数理解析研究所講究録1108 巻 1999 年 44-61

44

NON-HILBERr CONDITIONINGAND
VALUE OF INFORMATION*
by Hiroyuki$\mathrm{O}\mathrm{z}\mathrm{a}\mathrm{k}\mathrm{i}^{1}$

Faculty of Economics
Tohoku University
Kawauchi, Aoba-ku, Sendai 980-8576

JAPANFirst version: January 24, 1995
This version: September 29, 1998

Abstract
This paper generalizes the conditional expectation framework by replacing the Hilbertnorm with a more general measure of approximation errors. This enables one to develop

the concept of conditioning for non-expectation certainty equivalents. Under this concept
a single-agent model, in which the optimal level of information is endogenously determinedthrough the agent’s optimization behavior, can be constructed. In this setting it is possible
for the anti-preference for information to dominate the benefits of better planning available
I show that the “biasedness” of this new conditioning
under more information. In
the choice of information.
is important in$\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}_{)}$$\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{z}\mathrm{i}\dot{\mathrm{n}}\mathrm{g}$eI am grateful to aeminar participants at University of Western Ontario, Chubu University and Fukushima

University for their helphl comments.
-mail address: ooaki@econ.tohoku.ac.jp$\tau_{\mathrm{e}-}$45

1. Introduction
This paper generalizes the conditional expectation&amework by replacing the Hilbert-

norm ( $i.e.,$ -norm) with a more general measure of approximation errors, which enables us todevelop the concept of conditioning for non-expectation oertainty equivalents, and to constructa-agent model in whi& the optimal level of information is endogenously determinedthrough the agent’s optimization behavior, and in which the anti-preference for information could
dominate the benefits of better planning which would become
able under more information.
We ffist develop the concept of conditioning
for non-expectation certainty equivalents. Here is a random utihty and is a-algebra which represents partial information
held by the agent. To this end, we assume that the agent is endowed with a functional, by
which she measures apprmimation errors. We then define
as the best approximation
to within the -measurable functions when approximation errors are measured by this error ffictional (its-definition will be proved in Proposition 2.2). If the error functionalis specified by the -norm,
coincides with the conditional expectation. We
thethus defined as non-Hd bert conditionin9. Given the error functional, we
can define the associated certainty equivalent as the best approximation within the constants.
Therefore, for any (non-expectation) certainty equivalent $M$ , defining its conditioning can be
reduced to finding the error hnctional which defines the given certainty equivalent. The sameerror functional is now used to define
for any . In this manner, Section 3 develops the$L^{2}$$\sin_{\mathrm{o}}^{g}1$$\mathrm{a}\mathrm{v}\mathrm{a}4$$M(x|\mathcal{G})$$x$$\mathcal{G}$$\mathrm{s}\mathrm{u}\mathrm{b}-\sigma$$M(x|\mathcal{G})$$x$$\mathcal{G}$$\mathrm{w}\mathrm{e}\mathrm{U}$$L^{2}$$M(x|\mathcal{G})$$E(x|\mathcal{G})$$\mathrm{c}\mathrm{a}\mathrm{U}$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{n}_{\Leftrightarrow}\sigma$$M(x|\mathcal{G})$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$\mathcal{G}$for a broad class of non-expectation certainty equivalents.

While the conditional expectation is unbiased, the non-Hilbert conditioning is typicallybiased in the sense that $M(M(x|\mathcal{G}))\neq M(x)$ . Here we interpret the LHS as the current “mean”
utihty when the agent expects partial information to be obtained in the future, and the
RHS as the current “mean” utihty when the agent expects no more information. Therefore, it
affects agent’s welfare whether or not she gets future information. In particular, obtaining some
information may decrease agent’s current utihty, which in turn implies that she may prefer
not to be informed (rather than be informed) although new information would promote better
planning. For example,
the person who hates to be informed of the result of a medical
examination regardless of a better medical treatment he could get if he found some disease. This
aspect is missing in the standard conditional expectation ffamework. In Section 4, a simple 2period model
be constructed in which the agent chooses the level of information (together
with control variables) in order to maximize her
utihty. We show that it could be thecase that being ignorant is preferred to being informed. This new approach would allow us toendogenize the evolution of agent’s information in dynamic models as a result of optimization
behavior. In Section 5, we briefly mention the future research along this line.
$\mathcal{G}$$\mathcal{G}$$\mathrm{i}\mathrm{m}\mathrm{a}_{\Leftrightarrow}\sigma\dot{\mathrm{m}}\mathrm{e}$$\mathrm{w}\mathrm{i}\mathrm{U}$$\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathbb{I}$46

2. General Theory, where $F$ is a -algebra
the states of the world by a probabihty spaceWeof all conceivable events on , and is a probabihty measure on $F$. We then define the spaoeof essentially bounded real-valued ffictions on by$\sigma$$(\Omega,\mathcal{F},\mu)$$\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}\theta$$\Omega$$\mu$$\Omega$$L^{\infty}(\mathcal{F})=${$x:\Omegaarrow\Re|x$

is -measurable and$\mathcal{F}$$\mathrm{e}\mathrm{s}\mathrm{s}\sup_{w\in\Omega}|x(\omega)|<+\infty$}.

, and simply write
everywhere
if
with
We
as the set of -equivalence cksses of hnctions. We denote
as $x=y$ . That is, we regard, we say that is almost-algebras of $F$. Whenthe space of allby
the space. We denote byif there aeists such that is -measurabk and.
of essentially bounded ahnost -meaeurable hnctions, which becomes a subspace of
may be regarded as a random utihty.
or
In the current paper, any element ofMore specifically, we may consider as $x(\omega)=u(w(\omega),\omega)$ , where is a state-dependent von
Neumann-Morgenstern utihty function and is a random income. We are more concerned with$y\in L^{\infty}(\mathcal{F})$$\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}q_{X}\in L^{\infty}(F)$$L^{\infty}(F)$$F^{*}$$\mu-$$y$$\mathcal{G}- measu7\mathrm{u}ble$$x$$\mathcal{G}\in F^{*}$$\mathrm{s}\mathrm{u}\mathrm{b}-\sigma$$y$$(\mathrm{a}.\mathrm{e}.)$$x=y\mu \mathrm{a}\mathrm{l}\mathrm{m}\mathrm{o}\mathrm{e}\mathrm{t}$$L^{\infty}(\mathcal{G})$$x=y\mu- \mathrm{a}.\mathrm{e}$$\mathcal{G}$$L^{\infty}(F)$$\mathcal{G}$$\mathrm{a}\mathrm{U}$$L^{\infty}(\mathcal{G})$$L^{\infty}(\mathcal{F})$

$u$

$x$

$w$

the attitude toward uncertainty (that is, how totoward wealth (that is, the curvature of u).
For any $x\in L^{\infty}(F)$ and for any$\mathrm{a}\mathrm{g}_{0}\emptyset \mathrm{T}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{e}x$$\mathcal{G}\in F^{*}$

over states) rather than the attitude, the conditional expectation$E[x|\mathcal{G}]$is definedby$E[x| \mathcal{G}]=\mathrm{a}x\mathrm{g}\min\{||x-z||_{2}|z\in L^{\infty}(\mathcal{G})\}$(1),

spaces, in which caseis a closed
the minimum is always uniquely attained by the orthogonal projection sincewhere$||\cdot||_{2}$is the$L^{2}$-norm. This definition is typicaUy applied to$L^{2}$$L^{2}(\mathcal{G})$is up to a -equivalence class and we can always, this definition can be directly
choose a version which is -measurable. Because
spacesapplied to the current context. Furthermore, if the above definition is extended tosubspace of $L^{2}(F)$ . The uniqueness of$E[x|\mathcal{G}]$$\mu$$L^{\infty}\subset L^{2}$$\mathcal{G}$$L^{1}$

by the standard approximation argument, it coincides with a more common definition of theconditional expectation.1 Among the implications of this definition are$E[x]=E[x|\{\phi, \Omega\}]$

(2)

.

(3)

and
$E[x]=E[E[x|\mathcal{G}]]$The conditional expectation (1) can be interpreted as the “best” apprmimation to-measurable functions when approximation errors are measured by the -norm.

$x$

within the$L^{2}$$\mathcal{G}$is defined as a -meaeurable integrable function which satisfies
ngsley (1986, p. 477, 34.15).
. In this regard, aee

lThat is,$\int_{G}xd\mu$$E[x|\mathcal{G}]$$\mathcal{G}$$\mathrm{B}M$$( \forall G\in \mathcal{G})\int_{G}E[x|\mathcal{G}]d\mu=$47

We now extend (1) to a more general(1) with a more general measure of errors, which we

first define a partial order on$L^{\infty}(F)$and$x_{+}$$x$$\mathrm{w}\mathrm{i}\mathrm{U}$$L^{2}$

byand $x_{-}\leq y-,$$x\preceq y\Leftrightarrow x_{+}\leq y+$

where-norm incall error functional. To this end, weconcept by replacing the$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\dot{\mathrm{m}}_{\mathrm{o}}\sigma$-are the positive part and the negative part of , respectively. This order$x$

measures the “closeness”. That is, is “closer” to than is if $x-z\preceq x-z’$ . In this case, both]of the overestimate and thennaate (at each state) by arethan those by . Ako
note that this order is a proper subset of the order by the absolute values:
$z$

$z’$

$x$

$\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}\dot{\mathrm{u}}$

$z$

$z’$

$\mathrm{l}\mathrm{e}\mathrm{r}$$\mathrm{s}\mathrm{m}\mathrm{a}$$x\preceq y\Rightarrow|x|\leq|y|$but$\mathrm{n}\mathrm{o}\mathrm{t}\Leftarrow \mathrm{i}\mathrm{n}$general.

Let $X$ be a convex cone ofstated. An ervvr functional on $X$ is a$L^{\infty}(F)$. In what$\mathrm{m}\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma\Phi$$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{s}$, we set $X=L^{\infty}(F)$ unless otherwise: $X\mathrm{x}Xarrow\Re$ which satisfies the$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}$sixaxioms.2El.

$(\forall x, z\in X)$

E2.

$\Phi(x, z)=0\Leftrightarrow x=z$E3.

$x-z\preceq x-z’\Rightarrow\Phi(x, z)\leq\Phi(x, z’)$E4.

$(\forall x\in X)$

E5.

$(\forall x, z_{1}, z_{2}\in X)(\forall\lambda\in(0,1))$$z_{1}\neq z_{2^{3}}$$\Phi(x, z)\geq 0$$\Phi(x, \cdot)$

and

;

;
;

is a convex function;$(\exists \mathcal{G}\in \mathcal{F}^{*})z_{1},$$z_{2}\in L^{\infty}(\mathcal{G})$and$x\not\in L^{\infty}(\mathcal{G})$and$\Rightarrow\Phi(x, \lambda z_{1}+(1-\lambda)z_{2})<\lambda\Phi(x, z_{1})+(1-\lambda)\Phi(x, z_{2})|$

E6.

$(\forall x\in X)(\exists p\in[1, +\infty))$$||z_{n}-z_{0}||_{\mathrm{p}}arrow 0\Rightarrow\varliminf_{narrow\infty}\Phi(x, z_{n})\geq\Phi(x, z_{0})$,-norm. The number given bymeasures the error when we
proximate by . El and E2 are normalization axioms which are imposed so that the best
approximation of is achieved by itself with no error. E3 requires consistency of the error
with the “closeness” discussed earlier.
are technical axioms which are sufficient for the
existence of the best approximation. E4 and E5 require thatis convex and strictly convex, and E6 requires thatis strongly lower semi-continuous.
resembles a metric,
we do not impose the symmetric axiom: $\Phi(x, z)=\Phi(z, x)$ .
We can construct an error functional&om the functional : $Xarrow\Re$ which satisfiae:where$||\cdot \mathrm{t}|_{p}$is the$x$$L^{p}$$\Phi(x, z)$$\mathrm{a}\triangleright$

$z$

$x$

$x$

$\mathrm{E}\not\subset \mathrm{E}6$$\Phi(x, \cdot)$$\Phi(x, \cdot)$$\mathrm{W}\mathrm{l}\dot{\mathrm{u}}\mathrm{l}\mathrm{e}\Phi$$\hat{\Phi}$

2The domain of$3_{Z_{1}}\neq z_{2}$implicitly requires that $\Phi(x’, z’)=\Phi(x, z)$ if
means that there does not exist $N$ such that$\Phi$$x’=x\mathrm{a}.\mathrm{e}$. and$z’=z\mathrm{a}.\mathrm{e}$$\{\mathrm{l}v|z_{1}(\omega)\neq z_{2}(\omega)\}\subset N$.

and $\mu(N)=0$

.

48

;

Nl.

$(\forall x\in X)\hat{\Phi}(x)\geq 0$N2.

$\hat{\Phi}(x)=0\Leftrightarrow x=0$

N3.

$x\preceq y\Rightarrow\hat{\Phi}(x)\leq\hat{\Phi}(y)$N4.

$(\forall x\in X)\lambda\geq 0\Rightarrow\hat{\Phi}(\lambda x)=\lambda\hat{\Phi}(x)$N5.

$(\forall x,y\in X)\hat{\Phi}(x+y)\leq\hat{\Phi}(x)+\hat{\Phi}(y)$N6.

$\hat{\Phi}(x+y)<\hat{\Phi}(x)+\hat{\Phi}(y)$N7.

$(\exists p\in[1,+\infty))$

N7 can be repkced by the

N7’.

;
;

;
;

unless $(\exists c>0)x=w$ ; and.

$||x_{\mathfrak{n}}-x_{0}||_{p}arrow 0\Rightarrow\underline{\mathrm{h}\mathrm{m}}_{narrow\infty}\hat{\Phi}(x_{\mathfrak{n}})\geq\hat{\Phi}(x_{0})$axiom.
$\cdot$$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}$$(\exists p\in[1, +\infty)$

and $K>0$ )$(\forall x)$$\hat{\Phi}(x)\leq K||x||_{\mathrm{p}}$that N5 and N7’ imply N7. By N4 and N5,
It immediately
While resembles a norm, it is not symmetric in the sense that$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{s}$$\hat{\Phi}$.

is a Minkkowski functional.

$\hat{\Phi}(-x)\neq\hat{\Phi}(x)$$\hat{\Phi}$

Lemma 2.1:

Define$\Phi(x, z)\equiv\hat{\Phi}(x-z)$.

. If satisfies Nl-N7, then satisfies El-E6.
$\hat{\Phi}$$\Phi$(Proof in Appendix)

Given an error functional , we define$\Phi$$M_{\Phi}(x| \mathcal{G})=\arg\min\{\Phi(x,z)|z\in L^{\infty}(\mathcal{G})\}$$(\forall x\in X)(\forall \mathcal{G}\in \mathcal{F}^{*})$.

(4)

to within the ahnost, if it exists, is the “best”
measurable functions where approximation errors are measured by the error functional . WhenThe class of functions$M_{\Phi}(x|\mathcal{G})$$x$$\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{r}\alpha \mathrm{i}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathcal{G}-$$\Phi$is (almost) -measurable, $z=x$ uniquely attains the minimum by El and E2. When is not(ahnost) -measurable, the next proposition guarantees the existence of the best approximation.

$x$

$x$

$\mathcal{G}$$\mathcal{G}$Proposition 2.2: Let

is well-defined and unique (up$\Phi$$io$

be an $e77vrfi\mathrm{z}nctional$. Then for any and for any
a -equivalence class). (Proof in Appendix)$x$$\mathcal{G},$$M_{\Phi}(x|\mathcal{G})$$\mu$

whi&is -measurable, and refer to it as the$non- Hilbe\tau t$ conditioning. The non-Hilbert conditioning (4) generalizes the conditional expecWe can always choose a version oftation (1), and coincides with

$11*$

when

$\Phi$$M_{\Phi}(x|\mathcal{G})$is specified by the$\mathcal{G}$$L^{2}$-metric. Any strictly

can generate the same conditioning concept.
We define the certainty equivalent 4 generated by an error functionaltransformation of$\Phi$$(\forall x\in X)$$x$$\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\sin_{\mathrm{o}}\sigma$$M_{\Phi}(x)=M_{\Phi}(x|\{\phi,\Omega\})$.

4The term “certainty equivalent” is used for $M$ only to mean that $M(x)=x$ whenrather than a rondom income.
is now a random$\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{h}\psi$$\Phi$

by
(5)

$x$

is constant. Recall that

49
The -equivalence class of functions$\mu_{r}$$M_{\Phi}(x)$is the “best” approximation to

constant functions where approximation errors are measured bysion of$M_{\Phi}(x)$whi&is constant. The certaintyequivalent-$\Phi$$x$

within the (almost). Again, we can chooee a vergenerahzes the expectation$M_{\Phi}(\cdot)$, and coincides with it when is specified by the -metric. Note that the equation (5) is the definition while the equation (2) is an implication of the conditional expectation.
Given a oertainty equivalent $M$ , we can develop the concept of conditioning for $M$ by seeking
for an error ffinctional such that $(\forall x)M(x)=M_{\Phi}(x|\{\phi, \Omega\})$ . Such a then would be used
to define the conditioning of $M$ against any
by means of (4). Section 3 develops thefunctional$E(\cdot)$$L^{2}$$\Phi$$\Phi$$\Phi$$\mathcal{G}\in F^{*}$

conditioning for a broad ckss of non-expectation certainty equivalents in this way. Henceforth,
we suppress and simply write as
and $M(x)$ when is understood.
$\Phi$$M(x|\mathcal{G})$An error hnctional$\varphi:\Rearrow\Re$$\Phi$

is of integml$\Phi$$fom\mathrm{L}$if there exist mappings:$\phi$$\Re^{2}arrow\Re$

andsuch that$( \forall x,z\in X)\Phi(x, z)=\varphi(\int_{\Omega}\phi(x(\omega), z(\omega))\mu(d\omega))$and if$\Phi$$M(x|\mathcal{G})$satisfies EI-E6. The error functionak of this form
and $M(x)$ , which$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{s}$convenient characterization of
immediately ffom the definition.
$\mathrm{a}\mathrm{U}\mathrm{o}\mathrm{w}$Lemma 2.3: Let be of integml form. If is continuously differentiable in its second
argument and if is strictly increasing, then, for any $x\in X,$ $M(x)$ is defined as the unique$\Phi$$\phi$$\varphi$solution$z$to$\int_{\Omega}\phi_{2}(x, z)d\mu=0$

and$M(x|\mathcal{G})$satisfies$(\forall h\in L^{\infty}(\mathcal{G}))$$\int_{\Omega}\phi_{2}(x, M(x|\mathcal{G}))hd\mu=0$

Lemma 2.4: Let$\Phi$

be of integml $fo7m$ . If

if is strictly increasing, then$\varphi$$M(\cdot)$$\phi$.

is twicely continuovsly differentiable andis Gateavx differentiable and$\partial M(x;h)=-\frac{\int_{\Omega}\phi_{21}(x,M(x))hd\mu}{\int_{\Omega}\phi_{22}(x,M(x))d\mu}$

For each $x\in X$ , the value of infomation$\mathcal{G}$$M(M(x|\mathcal{G}))$.

is a red number$V_{x}(\mathcal{G})\equiv..M(M(x|\mathcal{G}))-M(x)$Here$(\forall x, h\in X)$$V_{x}(\mathcal{G})$defined by.

is the current “mean” utility when the agent expects partial information

be obtained in the future, while $M(x)$ is the current “mean” utihty when the$\mathrm{a}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{t}$$\mathcal{G}$toexpects no50

more idformation. When $M$ coincides with the expectation $E,$$\mathcal{G}$because of (3). However,$V$$V_{x}(\mathcal{G})=0$

for any$x$and for any

could be either positive or negative in general because$M(x)\neq M(M(x\}\mathcal{G}))$.

(6)

The “biaeedness” (6) of non-Hilbert conditioning contrasts with the “unbiasedness” (3) of conerror ffinctional is indifferent to the paxtial infforditional expectation. The agent with themation she might attain in the future (apart ffom the benefits of better planning which would$L^{2}$

become available with the additional information). On the other hand, with the non-Hilbertconditioning, the agent may strictly prefer to be uninformed. The preferenoe over information
can be introduced only by the “biasedness” of non-Hilbert conditioning. This new feature of

non-Hilbert conditioning$\mathrm{w}\mathrm{i}\mathrm{U}$be further$\mathrm{i}\mathrm{U}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}$by the examples in Sections 3 and 4.

3. Examples
This section considers several examples of non-expectation certainty equivalent, for eachof which we develop the concept of conditioning. We ako see its implications to the value of
information.

\’A.

$L^{p}$

Error Rnctional

Let $p\in(1, +\infty)$ . The $IPe$ rvor functional is defined by:$\Phi(x,z)\equiv(\int_{\Omega}\phi(x(\omega), z(\omega))d\mu(\omega))^{1/p}$

This is the metric$(1, +\infty)$since$0\sigma \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}$$M(x|\mathcal{G})$, where $\phi(x, z)=|x-z|^{p}$ .

by $IP$-norm and clearly satisfies EI-E6. We need to assume that $p\in$

is not unique in general when

$p=1\mathrm{o}\mathrm{r}+\infty$. When $p=2$ , the conditionalbe recovered. Dependent upon the value of , the certainty equivalent generated
by this error functional exhibits different sensitivity to events’ probabilities. $M$ is very sensitive
expectation$\mathrm{w}\mathrm{i}\mathrm{U}$$p$

to probabihties when is close to one, while large shows the insensitivity of $M$ to probabihties.
, and let. When $p=1,$ $M(x)=x(\omega_{i})$ , where
To see this, let) On the. (When there is a tie, $M(x)$ is any convex combination of such$p$$\Omega=\{\omega_{1},\omega_{2}\}$$p$$(\forall i)p_{\dot{2}}\equiv\mu(\{\omega_{i}\})>0$$x(\omega_{i})’ \mathrm{s}.$$i=\arg \mathrm{m}\mathrm{a}s\mathrm{c}_{j}p_{\mathrm{j}}$.
other hand, when $p=+\infty,$ $M(x)=(’1/2)(x(\omega_{1})+x(\omega_{2}))$ regardless of
simple example shows. Let
For $p\neq 2,$ $M(x)\neq M(M(x|\mathcal{G}))$ in general as the$p_{i}’ \mathrm{s}$$\mathrm{f}\mathrm{o}\mathbb{I}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$\Omega=\{\omega_{1},\omega_{2},\omega_{3}\};(\forall i)p_{i}\equiv\mu(\{\omega_{i}\});\mathcal{G}=\{\phi, \{\omega_{1}\}, \{\omega_{2},\omega_{3}\}, \Omega\};x_{1}\equiv x(\omega_{1})=x(\omega_{2})<x_{3}\equiv$$x(\omega_{3})$; and $\gamma\equiv 1/(p-1)$ . Then
$M(x)= \frac{(p_{1}+p_{2})^{\gamma}}{(p_{1}+p_{2})^{\gamma}+p_{3}^{\gamma}}x_{1}+\frac{p_{3}^{\gamma}}{\mathrm{b}+p_{2})^{\gamma}+p_{3}^{\gamma}}x_{3}$

and

51

$M(M(x| \mathcal{G}))=\frac{p_{1}^{\gamma}(\mathrm{p}_{2}^{\gamma}+p_{3}^{\gamma})+p_{2}^{\gamma}(p_{2}+p_{3})^{\gamma}}{(p_{2}^{\gamma}+p_{3}^{\gamma})(p_{1}^{\gamma}+(p_{2}+p_{3})^{\gamma})}x_{1}+\frac{p_{3}^{\gamma}(p_{2}+\mathrm{p}_{3})^{\gamma}}{(p_{2}^{\gamma}+p_{3}^{\gamma})(p_{1}^{\gamma}+(p_{2}+p_{3})^{\gamma})}x_{3}$.

The several lines of algebra show that
$M(x)\geq M(M(x|\mathcal{G}))\Leftrightarrow p\geq 2$$\mathrm{A}\mathrm{l}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}_{\Leftrightarrow}\sigma \mathrm{h}$.

this example is very specific and general cases seem to be very complicated, some

intuition can be derived ffom this example. First, assume that $p>2$ . For this case, an intermediate
dampens (resp. promotes) the mean when the aggregation takes
place among better (resp. worse) outcomes. This is because the dues of outcomes (rather than
the associated probabihties) count more and the best (resp. worst) outcome is always averaged
downward (resp. upward). In the above example, the intermediate aggregation lowers the best
outcome and hence dampens the mean. Next, assume that $p<2$ . For this case, an intermediate aggregation $(i.e., M(x|\mathcal{G}))$ promotes (resp. dampens) the mean when the relatively highprobabihties are associated with the lower (resp. higher) outcomes. This is because the increaseof the probabihty of better (resp. worse) events by the aggregation may cause the reversal of
relative probabilities between better outcomes and worse outcomes. In the above example, theaggregation causes the increase in the probabihty of better outcome, which contributes increaseof the mean.
$\mathrm{a}\mathrm{g}\circ \mathrm{o}^{\mathrm{T}\mathrm{e}}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}(i.e., M(x|\mathcal{G}))$

B. Asymmetric Error ffinctional

Let $p\in(1, +\infty)$ and let $\gamma>0$ . The $asymmet7^{\cdot}icer7vr$ fimctional is defined by:$\gamma|x-z|^{\mathrm{p}}$$\Phi(x, z)\equiv(\int_{\Omega}\phi(x(\omega), z(\omega))d\mu(\omega))^{1/p}$, where $\phi(x, z)=\{$$|x-z|^{p}$if$x\geq z$

if $x<z$ .

When $\gamma<1$ , the certainty equivalent generated by this error functional is a parametric specification of preferences studied by Gul (1991), and aehibits disappointment aversion. This error

functional measures the approximation errors asymmetricaUy. That is, an underestimate is tolerated (resp. penalized) compared with an overestimate when $\gamma<1$ (resp. $\gamma>1$ ). This error
functional can be generated ffom the asymmetric “norm” which is defined in the next lemma.

Lemma 3.1: Let $p\in(1, +\infty)$ and let a, $b>0$ . Define$(\forall x\in\Re)$Finally, define$||x||_{\gamma}$$\gamma:\Rearrow\Re$

$a$

if

$x\geq 0$$b$if$x<0$ .

$\gamma(x)=\{$

by$||x||_{\gamma}=( \int_{\Omega}\gamma(x)|x|^{p}d\mu)^{1/p}$

by

52

Then $(\forall x, z\in X)$$(\exists c>0)x=cz$$||x+z||_{\gamma}\leq||x||_{\gamma}+||z||_{\gamma}$. (Proof in Appendix). Furthemort,$||x+z||_{\gamma}<||x||_{\gamma}+||z||_{\gamma}$

urdess

Proposition 3.2: The asymmetric $em\tau$ functional is wdl-defined.
Proof: By Lemma 2.1, it suffices to prove thatare obvious. N5 and N6$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}$When $p=2$ , this error

ffom Lemma 3.1.

$\mathrm{N}7’$

satisfies NI-N7. NI-N4

holds with $K=( \max\{a,b\})^{1/\mathrm{p}}$.

. ctional has a convenient$\mathrm{f}\mathrm{m}$as the next proposition shows.

$\hat{\Phi}(x)=||x||_{\gamma}$$\square$implication on the value of information

Proposition 3.3: Let $M$ be genemted by the asymmetric error fimctional urith $p=2$ .
Then for any and for any $x,$ $M(x)\geq M(M(x|\mathcal{G}))\Leftrightarrow\gamma\leq 1$ . (Proof in Appendix)$\mathcal{G}$C. Quasilinear Error Functional
For this example, aet $X$ to be the positive cone of$L^{\infty}(F)$, and let$\alpha>0$. The quasdinear

error $fi_{4}nctional$ is defined by:$\Phi(x, z)\equiv\int_{\Omega}\phi(x(\omega),z(\omega))d\mu(\omega)$, where$\phi(x, z)=\alpha x^{\alpha+1}-(\alpha+1)x^{\alpha}z+z^{\alpha+1}$

be recovered. The certainty equivalent generated
When $\alpha=1$ , the conditional expectation.
by this error functional is a parametric family of quasihnear mean,$\mathrm{w}\mathrm{i}\mathrm{U}$$(E[x^{a}])^{1/\alpha}$

Proposition 3.4: The quasilinear error functional
Proof: This is a special case of Proposition 3.5.

By Lemma 2.3, we can compute$M(x|\mathcal{G})$$\dot{u}$wdl-defined.

$\square$explicitly as$(\forall x)(\forall \mathcal{G})M(x|\mathcal{G})=(E[x^{\alpha}|\mathcal{G}])^{1/\alpha}$Hence, for this error functional, the value of information is always$(E[E[x^{\alpha}|\mathcal{G}]])^{1/\alpha}=(E[x^{\alpha}])^{1/\alpha}=M(x)$$0$.

because $M(M(x|\mathcal{G}))=$.

D. Implicit Error Functional
For this exampk, we first define the certainty equivalent, and then seek for the error
functional which generates this certainty equivalent. Given $x\in X,$ $M(x)$ is defined as theunique solution $z\in\Re$ to$\int_{\Omega}\varphi(x(\omega), z)d\mu(\omega)=0$

is strictly decreasing, and (\"ui)is continuous. Note that $z=M(x)$ is-defined by the mean value theorem Thisimplicit mean, was studied by Fishburn (1986). By the continuity
certainty equivalent,

where$\varphi$:$\Re^{2}arrow\Re$satisfies: (i) $\varphi(x, x)=0,$ (\"u)$(\forall x)\varphi(x, \cdot)$$(\forall x)\varphi(x, \cdot)$$\mathrm{w}\mathrm{e}\mathrm{U}$$\mathrm{c}\mathrm{a}\mathrm{U}\mathrm{e}\mathrm{d}$53such that $(\forall x, z)\phi_{2}(x, z)\wedge=-\varphi(x, z)$ . Furthermore,strictly convex by
and has a minimum at $z=x$ by (i). FinaUy, let
We define the implicit error fimctional by:(\"ui), there exists$\Re^{2}arrow\Re$$\phi\wedge:$is$\phi(x, \cdot)\wedge$$\phi(x, z)\equiv\phi(x, z)-\phi(x, x)\wedge\wedge$$(\ddot{\mathrm{n}})$$\Phi(x,z)\equiv\int_{\Omega}\phi(x(\omega), z(\omega))d\mu(\omega)$.

.

Note that the quasilinear error functional is recovered when $\varphi(x, z)=(\alpha+1)x^{\alpha}-(\alpha+1)z^{\alpha}$ .

Proposition 3.5:

The implicit error functional is $wdl$-defined, and it genemtes theimplicit mean $M(x)$ .

Proof: EI-E6
sinoe
is a strictly convex
which has the minimized
value at $z=x$ . E7 holds by Aubin and Ekeland (1984, p. 13) sinoe
is continuous. The$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}$$\phi(x, \cdot)$$\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$0$$\phi(x, \cdot)$

second statement in the proposition is obvious by Lemma 2.3.
By the construction ofwhich generates the$0\sigma \mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}$$\Phi$$\square$and Proposition 3.5, we can always find the error functional

implicit mean. The value of information is non-zero except when$\varphi$

is separabk (as in Example C).

E. Rank-dependent Error Functional

Let be a normalized capacity on, that is, 9: $Farrow[0,1]$ is a mapping which
satisfies
, and
. We
ako assume that is convex
in the sense that $\theta(A\cup B)+\theta(A\cap B)\geq\theta(A)+\theta(B)$ . A simple exampk of a convex capacity is$\theta$$(\Omega,\mathcal{F})$$\theta(\phi)=0,$$\theta(\Omega)=1$$(\forall A)\theta(A)=(\mu(A))^{\alpha}$, wherewith respect to a capacity$\theta$$A\subset B\Rightarrow\theta(A)\leq\theta(B)$$\alpha\geq 1$

. Let

$X$

$\mathrm{w}\mathrm{i}\mathrm{U}$be the positive cone of$\theta$$L^{\infty}(\mathcal{F})$. A Choquet integml

is defined by:$(\forall x\in X)$$\int_{\Omega}x(\omega)\theta(\ ) \equiv\int_{0}^{\infty}\theta(\{\omega\in\Omega|x(\omega)\geq t\})dt$,

where the integral in the right-hand side is an improper Riemann (or equivalently, Lebesgue)$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{a}1$$\theta’ \mathrm{s}$. The Choquet$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\circ\tau \mathrm{a}\mathrm{l}\circ$cruciaUy depends upon the ranking of outcomes, and (given

convexity) it captures the notion of uncertainty aversion (for exampk, see Gilboa, 1987,

Schmeidler, 1989, and Chateauxneuf, 1991).
We construct the error functional which generates the oertainty equivalent defined by
the Choquet$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\circ\tau \mathrm{a}\mathrm{l}\circ\cdot$

To this end, we introduce a couple of concepts. We denote by $ba$ the spaceof bounded charges. That is, $ba$ is the space of bounded finitely additive set fimctions on
which are$\mathrm{n}\mathrm{u}\mathrm{U}$at$\phi$

and absolutely continuous with respect tothe topological dual of
uniformly converges to$L^{\infty}(F)$$x$. Let {$\varphi_{n})$$\mu$. Note that$(\Omega,\mathcal{F})$$ba=(L^{\infty}(\mathcal{F}))^{*}$,

be a nondecreasing sequence of simpk functions which. Such a sequence always exists. Then the Dunford-Schwartz integml

54

of $x\in X$ with respect to $\nu\in ba$ is defined by$\int_{\Omega}x(\omega)\nu(h)\equiv\lim_{\mathrm{n}arrow\infty}\int_{\Omega}\varphi_{\mathfrak{n}}(\omega)\nu(h)$,where the integrak of simple functions in the RHS are defined in the same manner as for an
exists and is independent of the choioe of theintegral.with respect to a measure. The
sequence of simple fimctions (Danford and Schwartz, 1954, p. 111). Finally, the core of the$\mathrm{h}\iota\dot{\mathrm{m}}\mathrm{t}$normalized capacity 9 is defined bycore(9)It immediately(1986):$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{s}$

$=$

that

{

and $\nu(\Omega)=1$ }.

$\nu\in ba|(\forall A\in F)\mathit{9}(A)\leq\nu(A)$$\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}(\mathit{9})\backslash$is weak * compact. The$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}$Lemma 3.6: Let 9 be a normalized oepacity. Then$(\forall x\in X)$$\theta$lemma is due to Schmeidler

is convex if and only if$\int_{\Omega}xd\theta=\mathrm{m}\dot{\mathrm{m}}\{\int_{\Omega}xd\nu|\nu\in \mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}(\theta)\}$We now define the error functional. Let
ffom $X$ into core
Then define a mapping$xrightarrow\nu_{x}$$\theta$$(\theta)$.

be a normalized capacity which is convex.
such that$\int_{\Omega}xd\theta=\int_{\Omega}xd\nu_{x}$,. Such a mapping exists by Lemma 3.6 while itis not unique in general. We define the $mnk$-dependent rror functional .

where the RHS is the Dunford-Schwartz$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{r}\mathrm{a}1$$\mathrm{b}\mathrm{y}^{9}$$\mathrm{e}$$\Phi(x, z)\equiv\int_{\Omega}(x(\omega)-z(\omega))^{2}d\nu_{x}(\omega)$.

Note that the integral is again in the sense of Dunford and Schwartz since$\nu_{x}$is a charge rather

than a measure. We can prove

Proposition 3.7: The $mnk$-dependent error functional is well-defined, and it genemtes
the Choquet integml. (Proof in Appendix)

For this error functional, the value of information is in general non-zero. Lemma 2.3(and the similar argument to the proof of Proposition 3.7) shows$M(x)= \int_{\Omega}xd\nu_{x}=\int_{\Omega}M(x|\mathcal{G})d\nu_{x}$but the last expression does not equal $M(M(x|\mathcal{G}))$ since$\nu_{x}$and,$\nu_{M(x|\mathcal{G})}$

are in general different.

55

4. Application to 2-Period Dynamic Model: An Example
The general two-period model is daecribed as follows. Let
be the set of
possibk states of the world of tomorrow together withconoeivabk events and a probabihty$(\Omega,F,\mu)$$\mathrm{a}\mathrm{U}$$\mathrm{a}\mathrm{U}$measure on it. And let$c_{0}$$c_{0}$

andis a non-negative real and$c_{1}$

be a consumption of today and of tomorrow, respectively.

$c_{1}$:

is a function of tomorrow’s state. The feasibihty$\Omegaarrow\Re_{+}$

of a consumption plan is described by the compact-valued and $F$-measurabk correspondence. It

be required that
. A function $u:\Omegaarrow\Re$ is a
$F$-measurabk utihty
function which gives future’s utihty given tomorrow’s state . A function$W:\Re_{+}\cross\Rearrow\Re$ is an intertemporal aggregator which generates
today’s utihty given today’s
consumption and tomorrow’s “mean” utility. In order to caluculate tomorrow’s “mean” utihty,
the agent employs the certainty equivalent $M$ which is generated by some error ffinctional.
The information structure of the model is as. Let be the spaoe ofalgebras of $F$, and let
be some subset of . Today, the agent chooses any
togetherwith a feasibk consumption plan
. The information will be revealed to the agent
tomorrow, and must be chosen so as to be -measurable. Therefore, the finer is, the morevariety of functions the agent can consider as a possibk plan of tomorrow’s consumption .
This substantiates the idea that the
information contributes the utihty by
the more flexible planning. FinaUy, the utihty of tomorrow
be computed as a conditional
utihty given , and this conditional utihty
beagain over states by $M$ to generate“mean” utihty of tomorrow. When $M$ is given by the expectation operator and when $W$ islinear in its second argument, this doubk aggregation lea&to the identical “mean” utihty
of . But since $M$ need not be now, the “mean” utihty may be affected by the
choice of .
In a summary, the agent chooses, and to maximize:$F:\Omegaarrow\Re_{+}^{2}$$(\forall\omega)(c_{0},c_{1}(\omega))\in F(\omega)$$\mathrm{w}\mathrm{i}\mathrm{U}$$\omega$$F^{*}$$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{s}$$\mathrm{s}\mathrm{u}\mathrm{b}-\sigma-$$F^{*}$$\mathcal{G}^{*}$$\mathcal{G}\in \mathcal{G}^{*}$$(c_{0}, c_{1})$$\mathcal{G}$$\mathcal{G}$$\mathrm{c}_{1}$$\mathcal{G}$$c_{1}$$\mathrm{a}\mathrm{d}\mathrm{d}\backslash \mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}$$\mathrm{a}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}$$\mathrm{w}\mathrm{i}\mathrm{U}$$\mathcal{G}$$\mathrm{w}\mathrm{i}\mathrm{U}$$\mathrm{a}_{\Leftrightarrow}\sigma \mathrm{g}\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}$

$E$

$E$

$\mathcal{G}$$\mathrm{r}\mathrm{e}_{\mathrm{o}}\sigma\pi \mathrm{d}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{s}$$\mathcal{G}$$c_{0},$$\mathcal{G}$$c_{1}$$W(c_{0}, M[W(c_{1}(\omega), M[u(\omega)|\mathcal{G}])])$$(\forall\omega\in\Omega)$

subject to

$\{$

is

$(c_{0}, c_{1}(\omega))\in F(\omega)$and$\mathcal{G}\in g*$$c_{1}$(7)-measurable.

$\mathcal{G}$When $W(c, m)=v(c)+\beta m$ and $M=E$ , where $v:\Re_{+}arrow\Re,$ $\beta>0$ and $E$ is an expectation
operator, equation (7) is reduced to$v(c_{0})+\beta E[v(c_{1}(\omega))]+\beta^{2}E[,E[u(\omega)|\mathcal{G}]]$$=v(c_{0})+\beta E[v(c_{1}(\omega))]+\beta^{2}E[u(\omega)]$

56

Here

$\mathcal{G}$vanishes, and hence it cannot affect the overaU utihty. More idormation is always

preferred. Next suppose that $W(c,m)=m$ . Eqaution (7) is now reduced to$M[M[u(\omega)|\mathcal{G}]]$

Henoe if $M\neq E,$ could ffiect the overaU utihty even for this simple intertemporal aggregator.
not be preferred according to the antiA finer leads to better plannin , but it
preference for information itself. To illustrate this point, we now provide a specific exampk of
the above model in whi& the attainmment of new information (and hence more flexibk planning)
may not necessarily improve the agent’s overaU utihty.
;
Infomation Stmcture. Let. For this simple example, the agent has only
; and$\mathcal{G}$$\mathrm{m}\mathrm{a}\mathrm{y}\backslash$$\mathcal{G}$$\mathrm{g}$$\Omega=\{\omega_{1},\omega_{2},\omega_{3}\};\mathcal{F}=2^{\Omega};(\forall i)p_{i}\equiv\mu(\{\omega_{i}\});\mathcal{G}_{0}\equiv\{\phi, \Omega\}$$\mathcal{G}_{1}\equiv\{\phi,\{\omega_{1}\}, \{\omega_{2},\omega_{3}\},\Omega\}$$\mathcal{G}^{*}=\{\mathcal{G}_{0}, \mathcal{G}_{1}\}$or partial information which can’t distinguish state 2
two alternatives: no information at&om state 3. We assume that it takes no cost obtaining information.
. The feasibihty correspondence is given by:
Feasibility. Let
and let$\mathrm{a}\mathrm{U}$$\epsilon>0$$(\forall\omega)F(\omega)=\{$When$\overline{c}$$g_{0}${{$\overline{c}>0$}$(x_{0},x_{1})\in\Re_{+}^{2}|x_{0}=0$and$0\leq x_{1}\leq\overline{c}$$(x_{0},x_{1})\in\Re_{+}^{2}|x_{0}=0$and$0\leq x_{1}\leq\overline{c}+\epsilon$}

if$\omega=\omega_{1}$if$\omega=\omega_{2}$is chosen by the agent, the best consumption availabk tomorrow is given by. When$c_{1}(\omega_{1})=\overline{c}$$\mathcal{G}_{1}$or$\omega_{3}$$(\forall\omega)c_{1}(\omega)=$is chosen by the agent, the best consumption availabk tomorrow is given by

and$\mathrm{c}_{1}(\omega_{2})=c_{1}(\omega_{3})=\overline{c}+\epsilon$. The more information leads to a better consumption of

tomorrow as we expect.
Prejerence. Let $b>0$ . Define $W$ by $(\forall c,m)W(c, m)=c+m$ and
$0$

if

$b$

if

$\omega=\omega_{1}$

or

$u$

by

$\omega_{2}$$(\forall\omega)u(\omega)=\{$$\omega=\omega_{3}$The state 3 is special in the sense that the agent gets a bonus utility$b$only when this stateis realized. FinaUy, the certainty equivalent $M$ is the one generated by the$L^{p}$

error functional(which was introduced in Section 3. ). Under these specifications, equation (7) is reduced to$\mathrm{A}$$M(c_{1}(\omega)+M(u(\omega)|\mathcal{G}))$.

Let $\gamma=1/(p-1)$ . The overaU maximum utihty when the agent chooses$U_{0} \equiv\overline{c}+M(u(\omega))=\overline{c}+\frac{p_{3}^{\gamma}}{(p_{1}+p_{2})^{\gamma}+p_{3}^{\gamma}}b$$\overline{c}$$\equiv M(\mathrm{c}_{1}(\omega)+M(u(\omega)|\mathcal{G}))$

where $c_{1}(\omega)=\{$$\overline{c}+\epsilon$

is given by:.

On the other hand, the overaU maximum utility when the agent chooses$U_{1}$$\mathcal{G}_{0}$

if
if

.

$= \overline{c}+\frac{(p_{2}+p_{3})^{\gamma}}{(p_{1}^{\gamma}+(\mathrm{p}_{2}+p\mathrm{s})^{\gamma})}\epsilon+\frac{p_{3}^{\gamma}(p_{2}+p_{3})^{\gamma}}{(p_{2}^{\gamma}+p_{3}^{\gamma})(p_{1}^{\gamma}+(p_{2}+p_{3})^{\gamma})}b$$\mathcal{G}_{1}$is given by:$\omega=\omega_{1}$$\omega=\omega_{2}$or$\omega_{3}$57

As the exampk in Section 3. shows, $U_{0}=M(x)\leq M(M(x’|\mathcal{G}))=U_{1}$ when $p\leq 2$ . In this case,obtaining partial idformation merits theutihty (partly because of the better planningand partly because of the preference for information itself). On the contrary, when $p>2$ and$\mathrm{A}$$\mathrm{h}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{h}\mathrm{e}\mathrm{r}$isenough (or, is large enough), $U_{0}=\dot{M}(x)>M(M(x’|\mathcal{G}))=U_{1}$ . For instanoe, when$p>2$ (and hence) and $p_{1}=p_{2}=p_{3}$ , it immediatelythat $U_{0}>U_{1}$ if (and only if). In this case, the agent prefers to remain uninformed rather than become
idformed. This is becauae the anti-preference for information dominates the merit by the better
planning.
$\epsilon$$b$$\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{U}$$\gamma<1$$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{s}$$b>[2^{\gamma}/(1-2^{\gamma-1})]\epsilon$

5. Concluding Remarks
The simple exampk of 2–period model in Section 4 shows that the agent with a generalerror functional (which may be different&om$L^{2}$-norm) can exhibit a strict preference or a strictanii-preference for information, the latter of which sometimes dominates even the merit of better

planning. In this manner, the non-Hilbert conditioning
us to make more general modek inwhich the information
bejurisdictionaUy chosen by the agent and hence will be endogenized.
$\mathrm{a}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{s}$$\mathrm{w}\mathrm{i}\mathrm{U}$In the traditional dynamic modek, the accumulation of agent’s information is exogenously given
by some filtration of -algebras. In another word, the choice of variables the agent observes
to obtain information is out of her control (where the filtration is generated by the variables).
$\sigma$Multi-period (in particular, infinite-horizon) modek with the non-Hilbert conditionin contrast
with the traditional ones and would provide new economic implications by
theproblem of which variables to observe. In an attempt to analyze such modek (the$\mathrm{g}$$\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{o}_{\mathrm{O}}^{\sigma}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{z}i\mathrm{n}\mathrm{g}$$\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{d}\dot{\mathrm{m}}_{\mathrm{o}}\sigma$

general 2-period model described at the beginning of Section 4), it would be necessary to study
the existence of the optimal information level, which in turn requires the “continuity” of $M(x|\cdot)$with respect to some information topology on$F^{*}$. To this end, we would need to extend theargument developed by Allen (1983), which employs Boylan’s (1971) topology for information.

58

APPENDIX
implications are
Proof of Lemma 2.1: The. N4 and N5 imply E4 sinoe:; andmediate:$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}$E2;$\mathrm{N}1\Rightarrow \mathrm{E}1;\mathrm{N}2\Rightarrow$$\mathrm{N}7\Rightarrow \mathrm{E}6$$\mathrm{N}3\Rightarrow \mathrm{E}3$(8)$\Phi(x, \lambda z_{1}+(1-\lambda)z_{2})$$(\forall\lambda\in(0,1))$$=$$\hat{\Phi}(x-(\lambda z_{1}+(1-\lambda)z_{2}))$$=$$\hat{\Phi}(\lambda(x-z_{1})+(1-\lambda)(x-z_{2}))$$\leq$$\hat{\Phi}(\lambda(x-z_{1}))+\hat{\Phi}((1-\lambda)(x-z_{2}))$$=$$\lambda\hat{\Phi}(x-z_{1})+(1-\lambda)\hat{\Phi}(x-z_{2})$$=$$\lambda\Phi(x, z_{1})+(1-\lambda)\Phi(x, z_{2})$.

FinaUy we prove E5. The inequality in (8) is strict unless there exists $c\in\Re_{+}$ such thatis a linear$\lambda(x-z_{1})=c(1-\lambda)(x-z_{2})$ , which is the case only when $\lambda-c(1-\lambda)=0$ or
combination of and . But the both are impossible under the presupposition of E5.
$x$$\square$$z_{2}$$z_{1}$

Proof of Proposition 2.2: Let $x\in X$ . Then there exists $b\in\Re_{+}$ such that $|x|\leq b$ .

Define$(L^{\infty}( \mathcal{G}))_{b}\equiv\{x\in L^{\infty}(\mathcal{G})|\mathrm{e}\mathrm{s}\mathrm{s}\sup_{\omega\in\Omega}|x(\omega)|\leq b\}$.

because $x-(z\wedge b)\vee(-b)\preceq x-z$ and because it implies$\Phi(x, (z\wedge b)\vee(-b))\leq\Phi(x, z)$ by E3.
such
be a sequence inLet (We can look for

$z$

in

$(L^{\infty}(\mathcal{G}))_{b}$.

$\mathrm{t}\mathrm{h}\mathrm{a}^{\mathit{1}\sim}$$(L^{\infty}(\mathcal{G}))_{b}$$z_{n}\rangle_{n=1}^{\infty}$$\Phi(x,z_{n})arrow\dot{\mathrm{m}}\mathrm{f}\Phi(x, z)z\equiv\inf\{\Phi(x, z)|z\in(L^{\infty}(\mathcal{G}))_{b}\}$

.

(9)

such
and
is weak * compact, there exists a subsequence (in the restshow thatconverges to in the weak * topology. We
thatof this paragraph. Fix with which meets E6 and let be its conjugate (When $p=1$ , let. To see this,$q=+\infty)$ . We claim thatconverges to in the weak topology of*and live in $IP$ , and that, by the definition of the weak topology,
note thatSince$z_{0}\in(L^{\infty}(\mathcal{F}))_{b}$$z_{n}.\cdot\rangle_{i=1}^{\infty}$$(L^{\infty}(F))_{b}$$z_{0}\in(L^{\infty}(\mathcal{G}))_{b}$$\mathrm{w}\mathrm{i}\mathrm{U}$$z_{0}$$z_{n}\dot{.}$$\Phi$$p$$L^{p}(\Omega,F,\mu)$$z_{0}$$z_{n}.\cdot$$z_{n:}$$q$$z_{0}$$(\forall x\in L^{1})$$\int_{\Omega}xz_{n:}d\muarrow\int_{\Omega}xz_{0}d\mu$.

and $L^{q}=(L^{p})^{*}$ . Therefore, some sequenoe of convex
becauseThe claim
converges to in the $IP$-norm by the Mazur’s kmma. Then
combinations of the elements
some subsequenoe of this sequence converges to ahnost everywhere. Since each component ofis also ahnost -measurabk.
this subsequenoe is almost -measurable,. Because converges to in theThis paragraph proves thatis lower semi-continuous in the
weak topology by the aecond paragraph, and because$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{s}$$L^{q}\subset L^{1}$$z_{0}$$z_{n:}$$z_{0}$$\mathcal{G}$$\mathcal{G}$$z_{0}$$\Phi(x, z_{0})=\mathrm{i}\mathrm{d}_{z}\Phi(x, z)$$z_{n}.\cdot$$z_{0}$$\Phi(x, \cdot)$

weak topology by E4 and E6,
by (9).

$\underline{\mathrm{h}\mathrm{m}}_{narrow\infty}\Phi(x, z_{n:})\geq\Phi(x,z_{0})$, whi&implies$\Phi(x, z_{0})=\mathrm{i}\mathrm{d}_{z}\Phi(x,z)$59

FinaUy, we prove the uniqueness. If is ahnost -measurabk, $z=x$ attains the uniqueby El and E2. Assume that is not almost -measurable, and that ahnostminmeasurabk distinct functions
andboth attain the minimum. But this contradicts E5.
$x$$\mathrm{i}\mathrm{m}\iota \mathrm{m}$$\mathcal{G}$$x$$z_{1}$$\mathcal{G}-$$\mathcal{G}$$z_{2}$$\square$

Proof of Lemma 3.1: If $x=0$ or $z=0$ , the conclusion is trivial. Henoe assume that. Define $x_{0}=x/\alpha$ and $z_{0}=z/\beta$ . Thenandand(since). The same is true for . If we define
is positively,
then$||x||_{\gamma}\equiv\alpha>0$$||z||_{\gamma}\equiv\beta>0$$||\cdot||_{\gamma}$$||x_{0}||_{\gamma}=1$$\alpha x_{0}=x$$\mathrm{h}\mathrm{o}\mathrm{m}\mathrm{o}_{\circ}^{\sigma}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{o}\mathrm{u}\mathrm{s}$$\lambda=\alpha/(\alpha+\beta)$$z_{0}$$\gamma(x+z)|x+z|^{\mathrm{p}}$$=\gamma(\alpha x_{0}+\beta z_{0})|\alpha x_{0}+\beta z_{0}|^{\mathrm{p}}$$=\gamma(\lambda(\alpha+\beta)x_{0}+(1-\lambda)(\alpha+\beta)z_{0})|\lambda(\alpha+\beta)x_{0}+(1-\lambda)(\alpha+\beta)z_{0}|^{p}$$=$$(\alpha+\beta)^{p}\gamma(\lambda x_{0}+(1-\lambda)z_{0})|\lambda x_{0}+(1-\lambda)z_{0}|^{p}$$\leq$$(\alpha+\beta)^{p}(\lambda\gamma(x_{0})|x_{0}|^{p}+(1-\lambda)\gamma(z_{0})|z_{0}|^{p})$,where the last equality holds since $\alpha+\beta>0$ , and the inequality hol&(with an equality only
when $x_{0}=z_{0}$ ) since
is strictly convex. Therefore,$\gamma(\cdot)|\cdot|^{\mathrm{p}}$$||x+z||_{\gamma}^{p}$$=$$\int_{\Omega}\gamma(x+z)|x+z|^{p}d\mu$$\leq$$\int_{\Omega}(\alpha+\beta)^{p}(\lambda\gamma(x_{0})|x_{0}|^{p}+(1-\lambda)\gamma(z_{0})|z_{0}|^{p})d\mu$$=$$(\alpha+\beta)^{p}(\lambda||x_{0}||_{\gamma}^{p}+(1-\lambda)||z_{0}||_{\gamma}^{p})$$=$$(\alpha+\beta)^{p}$.

By taking -root, the proof is completed.

$\square$$p$

Proof of Proposition 3.3: Define$\varphi:\Re^{2}arrow\Re$by$\gamma(x-z)$

if$x\geq z$

$x-z$

if

$X<Z$

$\varphi(x, z)=\{$

Then $M(x)$ is defined as a unique solutionLet$x,$ $h\in L^{\infty}(\mathcal{F})$$(\forall\omega)$

$z$

to

$\int_{\Omega}\varphi(x(\omega), z)d\mu(\omega)$.

and let $\gamma>1$ . Then by the convexity of ,$\varphi$$\varphi(x(\omega)+h(\omega),M(x+h))-\varphi(x(\omega),M(x))$$-\varphi_{1}^{+}(x(\omega),M(x))h(\omega)-\varphi_{2}^{-}(x(\omega), M(x))[M(x+h)-M(x)]$$\geq 0$where$\varphi_{1}^{+}$

and$\varphi_{1}^{-}$

are right-hand-side and-side derivatives, respectively. Hence$\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}- \mathrm{h}\mathrm{a}\mathrm{n}\mathrm{d}$$\int_{\Omega}(\varphi_{1}^{+}(x(\omega), M(x))h(\omega)+\varphi_{2}^{-}(x(\omega), M(x))[M(x+h)-M(x)])d\mu\leq 0$.

60

By aetting $h=M(x|\mathcal{G})-x$ ,$M(M(x| \mathcal{G}))-M(x)\geq-\frac{\int_{\Omega}\varphi_{1}^{+}(x(\omega),M(x))[M(x|\mathcal{G})-x]d\mu}{\int_{\Omega}\varphi_{2}^{-}(x(\omega),M(x))d\mu}$Note that, and that$\int_{\Omega}\varphi_{2}^{-}(x(\omega),M(x))d\mu\leq 0$.

$\int_{\Omega}\varphi_{1}^{+}(x(\omega), M(x))[M(x|\mathcal{G})-x]d\mu\geq 0$

because$\int_{\Omega}\varphi_{1}^{+}(x(\omega), M(x))[M(x|\mathcal{G})-x]d\mu($$=$$\int_{\{x\geq M(x)\}}\gamma(M(x|\mathcal{G})-x)d\mu+\int_{\{x<M(x)\}}(M(x|\mathcal{G})-x)d\mu$$=$$( \gamma-1)\int_{\{x\geq M(x)\}}(M(x|\mathcal{G})-x)d\mu+\int_{\Omega}(M(x|\mathcal{G})-x)d\mu$$\geq$$( \gamma-1)\int_{\{x\geq M(x|\mathcal{G})\}}(M(x|\mathcal{G})-x)d\mu+\int_{\Omega}(M(x|\mathcal{G})-x)d\mu$

$=$

$0$

,

where the last equality holds by Lemma 2.3. This completes the proof for $\gamma>1$ . The similarargument applies for $\gamma<1$ .
$\square$

Proof of Proposition 3.7: EI-E6 are obvious. Yoshida-Hewitt theorem claims, where is a countably additive part of and is a pure charge, both ofwhich are nonnegative. Hence,, ffom whi&the lower semi-continuity
in a usual manner since is now a measure.
of in
To show that generates the Choquet, let
be a sequence of real numbers$((x(\omega)-z_{n})^{2}-(x(\omega)-z)^{2})/(z_{n}-z)$
whi&converges to . Thenconverges to $(d/dz)(w(\omega)-z)^{2}$in “&arge’’ because$(\forall x)\nu_{x}=\nu_{x}^{\mathrm{c}}+\nu_{x}^{\mathrm{p}}$$\nu_{x}^{\mathrm{c}}$$\nu_{x}$$\nu_{x}^{p}$$\Phi(x, z)\geq\int_{\Omega}(x-z)^{2}d\nu_{x}^{\mathrm{c}}$$\Phi$$z\mathrm{f}\mathrm{o}\mathbb{I}\mathrm{o}\mathrm{w}\mathrm{s}$$\nu_{x}^{\mathrm{c}}$$\Phi$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}_{\mathrm{o}}\circ\tau \mathrm{a}1$$\langle$$z_{n})_{n}$$z$$(\forall\omega)$$| \frac{(x(\omega)-z_{n})^{2}-(x(\omega)-z)^{2}}{z_{n}-z}-\frac{d}{dz}(w(\omega)-z)^{2}|=|z_{n}-z|$.

Therefore, by the dominated convergence theorem for charges (Rao and Rao, 1983, p. 131),$(d/dz) \int_{\Omega}(x(\omega)-z)^{2}d\nu_{x}=\int_{\Omega}(d/dz)(x(\omega)-z)^{2}d\nu_{x}$$\nu_{x}$.

$\square$, which proves the claim by the definition of

61

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