Discussion Paper No.

610

PORTFOLIO INERTIA AND

Ɛ -CONTAMINATIONS

Takao Asano

June 2004

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Portfolio Inertia and ε-Contaminations ∗Takao Asano †Institute of Social and Economic Research
Osaka University
Mihogaoka 6-1, Ibaraki, Osaka 567-0047
JAPANFirst version: October 7, 2003
This version: June 13, 2004

Abstract
This paper analyzes investors’ portfolio selection problems in a two-period dynamic model of Knightian uncertainty. We account for the existence of portfolio
inertia in this two-period framework. Furthermore, by incorporating investors’ updating behavior, we analyze how new observation in the first period will affectinvestors’ behavior. By this analysis, we show that new observation in the first
period will expand portfolio inertia in the second period compared with the casein which new observation has not been gained in the first period if the degree of
Knightian uncertainty is sufficiently large.

∗

I would like to thank Hiroyuki Ozaki for his encouragement. I am also grateful to Akihiko Matsui for
his comments and discussions on this work. I appreciate Satoru Takahashi as well as seminar participants
at Macroeconomics Workshop (Osaka University) for their comments on this work. Needless to say, I
am responsible for any remaining errors. I gratefully acknowledge the financial support from the 21st
century COE program (Osaka University).
†
e-mail: tasano@iser.osaka-u.ac.jp

1. Introduction
In stock markets, it is often observed that portfolio inertia exists, that is, a situation
where some stocks are not traded or not priced for several minutes. While the existenceof portfolio inertia cannot be accounted for1 under the standard expected utility theory,2
it can be explained under non-expected utility theories.3 In this paper, we extend theirstatic frameworks to a two-period dynamic framework in order to incorporate decision
maker’s updating behavior. Within such a framework, we can consider the following
question: Does obtaining new information affect her portfolio inertia, which is derived
from her optimization behavior?
As Gilboa and Schmeidler (1993) point out, it is intriguing to cosider whether obtaining new information will shrink portfolio inertia or expand. In order to analyze sucha problem, we consider two updating rules, the Fagin-Halpern rule and Dempster-Shafer
rule, which are frequently adopted in economics and statistics.4 Recently, Epstein andSchneider (2003) and Wang (2003) axiomatize rational decision makers’ behaviors incorporating their belief-updatings in the multi-period dynamic framework. In this paper,
we consider the condition under which obtaining new information will expand portfolio
inertia by way of decision makers’ belief-updating.
The organization of this paper is as follows. In Section 2 we provide a review of
non-expected utility theories, in particular, the Choquet Expected Utility and the Maxmin
Expected utility. Section 3 provides the stochastic environment of a two-period model.
Section 4 presents the two updating rules, the Fagin-Halpern and Dempster-Shafer rules.
In Section 5, we define the dilation of Knightian uncertainty. Section 6 defines the εcontamination, which is a restriction of the set of decision makers’ beliefs. Section 7
analyses a portfolio selection problem à la Arrow (1965) under the non-expected utility
1

See Arrow (1965).
We explain the standard expected utility theory including non-expected utility theories in the next
section.
3
See Dow and Werlang (1992) that first account for the existence of portfolio inertia under the Choquet
Expected Utility Theory. Also see Asano (2003) that explains the existence of portfolio inertia under theconcept of ambiguity. For the concept of ambiguity, see Epstein (1999).
4
See Section 4 in details.
2

1

framework. Section 8 provides the main theorem of this paper. Section 9 concludes thispaper. Definitions of Choquet integrals and non-additive measures (or capacities) as wellas mathematical results about Choquet integrals are relegated to Appendix.
2. CEU and MMEU
Since Ellsberg (1961) first cast doubt on the validity of the Subjective Expected
Utility (henceforth SEU) theory axiomatized by Savage (1954),5 a number of generalizations have been proposed in order to overcome such a shortcoming and to analyze problems
that are not well explained within the framework of SEU.6 While risk is the situation inwhich decision maker’s beliefs are captured by a single probability measure, uncertainty is
the situation in which decision maker’s beliefs are represented by a set of probability measures or a non-additive probability measure. The difference between risk and uncertainty
is crucial in analyzing decision maker’s behavior under “non-deterministic” environment.
The Maxmin Expected Utility (henceforth MMEU) and the Choquet Expected
Utility (henceforth CEU) are two alternatives to SEU that are extensively investigated ineconomics,7 finance,8 game theory,9 and so forth. MMEU states that if a certain set of
axioms is satisfied, then DM’s beliefs are captured by a set of finitely additive measuresand her preferences are represented by the minimum of expected utilities over the setof these measures.10 On the other hand, CEU states that if a certain set of axioms is
satisfied, then DM’s beliefs are captured by a non-additive measure and her preferences
are represented by Choquet integrals.11 These two theories are axiomatized by different
sets of axioms, respectively. However, it can be shown that CEU with a convex non5
SEU states that if a certain set of axioms is satisfied, then DM’s beliefs are captured by a uniqueprobability measure and her preferences are represented by the expected utility.
6
Using an easy-to-understand example, Ellsberg shows that people often violate Savage’s crucial axiom,
the Sure Thing Principle.
7
For example, see Nishimura and Ozaki (2002b) (job search) or Ghirardato (1994) (agency theory).
8
For example, see Dow and Werlang (1992), and Epstein and Wang (1994, 1995).
9
For example, see Dow and Werlang (1994), Lo (1996, 1998, 1999), and Marinacci (2000).
10
MMEU is axiomatized by Gilboa and Schmeidler (1989) in the Anscombe and Aumunn (henceforthAA) framework and axiomatized by Casadesus-Masanell, Klibanoff and Ozdenoren (2000) in the Savage
framework.
11
CEU is axiomatized by Schmeidler (1989) in the AA framework and axiomatized by Gilboa (1987)
in the Savage framework.

2

additive measure µ is equivalent to MMEU with the set core(µ) as a set of probability
measures.12 In this paper, we further restrict the set of probability measures P to the
ε-contamination. The reason of its restriction to the ε-contamination is provided in thefollowing sections.
3. Stochastic Environment
In this section, we provide the formal description of our two-period dynamic model.
Let S be a state space. Let (s1 , s2 ) denote a generic element of S × S. Let
F = hFt it=0,1,2 , where F0 ≡ {∅, S × S}, and F1 is defined to be the algebra generated
by the set of finite partitions of S × S of the form: hEi × Sii for some finite partitionhEi imi=1 of S, and F2 is defined to be the algebra generated by the set of finite partitionsof S × S of the form: hEi × Fj ii,j for some finite partitions hFj inj=1 of S.
For notational abuse, we denote the first-period measurable space by (S, hEi ii),
where hEi ii is the algebra generated by the partition hEi i. We denote the set of all the
probability measures on it by M (S, hEi i). Similar notational abuse applies to the secondperiod measurable space denoted by (S, hFj ij ). Furthermore, we denote the set of all the
probability measures on (S × S, F2 ) by M (S × S, F2 ), and denote decision maker’s setof probability measures by P ⊆ M (S × S, F2 ). In this paper, we consider that decision
maker’s Knightian uncertainty is represented by the set P.
12

Let S be a set, 2S be the set of subsets of S, µ be a convex non-additive measure, P be the set offinitely additive measures on (S, 2S ), and let B(S, R) denote the space of bounded functions from S into
R. ThenZZ
X(s)µ(ds) = min
X(s)P (ds) | P ∈ core(µ) ,where X ∈ B(S, R) and the uppercore of µ is the set of finitely additive measures on (S, 2S ) that dominateµ for all E ∈ 2S , i.e., core(µ) = P ∈ P|(∀E ∈ 2S ) P (E) ≥ µ(E) , and the integral of the left-hand side
is in the sense of Choquet integrals. Note thatZ
Z
X(s)µ′ (ds) = max
X(s)P (ds) | P ∈ core(µ′ ) ,where X ∈ B(S, R), µ is a convex capacity on (S, 2S ), µ′ is the conjugate of µ, and the set core(µ′ ) is thelower core of µ′ . We provide the definitions of Choquet integrals, capacities, convex capacities, and thelower core of capacities in Appendix.

3Let p ∈ M (S × S, F2 ). Then we define p|1 (·) ≡ p(· × S). Formally speaking,p|1 is not a measure on (S, hEi i), but it can be considered as the first-period marginal
probability measure of p on (S × S, F2 ). In a similar way, we define the second-period
marginal probability measure, denoted by p|2 , and we consider it as a measure on (S, hFj i).
Let P ⊆ M (S × S, F2 ). Then we define the first and second period marginal
Knightian uncertainty by
P|1 = {p|1 | p ∈ P}
and P|2 = {p|2 | p ∈ P} ,respectively.
In order to describe updating rules after observing an event E in the first period,
we provide the definition of the Bayesian updating procedure. Let p ∈ P and let E ∈ hEi isuch that p(E ×S) > 0. Then we define the probability measure on (S ×S, F2 ) conditionalon the event E × S by(∀E ∈ hEi i)(∀F ∈ hFj i) p|2 (E × F |E) ≡p(E × F ).
p(E × S)We define the Bayesian procedure by a function (p, E) 7→ p|2 (·|E), where p|2 (·|E) ≡p|2 (E × ·|E). That is, it maps a pair of a probability measure p on (S × S, F2 ) and an
event E in hEi i in the first period, to p|2 (·|E), which can be considered as the probability
measure on (S, hFj i).
Let P be decision maker’s set of beliefs, and let E be an hEi i-measurable set suchthat p(E × S) > 0 for all p ∈ P. Then we define an updating rule by a function that maps
a pair (P, E) to a set of probability measures on (S, hFj i). In this paper, an updating
rule is denoted by φ.
4. Updating Rules: The Fagin-Halpern and Dempster-Shafer Rules
In this section, we provide the definitions of two updating rules that are heavily investigated in statistics13 and economics,14 the Fagin-Halpern updating rule (we abbreviate it
13
14

For example, see Dempster (1967, 1968), Shafer (1976), or Fagin and Halpern (1990).
For example, see Gilboa and Schmeidler (1993), Denneberg (1994), or Nishimura and Ozaki (2002a).

4

as the FH rule) and the Dempster-Shafer updating rule (we abbreviate it as the DS rule).
We denote the FH rule as φF H and define it by(∀P ⊆ M (S × S, F2 ))(∀E ∈ hEi ii ) φF H (P, E) = {p|2 (·|E) | p ∈ P} .

(1)

Note that a decision maker who adapts the FH rule updates all the probability measuresp ∈ P following the Bayesian procedure. Also note that when E = S, φF H (P, S) = P|2
for any P. Thus the second period marginal Knightian uncertainty P|2 can be derived
from the FH rule for E = S.
Before we define the DS rule, one more definition is in order. Define a set offirst-period probability measures by(∀E ∈ hEi ii ) P ∗ (E) = argmax {p|1 (E) | p ∈ P} .
Then we are in a position to define the DS rule by(∀P ⊆ M (S × S, F2 ))(∀E ∈ hEi ii ) φDS (P, E) = {p|2 (·|E) | p ∈ P ∗ (E)} .

(2)

Note that in the first period, a decision maker who follows the DS rule survives probability
measures p ∈ P that assign the maximum value to an event E, and in the second period,
she follows the Bayesian procedure for such a set of probability measures denoted byP ∗ (E).
From the definitions of FH and DS rules, we can immediately prove that(∀P ⊆ M (S × S, F2 ))(∀E ∈ hEi ii ) φDS (P, E) ⊆ φF H (P, E).
Although this result holds for any set of probability measures P, the converse set inclusion
does not necessarily hold for any P. A question is worth considering: Does the converse
set inclusion hold by restricting the set of probability measures P? That is the topic of
Section 5. Before we investigate this topic, the definition of the dilation of Knightian
uncertainty is in order.

55. Dilation of Knightian Uncertainty
In this section, we define the concept of dilation.15 Let P ⊆ M (S × S, F2 ) be the set ofprobability measures. We say that the dilation of Knightian uncertainty occurs if for any
E ∈ hEi i, the next strict set inclusion holds:
φ(P, E) ⊃ P|2 .
This means that if a decision maker has observed an event E ∈ hEi i in the first
period and updates her set of beliefs P by an updating rule φ, then the updated set ofprobability measures φ(P, E) expands compared with the case in which she observes no
event in the first period, that is, E = S. In this paper, a set of probability measures P isassumed to represent decision maker’s Knightian uncertainty. Thus, the occurrence of thedilation of Knightian uncertainty in the sense of this paper forces her to make decisions
based on “more ambiguous environment.”
6. The ε-Contamination
In this section, we restrict the set of priors P to the ε-contamination. At first, we definethe ε-contamination of some probability measure p0 on (S × S, F2 ).
15

Seidenfeld and Wasserman (1993) provide another definition of dilation based on upper and lower
probability measures. Let P ⊆ M (S × S, F2 ) and let B ∈ F2 such that (∀p ∈ P) p(B) > 0. Define theupper probability and the lower probability by(∀A ∈ F2 ) P (A) ≡ sup p(A), andp∈P(∀A ∈ F2 ) P (A) ≡ inf p(A),p∈Prespectively, and define the upper conditional probability and the lower conditional probability(∀A ∈ F2 ) P (A|B) ≡ sup p(A ∩ B)/p(B), andp∈P(∀A ∈ F2 ) P (A|B) ≡ inf p(A ∩ B)/p(B),p∈Prespectively. They define the concept of dilation as follows: B dilates A if
P (A|B) < P (A) ≤ P (A) < P (A|B).
Their necessary and sufficient conditions for the dilation to occur depend on a particular event A.
On the other hand, in this paper, we capture the concept of dilation by way of expantions of sets ofmeasures. Our concept of dilation is more appropriate than their concept when it comes to analyzing
economic problmes since our concept corresponds to the notion of Knightian uncertainty.

6

Let p0 be a probability measure on (S × S, F2 ) such that p0 (E × S) > 0, and letε ∈ (0, 1). We assume that the decision maker’s set of beliefs P is represented by the
ε-contamination of p0 on (S × S, F2 ) such that ε P = p0 ≡ (1 − ε)p0 + εq | q ∈ M (S × S, F2 ) .
ε

If ε = 0, then P = {p0 } reduces to P = {p0 }. This corresponds to the case inwhich decision maker’s belief is captured by a single probability measure. The larger thisnumber ε is, the more uncertain decision makers are about the true probability measurep0 .16 Thus, the number ε can be considered to be the value that captures the degree of
Knightian uncertainty. For further discussions on ε-contaminations, see Berger (1985),
Wasserman and Kadane (1990) or Nishimura and Ozaki (2002a, b).
For later use, we define the ε-contamination of p0 |2 (·|E) by(∀ε ∈ (0, 1))(∀E ∈ hEi i) 0
εp |2 (·|E) = (1 − ε)p0 |2 (·|E) + εq2 | q2 ∈ M (S, hFj i) .

As we have already pointed out, the converse inclusion φDS (P, E) ⊇ φF H (P, E)
is not necessarily true except for the trivial case in which P is a singleton. However,when we restrict the set of probability measures P to the ε-contamination, the converse
inclusion holds. That is, sets of probability measures updated by FH and DS rules are
identical. Moreover, the two updating rules, DS and FH rules can be represented bysome ε′ -contamination (ε′ is defined in the next theorem) of second-period probability
measures.
Theorem 1 (Nishimura and Ozaki (2002a)). Let ε ∈ (0, 1) and let E ∈ hEi ii . Then
φF H
where ε′ is defined by 0 ε  ε  ε′p
, E = φDS p0 , E = p0 |2 (·|E) ,ε′ = ε′ (ε, E) ≡ε
(1 − ε)p0 |1 (E) + ε

16

.

This
observation can be checked by way of expansions of sets of beliefs that are represented byεP = p0 .

7

Proof. See Nishimura and Ozaki (2002a).
It can be proved that the next equality holds:ε′core(θ1 ) = p0 |2 (·|E) ≡ (1 − ε′ )p0 |2 (·|E) + ε′ q | q ∈ M (S, hFj ij ) ,where θ1 : hFj ij → [0, 1] is defined by(∀A ∈ hFj ij ) θ1 (A) =(1 − ε′ )p0 |2 (A|E) if A 6= S
1
if A = S(3)From Fact 4, it can be shown that θ1 is a convex capacity. Thus it follows thatZ
Z(∀s1 ∈ E)
I(s1 , s2 )θ1 (ds2 ) = minI(s1 , s2 )P (ds2) | P ∈ core(θ1 )
S
S

Z
 0
ε′
,
= min
I(s1 , s2 )P (ds2) | P ∈ p |2 (·|E)
S

where the integral of the left-hand side is in the sense of the Choquet integral.

Next, we investigate the relation between the set of the first-period marginals ofthe ε-contamination, P|1 , and the ε-contamination of the first-period marginal probabilityεmeasures, {p0 |1 } .
εLemma 1. Let ε ∈ (0, 1). Let P ≡ {p0 } = {(1 − ε)p0 + εq | q ∈ M (S × S, F2 )}, and letεP|1 ≡ p|1 | p ∈ {p0 } . ThenεP|1 = p0 |1 = (1 − ε)p0 |1 + εq | q ∈ M (S, hEii) .
εεε

Proof. At first, we show that {p0 } |1 ⊆ {p0 |1 } . Let p1 ∈ {p0 } |1 . Then there existsεεp ∈ {p0 } such that p1 = p(· × S). Since p ∈ {p0 } , there exists q ∈ M (S × S, F2 ) suchthat p = (1 − ε)p0 + εq. Thus, p1 = p(· × S) = (1 − ε)p0 |1 (·) + εq|1 (·), which implies thatεp1 ∈ {p0 |1 } .
εεεNext, we show that {p0 |1 } ⊆ {p0 } |1 . Let p1 ∈ {p0 |1 } . Then, there exists
q1 ∈ M (S, hEiii ) such that p1 = (1 − ε)p0 |1 + εq1 . Let q2 ∈ M (S, hFj ij ) and let p =ε(1 − ε)p0 + ε(q1 × q2 ). Then, p ∈ {p0 } and p|1 = (1 − ε)p0|1 + εq1 = p1 , which impliesεthat p1 ∈ {p0 } |1 .
8

By this lemma, we can show thatεP|1 = p0 |1 = core(θ0 ),where θ0 : hEi ii → [0, 1] is defined by(∀E ∈ hEi ii ) θ0 (E) =(1 − ε)p0 |1 (E) if E 6= S
1
if E = S(4)

From Fact 4, it can be shown that θ0 is a convex capacity. Thus it follows that
Z

Z
(∀I)
I(s1 , s2 )θ0 (ds1 ) = minI(s1 , s2 )P (ds1) | P ∈ core(θ0 )
S
S
Z

 0 ε
= min
I(s1 , s2 )P (ds1) | P ∈ p |1.
S

Again, the integral of the left-hand side is in the sense of the Choquet integral.

Similarly, we can characterize the set of second-period marginals of the ε-contamination,
ε

P|2 , by the ε-contamination of the second-period marginal probability measures, {p0 |2 } .
εLemma 2. Let ε ∈ (0, 1). Let P|2 ≡ p|2 | p ∈ {p0 } . Then,εP|2 = p0 |2 = (1 − ε)p0 |2 + εq | q ∈ M (S, hFj i) .

By this lemma, it follows thatεP|2 = p0 |2 = core(θ2 ),where θ2 : hFj ij → [0, 1] is defined by(∀F ∈ hFj ij ) θ2 (F ) =

(1 − ε)p0 |2 (F ) if F 6= S
1
if F = S.

(5)

From Fact 4, we can show that θ2 is a convex capacity.
7. Portfolio Selection Problem
In this section, we consider a portfolio selection problem à la Arrow (1965) in a two
period framework. At first, the setup is in order. After that, we analyze the portfolio
selection problem in two different setting. The first setting is the one in which a decision9

maker observes an event E in the first period and updates her set of probability measuresε

P = {p0 } by the FH rule. The second is the one in which she observes no event in thefirst period and updates P by the FH rule, which is equal to the second-period marginalof P, P|2 , as we have already pointed out.
Let W ∈ R+ be the wealth at t = 0, N ∈ R be the amount of money invested at
t = 0, q > 0 be the price of a risky asset at t = 0, and X be the random payoff of the asset
at t = 2, where X is F2 -measurable. Moreover, let u : R → R be a monotonic increasing
concave function. We assume that an investor obtains new information at t = 1, but she
does not gain any payment at t = 1. We are in a position to define her objective function.
The objective function is

=
=
=
=
=
=

V (X(s1 , s2 ))pp
min E min E [X| F2] F1
p∈P
p∈P

Z 
p
min E [X| F2 ] p|1 (ds1 )
min
p∈P
p∈P

Z 
Z
min
min
u (W − N + (N/q)X(s1 , s2 )) q(ds2 ) p|1 (ds1 )p∈Pq∈φF H (P,E)

Z 
Z
min
min
u (W − N + (N/q)X(s1 , s2 )) q(ds2 ) p(ds1 )q∈φF H (P,E)
p∈P|1
#
Z "
Z
min

p∈{p0 |1 }ε

Z Z
S

min

(6)

u (W − N + (N/q)X(s1 , s2 )) q(ds2 ) p(ds1 )′q∈{p0 |2 (·|E)}εu (W − N + (N/q)X(s1 , s2 )) θ1 (ds2 ) θ0 (ds1 ),S(7)where θ0 : hEi ii → [0, 1] is defined by(∀E ∈ hEi ii ) θ0 (E) =(1 − ε)p0 |1 (E) if E 6= S
1
if E = S,

θ1 : hFj ij → [0, 1] is defined by(∀A ∈ hFj ij ) θ1 (A) =

(1 − ε′ )p0 |2 (A|E) if A 6= S
1
if A = S,and ε′ is defined in Theorem 1. Note that DM’s set of beliefs P is captured by the
ε-contamination of p0 , and her updating rule φ(P, E) is characterized by the FH rule10

φF H (P, E). Also note that the penultimate equality holds by Lemma 1 and Theorem
1. For axiomatizations of this preference (6), see Epstein and Schneider (2001) or Wang
(2003).
An investor is supposed to invest in two assets, a riskless asset and a risky asset.
The problem is to choose the amount of money N so as to maximizeZ Zu (W − N + (N/q)X(s1 , s2 )) θ1 (ds2 ) θ0 (ds1 ).
S

SThat is, she is supposed to choose the amount of money N to invest the risky asset so as
to maximize her non-expected utility of the terminal payoff.
Theorem 2. The investor will neither buy nor sell the risky asset, if q satisfies:
Z Z
S

X(s1 , s2 )θ1 (ds2 )θ0 (ds1 ) < q <

S

Z Z
S

X(s1 , s2 )θ1′ (ds2 )θ0′ (ds1 ),S

where θ0′ and θ1′ are the conjugate of θ0 defined by Equation (4) and the conjugate of θ1
defined by Equation (3), respectively. 17
This theorem states that there exists portfolio inertia in a two-period setting, inεwhich DM’s beliefs are represented by the ε-contamination P = {p0 } , and her set ofprobabilities P is updated by the FH rule (DS rule).
Proof. Assume that
q>

Z Z
S

17

X(s1 , s2 )θ1 (ds2 )θ0 (ds1 ).

S

Observe that
Z Z

X(s1 , s2 )θ1′ (ds2 )θ0′ (ds1 )

Z 
Z
=
max
max
X(s1 , s2 )q(ds2 ) p(ds1 )p∈core(θ0′ )q∈core(θ1′ )

Z 
Z
=
max
max
X(s1 , s2 )q(ds2 ) p(ds1 )p∈core(θ0 )q∈core(θ1 )
"
#
Z
Z
S

=

S

max

p∈{p0 |1 }εmax′q∈{p0 |2 (·|E)}ε11

X(s1 , s2 )q(ds2 ) p(ds1 ).

(8)

Then, for all N ≥ 0,
Z Zu (W − N + (N/q)X(s1 , s2 )) θ1 (ds2 ) θ0 (ds1 )Z Zu(W − N + (N/q)X(s1 , s2 )) θ1 (ds2 ) θ0 (ds1 )
S
S

Z 
Z
u W − N + (N/q) X(s1 , s2 )θ1 (ds2 ) θ0 (ds1 )
S
S
Z 


Z
u
W − N + (N/q) X(s1 , s2 )θ1 (ds2 ) θ0 (ds1 )Z Z
u W − N + (N/q)
X(s1 , s2 )θ1 (ds2 )θ0 (ds1 )
S

≤
=
≤
=



S

S

S

< u(W − N + (N/q)q)= u(W ),

where the first and second inequalities follow from Jensen’s inequality, the first and second
equalities follow from Fact 2, and the strict inequality follows from Equation (8).
Suppose that
q<

Z Z
S

X(s1 , s2 )θ1′ (ds2 )θ0′ (ds1 ).

Then, for all N ≤ 0,
Z Z

=
=
≤
=
=



u (W − N + (N/q)X(s1 , s2 )) θ1 (ds2 ) θ0 (ds1 )Z Zu(W − N + (N/q)X(s1 , s2 )) θ1 (ds2 ) θ0 (ds1 )
S
S

Z 
Z
u W − N − (N/q) −X(s1 , s2 )θ1 (ds2 ) θ0 (ds1 )
S
S Z

Z 
′
u W − N − (N/q) − X(s1 , s2 )θ1 (ds2 )θ0 (ds1 )
S
S
Z 


Z
′
W − N + (N/q) X(s1 , s2 )θ1 (ds2 ) θ0 (ds1 )
u
S

SZ


Z
′
u W − N − (N/q) −X(s1 , s2 )θ1 (ds2 ) θ0 (ds1 )
S
S

 Z Z

′
′
u W − N − (N/q) −X(s1 , s2 )θ1 (ds2 )θ0 (ds1 )
S

≤

(9)

S

S

S

< u (W − N − (N/q)(−q))= u(W ),

12S

where the first and second inequalities follow from Jensen’s inequality, the first and third
equalities follow from Fact 2, the second and fourth equalities follow from Fact 3, and thestrict inequality follows from Equation (9).
Next, we consider the case in which she does not update her set of belifes P,
which is still assumed to be characterized by the ε-contamination of p0 . That is, it isassumed that her set of second-period beliefs is represented P|2 , which is the set of her
second-period marginal probability measures defined by
P|2 = {p|2 | p ∈ P} .
As we have already pointed out, P|2 is equal to the FH rule, φF H (P, E) for E = S.
In this case, her objective function turns out to be

=
=
=
=
=

V (X(s1 , s2 ))

Z 
Z
min
min
u (W − N + (N/q)X(s1 , s2 )) q(ds2) p|1 (ds1 )p∈Pq∈φF H (P,S)

Z 
Z
min
min
u (W − N + (N/q)X(s1 , s2 )) q(ds2 ) p(ds1 )p∈P|1q∈φF H (P,S)

Z 
Z
min
min u (W − N + (N/q)X(s1 , s2 )) q(ds2 ) p(ds1 )
p∈P|1
q∈P|2

Z
Z 
u (W − N + (N/q)X(s1 , s2 )) q(ds2 ) p(ds1 )minminq∈{p0 |2 }εp∈{p0 |1 }εZ Zu (W − N + (N/q)X(s1 , s2 )) θ2 (ds2 ) θ0 (ds1 ),S

S

where θ0 : hEi ii → [0, 1] is defined by(∀E ∈ hEi ii ) θ0 (E) =(1 − ε)p0 |1 (E) if E 6= S
1
if E = S,(1 − ε)p0 |2 (F ) if F 6= S
1
if F = S.

and θ2 : hFj ij → [0, 1] is defined by(∀F ∈ hFj ij ) θ2 (F ) =Note that the fourth equality holds by Lemmas 1 and 2.
The problem is to choose the amount of money N so as to maximizeZ Zu (W − N + (N/q)X(s1 , s2 )) θ2 (ds2 ) θ0 (ds1 ).
S

S

13

(10)

Similarly, we can prove the following theorem.
Theorem 3. The investor will neither buy nor sell the risky asset, if q satisfies:
Z Z
S

X(s1 , s2 )θ2 (ds2 )θ0 (ds1 ) < q <

S

Z Z
S

X(s1 , s2 )θ2′ (ds2 )θ0′ (ds1 ),S

where θ0′ and θ2′ are the conjugate of θ0 defined by Equation (4) and the conjugate of θ2
defined by Equation (5), respectively.
Proof. The proof is omitted.
This theorem states that there also exists portfolio inertia in a two-period model,εin which DM’s beliefs are represented by the ε-contamination P = {p0 } , and she does
not obtain any information at t = 1, that is, P is restricted to the second period, P|2 .
Now, a question is worth raising: whether portfolio inertia expands or shrinks
when she obtains new information in the first period compared with the case in which
she has not gain any information in the first period? In the next section, we provide ananswer to this question.
8. Expansion of Portfolio Inertia
In this section, we consider the problem posed in the last section: whether obtaining new information will shrink or expand portfolio inertia?
In a two-period setting, we consider the following two cases. The first case is theone in which a decision maker observes an event E ∈ hEi ii in the first period, and she
evaluates a second-period event by the probability measures updated by the Bayesian
procedure. The second case is the one in which she observes no event in the first period,that is, E = S, and she evaluates the second-period event similar to the first case.
In order to analyze this problem, we define the “informational value” according to
Nishimura and Ozaki (2002a). Let E ∈ hEi ii and let δ(E) be defined by
δ(E) = max p0 |2 (Fj |E) − p0 |2 (Fj ) ,j=1,··· ,n14where δ(E) ∈ [0, 1]. This number δ(E) is one of the measures to capture the informational
value of knowing that an event E has occurred in the first period. Note that δ(E) = 0
implies that observing an event E in the first period will not affect decision maker’s
evaluation of an event F in the second period.
Some comments are in order before we provide the main result of this paper. In
the following theorem, we assume two additional conditions. The first is the condition
that m ≥ 2. This condition together with the assumption that p0 (E × S) > 0 imply that0 < p0 |1 (E) < 1. The second is the condition that F ∈ hFj inj=1 is such that p0 |2 (F ) > 0.
These two conditions imply that the first and the fourth inequalities in the following
theorem hold with strict inequality.
Now we are in a position to present the main result of this paper. If ε, which is thedegree of contamination of p0 , and which can be considered as the degree of Knightian
uncertainty, is sufficiently large with respect to the informational value of the observationE, δ(E), then obtaining new information expands portfolio inertia compared with the casein which an investor does not obtain new information.
Theorem 4. An investor neither buys nor sells the risky asset if q satisfies:
Z Z
Z Z
X(s1 , s2 )θ2 (ds2 )θ0 (ds1 ) < q <X(s1 , s2 )θ2′ (ds2 )θ0′ (ds1 ),
S

S

S

S

where θ0′ and θ2′ are the conjugate of θ0 defined by (4) and the conjugate of θ2 defined by
ε(5), respectively. Let P = {p0 } and let m ≥ 2. Furthermore, let F ∈ hFj inj=1 such thatp0 |2 (F ) > 0. Suppose that the following inequality holds:
ε>p0 |1 (E)δ(E).
(1 − p0 |1 (E)) minj p0 |2 (Fj )(11)Then, for such an asset price, the following inequalities hold:
Z Z
Z Z
X(s1 , s2 )θ1 (ds2 )θ0 (ds1 ) <X(s1 , s2 )θ2 (ds2 )θ0 (ds1 ) <
S
S
S
S
Z Z
Z Z
′
′
q <
X(s1 , s2 )θ2 (ds2 )θ0 (ds1 ) <X(s1 , s2 )θ1′ (ds2 )θ0′ (ds1 ),
S

S

S

S

where θ0′ , θ1′ and θ2′ are the conjugate of θ0 defined by (4), the conjugate of θ1 defined by(3) and the conjugate of θ2 defined by (5), respectively.
15

This theorem states that obtaining new information expands portfolio inertia compared with the case in which she does not obtain new information as long as ε is sufficiently
large.
Proof. In order to prove the statement, it suffices to show that
θ1 (F ) < θ2 (F ),for any F ∈ hFj ij such that p0 |2 (F ) > 0.
When δ(E) = 0, it follows that
(1 − ε)p0 |2 (F ) − (1 − ε′ )p0 |2 (F |E)= (1 − ε) p0 |2 (F ) − p0 |2 (F |E) − (ε − ε′ )p0 |2 (F |E)= (ε′ − ε)p0 |2 (F |E) > 0,

where the second equality holds since δ(E) = maxj |p0 |2 (F |E) − p0 |2 (Fj )| = 0 and ε′ > ε,and the inequality holds since ε′ > ε and p0 |2 (F |E) > 0.
When δ(E) > 0, it follows that

=
=
≥
=
≥
>
=

(1 − ε)p0 |2 (F ) − (1 − ε′ )p0 |2 (F |E)


1−ε 0
1 − ε′ 0′εp |2 (F ) −p |2 (F |E)
ε′
ε′



1 0
1−ε 0
0
0
′
p |1 (E) + 1 p |2 (F ) − p |1 (E)p |2 (F |E)(1 − ε)ε
ε
ε




1−ε 0
1 0
′
0
0
(1 − ε)εp |1 (E) + 1 p |2 (F ) − p |1 (E) p |2 (F ) + δ(E)
ε
ε


 0
δ(E) 0′0(1 − ε)ε 1 − p |1 (E) p |2 (F ) −p |1 (E)εδ(E) 000′
p |1 (E)(1 − ε)ε 1 − p |1 (E) min p |2 (Fj ) −
j
ε
"

−1 #
0

p
|
(E)
1
(1 − ε)ε′ 1 − p0 |1 (E) min p0 |2 (Fj ) − δ(E)p0 |1 (E)δ(E)
j(1 − p0 |1 (E)) minj p0 |2 (Fj )0,where the second equality follows from the definition of ε′ , the first inequality follows fromthe definition of δ(E), and the strict inequality follows from the assumption.
16

9. Conclusion
In this paper, we investigate a portfolio selection problem à la Arrow (1965) in a two period
dynamic model of Knightian uncertainty. We explain the existence of portfolio inertia in
the two-period framework in which decision maker’s preference is represented by MMEU
and her beliefs are captured by the ε-contamination. Moreover, by incorporating her
updating behavior (the FH rule or the DS rule), we show that new observation in thefirst period will expand portfolio inertia in the second period compared with the case inwhich new observation has not been gained in the first period if the degree of Knightian
uncertainty is sufficiently large.
In this paper, we provide the results in which decision maker’s beliefs are capturedby the ε-contamination. However, this set of beliefs is more restrictive than the set core(µ),
which is the set of probability measures dominating a convex non-additive measure µ,
and which is extensively investigated in the literature on non-expected utility theories.
Thus, one of the interesting topics for future research is to extend the results based on
ε-contaminations to the ones based on sets of probability measures, core.

17

Appendix
A.1 Choquet Integrals
Let S be a set and let 2S be the power set of S. A set function µ : 2S → [0, 1] isa non-additive measure (or capacity) if (a) µ(∅) = 0 and (b) E, F ∈ 2S and E ⊂ F
imply µ(E) ≤ µ(F ), where ∅ denotes the empty set. The conjugate of a capacity µ is
defined by µ′ (E) = 1 − µ(E c ) for all E ∈ 2S . A non-additive measure µ is convex ifµ(E ∪ F ) + µ(E ∩ F ) ≥ µ(E) + µ(F ) for all E, F ∈ 2S . The upper core and lower core ofa capacity µ, denoted by core(µ) and core(µ), respectively, are defined bycore(µ) =core(µ) =p ∈ P| (∀E ∈ 2S ) p(E) ≥ µ(E) ,p ∈ P| (∀E ∈ 2S ) p(E) ≤ µ(E) ,where P is the set of probability measures on (S, 2S ). If a capacity µ is convex (concave),then core(µ) (core(µ)) is non-empty. See Fact 6.
Let B(S, R) denote the space of bounded functions from S into R and let X ∈B(S, R). The integral of X with respect to a non-additive measure µ is called the Choquet
integral, and is defined byZ
X(s) µ(ds)
Z ∞
Z 0
=
µ ({s ∈ S| X(s) ≥ α}) dα +[µ ({s ∈ S| X(s) ≥ α}) − 1] dα,0−∞where integrals on the right hand side are in the sense of Riemann integrals.
A.2 Mathematical Results
Fact 1.
(∀X, Y ∈ B(S, R)) X ≥ Y ⇒Z

X(s) µ (ds) ≥Z

Y (s) µ (ds).

Fact 2.
(∀X ∈ B(S, R))(∀a ∈ R)(∀b ∈ R+ )Z

(a + bX(s))µ (ds) = a + b
18Z

X(s)µ (ds).

Fact 3.
(∀X ∈ B(S, R))Z

X(s) µ(ds) = −Z−X(s) µ′ (ds),where µ′ is the conjugate of µ.
Fact 4. Let P be a probability measure, and let f : [0, 1] → [0, 1] be a monotonic increasing function satisfying f (0) = 0 and f (1) = 1. Furthermore, define a real valued functionf ◦ P : 2S → [0, 1] such that(∀A ∈ 2S ) f ◦ P (A) = f (P (A)).
Then f ◦ P is a non-additive measure. If f is a concave (convex) function, then f ◦ P isconcave (convex). In particular, any real valued function P α : 2S → [0, 1], defined by(∀A ∈ 2S ) P α (A) = (P (A))α ,is a concave non-additive measure when α ∈ (0, 1) and is a convex non-additive measure
when α ∈ (1, ∞).
Fact 5. Let µ and ν be probability capacities on (S, 2S ). If µ(E) ≤ ν(E) for all E ∈ 2S ,then(∀X ∈ B(S, R))Z

X(s)µ(ds) ≤Z

X(s)ν(ds).

Fact 6. If µ is a convex capacity, then core(µ) is non-empty.
Proof. See Kelley (1959).
Fact 7. If µ is a capacity, thencore(µ) = core(µ′ ),where µ′ is the conjugate of µ.
Theorem 5. If µ is a convex capacity, then the following equality holds:
ZZ
X(s)P (ds) P ∈ core(µ) .
(∀X ∈ B(S, R))
X(s)µ(ds) = min19

Proof. See Schmeidler (1989).
Corollary 1. If µ is a concave capacity, then the following equality holds:ZZ(∀X ∈ B(S, R))
X(s)µ(ds) = max
X(s)P (ds) P ∈ core(µ) .
Theorem 6 (Jensen’s Inequality). Let (S, 2S , µ) be a measure space with µ(S) = 1,
where µ is a non-additive measure. Let X : S → (a, b) be in B(S, R), and u be a concave,increasing and real-valued function on (a, b). Then for all non-additive measures µ,ZZ
X(s) µ(ds) .
u(X(s)) µ(ds) ≤ u
Proof. See Asano (2003).

20

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23