omega))$

as the set of optimal actions associated with EU maximization.

Now, let
$\tilde{u}(a,\omega)=\lambda_{a}^{\star}+\delta\mathbb{E}_{q}(\phi^{\ast}((u(a,\omega)-\lambda_{a}^{\star})/\delta))$
and $\tilde{U}(a)=(\tilde{u}(a,\omega))_{\omega\in\Omega}$, and
$\tilde{U}=(\tilde{U}(a))_{a\in A}$, for all $a\in A,\omega\in\Omega$. In
addition, let

$A_{wt}(U)\triangleq\arg\max_{a\in
A}\max_{p\in\Delta(\Omega)}\left\\{\mathbb{E}_{p}(u(a,\omega))-\delta
C_{\phi}(p\|q)\right\\}.$

define the set of optimal actions for the WT problem. The next result
establishes the equivalence between both models.

######  Proposition 3.

A WT agent with utility function $u(a,\omega)$ is behaviorally equivalent to
an EU maximizer agent with the transformed utility function
$\tilde{u}(a,\omega)$. In particular,

$A_{wt}(U)=A_{eu}(\tilde{U})$

Two remarks are in order. First, Proposition 3 establishes a general
behavioral equivalence between WT models using $\phi$-divergences and EU
maximization. It shows that WT behavior can always be interpreted as the
outcome of EU maximization under a distorted utility function. An important
implication of this equivalence is that we cannot distinguish between these
models based on choice data alone. However, combining choice and belief data
may determine which framework is more appropriate. This offers a potential
avenue for practical model selection.

Second, the behavioral equivalence presented in Proposition 3 extends the
findings of Robson et al. (2022) non-trivially. The focus of Robson et al.
(2022) is on the specific case of Kullback-Leibler divergence, where the
distorted utility function is given by
$\tilde{u}(a,\omega)=\lambda_{a}^{\star}+\delta\exp\left(\frac{u(a,\omega)-\lambda_{a}^{\star}}{\delta}\right)$.
By leveraging the relationship
$\lambda_{a}^{\star}=\delta\log\mathbb{E}_{q}(e^{u(a,\omega)/\delta})$, it can
be shown that
$\mathbb{E}q(\tilde{u}(a,\omega))=\delta\log\mathbb{E}q(e^{u(a,\omega)/\delta})$,
leading to $A_{eu}(\tilde{U})=\arg\max_{a\in
A}\delta\log\mathbb{E}_{q}(e^{u(a,\omega)/\delta})$. The equivalence
established in Proposition 3 goes beyond the specific case of Kullback-Leibler
distance and holds for a broader class of $\phi$-divergences.

###  2.4. WT as an intrapersonal game

We close this section discussing a final and important implication of Lemma 2
and Proposition 1. Together with these results, we can represent optimal WT
behavior as the solution to a saddle point problem. The following result
formalizes this observation.

######  Corollary 3.

For each $a\in A$, let
$\Psi(a,\lambda_{a})\triangleq\lambda_{a}+\delta\mathbb{E}_{q}(\phi^{*}((u(a,\omega)-\lambda_{a})/\delta)$.
Then the pair $(a^{\star},p^{\star}(a))$ solves (2) iff the pair
$(a^{\star},\lambda_{a}^{\star})$ solves the saddle point problem:

$\max_{a\in A}\min_{\lambda_{a}\in\Lambda(a)}\Psi(a,\lambda_{a})$

where
$\Lambda_{a}\triangleq\\{\lambda_{a}:min_{\omega\in\Omega}{u(a,\omega)}\leq\lambda_{a}\leq\max_{\omega\in\Omega}u(a,\omega)\\}.$

The previous corollary is useful to understand WT decision-making in terms of
a two-player intrapersonal game in the spirit of Bracha and Brown (2012). In
their language, the player choosing the optimal action $a^{\star}\in A$
corresponds to the rational process while choosing the optimal belief
$p^{\star}(a^{\star})$ corresponds to the emotional process. In particular,
the rational agent solves $\max_{a\in A}\mathbb{E}_{p}(u(a,\omega))$ while the
emotional agent solves
$\max_{p\in\Delta(\Omega)}\mathbb{E}_{p}(u(a,\omega))-\delta C_{\phi}(p\|q)$.

Intuitively, the rational agent is an EU maximizer, while the emotional agent
is a subjective expected utility maximizer with a taste for higher payoff
states. Bracha and Brown (2012) show that the objective function in our WT
problem (2) is a potential function for this intrapersonal game. Thus, the
solution of this intrapersonal game is identical to the solution to the
problem (2). Corollary 3 adds a useful interpretation, emphasizing that the
tension between the rational and emotional agents can be represented as a max-
min problem incorporating an appropriate notion of risk.

##  3\. Extreme Beliefs

In this section, we delve into how our WT model can capture “extreme” beliefs
by relaxing the assumption of absolute continuity, which states that a state
has a zero objective probability if and only if it has a zero subjective
probability. This assumption is commonly made in the literature on motivated
reasoning, with a few exceptions, such as Bury et al. (2016) and Korner
(1970).

We first examine optimal beliefs in the case of cognitive suppression, where
the DM subjectively ignores states with low payoffs. To exhibit this behavior,
the DM selects a utility cutoff such that any state generating a utility below
this cutoff is subjectively disregarded. This cognitive suppression reflects
the DM’s tendency to ignore or downplay unfavorable outcomes, focusing only on
states with sufficiently high utilities. By doing so, the DM shapes their
beliefs to align with their desired outcomes, exhibiting WT.

Next, we investigate optimal beliefs in the context of cognitive emergence,
where the DM subjectively believes that a state with zero objective
probability (or an impossible state) can still be realized or observed with
some positive subjective probability. However, this bias is only exhibited in
states with the highest payoffs. In other words, the DM assigns positive
subjective probabilities to these extreme states, despite their objective
unlikelihood. This behavior reflects the DM’s inclination to perceive even
highly improbable outcomes as possible or likely when those outcomes align
with their desired goals or aspirations.

By exploring these variations of belief formation, our WT model captures the
cognitive processes of suppression and emergence. These behaviors highlight
how the DM’s subjective beliefs can deviate from objective probabilities,
leading to extreme beliefs influenced by optimism.

###  3.1.  Cognitive suppression

Optimal beliefs display cognitive suppression when, for some state
$\omega\in\Omega$ with $q(\omega)>0$, the DM chooses
$p^{\star}(\omega|a^{\star})=0$.444In this section, for ease of notation, we
set $\delta=1$. Results can be easily extended to an arbitrary $\delta$.

We aim to characterize the class of $\phi$-divergence functions that implies
WT behavior consistent with cognitive suppression. The following proposition
provides necessary conditions on $\phi$.555These conditions draw from similar
results in a data-driven problem in Bayraksan and Love (2015a).

######  Proposition 4.

Let $a^{\star}$ and $p^{\star}(a^{\star})$ be an optimal solution to the WT
problem (2). Then $p^{\star}(a^{\star})$ can generate cognitive suppression
only if

  * (i)

$\lim_{t\rightarrow 0+}\phi(t)<\infty$,

  * (ii)

$\lim_{t\rightarrow 0+}\phi^{\prime}(t)>-\infty$.

Furthermore, if these conditions hold, then there exists a cutoff
$\tilde{u}(a)$ such that $p^{\star}(\omega|a^{\star})=0$ if and only if
$u(a^{\star},\omega)\leq\tilde{u}(a^{\star})$.

These conditions ensure that the $\phi$-divergence function exhibits the
necessary behavior to accommodate corner solutions in the WT model. The
boundedness of the cost function when $p^{\star}(\omega|a^{\star})=0$ is
crucial for such solutions, allowing the DM to selectively ignore certain
states and assign them zero probabilities based on their utilities.

In summary, Proposition 4 provides conditions on the $\phi$-divergence
function that allows for optimal beliefs with corner solutions, where some
states are assigned zero probabilities. These conditions capture the DM’s
cognitive suppression behavior, where certain states are subjectively
disregarded based on their utilities.

The second assertion of Proposition 4 describes how the cognitive suppression
behavior takes place in the WT model. The DM determines a cutoff value,
denoted as $\tilde{u}(a^{\star})$, representing the threshold utility below
which states are subjectively suppressed or ignored. Intuitively, the DM
starts by suppressing the state with the lowest utility, then proceeds to
suppress the state with the second-lowest utility, and so on.

By setting the cutoff value $\tilde{u}(a^{\star})$, the DM effectively ignores
states that provide utilities below this threshold, treating them as having
zero probability. This cognitive suppression behavior allows the DM to focus
on states perceived as more desirable or relevant based on their utilities
while disregarding states considered less favorable.

In summary, the second assertion of Proposition 4 clarifies that in the
context of cognitive suppression, the DM establishes a cutoff value
$\tilde{u}(a^{\star})$ and selectively suppresses states with utilities below
this threshold. This behavior allows the DM to prioritize and focus on states
deemed more favorable or significant while ignoring less desirable states.

To see how the result works, we analyze an environment where
$\Omega=\\{\omega_{H},\omega_{L}\\}$ and $A=\\{1,\ldots,n\\}$. Let
$(a^{\star},p^{\star}(a^{\star}))$ be an optimal solution where the state
contingent utilities are $u(a^{\star},\omega_{H})=4$ and
$u(a^{\star},\omega_{L})=0$.

Figure 1. Subjective Probabilities vs. Baseline Probabilities

(a) Kullback-Leibler

(b) Modified $\chi^{2}$

(c) Burg Entropy

A mode of comparison, we first explore the Kullback-Leibler case. Figure 1(a)
illustrates the relationship between objective and subjective beliefs in the
case of the Kullback-Leibler distance. It is important to note a
characteristic of the Kullback-Leibler divergence:
$p^{\star}(\omega|a^{\star})>0$ if and only if $q(\omega)>0$. This means that
the Kullback-Leibler divergence does not generate cognitive suppression.

In the case where the $\phi$-divergence is the modified $\chi^{2}$ distance,
cognitive suppression can be generated by the WT agent.666The details of this
case can be found in §B.1 in Appendix B.  Specifically, if $q(\omega_{H})\geq
1/2$, the agent suppresses the state $\omega_{L}$ and assigns
$p^{\star}(\omega_{H}|a^{\star})=1$ and $p^{\star}(\omega_{L}|a^{\star})=0$
for the optimal action $a^{\star}$. This behavior is depicted in Figure 1(b).

Figure 1(b) illustrates the optimal subjective belief $p^{\star}(a^{\star})$
for different values of the objective probability $q(\omega_{H})$ in the case
of the modified $\chi^{2}$ distance. It shows that when $q(\omega_{H})\geq
1/2$, the WT agent completely disregards the possibility of the low state
$\omega_{L}$ and believes with certainty that the high state $\omega_{H}$ will
occur. This type of belief structure, where a state is completely ignored, and
another state is believed with certainty, results from optimism and cannot be
generated using the Kullback-Leibler divergence.

This example highlights how different $\phi$-divergences can lead to distinct
cognitive behaviors in the WT model. In this case, the modified $\chi^{2}$
distance allows for cognitive suppression, where the agent selectively ignores
certain states based on their objective probabilities.

Our result on cognitive suppression expands in a nontrivial way the analysis
in Mayraz (2019) and Caplin and Leahy (2019) who study the properties of WT
behavior in the case of the Kullback-Leibler distance.

###  3.2. Cognitive emergence

The optimal belief $p^{\star}(a^{\star})$ exhibits cognitive emergence when
for some state $\omega$ , the associated objective probability is
$q(\omega)=0$ and $p^{\star}(\omega|a^{\star})>0$. Intuitively, cognitive
emergence occurs when a WT agent believes that an “impossible” state $\omega$
is possible.

The following result provides a necessary condition for a DM to exhibit
cognitive emergence.

######  Proposition 5.

Let $a^{\star}$ and $p^{\star}(a^{\star})$ be an optimal solution to the WT
problem (2). Then:

  * (i)

$p^{\star}(a^{\star})$ can generate cognitive emergence only if the following
condition holds:

(11) $\lim_{t\rightarrow\infty}\frac{\phi(t)}{t}=b<\infty$

  * (ii)

If condition (11) holds, then state $\omega$ can emerge only if
$u(a^{\star},\omega)=\bar{u}_{a^{\star}}$, where
$\bar{u}_{a^{\star}}=\max_{\omega^{\prime}\in\Omega}u(a^{\star},\omega^{\prime})$.

  * (iii)

Finally if a state emerges it must hold that
$\lambda_{a^{\star}}^{\star}=\bar{u}_{a^{\star}}-b$

Part (i) of the proposition provides condition (11), which ensures that the
cost $C_{\phi}(p^{\star}(a^{\star})\|q)$ remains finite (bounded) when the DM
exhibits cognitive emergence. This condition guarantees that the cost function
remains well-defined even when $p^{\star}(\omega|a^{\star})=0$, allowing for a
meaningful analysis of cognitive emergence. Part (ii) establishes that a state
$\omega$ can emerge if associated with the highest payoff utility of the
optimal choice $a^{\star}$. This means the DM assigns positive probability to
states with the highest utility among all available choices. This behavior
captures the DM’s preference for skewness, as they are willing to assign
positive probability to unlikely states that offer a potentially high utility.
This aligns with the findings of Brunnermeier and Parker (2005) regarding the
preference for skewness in optimistic decision-making.

Part (iii) provides the exact value for $\lambda^{\star}_{a^{\star}}$ when
cognitive emergence occurs. This value represents the cutoff utility level
beyond which the DM assigns positive probability to a state. It characterizes
the DM’s threshold for cognitive emergence, indicating the point at which the
DM starts considering a state as a possibility.

To show how cognitive emergence operates, we revisit the environment where
$\Omega=\\{\omega_{H},\omega_{L}\\}$ and $A=\\{1,\ldots,n\\}$. Let
$(a^{\star},p^{\star}(a^{\star}))$ be an optimal solution where the state
contingent utilities are $u(a^{\star},\omega_{H})=4$ and
$u(a^{\star},\omega_{L})=0$. Assume that $\phi$ corresponds to the Burg
Entropy (see Table 1). For $q(\omega_{H})\in(0,1)$, and applying Lemma 1 and
Proposition 1, we get:

$p(\omega_{H}|a^{\star})=\frac{2q(\omega_{H})}{\sqrt{9+16q(\omega_{H})}-3}$

Now, we show how the state $\omega_{H}$ emerges when
$q(\omega_{H})\longrightarrow 0^{+}$, note that:

$\lim_{q(\omega_{H})\rightarrow
0^{+}}\frac{2q(\omega_{H})}{\sqrt{9+16q(\omega_{H})}-3}=\frac{3}{4}$

Therefore, when $q(\omega_{H})=0$ the DM’s optimal belief choice implies that
the state $\omega_{H}$ emerges with
$p^{\star}(\omega_{H}|a^{\star})=\frac{3}{4}$. Figure 1(c) displays the
relationship between $q$ and $p^{\star}(a^{\star})$.

From a behavioral standpoint, the previous analysis captures a situation where
an optimistic DM ignores the “impossibility” of $\omega_{H}$, and instead, she
believes its probability is $p^{\star}(\omega_{H}|a^{\star})=\frac{3}{4}$.

In simple terms, the pattern of cognitive emergence establishes that optimism
drives the DM to disregard impossibility. We explore this further in §B.2 in
the Appendix. In particular, we study a situation where a DM must decide
between a highly risky and riskless asset. In this regard, the cognitive
emergence can be useful to explain the recent surge in digital asset scams
with promises of exorbitant returns (Cryptocurrencies and NFTs). There are
serious doubts about these types of assets’ fundamental market values (Cheah
and Fry (2015)). When the DM is an EU maximizer, experts’ opinions (objective
beliefs) can dissuade investment in these assets. The Appendix shows this is
not necessarily true when investors display WT behavior. Expert advice can be
ignored, and many investors hold highly unrealistic beliefs. In this case, we
can say that the DM’s behavior is consistent with a preference for skewness.
In a different setting, Brunnermeier and Parker (2005) derives a similar
result in portfolio allocation problems.

###  3.3. Extreme Cognitive Dissonance

In this section, we delve into a fundamental characteristic of motivated
reasoning and WT models, drawing a connection between our concept of extreme
subjective beliefs and actions. Due to the intimate relationship between
actions and beliefs, subjective beliefs are treated as contingent upon the
chosen action. Consequently, a DM assigns distinct subjective beliefs to each
action throughout the decision-making process.

Action-contingent beliefs have been observed in various contexts. For
instance, studies have documented situations where well-educated employees
working with hazardous chemicals significantly underestimate the risks
associated with their work (Akerlof and Dickens (1982)). Similarly, in the
context of the COVID-19 pandemic, research has shown that employees’ beliefs
about the safety of returning to work can be influenced by their motivated
reasoning and can align with their preferred course of action (Orhun et al.
(2021)). These examples illustrate that a DM’s beliefs tend to align with
their chosen action, and if they were required to take a different action,
their beliefs might change accordingly.

Because our model can generate extreme beliefs, it is crucial to emphasize
that it enables the association of substantially divergent beliefs with
different actions. This becomes especially intriguing when a DM has a personal
stake in the outcome (Granberg and Holmberg (1988)). The field of politics
provides a fitting illustration, as DMs often maintain significantly disparate
beliefs that strongly correspond to their voting patterns. To illustrate this
phenomenon within our model, we present the following example:

######  Example 3.

Let $A=\\{a_{1},a_{2},a_{3}\\}$ and $\Omega=\\{\omega_{1},\omega_{2}\\}$. The
payoff structure is summarized in the following table:

| $a_{1}$ | $a_{2}$ | $a_{3}$
---|---|---|---
$\omega_{1}$ | $4$ | $3$ | $0$
$\omega_{2}$ | $0$ | $3$ | $4$

Assume that $q(\omega_{1})=q(\omega_{2})={1\over 2}$ and the cost
$C_{\phi}(p\|q)$ determined by the modified $\chi^{2}$ distance with
$\delta=1$. Using Lemma 1, we find $\lambda_{a_{1}}^{*}=\lambda_{a_{3}}^{*}=2$
and $\lambda_{a_{2}}^{*}=3$.777Note that for $a_{1}$ $(a_{3})$ any
$\lambda\geq 2$ implies that the DM would suppress $\omega_{2}$
($\omega_{1})$. The previous fact implies that

$V_{\phi}(U(a_{1}))=V_{\phi}(U(a_{2}))=V_{\phi}(U(a_{3}))=3$

From the preceding equation, we can deduce that the DM is indifferent among
$a_{1}$, $a_{2}$, and $a_{3}$. Nevertheless, each action is associated with a
distinct subjective probability vector, which can be obtained by applying
Proposition 1. Consequently, there exist three solutions corresponding to the
optimal set:

$\\{(a_{1},(1,0)),(a_{2},(1/2,1/2)),(a_{3},(0,1))\\}.$

$\square$

In the preceding example, the DM suppresses one state for $a_{1}$ and $a_{3}$,
whereas for $a_{2}$, the DM does not distort her beliefs. It is evident in
this example that $a_{2}$ yields the highest EU, yet the DM remains
indifferent. This outcome arises because $a_{1}$ and $a_{3}$ have more skewed
distributions across states.

##  4\. Risk and preference for skewness

In this section, we explore how our approach can establish a connection
between WT behavior, quantile-utility maximization, and a preference for
skewness by considering a specific choice of $\phi$. To simplify the analysis,
we assume that $\Omega$ is a continuous set of states throughout this section.
Let $Q$ represent the objective distribution and assume it has a well-defined
density function $q$. For each action, $a\in A$, the utility $u(a,\omega)$ is
a continuous random variable with distribution in $\mathbb{R}$ induced by the
distribution of $\omega$. We also assume that
$\mathbb{E}_{Q}(|u(a,\omega)|)<\infty$. The induced distribution of
$u(a,\omega)$ can be defined as follows:

$\Psi(a,z)\triangleq\mathbb{P}(\\{\omega\in\Omega:u(a,\omega)\leq z\\}).$

Following Rockafellar and Uryasev (2000), we assume that $\Psi(a,z)$ is
everywhere continuous and strictly increasing in $z$. We make this assumption,
similar to the previous assumption regarding the density in $\omega$, for the
sake of simplicity.

######  Definition 3.

For each $a\in A$, define the $\alpha$-quantile of the random variable
$u(a,\omega)$ as

(12)
$\tau_{\alpha}(a)\triangleq\min\\{z\in\mathbb{R}\mid\Psi(a,z)\geqslant\alpha\\},\alpha\in(0,1).$

The previous definition states that under the assumption that $\Psi(a,z)$ is a
strictly increasing and continuous random variable on $\mathbb{R}$, the
quantity $\tau_{\alpha}(a)$ is uniquely defined. We now introduce the concept
of _Conditional Tail Expectation_ (CTE).

######  Definition 4.

For each $a\in A$ and $\alpha\in(0,1)$ the $\alpha$-CTE is given by:

(13)
$\mu_{\alpha}(a)\triangleq\mathbb{E}(u(a,\omega))|u(a,\omega)\geq\tau_{\alpha}(a)),$

where $\tau_{\alpha}(a)$ is the $\alpha$-quantile in (3).

With Definitions 3 and 4 established, our attention turns to modeling the cost
$C_{\phi}(p\|q)$. In this regard, we specifically focus on the following
$\phi$-divergence

(14) $\phi(t)=\begin{cases}0,&0\leq t\leq{1\over 1-\alpha}\\\
+\infty&\text{otherwise},\end{cases}$

with $\alpha\in(0,1).$

The previous expression warrants some comments. First, the $\phi$-divergence
(14) establishes that when the ratio ${p(\omega)/q(\omega)}$ falls between $0$
and $(1-\alpha)^{-1}$, the associated cost of belief distortion is zero.
Conversely, if ${p(\omega)/q(\omega)}$ lies outside this range, then
$C_{\phi}(p\|q)=+\infty$. Because of this structure, we refer to
$C_{\phi}(p\|q)$ as the _threshold beliefs distortion cost_ induced by (14).
Second, it is evident that the conjugate $\phi^{\ast}(s)$ corresponds to
$\phi^{\ast}(s)={1\over 1-\alpha}\max\\{s,0\\}$. This latter fact, combined
with Proposition 1, yields the following result:

(15)
$V_{\phi}(U(a))=\min_{\lambda_{a}\in\Lambda(a)}\left\\{\lambda_{a}+{1\over
1-\alpha}\mathbb{E}_{Q}(\max\\{u(a,\omega)-\lambda_{a},0\\})\right\\}$

In the financial literature on risk measures, the expression (15) is commonly
referred to as the Conditional Value-at-Risk (CVaR), which was introduced by
Rockafellar and Uryasev (2000).

Now, we are prepared to state the main result of this section.

######  Proposition 6.

Consider the WT problem (2) with the $\phi$-divergence defined as (14). Then,
for an optimal solution $(a^{\star},p^{\star}(a^{\star}))$, the following
statements hold:

  * (i)

The optimal $\lambda_{a^{\star}}^{\star}$ in (15) satisfies
$\lambda_{a^{\star}}^{\star}=\tau_{\alpha}(a^{\star}).$

  * (ii)

A WT agent solves the following optimization problem:

(16) $\max_{a\in
A}\mathbb{E}_{Q}(u(a,\omega)|u(a,\omega)\geq\tau_{\alpha}(a)).$

The result presented in Proposition 6 establishes that when the
$\phi$-divergence is defined by the expression (14), the WT optimization
problem simplifies to selecting the action with the highest CTE, as measured
by $\mathbb{E}_{Q}(u(a,\omega)|u(a,\omega)\geq\tau_{\alpha}(a))$. From a
behavioral standpoint, this implies that a WT agent focuses primarily on the
upper tail outcomes of each $u(a,\omega)$. As a consequence, the DM faces a
cognitive bias that leads her to rely on a _truncated distribution_ ,
effectively disregarding payoffs below the $\alpha$-quantile
$\lambda_{a^{\star}}^{\star}$. Consequently, WT behavior implies that the DM
concentrates exclusively on the tail segment and chooses the action with the
highest _expected upper-tail utility_. Truncated beliefs have been proposed by
Deligonul et al. (2008) as a possible explanation for entrepreneurial activity
in situations where risk-return levels are significantly lower than those of
private and public equity indexes. Notably, the result in Proposition 6
represents the first formalization of this type of truncated distribution
cognitive bias in economic behavior.

Secondly, the characterization provided in Proposition 6(ii) formalizes the
preference for skewness exhibited by an optimistic agent. This preference can
be observed by examining the formula (16), which captures the behavior of a WT
agent who may favor alternatives with high utility levels despite their low
probability of occurrence. In other words, Proposition 6(ii) elucidates the
DM’s inclination towards skewness by explicitly addressing the DM’s
considerations regarding tail performance comparisons.

A third observation pertains to the relationship between the expression
$\mathbb{E}_{Q}(u(a,\omega)|u(a,\omega)\geq\tau_{\alpha}(a))={1\over
1-\alpha}\int_{\alpha}^{1}\tau_{\theta}(a)d\theta$ and the DM’s problem. By
rewriting the optimization problem, we have the following:

$\max_{a\in A}\left\\{{1\over
1-\alpha}\int_{\alpha}^{1}\tau_{\theta}(a)d\theta\right\\}$

This equation explicitly reveals that under the $\phi$-divergence (14), WT
behavior can be interpreted as optimizing the _average quantile gain_ for
$\theta\in(\alpha,1]$. Therefore, Proposition 6 establishes a connection
between the WT approach and the concept of _quantile preferences_ found in the
literature on quantile-utility maximization (Chambers (2009), de Castro and
Galvao (2019, 2022), Rostek (2010), and Manski (1988)). However, unlike
quantile preference models, the result in Proposition 6 considers not only the
utility _at the quantile level_ , but also the average expected utility
associated with it.

Finally, we mention that the parameter $\alpha$ can be interpreted a the DM’s
degree of optimism. In particular, the value of $\alpha$ tells us how much
weight the DM will assign to events associated with the upper tail of the
distribution.

###  4.1. Applications

In this section, we leverage the previous results to analyze two specific
scenarios: entry market decisions and discrete choice models. By applying the
insights gained from Proposition 6, we can shed light on the behavior of DMs
in these contexts.

####  4.1.1. Entry market decisions

Let us consider an entry decision problem inspired by Bresnahan and Reiss
(1991). Suppose a firm needs to decide whether to enter a particular market.
Formally, we define the action set as $A=\\{a_{1},a_{2}\\}$, where
$a_{1}=\text{out}$ represents the choice of staying out of the market, and
$a_{2}=\text{enter}$ represents the choice of entering the market. The utility
of staying out, denoted as $u(\text{out},\omega)$, is a positive value $\pi$
independent of the state realizations. For the enter option, we assume that
the utility function is given by $u(\text{enter},\omega)=\pi-k+\omega$, where
$k>0$ represents the entry cost and $\omega$ denotes an additive profit shock.
We assume that $\omega$ follows a zero-mean distribution with finite variance.

In this setting, a decision-maker who seeks to maximize the EU would never
choose to enter the market. This is because
$\mathbb{E}_{Q}(u(a_{1},\omega))=\pi>\pi-k>\mathbb{E}_{Q}(u(a_{2},\omega))$.
However, by applying the insights from Proposition 6, we can show that the
firm may choose to enter. Concretely, the firm may enter the market under
specific parameter configurations driven by the firm’s optimistic behavior. To
demonstrate this, we observe that for $a_{2}$ (enter), the optimal
$\lambda_{a_{2}}^{\star}$ is given by
$\lambda_{a_{2}}^{\star}=Q^{-1}(\alpha)+\pi-k$. Therefore, the firm chooses to
enter if and only if the following condition holds:

$\mathbb{E}(\omega|\omega\geq Q^{-1}(\alpha))>k.$

Therefore, even though the actual mean of the profit shock distribution is
zero, the firm’s perception, influenced by the truncated distribution, leads
it to consider an effective mean that is positive. This highlights the impact
of truncated beliefs and WT behavior on the firm’s market entry decision.

####  4.1.2. Optimistic discrete choice

In this variant of discrete choice models, we consider a choice set
$A=\\{a_{1},\ldots,a_{n}\\}$ and assume that the outcome space $\Omega$ is
$\mathbb{R}^{n}$, where $Q$ is a fully supported $n$-dimensional distribution
over $\Omega$. In this context, each $\omega$ corresponds to an
$n$-dimensional vector $\omega=(\omega_{a_{1}},\ldots,\omega_{a_{n}})$, where
$\omega_{a}$ represents the realization associated with alternative $a$.
Without loss of generality, for all $a\in A$ we assume that
$\mathbb{E}_{Q_{a}}(\omega_{a})=0$, where $Q_{a}$ denotes the marginal
distribution associated with alternative $a$.

The utility associated with $a$ is:

(17) $u(a,\omega)=u(a)+\omega_{a}.$

From Proposition 6(i), for each alternative $a\in A$ we get:

$\lambda_{a}^{*}=u(a)+Q_{a}^{-1}(\alpha).$

Then the WT agent will select the optimal action as the solution of the
problem

(18) $\max_{a\in A}\\{u(a)+\mathbb{E}_{Q_{a}}(\omega_{a}|\omega_{a}\geq
Q_{a}^{-1}(\alpha))\\}.$

In the latter expression, the wishful thinking (WT) agent considers both the
deterministic utility $u(a)$ and the average upper tail utility value
associated with realizations of $\omega_{a}$. Specifically, in equation (18),
the term $\mathbb{E}_{Q_{a}}(\omega_{a}|\omega_{a}\geq Q_{a}^{-1}(\alpha))$
captures the level of optimism associated with alternative $a$. From a
behavioral perspective, this implies that the WT agent overestimates the
utility of each alternative.

This upward bias arises because $\mathbb{E}_{Q_{a}}(\omega_{a}|\omega_{a}\geq
Q_{a}^{-1}(\alpha))\geq\mathbb{E}_{Q_{a}}(\omega_{a})=0$, indicating that the
WT agent assigns higher perceived utility to the alternative $a$ by focusing
on the positive upper tail outcomes of $\omega_{a}$. By selectively
considering the upper tail realizations and neglecting the lower tail or
average outcomes, the WT agent exhibits a biased perception of the actual
utility of each alternative. This bias reflects the optimistic belief that the
upper tail outcomes will occur more frequently or have a more significant
impact than they do in reality.

It is important to note that the additive utility structure presented in
equation (17) bears similarities to the well-known additive random utility
model (RUM) (McFadden (1978, 1981)). However, there are two significant
differences between the two approaches. First, the RUM framework does not
incorporate optimistic behavior explicitly. In contrast, our model allows for
WT by considering each alternative’s average upper tail utility values. This
introduces an element of optimism into the decision-making process, leading to
potentially different choice outcomes compared to the RUM. Second, the RUM
approach provides an optimal stochastic choice rule describing the
probabilities of each alternative $a\in A$. In contrast, our discrete choice
model with wishful thinking provides a rule specifying how the DM selects a
particular alternative based on WT behavior. The expression (18) serves as a
prescription for decision-making under wishful thinking.

Thus, by incorporating wishful thinking into the model, we capture a
behavioral bias that deviates from the traditional RUM framework. The WT model
allows for a more nuanced understanding of decision-making by considering the
influence of optimistic beliefs on the selection of alternatives.

######  Example 4.

To see how the previous model provides new insights let us assume that for
each $a\in A$, the random variable $\omega_{a}$ follows a normal distribution
with mean zero and variance $\sigma_{a}^{2}$. Using the results in Norton et
al. (2018), the problem (18) can be expressed as:

$\max_{a\in A}\left\\{u(a)+\sigma_{a}{q(Q_{a}^{-1}(\alpha))\over
1-\alpha}\right\\}$

Thus, when the $\omega_{a}$s are normally distributed, the distorted utilities
can be interpreted as a mean-variance term. Specifically, for each alternative
$a$, the distorted utility increases with the standard deviation $\sigma_{a}$.
Consequently, in this environment, the WT agent assigns value to the risk
associated with each alternative. This example highlights that a WT agent,
from a behavioral standpoint, prefers taking more risks than a traditional
rational DM.

####  4.1.3. Skewed discrete choice

To understand the influence of the factors
$\mathbb{E}_{Q_{a}}(\omega_{a}|\omega_{a}\geq Q_{a}^{-1}(\alpha))$ on the
choice process, we focus on the generalized Pareto distribution (GPD) family.
The GPD is characterized by two parameters: $\xi_{a}\in\mathbb{R}$ and
$\beta_{a}>0$. The GPD function can be expressed as follows:

(19)
$G_{\xi_{a},\beta_{a}}(\omega_{a})=\begin{cases}1-\left(1+\frac{\xi_{a}\omega_{a}}{\beta_{a}}\right)^{-1/\xi}&\xi_{a}\neq
0,\\\ 1-\exp\left(-\frac{\omega_{a}}{\beta_{a}}\right)&\xi_{a}=0\end{cases}$

where $\omega_{a}\in[0,\infty)$ for $\xi_{a}\geq 0$ and
$\omega_{a}\in[0,-\beta_{a}/\xi_{a}]$ for $\xi_{a}<0$.

The parameters $\xi_{a}$ and $\beta_{a}$ play a crucial role in determining
the shape and scale of the distribution. In particular, when $\xi_{a}=0$, the
GPD reduces to an exponential distribution. When $\xi_{a}>0$,
$G_{\xi_{a},\beta_{a}}$ represents a Pareto distribution. It is important to
note that for $\xi_{a}>0$, the $k$-th moment does not exist when $k\geq
1/\xi_{a}$, similar to the case of the Fréchet distribution. On the other
hand, when $\xi_{a}<0$, the expression (19) yields the Pareto Type II
distribution. However, this distribution is less useful in our context as it
has a fixed right endpoint.

We focus on the case where $\xi_{a}>0$, which offers two main advantages.
Firstly, the Pareto distribution allows for positive skewness in the utility
associated with specific alternatives. Secondly, in this case, we can provide
a closed-form expression for the terms
$\mathbb{E}_{Q_{a}}(\omega_{a}|\omega_{a}\geq Q_{a}^{-1}(\alpha))$. To
illustrate the latter point, let us compute the value associated with
$\lambda_{a}=\tau_{\alpha}(a)$. Using the expression (19), we have:

$\lambda_{a}={\beta_{a}\over\xi_{a}}(\alpha^{-\xi_{a}}-1).$

Now, thanks to the fact that $\mathbb{E}_{Q_{a}}(\omega_{a}|\omega_{a}\geq
Q_{a}(\alpha)^{-1})={(1-\alpha)^{-1}}\int_{\alpha}^{1}\tau_{\theta}(U(a))d\theta$,
direct computation yields:

$\mathbb{E}_{Q_{a}}(\omega_{a}|\omega_{a}\geq
Q^{-1}_{a}(\alpha))={Q^{-1}_{a}(\alpha)\over 1-\xi}+{\beta_{a}\over\xi_{a}}.$

The following proposition formalizes the previous analysis

######  Proposition 7.

Consider a WT problem with the additive payoff structure (17). Assume that
$\omega_{a}$ follows a GPD with parameters $\xi_{a}>0$ and $\beta_{a}>0$ for
each $a\in A$. Then a WT agent solves the following problem

(20) $\max_{a\in A}\left\\{u(a)+{Q^{-1}_{a}(\alpha)\over
1-\xi_{a}}+{\beta_{a}\over\xi_{a}}\right\\}.$

The expression (20) reveals that an optimistic DM will modify her utilities by
incorporating a term that depends on the quantile $Q^{-1}_{a}(\alpha)$ and the
shape and location of the associated distribution, represented by the
parameters $\xi_{a}$ and $\beta_{a}$ respectively. This modification allows
the DM to capture the effects of WT and tailor her preferences based on the
specific characteristics of each alternative.888In Appendix C we discuss
several distributions that yield closed-form formulas to
$\mathbb{E}_{Q_{a}}(\omega_{a}|\omega_{a}\geq Q_{a}^{-1}(\alpha)).$

To gain economic insights from the problem (20), we revisit the case of free
entry discussed in Section 4.1.1. In this scenario, a firm decides whether to
enter a particular market. The choice set is denoted by $A=\\{a_{1},a_{2}\\}$,
where $a_{1}=\text{out}$ represents the decision to stay out of the market,
and $a_{2}=\text{enter}$ represents the decision to enter the market.

Recall that the associated payoffs for the firm are given by
$u(out,\omega)=\pi>0$, which represents the utility of staying out, and
$u(enter,\omega)=\pi-k+\omega$, where $k>0$ denotes the fixed entry cost and
$\omega$ represents an additive profit shock.

Let us consider the case where the profit shock $\omega$ follows a GPD denoted
by $G_{\xi,\beta}$. We can analyze the decision-making process under WT in
this context by utilizing the solution provided by the problem (20).
Specifically, the WT firm will choose to enter the market if and only if the
following condition is satisfied:

${Q^{-1}(\alpha)}\geq(1-\xi)\left(k-{\beta\over\xi}\right).$

The entry rule described above provides a specific cutoff value determining
the firm’s decision to enter the market. This cutoff value depends on the
fixed entry cost $k$ and the parameters $\beta$ and $\xi$ of the GPD
representing the profit shock.

The parameter $\xi$ is particularly interesting, representing the degree of
skewness in the distribution. This parameter has a significant influence on
the decision-making process of the firm. Specifically, a higher value of $\xi$
indicates a greater degree of positive skewness, meaning the distribution has
a longer right tail. As a result, a firm with WT behavior may be more inclined
to enter the market when facing a positively skewed profit distribution, as it
becomes more optimistic about the potential for high profits.

##  5\. Related Literature

Our paper is situated within several strands of literature, addressing various
aspects of wishful thinking (WT) in economic decision-making models.

First, our paper is related to WT in economic decision-making models. The
papers by Mayraz (2019) and Kovach (2020) provide an axiomatic foundation for
WT behavior.

The paper by Caplin and Leahy (2019) shares similarities with our work and is
the closest related study to ours. Like us, they consider a decision-making
model where the DM selects a probability distribution over states based on the
associated EU and the cost of distorting baseline beliefs. They quantify this
cost using the Kullback-Leibler distance between subjective and objective
beliefs. However, several essential distinctions exist between Caplin and
Leahy (2019)’s results and ours. In particular, our framework encompasses the
Kullback-Leibler distance as a specific case within a broader framework of WT.
This allows us to explore cognitive emergence and suppression, the connection
between WT and risk measures, wishful thinking as quantile-utility
maximization, and the preference for skewness arising from WT. These aspects
go beyond the scope of Caplin and Leahy (2019).

Second, our paper is related to the literature on optimal expectations
specifically, the work of Brunnermeier and Parker (2005). They study a dynamic
model of belief choice, where the decision-maker selects their beliefs at the
beginning of their life before making other decisions. Subsequently, the
decision-maker behaves as a Bayesian, updating their beliefs based on new
information. Notably, Brunnermeier and Parker (2005) evaluate belief choice
using the objective probability distribution and proceed with the chosen
beliefs.

While there are similarities between their model and ours, there are important
structural differences. In our approach, objective beliefs anchor the cost of
distorting beliefs, but we do not utilize them to quantify the benefits and
costs of the decision-makers choices. We focus on WT, cognitive suppression,
cognitive emergence, the connection between WT and risk measures, and WT as
quantile-utility maximization. These aspects are not explicitly addressed in
Brunnermeier and Parker (2005) model. While they demonstrate how their model
can generate a preference for skewness in a portfolio allocation problem, it
is important to note that our model does not imply their result, and vice
versa, due to the fundamental differences in the nature and structure of the
two models.

Finally, our paper is also related to the literature on robustness in economic
models. Specifically, Hansen and Sargent (2001, 2008) introduce a robustness
approach to address the concern of model misspecification. They adopt a max-
min approach, akin to multiple priors models as in Gilboa and Schmeidler
(1989) and Maccheroni et al. (2006), to make decisions under ambiguity.

While robustness and ambiguity models typically take a pessimistic approach to
decision-making, our paper focuses on WT behavior and adopts an optimistic
approach by studying a max-max problem. By doing so, our paper draws on the
active and rapidly growing literature on distributionally robust optimization
problems Shapiro (2017) and Kuhn et al. (2019). This framework provides a
powerful tool to handle decision problems under uncertainty, allowing for a
consideration of a range of possible distributional assumptions. In our case,
we leverage this framework to capture WT behavior and its implications for
decision-making.

Thus, while our paper is related to the robustness literature, it differs in
terms of its focus on WT, adopting an optimistic approach, and using
distributionally robust optimization techniques. These distinctions enable us
to study the specific biases and preferences associated with WT and its impact
on decision outcomes.

##  6\. Conclusions

In this paper, we develop a tractable model of WT and optimism. By
incorporating the costs and benefits of biased beliefs, we establish
connections between WT behavior and risk measures, showing how an optimistic
agent chooses riskier alternatives. Our model captures extreme belief
behaviors such as cognitive suppression and emergence, optimism, quantile-
utility maximization, and the relationship between WT and preferences for
skewness.

Moving forward, there are several avenues for extending and enriching our
model. One important direction is considering dynamic environments where
decision-making occurs over multiple stages. Understanding how WT behavior
unfolds and evolves over time can provide valuable insights into biased belief
formation and decision-making dynamics.

Additionally, our current model focuses on single-agent WT behavior. Extending
the analysis to strategic interactions and WT can offer a deeper understanding
of how optimism and biased beliefs influence strategic decision-making and
outcomes in interactive settings.

Finally, experimental testing of the theoretical predictions derived from your
model can further validate and refine the insights obtained.

By pursuing these extensions and empirical investigations, further
advancements can be made in understanding WT behavior, its underlying
mechanisms, and its implications for various decision-making contexts.

##  Appendix A Proofs

### Proof of Lemma 1

In the optimization problem

$V_{\phi}(U(a))=\max_{p\in\Delta(\Omega)}\left\\{\sum_{\omega\in\Omega}p(\omega)u(a,\omega)-\delta
D_{\phi}(p\|q)\right\\},$

the objective is concave in $p$, and the constraints are linear. Therefore,
the optimal value is equal to the optimal value of the Lagrangian dual
problem,

(21) $\displaystyle\mathcal{L}(\lambda)$ $\displaystyle=$
$\displaystyle\inf_{\lambda_{a}}\max_{p\in\Delta(\Omega)}\left\\{\sum_{\omega\in\Omega}p(\omega)u(a,\omega)-\delta
D_{\phi}(p\|q)+\lambda_{a}\left(1-\sum_{\omega\in\Omega}p(\omega)\right)\right\\}$
$\displaystyle=$
$\displaystyle\inf_{\lambda_{a}}\left\\{\lambda_{a}+\sum_{\omega\in\Omega}\sup_{p(\omega)\geq
0}\left\\{p(\omega)\left(u(a,\omega)-\lambda_{a}\right)-\delta\sum_{\omega\in\Omega}q(\omega)\phi\left(\frac{p(\omega)}{q(\omega)}\right)\right\\}\right\\}$
$\displaystyle=$
$\displaystyle\inf_{\lambda_{a}}\left\\{\lambda_{a}+\sum_{\omega\in\Omega}\sup_{p(\omega)\geqslant
0}\left\\{p(\omega)\left(u(a,\omega)-\lambda_{a}\right)-\delta\sum_{\omega\in\Omega}q(\omega)\phi\left(\frac{p(\omega)}{q(\omega)}\right)\right\\}\right\\}$
$\displaystyle=$
$\displaystyle\inf_{\lambda_{a}}\left\\{\lambda_{a}+\sum_{\omega\in\Omega}q(\omega)\max_{p(\omega)\geqslant
0}\left\\{\frac{p(\omega)}{q(\omega)}\left(u(a,\omega)-\lambda_{a}\right)-\delta\phi\left(\frac{p(\omega)}{q(\omega)}\right)\right\\}\right\\}$
$\displaystyle=$
$\displaystyle\inf_{\lambda_{a}}\left\\{\lambda_{a}+\delta\sum_{\omega\in\Omega}q(\omega)\sup_{t\geqslant
0}\left\\{t\left((u(a,\omega)-\lambda_{a})/\delta\right)-\phi(t)\right\\}\right\\}$
$\displaystyle=$
$\displaystyle\inf_{\lambda_{a}}\\{\lambda_{a}+\delta\mathbb{E}_{q}(\phi^{*}((u(a,\omega)-\lambda_{a})/\delta))\\}$

Now, solving for $\lambda_{a}$ we find that there exists a unique
$\lambda_{a}^{\star}$ that satisfies

(22)
$\sum_{\omega\in\Omega}\phi^{\ast\prime}((u(a,\omega)-\lambda_{a}^{\star})/\delta)q(\omega)=1.$

Furthermore, $\lambda_{a}^{\star}\in[\underline{u}_{a},\bar{u}_{a}]$. To see
this, recall that $\phi^{*\prime}$ is monotonically increasing and
$\phi^{*\prime}(0)=1$. Assume that there exists an optimal
$\tilde{\lambda}_{a}$ such that $\tilde{\lambda}_{a}>\bar{u}_{a}$, then
$\sum_{\omega\in\Omega}\phi^{\ast\prime}((u(a,\omega)-\tilde{\lambda}_{a}\
)/\delta)q(\omega)<1$. Therefore $\tilde{\lambda}_{a}$ cannot be optimal. Now
assume there exists an optimal $\tilde{\lambda}_{a}$ such that
$\tilde{\lambda}_{a}<\underline{u}_{a}$, then
$\sum_{\omega\in\Omega}\phi^{\ast\prime}((u(a,\omega)-\tilde{\lambda}_{a})/\delta)q(\omega)>1$.
Therefore $\tilde{\lambda}_{a}$ cannot be optimal. We conclude that it must
hold $\lambda_{a}^{\star}\in[\underline{u}_{a},\bar{u}_{a}]$. Then the infimum
in (21) is achieved, and we can write

$\max_{p\in\Delta(\Omega)}\left\\{\sum_{\omega\in\Omega}p(\omega)u(a,\omega)-\delta
D_{\phi}(p\|q)\right\\}=\min_{\lambda_{a}\in[\underline{u}_{a},\bar{u}_{a}]}\\{\lambda_{a}+\delta\mathbb{E}_{q}(\phi^{*}((u(a,\omega)-\lambda_{a})/\delta))\\}.$

The uniqueness of $\lambda_{a}^{\star}$ follows from Assumption 1, which
implies that problem (5) is strictly convex in $\lambda_{a}$.  $\square$

### Proof of Proposition 1

From Lemma 1, we know that the necessary and sufficient first-order conditions
in problem (5) yields
$p^{\star}(\omega|a)=\phi^{\ast\prime}((u(a,\omega)-\lambda_{a}^{\star})/\delta)q(\omega)$
for all $a\in A$, $\omega\in\Omega$. Then by a straightforward application of
the envelope theorem, we get ${\partial V_{\phi}(U)\over\partial
u(a,\omega)}=\phi^{*\prime}((u(a,\omega)-\lambda_{a}^{\star})/\delta)q(\omega)$
for all $\omega\in\Omega$. Thus we conclude $\nabla
V_{\phi}(U(a))=p^{\star}(a)$ for all $a\in A$. $\square$

### Proof of Corollary 1

From expression (7) we know that the weights
$\phi^{\ast\prime}((u(a,\omega)-\lambda_{a}^{\star})/\delta)$ are increasing
in $u(a,\omega)$. To see the bias, we note that when
$u(a,\omega)>u(a,\omega^{\prime})$, the strict monotonicity of the gradient
$\phi^{\ast\prime}$ implies that
$\phi^{\ast\prime}((u(a,\omega)-\lambda_{a}^{\star})/\delta)>\phi^{\ast\prime}((u(a,\omega^{\prime})-\lambda_{a}^{\star})/\delta)$,
which implies $p^{\star}(\omega|a)>p^{\star}(\omega^{\prime}|a)$.

Next, we note
$p^{\star}(\omega|a)=\phi^{\ast\prime}((u(a,\omega)-\lambda_{a}^{\star})/\delta)q(\omega)$
is linearly increasing in $q(\omega)$. Combining these facts implies that
given states $\omega$ $\omega^{\prime}$ with $q(\omega)>q(\omega^{\prime})$
and $u(a,\omega)>u(a,\omega^{\prime})$, we have
$p^{\star}(\omega|a)>p^{\star}(\omega^{\prime}|a)$.

$\square$

### Proof of Proposition 2

Proofs of (i)-(iv)

(i) For any $\phi\in\Phi$ and any $c\in\mathbb{R}$

$\displaystyle V_{\phi}(U(a)+c)$
$\displaystyle=\inf_{\lambda_{a}\in\mathbb{R}}\\{\lambda_{a}+\mathbb{E}(\phi^{\ast}(u(a,\omega)+c-\lambda_{a}))\\}$
$\displaystyle=c+\inf_{\lambda_{a}\in\mathbb{R}}\\{\lambda_{a}-c+\mathbb{E}(\phi^{\ast}(u(a,\omega)-(\lambda_{a}-c)))\\}=c+V_{\phi}(U(a)).$

(ii) Since $\phi\in\Phi$, then $\phi^{\ast}(0)=0,0\in\partial\phi^{\ast}(1)$
and the convexity of $\phi^{\ast}$ implies $\phi^{\ast}(t)\geq t$, and hence

$V_{\phi}(U(a))\geq\inf_{\lambda_{a}\in\mathbb{R}}\\{\lambda_{a}+(c-\lambda_{a})\\}=c.$

For the converse inequality, since $\phi^{\ast}(0)=0$, one has
$V_{\phi}(c)\leq\\{c+\phi^{\ast}(c-c)\\}=c$. Then we conclude that
$V_{\phi}(c)=c$.

(iii) If $U(a)\leq\tilde{U}(a)$, then
$U(a)-\lambda_{a}\leq\tilde{U}(a)-\lambda_{a}$, and since $\phi^{\ast}$ is
nondecreasing it follows that,

$V_{\phi}(U(a))=\inf_{\lambda_{a}\in\mathbb{R}}\\{\lambda_{a}+\mathbb{E}\phi^{\ast}(u(a,\omega)-\lambda_{a}))\\}\leq\inf_{\lambda_{a}\in\mathbb{R}}\\{\lambda_{a}+\mathbb{E}\phi^{\ast}(\tilde{u}(a,\omega)-\lambda_{a}))\\}=V_{\phi}(\tilde{U}(a)).$

(iv) Let $\alpha\in(0,1)$ and for any random variables $U_{1}(a),U_{2}(a)$,
let $U_{\alpha}(a):=\alpha U_{1}(a)+(1-\alpha)U_{2}(a)$. Since $\phi^{\ast}$
is convex, the function
$f(z,\lambda_{a}):=\lambda_{a}+\phi^{\ast}(z-\lambda_{a})$ is jointly convex
over $\mathbb{R}\times\mathbb{R}$. Therefore, for any
$\lambda_{a}^{1},\lambda_{a}^{2}\in\mathbb{R}$, and with
$\lambda_{a}^{\alpha}\triangleq\alpha\lambda_{a}^{1}+(1-\alpha)\lambda_{a}^{2}$,
one has,

$\mathbb{E}f\left(U_{\alpha}(a),\lambda_{\alpha}\right)\leq\lambda\mathbb{E}f\left(U_{1}(a),\lambda_{a}^{1}\right)+(1-\alpha)\mathbb{E}f\left(U_{2}(a),\lambda_{a}^{2}\right).$

Since
$V_{\phi}\left(U_{\alpha}(a)\right)=\inf_{\lambda_{a}\in\mathbb{R}}\mathbb{E}f\left(U_{\alpha}(a),\lambda\right)$,
it follows that,

$\displaystyle V_{\phi}\left(U_{\alpha}(a)\right)$
$\displaystyle\leq\inf_{\lambda_{a}^{1},\lambda_{a}^{2}}\left\\{\alpha\mathbb{E}f\left(U_{1}(a),\lambda_{a}^{1}\right)+(1-\alpha)\mathbb{E}f\left(U_{2}(a),\lambda_{a}^{2}\right)\right\\}$
$\displaystyle=\alpha
V_{\phi}\left(U_{1}(a)\right)+(1-\alpha)V_{\phi}\left(U_{2}(a)\right).$

$\square$

### Proof of Proposition 3

The WT agent’s optimal actions are given by:

$\displaystyle A_{eu}(\tilde{U})$ $\displaystyle=$
$\displaystyle\arg\max_{a\in A}\mathbb{E}_{q}(\tilde{u}(a,\omega)),$
$\displaystyle=$ $\displaystyle\arg\max_{a\in
A}[\lambda_{a}^{\star}+\delta\mathbb{E}_{q}(\phi^{*}((u(a,\omega)-\lambda_{a}^{\star})/\delta))]\text{(By
Definition)},$ $\displaystyle=$ $\displaystyle\arg\max_{a\in
A}\max_{p\in\Delta(\Omega)}\left[\sum_{\omega\in\Omega}p(\omega)u(a,\omega)-\delta
C_{\phi}(p\|q)\right],$ $\displaystyle=$ $\displaystyle A_{wt}(U),$

where the last equality follows from the definition of $A_{wt}(U)$.

$\square$

### Proof of Corollary 3

Thanks to Lemma 2, we know that for each $a\in A$

$V_{\phi}(U(a))=\max_{p\in\Delta(\Omega)}\sum_{\omega\in\Omega}p(\omega)u(a,\omega)-\delta
D_{\phi}(p\|q)=\max_{\lambda_{a}\in\Lambda(a)}\Psi(a,\lambda_{a}).$

Then it follows that problem (2) is equivalent to

$\max_{a\in A}\min_{\lambda_{a}\in\Lambda(a)}\Psi(a,\lambda_{a}).$

$\square$

### Proof of Proposition 4

To study a situation of cognitive suppression, we consider a state
$\hat{\omega}$ such that $q(\hat{\omega})>0$. Let $a^{\star}$ and
$p^{\star}(a^{\star})$ be an optimal solution to WT problem (2).

From Proposition 1, the subjective belief vector is given by
$p^{\star}(\hat{\omega}|a^{\star})=\phi^{\ast\prime}(s_{\hat{\omega}}^{\star})q(\hat{\omega})$
where
$s_{\hat{\omega}}^{\star}=u(a^{\star},\hat{\omega})-\lambda_{a^{\star}}^{\star}$.
Because $q(\hat{\omega})>0$, $p^{\star}(\hat{\omega}|a^{\star})=0$ if and only
if $\phi^{\ast\prime}(s_{\hat{\omega}}^{\star})=0$. For
$\phi^{\ast\prime}(s_{\hat{\omega}}^{\star})=0$, it must be the case that
$\phi^{\ast}(s_{\hat{\omega}}^{\star})=c$. $\phi^{*}$ is a monotone, non-
decreasing function, so there exists $s_{\hat{\omega}}^{\star}$ such that
$\phi^{\ast}(s_{\hat{\omega}}^{\star})=c$ only if
$\lim_{s_{\hat{\omega}}^{\star}\rightarrow-\infty}\phi^{\ast}(s_{\hat{\omega}}^{\star})=c>-\infty$.
Therefore $p^{\star}(\hat{\omega}|a^{\star})=0$ only if
$\lim_{s_{\hat{\omega}}^{\star}\rightarrow-\infty}\phi^{\ast}(s_{\hat{\omega}}^{\star})=c>-\infty$
.

We now can prove condition (i), $\lim_{t\rightarrow 0+}\phi(t)<\infty$, is a
necessary condition for cognitive suppression by contradiction. Assume
$\lim_{t\rightarrow 0+}\phi(t)=\infty$. Applying the definition of a convex
conjugate, this implies
$\lim_{s_{\hat{\omega}}^{\star}\rightarrow-\infty}\phi^{*}(s_{\hat{\omega}}^{\star})=-\infty$.
Therefore, $p^{\star}(\hat{\omega}|a^{\star})=0$ cannot hold if
$\lim_{t\rightarrow 0+}\phi(t)=\infty$. We can then say that
$\lim_{t\rightarrow 0+}\phi(t)<\infty$ is a necessary condition for cognitive
suppression. This proves Condition (i).

We now prove condition (ii), $\lim_{t\rightarrow 0+}\phi^{\prime}(t)>-\infty$
is a necessary condition for cognitive suppression by contradiction. Assume
$\lim_{t\rightarrow 0+}\phi^{\prime}(t)=-\infty$. Bayraksan and Love (2015b)
shows that this implies
$\lim_{s_{\hat{\omega}}^{\star}\rightarrow-\infty}\phi^{\ast}(s_{\hat{\omega}}^{\star})=c$
asymptotically, but there does not exist $s_{\hat{\omega}}^{\star}$ such that
$\phi^{*}(s_{\hat{\omega}}^{\star})=c$. Therefore for all
$\lambda_{a^{\star}}^{\star}$ it must hold
$\phi^{*\prime}(s_{\hat{\omega}}^{\star})>0$. This implies
$p^{\star}(\hat{\omega}|a^{\star})=0$ cannot hold if $\lim_{t\rightarrow
0+}\phi^{\prime}(t)=-\infty$. We can then say that $\lim_{t\rightarrow
0+}\phi^{\prime}(t)>-\infty$ is a necessary condition for cognitive
suppression. This proves condition (ii).

To prove the final assertion, note that $\phi^{*}$ is a monotone, non-
decreasing, convex function. Then, if there exists an
$s_{\hat{\omega}}^{\star}$ s.t.
$\phi^{\ast\prime}(s_{\hat{\omega}}^{\star})=0$, there exists a cutoff
$\tilde{s}(a^{\star})$ such that
$\phi^{\ast\prime}(s_{\hat{\omega}}^{\star})=0$ if and only if
$s_{\hat{\omega}}^{\star}\leq\tilde{s}(a^{\star})$. Equivalently, there exists
$\tilde{u}(a^{\star})=\tilde{s}(a^{\star})+\lambda^{\star}_{a^{\star}}$ such
that
$\phi^{\ast\prime}(u(a^{\star},\hat{\omega})-\lambda^{\star}_{a^{\star}})=0$
if and only if $u(a^{\star},\hat{\omega})\leq\tilde{u}(a^{\star})$. Finally,
we can see $p^{\star}(\hat{\omega}|a^{\star})=0$ if and only if
$u(a^{\star},\hat{\omega})\leq\tilde{u}(a^{\star})$. This proves the final
assertion.

$\square$

### Proof of Proposition 5

To study a situation of cognitive emergence, assume there exists a state
$\hat{\omega}$ such that $q(\hat{\omega})=0$. Let $a^{\star}$ and
$p^{\star}(a^{\star})$ be an optimal solution to WT problem (2).

Recall that we defined
$0\phi(\frac{c}{0})=clim_{t\rightarrow\infty}\frac{\phi(t)}{t}$ $\forall c>0$
and $0\phi(\frac{0}{0})=0$.

We first prove (i), $\lim_{t\rightarrow\infty}\frac{\phi(t)}{t}=b<\infty$ is a
necessary condition for cognitive emergence by contradiction. Assume
$\lim_{t\rightarrow\infty}\frac{\phi(t)}{t}=\infty$. Using the definition of
$V_{\phi}(U(a^{\star}))$ we know that:

$\inf_{\lambda_{a^{\star}}}\left\\{\lambda_{a^{\star}}+\sum_{\omega}\sup_{p(\omega|a^{\star})\geqslant
0}\left\\{p(\omega|a^{\star})\left(u(a^{\star},\omega)-\lambda_{a^{\star}}\right)-q(\omega)\phi\left(\frac{p(\omega|a^{\star})}{q(\omega)}\right)\right\\}\right\\}$

Consider the term inside the summation corresponding to state $\hat{\omega}$:

$\sup_{p(\hat{\omega}|a^{\star})\geqslant
0}\left\\{p(\hat{\omega}|a^{\star})\left(u(a^{\star},\hat{\omega})-\lambda_{a}^{\star}\right)-0\phi\left(\frac{p(\hat{\omega}|a^{\star})}{0}\right)\right\\}=\sup_{p(\hat{\omega}|a^{\star})\geq
0}\\{F(p(\hat{\omega}|a^{\star}))\\}$

where:

$F(p(\hat{\omega}|a^{\star}))=\begin{cases}-\infty&\text{ if
}p(\hat{\omega}|a^{\star})>0\\\ 0&\text{ if
}p(\hat{\omega}|a^{\star})=0\end{cases}$

This yields $p^{\star}(\hat{\omega}|a^{\star})=0$. Then
$p^{\star}(\hat{\omega}|a^{\star})>0$ cannot hold if
$\lim_{t\rightarrow\infty}\frac{\phi(t)}{t}=\infty$. We can then say
$\lim_{t\rightarrow\infty}\frac{\phi(t)}{t}=b<\infty$ is a necessary condition
for cognitive emergence.

To prove assertions (ii) and (iii), let
$\lim_{t\rightarrow\infty}\frac{\phi(t)}{t}=b<\infty$. Again consider the term
inside the summation associated with state $\hat{\omega}$:

$\sup_{p(\hat{\omega}|a^{\star})\geqslant
0}\left\\{p(\hat{\omega}|a^{\star})\left(u(a^{\star},\hat{\omega})-\lambda^{\star}_{a^{\star}}\right)-0\phi\left(\frac{p(\hat{\omega}|a^{\star})}{0}\right)\right\\}$

$\sup_{p(\hat{\omega}|a^{\star})\geqslant
0}\left\\{p(\hat{\omega}|a^{\star})\left(u(a^{\star},\hat{\omega})-\lambda^{\star}_{a^{\star}}-b\right)\right\\}$

The optimal choice depends on the sign of
$(u(a^{\star},\hat{\omega})-\lambda^{\star}_{a^{\star}}-b)$. Consider all
three cases:

  1. (1)

$(u(a^{\star},\hat{\omega})-\lambda^{\star}_{a^{\star}}-b)>0$: The supremum
yields $p^{\star}(\hat{\omega}|a^{\star})=\infty$, making the term unbounded.
Because $\lambda^{\star}_{a^{\star}}$ is an argument which minimizes it’s
objective function, $\lambda^{\star}_{a^{\star}}$ cannot admit this case

  2. (2)

$(u(a^{\star},\hat{\omega})-\lambda^{\star}_{a^{\star}}-b)<0$. The supremum
yields $p^{\star}(\hat{\omega}|a^{\star})=0$.

  3. (3)

$(u(a^{\star},\hat{\omega})-\lambda^{\star}_{a^{\star}}-b)=0$. There is no
restriction on $p^{\star}(\hat{\omega}|a^{\star})$, i.e.
$p^{\star}(\hat{\omega}|a^{\star})\in[0,\infty)$

Therefore only cases (2) and (3) will be possible with the optimal
$\lambda_{a^{\star}}^{\star}$:
$(u(a^{\star},\omega)-\lambda_{a^{\star}}^{\star}-b)\leq 0$ for all states
$\omega\in\Omega$. The state $\hat{\omega}$ can have subjective probability
$p^{\star}(\hat{\omega}|a^{\star})$ only if
$u(a^{\star},\hat{\omega})-\lambda_{a^{\star}}^{\star}-b=0$. In other words:
$\lambda_{a^{\star}}^{\star}=u(a^{\star},\hat{\omega})-b$. Furthermore, it
must hold that $u(a^{\star},\hat{\omega})=\bar{u}(a^{\star})$; otherwise, case
(1) would arise for another state. This proves assertions (ii) and (iii).
$\square$

### Proof of Proposition 6

In expression (15) define $F(a,\lambda_{a})\triangleq\lambda_{a}+{1\over
1-\alpha}\mathbb{E}_{Q}(\max\\{u(a,\omega)-\lambda_{a},0\\})$. Then expression
(15) can be rewritten as

(23) $V_{\phi}(U(a))=\min_{\lambda_{a}\in\Lambda(a)}F(a,\lambda_{a}).$

By (Rockafellar and Uryasev, 2000, Thm. 1), we know that $F(a,\lambda_{a})$ is
convex and continuously differentiable with respect to $\lambda_{a}$. Thus the
optimal $\lambda_{a}^{\star}$ the necessary and sufficient first-order
condition yields:

$1-{1\over 1-\alpha}\mathbb{P}(u(a,\omega)\geq\lambda_{a})=0.$

The previous expression yields
$\alpha=\mathbb{P}(u(a,\omega)\leq\lambda^{\star}_{a})$. Then from Definition
3 it follows that $\lambda_{a}^{\star}=\tau_{\alpha}(a)$. This proves part
(i). Now, plugging $\tau_{\alpha}(a)$ in expression (15), we get:

$V_{\phi}(a)=\mathbb{E}_{Q}(u(a,\omega)|u(a,\omega)\geq\tau_{\alpha}(a)).$

As the previous expression holds for each $a\in A$. Hence, we conclude that
the WT agent solves the problem (16). $\square$

##  Appendix B Details of cognitive suppression and emergence

###  B.1. Modified $\chi^{2}$ and cognitive suppression

We now provide the details of the modified $\chi^{2}$ distance generating
$p^{\star}(\omega_{H}|a^{\star})=1.$ We recall that
$\Omega=\\{\omega_{H},\omega_{L}\\}$ and $A=\\{1,\ldots,n\\}$. Let
$(a^{\star},p^{\star}(a^{\star}))$ be an optimal solution where the state
contingent utilities are $u(a^{\star},\omega_{H})=4$ and
$u(a^{\star},\omega_{L})=0$. To determine the optimal subjective belief vector
$(p^{\star}(\omega_{H}|a^{\star}),p^{\star}(\omega_{L}|a^{\star}))$, we apply
Lemma 1 to find:

$\lambda_{a^{\star}}^{\star}=\begin{cases}6-\frac{2}{q_{H}}&q_{H}\geq 1/2\\\
4q_{H}&q_{H}<1/2\end{cases}$

Notice that $\lambda_{a^{\star}}^{\star}\geq 2\rightarrow
p^{\star}(\omega_{L}|a^{\star})=0$ as
$\phi^{*\prime}(\lambda_{a^{\star}}^{\star})=0$. Then the optimistic DM sets
$p^{\star}(\omega_{H}|a)=1$ and $p^{\star}(\omega_{L}|a^{\star})=0$ if
$q(\omega_{H})\geq\frac{1}{2}$. For $q(\omega_{H})<\frac{1}{2}$, apply
Proposition 1 to recover subjective probabilities.

###  B.2. Cognitive emergence and risky assets

Consider a DM with access to a highly risky asset $a_{R}$ that returns a high
payoff of 4 in the high state $\omega_{H}$ with probability $q(\omega_{H})=0$
(or arbitrarily close). Otherwise, the asset returns 0 in low state
$\omega_{L}$ ($q(\omega_{L})$ = 1). The DM also has access to a safe asset
$a_{S}$, which returns a guaranteed normalized value of 1. The objective
probability tells us there is little chance the risky asset will be a good
investment. Assume that the DM has a cost $C_{\phi}(p\|q)$ determined by the
Burg divergence with $\delta=1$. From our analysis of emergence, we know that
the DM’s optimal beliefs imply that $p(\omega_{H}|a_{R})=\frac{3}{4}$. Using
this fact, it follows that

$V(U(a_{R}))=3-log4>V(U(a_{S}))=1$

Accordingly, the optimal solution is:

$(a^{\star},p^{\star}(a^{\star}))=(a_{R},(3/4,1/4))$

In this example, cognitive emergence implies that the DM will purchase the
risky asset.

##  Appendix C Closed-form expressions for
$\mathbb{E}_{Q}(\omega_{a}|\omega_{a}\geq Q_{a}^{-1}(\alpha))$

In this section we describe several distributions that yield closed-form
expressions for $\mathbb{E}_{Q_{a}}(\omega_{a}|\omega_{a}\geq
Q_{a}^{-1}(\alpha))$. In doing so, we use the results in Norton et al. (2018)

###  C.1. The Logistic distribution

For each $a\in A$, assume that
$\omega_{a}\sim\operatorname{Logistic}(\mu_{a},s_{a})$. Setting $\mu_{a}=0$
and $s_{a}>0$, for all $a\in A$, we obtain $\mathbb{E}_{Q_{a}}(\omega_{a})=0$
and $Var(\omega_{a})=\frac{s_{a}\pi^{2}}{3}$. Accordingly,

$Q_{a}(\omega_{a})=\frac{1}{1+e^{-{\omega_{a}\over s_{a}}}},$

Then $Q^{-1}_{a}(\alpha)=\ln\left(\frac{\alpha}{1-\alpha}\right).$ From
Proposition (Norton et al., 2018, Prop. 10) we know that

$\mathbb{E}_{Q_{a}}(\omega_{a}|\omega_{a}\geq
Q_{a}^{-1}(\alpha))=s_{a}\frac{H(\alpha)}{1-\alpha}$

where $H(\alpha)\triangleq-\alpha\ln(\alpha)-(1-\alpha)\ln(1-\alpha).$ Thus
the problem (20) can be expressed as:

$\max_{a\in A}\left\\{u(a)+s_{a}\frac{H(\alpha)}{1-\alpha}\right\\}.$

###  C.2. The Student-t distribution

Assume $\omega_{a}\sim$ Student $-t(\nu_{a},s_{a},\mu_{a})$. where
$\nu_{a}>0,s_{a}>0$, $\mu_{a}>0$ with $E(\omega_{a})=\mu_{a}$ and
$Var(\omega_{a})=\frac{s_{a}^{2}\nu_{a}}{v_{a}-2}$. Setting $\mu_{a}=0$, the
Student $t$ distribution corresponds to

$Q_{a}(\omega_{a})=1-\frac{1}{2}\mathcal{I}_{g(\omega_{a})}\left(\frac{\nu_{a}}{2},\frac{1}{2}\right)$

where
$g(\omega_{a})=\frac{\nu_{a}}{\frac{\omega_{a}}{s}+\nu_{a}},\mathcal{I}_{t}(a,b)$
is the regularized incomplete Beta function, and $\Gamma(a)$ is the Gamma
function.

From (Norton et al., 2018, Prop. 12), we know that

$\mathbb{E}_{Q_{a}}(\omega_{a}|\omega_{a}\geq
Q_{a}^{-1}(\alpha))=s_{a}\left(\frac{\nu_{a}+T^{-1}(\alpha)^{2}}{(v-1)(1-\alpha)}\right)l\left(T^{-1}(\alpha)\right)$

where $T^{-1}(\alpha)$ is the inverse of the standardized Student-t cumulative
distribution function and $l(\cdot)$ is standardized Student-t probability
density function.

Accordingly, the DM’s problem (20) can be rewritten as:

$\max_{a\in
A}\left\\{u(a)+s_{a}\left(\frac{\nu_{a}+T^{-1}(\alpha)^{2}}{(v-1)(1-\alpha)}\right)l\left(T^{-1}(\alpha)\right)\right\\}.$

###  C.3. The generalized extreme value distribution

Finally, we discuss the generalized extreme value (GEV) distribution case.
Formally, we assume that $\omega_{a}$ follows a GEV distribution, which we
denote as $\omega_{a}\sim GEV(\mu,s,\xi)$. Recall that GEV parameters have
range $\mu_{a}\in\mathbb{R},s_{a}>0,\xi_{a}\in\mathbb{R}$. The parameters
$\mu_{a},s_{a},\xi_{a}$ capture location, scale, and shape respectively. The
cumulative distribution corresponds to:

$Q_{a}(\omega_{a})=\left\\{\begin{array}[]{ll}e^{-\left(1+\frac{\xi_{a}(\omega_{a}-\mu_{a})}{s_{a}}\right)^{\frac{-1}{\xi_{a}}}}&\xi_{a}\neq
0,\\\
e^{-e^{-\left(\frac{\omega_{a}-\mu_{a}}{s_{a}}\right)}}&\xi_{a}=0\end{array},\right.$

From (Norton et al., 2018, Prop. 15) we know that

(24) $\mathbb{E}_{Q_{a}}(\omega_{a}|\omega_{a}\geq
Q_{a}^{-1}(\alpha))=\begin{cases}\mu_{a}+\frac{s_{a}}{\xi_{a}(1-\alpha)}\left[\Gamma_{L}\left(1-\xi_{a},\ln\left(\frac{1}{\alpha}\right)\right)-(1-\alpha)\right]&\xi_{a}\neq
0\\\
\mu_{a}+\frac{s_{a}}{(1-\alpha)}(y+\alpha\ln(-\ln(\alpha))-\operatorname{li}(\alpha))&\xi_{a}=0\end{cases}$

where $\Gamma_{L}(a,b)=\int_{0}^{b}p^{a-1}e^{-p}dp$ is the lower incomplete
gamma function, $\operatorname{li}(x)=$ $\int_{0}^{\alpha}\frac{1}{\ln p}dp$
is the logarithmic integral function, and $y$ is the Euler-Mascheroni
constant.

Using the expression (24) we can rewrite the DM’s problem (20) accordingly.

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