# Mean-preserving capacities: A tractable class of Choquet capacities

Richard Peter, Pascal Toquebeuf

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## To cite this version:

Richard Peter, Pascal Toquebeuf. Mean-preserving capacities: A tractable class of Choquet capacities. Economic Theory, 2025, 10.1007/s00199-025-01688-8. hal-05324776

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# Mean-preserving capacities: A tractable class of Choquet capacities

Richard Peter<fnref n="1" /> Ⓟ · Pascal Toquebeuf<fnref n="2" />

Received: 5 June 2024 / Accepted: 6 October 2025

© The Author(s) 2025

## Abstract

This paper introduces mean-preserving capacities for Choquet expected utility. A capacity is defined as mean-preserving when the value assigned to an event falls within an interval centered around its subjective probability, with the interval's size reflecting the perceived level of ambiguity. These capacities are based on the complementary independence axiom and feature a centrally symmetric core for decision-makers who are averse to ambiguity. To illustrate their practical value, we explore a series of economic applications including portfolio choice, self-insurance and self-protection, the value of a statistical life, and precautionary saving. Mean-preserving capacities offer intuitive predictions for both ambiguity-averse and ambiguity-loving individuals, often requiring fewer or no ancillary assumptions than alternative models. Moreover, they simplify the analysis of greater ambiguity considerably.

**Keywords** Mean-preserving capacity · Choquet expected utility · Ambiguity · Ambiguity attitude · Comparative statics

**JEL Classification** D80 · D81

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<footnote>We would like to thank Paul Thistle for his helpful comments and suggestions. We also thank participants at the Southern Economic Association meeting in 2020 and the World Risk and Insurance Economics Congress in 2020 for valuable feedback. We are grateful to two careful reviewers and the handling editor for their thoughtful comments and suggestions, which helped improve the exposition of the paper. All remaining errors are ours.</footnote>

<footnote>Richard Peter
richard-peter@uiowa.edu</footnote>

<footnote>Pascal Toquebeuf
pascal.toquebeuf@univ-grenoble-alpes.fr</footnote>

<footnote n="1">¹ Department of Finance, University of Iowa, Tippie College of Business, Iowa, USA</footnote>

<footnote n="2">² Grenoble Applied Economics Laboratory, University of Grenoble-Alpes, INRAE, CNRS, Grenoble, France</footnote>
# 1 Introduction

Uncertainty is ubiquitous. People face uncertainty over their earnings, health, liability, physical assets, financial assets, and longevity. Starting with the seminal works of David Schmeidler, a variety of decision models for choice under uncertainty have been brought forward in the past thirty-five years to accommodate Ellsberg-type phenomena. This research has generated many models of choice under ambiguity, each featuring some violation of the classic independence axiom. The most prominent models are Choquet expected utility (CEU, Schmeidler 1989), maxmin expected utility (MEU, Gilboa and Schmeidler 1989), alpha-MEU (Ghirardato et al. 2004), and the popular smooth ambiguity model (Klibanoff et al. 2005). These decision criteria generalize subjective expected utility (SEU) by featuring ambiguity and ambiguity attitude. Ambiguity refers to uncertainty about probability, for example, due to missing information (Camerer and Weber 1992). Ambiguity attitude refers to the decision-maker's sensitivity towards ambiguity.<fnref n="1" />

In economic applications, we need tools that allow us to distinguish ambiguity attitudes (i.e., averse, neutral, loving), vary their intensity, manipulate the level of ambiguity and, at the same time, facilitate these operations in a tractable way. The smooth ambiguity model is often the go-to place for the comparative statics of ambiguity (e.g., Gollier 2011). Ambiguity lovers are rarely considered in applied work because their objective functions may fail to be concave in the smooth model. Empirical evidence suggests that ambiguity aversion is not universal (Kocher et al. 2018; Trautmann and Van De Kuilen 2015; Wakker et al. 2007). Based on these findings, we argue that ambiguity lovers should not be forgotten. The smooth ambiguity model also provides a notion of greater ambiguity (Jewitt and Mukerji 2017), which is, however, difficult to operationalize in applications. Often additional restrictions on the primitives are required for definitive results (e.g., Huang and Tzeng 2018; Peter 2019; Bleichrodt et al. 2019; Peter and Ying 2020). Finally, while Cubitt et al. (2020) find greater support for the smooth ambiguity model, Baillon and Bleichrodt (2015) and Chew et al. (2017) observe that CEU and alpha-MEU better explain their data.

Against the background of these considerations, we introduce mean-preserving (MP) capacities for CEU as a tractable and theoretically well-grounded tool to study ambiguity. For MP-capacities, the value of an event lies in an interval that is centered around the subjective probability of that event. The size of the interval measures the decision-maker's perceived level of ambiguity. MP-capacities identify ambiguity attitude parametrically, and one can vary the level of ambiguity by shrinking or expanding the intervals around the subjective probability. In a series of applications, we show that MP-capacities make predictions for ambiguity averters and ambiguity lovers alike and that, as expected, these predictions often flip for ambiguity lovers. We also find that MP-capacities simplify the analysis of greater ambiguity considerably and generate intuitive results without the many ancillary assumptions that other models require.

<footnote n="1">¹ Gilboa and Marinacci (2016) provide an overview of models of decision-making under ambiguity.</footnote>
Axiomatically, MP-capacities rest on the complementary independence axiom (see Siniscalchi 2009). In this case, the decision-maker's CEU takes the following form:

$$V(f) = \mathbb{E}_m U(f) + \beta(f).$$

$\mathbb{E}_m U(f)$ is the SEU of act $f$ and $\beta(f)$ its associated measure of ambiguity.<fnref n="2" /> The most general approach of this kind is the one by Grant and Polak (2013), who characterize so-called mean-dispersion preferences axiomatically. The sign of $\beta(f)$ characterizes the decision-maker's ambiguity attitude in these models. We also show that MP-capacities for CEU preferences are equivalently characterized by central symmetry of the core in case of ambiguity aversion (or central symmetry of the dual capacity's core for ambiguity lovers). The adjustment functional $\beta(f)$ takes a particular form for MP-capacities, and we relate our approach to alpha-MEU and neo-additive capacities. MP-capacities solve an identification problem for alpha-MEU because preferences are ambiguity-neutral if and only if alpha is equal to one-half.

In terms of applications, we study portfolio choice, self-insurance and self-protection, the value of a statistical life (VSL), and precautionary saving. For portfolio choice, CEU with MP-capacities can explain the nonparticipation puzzle whereas the smooth model cannot. Furthermore, we easily obtain results for ambiguity lovers, which are all new in the literature, and the analysis of greater ambiguity is straightforward. For self-protection and VSL, we highlight our results for endogenous ambiguity. A simple necessary and sufficient condition separates the case where the degree of ambiguity aversion has a positive effect from the case where its effect is negative. Our condition reconciles conflicting results from the literature. As before, our findings for ambiguity lovers are all new in the literature and the study of greater ambiguity is simple and straightforward.

Overall, our series of applications demonstrates the usefulness of MP-capacities to study ambiguity and ambiguity attitude in various economic contexts of interest. MP-capacities add a great deal of tractability to CEU and make intuitive predictions. We argue that this simplicity holds promise for their applicability in more involved settings.

We proceed as follows. The next section presents the decision theoretic set-up, introduces MP-capacities and their properties, characterizes them axiomatically, and relates them to other approaches. The third section presents several applications many of which have only been studied for ambiguity averters and/or in the smooth ambiguity model. In particular, we analyze portfolio choice, self-insurance and self-protection, the value of a statistical life, and precautionary saving. A final section concludes.

<footnote n="2">² This representation is in line with Maccheroni et al. (2006), Siniscalchi (2009), Vicig and Seidenfeld (2012), Schneider and Nunez (2015) and Nunez and Schneider (2019).</footnote>
# 2 Mean-preserving capacities

## 2.1 Decision theoretic set-up and Choquet expected utility

Let $\Omega = \{\omega_1, \dots, \omega_n\}$ be a finite set of mutually exclusive and collectively exhaustive states of the world.<fnref n="3" /> Denote by $\Sigma = \mathcal{P}(\Omega)$ the power set of $\Omega$. An element $E \in \Sigma$ is called an event. Its complement is $E^c = \Omega \setminus E$. Events other than $\emptyset$ and $\Omega$ are called nontrivial, and we denote their collection by $\Sigma_{nt} = \Sigma \setminus \{\emptyset, \Omega\}$. Let $X$ be an interval in $\mathbb{R}$. An act $f$ is a function from $\Omega$ to $X$. We let $f^{-1}(x) = \{\omega \in \Omega : f(s) = x\}$ denote the preimage of $x$ under $f$ for $x \in X$. Then $f^{-1}(x) \in \Sigma$ holds trivially so any act is measurable with respect to $\Sigma$. The set $\mathcal{F}$ denotes the collection of all acts. The range of act $f$ is given by $f(\Omega) = \{f(\omega) : \omega \in \Omega\}$.

A binary relation $\geq$ over $\mathcal{F}$ gives the decision-maker's preferences over acts. As usual, $\sim$ denotes the symmetric part of $\geq$ and $>$ the asymmetric part. We assume throughout the paper that preferences can be represented by Choquet expected utility (CEU).<fnref n="4" /> The decision-maker's beliefs are formalized by a Choquet capacity $\nu : \Sigma \to \mathbb{R}$ in this model. The set function $\nu$ is:

(i) Normalized: $\nu(\Omega) = 1$ and $\nu(\emptyset) = 0$;

(ii) Monotonic with respect to set inclusion: $E \subseteq E' \implies \nu(E) \leq \nu(E')$.

We write $\nu(\omega)$ instead of $\nu(\{\omega\})$ for an elementary event consisting of a single state $\omega \in \Omega$. The decision-maker's preference representation contains a continuous and strictly increasing utility function of outcomes, $U : X \to \mathbb{R}$. The CEU of act $f \in \mathcal{F}$ is then given by the Choquet integral of $U(f)$ with respect to capacity $\nu$:

$$V(f) = \sum_{x \in f(\Omega)} U(x) \cdot [\nu(f \geq x) - \nu(f > x)]. \quad (1)$$

We write $\nu(f \geq x)$ for $\nu(\omega \in \Omega : f(\omega) \geq x)$, and likewise for other set functions. To simplify the exposition, we assume that the entire state space is the only universal set, $\nu(E) = 1 \implies E = \Omega$, and that the empty set is the only null set, $\nu(E) = 0 \implies E = \emptyset$. We thus focus on capacities with $\nu(E) \in (0, 1)$ for all $E \in \Sigma_{nt}$.<fnref n="5" />

A capacity $\nu$ is convex (concave) if $\nu(E) + \nu(E') \leq (\geq) \nu(E \cup E') + \nu(E \cap E')$ for all $E, E' \in \Sigma$. A capacity is additive if it is both convex and concave. In this case, $\nu$ is a subjective probability on $\Omega$, and $V(f)$ in Eq. (1) reduces to subjective expected utility (SEU):

$$\mathbb{E}_m U(f) = \sum_{\omega \in \Omega} m_{\omega} \cdot U(f(\omega)). \quad (2)$$

<footnote n="3">³ The assumption of a finite state space is not required for the mathematical results in this section, but it avoids measurability assumptions. Applications with a finite number of states are pervasive in economic theory, including those we discuss in Section 3.</footnote>

<footnote n="4">⁴ Gilboa (1987), Schmeidler (1989), Wakker (1989), Sarin and Wakker (1992) and Ghirardato et al. (2003), among others, provide axiomatic foundations of CEU. See Abdellaoui and Zank (2023) for recent work providing axiomatic foundations of rank-dependent preferences within the two-stage framework of Anscombe and Aumann.</footnote>

<footnote n="5">⁵ Alternatively, we can use the approach in Chateauneuf et al. (2007). They partition $\Sigma$ into null events, universal events and essential events, and then focus on capacities that are exactly congruent with the set of null events.</footnote>
The expectation is taken with respect to the subjective probability $m$ on $\Omega$ with $m_\omega = m(\omega) = \nu(\omega)$ for all $\omega \in \Omega$. The dual capacity of $\nu$ is defined as $\bar{\nu}(E) = 1 - \nu(E^c)$ for all $E \in \Sigma$. A capacity is convex if and only if its dual capacity is concave.

Ghirardato and Marinacci (2002) propose a definition of ambiguity attitude and comparative ambiguity aversion for the decision-maker's preference relation $\geq$. They also provide characterizations when preferences admit a CEU representation. We use their results as our definitions of ambiguity attitude, comparative ambiguity aversion, and comparative ambiguity loving for CEU.

**Definition 1 (Ambiguity attitude)** A decision-maker is called ambiguity-averse (ambiguity-loving, ambiguity-neutral) if there is a subjective probability $m$ such that the capacity in his preference representation satisfies $m \geq (\leq, =) \nu$.

**Definition 2 (Comparative ambiguity aversion/loving)** Suppose that $\geq_1$ and $\geq_2$ are two preference relations with CEU representations $(U_1, \nu_1)$ and $(U_2, \nu_2)$. We say that $\geq_1$ is more ambiguity-averse (more ambiguity-loving) than $\geq_2$ if $U_1$ and $U_2$ differ by a positive affine transformation and $\nu_1 \leq (\geq) \nu_2$.

The part about ambiguity aversion in Definition 1 is Ghirardato and Marinacci's Corollary 13. We define ambiguity loving symmetrically. Definition 2 is Ghirardato and Marinacci's Theorem 17(i) for comparative ambiguity aversion. We define comparative ambiguity loving symmetrically. Definition 1 is a special case of Definition 2 when $\geq_2$ has an SEU representation. The associated capacity $\nu_2$ is then a subjective probability.

## 2.2 Mean-preserving capacities and their properties

We now introduce a class of capacities that makes Choquet integration more tractable.

**Definition 3 (MP-capacity)** A mean-preserving (MP) capacity $\nu$ is such that, for all $E \in \Sigma$,

$$ \nu(E) = m(E) + \left(\frac{1}{2} - \alpha\right) a(E), \quad (3) $$

where the triplet $(m, a, \alpha)$ consists of a subjective probability $m$ with $m(E) \in (0, 1)$ for $E \in \Sigma_{nt}$, a scalar $\alpha \in [0, 1]$, and a nonnegative set function $a: \Sigma \to \mathbb{R}$, with the following properties:

(a) Normalization and complement-symmetry: $a(\emptyset) = a(\Omega) = 0$ and $\forall E \in \Sigma: a(E) = a(E^c)$.

(b) Bounded changes: $\forall E, E' \in \Sigma: E \subseteq E' \implies |a(E') - a(E)| \leq 2(m(E') - m(E))$ with a strict inequality for $E = \emptyset$ or $E' = \Omega$.

(c) Submodularity: $\forall E, E' \in \Sigma: a(E) + a(E') \geq a(E \cup E') + a(E \cap E')$.

We show in Appendix B that normalization and bounded changes of $a$ ensure that $\nu$ in Eq. (3) is indeed a Choquet capacity for all $\alpha \in [0, 1]$ and that it satisfies $\nu(E) \in (0, 1)$ for all $E \in \Sigma_{nt}$. MP-capacities are centered around the subjective probability $m$, which motivates the terminology “mean-preserving.” Denneberg (2000) refers to $m$ as the central component and to the remainder as the ambiguity component.
We interpret $\alpha$ as a measure of (comparative) ambiguity attitude and $\alpha$ as the level of ambiguity. When $\nu$ is an MP-capacity, the CEU of act $f$ in Eq. (1) may be rewritten as follows:

$$V(f) = \mathbb{E}_m U(f) + (\frac{1}{2} - \alpha) \cdot \sum_{x \in f(\Omega)} U(x) \cdot [\mathfrak{a}(f \ge x) - \mathfrak{a}(f > x)]. \quad (4)$$

Intuitively, the decision-maker evaluates act $f$ according to its SEU and then adjusts this evaluation based on his attitude towards and perception of ambiguity.

Property (a) in Definition 3 rules out ambiguity when an event is certain to happen or certain not to happen. Property (b) relates the level of ambiguity to the decision-maker's subjective probability. When comparing a nontrivial event $E$ to certainty or impossibility, the level of ambiguity is bounded by $2 \min(m(E), 1 - m(E))$ to ensure $\nu(E)$ between zero and one. For the comparison of two nontrivial events, Property (b) restricts the change in ambiguity when moving from smaller to larger events. It is a continuity requirement because it rules out abrupt changes in the level of ambiguity. The submodularity of set function $\alpha$ in Property (c) means intuitively that the level of ambiguity for a larger event cannot exceed the sum of the ambiguity levels of constituent events. Properties (a) and (c) correspond to Marinacci's Definition 1 of $\alpha$ being an ambiguity level, consistent with our interpretation.

The following proposition collects properties of MP-capacities.

**Proposition 1** Let $\nu = (m, \alpha, \alpha)$ be an MP-capacity in the sense of Definition 3.

(i) Capacity $\nu$ is convex for $\alpha \ge 1/2$, concave for $\alpha \le 1/2$, and additive for $\alpha = 1/2$.

(ii) An increase in $\alpha$ increases ambiguity aversion in the sense of Definition 2.

Let $\nu_1 = (m, \alpha_1, \alpha)$ and $\nu_2 = (m, \alpha_2, \alpha)$ be two MP-capacities that differ only in their set functions.

(iii) When $\alpha_2 \ge \alpha_1$, the change from $\alpha_1$ to $\alpha_2$ decreases the decision-maker's CEU for $\alpha \ge 1/2$, increases it for $\alpha \le 1/2$, and has no effect on it for $\alpha = 1/2$.

Appendix A.1 provides a proof. Insight (i) means that a decision-maker with MP-capacity $\nu$ is either ambiguity-averse or ambiguity-loving, depending on whether $\alpha \ge 1/2$ or $\alpha \le 1/2$. We can verify Definition 1 directly by noting that $m \ge \nu$ for $\alpha \ge 1/2$ and $m \le \nu$ for $\alpha \le 1/2$ because $\alpha$ is nonnegative. The knife-edge case of $\alpha = 1/2$ represents ambiguity neutrality. For MP-capacities the definitions of ambiguity aversion proposed by Ghirardato and Marinacci (2002) and Schmeidler (1989) coincide. Similarly, the degree of uncertainty aversion proposed by Dow and da Costa Werlang (1992) is positive for $\alpha > 1/2$, zero for $\alpha = 1/2$, and negative for $\alpha < 1/2$. When keeping ambiguity level $\alpha$ fixed, MP-capacities facilitate a simple parametric handling of ambiguity attitude. Result (ii) then says that for a given set function $\alpha$, parameter $\alpha$ may serve to compare ambiguity attitude in the sense of Definition 2. An interesting direction for future work is to allow $\alpha$ to depend on the act itself; see Xue (2020).

Observation (iii) suggests that MP-capacities are useful to analyze greater ambiguity as well. Keeping parameter $\alpha$ fixed and moving to a set function $\alpha$ which is uniformly
larger has no effect on ambiguity-neutral decision-makers, but makes all ambiguity-averse decision-makers worse off (and all ambiguity-loving decision-makers better off). Such a change satisfies Jewitt and Mukerji's (2017) definition of “more ambiguous (I).” For a fixed $\alpha$-parameter, we can interpret the change from $\alpha_1$ to $\alpha_2$ as an increase in the level of ambiguity perceived by the decision-maker.

These interpretations assume that we start with a decision-maker with MP-capacity $v = (m, \alpha, \alpha)$, and then keep either $\alpha$ or $\alpha$ fixed as we vary the other ambiguity parameter. The reason is that only $m$ is uniquely identified from the capacity $v$, whereas different combinations of $\alpha$ and $\alpha$ are consistent with the same MP-capacity $v$. Unless there is another instrument to identify $\alpha$ or $\alpha$ by itself, their values are not unique based on observable choices, see also Section 2.5.

## 2.3 Characterizing MP-capacities

When there are only two states, MP-capacities involve no restrictions on a decision-maker's preferences under ambiguity, and any Choquet capacity is an MP-capacity. This is no longer the case with three or more states. In this section we analyze what restrictions MP-capacities place on the decision-maker's preferences over uncertainty. We provide an axiomatic and a geometric answer to this question. To formulate them, we first state a definition due to Siniscalchi (2009).

**Definition 4** Two acts $f, f' \in \mathcal{F}$ are complementary if, for any two states $\omega_i, \omega_j \in \Omega$,

$$ \frac{1}{2} f(\omega_i) + \frac{1}{2} f'(\omega_i) \sim \frac{1}{2} f(\omega_j) + \frac{1}{2} f'(\omega_j). $$

Complementary acts are mirror images of one another. Consider an urn with an unknown composition of red, black and yellow balls. Act $f$ represents winning a prize of $100 when drawing a red ball, or zero otherwise. Act $f'$ represents winning a prize of $100 when drawing a black or yellow ball, or zero otherwise. Acts $f$ and $f'$ are complementary because, when facing the 50-50 mixture between them, a decision-maker is indifferent over the color of the ball drawn: he receives $50 regardless. In general, two acts $f$ and $f'$ are complementary if there is a constant $c$ so that their utility profiles satisfy $U \circ f = c - U \circ f'$, see Siniscalchi (2009). These utility profiles have the same dispersion as measured, for example, by the standard deviation.

Preferences satisfy complementary independence if the independence axiom holds for complementary pairs of acts.

**Axiom (Complementary independence)** For any two complementary pairs of acts $(f, f')$ and $(g, g')$ in $\mathcal{F}$, and all $\lambda \in [0, 1]$,

$$ f \geq f' \text{ and } g \geq g' \quad \Rightarrow \quad \lambda f + (1 - \lambda)g \geq \lambda f' + (1 - \lambda)g'. $$

An alternative way to characterize MP-capacities involves the shape of their core. Let $\Delta(\Sigma)$ be the set of probabilities with domain $\Sigma$, and let $\mathcal{P}_v = \{p \in \Delta(\Sigma) : v \leq p\}$ be the core of Choquet capacity $v$. $\mathcal{P}_v$ is nonempty when $v$ is convex (see Shapley 1971), so for $\alpha \geq 1/2$ when $v$ is an MP-capacity. In case of $\alpha \leq 1/2$, we look at the
core of the dual capacity $\bar{v}$ instead. Convexity of $v$ is sufficient but not necessary for $P_v$ being nonempty (see Ghirardato and Marinacci 2002).

The following definition introduces the particular shape of the core we are interested in.

**Definition 5** The core of $v$ is centrally symmetric if there exists an $m_c \in P_v$, called the center of $P_v$, such that, for any $p \in \Delta(\Sigma)$, we have $p \in P_v$ if and only if $(2m_c - p) \in P_v$.

We are now ready to characterize MP-capacities in two ways.

**Proposition 2** Let the decision-maker's preferences $\geq$ be represented by CEU with utility function $U$ and convex (concave) capacity $v$. The following are equivalent:

(i) Capacity $v$ is an MP-capacity.

(ii) Preferences satisfy complementary independence.

(iii) The core of $v$ ($\bar{v}$) is centrally symmetric.

Appendix A.2 provides a proof. As shown by Siniscalchi (2009), complementary independence imposes an additivity requirement on the capacity. When preferences have a CEU representation with Choquet capacity $v$, they satisfy complementary independence if and only if, for all $E, E' \in \Sigma$,

$$v(E) - v(E^c) + v(E') - v(E^{c'}) = v(E \cup E') - v(E^c \cap E^{c'}) + v(E \cap E') - v(E^c \cup E^{c'}).$$

Result (iii) gives a geometric interpretation in terms of the central symmetry of the core of $v$ or the core of $\bar{v}$, depending on whether $v$ is convex or concave.<fnref n="6" /> A centrally symmetric subset of subjective probabilities also plays a role in the characterization of more ambiguous events in the alpha-MEU model, see Jewitt and Mukerji's Proposition 3.1. The set of probabilities for an event expands while retaining the same center as the event is substituted for a more ambiguous one. For MP-capacity $v$, the center of $P_v$ is given by $(v + \bar{v})/2$. The proof of Proposition 2 shows that it coincides with the subjective probability $m$ in Definition 3.

## 2.4 Relation to other approaches

CEU with MP-capacities lies at the intersection of several popular approaches of decision-making under uncertainty. First, it is a special case of vector expected utility proposed by Siniscalchi (2009). When preferences satisfy the complementary independence axiom, the decision-maker assesses an act by its SEU baseline evaluation and its utility variability around this baseline. Complementary acts receive the same

<footnote n="6">⁶ Ellsberg's three-color urn $\Omega = \{R, B, Y\}$ (red, black, yellow) gives a particularly tangible illustration. Let $a_i$ be the ambiguity level of event $i = R, B, Y$. The mean-preserving requirement is equivalent to the ambiguity levels $a_i$ describing a triangular bipyramid with vertices (0, 0, 0), (2/3, 2/3, 0), (2/3, 0, 2/3), (0, 2/3, 2/3) and (2/3, 2/3, 2/3). The subjective probability $m = (1/3, 1/3, 1/3)$ is located at its center.</footnote>
adjustment in the decision-maker's preference representation. Using Siniscalchi's terminology, we can call the term

$$ (\frac{1}{2} - \alpha) \cdot \sum_{x \in f(\Omega)} U(x) \cdot [\mathfrak{a}(f \ge x) - \mathfrak{a}(f > x)], $$

in Eq. (4) the “adjustment functional” for act $f$. It reflects both the decision-maker’s perception of ambiguity and his attitude towards it. MP-capacities supply an additional specification of this adjustment functional, which turns out to be quite tractable for comparative statics.

Second, CEU with MP-capacities is also a special case of $\alpha$-MEU. For MP-capacity $v = (m, \mathfrak{a}, \alpha)$, let $P_w = \{p \in \Delta(\Sigma) : w \le p\}$ be the core of Choquet capacity $w$ defined by $w := m - \mathfrak{a}/2$. According to insight (i) in Proposition 1, $w$ is convex and $P_w$ therefore nonempty. Similarly, $\bar{w} := m + \mathfrak{a}/2$ is concave and, by construction, the dual capacity of $w$. Then, the CEU of act $f$ with respect to MP-capacity $v$ has the following $\alpha$-MEU representation :

$$ V(f) = \alpha \min_{\hat{m} \in P_w} \mathbb{E}_{\hat{m}} U(f) + (1 - \alpha) \max_{\hat{m} \in P_w} \mathbb{E}_{\hat{m}} U(f). $$

By construction, we have $v = \alpha w + (1 - \alpha) \bar{w}$. Hence, MP-capacities are a special case of the class studied by Jaffray and Philippe (1997), named JP-capacities. It turns out that MP-capacities are precisely the special case of JP-capacities for which $\alpha = 1/2$ implies ambiguity neutrality.<fnref n="7" /> Other subclasses of JP-capacities, like neo-additive capacities proposed by Chateauneuf et al. (2007), do not share this property. In general, neo-additive capacities differ from MP-capacities: with three or more states, a capacity is both mean-preserving and neo-additive if and only if it is a subjective probability, see Peter and Toquebeuf (2020). Neo-additive capacities and MP-capacities thus offer two distinctive extensions of SEU.

## 2.5 Uniqueness of parameters

We already observed after Proposition 1 that subjective probability $m$ is uniquely identified, whereas $\alpha$ and $\alpha$ are not. In particular, uniqueness of $m$ follows directly from the uniqueness of $v$ in the CEU representation of preferences because $m = (v + \bar{v})/2$. As Siniscalchi (2009) observes, the decision-maker’s preferences over complementary acts identify the baseline prior $m$ behaviorally when the complementary independence axiom holds.

Set function $\mathfrak{a}$ and scalar $\alpha$ are, however, not unique. For instance, ambiguity neutrality is indistinguishable from the absence of ambiguity from a strictly behavioral point of view. This situation is common in the literature on decision-making under

<footnote n="7">⁷ When there is a nontrivial event $E$ with $w(E) < \bar{w}(E)$, then $\alpha = 1/2$ is necessary for a JP-capacity to be ambiguity-neutral. But some JP-capacities with $\alpha = 1/2$ are not ambiguity-neutral. The focus on MP-capacities rules those out and makes $\alpha = 1/2$ an if-and-only-if condition, see Peter and Toquebeuf (2020).</footnote>
ambiguity and also arises in other approaches because every representation of pref-
erences in decision theory is merely attributed to the decision-maker (see Ghirardato
et al. 2004).

The nonuniqueness of α and α may seem more problematic when preferences are
not ambiguity-neutral. For example, the same decision-maker may be seen as very
ambiguity-averse but only perceiving a moderate level of ambiguity, or, alternatively,
only moderately ambiguity-averse but perceiving a lot of ambiguity. Given this inde-
terminacy, we always keep one of the parameters fixed as we vary the other one in
applications. Ghirardato et al. (2004) accomplish uniqueness in the alpha-MEU model
by scaling down the ambiguity level as much as possible, but, as shown by Eichberger
et al. (2011), this implies α ∈ {0, 1} for a finite state space. The evidence suggests
different degrees of ambiguity aversion across decision-makers (see Dimmock et al.
2015; Berger and Bosetti 2020), which makes this approach unappealing from a prac-
tical standpoint. We thus dispense with the uniqueness of α and α in Eq. (3), but
recognize this issue as a caveat to the interpretation of our results.

3 Applications

To show the practical value of MP-capacities, we apply them to five economic prob-
lems: portfolio choice, self-insurance, self-protection, the value of a statistical life,
and precautionary saving. In each case, we examine how ambiguity attitude affects
decision-making, vary the degree of ambiguity aversion or ambiguity loving, and
analyze the effects of greater ambiguity. The nonuniqueness of α and α is a key
consideration in interpreting our results. We stress that, in each application, we vary
only one of these parameters at a time. As a result, our findings are best understood
intrapersonally—how does a decision-maker adjust his behavior when he becomes
more sensitive to ambiguity? And how does he respond when he perceives greater
ambiguity?

We compare our findings to existing results in the literature, which are often based
on the smooth ambiguity model. Prior studies tend to focus on ambiguity-averse indi-
viduals, whereas MP-capacities allow us to derive many novel insights for ambiguity
lovers, a group rarely studied in the smooth model. Furthermore, we highlight cases
where our predictions are more intuitive than those of existing models. When analyz-
ing the effect of greater ambiguity, the smooth model often relies on many ancillary
assumptions. MP-capacities allow us to eliminate those. This collection of results
substantiates our claim that MP-capacities significantly enhance the tractability of the
CEU model.

## 3.1 Portfolio choice

The standard portfolio problem dates back to Arrow (1963) and Pratt (1964) and has
been a subject of ongoing research. A decision-maker must choose between a safe
asset and an uncertain asset. The return of the safe asset is normalized to zero. The net
return of the uncertain asset is represented by a random variable z, which takes values
from the bounded interval $[z_{-}, z_{+}]$, where $z_{-} < 0 < z_{+}$. Without loss of generality, we assume that the finite state space orders the possible returns in ascending order, $z_{1} < \dots < z_{n}$, where $z_{i}$ is the return in state $\omega_{i}$. When the decision-maker invests $\rho$ in the uncertain asset, his final wealth is $W_{0} + \rho z$.

Let $U$ denote the decision-maker's utility function, and let $v = (m, a, \alpha)$ be his MP-capacity. The objective function is:

$$V(\rho; a, \alpha) = \sum_{i=1}^{n} U(W_{0} + \rho z_{i}) \cdot \left[ m_{i} + \left(\frac{1}{2} - \alpha\right) \left[a(\omega_{i}^{\cup}) - a(\omega_{i+1}^{\cup})\right] \right],$$

where $\omega_{i}^{\cup}$ is shorthand for $\{\omega_{i}, \dots, \omega_{n}\}$ with the convention that $\omega_{n+1}^{\cup} = \emptyset$. The decision-maker's optimal investment choice, $\rho^{*}$, maximizes $V(\rho; a, \alpha)$. If $U$ is concave in $W$, then $V$ is concave in $\rho$. The demand for the uncertain asset is positive if and only if $V_{\rho}(0; a, \alpha) > 0$, where the subscript stands for the partial derivative. We find:

$$ \begin{aligned} V_{\rho}(0; a, \alpha) &= U'(W_{0}) \cdot \left( \sum_{i=1}^{n} m_{i} z_{i} + \left(\frac{1}{2} - \alpha\right) \sum_{i=1}^{n} z_{i} \left(a(\omega_{i}^{\cup}) - a(\omega_{i+1}^{\cup})\right) \right) \\ &= U'(W_{0}) \cdot \left( \underbrace{\sum_{i=1}^{n} m_{i} z_{i}}_{\text{equity premium}} + \underbrace{\left(\frac{1}{2} - \alpha\right) \sum_{i=1}^{n-1} a(\omega_{i+1}^{\cup}) (z_{i+1} - z_{i})}_{\text{ambiguity adjustment}} \right). \end{aligned} $$

This decomposition shows the following result.

**Proposition 3** A decision-maker invests a positive (resp. negative) amount in the uncertain asset if and only if the ambiguity-adjusted equity premium is positive (resp. negative). The ambiguity-adjusted equity premium:

(i) Decreases (increases) with the degree of ambiguity aversion (ambiguity loving);

(ii) Decreases (increases) with the level of ambiguity for ambiguity averters (ambiguity lovers).

These findings align with intuition: as ambiguity aversion increases, the decision-maker becomes less likely to invest in the uncertain asset. Similarly, greater perceived ambiguity discourages investment for ambiguity-averse individuals. In contrast, the smooth ambiguity model predicts a different outcome. Gollier (2011) finds that the demand for the uncertain asset is positive if and only if the (unadjusted) equity premium is positive, regardless of the decision-maker's degree of ambiguity aversion. Therefore, MP-capacities provide a simple, ambiguity-based explanation for the nonparticipation puzzle, which suggests that some households do not invest in the stock market despite a positive equity premium.

The reason for these differences is that smooth ambiguity preferences exhibit second-order ambiguity aversion (Lang 2016). Then, positive exposure to ambiguity is optimal if and only if a subjective belief exists under which the act's expected outcome
is positive. Lang (2016) does not consider α-MEU preferences because they may fail to admit an uncertainty-averse representation (Cerreia-Vioglio et al. 2011). However, according to Proposition 3, investment behavior under CEU with MP-capacities is consistent with first-order ambiguity aversion. Ambiguity-averse decision-makers do not invest in the uncertain asset for some range of positive equity premia. The size of this range increases in the degree of ambiguity aversion and in the level of ambiguity, which is economically intuitive.<fnref n="8" />

For an interior solution $ρ^*$, the first-order condition is:

$$V_ρ(ρ^*; a, α) = \sum_{i=1}^{n} z_i U'(W_0 + ρ^* z_i) + (\frac{1}{2} - α) \sum_{i=1}^{n} z_i U'(W_0 + ρ^* z_i) (a(\omega_i^U) - a(\omega_{i+1}^U)) = 0.$$

Rearrange the second term as follows:

$$(\frac{1}{2} - α) \sum_{i=1}^{n-1} a(\omega_{i+1}^U) \cdot [z_{i+1}U'(W_0 + ρ^*z_{i+1}) - z_iU'(W_0 + ρ^*z_i)].$$

If the coefficient of relative risk aversion $-WU''(W)/U'(W)$ is below one, the square bracket is nonnegative, and we can sign the comparative static effects of $α$ and $a$ on $ρ^*$.

**Proposition 4** *When relative risk aversion is below unity, the demand for the uncertain asset:*

(i) Decreases (increases) with the degree of ambiguity aversion (ambiguity loving);

(ii) Decreases (increases) with the level of ambiguity for ambiguity averters (ambiguity lovers).

Gollier (2011) shows the result about ambiguity aversion in the smooth model, but his derivation relies on the additional assumption that priors are ordered by first-order stochastic dominance (FSD). In contrast, our result is simpler because it follows directly from the relative risk aversion assumption.<fnref n="9" /> Besides, MP-capacities provide new results for ambiguity-loving investors, who exhibit the reverse behavior. Huang and Tzeng (2018) study greater ambiguity in the smooth model, but require the ancillary assumption that relative ambiguity prudence be less than two. MP-capacities avoid this restriction and yield the intuitive result directly. Unlike in the smooth ambiguity model, where greater ambiguity aversion and greater ambiguity may paradoxically

<footnote n="8">⁸ This finding is similar in spirit to Mukerji and Tallon (2001), who show that ambiguity aversion can lead to a no-trade result under CEU preferences with idiosyncratic risk. Empirical evidence also supports the negative link between ambiguity aversion and stock market participation, see Dimmock et al. (2016).</footnote>

<footnote n="9">⁹ The evidence is more supportive of relative risk aversion above unity, see Meyer and Meyer (2005). The terms $z_iU'(w_0 + ρz_i)$ are not necessarily ordered in that case and the comparative static is indeterminate. Jouini et al. (2013) provide necessary and sufficient conditions for clear-cut risk effects under SEU.</footnote>
increase the demand for the uncertain asset (a Giffen-type effect), MP-capacities eliminate this paradox. Giffen-type effects due to ambiguity have not been documented empirically.

To provide intuition, let $\hat{m}_i = m_i + (\frac{1}{2} - \alpha)(a(\omega_i^U) - a(\omega_{i+1}^U))$ denote the multiplier on utility $U(w_0 + \rho z_i)$. Rewrite the objective function as $\sum_{i=1}^n \hat{m}_i U(w_0 + \rho z_i)$. We interpret $\hat{m}_i$ as a distorted probability or decision weight for state $\omega_i$. Property (b) in Definition 3 implies

$$|a(\omega_i^U) - a(\omega_{i+1}^U)| \le 2(m(\omega_i^U) - m(\omega_{i+1}^U)) = 2m_i,$$

with a strict inequality for $i = 1$ and $i = n$. These inequalities together with $\alpha \in [0, 1]$ ensure that $\hat{m}_i \ge 0$.<fnref n="10" /> Furthermore,

$$\sum_{i=1}^{n} \hat{m}_i = \underbrace{\sum_{i=1}^{n} m_i}_{=1} + (\frac{1}{2} - \alpha) \cdot \left( \underbrace{\sum_{i=1}^{n} a(\omega_i^U)}_{=0} - \sum_{i=1}^{n} a(\omega_{i+1}^U) \right) = 1.$$

This observation justifies our interpretation of $\hat{m}_i$ as a distorted probability. Let $M_i = \sum_{k=1}^i m_k$ denote the cumulative probabilities for $m$, and likewise $\hat{M}_i$ for $\hat{m}$. It holds that $\hat{M}_i = M_i - (\frac{1}{2} - \alpha)a(\omega_{i+1}^U)$ so that $\hat{M}_i$ and $M_i$ differ by a positive (negative) term under ambiguity aversion (ambiguity loving). The behavior of an ambiguity-averse decision-maker is then observationally equivalent to the behavior of an SEU maximizer, who has deteriorated his subjective belief in the sense of FSD compared to $m$. Ambiguity loving flips this distortion into an FSD improvement. Hadar and Seo (1990) analyze the effect of FSD shifts in the return distribution on investment behavior and identify the threshold value of unity on relative risk aversion.

According to Proposition 4, ambiguity aversion may explain why individuals allocate low amounts to equity (Dimmock et al. 2016). It also provides a possible explanation for the home-bias puzzle, where investors favor domestic over foreign stocks due to greater confidence in domestic return distributions (see also Bianchi and Tallon 2019; Asano and Osaki 2020). Beyond portfolio choice, our results have implications for insurance demand due to the formal equivalence between the portfolio problem and the coinsurance problem (see Schlesinger 2013; Menegatti and Peter 2022). Ambiguity aversion can explain why individuals demand full insurance even when premiums are actuarially unfavorable. Empirical evidence supports this behavior: laboratory studies show a strong preference for full insurance (Shapira and Venezia 2008), while field data reveal that households display a large willingness to pay to insure modest risks (Sydnor 2010). In contrast, the smooth ambiguity model does not explain these phenomena. In the smooth model, Mossin's Theorem holds, and ambiguity-averse decision-makers never purchase full insurance at unfair prices (see Alary et al. 2013). However, CEU with MP-capacities can rationalize the demand for full insurance at unfair prices, aligning more closely with observed behavior.

<footnote n="10">¹⁰ If $\alpha < 1/2$ and $a(\omega_i^U) - a(\omega_{i+1}^U) \ge 0$, then $\hat{m}_i > m_i \ge 0$. If $\alpha < 1/2$ and $a(\omega_i^U) - a(\omega_{i+1}^U) < 0$, then $\hat{m}_i > m_i + (\frac{1}{2} - \alpha) \cdot (-2m_i) = 2\alpha \cdot m_i \ge 0$. If $\alpha > 1/2$ and $a(\omega_i^U) - a(\omega_{i+1}^U) < 0$, then $\hat{m}_i > m_i \ge 0$. Finally, if $\alpha > 1/2$ and $a(\omega_i^U) - a(\omega_{i+1}^U) \ge 0$, then $\hat{m}_i > m_i + (\frac{1}{2} - \alpha) \cdot 2m_i = 2(1 - \alpha) \cdot m_i \ge 0$.</footnote>
## 3.2 Self-insurance

Ehrlich and Becker (1972) distinguish between self-insurance and self-protection, depending on whether the activity reduces the severity or the probability of loss. Both strategies help mitigate risk, but their comparative statics differ markedly, which is why they are usually studied separately in the literature (see Courbage et al. 2025, for a recent survey). Here, we analyze the marginal willingness to pay (WTP) and the optimal demand for each activity.

Willingness to pay. We use a binary loss model, which can be extended to multiple loss states. The decision-maker's initial wealth is $W_0$, and he faces a potential loss of $\ell < W_0$ with a subjective probability of $p$. Ambiguity is represented by an MP-capacity, where the ambiguity level is given by $\alpha \in (0, 2 \min\{p, 1-p\})$, and ambiguity attitude by $\alpha \in [0, 1]$. The decision-maker's CEU is:

$$V = [p - \alpha(\frac{1}{2} - \alpha)] U(W_0 - \ell) + [1 - p + \alpha(\frac{1}{2} - \alpha)] U(W_0).$$

Let $P(y)$ denote the willingness to pay to increase wealth in the loss state by $y$, defined via:

$$[p - \alpha(\frac{1}{2} - \alpha)] U(W_0 - \ell + y - P(y)) + [1 - p + \alpha(\frac{1}{2} - \alpha)] U(W_0 - P(y)) = V.$$

Since $y = 0$ represents the status-quo, we obtain $P(0) = 0$. The marginal willingness to pay for a small increase in $y$ is given by:

$$P'(0) = \frac{[p - \alpha(\frac{1}{2} - \alpha)] U'(W_0 - \ell)}{[p - \alpha(\frac{1}{2} - \alpha)] U'(W_0 - \ell) + [1 - p + \alpha(\frac{1}{2} - \alpha)] U'(W_0)} \quad (5)$$

$$= \left[ 1 + \frac{1 - p + \alpha(\frac{1}{2} - \alpha)}{p - \alpha(\frac{1}{2} - \alpha)} \cdot \frac{U'(W_0)}{U'(W_0 - \ell)} \right]^{-1}. \quad (6)$$

This leads to the following result, which also applies to non-infinitesimal reductions in loss severity.

**Proposition 5** *The marginal WTP for self-insurance:*

(i) Increases (decreases) with the degree of ambiguity aversion (ambiguity loving);

(ii) Increases (decreases) with the level of ambiguity for ambiguity averters (ambiguity lovers).

Result (i) for ambiguity lovers and Result (ii) are new in the literature. Alary et al. (2013) show Result (i) for ambiguity averters in the smooth ambiguity model, but they need an additional assumption on certainty-equivalent wealth. MP-capacities allow us to eliminate this restriction. Intuitively, self-insurance reduces the utility gap between the loss-state and the no-loss state. As the level of ambiguity increases, a self-insured individual experiences less impact from ambiguity compared to someone
who is not self-insured. Under ambiguity aversion, this rationale explains why self-insurance is more valuable the greater the level of ambiguity. The opposite holds for ambiguity lovers, who have a lower marginal WTP than if they were ambiguity-neutral. This effect is amplified as the level of ambiguity increases or as they become more ambiguity-loving. The asymmetry between ambiguity averters and ambiguity lovers suggests that, in aggregate, the effects of ambiguity might cancel each other out. Recent experimental evidence confirms such aggregation effects (see Couture et al. 2024).

The simplicity of MP-capacities allows us to identify a new interaction effect between risk and ambiguity preferences. The term $U'(W_0)/U'(W_0 - \ell)$ in Eq. (6) is negatively associated with the degree of risk aversion (see Pratt 1964). We can then interpret Proposition 5 by saying that ambiguity aversion reinforces risk aversion while ambiguity loving attenuates it. However, this relationship is not linear—changes in the ambiguity parameters have a stronger impact on $P'(0)$ when utility curvature is low. This interaction suggests that ambiguity plays a more important role when the degree of risk aversion is low.<fnref n="11" />

Optimal demand. We now analyze the demand for self-insurance. Let $y \in [0, \bar{y}]$ denote the level of self-insurance, where the potential loss $\ell(y)$ decreases with $y$ at a non-decreasing rate ($\ell' < 0, \ell'' \ge 0$). Self-insurance involves an upfront cost $c(y)$, which is increasing and non-concave ($c' > 0, c'' \ge 0$). Final wealth levels are $W_L = W_0 - \ell(y) - c(y)$ in the loss state and $W_N = W_0 - c(y)$ in the no-loss state. The decision-maker maximizes his CEU:

$$ \max_{y \in [0, \bar{y}]} V(y) = [p - \alpha(\frac{1}{2} - \alpha)] U(W_L) + [1 - p + \alpha(\frac{1}{2} - \alpha)] U(W_N). $$

The first-order condition for an interior solution $y^*$ is:

$$ \begin{aligned} & (-\ell'(y^*) - c'(y^*)) [p - \alpha(\frac{1}{2} - \alpha)] U'(W_L^*) \\ & -c'(y^*) [1 - p + \alpha(\frac{1}{2} - \alpha)] U'(W_N^*) = 0. \end{aligned} \quad (7) $$

Under risk aversion (i.e., $U'' < 0$), the second-order condition holds. We find the following result.

**Proposition 6** *The optimal level of self-insurance:*

(i) Increases (decreases) with the degree of ambiguity aversion (ambiguity loving);

(ii) Increases (decreases) with the level of ambiguity for ambiguity averters (ambiguity lovers).

The result holds by the Implicit Function Rule. The intuition comes from the effect of ambiguity on the marginal benefit and the marginal cost of self-insurance. The marginal benefit is:

$$ MB(y) = -\ell'(y) [p - \alpha(\frac{1}{2} - \alpha)] U'(W_L). $$

<footnote n="11">¹¹ Cleanly separating attitudes toward ambiguity from attitudes toward risk is conceptually challenging. See Zhang (2022) for recent work.</footnote>
It measures the increase in consumption utility in the loss state. Ambiguity aversion raises the marginal benefit, and greater ambiguity raises (lowers) it for ambiguity averters (ambiguity lovers). The marginal cost is:

$$MC(y) = c'(y) \cdot \left( \left[ p - a\left(\frac{1}{2} - \alpha\right) \right] U'(W_L) + \left[ 1 - p + a\left(\frac{1}{2} - \alpha\right) \right] U'(W_N) \right).$$

It measures the reduction in consumption utility from self-insurance expenditures. Ambiguity aversion raises the marginal cost because marginal utility in the loss state is higher than in the no-loss state. Similarly, greater ambiguity raises (lowers) the marginal cost for ambiguity-averse (ambiguity-loving) decision-makers. The net effect is therefore a priori indeterminate, because ambiguity affects the marginal benefit and the marginal cost in the same direction.

Since self-insurance is optimal when $MB(y^*) = MC(y^*)$, it follows that $-\ell'(y^*) > c'(y^*)$. As noted by Dionne and Eeckhoudt (1985), at the optimal level of self-insurance, final wealth in the loss state must be increasing in y. This observation implies:

$$-\ell'(y^*)U'(W_L^*) > c'(y^*) (U'(W_L^*) - U'(W_N^*)) .$$

Consequently, the effect of ambiguity aversion on the marginal benefit dominates its effect on the marginal cost, $MB_α(y^*) > MC_α(y^*) > 0$. Result (i) follows. Likewise, the effect of greater ambiguity on the marginal benefit dominates its effect on the marginal cost, $MB_α(y^*) > MC_α(y^*) > 0$ for $\alpha > 1/2$, and $MB_α(y^*) < MC_α(y^*) < 0$ for $\alpha < 1/2$, which explains Result (ii).

Our findings for ambiguity lovers are completely new in the literature. For ambiguity averters, Result (i) confirms and extends some existing results from the smooth ambiguity model to CEU with MP-capacities. In particular, Snow (2011) considers the presence versus the absence of ambiguity, whereas we are concerned with changes in the level of ambiguity, which is more general. Alary et al. (2013) need to make an ancillary assumption on the size of certainty-equivalent wealth to sign the comparative static. Once again, the simplicity of MP-capacities allows us to obtain economically intuitive results quickly and with ease.

If we interpret $c(y)$ as the insurance premium and $\ell(0) - \ell(y)$ as the insurance indemnity, optimal insurance demand is a special case of optimal self-insurance. It is then easy to show, along the lines of Huang and Tzeng (2018), that the comparative statics of greater ambiguity in the smooth model require an ordering assumption on the priors and a bound on relative ambiguity prudence of two. According to Result (ii) in Proposition 6, MP-capacities require no additional assumptions.

## 3.3 Self-protection

Willingness to pay. We modify the simple binary loss model from the previous section. Let $e \ge 0$ denote a reduction in the probability of loss, and let $P(e)$ represent the decision-maker's WTP for such a reduction. We determine $P(e)$ using the following indifference condition:

$$[p - e - a\left(\frac{1}{2} - \alpha\right)]U(W_0 - \ell - P(e)) + [1 - p + e + a\left(\frac{1}{2} - \alpha\right)]U(W_0 - P(e)) = V.$$

<figure><img src="image_4.png" /></figure>
<table><caption>**Table 1** Effect of greater ambiguity on the WTP for self-protection</caption><thead><tr><th></th><th>Risk-averse (U'' &lt; 0)</th><th>Risk-loving (U'' &gt; 0)</th></tr></thead><tbody><tr><td>Ambiguity-averse (α &gt; 1/2)</td><td>negative</td><td>positive</td></tr><tr><td>Ambiguity-loving (α &lt; 1/2)</td><td>positive</td><td>negative</td></tr></tbody></table>

From this, it follows that $P(0) = 0$. The marginal WTP at $e = 0$ is given by:

$$P'(0) = \frac{U(W_0) - U(W_0 - \ell)}{[1 - a(\frac{1}{2} - \alpha)]U'(W_0 - \ell) + [1 - p + a(\frac{1}{2} - \alpha)]U'(W_0)} \quad (8)$$

We establish the following result.

**Proposition 7** For risk averters (i.e., $U'' < 0$), the marginal WTP for self-protection:

(i) Decreases (increases) with the degree of ambiguity aversion (ambiguity loving);

(ii) Decreases (increases) with the level of ambiguity for ambiguity averters (ambiguity lovers).

For risk lovers (i.e., $U'' > 0$), these effects are reversed. For risk-neutral decision-makers (i.e., $U'' = 0$), ambiguity is inconsequential for the marginal WTP for self-protection.

Our results for ambiguity lovers and risk lovers are novel contributions to the literature. Proposition 7(i) for ambiguity-averse decision-makers confirms and simplifies Proposition 3 in Alary et al. (2013), who use the smooth model with several states. They find that the effect of ambiguity aversion is a priori indeterminate. They can only sign the comparative static by imposing restrictions on the decision-maker's utility function and on the wealth level in the state that is being self-protected, none of which are required for our result. As with self-insurance, the study of ambiguity lovers displays a reversal of effects. Our results for greater ambiguity are new as well. The tractability of MP-capacities reveals how risk and ambiguity attitudes jointly determine the effect of greater ambiguity on the WTP for self-protection. Table 1 summarizes.

When comparing the results for self-insurance and self-protection, two observations deserve further discussion. First, for risk- and ambiguity-averse decision-makers, greater ambiguity aversion and greater ambiguity positively affect the WTP for self-insurance, but negatively affect the WTP for self-protection. The reason is that self-insurance and self-protection affect the decision-maker's CEU differently. Self-insurance trades off an increase in consumption in the loss state against lower consumption in both states. Ambiguity and ambiguity aversion increase the marginal value of higher loss-state wealth (numerator in Eq. (5)), but also increase the marginal cost of lower expected consumption (denominator in Eq. (5)). The first effect dominates because marginal utility is positive, thus explaining the positive net effect. Self-protection instead trades off a reduction in the probability of loss against lower consumption in either state. Ambiguity and ambiguity aversion have no effect on the marginal value of a lower loss probability (numerator in Eq. (8)) because uncertainty
is about the level of the loss probability. A reduction of this probability has the same marginal effect on CEU at whatever level it occurs. The only effect of ambiguity is an increase in the marginal cost (denominator in Eq. (8)), which explains the negative effect.

The second observation is the reversal of effects for self-protection between risk averters and risk lovers. This reversal does not occur for self-insurance. For risk lovers who are ambiguity-averse, greater ambiguity lowers the marginal cost of both self-insurance and self-protection because $U'(W_0 - \ell) < U'(W_0)$. So the denominators in Eq. (5) and (8) both decrease. In case of self-insurance, this effect aligns with the positive effect of greater ambiguity on the marginal benefit under ambiguity aversion, so nothing changes compared to the case of risk aversion. For self-protection, the effect on the marginal cost is the only effect at work so the sign flips. This reversal implies that the comparative statics of ambiguity on the WTP for self-insurance and on the WTP for self-protection are identical for risk lovers. Differences only arise for risk averters. The analytical tractability of MP-capacities helps us uncover the important role of risk attitude, which prior literature has remained silent about.

**Optimal demand.** Let $e \in [0, \bar{e}]$ denote the level of self-protection. The probability of loss $p(e)$ is decreasing with self-protection at a non-decreasing rate ($p' < 0$, $p'' \ge 0$). Self-protection involves an upfront cost $c(e)$, which is increasing and non-concave ($c' > 0$, $c'' \ge 0$). Wealth levels are $W_L = W_0 - \ell - c(e)$ in the loss-state and $W_N = W_0 - c(e)$ in the no-loss state. Setting $\bar{e} = c^{-1}(W_0 - \ell)$ ensures nonnegative consumption. The decision-maker solves:

$$ \max_{e \in [0, \bar{e}]} V(e) = [p(e) - a(\frac{1}{2} - \alpha)] U(W_L) + [1 - p(e) + a(\frac{1}{2} - \alpha)] U(W_N). $$

Assuming an interior maximizer $e^*$, the first-order condition is:

$$ -p'(e^*)[U(W_N^*) - U(W_L^*)] -c'(e^*)([p(e^*) - a(\frac{1}{2} - \alpha)]U'(W_L^*) + [1 - p(e^*) + a(\frac{1}{2} - \alpha)]U'(W_N^*)) = 0. $$

If the objective function is concave in $e$, the interior maximizer is unique and corresponds to the global maximizer of the decision-maker's objective function.<fnref n="12" /> We obtain the following result.

**Proposition 8** *The optimal level of self-protection:*

(i) Decreases (increases) with the degree of ambiguity aversion (ambiguity loving);

(ii) Decreases (increases) with the level of ambiguity for ambiguity averters (ambiguity lovers).

<footnote n="12">¹² Convexity of $p$ in $e$ and concavity of $U$ in $W$ are not strong enough for global concavity of $V$ in $e$ (see Jullien et al. 1999; Peter 2025). Log-convexity of $p$ and nonincreasing absolute risk aversion are jointly sufficient under SEU (see Fagart and Fluet 2013). In our framework, these conditions remain jointly sufficient under ambiguity aversion but may no longer be strong enough under ambiguity loving. If $V$ is globally concave in $e$, the comparative statics hold in the large. If $V$ is not globally concave in $e$, the results hold in the small but may not hold in the large.</footnote>
The proof uses the Implicit Function Rule. Ambiguity affects the marginal cost but not the marginal benefit of self-protection. For ambiguity averters, the marginal cost is higher compared to SEU. Uncertainty associated with the probability of loss puts more weight on the loss state, in which marginal utility is high. This explains why greater ambiguity raises the marginal cost for ambiguity averters but lowers it for ambiguity lovers.

Our results for ambiguity lovers are entirely new in the literature. Result (i) for ambiguity averters is contrary to Snow (2011), who finds a positive effect of ambiguity and greater ambiguity aversion on self-protection. Alary et al. (2013), on the other hand, find that greater ambiguity aversion may increase or decrease the demand for self-protection, depending on how the wealth level in the state where ambiguity is concentrated relates to precautionary equivalent wealth. We find a definitive negative effect, without any ancillary assumptions, which is once again owed to the tractability of MP-capacities.

Our model delivers an ambiguity-based explanation for underprevention (see Bail-
lon et al. 2022), because ambiguity raises the marginal cost of self-protection for
ambiguity averters. This mechanism is reminiscent of Eeckhoudt and Gollier (2005),
who find a negative effect of prudence on self-protection. Prudent decision-makers
may prefer to invest less in self-protection relative to risk-neutral agents, because this
raises consumption in the loss state, which mitigates downside risk. We can interpret
Proposition 8 similarly. Ambiguity aversion introduces a precautionary motive to cut
back on self-protection. This preserves funds in all states, particularly the loss state.

Endogenous ambiguity. To explain why Snow (2011) finds the opposite result, we now endogenize the level of ambiguity. Self-protection may not only affect the decision-maker’s subjective probability of loss but also how much ambiguity he perceives. Berger et al. (2016) distinguish between an increasing, constant, or decreasing degree of model uncertainty (see their Figure 1). Let the level of ambiguity be a continuously differentiable function of effort, $a = a(e)$ for $e \in [0, \bar{e}]$. As $e$ changes, both the subjective probability of loss and the level of ambiguity change. We require $a(e) < 2 \min\{p(e), 1-p(e)\}$ for all $e \in [0, \bar{e}]$ for consistency with Definition 3. The decision-maker’s objective is now:

$$
\max_{e \in [0, \bar{e}]} V(e) = [p(e) - a(e)(\frac{1}{2} - \alpha)] U(W_L) + [1 - p(e) + a(e)(\frac{1}{2} - \alpha)] U(W_N).
$$

An interior solution $e^*$ is characterized by the following first-order condition:

$$
\begin{align*}
& \left[-p'(e^*) + a'(e^*)(\frac{1}{2} - \alpha)\right] \cdot \left[U(W_N^*) - U(W_L^*)\right] \\
& -c'(e^*) \left(\left[p(e^*) - a(e^*)(\frac{1}{2} - \alpha)\right] U'(W_L^*)\right. \\
& \qquad \left. + \left[1 - p(e^*) + a(e^*)(\frac{1}{2} - \alpha)\right] U'(W_N^*)\right) = 0.
\end{align*}
$$

With an endogenous ambiguity level, both the marginal cost and the marginal benefit of self-protection depend on the decision-maker’s ambiguity attitude. If self-protection reduces the level of ambiguity (i.e., $a' < 0$), the marginal benefit is increasing (decreasing) in the degree of ambiguity aversion (ambiguity loving). This introduces a countervailing effect on optimal self-protection compared to a constant ambiguity
level. If self-protection increases ambiguity (i.e., $a' > 0$), the effects of changes in the degree of ambiguity attitude on the marginal benefit and on the marginal cost are aligned. We obtain the following characterization.

**Proposition 9** Under endogenous ambiguity, the optimal level of self-protection decreases (increases) with the degree of ambiguity aversion (ambiguity loving) if and only if

$$ -\frac{a'(e^*)}{a(e^*)} \leq -c'(e^*) \frac{U'(W_N^*) - U'(W_L^*)}{U(W_N^*) - U(W_L^*)}. \quad (9) $$

The right-hand side of condition (9) is related to the decision-maker's degree of risk aversion. Let $\tilde{W}$ be uniformly distributed on $[W_L^*, W_N^*]$; then,

$$ -\frac{U'(W_N^*) - U'(W_L^*)}{U(W_N^*) - U(W_L^*)} = -\frac{\mathbb{E}U''(\tilde{W})}{\mathbb{E}U'(\tilde{W})}. $$

If $U$ is negative exponential or quadratic, this expression simplifies to Arrow-Pratt risk aversion. If $U$ is risk vulnerable (see Gollier and Pratt 1996), Arrow-Pratt risk aversion at the median wealth level $(W_L^* + W_N^*)/2$ is a lower bound. The ratio $-a'(e^*)/a(e^*)$ is the decay rate of the ambiguity level. If ambiguity increases in self-protection, we are in the situation of Proposition 8(i) because the left-hand side of (9) is negative and the right-hand side positive. If self-protection decreases the level of ambiguity, both sides of (9) are positive so the inequality may hold or flip. If it flips, all results are reversed and self-protection increases with the degree of ambiguity aversion. We thus obtain Snow's result if self-protection decreases ambiguity fast enough, which means faster than an endogenous threshold related to the decision-maker's degree of risk aversion.<fnref n="13" />

A parametric example simplifies condition (9) further. Let $U$ have constant absolute risk aversion $A > 0$, that is, $U(W) = 1 - \exp(-AW)/A$. Let the ambiguity level be negative exponential with decay rate $\xi > 0$, that is, $a(e) = a \cdot \exp(-\xi e)$. Assume a linear cost function $c(e) = e$. In this case, self-protection decreases (increases) with ambiguity aversion if $\xi \leq (\geq) A$. MP-capacities put us in a position to provide such a simple and intuitive criterion, that easily reconciles conflicting results from the smooth model and is amenable to empirical testing.

## 3.4 The value of a statistical life

The value of a statistical life (VSL) is used to assess the economic value of reducing mortality risks. The U.S. Department of Transportation currently values a statistical life at $13.2 million, according to its latest 2023 memorandum.<fnref n="14" /> To determine a decision-maker's willingness to pay for a reduction in mortality risk, economists typically

<footnote n="13">¹³ Alary et al. (2013) discuss the result by Snow (2011) and say that the difference depends on whether effort has a positive or negative effect on the ambiguity surrounding the probability of loss. The tractability of MP-capacities allows us to be much more specific and provide an easily interpretable condition that characterizes the effect of ambiguity attitude on the demand for self-protection.</footnote>

<footnote n="14">¹⁴ See https://www.transportation.gov/.</footnote>
multiply the change in the probability of death by the decision-maker's marginal rate of substitution between wealth and mortality risk (Drèze 1962; Jones-Lee 1974; Weinstein et al. 1980). Empirical estimates have been derived in various contexts (see Viscusi 1993; Viscusi and Aldy 2003).

Let $U_A$ be utility in the alive state and $U_D$ utility in the deceased state. Typical assumptions are $U'_A > 0$ and $U''_A < 0$ for the alive state, and $U'_D \ge 0$ (so that $U'_D = 0$ is possible) and $U''_D \le 0$ for the deceased state. Utility function $U_D$ reflects preferences regarding bequests and consumption during the part of the period that is survived. We also assume $U_A(W) > U_D(W)$ and $U'_A(W) > U'_D(W)$ for all relevant wealth levels (see Eeckhoudt and Hammitt 2001). Let $v = (m, a, \alpha)$ be the decision-maker's MP-capacity. His CEU is given by:

$$V = [p - a(\frac{1}{2} - \alpha)]U_D(W_0) + [1 - p + a(\frac{1}{2} - \alpha)]U_A(W_0).$$

Here, $p$ represents the individual's subjective baseline probability of death. The VSL is defined as the marginal rate of substitution between wealth and mortality risk:

$$VSL = \frac{dW_0}{dp}\Big|_{V=const} = \frac{U_A(W_0) - U_D(W_0)}{[p - a(\frac{1}{2} - \alpha)]U'_D(W_0) + [1 - p + a(\frac{1}{2} - \alpha)]U'_A(W_0)}.$$

The expression closely parallels Eq. (8). Since $U'_A(W_0) > U'_D(W_0)$, we obtain the following result as a corollary of Proposition 7.

**Corollary 1** The VSL:

(i) Increases (decreases) with the degree of ambiguity aversion (ambiguity loving);

(ii) Increases (decreases) with the level of ambiguity for ambiguity averters (ambiguity lovers).

Result (i) for ambiguity averters extends Treich's (2010) finding for the smooth ambiguity model to CEU. As usual we find the reverse effect for ambiguity lovers, which is new in the literature. Result (ii) states that greater ambiguity raises VSL for ambiguity averters. Under the smooth ambiguity model, greater ambiguity may either increase or decrease VSL, and restrictions on the magnitude of relative ambiguity prudence are needed to obtain a clear result (see Bleichrodt et al. 2019). The tractability of MP-capacities allows us to lift these restrictions.

As with self-protection, the ambiguity level can be easily endogenized. Let $a(p)$ denote ambiguity as a function of mortality risk. After adjusting the objective function, we find:

$$VSL = \frac{dW_0}{dp}\Big|_{V=const} = \frac{[U_A(W_0) - U_D(W_0)] \cdot [1 - a'(p)(\frac{1}{2} - \alpha)]}{[p - a(p)(\frac{1}{2} - \alpha)]U'_D(W_0) + [1 - p + a(p)(\frac{1}{2} - \alpha)]U'_A(W_0)}.$$

Denote by $\kappa = (U'_A(W_0) - U'_D(W_0))/U'_A(W_0)$ the percentage decline in marginal utility from death. Straightforward calculations then show the following result.
**Proposition 10** *Under endogenous ambiguity, the VSL increases (decreases) with the degree of ambiguity aversion (ambiguity loving) if and only if*

$$ \frac{p(U'_{A}(W_0) - U'_{D}(W_0))}{pU'_{D}(W_0) + (1-p)U'_{A}(W_0)} = \frac{p\kappa}{1-p\kappa} \geq -p \frac{a'(p)}{a(p)}. \quad (10) $$

If ambiguity increases with mortality risk (i.e., $a' > 0$), reductions in mortality reduce the perceived level of ambiguity. In this case, the right-hand side of criterion (10) is negative, leading to the same result as Corollary 1(i). If ambiguity decreases with mortality risk instead (i.e., $a' < 0$), condition (10) may still hold or flip. The left-hand side of criterion (10) is the mortality-elasticity of expected marginal utility, which we compare with the mortality-elasticity of the ambiguity level on the right-hand side. If ambiguity decreases faster than expected marginal utility as mortality risk declines, higher ambiguity aversion lowers the VSL.

With endogenous ambiguity, mortality reductions have two effects. First, the VSL decreases because the expected marginal utility-cost of funds increases. Pratt and Zeckhauser (1996) refer to this as the “dead-anyway” effect: when mortality is high, funds are more likely to be drawn from the state with low marginal utility, which reduces the utility cost of spending on risk reduction. Second, lower mortality may reduce perceived ambiguity, which increases the expected marginal utility-cost of funds for ambiguity-averse decision-makers. The elasticities in condition (10) compare the magnitudes of these effects.

A parametric example illustrates the usefulness of criterion (10). Let $U_D(W) = (1 - \kappa)U_A(W)$ for $\kappa \in (0, 1]$ and let $a(p) = a \cdot \exp(-\xi p)$ for a $\xi > 0$. The VSL is then increasing (decreasing) in ambiguity aversion if $\kappa/(1 - p\kappa) \geq (\leq) \xi$. The condition for a positive effect is more likely satisfied with higher baseline mortality (i.e., higher $p$), greater utility loss from death (i.e., higher $\kappa$), and slower ambiguity decay (i.e., lower $\xi$). Interestingly, as $\kappa$ approaches zero, the left-hand side approaches zero, and condition (10) flips. So when utility differences are small between the alive and the deceased state, ambiguity aversion always decreases the VSL. This insight aligns with Proposition 7(i) on self-protection.

How do these findings compare to the smooth model? Let $\varphi$ be the ambiguity function in the smooth ambiguity model. Bleichrodt et al. (2019) show that greater ambiguity increases the VSL for ambiguity averters when $\varphi''' = 0$ (their Result 2(ii)); if $\varphi''' > 0$, they find a sufficient condition (their Result 2(i)), while the effect remains indeterminate for $\varphi''' < 0$ (their Result 2(iii)). Greater ambiguity lowers the VSL for ambiguity lovers when $\varphi''' = 0$ (their Result 5(ii)); if $\varphi''' < 0$, they find a sufficient condition (their Result 5(i)), while the effect remains indeterminate for $\varphi''' > 0$ (their Result 5(iii)). This discussion makes it clear that MP-capacities simplify the analysis considerably. Their tractability renders a single necessary and sufficient condition that makes testable predictions and applies to ambiguity averters and ambiguity lovers alike.
## 3.5 Precautionary saving

When does income uncertainty generate a precautionary demand for saving? This classic question in economic theory dates back to Leland (1968) and Sandmo (1970), but it was not until Kimball (1990) that the theory of precautionary saving gained momentum. In the two-period consumption-saving model with time-separable utility, income risk leads to precautionary savings if and only if marginal utility is convex—a condition known as prudence.

Here, we use a nonseparable utility and treat separable utility as a special case. The decision-maker has a certain amount of income $W_1$ in the first period and uncertain income of $W_2$ in the second period. He can save an amount $s$ at a risk-free gross interest rate of $R \ge 0$. We represent income uncertainty with a random variable $\varepsilon$ supported on $[\varepsilon_-, \varepsilon_+]$. We assume without loss of generality that its realizations are ordered as $\varepsilon_1 < \dots < \varepsilon_n$. Let $U$ denote the decision-maker's utility function over timed consumption pairs, and let $v = (m, a, \alpha)$ be his MP-capacity. The objective function is:

$$V(s; a, \alpha) = \sum_{i=1}^{n} U(W_1 - s, W_2 + sR + \varepsilon_i) \cdot [m_i + (\frac{1}{2} - \alpha) [a(\omega_i^\cup) - a(\omega_{i+1}^\cup)]]$$

If $U$ is concave in the bivariate sense, then $V$ is concave in $s$ since $(W_1 - s, W_2 + sR + \varepsilon)$ is linear in $s$. To ensure nonnegative consumption, we restrict $s$ to $[-(W_2 + \varepsilon_1)/R, W_1]$. We assume an interior solution $s^*$, characterized by the following first-order condition:

$$V_s(s^*; a, \alpha) = -\sum_{i=1}^{n} U_1(W_1 - s^*, W_2 + s^*R + \varepsilon_i) \cdot [m_i + (\frac{1}{2} - \alpha) [a(\omega_i^\cup) - a(\omega_{i+1}^\cup)]] + R \sum_{i=1}^{n} U_2(W_1 - s^*, W_2 + s^*R + \varepsilon_i) \cdot [m_i + (\frac{1}{2} - \alpha) [a(\omega_i^\cup) - a(\omega_{i+1}^\cup)]] = 0.$$

Subscripts 1 and 2 on the utility function denote partial derivatives with respect to consumption in the first and second period. We define the following auxiliary function:

$$H(s, \varepsilon) = -U_1(W_1 - s, W_2 + sR + \varepsilon) + R U_2(W_1 - s, W_2 + sR + \varepsilon),$$

which is the first-order expression for an income shock of $\varepsilon$. It allows us to rewrite the first-order condition as follows:

$$V_s(s^*; a, \alpha) = \sum_{i=1}^{n} m_i H(s^*, \varepsilon_i) + (\frac{1}{2} - \alpha) \sum_{i=1}^{n-1} a(\omega_{i+1}^\cup) \cdot [H(s^*, \varepsilon_{i+1}) - H(s^*, \varepsilon_i)] = 0.$$

To determine how ambiguity and ambiguity aversion influence saving, we analyze the sign of the bracketed terms. If $H_\varepsilon(s^*, \varepsilon) < 0$, these terms are negative. A sufficient
condition for this is:

$$ -U_{12}(W_1 - s^*, W_2 + s^*R + \varepsilon) + RU_{22}(W_1 - s^*, W_2 + s^*R + \varepsilon) < 0. \quad (11) $$

This inequality holds under correlation-loving or correlation-neutrality (Epstein and Tanny 1980). The assumption $U_{12} \ge 0$ suffices because concavity of $U$ implies $U_{22} < 0$. Sandmo (1970) demonstrates that condition (11) is equivalent to first-period consumption being increasing with first-period income. In other words, this condition means that first-period consumption is a normal good.<fnref n="15" /> The following result holds.

**Proposition 11** If first-period consumption is a normal good, the demand for saving:

(i) Increases (decreases) with the degree of ambiguity aversion (ambiguity loving);

(ii) Increases (decreases) with the level of ambiguity for ambiguity averters (ambiguity lovers).

If first-period consumption is inferior, all effects are reversed.

When consumption is a normal good, ambiguity and ambiguity aversion naturally generate a precautionary demand for saving. The intuition follows from observational equivalence as in the standard portfolio problem. We can similarly define the distorted probabilities $\hat{m}_i$ and the associated cumulative probabilities $\hat{M}_i$. The behavior of an ambiguity-averse CEU maximizer with MP-capacity $\nu = (m, a, \alpha)$ is then indistinguishable from that of an SEU maximizer who has deteriorated his subjective belief in the sense of FSD relative to $m$. For an ambiguity lover, the belief distortion flips to an FSD improvement. Inequality (11) then allows us to sign the effects of FSD shifts in the distribution of future income on saving behavior.

The case of separable utility arises when $U_{12} = 0$. Condition (11) then follows directly from the concavity of $U$. Using Klibanoff et al.'s (2009) recursive smooth ambiguity model, Berger (2014) shows that a convex marginal ambiguity function is not sufficient for precautionary saving. Stronger conditions, such as nonincreasing absolute ambiguity aversion, are needed. Wang and Li (2020) use Hayashi and Miao's (2011) time-separable specification. They impose even more restrictions to obtain precautionary saving. None of these constraints are necessary with MP-capacities, which highlights their analytical tractability once again. Using the smooth model with nonseparable utility, Peter (2019) shows that ambiguity aversion alone can lead to precautionary saving when priors are ordered in the FSD sense. To sign the effect of greater ambiguity, he additionally imposes the typical bound of two on relative ambiguity prudence. Readers are by now familiar with the fact that MP-capacities liberate us from all of these ancillary assumptions.

According to Proposition 11 ambiguity may contribute to our understanding of cross-country differences in saving rates around the globe. The OECD reports saving

<footnote n="15">¹⁵ When first-period income increases, there is a positive direct effect on consumption in the first period and a potentially conflicting indirect effect because the level of saving may increase as well. Inequality (11) ensures that the direct effect dominates when the demand for saving is normal. The second-order condition implies that no more than one consumption level can be inferior with respect to the corresponding income level.</footnote>
rates of 6.0% in the U.S., 4.1% in the European Union and 0.7% in Japan.<fnref n="16" /> Our last result suggests that differential perception of ambiguity may be a determinant of differences in saving rates across countries.

## 4 Conclusion

In this paper, we introduced mean-preserving capacities for Choquet expected utility. A capacity is mean-preserving when the value of an event lies in an interval that is centered around the subjective probability of that event. The size of the interval measures the decision-maker's perceived level of ambiguity. We showed that MP-capacities rest on the complementary independence axiom. Under this axiom, a decision-maker assesses an act according to its SEU baseline and its utility variability around this baseline (Siniscalchi 2009). An equivalent characterization featured the central symmetry of the (dual) capacity's core. MP-capacities solve an identification problem for alpha-MEU: they are precisely those JP-capacities for which $α = 1/2$ characterizes ambiguity neutrality. MP-capacities extend SEU in a different direction from neo-additive capacities.

In the second part of the paper, we discussed a series of economic applications including portfolio choice, self-insurance and self-protection, the value of a statistical life, and precautionary saving. We made predictions for both ambiguity averters and ambiguity lovers. A lot of the existing applied work on ambiguity disregards ambiguity lovers, probably for technical reasons. MP-capacities simplify the analysis of greater ambiguity under CEU considerably. In the smooth ambiguity model, one needs to combine stochastic ordering assumptions on the priors with technical restrictions on the decision-maker's ambiguity function to control countervailing Giffen-type effects. None of this is needed for MP-capacities, and predictions are simple and intuitive.<fnref n="17" />

In our view, the main value of MP-capacities lies in their tractability. At the same time, MP-capacities are well-grounded in theory and sit at the intersection of popular decision-making models under ambiguity. We hope the simplicity of MP-capacities will propel their use in more involved applied settings and help revive the aging CEU model. We anticipate that MP-capacities will allow economists to explore the implications of ambiguity in setting that are not sufficiently tractable under existing approaches. We look forward to seeing the fruits of such endeavors.

<footnote n="16">¹⁶ They define net household savings as household disposable income minus household consumption expenditures plus the change in net equity of household pension fund holdings (see https://data.oecd.org/)</footnote>

<footnote n="17">¹⁷ One advantage of the smooth model is that the components of preferences can be uniquely identified from asset demand data, provided the market structure satisfies certain conditions, see Polemarchakis et al. (2025).</footnote>
# Appendix A Proofs

## A.1 Proof of Proposition 1

(i) For $E, E' \in \Sigma$, we obtain

$$m(E) + m(E') = m(E \cup E') + m(E \cap E')$$

because $m$ is a subjective probability. Due to Property (c) in Definition 3, we have

$$a(E) + a(E') \geq a(E \cup E') + a(E \cap E').$$

from the submodularity of $a$. The inequality

$$v(E) + v(E') \leq v(E \cup E') + v(E \cap E')$$

is then equivalent to

$$m(E) + (\frac{1}{2} - \alpha)a(E) + m(E') + (\frac{1}{2} - \alpha)a(E') \\ \leq m(E \cup E') + (\frac{1}{2} - \alpha)a(E \cup E') + m(E \cap E') + (\frac{1}{2} - \alpha)a(E \cap E'),$$

which is in turn equivalent to

$$(\frac{1}{2} - \alpha) \cdot [a(E) + a(E') - a(E \cup E') - a(E \cap E')] \leq 0.$$

Capacity $v$ is thus convex for $\alpha \geq 1/2$. It is concave for $\alpha \leq 1/2$ by reversing the last three inequalities. It is additive if it is both convex and concave, which holds for $\alpha = 1/2$.

(ii) The fact that an increase in $\alpha$ increases ambiguity aversion is directly evident from Eq. (3) because $a$ is nonnegative.

(iii) Let $v_1$ and $v_2$ be two MP-capacities that only differ by the level of ambiguity such that $a_2(E) \geq a_1(E)$ for all $E \in \Sigma$. It follows from Eq. (3) that $v_2 \leq v_1$ for $\alpha \geq 1/2$ and $v_2 \geq v_1$ for $\alpha \leq 1/2$. The Choquet integral is monotonic with respect to the capacity (see Denneberg 1994; Wang and Klir 1997). Therefore, we obtain

$$V_2(f) = \int_{\Omega} U(f) \, dv_2 \leq (\geq) \int_{\Omega} U(f) \, dv_1 = V_1(f) \quad \text{for } \alpha \geq (\leq) 1/2.$$

## A.2 Proof of Proposition 2

(i) $\implies$ (ii) : If $v$ is an MP-capacity, then $\bar{v}(E) = m(E) - (\frac{1}{2} - \alpha)a(E)$. It follows that $(v + \bar{v})/2 = m$ so $(v + \bar{v})/2$ is a subjective probability. According to Siniscalchi's Table II, preferences with a CEU representation satisfy complementary independence in this case.
(ii) $\implies$ (i): Siniscalchi (2009) shows in Section 4.6 that, if a preference relation with a CEU representation satisfies complementary independence, there is a subjective probability $m$ such that $1 - \nu(E^c) = 2m(E) - \nu(E)$ for all $E \in \Sigma$. So $(\nu + \bar{\nu})/2$ defines a subjective probability. Set $\beta = \nu - m$; if $\nu$ is convex, then $\nu \le \bar{\nu}$ so that $\nu \le m \le \bar{\nu}$ and therefore $\beta \le 0$. By setting $\alpha = 1$ and $\alpha = -2\beta$, we can rewrite $\nu$ in the desired form,

$$ \nu = m + (\nu - m) = m + \beta = m + (\frac{1}{2} - \alpha)\alpha. $$

We now show that set function $\alpha$ satisfies Properties (a) - (c) in Definition 3. We obtain $\alpha(\Omega) = -2\beta(\Omega) = -2\nu(\Omega) + 2m(\Omega) = 0$ and similarly $\alpha(\emptyset) = 0$ so $\alpha$ is normalized. For $E \in \Sigma$,

$$ \beta(E) = \nu(E) - m(E) = \nu(E) - \frac{1}{2}(\nu(E) + \bar{\nu}(E)) = \frac{1}{2}(\nu(E) + \nu(E^c) - 1) $$

and

$$ \beta(E^c) = \nu(E^c) - m(E^c) = \nu(E^c) - \frac{1}{2}(\nu(E^c) + \bar{\nu}(E^c)) = \frac{1}{2}(\nu(E) + \nu(E^c) - 1). $$

Therefore, $\beta(E) = \beta(E^c)$ for all $E \in \Sigma$ and $\alpha$ is symmetric. Set function $\alpha$ has bounded changes because capacity $\nu$ is monotonic with respect to set inclusion. It is submodular because $\nu$ is convex and $m$ is a subjective probability.

If $\nu$ is concave instead, then $\bar{\nu} \le \nu$ so that $\bar{\nu} \le m \le \nu$ and therefore $\beta \ge 0$. In this case, set $\alpha = 0$ and $\alpha = 2\beta$ to obtain $\nu = m + (\frac{1}{2} - \alpha)\alpha$. As before set function $\alpha$ satisfies the required properties in Definition 3.

(i) $\implies$ (iii): Let $\nu = (m, \alpha, \alpha)$ be an MP-capacity according to Definition 3. If $\nu$ is convex, we have $\alpha \ge 1/2$. Let $p \in \Delta(\Sigma)$ be an element in the core of $\nu$ so that $\nu \le p$. For $E \in \Sigma$ it follows that $\bar{\nu}(E) = 1 - \nu(E^c) \ge 1 - p(E^c) = p(E)$. But $\bar{\nu} \ge p$ is equivalent to $m - (\frac{1}{2} - \alpha)\alpha \ge p$, which is equivalent to $2m - p \ge m + (\frac{1}{2} - \alpha)\alpha = \nu$. So for any $p \in \Delta(\Sigma)$ we obtain $2m - p \in \Delta(\Sigma)$. The core of $\nu$ is thus centrally symmetric with center $m$. If $\nu$ is a concave MP-capacity, then $\bar{\nu}$ is a convex MP-capacity, and $P_{\bar{\nu}}$ is centrally symmetric with the same argument.

(iii) $\implies$ (i): Let the core of convex capacity $\nu$ be centrally symmetric with center $m_c \in P_{\bar{\nu}}$. Define $m = (\nu + \bar{\nu})/2$ and let $p \in \Delta(\Sigma)$ be an element in the core of $\nu$ so that $\nu \le p$. As before this implies $\bar{\nu} \ge p$, which is equivalent to $\nu + \bar{\nu} - p \ge \nu$ or $2m - p \ge \nu$. So $(\nu + \bar{\nu})/2$ is the center of $P_{\bar{\nu}}$, and therefore coincides with subjective probability $m_c$. But if $(\nu + \bar{\nu})/2$ is a subjective probability, we can proceed as in the only-if part above (i.e., $(ii) \implies (i)$). Therefore, $\nu$ is an MP-capacity in the sense of Definition 3. If $\nu$ is concave, the argument is analogous because the dual capacity of $\bar{\nu}$ is $\nu$ itself.

## Appendix B MP-capacities are Choquet capacities

Let $\nu = (m, \alpha, \alpha)$ be an MP-capacity with the properties specified in Definition 3. Then $\nu$ is normalized because $\alpha(\Omega) = \alpha(\emptyset) = 0$ implies $\nu(\Omega) = m(\Omega) = 1$ and
$v(\emptyset) = m(\emptyset) = 0$. For monotonicity with respect to set inclusion, let $E, E' \in \Sigma_{nt}$ with $E \subseteq E'$.<fnref n="18" /> We obtain

$$v(E') - v(E) = m(E') - m(E) + \left(\frac{1}{2} - \alpha\right) (a(E') - a(E)).$$

Property (b) ensures that $|a(E') - a(E)| \le 2(m(E') - m(E))$ for all $E, E' \in \Sigma$. If $\alpha < 1/2$ and $a(E') - a(E) \ge 0$, then $v(E') - v(E) \ge m(E') - m(E) \ge 0$. If $\alpha < 1/2$ and $a(E') - a(E) < 0$, then $v(E') - v(E) \ge 2\alpha(m(E') - m(E)) \ge 0$. If $\alpha > 1/2$ and $a(E') - a(E) < 0$, then $v(E') - v(E) \ge m(E') - m(E) \ge 0$. If $\alpha > 1/2$ and $a(E') - a(E) \ge 0$, then $v(E') - v(E) \ge 2(1 - \alpha)(m(E') - m(E)) \ge 0$. The desired inequality $v(E') \ge v(E)$ always holds. Therefore, $v$ is normalized and monotonic with respect to set inclusion, and hence a Choquet capacity.

The strict inequality in Property (b) for $E = \emptyset$ or $E' = \Omega$ guarantees that the entire state space is the only universal set and that the empty set is the only null set for MP-capacity $v$.

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