Discussion Paper No. 828

A THEORY OF MULTIDIMENSIONAL
INFORMATION DISCLOSURE

Wataru Tamura
The 14th ISER-Moriguchi Prize (2011)
Awarded Paper

January 2012

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

A Theory of Multidimensional Information Disclosure∗
Wataru Tamura†January 16, 2012

Abstract
We study disclosure of information about the multidimensional state of the world
when uninformed receivers’ actions aﬀect the sender’s utility. Given a disclosure rule,
the receivers form an expectation about the state following each message. Under
the assumption that the sender’s expected utility is written as the expected valueof a quadratic function of those conditional expectations, we identify conditions underwhich full and no disclosure is optimal for the sender and show that a linear transformation of the state is optimal if it is normally distributed. We apply our theory toadvertising, political campaigning, and monetary policy.
(JEL: D83, L15, M37, D72, E52)

Keywords: information disclosure, semideﬁnite programming, linear transformation.

∗

I am very grateful to Masaki Aoyagi for his most insightful comments and invaluable suggestions. This
paper (an earlier version of which had the title “Information Disclosure in Strategic Environments”) waspresented at YNU, Okayama University, Waseda University, Osaka Prefecture University, the Summer Workshop on Economic Theory 2010, Sapporo Applied Microeconomics Workshop, Applied Microeconomic Theory
Workshop, and the 2011 Japanese Economic Association Spring Meeting. I have beneﬁted from comments
and suggestions by the participants of these seminars and the workshops, and by Junichiro Ishida, Shigehiro
Serizawa, and Takashi Ui. I especially thank Takehito Masuda and Keiichi Morimoto for helpful discussions
which have signiﬁcantly improved earlier drafts of this paper. This paper was awarded the 14th Moriguchi
Prize by the ISER at Osaka University.
†
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. e-mail: jge005tw@mail2.econ.osaka-u.ac.jp.

11Introduction

Controlling “market expectations” about the state of the world is important in various
situations. For example, a central bank has to control market expectations in order tostabilize inﬂation and the output gap around the desired values. A manufacturing ﬁrm needs
to build a good reputation for the quality of its product. A ruling political party wants tomaintain a high approval rating by being sensitive to voters’ expectations about its policy
stance and competence. In order to control the information available to market participants,
a central bank designs a communication strategy (Blinder et al. (2008), Woodford (2005),
and others), a ﬁrm an advertising strategy (Anderson and Renault (2006), Johnson andMyatt (2006)), and a political party a campaign strategy (Prat (2002), Polborn and Yi(2006)).
In this paper, we analyze a model in which a privately informed sender discloses information about the realization of the state to uninformed receivers, who then engage in economic
activities that aﬀect the sender’s utility. Through the choice of a disclosure rule that speciﬁes
the information available to the receivers for each state of the world, the sender inﬂuences
the receivers’ belief. The question we address in this paper is, given the prior distribution of
the state, what disclosure rule maximizes the sender’s expected utility.
Formally, a disclosure rule assigns to each realization of the state a probability distribution
over messages. As such, the disclosure rule determines the joint distribution of the state andthe message, which in turn determines the distribution of the receivers’ belief. We assume
that the sender’s expected utility, which is originally a function of the joint distribution of
the receivers’ action proﬁle and the state, is reduced in equilibrium to the expected value ofa quadratic function of the receivers’ expectation of the state.1 Under this assumption, thesender’s problem is to control the distribution (more precisely, its second moments) of thereceivers’ expectation of the state. A suﬃcient condition on the underlying preferences forthis assumption is that both the sender and the receivers have quadratic utility functions
over the receivers’ action proﬁle and the state. Such a speciﬁcation is common in models ofoligopoly, network externalities, and so on, and in the recent studies of transparency policy
including Morris and Shin (2007) and Cornand and Heinemann (2008) among others. We
show through our applications presented in Section 6 that this assumption is satisﬁed in a
number of applications such as monopoly advertising, political campaigning, and monetary
policy making.
1

Chakraborty and Harbaugh (2010) study multidimensional cheap talk in a setting in which the sender’spreferences are described by a continuous (but not necessarily quadratic) utility function over the receivers’
conditional expectation. As illustrated in the next section, our formulation does not necessarily imply thatthe sender has state-independent preferences.

2Our ﬁrst result identiﬁes when full and no disclosure is optimal (Theorem 1). In order
to investigate the optimality of partial disclosure rule, we begin by establishing an upperbound of the sender’s expected utility. We show that the upper bound depends only on
the second moments of the state and is obtained by solving a semidefinite programming
problem (Theorem 2). To the best of our knowledge, the approach based on semideﬁnite
programming is novel in the context of information disclosure.2 With this preparation, weshow that the optimal disclosure rule is given by a linear transformation of the state when it
is normally distributed (Theorem 3). We should emphasize that this is the ﬁrst result that
presents a complete and systematic derivation of the optimal disclosure rule, whether partial
or full, in a continuous state space.
We next examine the implications of our results in three applications. In Subsection
6.1, we consider the optimal advertising strategy of a monopoly ﬁrm privately informedof its product quality and marginal cost, and show that its optimal advertising policy is
to reveal less information about its product quality than the socially optimal level. In
Subsection 6.2, we examine in a model of electoral competition the incentives of a politicalparty to reveal information about its candidate and show that incumbency advantage leads
to a socially ineﬃcient amount of information revelation to voters. In Subsection 6.3, weformulate a two-period model of monetary policy and characterize the optimal disclosure
rule and stabilization policy.
Optimal disclosure of information has been studied in a number of diﬀerent contexts,
including auctions (Milgrom and Weber (1982), Bergemann and Pesendorfer (2007), Board(2009), Ganuza and Penalva (2010)), corporate ﬁnance (Admati and Pﬂeiderer (2000), Bootand Thakor (2001)), interim performance evaluation (Aoyagi (2010), Goltsman and Mukherjee (2011)), transparency in policymaking (Gavazza and Lizzeri (2009), Prat (2005)), etc.
This paper is closely related to recent studies of Kamenica and Gentzkow (2011) andRayo and Segal (2010), who also investigate the optimal disclosure rule under alternative
speciﬁcations of the state space and the sender’s utility function. Kamenica and Gentzkow(2011) study a general setting in which the sender needs to control the distribution of posterior distributions and ﬁnd general properties of posteriors induced by the optimal disclosure
rule. They characterize the optimal disclosure rule in some simple settings, including when
the state space is binary. Rayo and Segal (2010) characterize the optimal (randomized) disclosure rule for the discrete state space when the sender has certain preferences. The main
contribution of our analysis is to identify optimal disclosure in the case where the state iscontinuously distributed. A detailed discussion of our contribution is provided in Section 3.
2

The same idea is also found in other applications of semideﬁnite programming such as minimal tracefactor analysis and optimal experiment design (see, for example Vandenberghe and Boyd (1996)).

3

In line with much of the literature including Kamenica and Gentzkow (2011) and Rayoand Segal (2010), we assume that the sender can commit to her disclosure rule. While this isa strong assumption, it is justiﬁable in situations where the sender’s information is veriﬁable
ex post in the form of survey data, estimation results, experts’ reports, and so on. In suchsituations, reputational and legal concerns would stop the principal from deviating fromthe pre-announced disclosure rule for a short-run gain. It is worth noting that the optimal
disclosure rule for the normally distributed state is a linear transformation, making it easy
to match the disclosed information with the private information. In this sense, it is morecredible than other complex rules. Although some recent papers assume that the sender cancommit to any disclosure rule (e.g., Goltsman and Mukherjee (2011)), most applied papers
assume a limited ability to commit to a disclosure rule and restrict the class of disclosurerules the sender can choose from. For example, some papers including Shapiro (1986) and
Ederer (2010) examine when full disclosure is superior to no disclosure while Gal-Or (1986)
and Admati and Pﬂeiderer (2000) among others assume that the sender is able to choose
only the precision of messages the receivers observe so that a closed-form solution for thesender’s expected utility is obtained.
This paper also contributes to the growing literature on the social value of information.3
In applications, we discuss the divergence between private and social incentives to disclose
information in terms of informativeness of the messages generated by each disclosure rule.
One advantage of our multidimensional analysis is that it allows us to examine not only
the level but also the type of information that is revealed in equilibrium and at the social
optimum.4 For example, the optimal disclosure rule for a political party may reveal lessinformation about its general competence and too much about its policy stance than thesocially optimal disclosure rule.
The paper is organized as follows. Section 2 presents a motivating example. In Section 3,
we set up the model and discuss our key assumptions. Section 4 identiﬁes conditions underwhich full and no disclosure is optimal and characterizes an upper bound of the sender’s
expected utility. In Section 5, the optimal disclosure rule is explicitly obtained when thestate is normally distributed. Section 6 provides applications, and Section 7 concludes thepaper.

3

For recent literature, see Morris and Shin (2002) and Angeletos and Pavan (2007) among others.
For the measure of informativeness of disclosure rules, we follow Ganuza and Penalva (2010), who
propose precision criteria based on the variability of conditional expectations. Especially, if the conditional
expectation induced by a disclosure rule is normally distributed, its variance can measure the precision of
the message generated.
4

4

2

Motivating Example

To provide a concrete example of what we analyze and what our assumption does anddoes not mean, we begin with a simple example in which a sender discloses information to tworeceivers who then take actions. This example illustrates how our reduced-form formulationarises in settings with quadratic preferences over the receivers’ actions and the state. This
motivating example is also useful in identifying the key issues that the theorems presented
below resolve.
An organization consists of a principal with private information and two agents (i = 1, 2)
who are hired by the principal to sell her products. The agents simultaneously choose thetarget consumers ai ∈ R and their proﬁts depend on agent-speciﬁc market condition xi ∈ R
as well as on the action proﬁle (a1 , a2 ). In particular, we suppose that agent i has a utility
functionui (a1 , a2 , x1 , x2 ) = −(ai − xi )2 − γ(ai − a−i )2
and that the principal hasv(a1 , a2 , x1 , x2 ) = −2∑(ai − xi )2 − δ(a1 − a2 )2i=1

where γ ≥ 0 and δ ≥ 0 measure the relative importance of coordination between two
agents.5 Each agent has incentives to adapt to the state in order to reduce the adaptation
loss, (ai − xi )2 , and to choose a similar target in order to reduce the coordination loss,(ai − a−i )2 , which may arise due to network externalities, reputations, economics of scale,and so on. The principal is privately informed about (x1 , x2 ), interpreted as the information
about her products and market conditions which may be estimated from past records. The
prior distribution of the state is common knowledge.6 How should the principal disclose her
private information? When is full disclosure optimal?
A (deterministic) disclosure rule is a mapping g : R2 → M that assigns to each realizationof the state a message m ∈ M where the message space M is also chosen by the principal.
Although we consider a more general class of (possibly randomized) disclosure rules in thefollowing sections, we begin by comparing the following three disclosure rules; full disclosureg f (x1 , x2 ) = (x1 , x2 ) ∈ R2 , which discloses full information; no disclosure g n (x1 , x2 ) = 0 ∈ R,
5

Use of a coordination game with quadratic preferences is common in organization economics. Our model
follows Alonso et al. (2008), Calvó-Armengol and de Martı́ (2009), Calvó-Armengol et al. (2009) and Dessein
and Santos (2006).
6
Our formulation allows any correlation between x1 and x2 .

5

which discloses a completely uninformative message; and average disclosure g a (x1 , x2 ) =(x1 + x2 )/2 ∈ R, which discloses a sample mean of two variables.
The timing of the game is as follows. First, the principal commits to a disclosure ruleg ∈ {g f , g n , g a }. Second, the state of the world is realized, and a message m = g(x1 , x2 ) isdisclosed according to the disclosure rule. Third, given the disclosure rule and the message,
the agents form a posterior belief and choose actions.
First, we derive the agents’ equilibrium strategies and the principal’s expected utility.
Given g and m ∈ M , two agents play a game whose unique Nash equilibrium is, for i = 1, 2,
a∗i (m) =(1 − ψ)E[xi |m] + ψE[x−i |m]where ψ = γ/(1 + 2γ). The equilibrium strategy is linear in the conditional expectations.
Therefore we denote the equilibrium strategy proﬁle by a∗ (x̂1 , x̂2 ) = (a∗1 (x̂1 , x̂2 ), a∗2 (x̂1 , x̂2 ))where x̂j ≡ E(xj |m) is the agents’ estimate of the state. Notice that a prior distribution of
the state and a disclosure rule specify the joint distribution of (x1 , x2 , x̂1 , x̂2 ), which determines the joint distribution of (x1 , x2 , a∗1 , a∗2 ) in equilibrium. We now rewrite the principal’s
expected utility. The adaptation loss is written as[[]]E(a∗i − xi )2 =E(x̂1 ,x̂2 ) Exi (a∗i )2 − 2a∗i xi + x2i |x̂1 , x̂2
[
]
[ (
)]
=E(x̂1 ,x̂2 ) (a∗i )2 − 2a∗i x̂i + E(x̂1 ,x̂2 ) E x2i |x̂1 , x̂2[()]=E(x̂1 ,x̂2 ) −x̂2i + ψ 2 x̂21 − 2x̂1 x̂2 + x̂22 + Ex2i .
Similarly, the coordination loss is[()]
E(a∗1 − a∗2 )2 = E(x̂1 ,x̂2 ) (1 − 2ψ) x̂21 − 2x̂1 x̂2 + x̂22 .
Thus, the principal’s expected utility is written as
E(x1 ,x2 ,x̂1 ,x̂2 ) v(a∗ (x̂1 , x̂2 ), x1 , x2 )=−2∑E(a∗i (x̂1 , x̂2 ) − xi )2 − δE(a∗1 (x̂1 , x̂2 ) − a∗2 (x̂1 , x̂2 ))2

i=1

]
[
=E(x̂1 ,x̂2 ) (1 − ϕ)(x̂21 + x̂22 ) + 2ϕx̂1 x̂2 − Ex21 − Ex22
where ϕ ≡ (2γ 2 + δ)/(1 + 2γ)2 . Since Ex21 and Ex22 are independent of the disclosure rule, theprincipal’s problem is reduced to the maximization of Ev̂(x̂1 , x̂2 ) + c where c is a constant6

Table 1: Comparison among the three rules.
Rulevar(x̂1 )var(x̂2 )cov(x̂1 , x̂2 )No
Full
Average
0
1/12
1/24
0
1/12
1/24
0
0
1/24
3rd
1st
2nd
for δ < δRank of Ev for δ < δ < δ
3rd
2nd
1st
for δ > δ
2nd
3rd
1st
The thresholds are δ = (1 + 4γ)/2 and δ = 1 + 4γ + 2γ 2 .

and v̂ is a quadratic function deﬁned byv̂(x̂1 , x̂2 ) = (1 − ϕ)(x̂21 + x̂22 ) + 2ϕx̂1 x̂2 .

(1)Note that in general the disclosure rule cannot aﬀect the expected value of the estimates,that is Ex̂i = EE(xi |m) = Exi . Note also that the second moments are given by Ex̂2i =var(x̂i ) + (Exi )2 and Ex̂1 x̂2 = cov(x̂1 , x̂2 ) + (Ex1 )(Ex2 ). From these observations, we haveEv(a∗ , x1 , x2 ) =(1 − ϕ)(var(x̂1 ) + var(x̂2 )) + 2ϕcov(x̂1 , x̂2 )− var(x1 ) − var(x2 ) − ϕ (Ex1 − Ex2 )2 .
Since the second line is independent of the disclosure rule, the principal’s expected utility isalso expressed as a linear function of variance-covariances of the estimates (plus a constantterm).
It is worth noting that the expected utility conditional on each message is not necessarily
written as v̂(x̂1 , x̂2 ) + c. The former is expressed as
E [v(a∗ , x1 , x2 )|m] = v̂(x̂1 , x̂2 ) − E(x21 + x22 |m).
The second term in the right-hand side may change as what message is disclosed while its
average is determined by the prior distribution but not the disclosure rule. For any disclosure
rule, the expectation of the conditional expectation becomes the unconditional expectation
so that Em [E[(x21 + x22 )|m]] = E(x21 + x22 ).
We now evaluate the performances of the three disclosure rules. For ease of exposition,
we assume that x1 and x2 are independent and uniformly distributed on [−1/2, 1/2]. Table

7

1 gives the characteristics of the three deterministic disclosure rules. The ﬁrst three rows
report the variances and covariance of the estimates while the last three rows report the
ranking of the principal’s expected utility. Given γ > 0, there are two thresholds for δ;δ ≡ (1 + 4γ)/2 and δ ≡ 1 + 4γ + 2γ 2 . Full disclosure is superior to average disclosure forδ < δ while the reverse holds for δ > δ. Intuitively, when the principal puts a lower weighton coordination (δ < δ), the principal should disclose full information in order to induce
adaptation even if it may cause mis-coordination. In contrast, when the coordination isimportant (δ > δ), the principal induces similar decisions by disclosing the average state.
Several questions arise. What is the optimal disclosure rule? Is a higher covariance
between x̂1 and x̂2 incompatible with a higher variance of each x̂i ? What determines thelimit of information disclosure as a means of controlling expectations? In the following
sections, we will answer these questions and ﬁnd that in the above example full and average
disclosure is indeed optimal among the general class of disclosure rules when the state isuniformly distributed.7

3

The Model

A sender privately observes the multidimensional state x = (x1 , . . . , xk )′ ∈ Rk where
x has a density over a convex support in Rk with a non-empty interior, zero mean and apositive deﬁnite variance matrix Σ.8 The sender publicly discloses information about therealization of the state to uninformed receivers who then engage in economic activities suchas consumption, investment, etc. that aﬀect the sender’s utility.
For the inducement of preferred actions, the sender controls what information to makeavailable to the receivers by choosing a disclosure rule (α, M ), which consists of a measurable
set M of messages and a family of conditional probability distributions {α(·|x)}x over M .
We assume that the disclosure rule (α, M ) is such that the conditional expectation E[x|m]exists for every m ∈ M and so does its second moment var(E[x|m]).9 Let x̂ ≡ E[x|m]be the conditional expectation given m, call the estimates. A disclosure rule induces ajoint distribution of (x, m), and hence a joint distribution of (x, x̂). This deﬁnition includes
the following communication strategies that are common in the literature of information
economics; full disclosure, that reveals the realization of the state; no disclosure, that reveals
no information; noisy communication, that adds white noises to the sender’s observation;
partition, that reveals an element of the partition over the state space that contains the
7

In Section 5, we also characterize the optimal disclosure rule when the state is normally distributed.
We denote by A′ the transpose of matrix A. Throughout the paper, all untransposed vectors are column
vectors.
9
A suﬃcient condition is that the support of x is a compact set in Rk .
88

state. The disclosure rule is deterministic if there exists a function g : Rk → M such thatm = g(x) almost surely.10 As we have seen in the previous section, disclosing the average∑value of the state, m = g(x) =xi /k, is an example of a deterministic disclosure rule. In
contrast, disclosing noisy messages, mi = xi + εi where εi ∼ N (0, σi2 ), is a typical example
of a stochastic disclosure rule.
As discussed in the introduction, we consider situations in which the sender controls thereceivers’ actions, denoted by a, through information disclosure. Let v(a, x) be the sender’sutility function. In what follows, we make the following two assumptions. First, we supposethat the receivers’ behavior is simply a continuous function of their conditional expectationof the state. We denote their actions given x̂ = E[x|m] by a∗ (x̂). Under this assumption,
the sender’s problem is to control the joint distribution of x and x̂ so as to maximize her
expected utility E(x,x̂) v(a∗ (x̂), x). Second, there exists a k × k symmetric matrix V such thatfor any disclosure rule,
E(x,x̂) v(a∗ (x̂), x) = Ex̂ [x̂′ V x̂] + c(2)

where c is a constant that is independent of the disclosure rule. Let v̂(x̂) = x̂′ V x̂. A suﬃcientcondition is that the receivers’ equilibrium strategies are given by an aﬃne function of theirconditional expectations and the sender has a quadratic utility function over a and x. We
call Ev̂(x̂) the gain from a disclosure rule.11
As brieﬂy discussed in the previous section, the law of iterated expectations plays a keyrole in deriving such a representation. First, the expected value of the product a∗ (x̂) · xi is
expressed as the expected value of a function of the estimates. That is, E(x,x̂) [a∗ (x̂)xi ] =Ex̂ [a∗ (x̂)x̂i ]. Second and more importantly, the expected utility conditional on the message,E[v(a∗ (x̂), x)|m], is not necessarily written as v̂(x̂)+c. For example, suppose that the sender
has v(a, x) = −(a − x)2 and the receiver has u(a, x) = −(a − x)2 so that the receiver choosesa∗ = x̂. Then the sender’s expected utility conditional on m is E[v(a∗ , x)|m] = x̂2 − E[x2 |m]while v̂(x̂) = x̂2 . Although the disclosure rule aﬀects the distribution of Ex [x2 |m], it cannot
aﬀect its average value Em [Ex [x2 |m]] = Ex2 . Thus, our formulation may apply when thesender’s conditional expected utility given each m cannot be written as a quadratic functionof the estimates.
An important consequence of our assumptions is that the sender’s expected utility can
10Note that every deterministic disclosure rule can be represented by a partition and vice versa. For
example,a message}m under a deterministic rule g is equivalent to disclosing its inverse image g −1 (m) ={kx ∈ R : g(x) = m .
11This terminology is due to Kamenica and Gentzkow (2011), who deﬁne it to be the diﬀerence between
the sender’s expected utilities under a disclosure rule and no disclosure. In this paper, no disclosure inducesx̂ = 0 and hence Ev = c.

9be expressed as Ex̂ [x̂′ V x̂] + c = tr(V Σ̂) + c where Σ̂ ≡ E [x̂x̂′ ] denotes the second moment of
the estimates.12 There are two key features of the sender’s problem that immediately follow
from this equality: (i) if two disclosure rules induce the same Σ̂, they yield the same expectedutility, and (ii) given Σ̂, it is easy to compute the sender’s expected utility, tr(V Σ̂) + c.13
Although such a reduced-form formulation can be generated in diﬀerent ways from theunderlying preferences of the receivers and their equilibrium behaviors as illustrated in theprevious section, there are important cases that cannot be reduced to our model. The ﬁrst
case is where the receivers’ action space is discrete. In the above simple example, if the action
space is given by {−1, 1}, then the receiver’s action is not continuous in his expectation of
the state.14 Second, even when the receiver’s behavior is given by a continuous function of
the estimates, the sender’s utility function should not be too complex. For example, supposethat a∗ (x̂) = x̂ and v(a, x) = −(a + 1)2 (a − 1)2 . Then the sender’s expected utility cannot
be expressed in the form of (2).
Rayo and Segal (2010) study optimal information disclosure where the sender’s expectedutility is written as Ex̂1 x̂2 + c, and characterize the optimal randomized disclosure rule for aﬁnite state space. Since we assume that the state has a continuous distribution, our analysis
is based on diﬀerent techniques and applicable to common distributions such as uniform
and normal. In Section 5, we will ﬁnd that the disclosure of a weighted average becomes asolution to their problem when the state is normally distributed.15
Kamenica and Gentzkow (2011) consider a general problem where the sender’s expectedutility is expressed as the expected value of a function of the receiver’s posterior belief andcharacterize the posterior beliefs induced by the optimal disclosure rule. They provide theoptimal (partial) disclosure rule in simple settings, especially when the state space is binary.
They also analyze whether no disclosure is suboptimal for the setting in which the sender’sutility depends only on the receiver’s conditional expectation of the state. Although we
make a stronger assumption on the sender’s preferences, we provide a simple and completecharacterization of the optimal disclosure rule in the continuous state space, which is usefulfor applied research.

For any rule (α, M ), we have E[x̂′ V x̂] = E[tr(x̂′ V x̂)] = E[tr(V x̂x̂′ )] = tr(V E[x̂x̂′ ]) = tr(V Σ̂) wheretr(A) is the trace of matrix A.
13
Even though this operation itself does not rely on the assumption that x̂ is the receivers’ conditional
expectation of x, we often use it when we derive matrix V . In the above simple example, we use it when wecompute E(x,x̂) xa∗ (x̂) = Ex̂2 .
14
In such a case, the receiver takes a = 1 (= −1) if x̂ ≥ (<)0, respectively. That is, the ﬁrst assumption
is not satisﬁed. In this case, the sender’s expected utility is expressed as E(x,x̂) v(a∗ , x) = 1 + 2bE(|x̂|) − Ex2 .
15
One can also show that, by applying Theorem 2 presented below, the average disclosure is optimal if(x1 , x2 ) is uniformly distributed over [0, 1]2 .
12

10

4

Optimal Disclosure and Semidefinite ProgrammingFirst, we identify when full and no disclosure is optimal and when partial disclosure yields
a higher expected utility than full and no disclosure. Detailed proof is in Appendix A
Theorem 1 (i) Full disclosure is optimal if and only if v̂ is a convex function or, equivalently, V is positive semidefinite. (ii) No disclosure is optimal if and only if v̂ is a concave
function or, equivalently, V is negative semidefinite.
The if parts, which simply follow from Jensen’s inequality and the law of iterated expectations, are already known in the literature. The only if part of (ii) is also found in
Kamenica and Gentzkow (2011)[Proposition 3], who analyze a setting in which the sender
has a continuous but not necessarily quadratic utility function over conditional expectations.
In contrast, the only if part of (i) is novel and relies on the quadraticity of v̂. To illustrate()
1 2
the main idea behind the proof, consider the following example: suppose that V =
,
2 1
( )
1
which is not a positive semideﬁnite matrix so that there is a vector x− =that satisﬁes−1( )1
x′− V x− < 0. Then any x can be expressed as x = βx+ + γx− where x+ =. We argue
1
that a partial disclosure rule that reveals only β yields a higher gain than full disclosure thatreveals both β and γ. When only β is disclosed, the estimate is given byx̂ = βx+ + E(γ|β)x− ,and then the realization of the state is written as
x = x̂ + (γ − E(γ|β)) x− .

(3)Indeed, if the sender reveals the realization of γ in addition to β, an additional variation in
the receivers’ expectation, (γ − E(γ|β))x− , is generated. Using the expression of (3), thesender’s gain from full disclosure can be written as follows:[]
E [x′ V x] =E [x̂ + (γ − E(γ|β))x− ]′ V [x̂ + (γ − E(γ|β))x− ]
]
]]
[
[ [
=E [x̂′ V x̂] + 2Eβ Eγ (γ − E(γ|β))x′− V x̂|β + E (γ − E(γ|β))2 x′− V x−)(
=E [x̂′ V x̂] + x′− V x− Eβ [var(γ|β)] .
Since E [var(γ|β)] > 0 and x′− V x− < 0, we have E[x′ V x] < E[x̂′ V x̂]. Intuitively, comparedwith full disclosure, the partial disclosure speciﬁed above reduces an unfavorable variability11

of conditional expectations. Hence, full disclosure is suboptimal whenever V is not positive
semideﬁnite.
The proof of the only if part of (ii) in Appendix is based on a reverse argument that,compared with no disclosure, the sender can generate a favorable variability of conditional
expectations if V is not negative semideﬁnite. In particular, the sender controls information
so that the conditional expectations lie on a vector x+ such that x′+ V x+ > 0.
For the purpose of use in applications in Section 6, we restate Theorem 1 for the case ofk ≤ 2.
Corollary 1 For the one-dimensional state space (k = 1), full disclosure is optimal if andonly if V ≥ 0 and no disclosure is optimal if and only if V ≤ 0. For the two-dimensional
state space (k = 2), full disclosure is optimal if and only if V11 , V22 , det(V ) ≥ 0 and no
disclosure is optimal if and only if V11 , V22 , −det(V ) ≤ 0.
In the motivating example, V is positive semideﬁnite if and only if δ ≤ (1 + 4γ)/2, whichcoincides with the condition for the optimality of full disclosure among the three simple
rules. So far we know little about what disclosure rule is optimal when δ > (1 + 4γ)/2 inthe example.
To investigate the optimality of partial disclosure, we begin by establishing an upperbound of the sender’s expected utility (or equivalently the gain) attainable through information control. Recall that in general E[x̂′ V x̂] = tr(V Σ̂) where Σ̂ = E [x̂x̂′ ] = var(x̂).
Hereafter, we interpret the sender’s problem as the choice of a variance matrix of x̂ bychoosing a disclosure rule.
We now investigate conditions on Σ̂ that can be induced by a disclosure rule. First weknow that (i) Σ̂ must be positive semideﬁnite since it is a variance matrix. Furthermore, forany joint distribution of (x, m), the law of total variance holds;
var(x) =E[var(x|m)] + var(E(x|m))where var(x|m) = E[(x − E[x|m])(x − E[x|m])′ |m]. Since every variance matrix is positive
semideﬁnite, so is its expectation E[var(x|m)].16 This implies that (ii) Σ− Σ̂ must be positive
semideﬁnite. Let ≽ denote the Löwner partial ordering on the set of k ×k symmetric positive
semideﬁnite matrices. That is, for two symmetric positive semideﬁnite matrices A and B,
A ≽ B if A−B is positive semideﬁnite.17 Let O and I denote the zero and the identity matrix,
For all non-zero vector z ∈ Rk , we have z′ E[var(x|m)]z = E[z′ var(x|m)z] ≥ 0 since var(x|m) is positive
semideﬁnite for any m ∈ M .
17
For the standard notation in matrix algebra and some basic properties of matrices, see Horn and Johnson(1985) and Boyd and Vandenberghe (2004).
1612

respectively. Then, every Σ̂ must satisfy Σ ≽ Σ̂ ≽ O. Roughly speaking, variance matrices ofthe estimates induced by the disclosure rule are (partially) ordered by the matrix inequality
according to which full (no) disclosure attains the greatest (least) element. Therefore, anupper bound on the gain is given by solving the following semidefinite programming:
max tr(V Σ̂).
Σ≽Σ̂≽OTo simplify the notation in the problem and its solution, it is useful to change the variable1
1
1
1
in the above program. Let Z ≡ Σ− 2 Σ̂Σ− 2 and W ≡ Σ 2 V Σ 2 . Since the variance matrix
Σ is nonsingular, Z must be a symmetric positive semideﬁnite matrix. Then the condition11Σ ≽ Σ̂ ≽ O is equivalent to I ≽ Z ≽ O. It is straightforward to see that Z = Σ− 2 ΣΣ− 2 = I
1
1
for full disclosure and Z = Σ− 2 OΣ− 2 = O for no disclosure. The gain is also written interms of Z and W as tr(V Σ̂) = tr(W Z). Thus, an upper bound of the gain is characterized
as follows:
1

1

Lemma 1 Let W = Σ 2 V Σ 2 . Then the upper bound of the gain is given by solving thefollowing semidefinite programming:(SDP)

max tr(W Z).

I≽Z≽O

Before presenting the solution to SDP, it may be helpful to give an intuition of theconstraint Σ ≽ Σ̂ ≽ O in the context of the motivating example in Section 2. Recall thatmatrix V has entries V11 = V22 = 1 − ϕ and V12 = V21 = ϕ where ϕ = (2γ 2 + δ)/(1 + 2γ)2 (see(1)). Moreover, we have assumed that x1 and x2 are independent and uniformly distributed
over [−1/2, 1/2] so that Σ has σ11 = σ11 = 1/12 and σ12 = 0. It immediately follows from2Σ̂ ≽ O (i.e., Σ̂ is positive semideﬁnite) that σ̂11 ≥ 0, σ̂22 ≥ 0 and σ̂11 σ̂22 ≥ σ̂12
. That is,the variances of the estimates are nonnegative and the correlation between two estimates,√2/(σ̂11 σ̂22 ), is in [−1, 1]. Similarly, from Σ ≽ Σ̂ (i.e., Σ − Σ̂ is positivecorr(x̂1 , x̂2 ) = σ̂12
semideﬁnite), we have σ11 ≥ σ̂11 , σ22 ≥ σ̂22 , and(σ12 − σ̂12 )2 ≤(σ11 − σ̂11 )(σ22 − σ̂22 ).

(4)

The ﬁrst two conditions imply that the variance of the estimate cannot exceed that of
the underlying state. It may make sense that any message about the state cannot be
more informative than revealing the state itself. The condition (4) implies that to generate a certain covariance of the estimates that diﬀers from that of the underlying state(cov(x̂1 , x̂2 ) ̸= cov(x1 , x2 )), the sender must induce lower variances of the estimates thanthat under full disclosure (var(x̂1 ) < var(x1 ) and var(x̂2 ) < var(x2 )). In the context of the
13

z12

z11 = z22

O

Figure 1: Solution to the semideﬁnite programming
motivating example, this condition turns out to be an important trade-oﬀ for the principal.
Recall that both the adaptation and coordination losses are decreasing in σ̂12 = E(x̂1 x̂2 ). To
induce a higher covariance between the two estimates, the sender must reduce the varianceof the estimates, which is also valuable to the principal whenever δ < δ. Roughly speaking, to facilitate coordination between the two agents, the principal needs to withhold some
information and reduce the degree of adaptation.
We now apply Lemma 1 to the problem in the motivating example and obtain an upper1
1
bound of the gain. Since Σ = 12
I, we have Z = 12Σ̂ and W = 12
V . Moreover, the2constraint I ≽ Z ≽ O is expressed as (i) z11 , z22 ≥ 0, z12 ≤ z11 z22 , and (ii) z11 , z22 ≤ 1,2
z12
≤ (1 − z11 )(1 − z22 ). Thus, SDP for the example is written as(SDP)

1[(1 − ϕ)(z11 + z22 ) + 2ϕz12 ]z11 ,z12 ,z22 12
max

subject to z11 , z22 ∈ [0, 1]
2
z12≤ min{z11 z22 , (1 − z11 )(1 − z22 )}.

Notice that in order to relax the constraint on z12 , we have to choose z11 = z22 .18 Then theinequality constraint is reduced to z12 ≤ min{z11 , (1 − z11 )}. Figure 1 depicts the feasible
set of (z11 , z12 ) as a shaded area with the level curves of the objective function tr(W Z).
Recall that the solution Z to SDP corresponds to the variance matrix of the estimates as111Σ̂ = Σ 12 ZΣ 12 = 12
Z. If the slope w11 /w12 = (1 − ϕ)/ϕ of the level curves is greater than
1, the solution is (z11 , z22 , z12 ) = (1, 1, 0), or equivalently (σ̂11 , σ̂22 , σ̂12 ) = (1/12, 1/12, 0),
which is achieved by full disclosure (see the ﬁrst three rows in Table 1). On the otherFor any (z11 , z22 ) with z11 ̸= z22 , consider z̃11 = z̃22 = (z11 + z22 )/2. Since w11 = w22 = (1 − ϕ)/12, we′have w11 z11 +w22 z22 = w11 z̃11 +w22 z̃22
. Moreover, z11 z22 < z̃11 z̃22 and (1 −z11 )(1− z22 ) < (1− z̃11 )(1− z̃22 ).
Thus we can relax the inequality constraint without altering the objective value.
18

14

hand, if w11 /w12 is less than 1, the solution is (z11 , z12 , z22 ) = (1/2, 1/2, 1/2), or equivalently(σ̂11 , σ̂22 , σ̂12 ) = (1/24, 1/24, 1/24), which is, indeed, what average disclosure achieves. Thus,
we ﬁnd that average disclosure attains the upper bound for δ > (1 + 4γ)/2.
iid

Corollary 2 In the motivating example with a prior xi ∼ U [− 12 , 12 ] for i = 1, 2, full disclosure is optimal if δ ≤ (1 + 4γ)/2 and average disclosure is optimal if δ > (1 + 4γ)/2.
Although Lemma 1 provides a key insight into the control of conditional expectations,
we need some knowledge in matrix algebra to solve the program. Here we present a solution
to SDP and relegate its derivation to Appendix A.19
Theorem 2 Let Q+ = [q1 , . . . , qr ] consist of the eigenvectors associated with the nonnega1
1
tive eigenvalues of W = Σ 2 V Σ 2 . Then a projection matrix Z = PQ+ = Q+ (Q′+ Q+ )−1 Q′+
is a solution to SDP. Moreover, the upper bound achieved equals the sum of all positive
eigenvalues of W .
An important implication for k = 2 is that the two estimates x̂1 and x̂2 must be perfectly
correlated when x1 and x2 are independent. To see this, suppose that σi2 = var(xi ) for i = 1, 2
and that V (and hence W ) is neither positive nor negative semideﬁnite so that there exists
exactly one positive eigenvalue. Suppose also that V12 ̸= 0 so that the optimal disclosure rule
is nontrivial.20 Let (q1 , q2 ) be the eigenvector associated with the unique positive eigenvalue.
Then the solution to SDP is given by
(
Z=

q12
2
q1 +q22
q1 q2
q12 +q22

q1 q2
q12 +q22
q22
2
q1 +q22),and the second moment of the estimates is
(
1
2

1
2

Σ̂ = Σ ZΣ =σ12 q12
q12 +q22
σ1 q1 σ2 q2
q12 +q22

σ1 q1 σ2 q2
q12 +q22
σ22 q22.
q12 +q22)

Thus, the correlation between x̂1 and x̂2 equals q1 q2 /|q1 q2 |, which is either 1 or −1 wheneverboth x̂1 and x̂2 have positive variances.21 In other words, to achieve the upper bound, thetwo estimates must satisfy a linear restriction x̂2 = β x̂1 where the coeﬃcient is given by
β = σ2 q2 /σ1 q1 . Another implication is that the solution to SDP satisﬁes z11 + z22 = 1. This19

If all eigenvalues of W is negative, the zero matrix Z = O achieves the upper bound as established in
Theorem 1.
20
If x1 and x2 are independent and V12 = 0, then the optimal disclosure is g(x1 , x2 ) = x1 when V11 >0 > V22 and g(x1 , x2 ) = x2 when V22 > 0 > V11 .
( )
( )
x̂1
ŷ1
21
− 21
.
For mutually dependent states, we have corr(ŷ1 , ŷ2 ) ∈ {−1, 1} where
=Σ
x̂2
ŷ2

15

implies that two variables, z11 ∈ [0, 1] and ρ ∈ {−1, 1}, determine the other two variables√
z22 and z12 as z22 = 1 − z11 and z12 = ρ z11 (1 − z11 ). From these observations, the search
for the upper bound is characterized as a simple maximization problem that does not needto compute the positive eigenvalue: let ρ ∈ {−1, 1} and z ∈ [0, 1] be parameters and h(ρ, z)
be a function on {−1, 1} × [0, 1] deﬁned by√h(ρ, z) = σ12 V11 z + 2σ1 σ2 V12 ρ z(1 − z) + σ22 V22 (1 − z).
Then we have max h(ρ, z) = maxΣ≽Σ̂≽O tr(V Σ̂).
Although Theorem 2 tells us what distribution of the estimates should be induced by the
optimal disclosure rule, it tells little about how to construct a disclosure rule that inducessuch a distribution of the estimates. While the upper bound characterized in Theorem 2
depends only on the second moment of the underlying distribution of the state, the optimal
rule may depend on the entire distribution of x since we have to obtain the conditional
expectations for each m. In general, there is little hope of ﬁnding a disclosure rule thatattains the upper bound. This leaves us with two choices: one is to investigate necessary
conditions for the optimal disclosure rule as in Kamenica and Gentzkow (2011) and Rayoand Segal (2010), while the other is to characterize the optimal rules for some class of state
distributions. We take the second approach and identify optimal disclosure rules under theassumption that the state has a normal distribution.

5

Normally Distributed State and Linear Disclosure
Rule

In this section, we characterize the optimal disclosure rule when the state has a normal
distribution. Speciﬁcally, we suppose that x ∼ N (0, Σ) where Σ is symmetric and positivedeﬁnite.
A linear rule of rank l is a deterministic disclosure rule such that m = g(x) is a linear
transformation of rank l.22 For l ≥ 1, we can represent a linear rule by g(x) = A′ x ∈ Rl
where A is a k × l matrix of rank l.23 We can interpret a linear rule of rank l as a rule
publicizing l variables (m1 , . . . , ml ) none of which is redundant. Note that full disclosure isa linear rule of rank k such as g(x) = x and no disclosure rank zero such as g(x) = 0. Since
22

That is, the dimension of the range of g equals l.
If g maps each realization of the state into RL with L > l, there are L − l redundant variables, say(ml+1 , . . . , mL ), in the sense that E[x|m1 , . . . , mL ] = E[x|m1 , . . . , ml ]. Thus we can represent the linear ruleof rank l by a k × l matrix A without loss of generality.
23

16

we know that no disclosure is optimal if and only if V is negative semideﬁnite, so we supposethat V is not negative semideﬁnite.
When the state is normally distributed and a linear rule of rank l ≥ 1 is chosen, themessage m = A′ x ∈ Rl has a normal distribution with zero mean. The standard result of
the normal distribution gives the conditional expectation x̂ = E[x|m] = ΣA(A′ ΣA)−1 m.24
Since m = A′ x, we havex̂ = ΣA(A′ ΣA)−1 A′ x.
Intuitively, a linear rule of rank l projects the realization of the state onto an l dimensional
subspace in which the estimates are distributed. It follows from the analogue of Theorem 1
that the optimal disclosure rule must induce a distribution of conditional expectations suchthat V is positive semideﬁnite on its support. That is, x̂′ V x̂ ≥ 0 for every x̂ in its support.
Otherwise, we can reduce unfavorable variability of conditional expectations in a similarmanner to the only if part of Theorem 1 (i). This necessary condition eﬀectively narrows
the set of potential solutions.
1
Let B ≡ Σ 2 A and PB = B(B ′ B)−1 B ′ . The matrix PB is an orthogonal projection matrix
1
1that maps vectors in Rk onto the column space of B. Then the estimates are x̂ = Σ 2 PB Σ− 2 x,
and hence the second moment Σ̂ is written asΣ̂ = Ex̂x̂′ = Σ 2 PB Σ 2 .
1

1

1

1

Thus the gain of the disclosure rule is written as tr(V Σ̂) = tr(V Σ 2 PB Σ 2 ) = tr(W PB ) where1
1
W = Σ2 V Σ2 .
1
Consider a linear rule m = A′ x such that A = Σ− 2 Q+ where, as denoted in Theorem
2, Q+ = [q1 , . . . , qr ] is the eigenvectors associated with the nonnegative eigenvalues of W .
1Then B = Σ 2 A = Q+ and tr(W PB ) = tr(W PQ+ ), which equals the upper bound identiﬁed in
Theorem 2. Thus, we ﬁnd a linear rule that is optimal among the general class of disclosurerules.
Theorem 3 Suppose that the state is normally distributed, that is x ∼ N (0, Σ). Then a1linear rule g(x) = Q′+ Σ− 2 x is optimal where Q+ = [q1 , . . . , qr ] is the eigenvectors associatedConsider the joint distribution (x, m) ∼ N (µ̃, Σ̃) where the mean vector and variance matrix can bepartitioned as
( )
(
)
µ̃x
Σ̃x,x Σ̃x,mµ̃ =, Σ̃ =.
µ̃mΣ̃m,x Σ̃m,m
24

It follows (see, for example Vives (2008)) then the conditional density of x given m is normal with conditional−1′mean µ̃x + Σ̃x,m Σ̃−1m,m (m − µ̃m ) and variance matrix Σ̃x,x − Σ̃x,m Σ̃m,m Σ̃m,x . Now apply to m = A x, wehave Σ̃x,m = E[xx′ A] = ΣA and Σ̃m,m = E[A′ xx′ A] = A′ ΣA.

17

1

1

with the nonnegative eigenvalues of the symmetric matrix W = Σ 2 V Σ 2 .
The optimal linear rule in Theorem 3 is interpreted as the following information process1
ing. First, the sender adjusts the variance of the state as y ≡ Σ− 2 x so that var(y) = Ik .
Second, r variables (m1 , . . . , mr ) is disclosed each of which is a linear combination mi =q1i y1 + · · · + qki yk of y where the weights (q1i , . . . , qki ) constitute an eigenvector associatedwith a nonnegative eigenvalue of W .25
Here we brieﬂy discuss the role of two key properties in Theorem 3: the normality ofthe state and the linearity of the disclosure rule. The ﬁrst remark is that there exists
Z ∈ {Z̃ : I ≽ Z̃ ≽ O} such that it cannot be induced by the linear rule. Recall that1
for any linear rule A (translated into B = Σ 2 A), we have Z = PB = B(B ′ B)−1 B ′ . A
property of orthogonal projection matrices (that is, symmetric and idempotent) is that every
eigenvalue is either zero or one. Therefore, the linear rule cannot induce Z such that it has
an eigenvalue in (0, 1). Second, when the normality fails, the linear rule is no longer
able
(
) to
1 1
achieve the upper bound identiﬁed in Theorem 2. For example, consider V =
and
1 0
iid

1
1
I and W = 12
V . Then the upper bound isxi ∼ U [−1/2, 1/2] for i = 1, 2 so that Σ = 12√
given by the positive eigenvalue of W , which equals (1 + 5)/24 ≈ 0.1348, while the linear
86
1
rule yields at most 27· 24≈ 0.1327 under g(x1 , x2 ) = 3x1 + 2x2 .

5.1

The Two-dimensional Normal State

We now apply the theory developed above to obtain the optimal disclosure rule under
the normally distributed state for k = 2. Let v̂(x̂) = x̂′ V x̂ and (x1 , x2 ) ∼ N ((0, 0), Σ). From
Corollary 1, we focus on the case where V is indeﬁnite (i.e., neither positive nor negative
semideﬁnite) and ﬁnd the optimal linear rule A of rank 1 that maximizes tr(V Σ̂).26
1
1
We normalize the state and the estimates by y = Σ− 2 x and ŷ = Σ− 2 x̂. First we see
how an orthogonal projection PB determines the distribution of ŷ. For B = (b1 , b2 )′ ∈ R2 ,we have ŷ = PB y, or equivalentlyŷ1 = b1 y21 +b22 y2 b1
b1 +b2ŷ = b1 y21 +b22 y2 b .
2

b1 +b2

2

It is worth noting that if the rank of W is less than k, say k − n, then the rank of optimal linear
rule is indeterminate. Formally, letting Q++ = [q1 , . . . , qr−n ] be the eigenvectors associated with positive1
eigenvalues, any linear rule A = Σ− 2 B such that B contains every column of Q++ but orthogonal to Q−
attains the same value. Hence the rank of an optimal linear rule may be r − n or more but must be less thanor equal to r.
26
If V is positive (negative) semideﬁnite, full (no) disclosure which corresponds to a linear rule of rank 2(0, respectively) is optimal.
25

18

Thus we ﬁnd that (ŷ1 , ŷ2 ) are distributed on the line b2 ŷ1 = b1 ŷ2 . Moreover the variancematrix of (ŷ1 , ŷ2 ) is var(ŷ) = E[PB yy′ PB′ ] = PB , or equivalently(
var(ŷ1 , ŷ2 ) =where (c1 , c2 ) = ( √ b21

b1 +b22, √ b22b1 +b22c21 c1 c2
c1 c2 c22)

) is a point on the unit sphere in R2 . This implies that thesender’s choice variable is essentially one-dimensional, and hence the optimization problem
can be solved by standard calculus.
1

1

Corollary 3 Suppose that V is indefinite and W = Σ 2 V Σ 2 . Then the optimal linear rule1
is A = Σ− 2 B where B = (b1 , b2 )′ ∈ R2 is such that: (i) if w12 = 0, then B = (1, 0)′ forw22 < 0 < w11 , and B = (0, 1)′ for w11 < 0 < w22 ; (ii) if w12 = w21 ̸= 0, then√2
(w11 − w22 ) + (w11 − w22 )2 + 4w12
b1
=
.
b2
2w12
Corollary 3 characterizes the optimal linear rule when k = 2 as in the motivating example
and in Rayo and Segal (2010). In contrast to the ﬁnite state space case, the optimal disclosure
rule is deterministic and linear if the state has a bivariate normal distribution. For example,
for xi ∼ N (0, σi2 ) for i = 1, 2, the solution to the Rayo and Segal (2010)’s problem is thelinear rule g(x1 , x2 ) = σ1−1 x1 + σ2−1 x2 .

6

Applications

6.1

Optimal Advertising Policy

Suppose that a monopoly ﬁrm chooses an information disclosure policy about its new
product. In particular, the ﬁrm observes its cost shock xc and quality shock xa . Assume
that random variables xa and xc are independent and normally distributed with means zeroand variances σa2 and σc2 , respectively. A disclosure rule (α, M ) determines information mavailable to consumers. For example, a computer manufacturer discloses various information about its product including display resolution, battery life, processing speed, and soon. A production company releases on-line free music/movie clips. By controlling information revealed, the ﬁrm can induce a preferred distribution of the consumers’ conditional
expectations about the product quality and production cost.
Suppose that a representative consumer has a quadratic utility function u(q, xa ) =()(a + xa )q − 21 q 2 − pq where q is the quantity consumed and p is the unit price. It fol19

lows that the inverse demand function given message m is q = a + x̂a − p where x̂a = E[xa |m]is the consumer’s conditional expectation about the quality shock. The ﬁrm’s proﬁt functionis v(q, xc ) = pq − (c + xc )q. For simplicity, we assume that the price is exogenously ﬁxed.27
The timing of the game is as follows. First, the ﬁrm commits to a disclosure rule (α, M ).
Second, the ﬁrm observes the realization of the state (xa , xc ) and discloses information m.
Third, the consumer estimates the product quality and determines the demand quantity.
The ﬁrm’s expected proﬁt is written as
Ev =E [(p − c − xc )(a + x̂a − p)]= − Ex̂a x̂c + (p − c)(a − p).
Thus, we have v̂(x̂a , x̂c ) = −x̂a x̂c . An immediate implication is that the ﬁrm’s expected
proﬁt is a decreasing function of the covariance between x̂a and x̂c . Intuitively, the ﬁrm
is better oﬀ increasing the probability that demand expand when it has a lower cost. The
expected value of social welfare (i.e., consumer surplus plus the ﬁrm’s proﬁt) is
[(

)
]
1 2
Ew =E (a + xa )q − q − (c + xc )q
2
)
[
]
(
1 2
a+p
−c
=E x̂a − x̂a x̂c + (a − p)22
so that the socially optimal disclosure rule maximizes the expected value of a quadratic
function ŵ(x̂a , x̂c ) = 12 x̂2a − x̂a x̂c .
From Corollary 3, the optimal disclosure rule for the ﬁrm is g P (xa , xc ) = σc xa − σa xc
while the socially optimal disclosure rule is g S (xa , xc ) = κxa − σa xc where
κ=

σa +

√
σa2 + 4σc2
> σc .
2

The coeﬃcient on xa is interpreted as the amount of information revealed about the productquality.
Proposition 1 The monopoly advertisements contain less information about the productquality than the socially optimal advertisements.

Lewis and Sappington (1994) examine the amount of information a monopoly ﬁrm might
27

Milgrom and Roberts (1986) analyze a model in which the ﬁrm chooses its price and consumers draw
product-quality inferences from price as well as advertisement. Although such a “signaling eﬀect” of action
is important in a number of diﬀerent contexts such as monetary policy (see, Baeriswyl and Cornand (2010)),it requires diﬀerent techniques and is beyond the scope of the present paper.

20

provide to potential buyers. In their setting, each buyer privately observes an imperfectsignal about his valuation and the ﬁrm controls the precision of the signals. Anderson
and Renault (2006) and Johnson and Myatt (2006) distinguish the informational content ofadvertisements (e.g., price vs. attributes and hype vs. real information, respectively). In ourmodel, the ﬁrm should disclose a one-dimensional index constructed from its product quality
and marginal cost and control the variances and covariance of the “market expectations.”

6.2

Campaign Advertising and Incumbency Advantage

The population consists of two groups of voters, indexed by i ∈ {1, 2}. These groups
diﬀer in their policy preferences over a one-dimensional policy space. Let q1 = − 21 and q2 = 12
be the preferred policies of groups 1 and 2, respectively. As in Prat (2002), voters also judge
candidates in another dimension, say valence, which represents the general competence of acandidate such as negotiating ability, leadership, and integrity. Unlike policy preferences, allvoters’ preferences are the same in the valence dimension.
Two parties compete against each other in an election. An incumbent runs from the rulingparty and a challenger from the opposition party. The ruling party is privately informed
about the characteristics of the incumbent and makes campaign advertising that may reveal
information about him. The incumbent is characterized by two parameters (x, y) wherex ∈ [− 21 , 12 ] represents his policy stance and y ∈ [− 12 , 12 ] represents his valence. Assume thatx and y have zero means and are independent of each other. The ex ante distribution of(x, y) is common knowledge, but its realization is observed only by the ruling party. We
address the optimal campaign policy for the ruling party that maximizes the probability ofreelection in the absence of the opposition party’s campaign.28 A possible interpretation of
the campaign strategy is such that the party chooses topics discussed in a meeting and incandidates’ speeches.
When the incumbent of type (x, y) is elected, voters in group i ∈ {1, 2} receive utility
ui = −|x − qi | + y. On the other hand, when the challenger is elected, they receive ui =−|0 − qi | + ti where ti is a private information of group i that represents an ideological bias
toward the challenger. We assume that ti is independent of t−i and is uniformly distributed
1
1over [− 2h, 2h] for a suﬃciently small h > 0.29 Thus, given m and ti , voters in group i votefor the ruling party if E[ui (x, y)|m] ≥ −|0 − qi | + ti .
The timing of the game is summarized as follows. The ruling party commits to a disclosure
rule. The ruling party observes the incumbent’s type (x, y) and publicly discloses a message
28

Information disclosure by multiple senders raises a new issue and is beyond the scope of the paper. A
recent work of Gentzkow and Kamenica (2011) tackles such a problem.
29
For h ∈ (0, 12 ), the interior solution is guaranteed.

21

m according to the disclosure rule. The noise variables (t1 , t2 ) are realized. Given m and ti ,each voter votes for the candidate he prefers. If both groups vote for the same candidate, hewins with probability one. If two groups disagree, then the incumbent wins with probabilityψ ∈ [0, 1].
Let x̂ = E[x|m] and ŷ = E[y|m]. The voters’ expected utility from the incumbent
conditional on m is written as−( 1 + x̂) + ŷ for group 1
2
E[ui (x, y)|m] =−( 1 − x̂) + ŷ for group 2
2

The probability P1 (P2 ) that voters in group 1 (group 2) vote for the incumbent is given
by P1 = 12 + h(ŷ − x̂) (P2 = 21 + h(ŷ + x̂), respectively). The conditional probability P (m)that the incumbent wins given message m is
P (m) =P1 P2 + ψ(1 − P1 )P2 + ψP1 (1 − P2 )1 + 2ψ
=+ hŷ + (1 − 2ψ)h2 (ŷ + x̂)(ŷ − x̂).
4Thus, we have v̂(x̂, ŷ) = (1 − 2ψ)h2 (ŷ 2 − x̂2 ). Since we assume that x and y are independent,
we ﬁnd the following result.30
Proposition 2 When the incumbent has an advantage (ψ ≥ 21 ), then it is optimal for theruling party to reveal only the incumbent’s policy stance (g(x, y) = x).
Intuitively, when the incumbent has an advantage, the ruling party has an incentive toincrease the probability that at least one group prefers the incumbent to the challenger eventhough it decreases the probability of unanimity. Consequently, the incumbency advantage
impairs the selection of a competent candidate through the campaign strategy that reveals
no information about the valence characteristics. We also predict that the opposition party
needs to attract both groups of voters and has an incentive to reveal the valence dimensionof the candidates; for example, revealing scandals involving the incumbent and emphasizing
his inconsistent statements.
Similar situations arise in diﬀerent contexts. For example, in a criminal court, a defense
attorney who needs to persuade only a part of juries is better oﬀ making an emotional appeal
to them while a prosecutor who needs to avoid a conﬂict among juror is better oﬀ gathering
objective evidence of guilt. In this case, the voting procedure determines the incentives of
information revelation by the prosecutor and the attorney.
30

Note that the result does not depend on the marginal distributions of x and y.

22

Polborn and Yi (2006) analyze information revelation in a political campaign assumingthat each candidate must truthfully reveal either positive or negative information. Coate(2004) and Galeotti and Mattozzi (2011) analyze informative campaign which perfectly reveals the candidate’s policy position to a fraction of voters while Prat (2002) analyzes campaign advertising when the campaign expenditures signal the candidate’s valence.

6.3

Central Bank Transparency

We examine how central bank transparency aﬀects the volatility of the output gap andinﬂation and characterize the optimal disclosure rule and monetary policy. As in Geraats(2002) and Jensen (2002), we consider a simple two-period model where period 1 is regarded
as the present and period 2 as the future.
The private sector behavior is summarized by a standard Phillips curve31
lt = πt − EPt−1 πt + εtwhere lt is (log) employment in period t, πt is the inﬂation rate in period t (the change in thelog price level between period t − 1 and t), and εt is an employment shock (a supply shock).
The expectation operator EPt−1 [·] denotes the market expectation formed in period t − 1.
The central bank has perfect control over inﬂation πt = it where it is the central bank’s
intended inﬂation.32 We assume that the central bank can commit to a contingent monetary
policy in the short-run, but cannot commit to the future policy. For example, career concernsof the policymakers may prevent discretionary policymaking in the short-run, but in the
future the composition of the policymaker board may alter and an alternative policy plan
may be chosen. Moreover, an unpredictable change in economic and political conditions may
make the initial plan totally inadequate.
The central bank’s loss function is E[L1 + βL2 ] where β ∈ (0, 1) is the discount factor
and Lt is the period t loss function
Lt = πt2 + λ(lt − lt∗ )2for some λ > 0. The employment target lt∗ can be interpreted as a demand shock due tostochastic preferences of the representative household or as the central bank’s preference
shock due to a change in the degree of central bank independence.
31

The description of the economy is based on Faust and Svensson (2001).
Faust and Svensson (2001) assume that the central bank has imperfect control over inﬂation so thatπt = it + ηt where ηt is a control error.
32

23

We assume that the supply shock and the employment target are independent and normally distributed with mean zero. In particular, we assume that l1∗ ∼ N (0, σl2 ) and ε1 ∼∗N (0, σε2 ), and that these shocks evolve according to εt+1 = ρε εt + ξt+1 and lt+1
= ρl lt∗ + ζt+1
where ξt+1 and ζt+1 are independent shocks with mean zero and ρl , ρε ∈ (−1, 1).
The timing of events is as follows. The central bank chooses a disclosure rule (α, M )and a short-run monetary policy plan i1 : R2 × M → R which depends on the realizationof the supply and demand shocks and the message disclosed. The state of nature (l1∗ , ε1 ) isrealized and a message m ∈ M is publicly announced according to (α, M ). The central banksets a short-run monetary policy i1 (l1∗ , ε1 , m), and the inﬂation rate π1 is determined. Givenm and i1 (l1∗ , ε1 , m), the private sector forms an expectation about the future inﬂation rate
EP1 [π2 ]. In period 2, the central bank chooses a policy i2 given the realization of (l2∗ , ε2 ) andthe market expectation.
In period 2, the central bank’s problem is given bymin π22 + λ(l2 − l2∗ )2i2

subject to l2 = π2 − EP1 π2 + ε2π2 = i2 .
From the ﬁrst-order condition and the rational expectation, we have]
λ [ ˆ∗λ(l2 − ε̂2 ) + (l2∗ − ε2 )1+λ]1 [ ˆ∗λ(l2 − ε̂2 ) + (l2∗ − ε2 )l2 − l2∗ = −
1+λ
π2 =

where ŷ ≡ EP1 y denotes the market expectation of a random variable y formed in period 1.
The loss in period 2 is given byL2 =]2
λ [ ˆ∗λ(l2 − ε̂2 ) + (l2∗ − ε2 ) .
1+λ

We now consider the short-run monetary policy and optimal disclosure rule, which solve
the following problem
min E[π12 + λ(l1 − l1∗ )2 ] + βEL2 (ˆl2∗ , ε̂2 , l2∗ , ε2 )i1 (·),(α,M )subject to l1 = π1 − EP0 π1 + ε1
π1 = i1 (l1∗ , ε1 , m).

24

Note that we can restrict our search for optimal short-run policy plans to the class of mmeasurable functions without loss of generality. To see this, consider a disclosure rule (α, M )and a short-run policy plan i1 : R2 ×M → R. The private sector observes m and i1 (l1∗ , ε1 , m),
and forms an expectation (ˆl2∗ , ε̂2 ). Now consider a disclosure rule (α̃, M × R) where themessage is given by m̃ = (m, i1 (l1∗ , ε1 , m)), and a m̃-measurable policy ĩ1 (m̃) = i1 (l1∗ , ε1 , m).
This pair of a disclosure rule and a policy plan is essentially identical to the initial pair((α, M ), i1 ) in the sense that the information revealed to the private sector and the shortrun monetary policy in period 1 are the same almost surely. Therefore we ﬁrst characterize
the optimal short-run policy plan given each disclosure rule, and then ﬁnd the optimal
disclosure rule.
Fix a disclosure rule (α, M ). From the ﬁrst-order condition, the optimal short-run policy
i1 (·) is given byλ ˆ∗
i1 (ˆl1∗ , ε̂1 ) =(l − ε̂1 ),1+λ 1
and the ex ante expected loss is
λ2
EL1 + βEL2 =λE(l1∗ − ε1 )2 −E(ˆl1∗ − ε̂1 )2
1+λ
]
[
λ
λ2 (2 + λ) ˆ∗∗22(l − ε2 ) +E(l2 − ε̂2 ) .
+β
1+λ 2
1+λ
Recall that ˆl2∗ = EP1 [l2∗ ] = ρl ˆl1∗ and ε̂∗2 = EP1 [ε∗2 ] = ρε ε̂∗1 . Then we have
λ2
V =
1+λ

(

)
(
)
1 −1
ρ2l
−ρl ρεβλ2 (2 + λ)
−
.
1+λ
−1 1
−ρl ρερ2ε(5)4

Since det(V ) = − β(2+λ)λ(ρl − ρε )2 ≤ 0, the following statement holds.
(1+λ)2
Proposition 3 A linear rule of rank 1 is optimal whenever ρl ̸= ρε .
The central bank needs to respond to the shocks (l1∗ , ε1 ) to stabilize the output gap.
On the other hand, it should avoid information revelation about the future policy since it
weakens the policy eﬀectiveness in the future. This trade-oﬀ makes partial revelation optimalin the generic case where ρl ̸= ρε .
To make this point clear, suppose that ρl = 1 and ρε = 0. Thenλ2
W =Σ VΣ =
1+λ
1
2

1
2

(

σl2 (1 − β(2 + λ)) −σl σε−σl σεσε225).

From Corollary 3, the optimal linear rule g(l1∗ , ε1 ) = κl1∗ − ε1 where κ ∈ (0, 1) is a decreasing
function of β(2 + λ) and (σε /σl ).33 As β increases, the information revelation about l1∗
becomes more costly and then the amount of information contained in the message, which
can be measured by κ, should decrease.34
Given the optimal linear rule, the short-run policy is written as
i1 =

λ ˆ∗
λ κσl2 + σε2(l1 − ε̂1 ) =(κl∗ − ε1 ).
1+λ
1 + λ κ2 σl2 + σε2 1(6)As κ increases, the stabilization policy becomes more responsive to l1∗ , which increases Eπ12and decreases E(l1 − l1∗ )2 . Intuitively, as the central bank becomes more myopic, the costfrom the output stabilization due to information revelation becomes less important. This
comparative statics is summarized as follows.35
Proposition 4 As the central bank becomes myopic, the output gap is stabilized while theinflation becomes volatile.
As illustrated above, our framework is useful to characterize the optimal monetary policythat plays a signaling role as well as the stabilization role.36 Note that the optimal short-run
policy plan i1 is a linear function of the message m = κl1∗ − ε1 (see (6)). This implies that thepolicy outcome i1 (m) contains the same information as the message, and hence the optimal
disclosure rule and monetary policy are also implemented by committing to the optimal
short-run policy plan and making the policy outcome transparent.
A number of papers (e.g., Faust and Svensson (2001) and Jensen (2002) among others)
investigate the welfare eﬀect of central bank transparency and the optimal monetary policy indiﬀerent transparency regimes. Unlike these papers which restrict communication strategies
available to the central bank to noisy communications, we characterize the optimal disclosure
rule and monetary policy plan in the general class of policies. Our analysis suggests that the33

An exact expression for κ is1
κ =1 − (1 + (σε /σl )2 + β(2 + λ))2
1√(1 + (σε /σl )2 + β(2 + λ))2 − 4β(2 + λ).
+2

34

The amount of information about the supply shock revealed is measured by the variability of theconditional expectation of l1∗ , var(E[l1∗ |m]) = κ2 σl4 /(κ2 σl2 + σε2 ), which is increasing in κ ∈ (0, 1).
35
In a similar manner, one can examine how the policy maker’s preferences, parameterized by λ, aﬀect
the monetary policy and central bank transparency.
36
Recently, Baeriswyl and Cornand (2010) study this problem in the setting where the monetary policy
is imperfectly observed by the private sector.

26

central bank should control the covariance of market expectations rather than the varianceof private sector forecast errors of each variable, written as var(ε − ε̂) and var(l∗ − ˆl∗ ).

7Conclusion

We study multidimensional information disclosure where the sender’s expected utility
is expressed as the expected value of a function of the receivers’ expectations of the state.
The semideﬁnite programming is applied to identifying necessary conditions for the second
moment of the conditional expectations that can be induced by the disclosure rule and characterizing an upper bound of the sender’s expected utility. We characterize the optimal
disclosure rule among the general class of (possibly randomized) rules as a linear transformation of the state when it is normally distributed. Based on such a simple and tractablecharacterization, we study several applications and provide interesting implications. Possible
directions for future work include studying settings with multiple senders and with receivers’
private information.

AppendixA

Proofs

Proof of Theorem 1. The if parts (the optimality of full/no disclosure when V is deﬁnite)follow from Jensen’s inequality and the only if parts (the optimality of partial disclosure
when V is indeﬁnite) are shown by constructing a rule that yields a higher gain than full/no
disclosure.
(i): if part. Suppose that V is positive semideﬁnite. We will show that full disclosure is
optimal. If V is positive semideﬁnite, or equivalently if v̂ is a convex function, then, for any
disclosure rule (α, M ), we have from Jensen’s inequality that Ev̂(x̂) ≤ E[E[v̂(x)|m]] = Ev̂(x)but the last is equal to the gain under full disclosure. Hence full disclosure is optimal.
(ii): if part. Suppose that V is negative semideﬁnite. We will show that no disclosure is
optimal. If V is negative semideﬁnite, or equivalently if v̂ is a concave function, then, forany disclosure rule (α, M ), we have Ev̂(x̂) ≤ v̂(Ex̂) = v̂(Ex) but the last is equal to the gain
under no disclosure. Hence no disclosure is optimal.
(i): only if part. Suppose that V is indeﬁnite. We will construct a partial disclosure thatattains a higher gain than full disclosure. Since V is not positive semideﬁnite, there exists
a vector x− ∈ Rk such that x′− V x− < 0. Let Lx− be the linear subspace in Rk spanned by27

ka vector x− and L⊥
x− be the orthogonal complement of Lx− . Any point in R is represented⊥
by the sum of vectors in Lx− and L⊥
x− , as x = y + γx− for some γ ∈ R and y ∈ Lx− .
Now consider a disclosure rule (α, M ) such that the sender discloses vector y ∈ L⊥
x− foreach realization of x. Under this rule, the receivers know on which line x is realized, but they
are still uninformed about γ ∈ R. Thus x|m is distributed over a line through x̂ parallel tox− , and hence we can write x|m − x̂ as γx− where γ ∈ R is a corresponding random variable.
Now compare the gain Ev̂(x) under full disclosure with the gain Ev̂(x̂) under rule (α, M ):E[x′ V x] − E[x̂′ V x̂] =E[E[x′ V x|m] − E[x̂′ V x̂]]=Em [E[(x − x̂)′ V (x − x̂)|m]]][=Em E[γ 2 x′− V x− |m]=x′− V x− Eγ 2 < 0
where the inequality holds since γ ̸= 0 almost surely.37(ii): only if part. Suppose that V is indeﬁnite. We will construct a partial disclosure thatattains a higher gain than no disclosure, which equals zero. Let s ∈ S k−1 be a point in theunit sphere in Rk and consider a disclosure rule under which the sender discloses the sign ofs′ x ∈ R.38 Let x̂p = E[x|s′ x ≥ 0] and x̂n = E[x|s′ x < 0].
Since V is not negative semideﬁnite, there exists x+ ∈ Rk such that x′+ V x+ > 0. We
want to show that there exists s ∈ S k−1 such that x̂p = γps x+ and x̂n = γns x+ for some
γps , γns ∈ R.
Let B be a k × (k − 1) matrix such that all its column vectors are orthogonal to x+ .
Consider a function g : Rk−1 → Rk−1 deﬁned by g(s) = B ′ [E(x|s′ x ≥ 0) − E(x|s′ x < 0)].39
Since g is a continuous function from an (k −1)-sphere into Euclidean (k −1)-space, from theBorsuk-Ulam theorem, there exists s ∈ S k−1 such that g(s) = g(−s). However we know thatg(−s) = −g(s).40 Therefore there exists s ∈ S k−1 such that g(s) = −g(s), which must be
equal to zero. Using the fact that Pr(s′ x ≥ 0)E[x|s′ x ≥ 0] + Pr(s′ x < 0)E[x|s′ x < 0] = Ex =0, we have B ′ E[x|s′ x ≥ 0] = B ′ E[x|s′ x < 0] = 0. This implies that they are proportional tox+ ; so we can write x̂ = γx+ where γ ∈ {γps , γns } is a corresponding random variable. ThenEv̂(x̂) = E[γ 2 x′+ V x+ ] > 0 = v̂(Ex).
Proof of Theorem 2. First, we give a necessary condition for the solution, and thenNote also that the second equality holds since Em [E[x̂′ V x|m]] = Em [E[x′ V x̂|m]] = Em [x̂′ V E[x|m]] =Em [x̂′ V x̂].
38
Formally, for each s ∈ S k−1 ⊂ Rk , deﬁne (αs , {m+ , m− }) by αs (m+ |s′ x ≥ 0) = 1, αs (m− |s′ x ≥ 0) = 0,sα (m+ |s′ x < 0) = 0, and αs (m− |s′ x < 0) = 1.
39
Note that for all s ∈ S k−1 , E[x|m] ̸= 0.
40This follows from B ′ [E(x| − s′ x ≥ 0) − E(x| − s′ x < 0)] = B ′ [E(x|s′ x ≤ 0) − E(x|s′ x > 0)]. Note that
{x : s′ x = 0} has measure zero.
37

28

establish the result.
Step 1 : We will show that Z is an orthogonal projection matrix whenever Z is a solution
to SDP. Since Z ∈ Sk+ is a symmetric matrix, we have the eigenvalue decomposition Z =CΛC ′ where C is an orthogonal matrix and Λ is a diagonal matrix with real entries.41 Z ≽ O
implies that all eigenvalues are nonnegative (λi ≥ 0 for every i), and I −Z = C(I −Λ)C ′ ≽ O
implies that all eigenvalues must satisfy 1 − λi ≥ 0 for all i. From a property of the trace∑
operator, we have tr(W Z) = tr(W CΛC ′ ) = tr(ΛC ′ W C) = ki=1 λi δi where δi is the i-th
diagonal entry of C ′ W C. Now consider a matrix Z̃ = C Λ̃C ′ where Λ̃ is a diagonal matrix
with each entry λ̃i being equal to 0 if δi < 0 and 1 if δi ≥ 0. By construction, we have∑∑tr(W Z) =λi δi ≤
λ̃i δi = tr(W Z̃). Since Z̃ is a symmetric positive semideﬁnite matrix
and furthermore is idempotent,42 the solution to SDP must be an orthogonal projection
matrix.43
Step 2 : We now show that for any orthogonal projection matrix Z of rank l, there
exists a k × l matrix D such that Z = QD(D′ D)−1 D′ Q′ where Q = [q1 , . . . , qk ] consistsof all eigenvectors of W . Since W is symmetric, we have the eigenvalue decomposition∑
W = QΩQ′ = ωi qi q′i .
Note that every orthogonal projection matrix is characterized by its target subspace in
Rk . Fix an arbitrary subspace in Rk and suppose that it is spanned by column vectors ofsome k × l matrix B. Then the orthogonal projection matrix onto this subspace is written
as PB = B(B ′ B)−1 B ′ .44 Let D = Q′ B. Then B = QD, and hence
PB =QD[(QD)′ (QD)]−1 (QD)′=QD[D′ D]−1 D′ Q′=QPD Q′ .
Step 3 : Assume, without loss of generality, that each eigenvalue ωi is nonnegative for
i = 1, . . . , r, and negative for i = r, . . . , k. Let Q+ ≡ [q1 , . . . , qr ] and Q− ≡ [qr+1 , . . . , qk ].
Note that Q′ Q = Ik implies Q′+ Q+ = Ir and Q′+ Q− = Or,k−r . We will show that tr(W PQ+ ) ≥tr(W PB ) for any k×k orthogonal projection matrix PB = QPD Q′ . Recall that every diagonal
entry of PD satisﬁes 0 ≤ (PD )ii ≤ 1 since both PD and I − PD are positive semideﬁnite.

A matrix A is orthogonal if AA′ = I.
42
A matrix A is idempotent if A2 = A. Note that Z̃ 2 = C Λ̃C ′ C Λ̃C ′ = C Λ̃2 C ′ = Z̃ since Λ̃2 = Λ̃.
43
A matrix A is an orthogonal projection matrix if it is symmetric and idempotent.
44
For any k × l matrix B of rank l, PB is symmetric and idempotent. Check PB′ = [B(B ′ B)−1 B ′ ]′ =′B(B B)−1 B ′ and PB2 = B(B ′ B)−1 B ′ B(B ′ B)−1 B ′ = B(B ′ B)−1 B ′ .
41

29

Then
tr(W PB ) =tr(QΩQ′ QPD Q′ )=tr(ΩPD )=k∑ωi (PD )ii

i=1≤ω1 + · · · + ωr .
Finally we check tr(W PQ+ ) =∑r

i=1 ωi .

tr(W PQ+ ) =tr(ΩQ′ Q+ (Q′+ Q+ )−1 Q′+ Q)
( ( )
)
(
)
Q′+
=tr Ω
Q+ (Q′+ Q+ )−1 Q′+ Q+ Q−
′
Q−
( (
)
)
(
)
Ir
=tr Ω
Ir Or,k−r
Ok−r,r
ω1 0 · · · 0)( 0 ω2 · · · 0 IOrr,k−r 
=tr .. . .
..  ..

. .  Ok−r,r Ok,k .
 .

0 0 · · · ωk=ω1 + · · · + ωr .
Thus we conclude that tr(W PQ+ ) ≥ tr(W Z) for every orthogonal projection matrix Z.

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