Persuading Policy Makers

Christian Salas†

July 31, 2018

Abstract

Interest groups persuade policy-makers by publicly providing information about policies

−e.g. through commissioning scientific studies or piloting programs− or about constituents’

views −e.g. through opinion polls or organizing manifestations. By understanding these public

lobbying activities as public signals whose informational content can be strategically manipu-

lated, this paper studies the strategic use of these tools in order to persuade a policy-maker. A

game between a policy-oriented interest group who can design a public signal and an electorally

accountable executive who can implement a policy is used to analyze the equilibrium public

signal and policy, the underlying persuasion mechanism, and the consequences for voters. This

paper finds that, even when an interest group always wants the same policy regardless of the

state of the world, it can succeed at changing policy in its preferred direction and voters can

sometimes benefit from the group’s activity. Furthermore, voters may be best served by a worse

(less able or more cynical) policy-maker. This is because a-priori a worse policy-maker will

tend to herd on the prior relatively more than a better policy-maker; this will force interest

groups to release greater amounts of information in order to change the policy-maker’s mind,

which increases the probability that the voters’ best policy is implemented. Ideologically biased

policy-makers are not totally undesirable either, for they induce similar incentives to interest

groups of opposite ideology.

†e-mail: christian.h.salas@yale.edu

1 Introduction

On July 5, 2017, Greenpeace released the results of a study which showed that wind energy and

solar power will be the cheapest form of power generation in every G20 country by the year 2030.

Producing studies that publicly inform about a policy is a common tactic used by Greenpeace

and other pressure groups to persuade policy-makers. This study, however, was not conducted

by Greenpeace, it was commissioned by them to six researchers at Lappeenranta University of

Technology.1 This seems to be a growing trend: while 19 of 21 studies released before 2013

were conducted directly by Greenpeace, in the past five years 7 of 17 were commissioned to an

independent entity.

There are many other ways of attempting to persuade policy-makers through the provision of

public information, a notable example is mass protests. Unlike commissioned studies which provide

information about the policy, public protests provide information about constituents’ preferences

(Lohmann 1993). Like commissioned studies, however, organizing public manifestations entails a

risk, for the resultant information is outside of the control of those initiating the public intervention.

Other examples of public information persuasion include opinion polls on specific issues or the

piloting of policy interventions.

These tactics have three important features. First, persuader, policy-maker and the general

public share a common ignorance regarding an aspect relevant for policy making. Second, specially

for protests and government pilot studies and less so for opinion polls, once the information is

produced it is made public without the persuader being able to modify it.2 Third, the initiating

party, while not being able to manipulate the final result, has some room to design the production

technology of such information. For example, in recent student protests in Latin America organizers

voiced concerns with including concerts in the route of the manifestation because the press would

hold these activities, rather than support for the cause, responsible for the turnout. Factors such

as the time of the day and day of the week are usually considered by protest organizers as well, not

only in how they promote turnout but to calculate how this turnout will be interpreted by their

1Ram, Child, Aghahosseini, Bogdanov, Poleva and Breyer (2017)2Formally, this is the commitment assumption usually required in the literature on the design of public exper-

iments, that is, that once the information is produced it is communicated unaltered. In the case of commissionedstudies, a natural exception would be when the persuader can decide whether to release −or how much effort toput into communicating− the information just after observing the results of the study. In mass public protests,this is not an issue because the information −e.g. the turnout− is revealed to persuader, public and policy-makersimultaneously.

1

audience of interest. On the other hand, the range of possible results that a pilot intervention,

a scientific study or an opinion poll can produce depends crucially on the site and time they are

carried out, and thus again not only the result can be manipulated but the way the result will be

interpreted in light of the prior expectation held by the audience.

This paper interprets a public lobbying activity such as the ones described above as the strategic

design of a public signal, and in doing so analyzes the equilibrium intervention and policy conse-

quence. It will do so by formalizing the interaction between a policy-maker (to whom I reserve the

pronoun ‘he’) deciding whether to implement a policy, and a special interest group (SIG, to whom

I reserve the pronoun ‘she’) who wishes it to be implemented. The game has three stages: first,

the SIG chooses the design of the public signal, that is, how the outcome of the study or protest

will change the public’s belief about the suitability of the policy; second, the policy-maker chooses

a policy; finally, the public updates its beliefs about the policy-maker’s ability. Before making his

choice, the policy-maker receives private information about the state, information whose quality

depends on his ability. If the executive follows his signal, his action informs the public about his

ability, which drives the updating of beliefs.

Underlying the ideas of this paper is the fact that competitive elections create a relationship

of formal accountability between policy-makers and their constituents. While policy-makers often

have a genuine desire to maximize voters’ welfare, they sometimes hold interests of their own,

may those be policy preferences or the desire to hold office for its own sake. A vast literature

in economics and political science has modeled the incumbent policy-maker’s problem as a career

concerns one (Holmstrom 1999; Levy 2004), where incumbents take actions to signal to voters their

ability to execute policies or their congruence with voters’ preferences (Fearon 1999; Canes-Wrone,

Herron and Shotts 2001; Maskin and Tirole 2004; Ashworth 2005).3 In this paper, the focus is on

the voters’ uncertainty regarding the policy-maker’s ability.

It is important to distinguish persuasion of policy-makers through the design public signals from

lobbying activities from experts. Experts’ lobbying is characterized by the fact that SIGs possess

private information about the state of the world −i.e. have expertise or the resources to acquire

it. Persuasion in this context is formalized as costly signaling or cheap talk (Austen-Smith and

Wright 1994; Persson and Tabellini 2000; Grossman and Helpman 2001; Austen-Smith and Banks

3See Ashworth (2012) for a review of this literature.

2

2002; Boehmke, Gailmard, Patty et al. 2006; Hirsch and Montagnes 2015; Schnakenberg 2017). In

the cases studied in this paper, SIG, policy-maker and the public who evaluates the policy-maker’s

fitness for office have symmetric (lack of) information: no one observes the state of the world and

a-priori all share the same prior belief.

A general persuasion problem under symmetric information where the actor who has no decision

power can nevertheless design a public signal has been generalized by Kamenica and Gentzkow

(2011), (henceforth, KG) and this paper draws on their construction.4 In particular, this paper is

interested in understanding how the provision of public information can manipulate a policy-maker,

among others through manipulating accountability problem that policy-maker and voters need to

solve.

Notice that the SIG releases information about the policy, not the policy-maker’s ability. The

way in which the SIG manipulates the policy-maker is through changing the informational envi-

ronment he faces when making a decision; anticipating the policy-maker’s induced preferences as a

function of beliefs, the SIG can indirectly induce the policy-maker’s action. Studying a variety of

policy-maker’s preferences sheds light on the ability (or lack thereof) that the SIG has to manip-

ulate him, and on the consequences of this persuasion relation on voters’ welfare. Among others,

this paper will show that despite the fact that the SIGs have extreme policy preferences, in most

circumstances the ability to design public signals will improve voters’ welfare. This is true, for

example, when the policy-maker is either benevolent or has policy preferences opposed to those of

the interest group; the opposite is true when the policy-maker has preferences aligned with those of

the interest group. Interestingly, there are situations where voters would prefer to appoint a more

cynical −more worried about his reputation− and a less able −one who possesses worse private

information− policy-maker, because this would force the SIG to provide excess information to the

public in order to persuade him, which in turn increases the probability that the welfare-maximizing

policy is implemented.

This paper benefits from previous work in several ways. Part of the motivation is the information

that can be transmitted through public protests, which is formalized by Lohmann (1993, 1994)

and Battaglini (2017), but which does not consider that the policy-maker faces electoral incentives.

In this paper we abstract from the specific way in which the information is produced in order to

4See also Brocas and Carrillo (2007).

3

encompass other initiatives that can be interpreted as public signals, such as commissioned scientific

studies or piloting policies, and to focus the attention on the interaction between information design

and electoral accountability. Insights from the literature on strategic communication in politics

were enlightening. Theoretical work includes the study of pandering (Morris 2001; Canes-Wrone,

Herron and Shotts 2001), of campaign finance (Prat 2002; Coate 2004a,b; Ashworth 2006; Vanberg

2008; Daley and Snowberg 2009), of pre-election communication (Kartik and Van Weelden 2014;

Dziuda and Salas 2018), of bureaucracies and policy-making (Gailmard and Patty 2007; Patty 2009;

de Mesquita and Landa 2015), and of government manipulation of the media (Gehlbach and Sonin

2014). Closer to this paper, Alonso and Camara (2016), study the optimal design of a public signal

in order to persuade voters under various voting rules and distribution of voters’ preferences. They

show that with a non-unanimous voting rule and heterogenous voters, the public signal can be

designed so as to target the winning coalitions, hence exploiting the preference disagreement across

voters. They study the welfare consequences of this information control, and then characterize

the voting rule that voters themselves would prefer to choose. For example, voters may adopt a

supermajority voting rule because it will induce the designer of the public signal to supply a more

informative experiment. This insight is related to the present paper, where sometimes voters may

prefer to delegate policy-making power to a less able or more cynical politician because they are

harder to persuade.

The paper is organized as follows. Section 2 will outline a baseline model where the policy-

maker only cares about maximizing voters’ welfare. Section 3 describes the equilibrium of the

baseline model and its welfare consequences. Section 4 studies this interaction when executives

have interests of their own, namely career concerns and policy preferences. Section 5 concludes.

2 Model

In this section we present a simple model of persuasion, where a policy-driven interest group chooses

a public signal to persuade a policy-maker who faces a binary policy choice. For ease of exposition,

we start with a baseline model where the policy-maker is benevolent, that is, whose preferences are

aligned with the voters’.

Players, policy and state. Consider a model with two strategic actors, an interest group

(indexed by I) and an executive (henceforth, E), together with a policy that can be implemented

4

by E. A third player is a non-strategic median voter whose sole role will be to update his posterior

beliefs about the ability of E once the game has been played. There is a binary state of the world,

ω ∈ Ω = 0, 1, with common prior Pr(ω = 1) = p ∈ (0, 1), which is interpreted as whether the

policy in question will in net benefit voters (ω = 1) or not (ω = 0).

Actions. E chooses ρ ∈ P = 0, 1, interpreted as implementing the policy (ρ = 1) or not

(ρ = 0).5 Before E’s choice, I chooses the type of public intervention that she will carry out. By

type of intervention here it is meant what information the intervention will be able to convey. For

example, I may commission an opinion survey, choosing a larger (smaller) sample the more precise

(garbled) she wishes the information to be produced. In a study of the extinction of a species, I

may choose a site where such species is known to abound, thus finding abundance of individuals

and great capacity to breed would constitute no evidence of the lack of extinction in the world

while the opposite finding would be clear evidence that the species is in danger. That is, I designs

the intervention in order to choose the set of possible beliefs the public can hold after observing

the result of the intervention.

To formalize this, assume that I can choose a public information system, consisting of signal

realizations s ∈ S = 0, 1 with distribution Pr(s|ω)ω∈0,1. Two signal realizations are used

without loss of generality.6 Following KG, I’s problem of choosing an information system can be

reduced to choosing a distribution of posterior beliefs Pr(ω|s)s∈0,1 over Ω satisfying what they

call bayes plausibility, in this case

Pr(ω = 1|s = 1) Pr(s = 1) + Pr(ω = 1|s = 0) Pr(s = 0) = Pr(ω = 1)

that is, that the expected posterior beliefs equals the prior. I may choose to carry out no public

intervention, or equivalently, to design a completely uninformative signal.7

Types and information. There are two types of executives, able (t = 1) and unable (t = 0),

with common prior Pr(t = 1) = π ∈ (0, 1). Before deciding whether to implement the policy, E

receives a binary signal y ∈ 0, 1 about the state of world.8 If E is able, this signal is perfect, that

5That E chooses a pure strategy is without loss of generality, because this paper’s analysis is based exclusivelyon the a-priori perspective, that is, what I expects E to do, the ex-ante probability the policy matches the state, etc.

6To design the optimal information system, it is sufficient that |S| ≥ min|Ω|, |A|. For details see KG.7That is, one where any outcome provides no new information, or in the notation of this paper, where both signal

realizations s = 0 and s = 1 induce the same posterior, equal to the prior.8It is assumed that Pr(y|s) = Pr(y), that is that the generating process of E’s private signal is independent

of the process designed by I which generates public signals, even though they both produce information about the

5

is Pr(y = 1|ω = 1) = Pr(y = 0|ω = 0) = 1. If E is unable, the signal is imperfect in the following

sense:

Pr(y = 1|ω = 1) = q Pr(y = 0|ω = 1) = 1− q

Pr(y = 1|ω = 0) = 1− q Pr(y = 0|ω = 0) = q

where q ∈[

12 , 1

). E does not observe his own type.9

Payoffs. I is policy-driven and only gains utility if the policy is implemented, that is uI = ρ.

As baseline, we will assume that E is congruent with the voters, that is, wishes to match the state.

Thus, E’s preferences are defined by uE = 1ρ=ω, where 1ρ=ω is an indicator function equal to 1

when the action matches the state. Later we will explore the implications of having an executive

that has interests of his own.

Timing. The timing of the game is as follows:

0. Nature determines the state ω and E’s type t. No one observes either.

1. I designs the intervention, that is, designs a public signal, whose design and realization is

observed by everyone.

2. E observes the intervention that was designed, the signal realization coming from the inter-

vention, and his private signal y ∈ 0, 1, and takes an action ρ ∈ 0, 1.

3. The voter observes I’s intervention design, the intervention’s signal realization, and E’s action,

and updates her beliefs about the quality of the executive. Payoffs are realized.

The equilibrium concept is Subgame Perfect Nash Equilibrium (SPE) extended to moves of nature.

3 Optimal public signal

This section finds the equilibrium public signal and policy decision of the baseline model. First the

policy-maker’s best-response will be found, then that of the interest group, to end with a discussion

about voters’ welfare.

same underlying fundamental. This assumption simplifies calculations of ex-ante expected welfare. This assumptionis reasonable to the extent that the mechanisms through which policy-makers get informed about the state of theworld, experts and internal polls, is unrelated to those available to pressure groups.

9It is a common assumption in the political accountability literature that when the dimension of interest to votersis ability, uncertainty about it is symmetric. Formally, this assumption allows one to abstract from any signaling rolethat the choice of ρ may have, and focus solely on symmetric reputation concerns.

6

3.1 Policy-maker

Let’s start by looking at the decision taken by the executive.

Lemma 1. After observing the public signal’s realization s, E updates his belief about ω to Pr(ω|s) =

c. His choice then is

• When c < (1− π)(1− q): E chooses ρ = 0 regardless of his private signal

• When (1− π)(1− q) < c < π + (1− π)q: E follows his private signal (ρ = y)

• When c > π + (1− π)q: E chooses ρ = 1 regardless of his private signal

Proof. All proofs are in the appendix.

The executive’s best-response simply uses information in the most effective manner in order to

match the state. When his private signal is accurate enough, with respect to the decisiveness of

his beliefs c prior to observing his private signal, he follows his signal (the middle case). In the

opposite case, that is, when his private signal is too noisy (low π or q) and/or his previous beliefs c

are very decisive (close to 0 or 1), the executive decides to ignore his signal and herd on the prior :

implements the policy for high c and does not for low c.

The updated belief c that E faces before making his decision is the crucial choice that I must

make in order to manipulate him. In the absence of intervention, c = p, the prior. Notice that the

fact that for a set of priors p the executive’s private signal is decision-irrelevant is precisely the

leverage that I has to manipulate E, as noted in general by KG.10

3.2 Interest group

Knowing the executive’s best-response to different signal realizations, the interest group chooses

the specific design of her intervention. As indicated above, this is equivalent to choosing which

distribution of posterior beliefs Pr(ω|s)s∈0,1 she wishes to generate with the intervention. For

ease of presentation, denote

10KG state that so long as E’s action is not linear in his beliefs, the interest group may benefit from persuasion.In this case this is achieved by allowing priors where E follows his signal and others where he does not, a situationthat is assumed away in other contexts −see for example Prat, 2005.

7

Pr(ω = 1|s = 1) = a Pr(ω = 1|s = 0) = b

Pr(ω = 0|s = 1) = 1− a Pr(ω = 0|s = 0) = 1− b

Pr(s = 1) = x Pr(s = 0) = 1− x

I’s problem is then to maximize E (ρ = 1) subject to satisfying p = b(1−x) +ax. Notice the trade-

off that I faces: per the bayes plausibility restriction, generating a signal that induces a posterior

belief farther away from the prior requires this signal to occur with lower probability.

The result of the optimal intervention is described in the following proposition.

Proposition 1. (Kamenica and Gentzkow (2011), given E’s best-response in Lemma 1) There are

two cases.

If E is ex-ante relatively unable (π ≤ π), I chooses the following distribution of posterior beliefs

• When 0 < p < π + (1− π)q, then a = π + (1− π)q b = 0 x = pπ+(1−π)q

• When p > π + (1 − π)q, then I does not intervene (or designs a perfectly uninformative

intervention), thus posterior equals prior

If E is ex-ante relatively able (π ≥ π), I chooses the following distribution of posterior beliefs

• When p < (1− π)(1− q), then a = (1− π)(1− q) b = 0 x = p(1−π)(1−q)

• When (1− π)(1− q) < p < π + (1− π)q, then

a = π + (1− π)q b = (1− π)(1− q) x = p−(1−π)(1−q)1−2(1−π)(1−q)

• When p > π + (1 − π)q: then I does not intervene (or designs a perfectly uninformative

intervention), thus posterior equals prior

This result is illustrated in the top of Figure 1. The executive’s expected (prior to receiving

her private signal) best-response in the absence of intervention is graphed as a thick grey line, the

thin black line illustrates the expected best-response resulting from the intervention. Proposition 1

proves that I’s optimal intervention uses information efficiently in the sense that it induces poste-

riors that provides E with just enough incentives to do what I deems optimal, either implementing

the policy regardless of E’s signal or inducing E to follow his signal. Which of the two will be

optimal to induce depends on the prior p and E’s ability π in a way that will be explained below.

This optimal strategy grants I payoffs that form the concave closure of the payoffs in the absence

8

Figure 1: I’s payoff and Voters’ welfare in equilibriumRelatively unable E: π ≤ π Relatively able E: π ≥ π

I’s expected payoff, the probability that the policy is implemented, in the top, and Voters’ welfare, the probability thatthe policy matches the state, in the bottom. Thin black line plots the case with intervention, thick grey line without.Simulations in the left graphs use q = 1

2and π = 1

4and in the right graphs use q = 1

2and π = 3

4.

of intervention.11 Since I’s payoff is the executive’s expected best-response (i.e. the probability

that E implements the policy), the thin back line forms the concave closure of the thick grey line.

It is important to notice that the interest group’s optimal intervention changes with the ability

of the executive in two different ways. First, the thresholds defining E’s best-response grow apart

as π increases and therefore so do the posterior beliefs induced by the optimal intervention. Second,

when π < π, I runs a single experiment for all priors lower than π+ (1−π)q (i.e. priors lower than

the one that induces E to implement the policy regardless of additional information), where one

signal induces the posterior belief 0 and the other signal induces the lowest posterior belief that

11As in KG, for this special case.

9

make E implement the policy regardless of his private signal, that is, π+ (1− π)q. However, when

π > π, I runs two different interventions, one for low priors p and another for intermediate priors.

In the first intervention, one signal induces E to not implement the policy and the other to follow

his signal; in the second intervention, one signal induces E to follow his signal and the other to

implement the policy.

The reason behind this structure comes from the trade-off faced by I: signals that induce higher

posteriors come at the cost of being less likely to be realized. That E is highly able means that he

follows his signal for a wide range of priors p, from a very low prior to a very high prior (e.g. see

top-right quadrant of Figure 1). So, suppose p is very low so that E’s no-intervention action is to

not implement the policy; in this case, I has two alternatives: inducing E to follow his signal or

inducing him to implement the policy regardless of his signal. Since, for highly able E, the first

of these requires a very low belief and the latter a very high belief, the probability with which I

induces E to follow his signal (i.e. the probability that the signal inducing this is realized) is much

higher than that which induces E to implement the policy. This difference is lower the less able the

executive: when E is of very low ability, he follows his signal only for priors very close to 0.5, in

other words, the minimum belief than induces E to follow his signal and that which induces him to

implement the policy regardless of the signal are very similar, and therefore so are the probability

of realization of the signals that induce these beliefs. The trade-off between expected action and

probability with which this action is induced is thus solved as follows: inducing E to implement

the policy when E is of low ability (Figure 1, top-left), and inducing E to follow his signal when

E is of high ability (Figure 1, top-right). I is indifferent between the two tactics exactly at π = π.

The same intuition follows for intermediate priors p.

Notice that, at around π, carrying out the single intervention generates posteriors much closer

to 0 or 1 than with either of the two interventions reserved for high ability executives. In other

words, at least around π, the single intervention is relatively more informative than either of the

two interventions in the second tactic.12

The essential insight from KG, here adapted to a simple model of a privately informed receiver,

is that even though the policy-maker’s preferences are aligned with the voters’ and even though the

12This notion of informativeness is equivalent to Blackwell’s, where a more informative experiment is one thatgenerates higher utility for the benevolent executive; generating posteriors “closer to 0 or 1” is the opposite of garbling,which generates posteriors closer to 0.5.

10

interest group has no private information of their own, a well-designed intervention can increase

the probability that the policy is implemented.13

3.3 Welfare

Are voters better or worse off with the possibility that interest groups are allowed to design public

interventions? In this section we explore this issue, where we will assume that voters’ welfare

is maximized when the policy matches the state. This implicitly ignores the SIG from welfare

calculations, which can be justified considering the negligible weight a small group of nevertheless

influential individuals has in the whole of a society. Having said this, we have the following result.

Proposition 2. When policy-makers are benevolent, voters’ welfare −measured by the probability

that the policy chosen by the executive matches the state of the world− is weakly higher when

allowing public interventions.

At first glance this result may be surprising, because voters end up better off when allowing a

particular interest group to try to influence policy, regardless of the fact that the interest group has

extreme preferences. However, this result comes from two key features of this context. First, that

information is a-priori symmetric, hence the interest group lacks private information that could

prevent her from conveying the wrong information. Second, that the method of influence is the

public provision of information which, together with a benevolent policy-maker, can never hurt.

The bottom of Figure 1 illustrates the impact of public interventions on welfare. This result does

not always hold when the policy-maker is not fully benevolent, an issue that will be explored in

the next section.

It is clear from Lemma 1 that, if public interventions were not allowed, voters would be better

served the more able a benevolent policy-maker is, that is, the higher π.14 A follow-up question

is whether, when public interventions are allowed, voters’ welfare also increases with the policy-

maker’s valence. The following proposition states that this is not always the case.

13KG discuss this extension in general, but do not solve any specific model. The key issue, as they point out,is carefully constructing the sender’s (here, the interest group) expected utility function, which here corresponds toLemma 1 and the proof of Proposition 1.

14Ability is drawn from two dimensions, π and q. Even though the most relevant comparative statics are equivalentfor both parameters, both are included in the model because they have slight nuances in some cases. We use π becauseof its interpretation: the probability that the policy-maker is perfectly informed, vis-a-vis some form of imperfectlyinformed politician.

11

Figure 2: Voters’ welfare with policy-makers of different ability

Simulations using q = 12

, on the left using p = 0.25 and on the right using p = 0.54. Indicated is π =√

2− 1, before whichI runs a single type of intervention, after which the intervention depends on p.

Proposition 3. Voters’ welfare is almost always (weakly) increasing in the executive’s ability,

except at π. This is because the interest group changes tactics at this threshold, below which the

tactic involves more information revelation. Thus, voters’ welfare is higher as it approaches π from

the left (lower ability) than from the right (higher ability).

This result is specially surprising in a model where the policy-maker only wishes to match

the state. The reason welfare may be lower with a higher ability executive is that, as stated in

Proposition 1, the best way to persuade a low ability policy-maker is to design a very informative

public signal. To recall, when the ability of the executive falls below π, I switches from running one

of two relatively less informative interventions to only one relatively more informative intervention.

Figure 2 illustrates how welfare changes across different levels of executive ability.

A trade-off, thus, arises. Conditional on the type of intervention, voters do better choosing an

executive of the highest ability because they are more likely to choose the correct policy.15 However,

the interest group generates more informative interventions for lower types, and more information

increases the probability that the policy matches the state. In other words, if the set of executives

to choose from is a small interval around π, voters may be better off choosing a representative of

lower ability. This result holds when the executive’s preferences are more complex like the ones

15A second effect in the same direction is also present. When π < π, the lower the ability the less informative theoptimal intervention, because the executive’s threshold π+(1−π)q in Proposition 1 decreases when ability decreases,that is, the belief that the policy is correct above which the executive implements the policy regardless of his privatesignal decreases, in the limit to 0.5.

12

studied in the next section.16

4 Interested policy-maker

Policy-makers usually have a personal agenda. Most policy-makers wish to stay in office, and thus

care about their reputation. Many policy-makers have policy preferences of their own. In this

section we explore the influence public signals can have, and the implications for voters’ welfare,

when faced with an interested executive.

4.1 Career concerns

When executives have career concerns, it is usually assumed that deciders, either voters or other

politicians, choose whether E stays in office or is promoted. Following the accountability literature,

we model deciders as forward looking, that is, as voting for candidates based on their ability to

generate welfare in the future.17 This allows us to, when considering an office-driven executive,

reduce E’s payoff to equal the voters’ posterior belief about his ability.18 In addition to their career,

policy-makers usually do care about voters’ welfare, thus we will model the executive’s preferences

as

θ 1ρ=ω + (1− θ) Pr[t = 1|s, ρ]

with θ ∈ [0, 1], where 1ρ=ω is an indicator function equal to 1 when the action matches the state,

and Pr[t = 1|s, ρ] is voters’ updated belief that the executive is of high ability having observed

the intervention’s design and signal realization as well as the executive’s action. The rest of the

16The purpose of this paper is to highlight a new strategic consideration that arises when SIGs can influencethe accountability relationship arising between a politician and his voters. This paper takes a reduced form ofthis accountability relation, and does not consider the delegation problem involved in the election underlying thisrelationship. Among others, this delegation problem should consider the fact that a politician’s objective may notalways be to maximize the voter’s perception of his ability (as studied in section 4.1), specially if for a neighborhoodof executive’s ability voters would prefer a lower qualified representative. The exact equilibrium of such delegationproblem is outside the scope of this paper, but it is an interesting avenue for research for the way in which this wouldaffect the result is a-priori unclear.

17This means that voters will consider past actions of incumbents in their evaluation of them, not because theywish to reward or sanction them, but because these actions may provide useful information about their ability togenerate welfare in the future. See Ashworth (2012) for further discussion.

18More generally, we should model E’s payoff as an increasing function of the voter’s posterior beliefs, that is,uE = f (Pr[t = 1|π, p, ρ, ω]) with f ′ > 0. The version in the baseline model is a simplification which sets f to bethe identity function. Among others, this simplification assumes that an election is not close, and hence that E onlycares about enhancing his reputation for the moment. When elections are too close, f may not be continuous, a casethat we will explore as an extension.

13

baseline model is unchanged. We wish to focus now on the parameter θ, thus for ease of presentation

suppose that q = π = 12 .19

Given these new preferences, the executive’s best-response is described below.

Lemma 2. After observing the public signal’s realization s, E updates his belief about ω to Pr(ω|s) =

c. Then, there exists cutoffs cθ and cθ such that

• When c < cθ: E chooses ρ = 0 regardless of his private signal

• When cθ < c < cθ: E follows his private signal (ρ = y)

• When c > cθ: E chooses ρ = 1 regardless of his private signal

where cθ ∈ [14 ,

12) and cθ ∈ (1

2 ,34 ], with

∂cθ∂θ > 0, ∂cθ

∂θ < 0, and limθ→0 cθ = limθ→0 cθ = 12 . Finally,

limθ→1 cθ = 14 and limθ→0 cθ = 3

4 , the benevolent E’s thresholds.

The exact expression of cθ and cθ are defined in the appendix. Lemma 2 shows that the

executive’s decision has the same structure as that of the benevolent one. As expected, at θ = 1

we have that cθ = 1 − cθ = 14 , which coincides with (1 − π)(1 − q) evaluated at q = π = 1

2 .

Importantly, as E becomes more cynical (i.e. as θ grows) the interval where E follows his signal

(cθ, cθ) shrinks from the widest possible interval,(

14 ,

34

)(that of the benevolent E), to the point 1

2

where the executive is fully career concerned.20 This is true because the more cynical an executive,

the more he will act as herding on the prior, which is optimal for him to the extent that following

his private signal involves the risk of looking poorly informed. This behavior is similar to that of

the relatively unable executive who is benevolent, albeit for different reasons. On the previous case,

it was optimal for E to herd more on the prior the less able he was because following his signal was

less likely to produce the policy that matched the state.

The following proposition characterizes the optimal intervention in this case.

Proposition 4. There are two cases.

If E is ex-ante relatively cynical(θ ≤ θ

), I chooses the following distribution of posterior beliefs

• When 0 < p < cθ, then a = cθ b = 0 x = pcθ

19Allowing π and q to take values on the full range [0, 1] and [ 12, 1] respectively does not add any insight, and the

values q = π = 12

do not eliminate any implication nor do they highlight a rare case.20Actually to the interval

(12− ε, 1

2+ ε), with ε > 0 very small, because when the interval collapses equilibrium

strategies change.

14

• When p > cθ, then I does not intervene thus posterior equals prior

If E is ex-ante relatively benevolent(θ ≥ θ

), I chooses the following distribution of posterior beliefs

• When p < cθ, then a = cθ b = 0 x = pcθ

• When cθ < p < cθ, then a = cθ b = cθ x =p−cθ1−2cθ

• When p > cθ, then I does not intervene thus posterior equals prior

Notice that I’s strategy has the same structure as in the baseline model. In this case, the

strategy involving only one intervention design is optimal for relatively cynical policy-makers for

the same reasons it was optimal for relatively unable executives in the previous section: the more

cynical, the closer together the beliefs from and to which E followed his signal, thus the easier it is

to induce him to implement the policy.

Finally, similar to before, welfare may be higher with more cynical policy-makers, but as a

contrast to the baseline case, public interventions will not always benefit voters.

Proposition 5. Voters’ welfare can improve or deteriorate with public interventions. Voters’

welfare can be higher with more cynical executives.

Voters are sometimes hurt because interventions can leverage the fact that, for the appropriate

prior p, the cynical executive herds on the prior when an otherwise benevolent executive would

follow his signal. This leverage is profitable for the interest group because it allows the intervention

to generate more extreme posterior beliefs than those possible with a benevolent executive, the

whole point of organizing an intervention that alters the informational environment.

The fact that welfare can be higher with more cynical executives follows the same logic as

that explaining how an executive of less ability could generate higher welfare: holding the form

of intervention constant, voters are better off the less cynical a policy-maker is, but around the

threshold θ society benefits from I’s change of tactic to a more informative public intervention.

These results are illustrated in Figure 3.

4.2 Policy preferences

It is natural to assume that policy-makers have policy preferences of their own. In this section,

we explore the implications on the optimal intervention strategy as well as on welfare of facing an

executive with known policy preferences.

15

Figure 3: Voters’ welfare when executive has career concerns

Voters’ welfare, the probability that the policy matches the state, is graphed against the prior p on the right panel andagainst the degree of benevolence θ on the left. Both simulations use π = q = 1

2. On the right, θ = 3

4; on the left, p = 1

4.

Consider the simplest possible presentation of this case, where E’s preferences are described by

α 1ρ=ω + (1− α) f(ρ)

where α ∈ [0.5, 1] and f(ρ) is either ρ or 1− ρ, that is, where E’s policy preference is either aligned

or opposed to that of the SIG. Furthermore, for ease of exposition again assume π = q = 12 . In this

case, E’s equilibrium policy best-response is described in the following lemma.

Lemma 3. After observing the public signal’s realization s, E updates his belief about ω to Pr(ω|s) =

c. His choice then is

• When c < cα: E chooses ρ = 0 regardless of his private signal

• When c ∈ (cα, cα): E follows his private signal (ρ = y)

• When c > cα: E chooses ρ = 1 regardless of his private signal

where cα ∈ [0.25, 1] and cα ∈ [0.75, 1] when f(ρ) = 1− ρ, and cα ∈ [0, 0.25] and cα ∈ [0, 0.75] when

f(ρ) = ρ.

The exact expression of cα and cα are defined in the appendix. Just as when E has career

concerns, Lemma 3 shows that E’s best-response has the same structure as that of the benevolent

E. In this case, however, the length of the interval in which E follows his signal is equal to that

16

when E was fully benevolent; what the policy preference does is shift this interval towards 0 when

f(ρ) = ρ and towards 1 when f(ρ) = 1 − ρ. This is intuitive because E has a bias towards, for

example, implementing the policy when f(ρ) = ρ, so the set of beliefs for which he is willing to

implement the policy is larger.

How can the interest group manipulate this kind of executive? Knowing the policy-maker’s best-

response, the intervention strategy mimics that of the baseline model, as laid out in the following

proposition.

Proposition 6. When f(ρ) = 1− ρ, there are two cases.

If E is ex-ante relatively cynical (α ≤ 0.8), I chooses the following distribution of posterior beliefs

• When 0 < p < cα, then a = cα b = 0 x = pcα

• When p > cα, then I does not intervene thus posterior equals prior

If E is ex-ante relatively benevolent (α ≥ 0.8), I chooses the following distribution of posterior

beliefs

• When p < cα, then a = cα b = 0 x = pcα

• When cα < p < cα, then a = cα b = cα x =p−cα1−2cα

• When p > cα, then I does not intervene thus posterior equals prior

When f(ρ) = ρ, for all α ∈ [0.5, 1], I chooses the following distribution of posterior beliefs

• When p < cα, then a = cα b = 0 x = pcα

• When cα < p < cα, then a = cα b = cα x =p−cα1−2cα

• When p > cα, then I does not intervene thus posterior equals prior

Considering the fact that there are more cases to consider, the intervention strategy is analogous

to that of previous sections and follows the same strategic logic. Given this equilibrium, let’s explore

two implications for social welfare.

Proposition 7. Welfare.

17

Figure 4: Voters’ welfare when executive has policy preferenceI and E opposed: f(ρ) = 1− ρ I and E aligned : f(ρ) = ρ

The probability that the policy is implemented is graphed in the top. Voters’ welfare, the probability that the policymatches the state, is graphed against the prior p in the middle and against the degree of benevolence α in the bottom. Thecase of f(ρ) = 1− ρ is graphed in the left, and the case of f(ρ) = ρ in the right. All simulations use π = q = 1

2. In the top

and middle α = 0.7, and in the bottom p = 0.5.

18

1. When interest groups have preferences aligned with those of the executive, public intervention

can only hurt voters. In addition, the more biased a policy-maker in this situation, the lower

the expected voters’ welfare.

2. When interest groups have preferences opposed to those of the executive, public interventions

can only benefit voters. In addition, the more biased a policy-maker in this situation, the

higher the expected voters’ welfare.

Notice from the Lemma 3 that policy-makers fail to implement policies that they have a bias

for because they are also partly interested in matching the state. What interest groups can do for

policy-makers of similar preferences is to allow them to implement the policy in situations where

absent the intervention they wouldn’t, just as they have done in the previous scenarios that have

been studied so far. On the other hand, when the policy-maker has opposite preferences to those

of the interest group, two things can occur; when the policy-maker is biased enough, the act of

persuasion requires so much information revelation that the policy-maker is left with a clear-cut safe

choice, that is one that most likely will match the state; when the policy-maker is not too biased

with respect to voters, the act of persuasion does the opposite than the aligned-preferences case, it

smoothes out the bias of the policy-maker mimicking the choice of a fully benevolent policy-maker.

Figure 4 illustrates the expected policy decision and welfare. Among others, this analysis

reinforces the fact that persuasion through the provision of public information may benefit voters,

however biased the providers of this information are or however unfit or incongruent the policy-

maker is. In this particular case, voters benefit the more biased is a policy-maker whose preferences

are opposed to interest groups, precisely because this forces interest groups to release enormous

amounts of information in order to persuade him, which ultimately increases the probability that

the policy most benefiting voters is chosen.

5 Discussion

This paper has understood the organization of public interventions −such as commissioning scien-

tific studies and organizing public protests− as the design of public signals. This paper has provided

a number of useful results regarding what we should expect strategically designed public interven-

tions to look like, the mechanism through which they may influence policy, and the implications

19

for voters’ welfare. We can summarize these insights in three conclusions.

First, if public interventions are strategically designed, it is very likely that the interventions we

observe in the real world are either going to fail completely or succeed completely, specially when

the policy-maker is perceived to be of low valence. The more capable the policy-maker, the more

likely it is that we observe intermediate levels of success, as measured by the informative impact of

the intervention and the ultimate policy decision.

Second, even when the only mechanism through which an intervention can influence policy

is through the provision of unbiased information, voters will not always benefit from allowing

public interventions. Public interventions will always benefit voters when the policy-maker is a

benevolent planner. When the policy-maker has interests of his own −such as career concerns or

ideology− voters will benefit only if the policy-maker is hard to persuade, because this requires

more information to be revealed in the public signal. Example of contexts that usually require

more information to be released are when the prior favors the policy that SIGs don’t like, when

the policy-maker’s preferences are opposed to those of the organizers, and when the policy-maker,

however not completely inept, is of low ability. In the opposite cases, SIGs will play on the policy-

maker’s bias to release information only selectively hurting voters’ prospects of achieving a welfare

maximizing policy.

Third, contrary to common wisdom, it is not true that voters will always be better off with

a more able or more congruent policy-maker. This follows from the same logic as the second

conclusion: voters will be better off the harder the policy-maker is to persuade, because this requires

more information to be released. For example, if policy-makers who are extremely able or that have

very little career concerns are not available to choose from, voters will sometimes prefer to delegate

to a policy-maker of relatively low ability or of high reputation concerns because they are expected

to be more likely to choose safe bests (i.e. not follow their private information), which makes

changing their minds require highly informative public signals. In addition, when the intervention

designers are of different ideology than the sitting policy-maker, the more opposed they are, that

is, the more extreme the preferences of the policy preference of the executive, the more accurate

the information that public signals will have to release in order to change the policy-maker’s mind.

Underlying this paper are two key processes that are here treated in reduced form. First, the

exact way in which an interest group can design a mass protest, scientific study or other, in order

20

to produce signals in such a flexible way as required here. Second, the delegation problem faced by

voters which underlies the accountability relationship between the policy-maker and the voters. It

will be left for future research to incorporate these dimensions at full length.

To the extent that present elected officials are facing higher degrees of social pressure as well

as plummeting levels of popular approval, and insofar as voters have ever increasing access to

information, enhancing our understanding of the ways in which public interventions can and have

shaped policy and studying the effects of these on voters’ welfare is of increasing importance, and

the insights produced in this paper are a step in that direction.

References

Alonso, Ricardo and Odilon Camara. 2016. “Persuading voters.” American Economic Review

106(11):3590–3605.

Ashworth, Scott. 2005. “Reputational dynamics and political careers.” Journal of Law, Economics,

and Organization 21(2):441–466.

Ashworth, Scott. 2006. “Campaign finance and voter welfare with entrenched incumbents.” Amer-

ican Political Science Review 100(1):55–68.

Ashworth, Scott. 2012. “Electoral accountability: recent theoretical and empirical work.” Annual

Review of Political Science 15:183–201.

Austen-Smith, David and Jeffrey S Banks. 2002. “Costly signaling and cheap talk in models of

political influence.” European Journal of Political Economy 18(2):263–280.

Austen-Smith, David and John R Wright. 1994. “Counteractive lobbying.” American Journal of

Political Science pp. 25–44.

Battaglini, Marco. 2017. “Public protests and policy making.” The Quarterly Journal of Economics

132(1):485–549.

Boehmke, Frederick J, Sean Gailmard, John Wiggs Patty et al. 2006. “Whose ear to bend? In-

formation sources and venue choice in policy-making.” Quarterly Journal of Political Science

1(2):139–169.

21

Brocas, Isabelle and Juan D Carrillo. 2007. “Influence through ignorance.” The RAND Journal of

Economics 38(4):931–947.

Canes-Wrone, Brandice, Michael C Herron and Kenneth W Shotts. 2001. “Leadership and pander-

ing: A theory of executive policymaking.” American Journal of Political Science pp. 532–550.

Coate, Stephen. 2004a. “Pareto-improving campaign finance policy.” American Economic Review

94(3):628–655.

Coate, Stephen. 2004b. “Political competition with campaign contributions and informative adver-

tising.” Journal of the European Economic Association 2(5):772–804.

Daley, Brendan and Erik Snowberg. 2009. “Even if it is not bribery: the case for campaign finance

reform.” The Journal of Law, Economics, & Organization 27(2):324–349.

de Mesquita, Ethan Bueno and Dimitri Landa. 2015. “Political accountability and sequential

policymaking.” Journal of Public Economics 132:95–108.

Dziuda, Wioletta and Christian Salas. 2018. Communication with detectable deceit. Technical

report mimeo.

Fearon, James D. 1999. “Electoral accountability and the control of politicians: selecting good types

versus sanctioning poor performance.” Democracy, accountability, and representation 55:61.

Gailmard, Sean and John W Patty. 2007. “Slackers and zealots: Civil service, policy discretion,

and bureaucratic expertise.” American Journal of Political Science 51(4):873–889.

Gehlbach, Scott and Konstantin Sonin. 2014. “Government control of the media.” Journal of Public

Economics 118:163–171.

Grossman, Gene M and Elhanan Helpman. 2001. Special interest politics. MIT press.

Hirsch, Alexander V and B Pablo Montagnes. 2015. “The Lobbyists Dilemma: Gatekeeping and the

Profit Motive.” Working pa-per, California Institute of Technology, http://people. hss. caltech.

edu/ avhirsch/HirschMon-LobbyistsDilemma-Wall is. pdf .

Holmstrom, Bengt. 1999. “Managerial incentive problems: A dynamic perspective.” The Review

of Economic studies 66(1):169–182.

22

Kamenica, Emir and Matthew Gentzkow. 2011. “Bayesian persuasion.” American Economic Review

101(6):2590–2615.

Kartik, Navin and Richard Van Weelden. 2014. “Informative Cheap Talk in Elections.” Unpublished

mansucript .

Levy, Gilat. 2004. “Anti-herding and strategic consultation.” European Economic Review 48(3):503–

525.

Lohmann, Susanne. 1993. “A signaling model of informative and manipulative political action.”

American Political Science Review 87(2):319–333.

Lohmann, Susanne. 1994. “Information aggregation through costly political action.” The American

Economic Review 84(3):518–530.

Maskin, Eric and Jean Tirole. 2004. “The politician and the judge: Accountability in government.”

American Economic Review 94(4):1034–1054.

Morris, Stephen. 2001. “Political correctness.” Journal of Political Economy 109(2):231–265.

Patty, John W. 2009. “The politics of biased information.” The Journal of Politics 71(2):385–397.

Persson, Torsten and Guido Tabellini. 2000. Political economics: explaining public policy. Cam-

bridge, MA: The MIT Press.

Prat, Andrea. 2002. “Campaign advertising and voter welfare.” The Review of Economic Studies

69(4):999–1017.

Ram, Manish, Michael Child, Arman Aghahosseini, Dmitrii Bogdanov, Alena Poleva and Christian

Breyer. 2017. “Comparing electricity production costs of renewables to fossil and nuclear power

plants in G20 countries.”.

Schnakenberg, Keith E. 2017. “Informational lobbying and legislative voting.” American Journal

of Political Science 61(1):129–145.

Vanberg, Christoph. 2008. “One Man, One Dollar? Campaign contribution limits, equal influence,

and political communication.” Journal of Public Economics 92(3-4):514–531.

23

A Proofs

In this section I prove numbered lemmas and propositions, in the order in which they appear in the text.

A.1 Lemma 1

Let E’s belief that the policy is correct, after observing the public signal’s realization, be Pr(ω = 1|s) = c.

Let’s first calculate some useful statistics.

Pr(y = 0|t = 0) = Pr(y = 0, ω = 0|t = 0) + Pr(y = 0, ω = 1|t = 0)

= Pr(y = 0|ω = 0, t = 0) Pr(ω = 0|t = 0) + Pr(y = 0|ω = 1, t = 0) Pr(ω = 1|t = 0)

= Pr(y = 0|ω = 0, t = 0) Pr(ω = 0) + Pr(y = 0|ω = 1, t = 0) Pr(ω = 1)

= q(1− c) + (1− q)c

Pr(ω = 0|y = 0, t = 0) =Pr(y = 0|ω = 0, t = 0) Pr(ω = 0|t = 0)

Pr(y = 0|t = 0)

=Pr(y = 0|ω = 0, t = 0) Pr(ω = 0)

q(1− c) + (1− q)c

=q(1− c)

q(1− c) + (1− q)c

Pr(ω = 0|y = 1, t = 0) =Pr(y = 1|ω = 0, t = 0) Pr(ω = 0|t = 0)

Pr(y = 1|t = 0)

=Pr(y = 1|ω = 0, t = 0) Pr(ω = 0)

1− (q(1− c) + (1− q)c)

=(1− q)(1− c)

cq + (1− c)(1− q)

Pr(t = 1|y = 0) =Pr(y = 0|t = 1) Pr(t = 1)

Pr(y = 0|t = 1) Pr(t = 1) + Pr(y = 0|t = 0) Pr(t = 0)

=Pr(ω = 0) Pr(t = 1)

Pr(ω = 0) Pr(t = 1) + [q(1− c) + (1− q)c] Pr(t = 0)

=(1− c)π

(1− c)π + [q(1− c) + (1− q)c] (1− π)

Pr(t = 1|y = 1) =Pr(y = 1|t = 1) Pr(t = 1)

Pr(y = 1|t = 1) Pr(t = 1) + Pr(y = 1|t = 0) Pr(t = 0)

=Pr(ω = 1) Pr(t = 1)

Pr(ω = 1) Pr(t = 1) + [1− (q(1− c) + (1− q)c)] Pr(t = 0)

=cπ

cπ + [qc+ (1− c)(1− q)] (1− π)

24

After observing his private signal’s realization y, E’s updated belief regarding the state of the world is:

Pr(ω = 0|y = 0) = Pr(ω = 0, t = 1|y = 0) + Pr(ω = 0, t = 0|y = 0)

= Pr(ω = 0|t = 1, y = 0) Pr(t = 1|y = 0) + Pr(ω = 0|t = 0, y = 0) Pr(t = 0|y = 0)

= 1 · (1− c)π(1− c)π + [q(1− c) + (1− q)c] (1− π)

+q(1− c)

q(1− c) + (1− q)c ·[q(1− c) + (1− q)c] (1− π)

(1− c)π + [q(1− c) + (1− q)c] (1− π)

=(1− c)

(1− c)π + [q(1− c) + (1− q)c] (1− π)

[π +

q

q(1− c) + (1− q)c · [q(1− c) + (1− q)c] (1− π)

]=

(1− c) [π + (1− π)q]

(1− c) [π + (1− π)q] + c(1− π)(1− q)

Pr(ω = 1|y = 1) = Pr(ω = 1, t = 1|y = 1) + Pr(ω = 1, t = 0|y = 1)

= Pr(ω = 1|t = 1, y = 1) Pr(t = 1|y = 1) + Pr(ω = 1|t = 0, y = 1) Pr(t = 0|y = 1)

= 1 · cπ

cπ + [qc+ (1− c)(1− q)] (1− π)+

cq

cq + (1− c)(1− q) ·[qc+ (1− c)(1− q)] (1− π)

cπ + [qc+ (1− c)(1− q)] (1− π)

=c

cπ + [qc+ (1− c)(1− q)] (1− π)

[π +

q

cq + (1− c)(1− q) · [qc+ (1− c)(1− q)] (1− π)

]=

c [π + (1− π)q]

c [π + (1− π)q] + (1− c)(1− q)(1− π)

Now, if E receives y = 1, he chooses ρ = 1 if

Pr(ω = 1|y = 1) > Pr(ω = 0|y = 1)

c [π + (1− π)q]

c [π + (1− π)q] + (1− c)(1− q)(1− π)>

(1− c)(1− q)(1− π)

c [π + (1− π)q] + (1− c)(1− q)(1− π)

which is satisfied if and only if

c [π + (1− π)q] > (1− c)(1− q)(1− π)

cπ + c(1− π)q > (1− q)(1− π)− c(1− q)(1− π)

cπ + c(1− π)q > (1− q)(1− π)− c(1− π) + cq(1− π)

cπ + c(1− π) > (1− q)(1− π)

c > (1− q)(1− π).= cπ

If, however, E receives y = 0, he chooses ρ = 0 if

Pr(ω = 0|y = 0) > Pr(ω = 1|y = 0)

(1− c) [π + (1− π)q]

(1− c) [π + (1− π)q] + p(1− π)(1− q) >c(1− π)(1− q)

(1− c) [π + (1− π)q] + c(1− π)(1− q)

25

which is satisfied if and only if

(1− c) [π + (1− π)q] > c(1− π)(1− q)

π + (1− π)q − cπ − c(1− π)q > c(1− π)(1− q)

π + (1− π)q > c(1− π)(1− q) + c(1− π)q + cπ

cπ.= π + (1− π)q > c

Finally, notice that cπ < cπ and cπ = 1− cπ.

A.2 Proposition 1

Let’s first calculate a few useful statistics. Suppose that Pr(ω = 1|s) = c. Then:

Pr(y = 1|ω = 1) = Pr(y = 1|ω = 1, t = 1) Pr(t = 1) + Pr(y = 1|ω = 1, t = 0) Pr(t = 0)

= 1 · π + q(1− π)

Pr(y = 1|ω = 0) = Pr(y = 1|ω = 0, t = 1) Pr(t = 1) + Pr(y = 1|ω = 0, t = 0) Pr(t = 0)

= 0 · π + (1− q)(1− π)

Pr(y = 1) = Pr(y = 1|ω = 1) Pr(ω = 1) + Pr(y = 1|ω = 0) Pr(ω = 0)

= [π + (1− π)q]c+ (1− π)(1− q)(1− c)

= c[1− 2(1− π)(1− q)] + (1− π)(1− q)

Start by noticing that when p is high enough so that E wishes to implement the policy regardless of his signal,

organizing an intervention can only hurt I by revealing less support for the policy than originally expected. Hence,

no intervention will be run or, equivalently, a fully uninformative intervention will.

Let c be the belief above which E implements the policy regardless of his signal and c the belief below which E

does not implement the policy regardless of his signal. By Lemma 2, in the baseline model we have that c = π+(1−π)q

and c = (1−π)(1−q). The logic of the proof, however, does not depend on the specific characterization of the cutoffs,

and in the proof of other propositions we will work with other cutoffs, hence the proof is presented with c and c.

When p > c, I does not intervene. Consider then the segment of p < c.

Start supposing that I will design one type of intervention regardless of the specific prior inside the range [0, c].

Recall the notation Pr(ω = 1|s = 1) = a, Pr(ω = 1|s = 0) = b and Pr(s = 1) = x. From bayes plausibility, we have

that b ≤ p ≤ a and x = p−ba−b . Additionally, from the structure of E’s best-response (e.g. from Lemma 2) we know

26

that21

Pr(ρ = 1|s = 1) =

1 if a = c

Pr(y = 1|s = 1) if a ∈ (c, c)

0 if a ≤ c

and analogous for

Pr(ρ = 1|s = 0) =

1 if b = c

Pr(y = 1|s = 0) if b ∈ (c, c)

0 if b ≤ c

First let’s set b.

If a = c and b < c, then I’s expected payoff is

E[ρ] = Pr(s = 0)E[ρ|s = 0] + Pr(s = 1)E[ρ|s = 1]

=a− pa− b · 0 +

p− ba− b · 1

which is decreasing in b because p ≤ a. So I would choose b = 0, and E[ρ] = pc.

If a ∈ [c, c] and b < c, then I’s expected payoff is

E[ρ] = Pr(s = 0)E[ρ|s = 0] + Pr(s = 1)E[ρ|s = 1]

=a− pa− b · 0 +

p− ba− b · Pr(y = 1|s = 1)

=p− ba− b · [[π + (1− π)q]a+ (1− π)(1− q)(1− a)]

which is decreasing in b because p ≤ a. So in this range I would again choose b = 0.

If a, b ≤ c, together with b ≤ p ≤ a, we have that E[ρ] = 0 unless a = c, in which case we already know that

b = 0.

Suppose now that a, b ∈ [c, π + (1− π)q], then I’s expected payoff is

E[ρ] = Pr(s = 0)E[ρ|s = 0] + Pr(s = 1)E[ρ|s = 1]

=a− pa− b · Pr(y = 1|s = 0) +

p− ba− b · E[ρ|s = 1]

For ease of notation, let v = Pr(y = 1|s = 0) and w = E[ρ|s = 1]. If a < c then w = Pr(y = 1|s = 1) and if a = c

then w = 1. In both cases v ≤ w, in the former because b ≤ a. Then, we have

E[ρ] =(a− p)va− b +

(p− b)wa− b =

(a− p)v + pw

a− b − bw

a− b21The term Pr(y = 1|s) means the probability that y = 1 is observed having updated the prior belief regarding

the state according to the public signal s.

27

Differentiating with respect to b we have

∂E[ρ]

∂b=

(a− p)v + pw

(a− b)2 − w(a− b) + bw

(a− b)2

=(a− p)v + pw − wa

(a− b)2

=(a− p)(v − w)

(a− b)2

≤ 0

because p ≤ a and v ≤ w. So I would choose b = c.

So far we have that, for any a, I chooses b = 0 whenever b < c and b = c whenever b ∈ [c, c].

Let’s now study a. From above we know that when a ≤ c, I chooses a = c. Consider then the case when a ∈ [c, c].

Suppose a ∈ [c, c] and b = 0. We have two cases. When a = c, from above we know that

E[ρ] =p

c

When a ∈ [c, c), we have that

E[ρ] = Pr(s = 0)E[ρ|s = 0] + Pr(s = 1)E[ρ|s = 1]

=a− pa· 0 +

p

a· Pr(y = 1|s = 1)

=p

a· [[π + (1− π)q]a+ (1− π)(1− q)(1− a)]

=p

a· [a[π + (1− π)q − (1− π)(1− q)] + (1− π)(1− q)]

= p

[1− 2(1− π)(1− q) +

(1− π)(1− q)a

]This expression is decreasing in a, hence the largest I’s payoff can be is when a = p, that is, when no experiment is

run. In this case, the payoff is simply

E[ρ] =

0 if p < c

Pr(y = 1) = p [1− 2(1− π)(1− q)] + (1− π)(1− q) if p ∈ [c, c]

When p < c the intervention is clearly better than nothing. Whether an intervention is better for p ∈ [c, c] depends

on the specific prior and the specific characterization of cutoffs c and c.

However, notice that the intervention is least attractive when p = (1− π)(1− q), because the slope of Pr(y = 1)

with respect to p is less than that of pπ+(1−π)q .22 Hence, a sufficient condition for the experiment to be preferred by

I to not intervening is that pπ+(1−π)q > Pr(y = 1) when p = (1− π)(1− q), which solves to

π <

1√2− q

1− q = π

22We wish to show that 1π+(1−π)q > 1−2(1−π)(1− q). Replacing (1−π)(1− q) by 1− [π + (1− π)q], multiplying

through the denominator, moving everything to the left-hand and factorizing adequately, we get 1− [π + (1− π)q]2 +[π + (1− π)q] [1− π + (1− π)q] > 0, which is true because π + (1− π)q < 1.

28

Finally, suppose that a ∈ [c, c] and b = c.

If a = c I’s payoff is Pr(s = 0) Pr(y = 1|s = 0) + Pr(s = 1) · 1, whereas if a ∈ [c, c) I’s payoff is Pr(s = 0) Pr(y =

1|s = 0) + Pr(s = 1) Pr(y = 1|s = 1). Since Pr(y = 1|s = 1) < 1, I will set a = c. In this case, I’s payoff would be

x2z [1− x]− 1

z − x + p1− 2x(1− x)

z − x (1)

where for ease of notation we denote x = (1− π)(1− q) and z = π + (1− π)q.

There are two cases then. I will either set b = 0 and a = c for all p ∈ [0, c], or will set b = 0 and a = c for

p ∈ [0, c] and b = c and a = c for p ∈ [c, c].

Using now the specific characterization of cutoffs c and c: the payoff in the first case is pπ+(1−π)q and the payoff

in the second case is 2p(π + (1 − π)q) in the first range and (1) in the second range. We have that pπ+(1−π)q >

2p(π + (1− π)q) if

π <

1√2− q

1− q

Notice that the interventions for both cases in the range p ∈ [(1 − π)(1 − q), π + (1 − π)q] have a = π + (1 − π)q.

Since p enters linearly in both cases, we need only to compare the payoffs at any point in range p < π + (1 − π)q.

Use p = (1 − π)(1 − q). Again for ease of notation, call x = (1 − π)(1 − q) and z = π + (1 − π)q (and recall that

z = 1− x). Thus, we have that pπ+(1−π)q >(1) if and only if

x

1− x > x2z [1− x]− 1

z − x + x1− 2x(1− x)

z − x

Moving everything to the left-hand-side and factoring we have the equivalent condition

−1 + 4x− 2x2

1− x > 0

which is true if the numerator is positive, which happens if z2 < 12, which is equivalent to

π <

1√2− q

1− q.= π

A.3 Proposition 2

For ease of notation, call x = (1− π)(1− q) and z = π + (1− π)q (and recall that z = 1− x).

Without interventions, the probability that the policy matches the state is

Pr(ρ = ω) =

1− p if p < (1− π)(1− q)

Pr(y = ω) = π + (1− π)q if p ∈ [(1− π)(1− q), π + (1− π)q]

p if p > π + (1− π)q

29

When interventions are allowed, we have two cases. First suppose that π <1√2−q

1−q . Let’s calculate some useful

statistics first.

Pr(s = 0|ω = 0) =Pr(ω = 0|s = 0) Pr(s = 0)

Pr(ω = 0)=

1 ·(1− p

z

)1− p

Pr(s = 1|ω = 1) =Pr(ω = 1|s = 1) Pr(s = 1)

Pr(ω = 1)=z ·(pz

)p

= 1

When p ≤ π + (1− π)q, the probability that the policy matches the state is

Pr(ρ = ω) = Pr(ω = 0) Pr(ρ = 0|ω = 0) + Pr(ω = 1) Pr(ρ = 1|ω = 1)

= (1− p) Pr(s = 0|ω = 0) + pPr(s = 1|ω = 1)

= (1− p) ·(

1− pz

1− p

)+ p · 1

= 1− p(xz

)When p > π + (1− π)q, E always implements the policy, hence the probability that the action matches the state is

p. Notice then that, for π <1√2−q

1−q , welfare is strictly greater when allowing for interventions (because x < z).

Now suppose that π >1√2−q

1−q . Let’s start with p < (1− π)(1− q). Some useful statistics:

Pr(s = 0|ω = 0) =Pr(ω = 0|s = 0) Pr(s = 0)

Pr(ω = 0)=

1 ·(1− p

x

)1− p

Pr(s = 1|ω = 0) =Pr(ω = 0|s = 1) Pr(s = 1)

Pr(ω = 0)=z ·(px

)1− p

Pr(s = 1|ω = 1) =Pr(ω = 1|s = 1) Pr(s = 1)

Pr(ω = 1)=x ·(px

)p

= 1

Thus, the probability that the policy matches the state is23

Pr(ρ = ω) = Pr(ω = 0) Pr(ρ = 0|ω = 0) + Pr(ω = 1) Pr(ρ = 1|ω = 1)

= (1− p) [Pr(s = 0|ω = 0) · 1 + Pr(s = 1|ω = 0) Pr(y = 0|s = 1, ω = 0)] + p [Pr(s = 1|ω = 1) Pr(y = 1|s = 1, ω = 1)]

= (1− p)

[1− p

x

1− p +z(px

)1− p · z

]+ p [z]

= 1− p

x+z2p

x+ pz

= 1 +p

x

[−1 + z2 + xz

]= 1 +

p

x[−1 + z(z + x)]

= 1 +p

x[−1 + z]

= 1− p

23Crucially, recall that we assume that the process through which a signal s is realized and that which realizesa signal y, however informant of the same underlying state of the world, are independent. Therefore, Pr(y|s, ω) =Pr(y|ω).

30

Let’s continue now with p ∈ [(1− π)(1− q), π + (1− π)q]. Some useful statistics:

Pr(s = 0|ω = 0) =Pr(ω = 0|s = 0) Pr(s = 0)

Pr(ω = 0)=z ·(z−pz−x

)1− p

Pr(s = 0|ω = 1) =Pr(ω = 1|s = 0) Pr(s = 0)

Pr(ω = 1)=x ·(z−pz−x

)p

Pr(s = 1|ω = 1) =Pr(ω = 1|s = 1) Pr(s = 1)

Pr(ω = 1)=z ·(p−xz−x

)p

Thus, the probability that the policy matches the state is

Pr(ρ = ω) = Pr(ω = 0) Pr(ρ = 0|ω = 0) + Pr(ω = 1) Pr(ρ = 1|ω = 1)

= (1− p) [Pr(s = 0|ω = 0) Pr(y = 0|s = 0, ω = 0)] + p [Pr(s = 0|ω = 1) Pr(y = 1|s = 0, ω = 1) + Pr(s = 1|ω = 1) · 1]

= (1− p)

z ·(z−pz−x

)1− p · z

+ p

x ·(z−pz−x

)p

· z +z ·(p−xz−x

)p

=

z

z − x [(z − p)(z + x) + (p− x)]

=z

z − x [z − x]

= z

When p > π + (1− π)q, E always implements the policy, hence the probability that the action matches the state is

p. Notice that when π >1√2−q

1−q welfare coincides with the case where no intervention exists.

A.4 Proposition 3

First establish two points.

1. For a fix π, welfare is strictly higher under the single experiments than under two experiments as long as

p < π + (1− π)q (after that, welfare is equal). First, clearly 1− p(

(1−π)(1−q)π+(1−π)q

)> 1− p. Second, notice that

1− p(

(1− π)(1− q)π + (1− π)q

)> π + (1− π)q

1− p(

1− (π + (1− π)q)

π + (1− π)q

)> π + (1− π)q

1− (π + (1− π)q) > p

(1− (π + (1− π)q)

π + (1− π)q

)π + (1− π)q > p

2. When π changes, two things occur: welfare changes given the intervention strategy and the actual intervention

strategy changes. Within intervention strategy, welfare is always increasing in π. This is because 1− p and p

31

are fixed in π, π + (1− π)q is increasing in π and

1− p(

1

π + (1− π)q− 1

)is also increasing in π.

Now notice that, at π =1√2−q

1−q , I switches intervention strategy and, from point 1. above, we know that at the

switch point welfare will be higher under the sole intervention strategy. From point 2., decreasing π from π decreases

welfare. Thus, we look for the π below witch the sole-intervention strategy provides less welfare that under the

two-intervention strategy at π.

First of all, notice that π + (1− π)q = 1√2

and (1− π)(1− q) =√2−1√2

.

When p < (1− π)(1− q), we have

1− p(

1

π∗ + (1− π∗)q − 1

)= 1− p

1 =

(1

π∗ + (1− π∗)q − 1

)π∗ + (1− π∗)q =

1

2

π∗ =12− q

1− q

When p ∈ [(1− π)(1− q), π + (1− π)q], we have

1− p(

1

π∗∗ + (1− π∗∗)q − 1

)= π + (1− π)q

1− p(

1

π∗∗ + (1− π∗∗)q − 1

)=

1√2√

2− 1

p√

2=

(1

π∗∗ + (1− π∗∗)q − 1

)π∗∗ + (1− π∗∗)q =

p√

2

p√

2 +√

2− 1

π∗∗ =p√

2− q[√

2(1 + p)− 1]

(1− q)[√

2(1 + p)− 1]

Hence, we have

If p <√2−1√2

then welfare for π ∈ [π∗, π] is higher than at π

If p ∈[√

2−1√2, 1√

2

]then welfare for π ∈ [π∗∗, π] is higher than at π

If p > 1√2

then welfare is weakly increasing in π

32

A.5 Lemma 2

First, let’s calculate some useful statistics. Suppose that the signal realization from the intervention allows everyone

to update their belief regarding the state to Pr(ω = 1, s) = c. Based on calculations made above, we have

Pr(ω = 0|y = 0) =(1− c) 3

4

(1− c) 34

+ c 14

=3− 3c

3− 2c

Pr(ω = 1|y = 1) =c 34

c 34

+ (1− c) 14

=3c

2c+ 1

Pr(y = 1|t = 0) = Pr(y = 1|ω = 1, t = 0) Pr(ω = 1|t = 0) + Pr(y = 1|ω = 0, t = 0) Pr(ω = 0|t = 0)

= qc+ (1− q)(1− c)

Pr(y = 0|t = 0) = Pr(y = 0|ω = 1, t = 0) Pr(ω = 1|t = 0) + Pr(y = 0|ω = 0, t = 0) Pr(ω = 0|t = 0)

= (1− q)c+ q(1− c)

Denote the updated belief regarding the quality of the executive, given a signal realization from the intervention and

the executive’s choice a, as Pr(t = 1|a).

Suppose that, given the fundamentals of the world, society expects E to follow his signal. If he does, the action

E takes reveals the signal y he received and hence helps voters update their belief regarding the ability of E. Thus,

using bayes rule we have

Pr(t = 1|ρ = 1) =Pr(ρ = 1|t = 1) Pr(t = 1)

Pr(ρ = 1|t = 1) Pr(t = 1) + Pr(ρ = 1|t = 0) Pr(t = 0)

=Pr(y = 1|t = 1) Pr(t = 1)

Pr(y = 1|t = 1) Pr(t = 1) + Pr(y = 1|t = 0) Pr(t = 0)

=cπ

cπ + [qc+ (1− q)(1− c)] (1− π)

=c

c+ 12

And similarly we have

Pr(t = 0|ρ = 0) =[(1− q)c+ q(1− c)] (1− π)

[(1− q)c+ q(1− c)] (1− π) + (1− c)π =12

32− c

Pr(t = 1|ρ = 0) =(1− c)π

(1− c)π + [(1− q)c+ q(1− c)] (1− π)=

1− c32− c

Pr(t = 0|ρ = 1) =[qc+ (1− q)(1− c)] (1− π)

cπ + [qc+ (1− q)(1− c)] (1− π)=

12

c+ 12

33

Now, when E receives signal y = 0 he takes action ρ = 0 if and only if

θ Pr(ω = 0|y = 0) + (1− θ) E [Pr(t = 1|ρ = 0)|y = 0] > θ Pr(ω = 1|y = 0) + (1− θ) E [Pr(t = 1|ρ = 1)|y = 0]

θ3(1− c)3− 2c

+ (1− θ) 1− c32− c

> θc

3− 2c+ (1− θ) c

c+ 12

θ

[3− 4c

3− 2c

]> (1− θ)

[c

c+ 12

− 1− c32− c

]θ

[3− 4c

3− 2c

]> (1− θ)

[c− 1

212(3− 2c)(c+ 1

2)

]θ(3− 4c)

(c+

1

2

)> 2(1− θ)

(c− 1

2

)0 > 4θc2 + (2− 3θ)c− 1− θ

2

which solves to

c <3θ − 2 +

√4 + 4θ + 17θ2

8θ

.= cθ

Now, when E receives signal y = 1 he takes action ρ = 1 if and only if

θ Pr(ω = 1|y = 1) + (1− θ) E [Pr(t = 1|ρ = 1)|y = 1] > θ Pr(ω = 0|y = 1) + (1− θ) E [Pr(t = 1|ρ = 0)|y = 1]

θ3c

2c+ 1+ (1− θ) c

c+ 12

> θ1− c2c+ 1

+ (1− θ) 1− c32− c

(1− θ)[

c

c+ 12

− 1− c32− c

]> θ

1− 4c

2c+ 1

(1− θ)

[c− 1

212

(2c+ 1)(32− c)] > θ

1− 4c

2c+ 1

2(1− θ)(c− 1

2

)> θ (1− 4c)

(3

2− c)

0 > 4θc2 − (2 + 5θ)c+ 1 +θ

2

which solves to

c >5θ + 2−

√4 + 4θ + 17θ2

8θ

.= cθ

Notice that cθ < cθ, cθ = 1 − cθ, and when θ → 0 then cθ → 12

and cθ → 12, and when θ = 1 then cθ = 3

4and

cθ = 14.

Finally, when c < cθ and E deviates from ρ = 0 and when c > cθ and E deviates from ρ = 1 are both off-the-

equilibrium path. In both cases E will not deviate provided the right off-path beliefs Pr(t = 1|c < cθ, ρ = 1) and

Pr(t = 1|c > cθ, ρ = 0).

For what follows, note that outside the range [cθ, cθ] in equilibrium E does not follow his signal thus society does

not update their beliefs regarding E’s ability. That is, Pr(t = 1|c < cθ, ρ = 0) = Pr(t = 1|c > cθ, ρ = 1) = π.

34

Suppose that c < cθ. If E receives y = 0 he chooses ρ = 0 if

θ Pr(ω = 0|y = 0) + (1− θ) Pr(t = 1|c < cθ, ρ = 0) > θ Pr(ω = 1|y = 0) + (1− θ) Pr(t = 1|c < cθ, ρ = 1)

θ3(1− c)3− 2c

+ (1− π) π > θc

3− 2c+ (1− π) Pr(t = 1|c < cθ, ρ = 1)

θ

1− θ3− 4c

3− 2c+ π > Pr(t = 1|c < cθ, ρ = 1)

Similarly, if E receives y = 1 he chooses ρ = 0 if

θ

1− θ1− 4c

2c+ 1+ π > Pr(t = 1|c < cθ, ρ = 1) (2)

Suppose now that c > cθ. If E receives y = 0 he chooses ρ = 1 if

θ Pr(ω = 1|y = 0) + (1− θ) Pr(t = 1|c > cθ, ρ = 1) > θ Pr(ω = 0|y = 0) + (1− θ) Pr(t = 1|c > cθ, ρ = 0)

θc

3− 2c+ (1− π) π > θ

3(1− c)3− 2c

+ (1− π) Pr(t = 1|c > cθ, ρ = 0)

θ

1− θ4c− 3

3− 2c+ π > Pr(t = 1|c > cθ, ρ = 0) (3)

Similarly, if E receives y = 1 he chooses ρ = 1 if

θ

1− θ4c− 1

2c+ 1+ π > Pr(t = 1|c > cθ, ρ = 0)

Since 4c−12c+1

> 4c−33−2c

for c < 1, sufficient conditions on off-path beliefs in order to sustain this equilibrium are (2)

and (3).

A.6 Proposition 4

The structure of the intervention designs that will be used are already proved in proposition 1 above. All that is left

to prove is when I will switch intervention strategies.

I will choose a single-intervention design, as opposed to two-intervention design as long as

p

c

∣∣∣p=c

> Pr(y = 1|p = c)

c

c>

c

2+

1

4

2c2 − 7c+ 4 > 0

which solves to

θ <7 + 3

√17

26

.= θ

which is roughly 34.

35

A.7 Proposition 5

This proposition follows the same logic as proposition 2 and 3.

There are two strategies that I can ex-ante follow: (i) a sole-intervention, regardless of p ∈ [0, cθ], or (ii) two

different interventions depending on p.

When p > cθ both cases generate the same welfare p. When p ∈ [0, cθ], welfare under case (i) is

Pr(ρ = ω) = Pr(ω = 0) Pr(ρ = 0|ω = 0) + Pr(ω = 1) Pr(ρ = 1|ω = 1)

= (1− p) Pr(s = 0|ω = 0) + pPr(s = 1|ω = 1)

= (1− p) ·

(1− p

cθ

1− p

)+ p · 1

= 1− p(cθcθ

)Under case (ii), when p < cθ welfare is

Pr(ρ = ω) = Pr(ω = 0) Pr(ρ = 0|ω = 0) + Pr(ω = 1) Pr(ρ = 1|ω = 1)

= (1− p) [Pr(s = 0|ω = 0) · 1 + Pr(s = 1|ω = 0) Pr(y = 0|s = 1, ω = 0)] + p [Pr(s = 1|ω = 1) Pr(y = 1|s = 1, ω = 1)]

= (1− p)

1− pcθ

1− p +cθ(pcθ

)1− p ·

3

4

+ p

[3

4

]

=1

cθ

[cθ − p+

3

4cθp+

3

4pcθ

]=

1

cθ

[cθ − p+

3

4p

]= 1− p

4cθ

Finally, when p ∈ [cθ, cθ] welfare is

Pr(ρ = ω) = Pr(ω = 0) Pr(ρ = 0|ω = 0) + Pr(ω = 1) Pr(ρ = 1|ω = 1)

= (1− p) [Pr(s = 0|ω = 0) Pr(y = 0|s = 0, ω = 0)] + p [Pr(s = 0|ω = 1) Pr(y = 1|s = 0, ω = 1) + Pr(s = 1|ω = 1) · 1]

= (1− p)

cθ ·(cθ−pcθ−cθ

)1− p · 3

4

+ p

cθ ·(cθ−pcθ−cθ

)p

· 3

4+cθ ·

(p−cθcθ−cθ

)p

=

1

cθ − cθ

[cθ

(3

4− cθ

)− p

(3

4− cθ

)]

First, just as in proposition 3, let’s prove that the point of switch of strategy, θ = θ, strategy (i) produces strictly

higher welfare that strategy (ii). Start in the range p < cθ. We have that

1− p(cθcθ

)> 1− p

4cθ

if and only if 4cθ + cθ − 1 < 0, which at θ = θ is true. Now, compare welfare at the switch in strategy in the range

36

p ∈ [cθ, cθ]. We have that

1− p(cθcθ

)>

1

cθ − cθ

[cθ

(3

4− cθ

)− p

(3

4− cθ

)]if and only if (1− p) + c(p− 2) + c2(4p− 3) + 4c3 > 0, which at θ = θ in the range p ∈ [cθ, cθ) is indeed true.

Second, notice that welfare in the range p < cθ is always higher with an intervention than without, because

1 − p4cθ

> 1 − p is true if and only if cθ >14, which is always true. In the range p ∈ [cθ, cθ], however, the highest

welfare depends on θ as well as the exact p.

In the case where p ∈ [cθ, cθ] and θ > θ, welfare is always lower with interventions than without. To prove this

examine this expression3

4≥ 1

cθ − cθ

[cθ

(3

4− cθ

)− p

(3

4− cθ

)]Notice the right-hand-side is decreasing in p, hence its highest at p = cθ, which replaced solves to

1

cθ − cθ

[3

4(cθ − cθ)

]=

3

4

In the case where p ∈ [cθ, cθ] and θ < θ, welfare can be lower or higher with interventions than without, depending

on θ and p. First, notice that at p = cθ welfare is always higher without an intervention, because 1 − p cθcθ

= 1 − cθwhich is always less than or equal to 3

4. Second, notice that 1− p cθ

cθis highest at p = cθ, and at this point it can be

less or greater than 34. This is because the equation 1− c2θ

cθ− 3

4> 0 is equivalent to 4c2θ + cθ − 1 < 0, which at θ = θ

we already showed that is true, and notice that at θ → 0 (where cθ → 12) is false.

A.8 Lemma 3

First suppose that f(ρ) = 1− ρ.

When E receives signal y = 0 he takes action ρ = 0 if and only if

α Pr(ω = 0|y = 0) + (1− α) · 1 > α Pr(ω = 1|y = 0) + (1− α) · 0

α3(1− c)3− 2c

+ (1− α) · 1 > αc

3− 2c+ (1− α) · 0

which solves to

c <3

2α+ 2= cα

Now, when E receives signal y = 1 he takes action ρ = 1 if and only if

α Pr(ω = 1|y = 1) + (1− α) · 0 > α Pr(ω = 0|y = 1) + (1− α) · 1

α3c

2c+ 1+ (1− α) · 0 > α

1− c2c+ 1

+ (1− α) · 1

which solves to

c >1

6α− 2

.= cα

Now suppose that f(ρ) = ρ.

37

When E receives signal y = 0 he takes action ρ = 0 if and only if

α Pr(ω = 0|y = 0) + (1− α) · 0 > α Pr(ω = 1|y = 0) + (1− α) · 1

α3(1− c)3− 2c

+ (1− α) · 0 > αc

3− 2c+ (1− α) · 1

which solves to

c <6α− 3

6α− 2= cα

Now, when E receives signal y = 1 he takes action ρ = 1 if and only if

α Pr(ω = 1|y = 1) + (1− α) · 1 > α Pr(ω = 0|y = 1) + (1− α) · 0

α3c

2c+ 1+ (1− α) · 1 > α

1− c2c+ 1

+ (1− α) · 0

which solves to

c >2α− 1

2α+ 2

.= cα

A.9 Proposition 6

The structure of the intervention designs that will be used are already proved in proposition 1 above.

A.10 Proposition 7

The proof of this proposition follows the same logic as proposition 2, 3 and 5.

38