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Disclosure to a Psychological Audience
by Elliot Lipnowski & Laurent Mathevet∗
New York University†
June 7, 2015
Abstract
We provide an applied toolbox for the benevolent expert who discloses information
to a psychological agent. The expert faces a tradeoff if the agent is intrinsically averse to
information but has instrumental uses for it. While the Revelation Principle fails here,
simply recommending actions remains optimal; this speaks to the debate over simplified
disclosure. For agents who are neither information-averse nor loving, we develop general disclosure tools. Our method of posterior covers provides reductions, sometimes
drastic, in the complexity of the expert’s problem. Applications include disclosure prescriptions for agents facing temptation or cognitive dissonance.
Keywords: information disclosure, psychology, information aversion, temptation, persuasion, transparency.
JEL Classification: D03, D83, D82, D69, l18, M30.
∗
We thank Adam Brandenburger, Juan Dubra, Emir Kamenica, David Pearce, and Tomasz Strzalecki for
their comments. We also thank seminar audiences at Penn State University, Princeton University, Indiana
University, Harvard University and M.I.T., and the 2015 Canadian Economic Theory Conference. Email:
[email protected] and [email protected]
†
Address: Economics Department, 19 West 4th St., New York, NY 10012.
1
1
Introduction
Information disclosure choices —what is revealed and what is concealed—affect what people know and, in turn, the decisions that they make. In many contexts, information disclosure
is intended to improve people’s welfare. In public policy, for instance, regulators impose disclosure laws to this effect, such as required transparency by a seller of a good to a buyer. But
should a well-intentioned advisor always tell the whole truth? In standard economics, the
answer is ‘yes,’ because a better informed person makes better decisions (Blackwell (1953)).
However, parents do not always tell the whole truth to their children; people conceal embarrassing details about their personal lives from their family elders; doctors do not always reveal to their patients all the details of their health, when dealing with hypochondriac patients
for instance; future parents do not always want to know the sex of their unborn children; and
many of us would not always like to know the caloric content of the food that we eat.
In this paper, we study how an informed expert should disclose information to an agent
with psychological traits. It is hard to argue that our beliefs affect our lives only through the
actions that we take. The mere fact of knowing or not knowing something can be a source
of comfort or discomfort. How should we disclose information for the good of someone
whose state of mind has a direct impact on his well-being? This question is relevant to a
trusted advisor to the agent, but also to any benevolent entity that, by law, can enforce what
the informed party must disclose. Parents, family, and friends are examples of the first kind,
and may directly take into account the psychological well-being of their loved one when
deciding what to say. Certain professionals, such as teachers and doctors, also have interests
well-aligned with those they serve. The direct relevance of patients’ beliefs is well-known in
medical science. As Ubel (2001) puts it:
Mildly but unrealistically positive beliefs can improve outcomes in patients with
chronic or terminal diseases. . . Moreover, unrealistically optimistic views have
been shown to improve quality of life.
Many patients desire to be informed about their health only when the news is good, but
would rather not know how bad it is otherwise. Even in this simple case, it is not obvious
how optimally to disclose information. The first scheme that comes to mind—reveal the news
when it is good and say nothing otherwise—accomplishes little here, as no news reveals bad
news. However, in line with Benoˆıt and Dubra (2011), it is possible to leave many—but
not all—patients with unrealistically positive beliefs (even if they process information in a
Bayesian way).
Such deliberate information choices may even be reasonable when the expert and the
agent are one and the same. Indeed, we often act so as to control our inflow of information
2
for our own benefit; for instance, one might refrain from ordering a store catalog today so as
to prevent the possible temptation to overspend tomorrow.
As far as regulation is concerned, lawmakers often mandate disclosure with the uninformeds’ welfare in mind, in areas ranging from administrative regulations, loans, credit
cards, and sales, to insurance contracts, restaurants, doctor-patient relationships and more
(see Ben-Shahar and Schneider (2014)). Loewenstein, Sunstein, and Goldman (2014) argue
that psychology may lead one to rethink public disclosure policies. For example, it may be
counter-productive to offer detailed information to a person with limited attention, as she
may ignore information. Likewise, a comprehensive accounting of costs should include the
negative hedonic cost of dealing with the information. Think, for example, of giving information to a person who is fasting or dieting about the food which he could be eating; this
could create temptation if he resists and disappointment if he gives in.
The Model in Brief. In our model, a principal will learn the state of nature, and she must
choose what to tell the agent in every contingency. After receiving his information from the
principal, the agent updates his prior beliefs and then takes an action. The principal wants to
maximize the agent’s ex-ante expected utility by choosing the right disclosure policy.
We model psychology by assuming that the agent’s satisfaction depends not only on the
physical outcome of the situation, but also directly on his updated beliefs. In doing so, we
employ the framework of psychological preferences (Geanakoplos, Pearce, and Stacchetti
(1989)), which captures a wide range of emotions. In our model, this allows us to represent
various psychological phenomena, such as prior-bias and cognitive dissonance, temptation
and self-control problems (Gul and Pesendorfer (2001)), aversion to ambiguity (Hansen and
Sargent (2001)), and more.
We model communication by assuming that the principal can commit ex-ante, before
the state is realized, to an information policy. That is, to every state of nature corresponds
a (possibly random) message by the principal which, in turn, leads the agent to update his
belief. From an ex-ante perspective, the state follows some prior distribution; hence, an information policy induces a distribution over updated (or posterior) beliefs and actions. This
allows us to apply the method of random posteriors employed by Kamenica and Gentzkow
(2011) (KG hereafter). Alternative frameworks are available, most notably the ‘cheap talk’
model of Crawford and Sobel (1982), wherein the principal cannot commit ex-ante. Given
our focus on the impact of psychology on disclosure, we give the principal full commitment
power, abstracting from strategic interactions between different interim principal types. Depending on the application, we see at least two situations in which such commitment power is
a reasonable approximation. First, in many applications, a benevolent principal is some third
party who regulates information disclosure between a sender (e.g., a seller) and the agent,
with the agent’s interests in mind. Such a principal sets the disclosure policy in ignorance
3
of the realized state, and thus faces no interim opportunity to lie. Second, if the relationship
is subject to exogenous full transparency regulations (e.g., patients have the right to access
their lab test results), this legislative restriction creates commitment power. The principal
makes an ex-ante decision of which information to seek (e.g., the doctor decides which tests
to run) and is bound by law to reveal whatever she finds.
Psychological Preferences. Psychological preferences entail a number of new considerations. A first-order question is: “Does the agent like information?” A classical agent
is intrinsically indifferent to information. Holding his choices fixed, he derives no value
from knowing more or less about the world. But if he can respond to information, then he
can make better decisions and, thus, information is always a good thing (Blackwell (1953)).
This observation leads us to distinguish between two general standards of information preference. On the one hand, an agent may be psychologically information-loving [or -averse]
if he likes [dislikes] information for its own sake. On the other hand, he may be behaviorally information-loving [or -averse] if he likes [dislikes] information, taking its instrumental value into account. To see the distinction, imagine a person considering the possibility that his imminent seminar presentation has some embarrassing typos. If his laptop
is elsewhere, he may prefer not to know about any typos, since knowing will distract him.
He is psychologically information-averse. But if he has a computer available to change the
slides, he may want to know; such information is materially helpful. He is behaviorally
information-loving.
We characterize these four classes of preferences in Proposition 1, and optimal disclosure follows readily for three of them. Almost by definition, if the agent is behaviorally
information-loving [-averse], it is best to tell him everything [nothing]. If an agent is psychologically information-loving, then the potential to make good use of any information only
intensifies his preference for it, so that giving full information is again optimal. More interesting is the case in which information causes psychological distress, i.e., a psychologically
information-averse agent. This case entails a natural tradeoff between the agent’s need for
information and his dislike for it.
Information Aversion. Under psychological information aversion, the main finding is
that practical information policies are also optimal. In real life, mandated disclosure often
requires an avalanche of information, literally taking the form of probability distributions,
for example about possible illnesses and side effects (see Ben-Shahar and Schneider (2014)
and Section 5 for concrete examples). Although there are reasons for wanting such transparency, the benefits are being seriously questioned by professionals (ibid.). We contribute
to the debate by asking whether a simpler form of communication may suffice. A natural
alternative to such complex policies is a recommendation policy; that is, the principal can
simply tell the agent what to do. If the agent innately likes information, this minimalist
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class leaves something to be desired; but what if, to agents who like information only for its
instrumental value (i.e., psychologically information-averse agents), we only recommend a
behavior? Of course, the agent may not comply: we show that the Revelation Principle fails
in psychological environments. Despite this negative observation, Theorem 1 says that some
incentive-compatible recommendation policy is optimal when preferences display psychological information aversion.1 This suggests that a doctor may simply tell her patient which
medical action to take, instead of telling him what the test statistically reveals about the illness. This approximates many information policies that are actually observed in real life,
especially when a standard of full transparency is impractically cumbersome.
Still, psychologically information-averse agents can vary in their instrumental desire for
information. To one such agent, the principal could recommend a different action in every
state, thereby indirectly revealing the truth; and to another, she could recommend the same
action in all states, thereby revealing nothing. Clearly, recommendations can differ in their
informative content. In the same way that comparative risk aversion a` la Arrow-Pratt is
important in offering a person the right insurance policy, comparative information aversion
ought to be important in offering a person the right disclosure policy. Thus, we propose an
(apparently very weak) order on information aversion and study the implications of varying aversion. We arrive at a strong conclusion: when two agents are ranked according to
information aversion, their indirect utilities differ by the addition of a concave function. In
this model, concavity amounts to disliking information everywhere. In a binary-state environment, the principal will respond to this by providing a less informative policy to a more
information-averse agent. Interestingly, this is not true in environments with more than two
relevant states; a change in information aversion may optimally be met by a “change of
topic,” rather than by garbling information.
Method of Posterior Covers and its Applications. The world of psychological preferences is diverse, and many preferences escape our classification. Accordingly, we offer
general tools of optimal information disclosure. First, the concave envelope result of KG extends readily to our environment (Proposition 2), characterizing the optimal welfare of any
psychological agent. The concave envelope is a useful abstract concept, but, in principle,
deriving it (and hence an optimal disclosure policy) can be laborious. We exploit extra structure, suggested by the economics of the problem, to implement this program. This gives rise
to our method of posterior covers. This method, made possible by the benevolence of our
design environment, allows us to derive explicitly an optimal policy in meaningful situations.
An agent may not be globally information-loving, but may still be locally information-loving
at some beliefs. We leverage this to show that, in such regions, the principal should always
1
This result and its proof are different from the standard Revelation Principle proof, and the result is false
for general psychological preferences.
5
give more information. A posterior cover is a collection of convex sets of posterior beliefs
over each of which the agent is information-loving. By Theorem 3, the search for an optimal
policy can be limited to the outer points of a posterior cover of the indirect utility. By the
same theorem, this problem can be reduced to that of finding posterior covers of primitives
of the model—which hinges on the assumption of benevolence. Finally, in a broad class of
economic problems, all of these objects can be computed explicitly.
How should we talk to someone facing a prior-bias or cognitive dissonance? How should
we talk to someone who has temptation and self-control problems? We apply our method to
answer these questions and work out explicit examples. In the case of cognitive dissonance,2
the main message is that we should communicate with such an individual by an all-or-nothing
policy. Either the person can bear information, in which case we should tell him the whole
truth to enable good decision-making, or he cannot, in which case we should say nothing.
In the case of temptation and self-control, the optimal policy depends on the nature of information. We study a consumption-saving problem for an individual with impulsive desires
to consume rather than invest in some asset. We demonstrate how the method of posterior
covers can be brought to bear quite generally in the setting of temptation and self-control
a` la Gul and Pesendorfer (2001). In particular, the method captures the common precept
studied in psychology that a tempted agent does not want to know what he is missing. This
desire to limit information is something one observes in real life. By formalizing exactly in
which way the agent does not want to know what he is missing, our analysis explains the
precept as an optimal reaction to costly self-control. Specific prescriptions follow from additional assumptions. When information appeals only to the agent’s rational side, say when
it is about the asset’s return, full information is optimal. But when information appeals only
to the agent’s impulsive side, say when it is about the weather in a vacation destination, ‘no
information’ may not be optimal. By sometimes revealing good weather, an advisor induces
the agent to consume and avoid all temptation, or to invest and face reduced temptation;
partial revelation may then be optimal.
Related Literature. This paper is part of a recently active literature on information design (KG; Ely, Frankel, and Kamenica (2014); Kolotilin et al. (2015)). The most related
works, methodologically speaking, are KG and Gentzkow and Kamenica (2014). We adopt
the same overall method, study a different problem, and, most importantly, provide an applied toolbox that goes beyond the abstract formulation of optimality (as concavification)
and enables explicit prescriptions in important problems.
The relevance of psychological preferences in information contexts is exemplified by Ely,
Frankel, and Kamenica (2014), who study dynamic information disclosure to an agent who
2
In our model of cognitive dissonance, the agent pays a psychological penalty given by the distance between
his posterior belief and his prior position.
6
gets satisfaction from being surprised or feeling suspense. They define surprise as change
from the previous belief to the current one, define suspense as variance over the next period’s
beliefs, and derive optimal information policies to maximize each over time.
Our work does not belong to the cheap talk literature spanned by Crawford and Sobel
(1982), because our principal can commit to a communication protocol. A relevant contribution to that literature is Caplin and Leahy (2004), who study a detailed interaction between
a doctor and a patient with quadratic psychological preferences. Unlike us, they allow for
private information on the part of the agent. Our analysis concerns a strategically simpler
interaction, but encompasses a great range of applications.
There is a conceptual connection between psychological preferences and nonstandard
preferences over temporal resolution of uncertainty (e.g., Kreps and Porteus (1978)), which
can be reinterpreted as a psychological effect. The mathematical arguments behind our
classification of preferences are similar to Grant, Kajii, and Polak (1998), in their study
of intrinsic informational preferences (over lotteries of lotteries) absent the so-called reduction axiom. However, we make a conceptual distinction between psychological and behavioral information preferences that is critical for information disclosure. Moreover, information disclosure imposes additional constraints on the domain of the agent’s preferences—
specifically, Bayes-consistency—that are absent in that literature. Indeed, the beliefs of a
Bayesian agent cannot be manipulated at will; this restriction of the preference domain matters, for instance, when comparing information aversion across agents (Section 5.2).
The paper is organized as follows. Section 2 introduces psychological preferences and
the method of random posteriors. Section 3 presents the psychological agent, and studies
and characterizes various attitudes towards information. Section 4 lays out the benevolent
design problem. Section 5 studies information aversion. Section 6 supplies the general
tools of optimal information design. Section 7 applies these tools to prior-bias and cognitive
dissonance, and temptation and self-control.
2
The Environment
Consider an agent who must make a decision when the state of nature θ ∈ Θ is uncertain.3
The agent has (full support) prior µ ∈ ∆Θ, and receives additional information about the state
from the principal. After receiving said information, the agent forms a posterior belief by
updating his prior, and then he makes a decision.
3
In this paper, all spaces are assumed nonempty compact metrizable spaces, while all maps are assumed
Borel-measurable. For any space Y, we let ∆Y = ∆(Y) denote the space of Borel probability measures on Y,
endowed with the weak*-topology, and so itself compact metrizable. Given π ∈ ∆Y, let supp(π) denote the
support of π, i.e., the smallest closed subset of Y of full π-measure.
7
2.1
Psychological Preferences
An outcome is an element (a, ν) of A × ∆Θ where a is the action taken by the agent and
ν ∈ ∆Θ denotes the agent’s posterior belief at the moment when he makes his decision. We
assume that the agent has a continuous utility function u : A × ∆Θ → R over outcomes.
The true state is excluded from the utility function, without loss of generality: given any true
underlying
R preferences u˜ : A × Θ × ∆Θ → R, we can define the reduced preferences u via
u(a, ν) = Θ u˜ (a, θ, ν) dν(θ). The reduced preferences are the only relevant information, both
behaviorally and from a welfare perspective.4 Given posterior beliefs ν, the agent chooses
an action a ∈ A so as to maximize u(a, ν).
In the classical case, u˜ does not depend on beliefs, and so u(a, ·) is affine for every a.
We do not make this assumption here. In our environment, the agent’s satisfaction depends
not only on the physical outcome of the situation, but also on his posterior beliefs. In the
literature, such an agent is said to have psychological preferences (Geanakoplos, Pearce, and
Stacchetti (1989)).5
This formulation covers a wide range of phenomena. Consider the following examples
with Θ = {0, 1} and A = [0, 1], and note that we can write any posterior belief as ν =
Prob({θ = 1}):
(a) (Purely psychological agent) Let
u(a, ν) = −V(ν)
represent an agent whose only satisfaction comes from his degree of certainty, as quantified
by the variance V(ν) = ν(1 − ν).
(b) (Psychological and material tensions) For k ∈ R, let
h
i
u(a, ν) = kV(ν) − Eθ∼ν (θ − a)2
This agent wants to guess the state but he has a psychological component based on the
variance of his beliefs. When k > 0, the agent faces a tension between information, which
he dislikes per se, and his need for it to guess the state accurately.
4
Later, when we consider questions of information design by a benevolent principal, we will also work with
u rather than u˜ . This remains without loss, but only because the designer is assumed to have full commitment
power. This is in contrast to the setting of Caplin and Leahy (2004).
5
In a game-theoretic context, a player’s utility function could depend on the perceived beliefs of the other
players; preferences concerning fashion and gossip are examples of this phenomenon. In a dynamic context, a
player’s utility at some time could depend on the relationship between his current beliefs and any of his past
beliefs; surprise and suspense are examples (Ely, Frankel, and Kamenica (2014)).
8
(c) (Prior-bias and cognitive dissonance) Let
h
i
u(a, ν) = −|ν − µ| − Eθ∼ν (θ − a)2
represent an agent who wants to guess the state but experiences discomfort when information
conflicts in any way with his prior beliefs. This functional form is reminiscent of Gabaix
(2014)’s sparse-maximizer, though the economic interpretation is quite different: a distaste
for dissonance, rather than a cost of considering information.
(d) (Temptation and self-control) Given two classical utility functions uR (rational) and uT
(tempted), and action set A, let
ˆ
ˆ
u(a, ν) = Eθ∼ν uR (a, θ) − max Eθ∼ν
uT (b, θ) − uT (a, θ)
ˆ
b∈A
be the form proposed by Gul and Pesendorfer (2001). After choosing a menu A, the agent
receives some information and then chooses an action. His non-tempted self experiences
ˆ − uT (a, θ)]
ˆ is a cost of self-control
utility uR (a, θ) from action a, while maxb∈A Eθ∼ν
ˆ [uT (b, θ)
faced by the tempted side.
(e) (Aversion to ambiguity: multiplier preferences) For k > 0, let
(Z
)
0
0
u(a, ν) = min
uC (a, θ)dν (θ) + kR(ν ||ν)
0
ν
where R(ν0 ||ν) is the relative entropy of ν0 with respect to ν.6 These are the multiplier preferences of Hansen and Sargent (2001). The agent has some best guess ν of the true distribution,
but he does not fully trust it. Thus, he considers other probabilities ν0 whose plausibility decreases as the distance to ν increases.
With the tools that we develop, we will characterize optimal disclosure for each of the
above examples and more.
2.2
Signals and Random Posteriors
The principal chooses the agent’s information. Typically, one models information as a signal
(S , σ), which consists of a space S of messages and a map σ : Θ → ∆S . In any state θ, the
agent sees a message s ∈ S , drawn according to σ(·|θ) ∈ ∆S . A signal is then an experiment
concerning the state, whose results the agent will observe. We assume here that the principal
fully controls the signal, which is the only information obtained by the agent.
6
For any ν, ν0 ∈ ∆Θ, recall that R(ν0 ||ν) =
P
θ
ln [ν0 (θ)/ν(θ)] ν0 (θ).
9
Given the signal (S , σ), any realized message s leads the agent to form a posterior belief
of θ via Bayesian updating. As the state is ex-ante uncertain, a signal induces a distribution
over the agent’s posterior beliefs when seen from an ex-ante perspective. As far as payoffs
are concerned, this distribution over posterior beliefs, called a random posterior, is all that
matters. The situation in which the principal chooses a signal is equivalent to that in which
she chooses an information policy, as defined below.
Definition 1. An information policy (given prior µ) is an element of
R(µ) := {p ∈ ∆∆Θ : Eν∼p [ν] = µ}
(
)
Z
ˆ dp(ν) = µ(Θ)
ˆ for every Borel Θ
ˆ ⊆Θ .
=
p ∈ ∆∆Θ :
ν(Θ)
∆Θ
While there is room for manipulation—even in binary-state environments, R(µ) is an
infinite-dimensional object—the posterior beliefs of a Bayesian agent must on average equal
his prior, and thus, any feasible information policy must satisfy the Bayesian condition
Eν∼p [ν] = µ. As shown in Kamenica and Gentzkow (2011), all random posteriors obeying this law can be generated by some signal, and all signals generate random posteriors
obeying this law. In particular, one could employ a direct signal (also known as a posterior
policy) (S p , σ p ) to produce p, i.e., one for which (i) every message is a posterior in ∆Θ,
i.e., S p := supp(p) ⊆ ∆Θ; and (ii) when the agent hears message ‘s’, his update yields a
posterior belief equal to s. This signal tells the agent what his posterior belief should be, and
his beliefs conform.7
As preferences over information will play a large role in our paper, we now record a
natural definition of what it means for one information policy to be more informative than
another.
Definition 2. Given two random posteriors, p, q ∈ R(µ), p is more (Blackwell-) informative
than q, denoted p µB q, if p is a mean-preserving spread of q.8
Intuitively, a signal is more informative if the message is more correlated with the state,
in which case it is (from an ex-ante perspective, θ being random) more dispersed. The
mean-preserving spread is, it turns out, exactly the right notion of dispersion to capture more
7
One can show that the signal with the following law works:
Z
ds
σ p (Sˆ |θ) =
(θ) dp(s) for every θ ∈ Θ and Borel Sˆ ⊆ S p .
Sˆ dµ
That is, if there is a map r : S q → ∆(S p ) ⊆ ∆∆Θ such that (i) for every Borel S ⊆ ∆Θ, p(S ) =
and (ii) for every t ∈ S q , r(·|t) ∈ R(t).
8
10
R
Sq
r(S |·) dq
information. It is well-known that this definition is equivalent to the usual Blackwell (1953)
garbling definition (Lemma 1, in the appendix).
It is apparent from this definition that information looks a lot like ex-ante risk concerning
the agent’s eventual posterior beliefs. Proposition 1 below will sharpen this intuition.
3
The Psychological Agent
Some agents like to be informed and others do not (see Frankl, Oye, and Bellamy (1989)
for attitudes of patients toward information about life support). Clearly, attitudes toward
information, by driving the agent’s welfare, are a critical aspect of information disclosure. In
this section, we study the formal underpinnings of information preferences in our model, and
offer a classification of psychological agents. In particular, we make a distinction between
psychological and behavioral attitudes toward information.
Definition 3. The agent is psychologically information-loving [resp. -averse, -neutral] if
given any action a ∈ A and any random posteriors p, q ∈ R(µ) with p µB q,
Z
Z
u(a, ·) dp ≥ [resp. ≤, =]
u(a, ·) dq.
∆Θ
∆Θ
An agent is psychologically information-loving [-averse] if he likes [dislikes] informativeness for its own sake, in the hypothetical event that he cannot adapt his decisions to it.
That is, information is intrinsically valuable [damaging], abstracting from its instrumental
value.9
For every posterior belief ν ∈ ∆Θ, define the indirect utility associated with ν as
U(ν) = max u(a, ν).
a∈A
(1)
In the next definition, the agent is assumed to respond optimally to information, which defines his behavioral attitude towards it.
Definition 4. The agent is behaviorally information-loving [resp. -averse, -neutral] if, given
any random posteriors p, q ∈ R(µ) with p µB q,
Z
Z
U dp ≥ [resp. ≤, =]
U dq
(2)
∆Θ
∆Θ
9
Grant, Kajii, and Polak (1998) define what it means to be single-action information-loving for an agent
who makes no decision. In their model, there is no distinction to be made between psychological and behavioral
information preferences.
11
The distinction between psychological information preferences and behavioral information preferences is important, though the two are closely linked.
Proposition 1. !!!
1. The agent is psychologically information-loving [resp. -averse, -neutral] if and only if
u(a, ·) is convex [resp. concave, affine] for every a ∈ A.
2. The agent is behaviorally information-loving [resp. -averse, -neutral] if and only if U
is convex [resp. concave, affine].
3. If the agent is psychologically information-loving, then he is behaviorally informationloving as well.
4. If u is concave (for A ⊆ Rk convex), then the agent is behaviorally information-averse.
The proposition makes clear a sense in which information is a form of risk. Getting no
information entails a deterministic posterior belief: the posterior equals the prior for sure.
But getting some information about the state entails stochastic posterior beliefs. In this context, where beliefs are payoff relevant, it is appropriate to think of information preferences
as a particular instance of risk preferences.
The first two points, included for the sake of completeness, are familiar to readers of the
literature on temporal resolution of uncertainty, as in Kreps and Porteus (1978). For instance,
in the special case where no action is taken (A = {a}, so that u(a, ·) = U), the main result of
Grant, Kajii, and Polak (1998, Proposition 1) is a substantial generalization of the first two
points of Proposition 1.
An immediate consequence of these first two points is that a classical agent is psychologically information-neutral and, in line with Blackwell’s theorem, behaviorally informationloving, because the linearity of u in beliefs implies that U is convex.
It is straightforward from the proposition (and given concavity of V) that, for k ∈ (0, 1),
the agent in Example (b) of Section 2.1 is psychologically information-averse but behaviorally information-loving. He intrinsically dislikes information, yet unequivocally prefers
to have it. This also shows that an exact aversion counterpart to the proposition’s third point
is unavailable. Even so, the fourth part tells us that behavioral information aversion follows if the agent’s utility is jointly concave in (a, ν), a condition that imposes psychological
aversion to information and limits its instrumental use (via concavity in action and limited
complementarity with beliefs). Note that such a concavity condition is never satisfied in the
classical world, excepting trivial cases.10
10
In the classical world, where, say A is a set of mixed actions, the utility is bilinear. Bilinear functions have
saddle points.
12
4
Designing Information
We now study the choices of information disclosure by a benevolent principal. As argued in
the Introduction, this problem is relevant when the principal is a trusted advisor to the agent,
but also when the principal is a benevolent entity that, by law, can enforce what the informed
party must disclose. Parents, family, professionals such as doctors, and our own selves often
act as benevolent principals. In technical terms, the principal’s goal is to maximize the
agent’s ex-ante expected welfare.
R
R
Definition 5. An information policy p ∈ R(µ) is optimal if ∆Θ U dp ≥ ∆Θ U dq for all
q ∈ R(µ). A signal (S , σ) is optimal if it generates an optimal information policy.
For purposes of characterizing optimal policies, the designer need only consider the design environment hΘ, µ, Ui. As we shall see later, however, the extra data A and u can be
relevant for constructing practical indirect signals (in Section 5), and for deducing designrelevant properties of U (in Section 6).
An optimal policy is immediate for certain classes of agents. If the agent is behaviorally
information-loving, then the most informative policy is optimal: one should tell him everything. Conversely, if the agent is behaviorally information-averse, then one should tell him
nothing. Finally, we know from Proposition 1(3) that fully revealing the state is also optimal
for psychologically information-loving agents, as in case (a) in Section 2.1. Full information is also optimal for an agent with multiplier preferences (case (e) in Section 2.1), who is
psychologically information-loving due to (joint) convexity of the relative entropy. With no
residual uncertainty, he will face no negative consequences from ambiguity.
Things are not so simple for agents who exhibit psychological information aversion.
They face a conflict between their innate dislike for information and their instrumental need
for it, which is the subject of the next section.
5
Information Aversion
In this section, we prescribe the use of practical information policies for psychologically
information-averse agents. Further, we compare such policies across individuals. Just as
an insurance policy should be catered to an agent’s degree of risk aversion, an information
policy should be catered to the degree of information aversion.
13
5.1
Practical Policies
Posterior policies (also called direct signals in Section 2.2) demand that the principal send
entire probability distributions to the agent. In a world with binary states, this can be plausible. For example, a doctor who uses a blood test to check for some disease may report the
updated probability that the patient has the illness rather than not. When the state space is
larger and more complicated, however, posterior policies are cumbersome and preposterous.
This is the point of view of Ben-Shahar and Schneider (2014), who argue forcefully against
Informed Consent and the Clery Act. The former demands that a doctor include a detailed
statistical description of all side effects of a treatment or drug, and the latter requires institutions of higher education to publish an annual campus security report about crimes that
have occurred in the past three years—in either case, something akin to posterior policies.
Ben-Shahar and Schneider (2014) conclude:
Doctors properly think their principal task is to cure illness and soothe suffering,
and that alone wholly outstrips their available time. A typical doctor lacks time
to provide just the preventive care the U.S. Preventive Services Task prescribes.
and:
Even if [prospective and current students] read the reports, which are longer
than insurance policies and full of tables and charts. . . the money colleges spend
complying with the Clery Act—and pass on to their students in the form of
higher tuition—is simply wasted.
Given the cost of superfluous disclosure, it seems natural to ask whether a simpler form
of communication can deliver the same benefit. If an agent might want information for its
own sake, then, of course, giving extraneous information will serve him best. But psychologically information-averse agents have no such desire. In the above examples, the obvious
alternative would be simply to recommend an action: define a recommendation policy to
be a signal of the form (A, σ). For example, the doctor could simply recommend dietary
choices or a particular exercise routine if her analysis reveals a pathology. In the case of
campus safety, the local police could recommend to avoid walking in certain areas if they
notice some potential danger during their patrol. Informed Consent requires doctors to give
patients all the information a reasonable person might want. By contrast, recommendation
policies tell the decision-maker only the information he needs.
As with any recommendation, the problem is that the person may not follow that advice.
In our framework, this is captured by the failure of the Revelation Principle. Every information policy admits a corresponding recommendation policy in which the agent is simply
14
u(1, ν)
u
0.1
0
0.6
0.9
ν
1
u(0, ν)
Figure 1: The set {ν : a = 0 is optimal at ν} is not convex.
told how he would have optimally responded to his posterior. But there is no guarantee a
priori that the agent will want to follow this recommendation. The Revelation Principle (as
in Myerson (1979), Aumann (1987), and KG) is the result that usually guarantees it.
The Revelation Principle says that, for any signal, the corresponding recommendation
policy is incentive-compatible.11 Under psychological information aversion, and a fortiori in
general psychological
environments,
it is no longer true. For example, take Θ = A = {0, 1}
h
i
1
and u(a, ν) = a V(ν) − 9 , so that the agent is psychologically information-averse. Let µ = .6
and consider a signal that generates random posterior p = 38 δ.1 + 85 δ.9 .As shown in Figure 1,
whether the posterior is .1 or .9, the agent plays action 0. If, instead, the designer told the
agent what he would have played, i.e., always sent the message “play a = 0,” the agent would
receive no information and his posterior would be µ = .6. The agent would then optimally
play action 1 and the recommended action, a = 0, would not be incentive-compatible.12
Recommendation policies may be useful in practice, but they cannot generate all joint
distributions over actions and beliefs that posterior policies can attain, given the failure of
the Revelation Principle. Even so, our next theorem says that this intuitive class is always
optimal for psychologically information-averse agents.
Theorem 1. If the agent is psychologically information-averse, then some incentive-compatible
recommendation policy is optimal.
It may be surprising at first that a recommendation policy can serve the needs of all psychologically information-averse agents—even the behaviorally information-loving ones, as
11
In its standard form, the Revelation Principle (Myerson (1991, page 260)) has two sides. When a principal
has full commitment power, it says that all attainable outcomes (in classical settings) are attainable when (i)
players report all private information to the principal, and (ii) the principal simply makes incentive-compatible
action recommendations to the players. In the present setting, with the agent having no private information, we
focus on the failure of the latter.
12
The Revelation Principle fails here because the set of posterior beliefs for which a given action is optimal
may not be convex. In classical environments, if an action is optimal for certain messages, then it is without
loss to replace each of the messages with a recommendation of said action. By doing so, we “pool” these
messages, thereby also pooling the posteriors attached to these messages. In a psychological world, the action
may not remain optimal under the averaged posteriors.
15
in Example (b) from Section 2.1 with k ∈ (0, 1)—because passing from a policy to its associated recommendations degrades information. When full revelation is optimal and the agent
is psychologically information-averse, this too can be implemented by a recommendation
policy; then, the optimal action necessarily reveals the state. In Example (b), for instance,
the expert recommends action a = θ with probability one.
The theorem shows that, for any optimal signal (S , σ), the corresponding recommendation policy is incentive-compatible. The intuition is that giving an information-averse agent
only instrumental information is helpful, and, if the policy indeed maximizes the agent’s
utility, then following its advice should serve the agent’s interests (i.e., should be incentivecompatible). This argument is specific to benevolent design: the result may not hold under
psychological preferences if the expert and the agent also have conflicting interests.
In general, the result breaks down for information-loving agents. Consider A = {¯a},
θ ∈ {0, 1}, and u(¯a, ν) = −V(ν). The only recommendation is a¯ in all states, which gives the
agent no information to update his prior, and so V(ν) = V(µ). Clearly, it would be preferable
to tell the agent which state has occurred, so that V(ν) = 0.
5.2
Comparative Information Aversion
How should one tailor recommendations to agents with differing degrees of information
aversion? This question has real prescriptive implications: if we find that a well-intentioned
expert should provide a less informative policy to a more information-averse agent, the more
information-loving agent is then equipped to provide information to the other. If a doctor
considers whether to tell parents the sex of their child, and the father is more informationaverse than the mother, the doctor can simply provide optimal information to the mother,
who will then be equipped to inform optimally the father. Likewise, if a child experiences
only psychological consequences of discovery of some illness, while the parents care about
the former and about making informed medical decisions, then a pediatrician may simply
choose an optimal information policy for the parents, and leave the parents to tell their child
(however much they see fit) about the diagnosis.
We compare two individuals via the following behavioral definitions. Let U1 and U2 be
their indirect utilities, as defined in (1).
Definition 6. Agent 2 is more (behaviorally) information-averse than agent 1 if for any p
and q such that p µB q,
Z
Z
Z
Z
U1 dq ≥ [resp. >] U1 dp =⇒
U2 dq ≥ [resp. >] U2 dp.
16
An agent is more information-averse than another if any time the latter would prefer a
less informative policy over a more informative one (with the same prior), so would the
former agent.
This definition is qualitative rather than quantitative; it compares when two decisionmakers prefer more/less information rather than the degree of such preference (so that, for
instance, two different strictly information-averse agents are vacuously always ranked). This
is quite different from the standard literature on comparative risk aversion (started by Pratt
(1964) and Arrow (1971), including Kihlstrom and Mirman (1974), etc.), in which risk aversion is measured by comparison to risk-free (i.e., degenerate) lotteries, in search of certainty
equivalents.13 Usually, an agent is said to be more risk-averse than another if all risky lotteries are worth no more to him than to the other individual, in risk-free terms. As such,
certainty equivalents provide an objective “yardstick” to quantify risk aversion.
In the context of information disclosure, where random posteriors can be seen as lotteries
over beliefs, the only available risk-free lottery is the ‘no information’ policy. Indeed, all our
lotteries must have the same mean (the prior, as in Definition 1), and hence the only riskfree lottery puts probability one on the prior. We therefore have no analogous objective
yardstick. Of course, we could ask each agent which prior belief would hypothetically make
him indifferent to any given policy, but this seems an unrealistic exercise.
Note, also, that the definition only imposes restrictions on a narrow set of comparisons—
how two agents rank Blackwell-comparable policies—since Blackwell-incomparability is
generic.14
One application of our definition is to the mixture between classical and psychological
preferences:
Z
uλ (a, ν) = λuP (ν) + (1 − λ)
Θ
uC (a, θ) dν(θ),
(3)
where uP is any purely psychological motive, uC is any classical motive, and λ ∈ (0, 1)
measures the departure from the classical case. The more psychological (3) becomes, i.e.,
the larger λ, the more information-averse the agent becomes.15 This is true regardless of the
shape of the psychological component.
While weak, Definition 6 has strong implications for utility representation and, in two13
There are many definitions of comparative risk aversion, not all of which directly mention certainty equivalents. That said, all behavioral definitions that we are aware of involve comparing risky lotteries to risk-free
ones, without the added restriction that the risk-free lottery delivers the same expectation as the risky one.
14
See Proposition 3 in the Appendix, which states that almost all (i.e., an open dense set of) pairs of information policies are incomparable with respect to (Blackwell-) informativeness.
15
To see why, let Uλ be the indirect utility associated with uλ . For any λ00 > λ0 and letting γ = λ00 /λ0 , we
have Uλ00 = γUλ0 + (1 − γ)U0 . Note that (1 − γ)U0 is concave—because U0 is convex by Proposition 1. Then
Theorem 2 completes the argument.
17
state environments, is enough to obtain comparative statics of optimal policies.
Theorem 2. Assume agents 1 and 2 are neither behaviorally information-loving nor -averse.
Then, 2 is more information-averse than 1 if and only if U2 = γU1 + C for some γ > 0 and
concave C. Moreover, in binary-state environments, if 2 is more information-averse than 1,
then p∗1 µB p∗2 for some optimal policies p∗1 and p∗2 .
Unlike the classic Arrow–Pratt result, which ties comparative risk aversion to concave
transformation, our result shows the equivalence of comparative information aversion and
concave addition: when two agents are ranked according to information aversion, their indirect utilities differ by a concave function. This distinction is significant. In general, neither
definition of comparative information/risk aversion implies the other, and the two results
derive their cardinal content from different sources.16
In this model, the addition of a concave function amounts to adding a global distaste for
information. In a world of binary uncertainty, the principal will respond to this by providing
a less informative policy to a more information-averse agent. Considering that almost all
pairs of policies are informatively incomparable—by Proposition 3 in the Appendix, which
holds even in binary-state environments—this result considerably narrows the search for an
optimal policy, once one is known for one of the agents.
One might suspect that this conclusion generalizes, in that a principal should always
provide unambiguously less information to a more information-averse agent. Yet, with richer
state spaces, a change in information aversion can be met by a “change of topic,” rather than
by uniformly garbling information. To be clear, adding dislike for information everywhere
does not mean everywhere with the same intensity; two agents can be ranked by information
aversion, but, due to psychological substitution effects, it can be optimal to inform them on
different dimensions. We provide a formal example in Section 8.5.1.
6
Optimal Policies: The General Case
The world of psychological preferences is greatly diverse, with many preferences escaping
our classification. Equivocal information preference quickly comes to mind, as in the case
of patients who desire to be informed about their health when the news is good, but would
rather not know how bad it is otherwise. Such an agent is neither information-averse nor
-loving. For those general psychological agents, the concavification result of KG extends
readily, providing an abstract characterization of the optimal value.
To easily compare the two results: consider the binary-state world, and assume that U1 , U2 are both strictly
increasing on [0, 1] and twice-differentiable. The Arrow–Pratt result comes from ranking Ui00 (ν)/Ui0 (ν) at each
ν; ours comes from ranking sign[Ui00 (ν)] at each ν, and ranking Ui00 (ν1 )/Ui00 (ν0 ) at each (ν1 , ν0 ).
16
18
In applications, however, concavification is not a trivial affair; it remains largely a theoretical construct (for some discussion of the difficulty involved, see Tardella (2008)). For
this reason, we develop an applied method, the method of posterior covers, that allows us
to simplify the computation of an optimal policy and to arrive at clear-cut prescriptions in
economic applications of interest.
6.1
Optimal Welfare
A key result in KG is the full characterization of the maximal ex-ante expected utility. The
characterization is based on the concave envelope of U, defined via
U(ν) = inf{φ(ν) : φ : ∆Θ → R affine continuous, φ ≥ U}.
This result carries over to our environment for all psychological preferences:
Proposition 2. For any prior µ, there is an information policy that induces indirect utility
U(µ) for the agent, and no policy induces a strictly higher value.
6.2
Optimal Policies
The concave envelope of U describes the optimal value and implicitly encodes optimal policies. Unfortunately, it is a difficult task to derive qualitative properties of the concave envelope of a general function, even more so to compute it. The main issue is that the concave
envelope is a global construct: evaluating it at one point amounts to solving a global optimization problem (Tardella (2008)). While concavifying a function is difficult (and a currently active area of mathematical research), we are able to exploit the additional structure
that the economics puts on the problem, simplifying the exercise.
We approach the problem by reducing the support of the optimal policy based on local
arguments, a method that will be successful in a class of models that includes economic
applications of interest.
Oftentimes, the designer can deduce from the primitives of the problem that the indirect
utility U must be locally convex on various regions of ∆Θ. In every one of those regions, the
agent likes (mean-preserving) spreads in beliefs, which correspond to more informative policies. As a consequence, an optimal policy need not employ beliefs inside of those regions,
regardless of its general shape.
The central concept of our approach is that of the posterior cover, which is a collection
of such regions.
19
Definition 7. Given a function f : ∆Θ → R, an f -(posterior) cover is a family C of closed
S
convex subsets of ∆Θ such that C is Borel and f |C is convex for every C ∈ C.
A posterior cover is a collection of sets of posterior beliefs, over each of which a given
function is convex. Given a posterior cover C, let
out(C) = {ν ∈ ∆Θ : ∀C ∈ C, if ν ∈ C then ν ∈ ext(C)}
be its set of outer points.17 The next result explains why posterior covers and their outer
points play an important role in finding an optimal policy.
Theorem 3 (Method of Posterior Covers).
1. If C is a countable U-cover, then p(out(C)) = 1 for some optimal policy p.
W
18
2. If Ca is a u(a, ·)-cover for every a ∈ A,
R then C := a Ca is a U-cover.
If u takes the form u(a, ν) = uP (ν) + Θ uC (a, ·) dν, then any uP -cover is a U-cover.
3. If Θ is finite and f is the pointwise minimum of finitely many affine functions { fi : i ∈ I},
then C := {{ fi = f } : i ∈ I} is an f -cover.
The method of posterior covers consists of three parts. From the first part, it follows that
the search for an optimal policy can be limited to the outer points of a U-cover. Given that
the principal wants to spread the posterior beliefs as much as possible within each set of a
U-cover, beliefs supported by some optimal policy will be extreme points of those sets or
will lie outside. This explains our interest in posterior covers of the indirect utility. But, the
indirect utility is a derived object—the computation of which may be demanding in its own
right—and so it is important to tie its cover to primitive features of the model. This is the
next step.
The second part shows that the problem of finding a U-cover can be reduced to that
of finding posterior covers of primitives of the model. Our ability to do so comes from
the benevolent feature of our design problem. For one, if we determine a useful posterior
cover of u(a, ·) for each a, then it is easy to name a useful U-cover, namely the coarsest
common refinement of those u(a, ·)-covers. Moreover, for agents who can fully separate their
psychological effects from their classical motive, the determination of a U-cover reduces to
finding a posterior cover of the psychological component alone. This includes all preferences
in Section 2.1 from (a) to (d).
17
A point ν ∈ ext(C) if there is no nontrivial (i.e., of positive length) segment for which ν is an interior point.
T
Ca = { a∈A Ca : Ca ∈ Ca ∀a ∈ A} , the coarsest common refinement of {Ca }a .
18 W
a∈A
20
Having connected the problem to primitives, it remains now to actually compute a useful
cover of those primitive functions. In general, this task depends on the exact nature of the
function at hand and can be difficult. The third part points to a flexible class of functions
for which such a computation is particularly straightforward. For the class of preferences
described in (3), namely the minimum of affine functions, we can explicitly compute a useful
U-cover and its (finite) set of outer points. This class includes many economic situations of
interest. In the additive case alone,
Z
u(a, ν) = uP (ν) +
uC (a, ·) dν,
Θ
it is common that uP (ν) := − max{ fi (ν)} = min{− fi (ν)} with affine fi ’s. This is the case for
cognitive dissonance (Section 7.1) and for temptation and self-control a` la Gul and Pesendorfer (2001) (Section 7.2 also treats a temptation model without such separability). Then, uP is
piecewise linear; the pieces form a U-cover; and the outer points are easy to find. In Section
8.7 of the Appendix, we supplement Theorem 3 with a “hands-on” computation of the outer
points.
Another important benefit of the method of posterior covers is that the principal need
not solve the agent’s optimization problem at every conceivable posterior belief. Indeed,
Theorem 3 enables us to compute a posterior cover (and its outer points) without solving
the agent’s problem, and lets us derive an optimal policy while computing the indirect utility
only on those outer points. This simplification—the possibility of which stems from our
restriction to aligned interests—makes our task considerably easier. For instance, we shall
see in the cognitive dissonance application that we need only solve the agent’s problem at
three different posterior beliefs in order to derive an optimal policy.
7
7.1
Applied Disclosure
Cognitive Dissonance
The preferences given in Example (c) of Section 2.1 can represent various sources of priorbias, including cognitive dissonance and reputational concerns. Our methods deliver clear
prescriptions for such an agent.
Consider a ruling politician who must make a decision on behalf of her country—e.g.,
deciding whether or not to go to war—in the face of an uncertain state θ ∈ {0, 1}. She will
first seek whatever information she decides and then announce publicly the country’s action
a ∈ A and (updated) belief ν that justifies it. The decision must serve the country’s interests
given whatever information is found.
21
μ
0
1
ν
-|ν -μ|
Figure 2: The Psychological Component: Concern for Reputation.
On the one hand, the politician wishes to make decisions that serve the country’s interest
uC (a, θ). On the other hand, she campaigned on her deeply held beliefs µ ∈ (0, 1) and her
political career will suffer if she is viewed as a ‘flip-flopper’. Publicly expressing a new
belief ν entails a reputational cost, uP (ν) = −ρ|ν − µ| for ρ > 0. When the politician decides
which expertise to seek, she behaves as if she has psychological preferences
u(a, ν) = uP (ν) + Eθ∼ν [uC (θ, a)].
Even without any information about uC , we can infer that the optimal information policy
takes a very special form: the politician should either seek full information or no information
at all. She finds no tradeoff worth balancing.
The psychological component uP is affine on each member of C := {[0, µ], [µ, 1]}, hence
C is a uP -cover. By Theorem 3, it is also a U-cover and hence some optimal policy p∗ is
supported on out(C) = {0, µ, 1}. By Bayesian updating, it must be that p∗ = (1 − λ)δµ + λ[(1 −
µ)δ0 + µδ1 ] for some λ ∈ [0, 1]. That is, the politician either seeks full information, seeks
none, or randomizes between the two. But in the latter, she must be indifferent, and so full
information or no information must be optimal.
More can be said from our comparative static result. If we consider increasing ρ, then
the politician becomes more information-averse—as the increase corresponds to adding a
concave function. Then, by Theorem 2, there is some cutoff ρ∗ ∈ R+ such that, fixing µ, full
information is optimal when ρ < ρ∗ and no information is optimal when ρ > ρ∗ .
22
7.2
Temptation and Self-Control
Individuals often face temptation. Economists and psychologists have presented evidence
that temptation influences consumers’ behavior, especially in consumption-saving decisions
(Baumeister (2002), Huang, Liu, and Zhu (2013)). Motivated by this evidence, a recent
literature in macroeconomics has asked how tax policies might be designed to alleviate the
effects of temptation (e.g., Krusell, Kuruscu, and Smith (2010)). In this section, we ask how
information disclosure policies could do the same.
As will become clear, the method of posterior covers applies directly to the Gul and Pesendorfer (2001) (GP hereafter) model. The conclusions will capture the common precept
that “you don’t want to know what you’re missing” and make it precise. This has also been
studied in psychology (Otto and Love (2010)) to describe how information affects temptation. The idea is that, when a person does not succumb to temptation, information may
exacerbate the appeal of forgone choices and cause pain. Naturally, the desire to limit information in these situations is something one observes in real life; our analysis explains it as
an optimal reaction to costly self-control.
Consider a consumer who decides whether to invest in a risky asset or to consume. (Later,
the consumer will have more actions.) The state of the world, θ ∈ Θ—which will be interpreted in several ways—is unknown to him, distributed according to prior µ. Before making
his decision, the agent learns additional information from an advisor. Investing draws its
value from a higher consumption tomorrow but prevents today’s consumption. When the
agent deprives himself of consumption, it creates mental suffering.
The consumer is a standard GP agent. Given two classical utility functions uR and uT ,
and action set A, the consumer faces a welfare of
n
o
ˆ − uT (a, θ)
ˆ ,
u(a, ν) = Eθ∼ν {uR (a, θ)} − max Eθ∼ν
uT (b, θ)
ˆ
b∈A
(4)
when he chooses a ∈ A at posterior belief ν. It is as if the consumer had two “sides”: a
rational side uR and a tempted side uT . Given action a, the rational side faces expected value
of Eθ∼ν {uR (a, θ)}, while exerting self-control entails a personal cost of maxb∈A {Eθ∼ν [uT (b, θ)]−
Eθ∼ν [uT (a, θ)]}. The psychological penalty is the value forgone by the tempted side in choosing a. The consumer makes decisions to balance these two motives.
How should a financial advisor—who may or may not be the consumer himself—inform
the consumer about θ when his interests align with the sum welfare u?
Method. Preferences (4) can be written as
u(a, ν) = uP (ν) + Eθ∼ν [uR (a, θ) + uT (a, θ)],
23
where
uP (ν) = −UT (ν) = min {−Eθ∼ν [uT (b, θ)] : b ∈ A} .
By linearity of expectation, uP is a minimum of affine functions, so that all parts of Theorem
3 apply. Finding a uP -cover with a small set of outer points, and then naming an optimal
policy, is a straightforward exercise. Let us illustrate this in an example.
Let θ ∈ {`, h} be the exchange rate of $1 in euros tomorrow, where h [resp. `] stands
for ‘higher’ [‘lower’] than today. Let A = {a1 , a2 , a3 } where ai means “invest in firm i” for
i = 1, 2, and a3 means “buy a trip to Europe.” Firm 1 uses domestic raw material to produce
and mostly sells to European customers. Firm 2 imports foreign raw material from Europe
and mostly sells domestically.
The consumer’s rational and impulsive utilities are given in the following matrices:
a1
a2
a3
`
α
−β
0
h
−γ
δ
0
a1
a2
a3
`
0
-1
T
h
-1
0
1
Function uR
Function uT
The rational side gets no value from the trip, but likes to invest in the firm that benefits
from the exchange rate. The impulsive side gets no value from investing in the right firm, but
has a disutility from investing in the wrong one. Moreover, it experiences great temptation
when the exchange rate is high, and temptation governed by T when the rate is low.
From Theorem 3 and Corollary 2 (in the Appendix), we know that C = {C1 , C2 , C3 } is a
U-cover with
Ci = {ν : Eθ∼ν [uT (ai , θ)] ≤ Eθ∼ν [uT (a j , θ)] for all j}
and that the outer points are contained in19
S = {0, 1} ∪ {ν : Eθ∼ν [uT (ai , θ)] = Eθ∼ν [uT (a j , θ)] for some i, j}
o
n
T
,
∩ [0, 1].
= 0, 1, 21 , 1+T
T
T −2
From here, an optimal policy comes from choosing the best policy with support in S , which
is a simple linear program:
X
X
X
max
pν U(ν) s.t.
pν = 1 and
pν ν = µ.
p∈RS+
19
ν∈S
ν∈S
If one of the fractions has zero denominator, just omit it from S .
24
ν∈S
Just as in this example, our toolbox greatly simplifies the analysis for any GP agent. In what
follows, we spell out some of the lessons that the method arrives at regarding how one should
speak to a tempted individual.
Prescriptions. In the GP model, information is critical for rational decision making, more
precisely for maxa E[uR + uT ], but it also induces more temptation, as information increases
the value to the impulsive side, maxb E[uT ]. Optimal disclosure is about balancing these two
parts; below, we explain a precise sense in which optimal information leaves the agent not
knowing what he is missing. First we consider two extreme cases.
If uT is state-independent, say if θ is the rate of return of the asset, the agent is psychologically information-loving. By Proposition 1, the consumer is behaviorally information-loving
too, and so the advisor optimally gives full information.
If uR is state-independent, say if θ is the weather in a vacation destination, the consumer’s
welfare is given by
u(a, ν) = uR (a) + Eθ∼ν [uT (a, θ)] − UT (ν),
and he exhibits psychological information aversion (by Proposition 1); knowing more is
unequivocally harmful him if his vacation choices are fixed. Yet, the financial advisor may
still want to convey some information. We give the intuition in an example. Say A = Θ =
{0, 1}, uT (a, θ) = 1{a=θ} and uR (a) = 31 a. If the weather in Hawaii is perfect (θ = 0), the
consumer is tempted to splurge on a vacation (a = 0); otherwise he faces no temptation. To
the rational side, weather is irrelevant and investing is always better. One can show (using
Theorem 3 and Corollary 2) that it is optimal to give information when µ < .5. Suppose
µ = .4, so that there is 40% chance of rainy weather. If the advisor said nothing, the agent
would invest and face a large cost of self-control, with a 60% chance that he is missing out
on perfect weather. But (based on Theorem 1) the advisor could do better by recommending
her client when to invest and when to go on a vacation: if the weather is perfect, then with
probability 31 she tells her client to take a vacation, and with complementary probability or
when the weather in Hawaii is bad, she tells him to invest. This recommendation policy
effectively harnesses temptation. When the consumer is told to take a vacation, he goes (and
so faces no temptation); and when told to invest, he faces a reduced cost of self-control, now
believing that it is raining in Hawaii with 50% probability.
The general advice delivered by the method of posterior covers is that some optimal
information policy must be supported on beliefs at which the tempted self has, in some
sense, as little instrumental information as possible. Say that an agent does not know what
25
he is missing at ν if for all q ∈ R(ν) and all ν0 ∈ S q \{ν},
argmaxb Eθ∼ν [uT (b, θ)] * argmaxb Eθ∼ν0 [uT (b, θ)].
(5)
When an agent does not know what he is missing, any further information would serve to
eliminate some temptations from the agent’s mind, and give him a better idea, qualitatively
speaking, of precisely what he is missing. An information policy is optimal if either the
agent knows the state, and hence is not really missing out on anything, or his tempted self is
kept indifferent between as many actions as can be.20 In that state of confusion, temptation
must indeed be minimized. The formal argument for why, in some optimal policy, the agent
does not know what he is missing comes from Claim 1 in the Appendix, and the equivalence
between (5) and the condition S (ν∗ ) = {ν∗ }.
Extension: Non-separable Temptations. Thus far, we have applied our method only to
agents who balance two separate motives, a purely psychological one (with no action) and a
classical one (that takes action). For those, it should now be clear how to derive a U-cover.
We stress, however, that the methods of Theorem 3 carry over readily to serve agents without
such separable motives, which is also quite natural. The temptations of Salt Lake City differ
from the temptations of Atlantic City; the actions we choose may alter the allure of those
we avoid. Thus, the information a person gets, by virtue of affecting his chosen action, will
affect which temptations he suffers.
Consider now a natural variant of the Gul and Pesendorfer (2001) model, in which the
nature of each temptation faced by the agent can vary with the action he takes. So, the agent
still has both rational and tempted motives uR , uT ; but given any chosen action a ∈ A, the
tempted side values each possible temptation b ∈ A at uT (b, θ | a). That is, preferences take
the form:
ˆ
ˆ
u(a, ν) = Eθ∼ν uR (a, θ) − max Eθ∼ν
uT (b, θ | a) − uT (a, θ | a) .
ˆ
b∈A
In this more complex model, the agent’s optimal choices and the psychological effects
(temptation) that he faces are intertwined. Even in this non-separable setting, the method of
posterior covers remains useful. For each a, the function u(a, ·) is of the form
n
o
ˆ | a) + Eθ∼ν [uR (a, θ) + uT (a, θ | a)] .
ν 7→ min −Eθ∼ν
ˆ uT (b, θ
b∈A
Then Theorem 3 can be used to find a small posterior cover Ca for u(a, ·), at which point it
20
This is correct in a binary-state environment. More generally, given the set of states the agent views as
possible after a message, his tempted side is as confused as possible.
26
delivers a small U-cover C; then the search for an optimal policy reduces to those supported
on C’s set of outer points.21
Return to the consumption-saving setting. Suppose a financial advisor must choose how
much to tell the consumer about upcoming liquidity needs, which are captured by some
binary state. The consumer can choose to invest in a liquid asset, invest in an illiquid asset, or
consume. Only if he invests in the liquid asset—and thus always has the option to consume
now—he faces temptation (and an ensuing cost of self-control). The method of posterior
covers delivers something in the spirit of not telling the consumer what he is missing. Letting
ν be the belief at which his tempted side is indifferent22 between consuming and investing
in the liquid asset, we know that some optimal policy either reveals the state or keeps his
belief at this ν. We can actually deduce even more from the method of posterior covers in
this non-separable setting: if, at this ν, he optimally chooses only the illiquid asset, then full
information is an optimal policy. Note that the latter argument does not require (or imply) a
psychologically information-loving agent.
21
This procedure delivers a finite set of outer points if the set of temptations is finite;
by Sperner’s theorem
|A|
|Θ|
and the work in Section 8.7 of the Appendix, an easy upper bound is (2 − 1)|A| b 1 |A|c .
2
22
In the nontrivial case, there is exactly one such ν. The only other two possibilities are when he is always
indifferent or never indifferent between consuming and investing in the liquid asset. In either of the latter cases,
he is psychologically information-loving.
27
8
Appendix
8.1
Lemma 1
The next lemma formalizes the connection between our ‘mean-preserving spread’ definition
of informativeness and the usual garbling definition.23
Lemma 1. Given two random posteriors p, q ∈ R(µ), the ranking p µB q holds if and only if
(S q , σq ) is a garbling of (S p , σ p ), i.e., if there is a map g : S p → ∆(S q ) such that
Z
ˆ
σq (S |θ) =
g(Sˆ |·)dσ p (·|θ)
(6)
Sp
for every θ ∈ Θ and Borel Sˆ ⊆ S q .
In both directions, the idea of the proof is to choose g (given r) or r (given g) to ensure
dg(t|s) dσ p (s|θ) = dr(s|t) dσq (t|θ).
Everything else is just bookkeeping.
Proof. First suppose such r exists, and define g : S p → ∆(S q ) via
Z
dr(·|t)
g(T |s) :=
(s) dq(t),
dp
T
for any s ∈RS p and Borel T ⊆ S q . Then for θ ∈ Θ and Borel T ⊆ S q , it is straightforward to
verify that S g(T |s) dσ p (s|θ) = σ(T |θ), so that g witnesses p µB q.
p
Conversely, suppose p µB with the map g : S p → ∆(S q ) as in the definition of µB . Let
Z
dg(·|s)
(t) dp(s)
r(S |t) :=
dq
S
Rfor t ∈ S q and Borel S ⊆ S p . Then for Borel S ⊆ S p , it is again straightforward to verify that
r(S |·) dq = p(S ), which verifies the first condition.
Sq
Now, given Borel T ⊆ S q and θ ∈ Θ, one can check that
Z
Z Z
dt
ds
(θ) dq(t) =
(θ) dr(s|t) dq(t),
T dµ
T ∆Θ dµ
23
For finite-support random posteriors, the result follows from Grant, Kajii, and Polak (1998, Lemma A.1).
Ganuza and Penalva (2010, Theorem 2) prove a related result in the special case of Θ ⊆ R.
28
dt
so that dµ
(θ) =
ˆ
Θ ⊆ Θ,
R
ds
(θ)
∆Θ dµ
dr(s|t) for a.e.-µ(θ)q(t). From this we may show that, for any Borel
Z
∆Θ
ˆ dr(s|t) = t(Θ),
ˆ
s(Θ)
which verifies the second condition.
8.2
Proof of Proposition 1
Lemma 2. The set M := {γ ∈ ∆Θ : ∃ > 0 s.t. γ ≤ µ} is w∗ -dense in ∆Θ.
)
(
dγ
is (essentially) bounded is conProof. First, notice that M = γ ∈ ∆Θ : γ << µ and
dµ
¯ is closed (and so compact, by
vex and extreme (i.e. a face of ∆Θ). Thus its w∗ -closure M
the Banach-Alaoglu theorem (Aliprantis and Border (1999, Theorem 6.21))), convex, and
¯ Because M
¯ is extreme, E is a subextreme. Now, let E be the set of extreme points of M.
ˆ ⊆ Θ. By the Krein-Milman theorem
set of ext(∆Θ) = {δθ }θ∈Θ . So E = {δθ }θ∈Θˆ for some Θ
¯ = coE = ∆(Θ0 ), where Θ0 is the closure of
(Aliprantis and Border (1999, Theorem 7.68)), M
ˆ Finally, notice that µ ∈ M implies Θ0 ⊇ supp(µ) = Θ. Thus M
¯ = ∆Θ as desired.
Θ.
Lemma 3. Fix a continuous function f : ∆Θ → R. Then the following are equivalent (given
µ is of full support):
R
1. For all ν ∈ ∆Θ and p ∈ R(ν), we have ∆Θ f dp ≥ f (ν)
R
R
0
2. For all µ0 ∈ ∆Θ and p, q ∈ R(µ0 ) with p µB q, we have ∆Θ f dp ≥ ∆Θ f dq.
R
R
3. For all p, q ∈ R(µ) with p µB q, we have ∆Θ f dp ≥ ∆Θ f dq.
4. f is convex.
Proof. Suppose (1) holds, and consider anyR µ0R ∈ ∆Θ and q ∈ R(µ0 ).RIf r : S q → ∆∆Θ satisfies
r(·|ν) ∈ R(ν) for every ν ∈ S q , (1) implies S ∆Θ f dr(·|ν) dq(ν) ≥ S f dq. Equivalently (by
q
q
R
R
Definition 2), any p more informative than q has f dp ≥ f dq, which yields (2).
That (2) implies (3) is immediate.
Now, suppose (4) fails. That is, there exist γ, ζ, η ∈ ∆Θ and λ ∈ (0, 1) such that
(1 − λ)ζ + λη = (1 − λ)ζ + λη.
f ((1 − λ)ζ + λη) < (1 − λ) f (ζ) + λ f (η).
29
Now, we want to exploit the above to construct two information-ranked random posteriors such that f has higher expectation on the less informative of the two.
To start, let us show how to do it if γ ≤ µ for some ∈ (0, 1). In this case, let
ν :=
1
(µ−γ) ∈ ∆Θ, p := (1−)δν +(1−λ)δζ +λδη ∈ R(µ), and q := (1−)δν +δγ ∈ R(µ).
1−
Then p µB q, but
Z
Z
f dp −
∆Θ
∆Θ
f dq = (1 − λ) f (ζ) + λ f (η) − f (γ) < 0,
as desired.
Now, notice that we need not assume that γ ≤ µ for some ∈ (0, 1). Indeed, in light
of Lemma 2 and continuity of f , we can always pick γ, ζ, η ∈ ∆Θ
R to ensureR it. So given
µ
continuous nonconvex f , we can ensure existence of p B q with ∆Θ f dp < ∆Θ f dq. That
is, (3) fails too.
Finally, notice that (4) implies (1) by Jensen’s inequality.
Proof of Proposition 1.
1. This follows immediately from applying Lemma 3 to u(a, ·) and −u(a, ·) for each a ∈ A.
2. This follows immediately from applying Lemma 3 to U and −U.
3. This follows from the first two parts, and from the easy fact that a pointwise maximum
of convex functions is convex.
4. This follows from part (2) and the fact that U is concave by the (convex) maximum
theorem.
8.3
Proof of Theorem 1
Proof. Suppose the agent is psychologically information-averse.
Fix some measurable selection24 a∗ : ∆Θ → A of the best-response correspondence ν 7→
arg maxa∈A u(a, ν). In particular, given any q ∈ R(µ), a∗ |S q is an optimal strategy for an agent
with direct signal (S q , σq ).
Toward a proof of the theorem, we first verify the following claim.
Claim: Given any random posterior p ∈ R(µ), we can construct a signal (A, α p ) such that:
24
One exists, by Aliprantis and Border (1999, Theorem 8.13).
30
1. The random posterior q p induced by (A, α p ) is less informative than p.
2. An agent who follows the recommendations of α p performs at least as well as an agent
who receives signal (S p , σ p ) and responds optimally, i.e.
Z Z Z
A,α p
u a, β (·|a) dα p (a|θ) dµ(θ) ≥
U dp.
Θ
∆Θ
A
To verify the claim, fix any p ∈ R(µ), and define the map
αp : Θ
∆(A)
→
θ 7−→ α p (·|θ) = σ p (·|θ) ◦ a∗−1 .
Then (A, α p ) is a signal with
o n
ˆ = σ p s ∈ S p : a∗ (s) ∈ Aˆ θ
α p (A|θ)
for every θ ∈ Θ and Borel Aˆ ⊆ A. The signal (A, α p ) is familiar: replace each message in
(S p , σ p ) with a recommendation of the action that would have been taken.
Let q p ∈ R(µ) denote the random posterior induced by signal (A, α p ). Now, let us show
that q p delivers at least as high an expected value as p.
By construction,25 q is a garbling of p. . Therefore, by LemmaR 1, p µB q, so that there is
a map r : S q → ∆(S p ) such that for every Borel S ⊆ ∆Θ, p(S ) = S r(S |·) dq, and for every
q
t ∈ S q , r(·|t) ∈ R(t). Then, appealing to the definition of psychological information aversion,
Z
Z
A,α p
u a, β (·|a) dα p (a|θ) =
u a∗ (s), βA,α p (·|a∗ (s)) dσ p (s|θ)
A
S
Z pZ
∗
A,α p
∗
≥
u (a (s), ν)) dr νβ (·|a (s)) dσ p (s|θ)
Sp Sp
Z Z
=
U(ν) dr νβA,α p (·|a∗ (s)) dσ p (s|θ)
S
S
Z pZ p
=
U(ν) dr νβA,α p (·|a) dα p (a|θ).
A
25
Sp
Indeed we can define g in (6) via: g(a|s) = 1 if a∗ (s) = a and 0 otherwise.
31
Therefore,
Z Z Z Z Z
A,α p
A,α p
u a, β (·|a) dα p (a|θ) dµ(θ) ≥
U(ν) dr νβ (·|a) dα p (a|θ) dµ(θ)
Θ A
Θ A Sp
Z
=
U(ν) dr(ν|t) dq(t)
Sq
Z
=
U dp,
Sp
which verifies the claim.
Now, fix some optimal policy p∗ ∈ R(µ), and let α = α p∗ and q = q p∗ be as delivered by
the above claim. Let the measure Q ∈ ∆(A × ∆Θ) over recommended actions and posterior
beliefs be that induced by α p . So
Z Z
Q(Aˆ × Sˆ ) =
1βA,α (·|a)∈S dα(a|θ) dµ(θ)
Θ
for Borel Aˆ ⊆ A, Sˆ ⊆ ∆Θ.
Then,26
Z
Z
U dp ≤
∆Θ
Aˆ
Z
u dQ ≤
U dq ≤
∆Θ
A×∆Θ
Z
U dp,
∆Θ
so that:
Z
U dq =
Z
u(a, ν) dQ(a, ν) =
Z
∆Θ
Z
A×∆Θ
∆Θ
∆Θ
U dp, i.e. q is optimal; and
Z
U dq =
max u(˜a, ν) dQ(a, ν).
A×∆Θ a˜ ∈A
The latter point implies that a ∈ argmaxa˜ ∈A u(˜a, ν) a.s.-Q(a, ν). In other words, the recommendation (A, α) is incentive-compatible as well. This completes the proof.
We note that the claim in the above proof delivers something more than the statement
of Theorem 1. Indeed, given any finite-support random posterior p, the claim produces
a constructive procedure to design an incentive-compatible recommendation policy which
outperforms p. The reason is that (in the notation of the claim):
1. If a∗ |S p is injective, then q p = p, so that (A, α p ) is an incentive-compatible recommendation policy inducing random posterior p itself.
26
Indeed, the inequalities follow from the above claim, the definition of U along with the property
marg∆Θ Q = q, and optimality of p, respectively.
32
2. Otherwise, |S q p | < |S p |.
In the latter case, we can simply apply the claim to q p . Iterating in this way—yielding a new,
better policy at each stage—eventually (in fewer than |S p | stages) leads to a recommendation
policy which is incentive-compatible and outperforms p.
8.4
8.4.1
Comparative Statics
Proposition 3: Most policies are not Blackwell-comparable
Proposition 3. Suppose |Θ| > 1. Then Blackwell-incomparability is generic, i.e. the set
Nµ = {(p, q) ∈ R(µ)2 : p µB q and p µB q} is open and dense in R(µ)2 .
Proof. To show that Nµ is open and dense, it suffices to show that µB is. Indeed, it would then
be immediate that µB is open and dense (as switching coordinates is a homeomorphism), so
that their intersection Nµ is dense too.
Given Definition 2, it is straightforward27 to express µB ⊆ (∆Θ)2 as the image of a continuous map with domain ∆∆∆Θ. Therefore µB is compact, making its complement an open
set.
Take any p, q ∈ R(µ). We will show existence of {p , q }∈(0,1) such that p µB q and
(p , q ) → (p, q) as → 0. For each fixed ∈ (0, 1), define
g : ∆Θ
→
∆Θ
ν 7−→ (1 − )ν + µ.
Because g is continuous and affine, its range G is compact and convex. Define
p := p ◦ g−1
and q := (1 − )q + f,
where f ∈ R(µ) is the random posterior associated with full information.28 It is immediate,
say by direct computation with the Prokhorov metric and boundedness of Θ, that (p , q ) →
(p, q) as → 0. Moreover,
co(S p ) ⊆ co(G ) = G ( ∆Θ = co(S q ).
In particular, co(S p ) + co(S p ). Then, from Definition 2, p µB q as desired.
Indeed, let β1 : ∆∆Θ → ∆Θ and β2 : ∆∆∆Θ → ∆∆Θ be the maps taking each measure to its barycenter.
µ
2
−1
−1
Then define
B : ∆∆∆Θ →
(∆∆Θ) via B(Q) = (Q ◦ β1 , β2 (Q)). Then B = B((β1 ◦ β2 ) (µ)).
28
ˆ
ˆ
ˆ
i.e. f {δθ : θ ∈ Θ} = µ(Θ) for every Borel Θ ⊆ Θ.
27
33
8.5
Theorem 2
Lemma 4. Given p, q ∈ ∆∆Θ, the following are equivalent:
1. There is some ν ∈ ∆Θ such that p, q ∈ R(ν) and p νB q.
R
R
2. For every convex continuous f : ∆Θ → R, we have ∆Θ f dp ≥ ∆Θ f dq.
Proof. That (1) implies (2) follows from Lemma 3. Now, suppose (2) holds. The Theorem
of Blackwell (1953), reported in Phelps (2001, main Theorem, Section 15), then says that p
is a mean-preserving spread of q, yielding (1).
Notation 1. Given any F ∪ G ∪ {h} ⊆ C(∆Θ):
• Let %F be the (reflexive, transitive, continuous) binary relation on ∆∆Θ given by
Z
Z
p %F q ⇐⇒
f dp ≥
f dq ∀ f ∈ F .
∆Θ
∆Θ
• Let hF i ⊆ C(∆Θ) be the smallest closed convex cone in C(∆Θ) which contains F and
all constant functions.
• Let F + G denote the Minkowski sum { f + g : f ∈ F , g ∈ G}, which is a convex cone
if F , G are.
• Let R+ h denote the ray {αh : α ∈ R, α ≥ 0}.
We now import the following representation uniqueness theorem to our setting.
Lemma 5 (Dubra, Maccheroni, and Ok (2004), Uniqueness Theorem, p. 124). Given any
F , G ⊆ C(∆Θ),
%F = %G ⇐⇒ hF i = hGi.
As a consequence, we get the following:
Corollary 1. Suppose F ∪ {g} ⊆ C(∆Θ). Then
%F ⊆ %{g} if and only if g ∈ hF i.
Proof. If g ∈ hF i, then %{g} ⊇ %hF i , which is equal to %F by the uniqueness theorem.
If %F ⊆ %{g} , then %F ⊆ %F ∩ %{g} = %F ∪{g} ⊆ %F . Therefore, %F ∪{g} = %F . By the
uniqueness theorem, hF ∪ {g}i = hF i, so that g ∈ hF i.
34
Lemma 6. Suppose F ⊆ C(∆Θ) is a closed convex cone that contains the constants and
g ∈ C(∆Θ). Then either g ∈ −F or hF ∪ {g}i = F + R+ g.
Proof. Suppose F is as described and −g < F .
Since a closed convex cone must be closed under sums and nonnegative scaling, it must
be that hF ∪ {g}i ⊇ F + R+ g. Therefore, hF ∪ {g}i = hF + R+ gi, which leaves us to show
hF + R+ gi = F + R+ g. As F + R+ g is a convex cone which contains the constants, we need
only show it is closed.
∞
∞
To show it, consider sequences {αn }∞
n=1 ⊆ R+ and { fn }n=1 ⊆ F such that ( fn + αn g)n=1
converges to some h ∈ C(∆Θ). We wish to show that h ∈ F + R+ g. By dropping to a
subsequence, we may assume (αn )n converges to some α ∈ R+ ∪ {∞}.
If α were equal to ∞, then the sequence αfnn from F would converge to −g, implying
n
(since F is closed) −g ∈ F . Thus α is finite, so that ( fn )n converges to f = h − αg. Then,
since F is closed, h = f + αg ∈ F + R+ g.
Proof of Theorem 2, Part (1)
R
R
µ
Proof. (⇒) If C is a concave
function
and
p
q,
then
by
Lemma
3,
C
dq
≥
C dp.
B
R
R
R
R
µ
RAssuming Rp B q and U1 dq ≥ U1 dp, we obtain γU1 dq ≥ γU1 dp and, therefore,
U2 dq ≥ U2 dp.
(⇐) In light of Lemma 4, that 2 is more information-averse than 1 is equivalent to the
pair of conditions:
%C∪{U2 }
⊆
%{U1 }
%(−C)∪{U1 }
⊆
%{U2 } ,
where C is the set of convex continuous functions on ∆Θ; notice that C is a closed convex
cone which contains the constants.
Then, applying Corollary 1 and Lemma 6 twice tells us:
• Either U2 is concave or U1 ∈ C + R+ U2 .
• Either U1 is convex or U2 ∈ −C + R+ U1 .
By hypothesis,29 1 is not behaviorally information-loving, i.e. U1 is not convex. Then U2 ∈
−C + R+ U1 . That is, U2 = γU1 + C for some γ ≥ 0 and concave C. By hypothesis, 2 is not
behaviorally information-averse, so that γ > 0, proving the proposition.
29
Note: this is the first place we have used this hypothesis.
35
The proof requires that 30 1 not be information-loving and 2 not be information-averse.
We cannot dispense with these conditions. Indeed, consider the Θ = {0, 1} world with
U1 = H (entropy) and U2 = V (variance). Both are strictly concave, so that in particular
U2 is more information-averse than U1 . However, any sum of a concave function and a
strictly positive multiple of U1 is non-Lipschitz (as H 0 (0) = ∞), and so is not a multiple of
U2 . There are analogous counterexamples in the case where both agents are informationloving.
Proof of Theorem 2, Part (2)
Proof. First, notice that we are done if either U1 is information-loving (in which case p∗1
can just be full information) or U2 is information-averse (in which case p∗2 can just be no
information). Thus focus on the complementary case.
Next, we apply the first part of the theorem: positively scaling if necessary, it is without
loss to assume f := U1 − U2 is convex.
For i = 1, 2, fix an optimal policy p∗i ∈ extR(µ). Notice that a random posterior (in
the present binary-state setting) is extreme if and only if it has support of size ≤ 2. Say
S p∗i = {νiL , νiH }, where 0 ≤ νiL ≤ µ ≤ νiH ≤ 1. Because U1 is continuous, we may further
assume p∗1 is Blackwell-maximally optimal.
R
For each z ∈ {H, L}, we can construct some pz ∈ R(ν1z ) such that U1 dpz ≥ U1 (ν1z ) and
pz ((ν2L , ν2H )) = 0. Indeed, if ν1z < (νR2L , ν2H ), just let pz be the point mass on νz . Otherwise, let pz
put weight
on S p∗2 . We know that U2 dpz ≥ U2 (ν1z ) by optimality of p∗2 , and it then follows
R
that U1 dpz ≥ U1 (ν1z ) by convexity of f.
Replacing each ν1z with pz , gives a more informative optimal policy for U1 . By our
assumption that p∗1 is Blackwell-maximal, it must be that we in fact haven’t changed p∗1 at
all; that is, ν1L , ν1H < (ν2L , ν2H ). So
ν1L ≤ ν2L ≤ ν2H ≤ ν1H .
It follows readily that p∗1 is more informative than p∗2 .
8.5.1
Example
Suppose that θ = (θ1 , θ2 ) ∈ {0, 1}2 is distributed uniformly. Let x1 (ν) and x2 (ν) ∈ [−1, 1]
be some well-behaved measures of information about θ1 , θ2 respectively (the smaller xi , the
30
For brevity, the statement of the theorem required that each agent is neither behaviorally information-loving
nor -averse, which is stronger than required.
36
more dispersed the ith dimension of distribution ν).31 Then define U1 , U2 via
1
x1 (ν)2 + x2 (ν)2 + x1 (ν)x2 (ν) ;
2
1
U2 (ν) := U1 (ν) − x1 (ν).
5
U1 (ν) := −
Agent 2 receives extra utility from being less informed and so he is more information-averse
than 1. Meanwhile, we can easily verify (via first-order conditions to optimize (x1 , x2 )) that
any optimal policy for 1 will give more information concerning θ1 , but less information
concerning θ2 , than will any optimal policy for 2. Therefore, the optimal policies for 1 and 2
are Blackwell-incomparable, although 2 is more information-averse than 1. Intuitively, the
penalty for information about θ1 induces a substitution effect toward information about θ2 .
Technically, the issue is a lack of supermodularity.
8.6
Proof of Proposition 2
Below, we will prove the slightly stronger result. Fix any p¯ ∈ R(µ), and let S := co(S p¯ ) and
Φ := {φ : S → R : φ is affine continuous and φ ≥ U|S }. We will show that
Z
max
U dp = cav p¯ U(µ),
p∈∆(S )∩R(µ)
where cav p¯ U(ν) := inf φ∈Φ φ(ν).
Proof. Let β : ∆(S ) → S be the unique map such that Eν∼p ν = β(ν). Such β is well-defined,
continuous, affine, and surjective, as shown
R in Phelps (2001).
For every p ∈ ∆(S ), define EU(p) = U dp. The map EU is then affine and continuous.
For every ν ∈ S , define U ∗ (ν) = max p∈∆(S ),β(p)=ν EU(p), which (by Berge’s theorem) is welldefined and upper-semicontinuous.
For any ν ∈ S :
• U ∗ (ν) ≥ EU(δν ) = U(ν). That is, an optimal policy for prior ν does at least as well as
giving no information.
• For all p ∈ ∆(S ) and all affine continuous φ : S → R with φ ≥ U|S
!
Z
Z
Z
EU(p) =
U dp ≤
φ(s) dp(s) = φ
s dp(s) = φ(β(p)).
∆Θ
For instance, let xi (ν) := 1 − 8V(margi ν), i = 1, 2. What we need is that x1 (ν), x2 (ν) be convex [−1, 1]valued functions of marg1 ν, marg2 ν respectively, taking the priors to −1 and atomistic beliefs to 1.
31
37
So, U ∗ (ν) can be no higher than cav p¯ U(ν).
Moreover, if ν, ν0 ∈ S and λ ∈ [0, 1], then for all p, q ∈ ∆(S ) with β(p) = ν and β(q) = ν0 , we
know β((1 − λ)p + λq) = (1 − λ)ν + λν0 , so that
U ∗ ((1 − λ)ν + λν0 ) ≥ EU((1 − λ)p + λq) = (1 − λ)EU(p) + λEU(q).
Optimizing over the right-hand side yields
U ∗ ((1 − λ)ν + λν0 ) ≥ (1 − λ)U ∗ (ν) + λU ∗ (ν0 ).
That is, an optimal policy for prior (1 − λ)ν + λν0 does at least as well as a signal inducing
(interim) posteriors from {ν, ν0 } followed by an optimal signal for the induced interim belief.
So far, we have established that U ∗ is upper-semicontinuous and concave, and U|S ≤
U ∗ ≤ cav p¯ U. Since cav p¯ U is the pointwise-lowest u.s.c. concave function above U|S , it must
be that cav p¯ U ≤ U ∗ , and thus U ∗ = cav p¯ U.
8.7
On Optimal Policies: Proofs and Additional Results
Proof of Theorem 3
Proof of part 1. By continuity of Blackwell’s order, there is a µB -maximal optimal policy
p ∈ R(µ).
For any C ∈ C, it must be that p(C \ extC) = 0. Indeed, Phelps (2001, Theorem 11.4)
provides a measurable map r : C → ∆(extC) with Rr(·|ν) ∈ R(ν) for every ν ∈ C. Then
we can define p0 ∈ R(µ) via p0 (S ) = p(S \ C)R + C r(S |·) Rdp for each Borel S ⊆ ∆Θ.
Then U-covering and Jensen’s inequality imply U dp0 ≥ U dp, so that p0 is optimal
too. By construction, p0 µB p, so that (given maximality of p) the two are equal. Therefore
p(C \ extC) = p0 (C \ extC) = 0. Then, since C is countable,


[

p(out(C)) = 1 − p  [C \ extC] = 1.
C∈C
Given the above, it is useful to have results about how to find posterior covers. Below,
we prove two useful propositions for doing just that.
38
Proof of part 2. As an intersection of closed convex sets is closed convex, and a sum or
supremum of convex functions is convex, the following are immediate.
1. Suppose f is the pointwise supremum of a family of functions, f = supi∈I fi . If Ci is
an fi -cover for every i ∈ I, then




_
\




C :=
Ci = 
C
:
C
∈
C
∀i
∈
I

i
i
i




i∈I
i∈I
is an f -cover.
2. Suppose f is the pointwise supremum of a family of functions, f = supi∈I fi . If C is an
fi -cover for every i ∈ I, then C is an f -cover.
3. If C is a g-cover and h is convex, then C is a (g + h)-cover.
The first part follows from (1), letting I = A and fa = u(a, ·). The second part follows
from (2) and (3), since U = uP + UC for convex UC .
We note in passing that the previous proof generates a comparative statics result for
posterior covers. Given Theorem 2, if U2 is more information-averse than U1 , then any
U2 -cover is a U1 -cover.
Proof of Part 3. By finiteness of I, the collection C covers ∆Θ. For each i ∈ I, note that
T
Ci = j∈I {ν ∈ ∆Θ : fi (ν) ≥ f j (ν)}, an intersection of closed convex sets (since { f j } j∈I are
affine continuous), and so is itself closed convex. Restricted to Ci , f agrees with fi and so is
affine, and therefore convex.
Now, under the hypotheses of Theorem 3, we prove a claim that fully characterizes the
set of outer points of the f -cover, reducing their computation to linear algebra.
Claim 1. The f -cover C = {Ci : i ∈ I} given by Theorem 3(3) satisfies out(C) = {ν∗ ∈ ∆Θ :
{ν∗ } = S (ν∗ )} where
S (ν∗ ) := {ν ∈ ∆Θ : supp(ν) ⊆ supp(ν∗ )
and ∃J ⊆ I such that fi (ν) = f j (ν) = f (ν) ∀i, j ∈ J}. (7)
Proof. Fix some ν∗ ∈ ∆Θ, for which we will show {ν∗ } , S (ν∗ ) if and only if ν∗ < out(C).
Let us begin by supposing {ν∗ } , S (ν∗ ); we have to show ν∗ < out(C). Since ν∗ ∈ S (ν∗ )
no matter what, there must then be some ν ∈ S (ν∗ ) with ν , ν∗ . We will show that S (ν∗ ) must
then contain some line segment co{ν, ν0 } belonging to some Ci , in the interior of which lies
39
ˆ be the support of ν∗ , and let J := {i ∈ I : ν∗ ∈ Ci }.
ν∗ ; this will then imply ν∗ < outC. Let Θ
ˆ with fi (ν) = f j (ν) = f (ν) ∀i, j ∈ J. Now, for
Given that ν ∈ S (ν∗ ), we have ν ∈ ∆Θ
ˆ Then
sufficiently small > 0, we have (ν − ν∗ ) ≤ ν∗ .32 Define ν0 := ν∗ − (ν − ν∗ ) ∈ ∆Θ.
fi (ν0 ) = f j (ν0 ) = f (ν0 ) ∀i, j ∈ J too and, by definition of ν0 , we have ν∗ ∈ co{ν, ν0 }. If i < J,
then it implies fi (ν∗ ) > f (ν) because ν∗ < Ci . Therefore, by moving ν, ν0 closer to ν∗ if
necessary, we can assume f (ν) = f j (ν) < fi (ν) and f (ν0 ) = f j (ν0 ) < fi (ν0 ) for any j ∈ J and
i < J. In particular, fixing some j ∈ J yields ν, ν0 ∈ C j , so that ν∗ is not in out(C).
To complete the proof, let us suppose that ν∗ < out(C), or equivalently, ν∗ ∈ Ci but
ν∗ < ext(Ci ) for some i ∈ I. By definition of Ci , we have that fi (ν∗ ) = f (ν∗ ). The fact that
ν∗ < ext(Ci ) implies that there is a non-trivial segment L ⊆ Ci for which ν∗ is an interior
point. It must then be that supp(ν) ⊆ supp(ν∗ ) and fi (ν) = f (ν) for all ν ∈ L. As a result,
L ⊆ S (ν∗ ) so that {ν∗ } , S (ν∗ ), completing the proof.
Corollary 2. Suppose Θ = {0, 1}; A is finite; and for each a ∈ A, u(a, ·) = mini∈Ia fa,i , where
{ fa,i }i∈Ia is a finite family of distinct affine functions for each a. Then, there exists an optimal
policy that puts full probability on
[
S := {0, 1} ∪ {ν ∈ [0, 1] : fa,i (ν) = fa, j (ν) for some distinct i, j ∈ Ia }.
a∈A
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