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# Regret aversion and information aversion

Emmanuelle Gabillon

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Emmanuelle Gabillon. Regret aversion and information aversion. 2022. hal-03898012

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Distributed under a Creative Commons Attribution 4.0 International License
Regret aversion and information
aversion

Emmanuelle GABILLON

Univ. Bordeaux, CNRS, BSE, UMR 6060, F-33600 Pessac, France
Abstract

Regret is a negative and counterfactual emotion that occurs when a decision maker believes her past decision, if changed, would achieve a better outcome. Regret is intrinsically related to the comparison of the chosen alternative outcome with the foregone alternative outcomes. The result of this comparison is influenced by the decision maker's information about the foregone alternative outcomes (feedback structure). In this paper, we use Gabillon (2020)'s model, which generalizes regret theory to any feedback structure. We show that a regretful decision maker exhibits information aversion. The anticipation of learning about the payoffs of the foregone alternatives decreases her expected utility. We use the concept of statistical sufficiency in order to classify the feedback structures according to their informational content. We show that the less informative the feedback structure is, the higher the utility of a regretful decision maker.

Keywords: Regret, Emotion, Information.

JEL: D03, D81, D82.

**To cite this paper:** GABILLON Emmanuelle (2022), Regret aversion and information aversion, Bordeaux Economics Working Papers, BxWP2022-12

https://ideas.repec.org/p/grt/bdxewp/2022-12.html

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# Regret aversion and information aversion

Emmanuelle Gabillon*

August 12, 2022

## Abstract

Regret is a negative and counterfactual emotion that occurs when a decision maker believes her past decision, if changed, would achieve a better outcome. Regret is intrinsically related to the comparison of the chosen alternative outcome with the foregone alternative outcomes. The result of this comparison is influenced by the decision maker's information about the foregone alternative outcomes (feedback structure). In this paper, we use Gabillon (2020)'s model, which generalizes regret theory to any feedback structure. We show that a regretful decision maker exhibits information aversion. The anticipation of learning about the payoffs of the foregone alternatives decreases her expected utility. We use the concept of statistical sufficiency in order to classify the feedback structures according to their informational content. We show that the less informative the feedback structure is, the higher the utility of a regretful decision maker.
**JEL classification** D03 . D81 . D82

<footnote>*BSE - UMR CNRS 6060, Bâtiment H, Université de Bordeaux, Avenue Léon Duguit, 33608 Pessac cedex, France. Email: emmanuelle.gabillon@u-bordeaux.fr.</footnote>
Regret is a counterfactual emotion (Kahneman and Miller 1986), which can occur when a decision maker (DM) compares the result of her choice to what she would have obtained had she made another decision. Regret occurs when the DM concludes that things would have been better under a different choice.

In decision theory, anticipated regret was first considered by Savage (1951) and further explored by Luce & Raiffa (1956). Seminal contributions in regret modeling can be found in Bell (1982, 1983) and Loomes and Sugden (1982, 1987). Axiomatic foundations of preferences are provided by Fishburn (1989), Sugden (1993), Quiggin (1994) and, more recently, by Diecidue and Somasundaram (2017).

A large part of regret theory is established under perfect information, where the payoffs of the foregone alternatives are perfectly observable. Among the exceptions, Bell (1983) considers the choice between a sure thing and a risky lottery and asks whether the choice of the sure thing would not be more attractive if the foregone lottery were unresolved. More recently, Gabillon (2020) proposes a model in which a DM receives signals about the foregone alternative payoffs. This approach allows considering any level of information about the foregone payoffs (feedback structure). The feedback structure (FS) can be perfectly informative when signals perfectly reveal the payoffs of the foregone alternatives, non-informative when signals do not disclose any information, and imperfectly informative when signals partially reveals the payoffs of the foregone alternatives. Gabillon (2020) shows that anticipated regret does depend on anticipated feedback.

In this paper, we use the model of Gabillon (2020) to compare the different FSs according to the preferences of a regretful DM. We show that a regretful DM prefers to be as little informed as possible about the rejected alternative payoffs. Information about the foregone alternative payoffs makes anticipated regret more salient and decreases the DM expected utility. In order to obtain this result, we use Fisher (1920)'s statistical criteria of sufficiency in order to compare the FS informational contents. FS A is said to be more informative than FS B if signals in A are sufficient statistics for signals in B about the foregone alternative payoffs. We show that, if FS A is more informative than FS B, a regretful DM prefers FS B. Moreover, we show that, among all possible FS, a regretful DM always prefers the non-informative FS. Regret aversion leads to aversion to *ex-post* information, information which occurs after the choice. Furthermore, Gabillon (2020) shows that information received before the choice can also be harmful to a regretful DM, even though this information improves the DM's knowledge about the different alternatives among which she has to choose.

On the experimental side, our results about regret and information are consistent with the study of Zeelenberg et al. (1996), which shows that people tend to avoid having information about foregone alternatives. The authors performed an experiment where they set up two risky lotteries to which participants are indifferent. Indifference as regards the two lotteries is established when there is no feedback on the foregone lottery. Stated otherwise, people exclusively obtain feedback on the lottery of their choice. One of the two lotteries is relatively
risky, the other relatively safe (the probability of winning is higher but the gain
is lower). Zeelenberg et al. (1996) modify the feedback context and observe
the behavioral consequences. When people know that the result of the risky
lottery will be systematically revealed, they are no longer indifferent to the two
lotteries, tending to prefer the risky one. They abandon the safe lottery because
they try to protect themselves against the regret which may arise from having
information about the foregone lottery (information about the risky lottery if
they choose the safe lottery). Zeelenberg et al. (1996) show that regret aversion
induces risk-seeking behavior (when people anticipate feedback on the risky lot-
tery), or risk-avoiding behavior (when people anticipate feedback on the safe
lottery). These types of behavior, which consist in avoiding information about
the foregone lottery, are consistent with our result. Many other experimental
studies (Josephs et al. 1992; Larrick and Boles 1995; Ritov 1996; Zeelenberg and
Beattie 1997; Zeelenberg 1999) also reveal the sensitivity of choices to the feed-
back context, and demonstrate that people try to protect themselves against
information about what they could have obtained by making a different choice.

In regret theory, Bell (1983) introduces the concept of cancellation price, which corresponds to the sure thing which makes a regretful DM indifferent about choosing the sure thing or choosing a risky lottery. Bell (1983) assumes that the foregone lottery is not resolved if the sure thing is selected. In our vocabulary, Bell (1983) defines the cancellation price under a non-informative FS. Gabillon (2020) generalizes Bell (1983)'s cancellation price to a large range of utility functions and refers to the cancellation price as the *regret certainty equivalent*.

In this paper, we generalize the regret certainty equivalent of a risky lot-
tery to any FS. We show that the regret certainty equivalent of a risky lottery
increases with the informativeness of the FS. The certainty equivalent is thus
maximal when the FS is perfectly informative and minimal when the FS is
non-informative. Under a non-informative FS, the choice of the sure thing fully
protects the DM against anticipated regret. After the choice, since the lottery
payoff is not observable, it cannot be compared to the sure thing and regret
cannot be experienced. When, however, some information about the foregone
choice is available after the choice, the sure thing no longer offers a full pro-
tection against regret and becomes less attractive. Consequently, the regret
certainty equivalent of the risky lottery increases with the informativeness of
the FS.

The paper is organized as follows. Sections 1 and 2 are dedicated to a short
presentation of the model of Gabillon (2020): Section 1 presents the concept
of FS and Section 2 presents a generalization of preferences to any FS. Section
3 introduces the concept of statistical sufficiency in regret theory. Section 4 is
devoted to information aversion.
# 1 Feedback Structures

In this section, we briefly recall the concept of FS introduced in Gabillon (2020). Let $\Phi = \{Y_1, \dots, Y_{N+1}\}$ denote the set of $N+1$ risky alternatives. A risky alternative $Y_n$ is a random variable, which takes its values on a set $\Omega$, which contains a finite number of positive values. A risky alternative $Y_n$ is characterized by a probability distribution on $\Omega$, denoted by its generic term $p(y_n)$.

Without loss of generality, the chosen alternative is denoted by $X$ and the foregone alternatives by $Y_1, \dots, Y_N$. In order to deal with any FS, we assume that, at the feedback stage (i.e., after the choice), a DM observes not only the chosen alternative's outcome $x$ but also a realization $m$ of a signal $M_X$ about the foregone alternative outcomes $y_1, \dots, y_N$. Signal $M_X$ is assumed to be a random variable (possibly multidimensional), which takes its values on a set $M$, which contains a finite number of elements. In order to shorten our notations, the foregone alternative outcomes $y_1, \dots, y_N$ will be denoted by $\theta_{-x}$. The random variable $M_X$ is a signal on the feedback stage's state of nature $\theta_{-x}$ and is characterized by a conditional probability distribution on $M$, denoted by its generic term $p(m|\theta_{-x}, x)$. This conditional probability distribution depends on the chosen alternative $X$, but can also differ according to the observed payoff $x$. Probability $p(m|\theta_{-x}, x)$ represents the probability of observing signal $m$ given that alternative $X$ has been chosen, given payoff $x$ and given foregone alternative payoffs $\theta_{-x}$.

In what follows, we first introduce the concept of FS at the level of one alternative $X$ and then, we define a FS associated with an entire choice set $\Phi = \{Y_1, \dots, Y_{N+1}\}$.

Let $FS_X = (X, M_X)$ denote the alternative $X$ FS. $FS_X$ contains all available information at the feedback stage, after alternative $X$ has been selected.

**Definition 1.** $FS_X$ is said to be non-informative if the probability distribution of $M_X$ is the same for all $\theta_{-x}: \forall x \in \Omega, \forall \theta_{-x} \in \Omega^N, p(m|\theta_{-x}, x) = p(m|x)$. One cannot learn about $\theta_{-x}$ by observing from $M_X$.

At the other end, $FS_X$ is said to be perfectly informative if for every $x \in \Omega$, for every pair $(\theta_{-x}^i, \theta_{-x}^j) \in \Omega^N \times \Omega^N$, the intersection of the support sets on which $p(m|\theta_{-x}^i, x)$ and $p(m|\theta_{-x}^j, x)$ are strictly positive is an empty set. After observing $M_X$, the state of nature $\theta_{-x}$ that generated $M_X$ can be identified with certainty.

$FS_X$ is said to be imperfectly informative in all other situations.

We point out that, even when $FS_X$ is non-informative, a DM can learn about $\theta_{-x}$ by observing $x$. This happens when $X$ and $Y_1, \dots, Y_N$ are not statistically independent, a situation that we do not exclude from our analysis.

In the rest of the paper, either $\{Y_1, \dots, Y_{N+1}\}$ or $\{X, Y_1, \dots, Y_N\}$ will refer to the choice set $\Phi$ depending on whether we need or not to distinguish the chosen alternative $X$ from the other alternatives. Hereafter, we give the definition of a FS associated to a choice set $\Phi$:
**Definition 2.** The feedback structure FS, associated to the choice set $\Phi = \{Y_1, \dots, Y_{N+1}\}$, is the set of all alternative FSs:

$$FS = \{FS_{Y_1}, \dots, FS_{Y_{N+1}}\}$$

**Definition 3.** A FS is said to be non-informative if $FS_{Y_n}$ is non-informative for every $Y_n \in \Phi$.

A FS is said to be perfectly informative if $FS_{Y_n}$ is perfectly informative for every $Y_n \in \Phi$.

A FS is said to be imperfectly informative in all other situations.

## 2 Preferences

We use the preferences developed in Gabillon (2020). The regret utility function (r-utility) $u(x, r)$ depends on the payoff $x$ of the chosen alternative $X$ and on a reference point. The reference point represents the impact of anticipated regret on the DM's utility. When $r > x$, we will see in what follows that a foregone alternative performs better than the chosen alternative, and regret is anticipated<fnref n="1" />. In order to give a definition of the reference point, we need to define the concept of choiceless utility (c-utility):

**Definition 4.** The c-utility function is defined as $v(x) = u(x, x)$ and measures the satisfaction generated by the consumption of payoff $x$, independently of any choice-related feeling.

In Definition 4, the c-utility function represents preferences in which sensitivity to regret has been removed ($r = x$) and corresponds to the DM's preferences if she were not regret averse. The c-utility $v(x)$ represents the utility of $x$ when the payoff $x$ is evaluated in a no-choice setting. At the feedback stage, we assume that the $N+1$ alternatives are evaluated with the c-utility function since the choice has already been made and cannot be modified.

Gabillon (2020) makes some assumptions about the r-utility function $u(x, r)$. Let $u_1(x, r)$ denote $\frac{\partial u(x, r)}{\partial x}$, $u_2(x, r)$ denote $\frac{\partial u(x, r)}{\partial r}$ and $v'(x)$ denote $\frac{\partial v(x)}{\partial x}$.

A0. The r-utility $u(x, r)$ is differentiable on $\mathbb{R}^{+2}$.

A1. $v'(x) = u_1(x, x) + u_2(x, x) > 0$.

A2. $u_1(x, r) > 0$.

A3. $u_2(x, r) < 0$.

<footnote n="1">¹Gabillon (2020) shows that $r > x$ does not necessarily imply that regret is anticipated. A reference point greater than the chosen action payoff can also mean that the DM experiences a psychological opportunity cost. In this paper, we will, however, do not make the distinction, always referring to anticipated regret when $r > x$. See Gabillon (2020) for a formal distinction between anticipated regret and psychological opportunity cost.</footnote>
Assumptions A1 and A2 state that utility increases with payoff $x$. Assumption A3 states that the r-utility decreases with the reference point and characterizes regret aversion. Given payoff $x$, anticipated regret increases with the reference point and utility decreases.

We assume that a DM has some priors about the foregone alternative payoffs denoted by $p(\theta_{-x})$. At the feedback stage, each foregone alternative $Y_n$ is characterized by a posterior probability distribution $p(\theta_{-x} | x, m)$ after the information has been processed. The posterior probability distribution represents the DM's knowledge about $Y_n$ at the feedback stage. We recall that, at the feedback stage, a DM evaluates the $N+1$ alternatives with the c-utility function. We can compute the posterior certainty equivalent of each alternative $Y_n$:

$$v(CE_{Y_n}^{v,FS_x}) = E[v(Y_n)|FS_x], \qquad (1)$$

where $FS_x = \{x, m\}$ and where the operator $E[\cdot|FS_x]$ represents the conditional expectation, given the realizations of $X$ and $M_X$. The notation $CE_{Y_n}^{v,FS_x}$ indicates, in superscript, that the certainty equivalent is computed with the c-utility function $v(\cdot)$, given information $FS_x = \{x, m\}$.

The posterior certainty equivalent of the chosen alternative is equal to the realization of the payoff itself.

$$CE_X^{v,x,m} = x. \qquad (2)$$

We can now give the definition of a reference point which accommodates any FS.

**Definition 5.** The reference point $R^{FS_x}$ is the highest posterior certainty equivalent:

$$R^{FS_x} = \text{Max} \{X, CE_{Y_1}^{v,FS_x}, \dots, CE_{Y_N}^{v,FS_x}\} \text{ where } FS_x = (X, M_X).$$

The notation $R^{FS_x}$ indicates, in superscript, the variables that a DM observes at the feedback stage. Under assumption A1, the reference point is the certainty equivalent of the alternative which maximizes the expected c-utility, given available information at the feedback stage<fnref n="2" />. In the event $R^{FS_x} > X$, the DM regrets her choice since, given her information, a foregone alternative proves to be more attractive than the chosen alternative.

We can rewrite the reference point (see Definition 5) as

$$R^{FS_x} = \text{Max} \{X, CE_{Max}^{v,FS_x}\}, \qquad (3)$$

$$\text{with } CE_{Max}^{v,FS_x} = \text{Max} \{CE_{Y_1}^{v,FS_x}, \dots, CE_{Y_N}^{v,FS_x}\}.$$

<footnote n="2">²The definition of the reference point implies that $R^{FS_x}$ cannot be lower than $X$, excluding the feeling of rejoicing, when a DM learns that the chosen alternative turns out to be the best choice.</footnote>
We obtain the preferences of a regretful DM :

$$
E[u(X, R^{FS_x})] = E[u(X, Max\{X, CE_{Max}^{v,FS_x}\})]. \quad (4)
$$

The properties of these preferences are analyzed in Gabillon (2020).

In this paper, we introduce two additional assumptions about the r-utility function:

$$
A4. \quad u_{22}(x, r) \le 0
$$

A5. $v''(x) \le 0$

Assumptions A4 and A5 are “concavity assumptions”. Concavity does not need to be strict: our results are compatible with $u_{22}(x, r) = 0$ and $v''(x) = 0$. (see Example 1 in the next section).

3 Statistical Sufficiency and Feedback Structures

In this section, we compare the informativeness of different FSs. We begin
our analysis at the level of one alternative $X$, by comparing the informational
content of two different alternative $X$ FSs. Then, we generalize our comparison
to a choice set $\Phi$, by comparing the informational content of two different FS.
For this purpose, we use the concept of statistical sufficiency of Fisher (1920).

Consider two different alternative X FSs, $FS_X^a = (X, M_X^a)$ and $FS_X^b = (X, M_X^b)$. Let $p(m_a | \theta_{-x}, x)$ denote the conditional probability of signal $M_X^a$ and $p(m_b | \theta_{-x}, x)$ denote the conditional probability of signal $M_X^b$.

**Definition 6.** $FS_X^a$ is sufficient for $FS_X^b$ about the foregone alternative payoffs if, for every $x \in \Omega$, there exists a stochastic transformation $\pi (m_a | m_b, x)$ such that

$$
\forall m_b \in M, \forall \theta_{-x} \in \Omega^N, p(m_b | \theta_{-x}, x) = \sum_{m_a \in M} \pi(m_b | m_a, x) p(m_a | \theta_{-x}, x)
$$

$$
\text{with } \sum_{m_b \in M} \pi(m_b | m_a, x) = 1.
$$

Definition 6 can be reformulated as follows: $FS_X^a$ is sufficient for $FS_X^b$ if there exists a joint probability distribution $\pi (m_a, m_b | \theta_{-x}, x)$ (possibly different from the real joint probability distribution $p (m_a, m_b | \theta_{-x}, x)$), whose marginal distributions coincide with the true marginal distributions:

$$
\sum_{m_b \in M} \pi(m_a, m_b | \theta_{-x}, x) = p(m_a | \theta_{-x}, x) \quad \text{and} \quad \sum_{m_a \in M} \pi(m_a, m_b | \theta_{-x}, x) = p(m_b | \theta_{-x}, x). \tag{5}
$$

and which satisfies
$$ \forall \theta_{-x} \in \Omega^N, \pi(m_b|m_a, \theta_{-x}, x) = \pi(m_b|m_a, x). \qquad (6) $$

Given distribution $\pi(m_a, m_b | \theta_{-x}, x)$, signal $M_X^b$ is garbled from signal $M_X^a$ in the sense of Blackwell (1951). In other words, we move from signal $M_X^a$ to signal $M_X^b$ by adding noise. A DM who observes $M_X^a$ can generate $M_X^b$ with the stochastic process $\pi(m_b|m_a, x)$, which is independent of $\theta_{-x}$.

In order to compare different FSs with the concept of sufficiency, we introduce the following definition:

**Definition 7.** $FS^a$ is sufficient for $FS^b$ if, for at least one $Y_n \in \Phi$, $FS_{Y_n}^a$ is sufficient for $FS_{Y_n}^b$, other alternative FSs being identical.

Let $FS^{ni}$ denote a non-informative FS and $FS^i$ a perfectly informative FS. We obtain the following proposition:

**Proposition 1.** Any FS is sufficient for $FS^{ni}$. $FS^i$ is sufficient for any FS.

*Proof.* See Appendix A. □

# 4 Information Aversion

In this Section, we show that a regretful DM exhibits feedback aversion or *ex-post* information aversion. *Ex-post* information refers to information about the foregone alternative payoffs, which arises after the choice has been made. In order to go further, we introduce the following definition:

**Definition 8.** A regretful DM prefers $FS^b$ to $FS^a$ if her expected utility under $FS^b$ is higher than under $FS^a$:

$$ \underset{Y_n \in \Phi}{\operatorname{Max}} E \left[ u \left( Y_n, \underset{Max}{\operatorname{Max}} \left( Y, CE_{Max}^{v,FS_{Y_n}^b} \right) \right) \right] \geq \underset{Y_n \in \Phi}{\operatorname{Max}} E \left[ u \left( Y_n, \underset{Max}{\operatorname{Max}} \left( Y_n, CE_{Max}^{v,FS_{Y_n}^a} \right) \right) \right]. $$

We obtain the following Proposition, which characterizes information aversion:

**Proposition 2.** If $FS^a$ is sufficient for $FS^b$ then a regretful DM prefers $FS^b$ to $FS^a$.

*Proof.* See Appendix B. □

Appendix B shows that the expected utility of each alternative $Y_n$ is higher under $FS^b$. According to Proposition 2, a DM prefers to minimize her exposure to *ex-post* information in order to protect herself against future regret. Concavity assumptions A4 and A5, which include the case $u_{22}(x, r) = 0$ and $v''(x) = 0$, are sufficient conditions for *ex-post* information aversion<fnref n="3" />.

<footnote n="3">³In an alternative setting, in which the reference point is defined as $R^{FSX} = \underset{Max}{\operatorname{Max}} \{v(X), v(CE_{Y_1}^{v,FSX}), \dots, v(CE_{Y_N}^{v,FSX})\}$, assumption A4 alone leads to information aversion. There is no need to formulate assumption A5.</footnote>
To give an intuition about information aversion, let us consider the following property, which is demonstrated when $FS^a$ is sufficient for $FS^b$ (see Equation B.15 in Appendix B):

$$ \forall X \in \Phi, \forall Y_n \in \Phi / \{X\}, \forall x \in \Omega, \forall m_b \in M, v(CE_{Y_n}^{v,FS_x^b}) = \sum_{m_a \in M} k_{m_b m_a}^x v(CE_{Y_n}^{v,FS_x^a}), \quad (7) $$

$$ \text{with } \sum_{m_a \in M} k_{m_b m_a}^x = 1. $$

Under $FS^b$, at the feedback stage, the expected c-utility derived from a foregone alternative $Y_n$ is a convex combination of the expected c-utilities of $Y_n$ under $FS^a$. In other words, expected c-utilities under $FS^b$ are mean preserving contractions of expected c-utilities under $FS^a$.

To give an intuition about the information aversion result, let us consider a particular case where $v(x)$ is linear. Equation 7 becomes

$$ \forall X \in \Phi, \forall Y_n \in \Phi / \{X\}, \forall x \in \Omega, \forall m_b \in M, CE_{Y_n}^{v,FS_x^b} = \sum_{m_a \in M} k_{m_b m_a}^x CE_{Y_n}^{v,FS_x^a}, \quad (8) $$

Given the chosen alternative payoff $x$, the reference point $R^{FS_x} = \text{Max}\{x, CE_{Y_1}^{v,FS_x}, \dots, CE_{Y_N}^{v,FS_x}\}$ is a convex function and the r-utility, which decreases with the reference point, is concave. Mean preserving contraction and concavity explain the information aversion result. This property of concavity is not related to the shape of the r-utility function. Information aversion exists as soon as the r-utility does not exhibit too much convexity, which could then counteract the concavity property. The same goes for the c-utility function. In particular, information aversion is compatible with $u_{22}(x,r) = 0$ and $v(x) = 0$.

**Example 1.** The r-utility function is $u(x,r) = x - \frac{r}{2}$ and the c-utility function is $v(x) = u(x,x) = \frac{x}{2}$. We consider a choice set $\Phi = \{Y_1, Y_2\}$, containing two statistically independent alternatives. Each alternative takes its values on the set $\Omega = \{6, 8, 16, 18\}$. Alternative $Y_1$ is characterized by the probability distribution $(\frac{1}{2}, 0, \frac{1}{2}, 0)$ and alternative $Y_2$ by the probability distribution $(0, \frac{3}{4}, 0, \frac{1}{4})$. Signals $M_{Y_1}$ (signal on $Y_2$ when $Y_1$ is chosen) and $M_{Y_2}$ (signal on $Y_1$ when $Y_2$ is chosen) take values on $M = \{0, 1\}$.

We assume that $FS^a$ is perfectly informative. Signals $M_{Y_1}^a$ and $M_{Y_2}^a$ are characterized by the following conditional probability distributions:

<table><thead><tr><th>Signal M<sup>a</sup><sub>Y1</sub></th><th>y<sub>2</sub> = 8</th><th>y<sub>2</sub> = 18</th></tr></thead><tbody><tr><td>m<sub>a</sub> = 0</td><td>p(m<sub>a</sub> = 0 | y<sub>2</sub> = 8) = 1</td><td>p(m<sub>a</sub> = 0 | y<sub>2</sub> = 18) = 0</td></tr><tr><td>m<sub>a</sub> = 1</td><td>p(m<sub>a</sub> = 1 | y<sub>2</sub> = 8) = 0</td><td>p(m<sub>a</sub> = 1 | y<sub>2</sub> = 18) = 1</td></tr></tbody></table>

and

<table><thead><tr><th>Signal M<sup>a</sup><sub>Y2</sub></th><th>y<sub>1</sub> = 6</th><th>y<sub>1</sub> = 16</th></tr></thead><tbody><tr><td>m<sub>a</sub> = 0</td><td>p(m<sub>a</sub> = 0 | y<sub>1</sub> = 6) = 1</td><td>p(m<sub>a</sub> = 0 | y<sub>1</sub> = 16) = 0</td></tr><tr><td>m<sub>a</sub> = 1</td><td>p(m<sub>a</sub> = 1 | y<sub>1</sub> = 6) = 0</td><td>p(m<sub>a</sub> = 1 | y<sub>1</sub> = 16) = 1</td></tr></tbody></table>
Under FS<sup>b</sup>, signal probability distributions are the following:

<table><thead><tr><th>Signal M<sub>Y<sub>1</sub></sub><sup>b</sup></th><th>y<sub>2</sub> = 8</th><th>y<sub>2</sub> = 18</th></tr></thead><tbody><tr><td>m<sub>b</sub> = 0</td><td>p(m<sub>b</sub> = 0 | y<sub>2</sub> = 8) = <sup>3</sup>&frasl;<sub>4</sub></td><td>p(m<sub>b</sub> = 0 | y<sub>2</sub> = 18) = <sup>1</sup>&frasl;<sub>4</sub></td></tr><tr><td>m<sub>b</sub> = 1</td><td>p(m<sub>b</sub> = 1 | y<sub>2</sub> = 8) = <sup>1</sup>&frasl;<sub>4</sub></td><td>p(m<sub>b</sub> = 1 | y<sub>2</sub> = 18) = <sup>3</sup>&frasl;<sub>4</sub></td></tr></tbody></table>

and

<table><thead><tr><th>Signal M<sub>Y<sub>2</sub></sub><sup>b</sup></th><th>y<sub>1</sub> = 6</th><th>y<sub>1</sub> = 16</th></tr></thead><tbody><tr><td>m<sub>b</sub> = 0</td><td>p(m<sub>b</sub> = 0 | y<sub>1</sub> = 6) = <sup>3</sup>&frasl;<sub>4</sub></td><td>p(m<sub>b</sub> = 0 | y<sub>1</sub> = 16) = <sup>1</sup>&frasl;<sub>4</sub></td></tr><tr><td>m<sub>b</sub> = 1</td><td>p(m<sub>b</sub> = 1 | y<sub>1</sub> = 6) = <sup>1</sup>&frasl;<sub>4</sub></td><td>p(m<sub>b</sub> = 1 | y<sub>1</sub> = 16) = <sup>3</sup>&frasl;<sub>4</sub></td></tr></tbody></table>

First, we note that FS<sup>a</sup> is sufficient for FS<sup>b</sup>:

$$ \forall m_b \in M = \{0, 1\}, \forall \theta_{-x} \in \{6, 16\} \text{ or } \{8, 18\}, p(m_b | \theta_{-x}) = \sum_{m_a \in M} \pi(m_b | m_a) p(m_a | \theta_{-x}), \quad (9) $$

$$ \text{with } \begin{array}{c|c|c} \pi(m_b | m_a) & m_a = 0 & m_a = 1 \\ m_b = 0 & \frac{3}{4} & \frac{1}{4} \\ m_b = 1 & \frac{1}{4} & \frac{3}{4} \end{array} \quad \text{and} \quad \sum_{m_b \in M} \pi(m_b | m_a, x) = 1. $$

Annexe C shows that the expected r-utilities of both alternatives are higher under FS<sup>b</sup>.

From propositions 1 and 2, we obtain the following corollary, which also characterizes information aversion :

**Corollary 1.** A regretful DM prefers a non-informative FS to any other FS. On the other hand, the perfectly informative FS is the worst FS for a regretful DM.

While Proposition 2 is restricted to the comparison of FSs that can be ordered with the criteria of sufficiency, we stress the generality of Corollary 1. Among all FSs (without any restrictions), a regretful DM prefers the non-informative FS. Similarly, among all FS, the perfectly informative FS represents the least desirable FS.

In what follows, we generalize to any FS the definition of the regret certainty equivalent, which was developed under a non informative FS in Gabillon (2020) and first introduced by Bell (1983) under the name of cancellation price.

**Definition 9.** The regret certainty equivalent CE<sub>Y</sub><sup>u,FS</sup> of a risky alternative Y corresponds to the sure payoff which makes the DM indifferent about choosing CE<sub>Y</sub><sup>u,FS</sup> or Y.

The regret certainty equivalent CE<sub>Y</sub><sup>u,FS</sup> is the Z-solution of the following Equation:

$$ E \left[ u \left( Z, \text{Max} \left( Z, CE_{Y}^{v,FS_{Z}} \right) \right) \right] = E \left[ u \left( Y, \text{Max} \left( Y, Z \right) \right) \right], \quad (10) $$
where $FS_Z = \{Z, M_Z\}$ contains a signal $M_Z$ about lottery $Y$.

When a DM chooses $Y$, she observes the result of her choice (the result of the risky lottery $Y$) and she knows the result of the foregone choice (the value of the sure payoff). When she chooses the sure payoff, she obviously knows the result of her choice and she receives a signal $M_Z$ on the result of the foregone risky alternative $Y$. In this setting, the informativeness of $M_Z$ determines the level of informativeness of the FS.

Let $CE_Y^{u,ni}$ denote the regret certainty equivalent of $Y$ under a non-informative FS ($M_Z$ conveys no information). Gabillon (2020) shows that, $\underline{y} < CE_Y^{u,ni} < CE_Y^v$, where $\underline{y}$ denotes the minimum value that $Y$ can take on its support $\Omega$ and $CE_Y^v$ denotes the Arrow-Pratt certainty equivalent of $Y$, computed with the c-utility function$^4$. Gabillon (2020) shows that, under a non-informative FS, the regret certainty equivalent is lower than the Arrow-Pratt certainty equivalent. This result is independent of the shape of the r-utility function. Under a non-informative FS, for a DM who chooses between a sure payoff and a risky lottery, the sure payoff is more attractive when anticipated regret is taken into account in decision-making. The sure payoff offers a protection against anticipated regret. When the sure payoff is chosen under a non-informative FS, the DM does not learn the result of the foregone risky alternative after her choice and regret cannot be felt. On the contrary, when she chooses the risky alternative $Y$, she can compare the obtained payoff to the sure payoff. The difference $\Pi_Y = CE_Y^v - CE_Y^u$ is a generalization to a large range of utility function $u(x, r)$ (satisfying A0 to A3) of the regret premium introduced by Bell (1983). Gabillon (2020) show that the regret premium is always positive when rejoicing is not taken into account. The author also proposes a new interpretation of the regret premium, as the maximum psychological opportunity cost a DM is willing to endure to avoid regret$^5$.

**Proposition 3.** Whatever the FS, the regret certainty equivalent $CE_Y^{u,FS}$ exists and is unique.

*Proof.* See Appendix D. $\square$

**Proposition 4.** If $FS^a$ is sufficient for a $FS^b$ then for any risky alternative $Y$, we have $CE_Y^{u,FS^b} \le CE_Y^{u,FS^a}$.

*Proof.* See Appendix D. $\square$

When the FS becomes more informative, choosing the sure payoff offers less protection against feedback and anticipated regret. The attractiveness of the sure thing decreases with the informativeness of the FS. In order to remain as attractive as the lottery, the certainty equivalent must increase as the informativeness of the FS intensifies.

<footnote>$^4$Under a non-informative FS, the Arrow-Pratt certainty equivalent satisfies $v(CE_Y^v) = E[v(Y)]$.</footnote>

<footnote>$^5$A psychological opportunity cost is defined as the negative emotional counterpart of a strategy of regret avoidance.</footnote>
$FS^i$ is perfectly informative if $\forall X \in \Phi, \forall x \in \Omega$, for each $\theta_{-x} \in \Omega^N$, there exist a subset $F_{\theta_{-x}} \subset M$ such that

$$ \forall m_i \in F_{\theta_{-x}}, p(m_i | \theta_{-x}, x) > 0 \text{ and } \sum_{F_{\theta_{-x}}} p(m_i | \theta_{-x}, x) = 1. \quad (A.7) $$

Observing $M_X^i$ is equivalent to observing $\theta_{-x}$ and thus $p(m|m_i, \theta_{-x}, x) = p(m|m_i, x)$. A DM who observes the value $m_i$ taken by signal $M_X^i$ knows the state of nature $\theta_{-x}(m_i)$ and can generate signal $M_X$ with the stochastic process $p(m|\theta_{-x}(m_i), x)$ and obtain a result equivalent to the result of observing both $M_X^i$ and $M_X$. Observing $M_X^i$ is sufficient.

We have

$$ p(m|\theta_{-x}, x) = \sum_{m_i \in M} p(m|m_i, \theta_{-x}, x) p(m_i|\theta_{-x}, x) = \sum_{m_i \in M} p(m|m_i, x) p(m_i|\theta_{-x}, x) \quad (A.8) $$

This ends the proof of the second part of Proposition 1.

# Appendix B

Let $p(\theta_x|x)$ the posterior probability of $\theta_{-x}$ given the observation of the chosen alternative payoff. Definition 6 gives:

$$ \begin{aligned} & \forall x \in \Omega, \forall m_b \in M, \forall \theta_{-x} \in \Omega^N, p(m_b|\theta_{-x}, x) p(\theta_{-x}|x) \quad (B.1) \\ &= \sum_{m_a \in M} \pi(m_b|m_a, x) p(m_a|\theta_{-x}, x) p(\theta_{-x}|x), \end{aligned} $$

with $\sum_{m_b \in M} \pi(m_b|m_a, x) = 1$.

By summing over $\theta_{-x}$, we have

$$ \begin{aligned} & \forall x \in \Omega, \forall m_b \in M, \sum_{\theta_{-x} \in \Omega^N} p(m_b|\theta_{-x}, x) p(\theta_{-x}|x) \quad (B.2) \\ &= \sum_{\theta_{-x} \in \Omega^N} \sum_{m_a \in M} \pi(m_b|m_a, x) p(m_a|\theta_{-x}, x) p(\theta_{-x}|x), \end{aligned} $$

with $\sum_{m_b \in M} \pi(m_b|m_a, x) = 1$.

Which gives

$$ \forall x \in \Omega, \forall m_b \in M, p(m_b|x) = \sum_{m_a \in M} \pi(m_b|m_a, x) p(m_a|x), \quad (B.3) $$
with $\sum_{m_b \in M} \pi(m_b|m_a, x) = 1$.

Besides, Equation (B.1) can be rewritten as follows:

$$ \forall x \in \Omega, \forall m_b \in M, \forall \theta_{-x} \in \Omega^N, p(m_b, \theta_{-x} | x) = \sum_{m_a \in M} \pi(m_b | m_a, x) p(m_a, \theta_{-x} | x), \qquad (B.4) $$

with $\sum_{m_b \in M} \pi(m_b|m_a, x) = 1$.

Or else

$$ \begin{aligned} \forall x & \in \Omega, \forall m_b \in M, \forall \theta_{-x} \in \Omega^N, p(\theta_{-x} | m_b, x) p(m_b | x) & (B.5) \\ & = \sum_{m_a \in M} \pi(m_b | m_a, x) p(\theta_{-x} | m_a, x) p(m_a | x). \end{aligned} $$

with $\sum_{m_b \in M} \pi(m_b|m_a, x) = 1$.

We obtain

$$ \begin{aligned} \forall x & \in \Omega, \forall m_b \in M, \forall \theta_{-x} \in \Omega^N, p(\theta_{-x} | m_b, x) & (B.6) \\ & = \sum_{m_a \in M} \frac{\pi(m_b|m_a, x) p(m_a|x)}{p(m_b|x)} p(\theta_{-x}|m_a, x). \end{aligned} $$

with $\sum_{m_b \in M} \pi(m_b|m_a, x) = 1$.

Let us introduce a new variable:

$$ k_{m_b m_a}^x = \frac{\pi(m_b|m_a, x) p(m_a|x)}{p(m_b|x)} \qquad (B.7) $$

Equation (B.3) implies that

$$ \sum_{m_a \in M} k_{m_b m_a}^x = \frac{\sum_{m_a \in M} \pi(m_b|m_a, x) p(m_a|x)}{p(m_b|x)} = \frac{p(m_b|x)}{p(m_b|x)} = 1. \qquad (B.8) $$

Equations (B.6), (B.7) and (B.8) give

$$ \forall x \in \Omega, \forall m_b \in M, \forall \theta_{-x} \in \Omega^N, p(\theta_{-x} | m_b, x) = \sum_{m_a \in M} k_{m_b m_a}^x p(\theta_{-x} | m_a, x), \qquad (B.9) $$
with $\sum_{m_a \in M} k_{m_b m_a}^x = 1$.

Given that $\theta_{-x} = \{y_1, ..., y_N\}$, it is easy to obtain from Equation (B.9) that

$$ \forall x \in \Omega, \forall m_b \in M, \forall Y_n \in \Phi / \{X\}, \forall y_n \in \Omega, p(y_n | m_b, x) = \sum_{m_a \in M} k_{m_b m_a}^x p(y_n | m_a, x), \quad (B10) $$

with $\sum_{m_a \in M} k_{m_b m_a}^x = 1$.

Besides (see Equation 1), we recall that $\forall Y_n \in \Phi / \{X\}$

$$ \forall x \in \Omega, \forall m_b \in M, v(CE_{Y_n}^{v,FS_x^b}) = E[v(y_n) | FS_x^b], \quad (B.11) $$

where $FS_x^b = \{x, m_b\}$.

Or, equivalently

$$ \forall x \in \Omega, \forall m_b \in M, v(CE_{Y_n}^{v,x,m_b}) = \sum_{y_n \in \Omega} v(y_n) p(y_n | m_b, x). \quad (B.12) $$

From Equation (B10) and Equation (B.12), we obtain

$$ \forall x \in \Omega, \forall m_b \in M, v(CE_{Y_n}^{v,x,m_b}) = \sum_{y_n \in \Omega} v(y_n) \sum_{m_a \in M} k_{m_b m_a}^x p(y_n | m_a, x). \quad (B.13) $$

Or, equivalently

$$ \forall x \in \Omega, \forall m_b \in M, v(CE_{Y_n}^{v,x,m_b}) = \sum_{m_a \in M} k_{m_b m_a}^x \sum_{y_n \in \Omega} v(y_n) p(y_n | m_a, x). \quad (B.14) $$

We obtain the following relationship between $CE_{Y_n}^{v,x,m_a}$ and $CE_{Y_n}^{v,x,m_b}$:

$$ \forall X \in \Phi, \forall Y_n \in \Phi / \{X\}, \forall x \in \Omega, \forall m_b \in M, v(CE_{Y_n}^{v,x,m_b}) = \sum_{m_a \in M} k_{m_b m_a}^x v(CE_{Y_n}^{v,x,m_a}), \quad (B.15) $$

with $\sum_{m_a \in M} k_{m_b m_a}^x = 1$.

We thus have
$$ \forall X \in \Phi, \forall Y_n \in \Phi / \{X\}, \forall x \in \Omega, \forall m_b \in M, v(CE_{Y_n}^{v,x,m_b}) \le \sum_{m_a \in M} k_{mbm_a}^x v(CE_{Max}^{v,x,m_a}), \quad (B.16) $$

$$ \text{with } CE_{Max}^{v,x,m_a} = \text{Max} \{CE_{Y_1}^{v,x,m_a}, \dots, CE_{Y_N}^{v,x,m_a}\}. $$

And thus, we also have

$$ \forall X \in \Phi, \forall x \in \Omega, \forall m_b \in M, v(CE_{Max}^{v,x,m_b}) \le \sum_{m_a \in M} k_{mbm_a}^x v(CE_{Max}^{v,x,m_a}), \quad (B.17) $$

$$ \text{with } CE_{Max}^{v,x,m_b} = \text{Max} \{CE_{Y_1}^{v,x,m_b}, \dots, CE_{Y_N}^{v,x,m_b}\}. $$

Under assumption A5, equations (B.8) and (B.17) imply

$$ \forall X \in \Phi, \forall x \in \Omega, \forall m_b \in M, v(CE_{Max}^{v,x,m_b}) \le v \sum_{m_a \in M} k_{mbm_a}^x CE_{Max}^{v,x,m_a}). \quad (B.18) $$

Which implies, under A1,

$$ \forall X \in \Phi, \forall x \in \Omega, \forall m_b \in M, CE_{Max}^{v,x,m_b} \le \sum_{m_a \in M} k_{mbm_a}^x CE_{Max}^{v,x,m_a}. \quad (B.19) $$

Which implies, under A3,

$$ \forall X \in \Phi, \forall x \in \Omega, \forall m_b \in M, u, x, Max \ x, \sum_{m_a \in M} k_{mbm_a}^x CE_{Max}^{v,x,m_a}) \le u(x, Max(x, CE_{Max}^{v,x,m_b})). \quad (B.20) $$

Moreover, since the Max function is convex, we have

$$ \text{Max} \sum_{m_a \in M} k_{mbm_a}^x x, \sum_{m_a \in M} k_{mbm_a}^x CE_{Max}^{v,x,m_a}) \le \sum_{m_a \in M} k_{mbm_a}^x \text{Max}(x, CE_{Max}^{v,x,m_a}). \quad (B.21) $$

Given equations (B.20) and (B.21), we obtain, under A3,

$$ \forall X \in \Phi, \forall x \in \Omega, \forall m_b \in M, u, x, \sum_{m_a \in M} k_{mbm_a}^x \text{Max}(x, CE_{Max}^{v,x,m_a}) \le u(x, \text{Max}(x, CE_{Max}^{v,x,m_b})). \quad (B.22) $$
Which implies, under A4,

$$ \forall X \in \Phi, \forall x \in \Omega, \forall m_b \in M, \sum_{m_a \in M} k_{m_b m_a}^x u(x, Max(x, CE_{Max}^{v,x,m_a})) \le u(x, Max(x, CE_{Max}^{v,x,m_b})). \quad (B.23) $$

Equations (B.7) and (B.23) give

$$ \forall X \in \Phi, \forall x \in \Omega, \forall m_b \in M, $$

$$ \sum_{m_a \in M} \pi(m_b | m_a, x) p(m_a | x) u(x, Max(x, CE_{Max}^{v,x,m_a})) \le p(m_b | x) u(x, Max(x, CE_{Max}^{v,x,m_b})). \quad (B.24) $$

Which implies that

$$ \forall X \in \Phi, \forall x \in \Omega, $$

$$ \sum_{m_b \in M} \sum_{m_a \in M} \pi(m_b | m_a, x) p(m_a | x) u(x, Max(x, CE_{Max}^{v,x,m_a})) \le \sum_{m_b \in M} p(m_b | x) u(x, Max(x, CE_{Max}^{v,x,m_b})). \quad (B.25) $$

Or, equivalently

$$ \forall X \in \Phi, \forall x \in \Omega, $$

$$ \sum_{m_a \in M} p(m_a | x) u(x, Max(x, CE_{Max}^{v,x,m_a})) \sum_{m_b \in M} \pi(m_b | m_a, x) \le \sum_{m_b \in M} p(m_b | x) u(x, Max(x, CE_{Max}^{v,x,m_b})). \quad (B.26) $$

Given that $\sum_{m_b \in M} \pi(m_b | m_a, x) = 1$, we obtain

$$ \forall X \in \Phi, \forall x \in \Omega, $$

$$ \sum_{m_a \in M} p(m_a | x) u(x, Max(x, CE_{Max}^{v,x,m_a})) \le \sum_{m_b \in M} p(m_b | x) u(x, Max(x, CE_{Max}^{v,x,m_b})). \quad (B.27) $$

We thus have

$$ \forall X \in \Phi, $$

$$ \sum_{x \in \Omega} p(x) \sum_{m_a \in M} p(m_a | x) u(x, Max(x, CE_{Max}^{v,x,m_a})) \le \sum_{x \in \Omega} p(x) \sum_{m_b \in M} p(m_b | x) u(x, Max(x, CE_{Max}^{v,x,m_b})). \quad (B.28) $$
Which allows us to conclude that

$$ \forall X \in \Phi, E[u(X, Max(X, CE_{Max}^{v,FS_X^a}))] \le E[u(X, Max(X, CE_{Max}^{v,FS_X^b}))] \quad (B.29) $$

And thus

$$ \underset{Y_n \in \Phi}{MaxE} [u(Y_n, Max(Y_n, CE_{Max}^{v,FS_n^q}))] \le \underset{Y_n \in \Phi}{MaxE} [u(Y_n, Max(Y, CE_{Max}^{v,FS_n^b}))] \quad (B.30) $$

# Appendix C

We compute the certainty equivalents of $Y_1$ and $Y_2$ under $FS^a$ (see equations 1 and 2):

<table><thead><tr><th></th><th>Y<sub>1</sub></th><th>Y<sub>2</sub></th></tr></thead><tbody><tr><td>When Y<sub>1</sub> is chosen, given m<sub>a</sub> = 0</td><td>CE<sup>v,y1,m<sub>a</sub>=0</sup><sub>Y<sub>1</sub></sub> = y<sub>1</sub></td><td>CE<sup>v,y1,m<sub>a</sub>=0</sup><sub>Y<sub>2</sub></sub> = 8</td></tr><tr><td>When Y<sub>1</sub> is chosen, given m<sub>a</sub> = 1</td><td>CE<sup>v,y1,m<sub>a</sub>=1</sup><sub>Y<sub>1</sub></sub> = y<sub>1</sub></td><td>CE<sup>v,y1,m<sub>a</sub>=1</sup><sub>Y<sub>2</sub></sub> = 18</td></tr><tr><td>When Y<sub>2</sub> is chosen, given m<sub>a</sub> = 0</td><td>CE<sup>v,y1,m<sub>a</sub>=0</sup><sub>Y<sub>1</sub></sub> = 6</td><td>CE<sup>v,y2,m<sub>a</sub>=0</sup><sub>Y<sub>2</sub></sub> = y<sub>2</sub></td></tr><tr><td>When Y<sub>2</sub> is chosen, given m<sub>a</sub> = 1</td><td>CE<sup>v,y1,m<sub>a</sub>=1</sup><sub>Y<sub>1</sub></sub> = 16</td><td>CE<sup>v,y2,m<sub>a</sub>=1</sup><sub>Y<sub>2</sub></sub> = y<sub>2</sub></td></tr></tbody></table>

Easy computations give the probability distribution of $M_{Y_1}^a$ ($p(m_a = 0) = \frac{3}{4}$ and $p(m_a = 1) = \frac{1}{4}$) and the probability distribution of $M_{Y_2}^a$ ($p(m_a = 0) = \frac{1}{2}$ and $p(m_a = 1) = \frac{1}{2}$). We use these probability distributions and the certainty equivalents of $Y_1$ and $Y_2$ to compute the expected r-utilities (see Equation 4):

$$ \begin{aligned} E[u(Y_1, R^{FS_1^a})] &= E[u(Y_1, Max\{Y_1, CE_{Y_2}^{v,y_1,m_a}\})] \\ &= \frac{1}{2} \left[ \frac{3}{4} u\left(6, \underbrace{Max\{6, 8\}}_{\text{regret}}\right) + \frac{1}{4} u\left(6, \underbrace{Max\{6, 18\}}_{\text{regret}}\right) \right] \\ &\quad + \frac{1}{2} \left[ \frac{3}{4} u(16, Max\{16, 8\}) + \frac{1}{4} u\left(16, \underbrace{Max\{16, 18\}}_{\text{regret}}\right) \right] \\ &= \frac{17}{4} = 4,25. \end{aligned} $$
$$
\begin{align*}
E[u(Y_2, R^{FS_{Y_2}^a})] &= E[u(Y_2, \mathrm{Max}\{Y_2, CE_{Y_1}^{v,y_2,m_a}\})] \\
&= \frac{3}{4} \left[ \frac{1}{2}u(8, \mathrm{Max}\{8, 6\}) + \frac{1}{2}u\left(8, \underbrace{\mathrm{Max}\{8, 16\}}_{\text{regret}}\right) \right] \\
&\quad + \frac{1}{4} \left[ \frac{1}{2}u(18, \mathrm{Max}\{18, 6\}) + \frac{1}{2}u(18, \mathrm{Max}\{18, 16\}) \right] \\
&= \frac{15}{4} = 3,75.
\end{align*}
$$

We also compute the certainty equivalents of $Y_1$ and $Y_2$ under $FS^b$ (see equations 1 and 2):

<table><thead><tr><th></th><th>Y<sub>1</sub></th><th>Y<sub>2</sub></th></tr></thead><tbody><tr><td>When Y<sub>1</sub> is chosen, given m<sub>b</sub> = 0</td><td>CE<sub>Y<sub>1</sub></sub><sup>v,y<sub>1</sub>,m<sub>b</sub>=0</sup> = y<sub>1</sub></td><td>CE<sub>Y<sub>2</sub></sub><sup>v,y<sub>1</sub>,m<sub>b</sub>=0</sup> = 9</td></tr><tr><td>When Y<sub>1</sub> is chosen, given m<sub>b</sub> = 1</td><td>CE<sub>Y<sub>1</sub></sub><sup>v,y<sub>1</sub>,m<sub>b</sub>=1</sup> = y<sub>1</sub></td><td>CE<sub>Y<sub>2</sub></sub><sup>v,y<sub>1</sub>,m<sub>b</sub>=1</sup> = 13</td></tr><tr><td>When Y<sub>2</sub> is chosen, given m<sub>b</sub> = 0</td><td>CE<sub>Y<sub>1</sub></sub><sup>v,y<sub>1</sub>,m<sub>b</sub>=0</sup> = 17/2</td><td>CE<sub>Y<sub>2</sub></sub><sup>v,y<sub>2</sub>,m<sub>b</sub>=0</sup> = y<sub>2</sub></td></tr><tr><td>When Y<sub>2</sub> is chosen, given m<sub>b</sub> = 1</td><td>CE<sub>Y<sub>1</sub></sub><sup>v,y<sub>1</sub>,m<sub>b</sub>=1</sup> = 27/2</td><td>CE<sub>Y<sub>2</sub></sub><sup>v,y<sub>2</sub>,m<sub>b</sub>=1</sup> = y<sub>2</sub></td></tr></tbody></table>

Easy computations give the probability distribution of $M_{Y_1}^b$ ($p(m_b = 0) = \frac{10}{16}$ and $p(m_b = 1) = \frac{6}{16}$) and the probability distribution of $M_{Y_2}^b$ ($p(m_b = 0) = \frac{1}{2}$ and $p(m_b = 1) = \frac{1}{2}$). We compute the expected r-utilities under $FS^b$:

$$
\begin{align*}
E[u(Y_1, R^{FS_{Y_1}^b})] &= E[u(Y_1, \mathrm{Max}\{Y_1, CE_{Y_2}^{v,y_1,m_b}\})] \\
&= \frac{1}{2} \left[ \frac{10}{16} u\left(6, \underbrace{\mathrm{Max}\{6, 9\}}_{\text{regret}}\right) + \frac{6}{16} u\left(6, \underbrace{\mathrm{Max}\{6, 13\}}_{\text{regret}}\right) \right] \\
&\quad + \frac{1}{2} \left[ \frac{10}{16} u(16, \mathrm{Max}\{16, 9\}) + \frac{6}{16} u(16, \mathrm{Max}\{16, 13\}) \right] \\
&= \frac{35}{8} = 4,375.
\end{align*}
$$
$$
\begin{align*}
E[u(Y_2, R^{FS_{Y_2}^b})] &= E[u(Y_2, \text{Max}\{Y_2, CE_{Y_1}^{v,y_2,m_b}\})] \\
&= \frac{3}{4} \left[ \frac{1}{2}u \left( 8, \text{Max}\left\{8, \frac{17}{2}\right\} \right) + \frac{1}{2}u \left( 8, \text{Max}\left\{8, \frac{27}{2}\right\} \right) \right] \\
&\quad + \frac{1}{4} \left[ \frac{1}{2}u \left( 18, \text{Max}\left\{18, \frac{17}{2}\right\} \right) + \frac{1}{2}u \left( 18, \text{Max}\left\{18, \frac{27}{2}\right\} \right) \right] \\
&= \frac{33}{8} = 4,125.
\end{align*}
$$

Expected r-utilities are higher under $FS^b$ than under $FS^a$.

# Appendix D

## Proof of Proposition 3:

First, let us show that the solution of Equation 10 exists and is unique. In order to do that, let $\underline{y}$ and $\bar{y}$ respectively denote the minimum value and the maximum value of $Y$.

If $Z = \underline{y}$ then the left-hand side of Equation 10 is

$$E[u(Z, Max(Z, CE_{Y}^{v,FSZ}))] = E[u(\underline{y}, Max(\underline{y}, CE_{Y}^{v,FSZ}))] . \quad (D.1)$$

The right-hand side is

$$E[u(Y, Max(Y, Z))] = E[u(Y, Y)] = E[v(y)]. \quad (D.2)$$

Equation 10 is not satisfied since, under A1 and A3, we have

$$E[u(\underline{y}, Max(\underline{y}, CE_{Y}^{v,FSZ}))] < u(\underline{y}, \underline{y}) = v(\underline{y}) < E[v(y)]. \quad (D.3)$$

The right-hand side of Equation 10 is greater than the left-hand side.

If $Z = \bar{y}$ then the left-hand side of Equation 10 is

$$E[u(Z, Max(Z, CE_{Y}^{v,FSZ}))] = u(\bar{y}, \bar{y}). \quad (D.4)$$

The right-hand side is

$$E[u(Y, Max(Y, Z))] = E[u(Y, \bar{y})]. \quad (D.5)$$
Equation 10 is not satisfied since, under A2, $u(\bar{y}, \bar{y}) > E[u(Y, \bar{y})]$. The left-hand side of Equation 10 is now greater than the right-hand side.

Moreover, under A1, function $E[u(Z, Max(Z, CE_{Y}^{v,FSZ}))]$ increases with $Z$ and under A3, function $E[u(Y, Max(Y, Z))]$ decreases with $Z$. Under A0, the solution of Equation 10 exists, is unique and belongs to $[\underline{y}, \bar{y}]$.

### Proof of Proposition 4:

If $Z_a$ and $Z_b$ respectively denote the Z-solution of Equation 10 under $FS^a$ and under $FS^b$, let us show that $Z_b \le Z_a$.

Proposition 1 states that if $FS^a$ is sufficient for a $FS^b$ then we have

$$ \forall Y \in \Phi, E[u(Y, Max(Y, CE_{Max}^{v,FS^a}))] \le E[u(Y, Max(Y, CE_{Max}^{v,FS^b}))]. \quad (D.6) $$

This property also holds for a sure thing $Z$:

$$ E[u(Z, Max(Z, CE_{Y}^{v,FS^a}))] \le E[u(Z, Max(Z, CE_{Y}^{v,FS^b}))]. \quad (D.7) $$

The left-hand side of Equation 10 is greater under $FS^b$ than under $FS^a$.

The right-hand side of Equation 10, $[u(Y, Max(Y, Z))]$, is independent of the FS and, under A3, decreases with $Z$. We thus have $Z_b \le Z_a$.

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