数理解析研究所講究録
第 1864 巻 2013 年 61-70

61

The Dual Theory of the Smooth Ambiguity Model*
Hideki Iwaki

Faculty of Business Administration,
Kyoto Sangyo University
Yusuke Osaki
Faculty of Economics,
Osaka Sangyo University

Abstract
This paper studies the dual” theory of the smooth ambiguity model introduced by Klibanoff, Marinacci and Mukerji (2005). Unlike the original model, we characterize the attitude toward ambiguity
captured by second-order beliefs in the dual model. First, we give a set of axioms to derive a dual
representation of the smooth ambiguity model. Second, we present a characterization of ambiguity
aversion. Last, as an application to our dual theory to a standard portfolio problem, we conduct comparative static predictions which give sufficient conditions to guarantee that an increase in smoothambiguity aversion decreases the optimal portfolio.

Keywords: Ambiguity, Ambiguity aversion, Comparative statics, Smooth ambiguity model
JEL Classification D800. D810

1Introduction

Many experiments provide clear evidence that the expected utility model cannot capture actual behavior
under risk and uncertainty. One of the most famous classical experiments was conducted by Ellsberg(1961). His experiment denied that probabilistic beliefs are not used for decision making under risk
and uncertainty. Many alternatives, including Gilboa and Schmeidler (1989), Schmeidler (1989) and soon, have been proposed to explain the choices in Ellsberg’s experiment. One explanation of the choicesthat is consistent with the Ellsberg experiment is based on second-order beliefs, which are probabilistic
beliefs over probabilistic beliefs. Klibanoff, Marinacci and Mukerji (2005) gave a preference functional
based on this idea, named smooth ambiguity aversion. The smooth ambiguity model has an advantage
in tractability, since it is easier to apply to results in the expected utility model under circumstances with
ambiguity. Gollier (2011) presented the systematic comparative statics technique to measure the effectsof smooth ambiguity aversion. This paper develops a “dual” theory of the smooth ambiguity model,
representation, comparison and comparative statics. We note that the term dual” apparently comes from
Yaari (1986, 1987).
The dual extension of the original smooth ambiguity model has not only theoretical but also behavioral contributions. The smooth ambiguity model represents the double expected utility form with respect
to first-order belief (risk) and second-order belief (uncertainty). Similarly, the dual model of the smooth*This is such a shorter version of the original paper with the same title that proofs of all of the propositions are abbreviated.
This paper is partly supported from the Grant-in-Aid for Scientific Research (c), for Young Scientists (B) and the 2010 Zengin
Foundation for Studies on Economics and Finance. Of course, all remaining errors are ours.

62ambiguity model also shares the analog form of the expected utility model. This basically means that experimental methods under risk can be applied to ambiguity. The cumulative prospect theory of Tversky
and Kahneman (1992) combines loss aversion with rank dependent probability. Since then, measuning
the rank dependent probability experimentally has been well developed. We note that Yaari’s dual theory
can also be seen as rank dependent probability with linear utility, so our development of the dual theory of
the smooth ambiguity model is also useful for measuring the attitude toward ambiguity experimentally.
Smooth ambiguity aversion is motivated by the descriptive viewpoint, so that the experimental application of the dual model has an essential meaning. systematic method of comparative statics is anothercontribution of this paper. By the similarity in form between the expected utility model and the smoothambiguity model, it is natural to apply comparative statics results from the expected utility model to thesmooth ambiguity model. Gollier (2011) developed a comparative statics method by transforming thedecision problem with ambiguity to that without ambiguity. The transformation is accomplished using
an artificial probability weighted by smooth ambiguity aversion. Unlike Gollier (2011), our method uses
second-order belief directly. We note that this tool gives another proof of Gollier’s result. We conduct
a comparative statics analysis under ambiguity applying a useful method for the expected utility modelfrom Jewitt (1987) and Athey (2002). As in the expected utility model, this method also becomes to bea powerful tool for comparative statics under ambiguity. The dual representation of smooth ambiguity
aversion also has merit from a theoretical viewpoint. Our results insist that smooth ambiguity aversion iseasier to analyze under the original smooth ambiguity model in some situations, but that it is much easier
to analyze under the dual model in other situations.
The organization of this paper is as follows. The next section introduces the notation and the setting. In Section 3, we derive the dual representation of the smooth ambiguity model. In Section 4, wecharacterize the notion of smooth ambiguity aversion and more smooth ambiguity aversion in our dual
theory. In Section 5, we display the effect of smooth ambiguity aversion on the portfolio choice throughcomparative static analysis.
$A$

2 Preliminaries
We begin with the setting and notation for our model. The setting is essentially constructed by combining
Yaari (1987) and Klibanoff, Marinacci and Mukerji (2005), henceforth KMM. Thus, we follow the same
notation as KMM. The state space $S=\Omega\cross(0,1]$ consists of the separable metric space, , and $(0,1].$. Let $f$ :be the Borel -algebra of $(0,1]$ andLet be the Borel -algebra ofpreference over the set ofis the set of consequences. We denote by
be a Savage act, where
Savage acts and by the set of all bounded -measurable Savage acts.
, we can assume thatwithout loss of generality. Thus we assume that
Since$\Omega$$\mathscr{A}$$\Omega,$$\sigma$$\Sigma=\mathscr{A}\otimes \mathscr{B}_{1}$$\sigma$$\mathscr{B}_{1}$$Sarrow \mathscr{C}$$\succeq a$$\mathscr{C}\subset \mathbb{R}$$\Sigma$$\mathscr{F}$$\mathscr{C}=$$\mathscr{C}\subseteq[0,1]$$f\in \mathscr{F}$$[0,1].$

An act$1\in \mathscr{F}$is called a lottery if$l(\omega_{1},r)=l(\omega_{2},r)$and is also Riemann integrable. Let$\mathscr{L}$$\forall$(1)の 1, $ob\in\Omega,$ $r\in(0,1],$be the set of all such lotteries. For$f\in \mathscr{L}$and $r\in(O, 1]$ , we define$f(r)=f(\omega,r) \forall\omega\in\Omega.$

We assume that the following assumption that is derived by Grandmont (1972) holds.

Assumption 1 (Expected Utility on Lotteries) There exists a uniqueincreasing, and normalized so that $u(O)=0$ and $u(1)=1$ such that$u:\mathscr{C}arrow \mathscr{C}$$f \succeq g\Leftrightarrow\int_{(0,1]}u(f(r))dr\geq\int_{(0,1]}u(g(r))dr \forall f,g\in \mathscr{L}.$, continuous, strictly

63
Let $\pi:\Sigmaarrow[0,1]$ be a probability measure satisfying$\pi(A\cross B)=\pi(A\cross(0,1])\lambda(B) \forall A\in \mathscr{A}, B\in \mathscr{B}_{1}$

where
Let

$\lambda$$\mathscr{C}(S)$:$\mathfrak{F}$(2)

is the Lebesgue measure. Let denote the set of all such probability measures.
denote the set of all continuous and bounded real-valued functions on$\mathscr{B}_{1}arrow[0,1]$$\Delta$$S.$

Definition 1

Let

,

$A$

second-order act is any bounded

be the set of all second-order acts and$\sigma(\Delta)$$\succeq^{2}$-measurablefunction$f:\Deltaarrow \mathscr{C}.$be the preference ordering on$\mathfrak{F}.$

Assumption2 (Subjective Expected Utmty on Second Order Acts) There exists a probability measure $\mu:\sigma(\Delta)arrow[0,1];0<\mu(J)<1\forall J\in\sigma(\Delta)$ , and a continuous, snictly increasingfunction
such that$v:\mathscr{C}arrow \mathbb{R}$$f\succeq^{2}\mathfrak{g}\Leftrightarrow\int_{\Delta}v(f(\pi))d\mu\geq\int_{\Delta}v(\mathfrak{g}(\pi))d\mu \forall f,\mathfrak{g}\in \mathfrak{F}.$As to the necessary and sufficient conditions such that the above assumption holds, we can refer to, e,g.,
Theorem 10.3 of Fishburn (1982).
For each$f\in \mathfrak{F},$$G_{f}(t):=\mu(f(\pi)>t) , t\in[O, 1].$

Axiom 1 (Neutrality) For each$\Gamma=$$f,\mathfrak{g}\in \mathfrak{F}$,if$G_{f}=G_{g}$, then$f\sim^{2}\mathfrak{g}.${ $G:[0,1]arrow[0,1]|G$ is nonincreasing, right-continuous and $G(1)=0$ }.

From Axiom 1, the preference relation$\succeq^{2}$on$\Gamma$

is consmlcted by$G=G_{f}\succeq^{2}H=G_{\mathfrak{g}}\Leftrightarrow f\succeq^{2}\mathfrak{g},$for $G,H\in\Gamma$ andFor each$f,\mathfrak{g}\in \mathfrak{F}.$$G\in\Gamma$

and each $t\in[O, 1]$ , let$\hat{G}(t)$be defined by$\hat{G}(t)=\{x|G(t)\leq x\leq G(t-)\}.$We define the inverse of$G$by$G^{-1}(p)= \min\{t|p\in\hat{G}(t)\}.$

We note thatand that $(G^{-1})^{-1}=G.$
We define a mixture of functions contained in . For each $G,H\in\Gamma$ and $\alpha\in[0,1]$$G^{-1}\in\Gamma$$\Gamma$$\alpha G$

ffl $(1-\alpha)H:=(\alpha G^{-1}+(1-\alpha)H^{-1})^{-1}$(3)

Axiom 2 (Dual Independence) If $G,G’,H\in\Gamma$ and $\alpha\in[0,1]$ , then$G\succeq^{2}G’\Rightarrow\alpha G$

ffl $(1-\alpha)H\succeq^{2}\alpha G’$

ee$(1-\alpha)H.$

We have the following lemma.

Lemmasatisfies Assumption2 and Axioms 1-2 if and only if there exists a continuous strictlyincreasingfunction $\varphi:[0,1]arrow[0,1]$ such that$1\succeq^{2}$$f\succeq^{2}\mathfrak{g}\Leftrightarrow\int_{(0,1]}\varphi(G_{f}(t))dJ\geq\int_{(0,1]}\varphi(G_{\mathfrak{g}}(t))dJ \forall f,\mathfrak{g}\in \mathfrak{F}.$

64
Let

:

$\pi_{f}$$\mathscr{B}_{1}arrow[0,1]$

be defined by$\pi_{f}(B)=\pi(f^{-1}(B)) \forall B\in \mathscr{B}_{1}.$Lemma 2 For all$f\in \mathscr{F}$and$\pi\in\Delta$, there exists a non-decreasing lottery act$l_{f}(\pi)\in \mathscr{L}$such that$\lambda(l_{f}(\pi)\in B)=\pi_{f}(B) \forall B\in \mathscr{B}_{1}.$Let be the constant act for.
such that$\delta_{X}$$x\in \mathscr{C}$. Let$c_{f}(\pi)$denote the certainty equivalent of the lottery act$l_{f}(\pi)$$\delta_{c_{f}(\pi)}\sim l_{f}(\pi)$Demtion $2Gi\nu enf\in \mathscr{F},$$f^{2}\in \mathfrak{F}$is a second-order act associated with $f$ if$f^{2}(\pi)=u(c_{f}(\pi)) \forall\pi\in\Delta.$

Assumption 3 (Consistency with preferences over associated second-order acts) Given$f,g\in \mathscr{F}$and$f^{2},g^{2}\in \mathfrak{F},$$f\succeq g\Leftrightarrow f^{2}\succeq^{2}g^{2}.$3 Representation Theory
Theorem 1 presents the dual theory of the smooth ambiguity model.

Theorem 1 Given Assumptions 1 and 2 and Axioms $1- 5,$$\succeq is$represented by$V(f)= \int_{0}^{1}\varphi[G_{u^{\pi}(f)}(t)]dt$

,

(4)

where
$u^{\pi}(f):\Piarrow \mathscr{C}$If we define$F_{u^{\pi}(f)}^{\varphi}(t)=1-\varphi[G_{u^{\pi}(f)}(t)]$;$V:\mathscr{F}arrow \mathbb{R},$$u^{\pi}(f)= \int_{S}u(f(s))d\pi(s)$.

, then integration by parts allows us to rewnite (4) as$V(f)= \int_{0}^{1}tdF_{u^{\pi}(f)}^{\varphi}(t)$.

(5), we can consider a probability measure
is a cumulative distribution function of
Since. Hence we obtain the following Corollary.
over induced by$u^{\pi}(f)$$F_{u^{\pi}(f)}^{\varphi}$$\Delta$$F_{u^{\pi}(f)}^{\varphi}$denote the expectation under the probability measure
Corollary 1 Let
is represented bytions and axioms as Theorem 1, thefunctional :$\mu^{\varphi}$$\mathbb{E}_{\mu}^{\varphi}$$V$$\mathbb{E}^{\varphi}$. Under the same assump-$\mathscr{F}arrow \mathbb{R}$$V(f)= \mathbb{E}_{\mu}^{\varphi}[\int u(f(s))d\pi(s)]=\mathbb{E}^{\varphi}[u\circ f]$where$\mu^{\varphi}$,denotes the expectation under the probability measure defined by$\mathbb{E}_{\mu}^{\varphi}[\mathbb{E}_{\pi}[1_{A\cross B}]] \forall A\in \mathscr{A}, B\in \mathscr{B}_{1}.$

(6)

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4 Ambiguity AttitudeLet

be the induced distribution defined by$\mu_{f}$$\mu_{f}(u(B))=\mu((f^{2})^{-1}(B)) \forall B\in \mathscr{B}_{1}.$We denote by the support of . In addition to the earlier assumptions, we impose the following
assumption in order to discuss ambiguity attitude of decision makers.
$\Pi$

$\mu$

Assumption 4 Fix a family ofpreference relationships
the restriction of
to lottery acts and the risk preferences derived$\{\succ\Pi, \succ_{\Pi}^{2}\}_{\Pi\subseteq\Delta}$$\succ\Pi$for a given decision maker. Bothremain the same for every$from\succ_{\Pi}^{2}$$\Pi\subseteq\Delta.$

This assumption guarantees that the same may be used to represent eachthe support of her subjective belief varies.
According to KMM, smooth ambiguity aversion is defined as follows.
$\varphi$$\succ\Pi$for a decision maker as

Definition 3 $ADM$ (Decision Maker) displays smooth ambiguity aversion at $(f,\Pi)$ if$\delta_{u^{-1}e(\mu_{f})}\succ\Pi f,$where$e( \mu_{f})=\int_{(0,1]}xd\mu_{f},$$ADM$ displays smooth ambiguity aversion if she displays smooth ambiguity
and has a support
aversion at $(f,\Pi)$ for all
and all supports$\Pi.$$\mu$$f\in \mathscr{F}$$\Pi\subseteq\Delta.$

Proposition 1 Under Assumptions 1-3, the following conditions are equivalent.

$\cdot$$(a)$

Thefunction$(b)$

The $DM$ displays smooth ambiguity

Let

$A$

and

each support$B$

is convex.
$a\nu$

ersion.

be two DMs whose families of preferences share the same probability measure. According to KMM, we define the statement“ is more ambiguity averse than$A$

is more ambiguity averse than

$A$

$B$

for

$\mu\Pi$$\Pi$

follows.
Definition 4$\varphi:[0,1]arrow[0,1]$

$B$

”

as

if

$f\succeq_{\Pi}^{A}l\Rightarrow f\succeq_{\Pi}^{B}l \forall f\in \mathscr{F}, l\in \mathscr{L}, \Pi\in\Delta$.

(7)

Theorem 2 $LetA$ and be two $DMs$ whose families ofpreferences share the same probability measurefor each support . Then is more ambiguity averse than if and only if they share the same (vonNeumman-Morgenstern) utility function and$B$$\mu\Pi$

$\Pi$

$A$

$B$

$u$

$\varphi_{A}=h\circ\varphi_{B}$for some strictly increasing, continuous and convex $h:[0,1]arrow[0,1].$

66
5 Application to the Portfolio Problem and Comparative Statics
5.1 Preliminaries
In this section, we apply the dual theory of the smooth ambiguity model to the simple portfolio problem
consisting of one risk-free asset and one risky asset. The risk-free rate of remm is nonnalized to zero
without loss of generality. The risky asset remm is ambiguous in the sense that its remm is indexed byis represented by a decumulative function. To avoid technical difficulties,
is denoted bytakes a value in a bounded interval. Given an endowment wealth , the investor
we assume thatchooses the share of the risky asset. Then she receives the (conditional) expected utility. Here is the (von Neumman-Morgenstem) utility function which is assumed to be
denotes the conditional expectation given byincreasing and concave, andis ranked by ascending order in . In other words,
We also assume that the expected utility
the second-order stochastic dominance for $\alpha>0.$
the excess asset retum is ranked by the firstfor each $t\in[O, 1]$ . Then, fromsuch that
By this assumption, there exists
through the transformation function :Theorem 1, the investor computes the welfare$\theta\in\Theta=[0,1]$$G(\theta)$. We assume that the investor’s belief over

. The excess retum of the risky asset indexed by$\Theta$$\theta$$\tilde{x}(\theta)$$w$$\tilde{x}(\theta)$$U(\alpha, \theta)=$$\alpha$$E_{\theta}[u(w+\alpha\tilde{x}(\theta))]$$u$$\theta.$$\mathbb{E}_{\theta}$$\theta$$U(\alpha, \theta)$$and\int or$$U(\alpha, \theta^{*}(\alpha,t))=t$$\theta^{*}(\alpha,t)\in\Theta$$V(\alpha)$$\varphi$$V( \alpha)=\int_{0}^{1}\varphi[G(\theta^{*}(\alpha,t))]dt$,(8)

where is increasing and satisfies $\varphi(0)=0$ and $\varphi(1)=1$ . From Proposition 1 the smooth ambiguity
aversion is captured by the convexity of . In this representation, comparison of smooth ambiguity
is more smoothly ambiguity averse thm , then there exists anaversion is defined as follows: if
increasing, convex transformation function such thatAt first, we confirm that the $sign$ of the risky asset holding coincides with that of the expectation of. The first-order condition is given by
are the same for all
the risky asset if the signs of$\varphi$$\varphi$$\varphi_{1}$$\varphi_{2}$$h$$\varphi_{2}=h\circ\varphi_{1}.$$\theta\in\Theta$$\mathbb{E}_{\theta}[\tilde{x}(\theta)]$(9)$V’( \alpha)=\int_{0}^{1}\varphi’(G(\theta^{*}(\alpha,t))G’(\theta^{*}(\alpha,t))\frac{\partial\theta^{*}(\alpha,t)}{\partial\alpha}dt=0.$

For each $t\in[O, 1]$ , totally differentiating$U(\alpha, \theta)$

gives$\frac{\partial\theta^{*}(\alpha,t)}{\partial\alpha}|_{\alpha=0} = -\frac{E_{\theta}[u’(w+\alpha\tilde{x}(\theta))\tilde{x}(\theta)]}{\partial U(\alpha,\theta)/\partial\theta}|_{\alpha=0}$$= - \frac{u’(w)\mathbb{E}_{\theta}[\tilde{x}(\theta)]}{\partial U(\alpha,\theta)/\partial\theta|_{\alpha=0}}.$Since$\varphi’(\cdot)G’(\cdot)\leq 0$, by the concavity of the objective function, the following relation holds:

.

(10)

$\mathbb{E}_{\theta}[\tilde{x}(\theta)]\geq(\leq)0\Rightarrow\alpha^{*}\geq(\leq)0$, that is, the risky
is positive for eachIn the remainder of this section, we assume that
asset holding is positive. We obtain similar results by changing the $sign$ for the case when the risky asset
holding is negative. Thus we omit them.
$\theta\in\Theta$$\mathbb{E}_{\theta}[\tilde{x}(\theta)]$

5.2 Two indexes caseIn this subsection, we consider a simple case in which the set of indexes consists of two elements$\{0,1\}$ .
The investor believes that $\theta=0$ and that $\theta=1$ are given by $1-p$ and respectively. The$\Theta$$p$

67
excess risky asset retums indexed by them are given by
andrespectively. We assume
that dominatesin the sense of first-order stochastic dominance (FSD), so that$\tilde{x}_{0}=\tilde{x}(0)$$\tilde{x}_{1}$$\tilde{x}_{1}=x(1)$$\tilde{x}_{0}$$\mathbb{E}[u(w+\alpha x_{1})]$$\mathbb{E}[u(w+\alpha\tilde{x}_{0})]\leq$for any $\alpha>0$ . In this setting, the investor maximizes the welfare:$\max_{\alpha}(1-\varphi(p))\mathbb{E}[u(w+\alpha\tilde{x}_{0})]+\varphi(p)E[u(w+\alpha\tilde{x}_{1})]$

The first-order condition is given by$\alpha=\alpha^{*}$

.

(11)

satisfying$(1-\varphi(p))\mathbb{E}[u’(w+\alpha^{*}\tilde{x}_{0})\tilde{x}_{0}]+\varphi(p)\mathbb{E}[u’(w+\alpha^{*}\tilde{x}_{1})\tilde{x}_{1}]=0$

.

(12)

The second-order condition is also satisfied by the concavity of the utility function. We examine theeffect of increasing smooth ambiguity aversion on the optimal portfolio in our dual theory.
Let an investor with her transformation function
be more ambiguity averse than an investor with
her transformation function . From Theorem 2, we can readily observe that
. It isnatural to conjecture that an increase in smooth ambiguity aversion decreases the holding of the risky
asset. However, as illustrated by Gollier (2011) for the original smooth ambiguity model, this does not
hold.
$\varphi_{1}$$\varphi_{2}(p)>\varphi_{1}(p)$$\varphi_{2}$Proposition2 In the two indexes case, if the investor has the preference that her relative risk aversion
is less than unity, an increase in smooth ambiguity aversion decreases the share of the risky asset.
We note that if the investor’s relative risk aversion is not less than unity, it may be possible thatan increase in smooth ambiguity aversion increases the share of the risky asset since it may occur thatas shown in an example of a quadratic utility function in Fishburn and Porter (1976).
$\infty>\alpha_{1}$5.3

The general caseIn this subsection, we consider a general case in which the set of indexes is defined by the interval$\Theta=[0,1]$ . Let an investor with transformation
function
be more ambiguity averse than an investor
with transformation function . The optimal portfolio of the risky asset for the investor with is givenby the following first-order condition:$\varphi_{1}$$\varphi_{2}$$\varphi_{1}$$V_{1}’( \alpha)=\int_{0}^{1}\varphi_{1}’(G(\theta^{*}(\alpha,t))G’(\theta^{*}(\alpha,t))\frac{\partial\theta^{*}(\alpha,r)}{\partial\alpha}dt=0$

.

(13)

To determine a condition such that an increase in smooth ambiguity aversion decreases the optimal
portfolio of the risky asset, we have to find a condition that satisfies the following inequality:$V_{2}’( \alpha)=\int_{0}^{1}\varphi_{2}’(G(\theta^{*}(\alpha,t))G’(\theta^{*}(\alpha,t))\frac{\partial\theta^{*}(\alpha,t)}{\partial\alpha}dx\geq 0$

.

(14)

To prove this inequality, we use the variation diminishing property introduced by Karlin and Novikoff
Jewitt (1987) and Athey (2002) have discussed its economic and financial
applications.
(1963) and Karlin (1968).

Property 1 Suppose that :satisfying $(x-x_{0})g(x)\geq 0$ for all
$g$

.
.

$\phi$:$\mathbb{R}^{2}arrow \mathbb{R}$$\mathbb{R}arrow \mathbb{R}$satisfies the single crossing condition that there exists a$x\in \mathbb{R}$. Then the following two conditions are equivalent.

is $log$ -supermodular (LSPM).

$\mathbb{E}[g(x)\phi(x, \theta_{L})]=0\Rightarrow \mathbb{E}[g(x)\phi(x, \theta_{H})]\geq 0$$\forall\theta_{H}\geq\theta_{L}.$$\cdot$$x_{0}\in \mathbb{R}$68
To use this property, we begin with the following lemma.
is differentiable with respect to its first argument. Then
Lemma 3 Suppose :is non-decreasing inand only if$\phi$$\mathbb{R}^{2}arrow \mathbb{R}^{+}$$\frac{\partial\phi(x,y)/\partial x}{\phi(x,y)}$$\phi$

is LSPM if$y.$

Proposition 3 Suppose that an irtvestor follows the dual theory of smooth ambiguity aversion. An increase in smooth ambiguity aversion reduces the optimal share of the risky asset if$g(t)=G’( \theta^{*}(\alpha,t))\frac{\partial\theta^{*}(\alpha,t)}{\partial\alpha}$

has the single crossing property at $t_{0};(t-t_{0})g(t)\geq 0.$

Theorem 3 Suppose that an investor follows the dual theory of smooth ambiguity aversion. If the investor has the preference that her relative risk aversion is less than unity and the risky asset retums areranked by first-order stochastic dominance, then an increase in smooth ambiguity aversion decreases theshare of the risky asset.
Theorem 4 Suppose that an investor follows the dual theory of smooth ambiguity aversion. If the investor has the preference that her relative prudence is greater than zero and less than two and the risky
asset returns are ranked by the second-order stochastic dominance, an increase in smooth ambiguity
aversion decreases the share of the risky asset.
A similar theorem holds for central dominance, which is necessary and sufficient for increasing theshare of the risky asset for all risk averse investors. In addition, a similar theorem also holds for any
stochastic dominance that is stronger than central dominance. See Chapter 6 in Gollier (2001) for an
explanation of central dominance and the related stochastic dominance.

Theorem 5 Suppose that an investorfollows the dual theory of smooth ambiguity aversion. An increasein smooth ambiguity aversion decreases the share of the risky asset when the risky asset returns areranked by central dominance.

6 Concluding Remarks
This paper studies the dual theory of the smooth ambiguity model introduced by KMM. This can beviewed as an extension of Yaari’s dual theory of the expected utility model for ambiguity. In our dual
theory, the preferences for ambiguity capmre second-order beliefs. Adding some axioms to the originalmodel, we present the preference functional for the dual representation of the smooth ambiguity model.
We characterized ambiguity aversion and its comparison in our dual theory. Lastly, we determined a setof sufficient conditions that guarantee that smooth ambiguity aversion decreases the optimal portfoho in
an application of our model to the standard portfolio problem.

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70
Hideki Iwaki
Faculty of Business Administration
Kyoto Sangyo University
Kyoto 603-8555
JAPAN-mail address: iwaki@cc.kyoto-su.acjp$E$

Yusuke Osaki
Faculty of Economics
Osaka Sangyo University
Osaka 574-8530

JAPAN
$E$

-mail: osaki@eco.osaka-sandai.ac.jp$\overline{H_{\backslash }}fflE\ovalbox{\tt\small REJECT}\star\Leftrightarrow\cdot\ovalbox{\tt\small REJECT} g\Leftrightarrow U_{0}g\Re \mathfrak{F}\mathbb{H}$xwrre$\ovalbox{\tt\small REJECT} x\Leftrightarrow\cdot\ovalbox{\tt\small REJECT}\Re\oplus\pi_{D}$Effi ffift