The Value of Confidential Policy Information: Persuasion, Transparency, and Influence

Abstract

Transparency of the lobbying process is hailed as an effective means to limit the influence of special interest groups, but should transparency also apply to the information obtained by policy makers (PMs)? This article extends theories of informational lobbying by explicitly modeling the choice of PMs to obtain information before interacting with lobbyists. This approach reveals a new channel for the value of confidentiality: extracting evidence from special interest groups. It shows that, counter-intuitively, the influence of special interest groups can increase as PMs become more expert. These results shed light on the relationship between confidentiality, good governance, and influence.

1. Introduction

Transparent policy making is often considered a defining feature of democracy. When the information available to policy makers (PMs) is easily accessible, the public can scrutinize policy decisions and hold elected representatives accountable.

While governments have started disclosing the identity of external sources of information and the interests they represent, through bills such as the Lobbying Disclosure Act in the USA in 1995 or the Transparency of Lobbying Act in the UK in 2014, there is an ongoing debate about whether their internal sources should remain confidential. For example, the Congressional Research Service (CRS) has defended the confidentiality of its research for many years until eventually agreeing to make its reports publicly available in 2018 (DeBonis 2017; Hayden 2018).¹ One of the arguments advanced by the CRS to defend the confidentiality of its reports is the risk of influence by outsiders: “Widespread public dissemination will almost certainly increase partisan and special interest pressure […]. Such pressure from the public […] could subtly affect the way CRS authors write their reports. Congress may ultimately benefit less from the information in CRS Reports.”² In parallel, a debate arose about how much influence legislators should have on the way this research is carried out. According to a former CRS researcher, following pressure from members of congress “CRS analysts were told not to end their reports with a section titled ‘conclusion.’ That sounded far too definitive and authoritative.” (Kosar 2015).³

In this article, I evaluate the effect of keeping internal information confidential on the influence that special interest groups exert on policies. I extend theories of informational lobbying—the influence of interest groups through the provision of information, rather than through monetary contributions—by explicitly considering PMs’ control over their internal information. This approach reveals a novel channel through which confidentiality can be beneficial: by keeping their own information confidential, PMs can induce special interest groups to provide more evidence. The value of confidentiality to PMs is not driven by reputational concerns or bargaining considerations, and hence can be socially beneficial. Therefore, while preventing the release of CRS reports might have been driven by more prosaic considerations such as the cost of disseminating them or legal protection for its authors, this confidentiality might have had positive effects on the quality of policies. I characterize the PMs’ strategic choices of internal information and show that they can benefit from limiting the precision of their information. To the extent that avoiding authoritative conclusions reduces the precision of the information available, legislators’ suggestions to CRS researchers may have led to better policy decisions.

The relationship between PMs’ choices of internal information and the provision of information by special interest groups is driven by a simple intuition. When interest groups can observe the information already available to PMs, they can produce evidence that is just sufficiently accurate to tilt the policy decision in their favor. When the information available to the government is not publicly available, interest groups form beliefs about the information PMs are most likely to have. These beliefs determine whether interest groups want to offer more or less information: if they believe that PMs are likely to be skeptical about their preferred policy, then they need to offer more evidence. Therefore, PMs should shape their preliminary investigations to let lobbyists believe that they are sufficiently skeptical and that more evidence is needed.

I formalize this intuition to address the following questions. First, when is confidentiality valuable to PMs, and therefore most likely to be used by governments? Second, how does the government’s control over internal investigations affect the influence that special interest groups exert on policy making?

To answer these questions, I consider a model with a single PM and a lobbyist. The PM has to decide whether to enact a new policy that is supported by the lobbyist, but faces uncertainty. In the first stage, the PM chooses the precision of a signal about an unknown state of nature that she receives confidentially. She would like to choose the welfare-maximizing policy given the state, but her limited expertise constrains the precision of her signal. The lobbyist also acquires some independent signals to share with the PM. His expertise is not limited and he can perfectly adjust the precision of his information to persuade the PM to choose his preferred policy.

I show that confidentiality is valuable as it forces the lobbyist to choose an investigation that reveals more evidence than necessary to persuade the PM. By keeping her own signal realizations confidential, the PM strategically creates a situation of asymmetric information which allows her to extract informational rent. This occurs even when her preliminary information would have no effect on her policy choice, in the absence of lobbying.

The value of confidentiality has limits, however. I show that the PM may need to distort her own information in order to induce the lobbyist to provide more evidence than he would like to. These distortions involve reducing the precision of certain conclusions of the investigation, and hence reduce its overall informativeness. There is thus a tradeoff between obtaining information internally and extracting it from external sources.

This tradeoff generates results relevant to two commonly studied policy questions. First, the value of confidentiality is non-monotonic in the PM’s expertise. When government expertise is low, the value of confidentiality increases in expertise, as more expertise allows the PM to extract more evidence from the lobbyist. However, when expertise is high, that value eventually declines as the additional gains from inducing the lobbyist to provide more information are relatively less important, compared to the costs of distorting internal information. In addition, the value of confidentiality is higher for PMs who are initially opposed to the lobbyist’s policy than for allies of the lobbyist.

Second, the influence of the lobbyist on policy can sometimes increase with the government’s expertise. This arises because more expertise can make the PM more likely to choose the lobbyist’s least-preferred policy and therefore makes the lobbyist’s presence even more important to overturn that choice. This occurs despite the fact that expertise improves policy choice and makes the PM better-off. The influence of the lobbyist is also higher on PMs who initially agree with them than on those who initially disagree with them, because the former is easier to re-convince, should they temporarily change their mind after acquiring their own information. These results have both positive implications for evaluating the influence of interest groups, and normative consequences for the optimal design of institutions.

Consider the fact that the resources spent on lobbying in the USA have significantly increased in real terms over the last 15 years, while the budget of the CRS has remained relatively constant (Figure 1). This could suggest that information generation is increasingly being outsourced to external groups whose influence is increasing. An alternative explanation is that even with limited resources, PMs can force lobbyists to provide additional information, and therefore expend more resources, so these changes in fact reflect a loss of influence. Which of these two explanations is correct depends on how these sources of information interact with one another. More generally, whether returns to lobbying expenditures can be interpreted as influence depends on the information available to PMs in a counterfactual world without lobbying. This article offers a framework to think about this counterfactual scenario.

The results also explain why preferences over confidentiality can vary over time and across policy areas as the policy agenda evolves or as more technocratic PMs get elected. These strategic preferences for confidentiality can be positively correlated with the quality of policy making even in the absence of a causal relationship between the two variables. This is consistent with empirical evidence that higher levels of transparency are associated with better governance (e.g., Islam 2006) but suggests that there can be other factors, such as government expertise, moving both of these variables. In fact, the model shows that shining too much light on the policy process can reduce the quality of policy making.

Finally, the article offers a new rationale for the observation that lobbyists often target friendly legislators who already support their policies.⁴ Lobbyists may want to provide information to PMs who already support their proposal because there is a risk that the PMs change their policy choice after acquiring their own information. Since these PMs are easier to re-convince, lobbyists have more to gain from targeting them.

1.1 Related Literature

This article relates to two strands of literature: models of informational lobbying and studies of transparency in political institutions. It shows that these two questions are linked: transparency determines how information flows between PMs and lobbyists which, in turn, affects the information that lobbyists provide.

A large literature has looked at how information is transmitted to legislators by lobbyists (e.g., Potters and van Winden 1992, Austen-Smith and Wright 1992, Rasmusen 1993, Austen-Smith 1993, and Lagerlof 1997). The most closely related papers study how informational lobbying is affected by information already held by PMs. Felgenhauer (2013) shows that expert politicians are not always better than non-experts in the presence of lobbyists. In his model, the expertise of the politician only has an effect when two lobbies compete. By allowing the information to be concealed, I show that even a single lobby can be induced to provide more information as the politician’s expertise increases. Cotton and Dellis (2016) show that informational lobbying can be detrimental if more information provided by lobbyists shifts the focus of a PM toward less important issues and thus reduces the information she collects. This substitution across the two sources of information relies on the existence of multiple policies and the limited capacity of the PM to act on these policies. Substitution arises in my model even with one policy dimension because information can be confidential, so that the PM’s choice of information affects the beliefs of lobbyists and the evidence they provide. Finally, in Ellis and Groll (2020), the tradeoff between acquiring costly information in-house or relying on that provided by lobbyists comes from the difference in their resource constraints. Information is costless in my model and the interaction between the two types of information depends on whether that information is made public or not.⁵

Another closely related paper, Cotton and Li (2018), studies the effect of internal information on monetary lobbying. They show that because a better-informed politician might be harder to sway through contributions, politicians might prefer to reduce the informativeness of their signals. Because they focus on the effect of internal information on monetary rather than informational lobbying, this article is complementary to theirs. In particular, I show that additional internal information can be detrimental even with a benevolent politician rather than one who maximizes contributions.

In the transparency literature, Felgenhauer (2010) and Gailmard and Patty (2019) study the effect of making PMs’ information public.⁶Felgenhauer (2010) finds that confidentiality can be socially beneficial as it can reduce monetary contributions from lobbyists and result in better policies. More internal information is unambiguously valuable as it reduces lobbyists’ influence. Instead, I show that with informational lobbying, more precise information can be detrimental. Gailmard and Patty (2019) find that transparency of the PM’s information can reduce the amount of information transmitted from a bureaucrat. Their focus on bureaucracies, rather than interest groups’ influence, leads them to study optimal delegation rules rather than distortions in the PM’s information.

From a technical perspective, this article builds on the persuasion literature with privately informed receivers, such as Kolotilin (2018) and Guo and Shmaya (2019). These papers show that the receiver’s payoff can be decreasing in the precision of her private information. This article extends their analysis in a simpler setting by showing that the receiver would indeed prefer to keep her information private as long as its precision is limited, and by characterizing the optimal investigation of the receiver. This article mainly differs in its objective, however: to propose a simple model of the relationship between PMs’ information choices and the influence of lobbyists, rather than to add to the theory of persuasion.

The rest of the article is organized as follows. Section 2 introduces the model. Section 3 characterizes the lobbyist’s and the PM’s choice of information and shows that the PM can gain from confidentiality. Section 4 shows that both confidentiality and interest group influence vary non-monotonically in the PM’s expertise and compares these outcomes between allied and opposed PMs. Section 5 discusses some empirical implications of these results and some extensions. Section 6 concludes. All proofs are presented in  Appendix.

2. Model

The lobbyist can credibly commit to this strategy and to revealing s to the PM. I denote the posterior belief of the PM following realizations r and s by $μsr:=P(ω=1|s,r)$⁠.

This bound implies that the PM cannot learn the state of the world perfectly: the posterior beliefs μ^r must belong to an interval $[μ̱,μ¯]⊂[0,1]$⁠. The lowest and highest posterior beliefs that she can induce are: $μ̱=μ0μ0+(1−μ0)B$ and $μ¯=Bμ0Bμ0+(1−μ0)$⁠. The parameter B captures the difference in expertise between the PM and the lobbyist and is public knowledge. The lobbyist’s advantage stems from facing no expertise bound as, in effect, B = + ∞ for him.

I refer to the investigation p such that $p(r0|ω=0)p(r0|ω=1)=p(r1|ω=1)p(r1|ω=0)=B$ as the most informative investigation available to the PM and denote it $p¯$⁠. This investigation induces interim beliefs $μr0=μ̱$ and $μr1=μ¯$⁠.

I consider three possible regimes. Under transparency, the lobbyist observes both the PM’s choice of investigation p and its outcome r. Under partial confidentiality, the lobbyist observes the choice of p but not its outcome r. Finally, under full confidentiality, the lobbyist observes neither p nor r.

Under partial confidentiality, the timing is as follows:

1. the PM publicly chooses a preliminary investigation p;

2. $r∈{r0,r1}$ is realized but only observed by the PM;

3. the lobbyist chooses a persuasion strategy π after observing p;

4. $s∈{s0,s1}$ is publicly realized; and

5. the PM updates her beliefs and chooses $x∈{0,1}$⁠.

Under transparency, the timing is the same, but the lobbyist can observe the realized r and therefore condition π on both p and r. The full confidentiality case is equivalent to both players choosing π and p simultaneously, before observing both realizations r and s.

The equilibrium concept is weak perfect Bayesian equilibrium: the players’ strategies are sequentially rational given their beliefs, and beliefs are updated according to Bayes rule whenever possible.⁸ I focus on pure strategy equilibria within this class.⁹

2.1 Discussion of Assumptions

The assumption that the lobbyist can commit to disclosing his information makes the model more tractable and allows me to focus on the type of evidence that both parties generate, and the influence this information has on policies. While there are some lobbying situations that satisfy this assumption,¹⁰ not all interactions between lobbyists and PMs necessarily involve truthful communication. In the Supplementary Appendix (Section A), I consider an extension of the model in which the lobbyist can lie about his information.¹¹ I show that the main results continue to hold: the PM can benefit from confidential information but the benefits from confidentiality disappear as her information becomes more precise, she can gain from reducing the precision of internal information, and an increase in her expertise can both force the lobbyist to spend more resources and decrease his influence. In the Supplementary Appendix (Section C), I also show that it is not necessary for the PM to commit to keeping that information confidential: if she had the possibility to disclose it, she would never choose to do so in equilibrium.

2.2 Policy Choice

These expected utilities are illustrated in Figure 2.

When the PM’s prior belief μ₀ is below 1/2, she needs to be persuaded to take action x = 1. When $μ0 ⩾1/2$⁠, the PM does not need to be persuaded absent any preliminary information. However, the information she acquires can push her belief below 1/2 and induce the lobbyist to provide additional information. In line with the literature on friendly lobbying (see e.g., Cotton and Dellis 2016; Schnakenberg 2017; Awad 2020), I refer to the PM in the first case as an enemy and in the second case as an ally. In both cases, I say that the PM becomes more sympathetic after receiving some preliminary information if her belief increases toward 1 and becomes more skeptical if her belief decreases.

3. The Role of Confidential Policy Information

In this section, I explain how the PM can extract information from the lobbyist. I first describe the players’ strategies under transparency. I then show how the lobbyist adapts his strategy to confidentiality and how the PM maximizes the benefits from her investigation.

3.1 Transparency

Suppose first that the PM’s information is not confidential. Following realization r from the PM’s investigation, both players share the belief μ^r.

3.1.1 Lobbyist’s Persuasion Strategy

The lobbyist’s strategy, π, induces a lottery over the PM’s posterior beliefs: with probability $Pπ(s0|r)$⁠, the PM observes s0 and updates her belief to $μs0r$⁠, while with probability $Pπ(s1|r)$ she updates her belief to $μs1r$⁠, where $μs0r ⩽ μr ⩽ μs1r$⁠. The lobbyist only gains when the PM chooses x = 1, which requires her belief to be above 1/2. If the PM already chooses the lobbyist’s preferred policy given her own information (⁠$μr⩾1/2$⁠), the lobbyist does not need to provide any evidence.

If $μr<1/2$⁠, a posterior belief above $1/2$ can only occur following realization s₁. The lobbyist’s problem is therefore to maximize the probability that s₁ occurs, while ensuring that the PM chooses policy x = 1 after observing that realization. As shown in Kamenica and Gentzkow (2011), this can be achieved with a persuasion strategy π such that the PM is just sufficiently persuaded following favorable evidence (⁠$μs1r=1/2$⁠), and such that unfavorable evidence is as precise as possible (⁠$μs0r=0$⁠). 

3.1.2 PM’s Preliminary Investigation

The PM chooses her investigation to maximize the total information that she receives. If the PM is an enemy and her expertise is limited (so that μ^r is always below $1/2$⁠), the lobbyist will optimally adjust his persuasion strategy to the PM’s belief. The PM cannot gain from her own information and is therefore indifferent between any investigation. If her expertise is sufficiently high (⁠$μ¯>1/2$⁠), the lobbyist provides no valuable information following r₁ if $μr1>1/2$⁠. The PM faces a sharp tradeoff: if she chooses the most informative investigation herself, and obtains some belief $μ¯>1/2$⁠, the lobbyist stops providing information. This tradeoff is resolved in favor of obtaining more preliminary information: by choosing the most informative investigation (and inducing $μ¯>1/2$⁠), she can become more confident in her policy decision than she would ever be with the lobbyist’s information. If the PM is an ally $(μ0⩾1/2)$ and her expertise is sufficiently high (⁠$μ̱<1/2$⁠), she can trigger the lobbyist to provide information following r₀ if $μr0<1/2$⁠, so choosing the most informative investigation is beneficial. An allied PM with limited expertise (⁠$μ̱⩾1/2$⁠) never obtains information from the lobbyist and is indifferent between any investigation.

In any of these cases, the PM does not gain from the lobbyist’s information, and it is optimal for her to obtain as much preliminary information as possible.¹² 

3.2 Partial Confidentiality

Suppose now that the PM’s information is confidential. The lobbyist observes the investigation commissioned by the PM (p), but not the conclusions of this investigation (r). This corresponds to situations where PMs can run pilot projects, or commission reports in visible ways without having to publicize the results. Each realization r of the investigation defines a type of the PM.

3.2.1 Lobbyist’s Persuasion Strategy

When the PM is an ally and her expertise is limited (⁠$μ̱⩾1/2$⁠), the lobbyist is indifferent between any persuasion strategy that keeps posterior beliefs above $1/2$ for any possible realization of r. Therefore, providing no information is an optimal strategy.

When the PM is an enemy and her expertise is limited, her interim beliefs (μ^r) are always below $1/2$⁠, so one realization of the lobbyist’s strategy (s₀) will not persuade any type. The lobbyist needs to choose between generating favorable evidence (s₁) that persuades both the skeptical type r₀ and the sympathetic type r₁ and favorable evidence that only persuades the sympathetic type r₁.

When the PM’s expertise is sufficiently high, her own information is sometimes persuasive (⁠$μr⩾1/2$⁠) and sometimes not (⁠$μr<1/2$⁠). This can happen both for allies and enemies. In this case, the favorable realization (s₁) should always persuade both types but the lobbyist chooses whether the unfavorable realization (s₀) should fully reveal the state to be ω = 0, or be sufficiently imprecise that the sympathetic type (r₁) still prefers policy x = 1, that is, $μs0r1⩾1/2$⁠.

In equilibrium, the lobbyist will choose one of two persuasion strategies. In particular, restricting the lobbyist’s persuasion strategy to binary signals is without loss of generality. This is illustrated in Figure 3 for the case where $μ¯<1/2$⁠. If the lobbyist’s favorable evidence is not very informative, and only induces a belief at point A, it only persuades the sympathetic type (⁠$μr1$⁠), and generates a payoff less than 1. If it is more informative, and induces a belief at B, it can persuade both types and generates a payoff of 1. The optimal persuasion strategy can be determined by the concavification of this step function.¹³

I refer to the optimal persuasion strategy which targets the sympathetic type r₁ as the targeted persuasion strategy (π_T). If the sympathetic type is not already persuaded (⁠$μr1<1/2$⁠), this persuasion strategy should be designed as if the lobbyist knew that the PM had observed r₁, that is $πT=πr1$⁠, as defined in Lemma 1.

When the sympathetic type is already persuaded (⁠$μr1>1/2$⁠), favorable evidence (s₁) should persuade the skeptical type (⁠$μs1r0=1/2$⁠) and unfavorable evidence (s₀) should leave the sympathetic type just persuaded to choose policy x = 1 (⁠$μs0r1=1/2$⁠). 

  

3.2.2 PM’s Preliminary Investigation

I first show under what conditions the PM strictly benefits from confidential information, and then show that these benefits are limited by the incentive constraint. I focus on the case where the PM is either an enemy, or an ally with sufficient expertise to generate some interim beliefs below $1/2$⁠. An ally with insufficient expertise never receives any information from the lobbyist and is indifferent between any investigation.

3.2.3 Gains from Confidentiality

Where, as before, $P(r)$⁠, is determined by the pair of interim beliefs $(μr0,μr1)$ and the Bayes plausibility constraint: $P(r0)μr0+P(r1)μr1=μ0$⁠.

The PM’s gain from confidential information arises when the lobbyist chooses a general persuasion strategy: π_G is designed to persuade the skeptical type r₀ who requires more evidence to be persuaded. This additional evidence is valuable to the sympathetic type r₁ who gains some informational rent. Had the lobbyist known that the PM was sympathetic, he would have provided less evidence.

The rent obtained from confidential information is captured by the distance between the belief that the lobbyist would like to induce the following favorable evidence (⁠$1/2$⁠), and the belief the sympathetic type actually has (⁠$μs1r1$⁠). When the latter is strictly above $1/2$⁠, the PM is more confident that choosing policy x = 1 is the correct thing to do: she is better off because her uncertainty is reduced. This intuition is illustrated in Figure 5.

When the lobbyist chooses a targeted strategy π_T, the PM gains no informational rent since the beliefs induced by the lobbyist never make her strictly prefer policy x = 1. A targeted persuasion strategy π_T, therefore, yields the same payoff as public information.¹⁵ These results are formalized in Proposition 2. 

3.2.4 Limits of Confidentiality and Optimal Investigation

Given a general persuasion strategy, the PM prefers to be as skeptical as possible as this forces the lobbyist to provide more evidence. This is illustrated in Figure 6: the lower the skeptical belief $μr0$⁠, the more evidence the lobbyist needs to produce under a general strategy π_G to ensure that the skeptical PM chooses policy x = 1. The PM’s expected utility given π_G is therefore decreasing in $μr0$⁠.¹⁸

However, to induce the lobbyist to choose that strategy, the PM may need to distort her information. When expertise B is high, making full use of expertise (choosing $p=p¯$⁠) induces beliefs $(μ̱,μ¯)$ that violate the incentive constraint (⁠$μ̱<m*(μ¯)$⁠). The lobbyist, therefore, chooses a targeted strategy π_T (Lemma 2) which provides less evidence than a general strategy π_G.

The PM, therefore, faces a tradeoff: to extract more information from the lobbyist, she needs to distort the preliminary information she obtains. The distortion needs to ensure that the skeptical type is sufficiently likely and not too hard to persuade, so the PM chooses an investigation that makes her skeptical type not too skeptical (⁠$μr0⩾m*(μr1)$⁠). This limits the value of confidential information.

These observations lead to the following characterization of the PM’s optimal strategy.

 

1. The PM chooses the most informative investigation $p¯$ if either $B⩽Ḇ$ or $B⩾B¯(μ0)$

2. She imposes distortions on her investigation if $Ḇ<B<B¯(μ0)$ and sets $p(r0|0)p(r0|1)=1−m*(μ¯)m*(μ¯)·μ01−μ0 and p(r1|1)p(r1|0)=B$

The precision of the PM’s investigation relative to her expertise B is therefore non-linear in expertise: at low expertise, the PM chooses the most informative investigation, at intermediate expertise she restricts her information to induce the lobbyist to choose a general persuasion strategy, and at higher expertise she chooses the most informative investigation. This is illustrated in Figure 7. The solid line represents the skeptical type’s belief $μr0$ induced by the PM’s equilibrium investigation. The dashed line represents the lowest possible belief $μ̱$ and the dash–dot line the incentive constraint $m*(μ¯)$⁠, both as functions of B.

When expertise (B) is low, the expertise constraint (⁠$μr0⩾μ̱$⁠) binds before the incentive constraint (⁠$μr0⩾m*(μ¯)$⁠). The PM does not need to distort her information to induce the lobbyist to choose π_G and generates the lowest possible skeptical belief: $μr0=μ̱$⁠.¹⁹ The solid line (equilibrium $μr0$⁠) coincides with the dashed line (lowest possible belief $μ̱$⁠).

When expertise (B) is intermediate, the incentive constraint binds so the PM faces a tradeoff between obtaining more preliminary information (setting $μr0=μ̱$⁠) and inducing the lobbyist to provide more evidence (choosing π_G). The loss in expected utility from distorting her information (setting $μr0=m*(μ¯)>μ̱$⁠) is small relative to the gain from extracting information from the lobbyist (inducing π_G instead of π_T). Conditional on inducing the lobbyist to play a general persuasion strategy π_G, she chooses her investigation to induce the lowest possible skeptical belief: $μr0=m*(μr1)$⁠. As the constraint $m*(μr1)$ is decreasing in the sympathetic type’s belief $μr1$⁠, it is optimal to induce $μr1=μ¯$⁠. The solid line, therefore, coincides with the dash–dot line (incentive constraint $m*(μ¯)$⁠).

As expertise (B) becomes sufficiently large, the PM is willing to give up inducing a general persuasion strategy π_G if the gains from making full use of her expertise (⁠$μr0=μ̱$ and $μr1=μ¯$⁠) are sufficiently large. The solid line (equilibrium $μr0$⁠) coincides again with the dashed line (lowest possible belief $μ̱$⁠). Intuitively, in the limit (as B becomes very large), the PM can learn the state almost perfectly and the lobbyist’s signal becomes negligible.

The interval $[Ḇ,B¯(μ0)]$ is nonempty as long as μ₀ is not too large,²⁰ so this characterization applies to any enemy and to an ally who faces sufficient uncertainty. An ally who is sufficiently certain that the state is 1 would always choose the most informative investigation, because the lowest belief which satisfies the incentive constraint, $m*(μ¯)$⁠, is never far enough below $1/2$ for a general strategy to be beneficial.

3.3 Full Confidentiality

Another possible regime is full confidentiality. Under this regime, neither the PM’s investigation p nor the results r are observable. This transforms the problem into a simultaneous move game.

3.3.1 Lobbyist’s Persuasion Strategy

In this modified game, the lobbyist best responds to the investigation that he expects the PM to choose in equilibrium. His best-response to a given equilibrium investigation p, inducing interim beliefs $(μr0,μr1)$ is determined exactly as in Lemma 2. If $(μr0,μr1)$ are such that $μr0⩾m*(μr1)$⁠, then the lobbyist chooses a general strategy, and if $μr0<m*(μr1)$ he chooses a targeted strategy.

3.3.2 PM’s Preliminary Investigation

The PM’s payoff in equilibrium (i.e., when the lobbyist has the correct beliefs about the PM’s choice of investigation) is the same as under the partial confidentiality case. However, her payoff following a deviation is now different: because the lobbyist cannot observe that deviation, deviating to a different investigation does not affect the choice of persuasion strategy. However, the posterior beliefs induced by the lobbyist’s strategy are now different than what the lobbyist intended.

As a result, starting from any candidate equilibrium investigation that induces beliefs $μr0>μ̱$ and $μr1<μ¯$⁠, the PM prefers to deviate to a more informative investigation. In particular, the PM would deviate from an investigation inducing $(m*(μ¯),μ¯)$ to one inducing $(μ̱,μ¯)$⁠. She can therefore no longer manipulate the beliefs of the lobbyist. Intuitively, the only reason the PM would deliberately limit the informativeness of her investigation is to induce the lobbyist to choose a general strategy. However, given that the lobbyist chooses such as a strategy, the PM would prefer to deviate and obtain more information. The lobbyist anticipates this behavior and chooses the optimal persuasion strategy corresponding to the PM’s most informative investigation.

Choosing the most informative investigation does constitute an equilibrium. Indeed, given this preliminary investigation, the lobbyist optimally chooses a targeted strategy (π_T) if expertise is high (⁠$B>Ḇ$⁠), and the PM would get a lower payoff by deviating to a less informative investigation. If expertise is low (⁠$B⩽Ḇ$⁠), the lobbyist would choose a general strategy and it is indeed optimal for the PM to choose the most informative investigation $p¯$⁠.²¹

The PM only gains from concealing r when the lobbyist plays a general persuasion strategy. If expertise is sufficiently low (⁠$B⩽Ḇ$⁠), the lobbyist chooses a general persuasion strategy (π_G) in equilibrium. But if expertise is too high (⁠$B>Ḇ$⁠), the lobbyist plays a targeted persuasion strategy in any equilibrium. We, therefore, obtain the following result. 

4. The Value of Confidentiality and Its Effect on Influence

In this section, I show that confidentiality improves the quality of policy making most when government expertise is intermediate and that the lobbyist’s influence (how often his preferred policy is passed) can increase in the PM’s expertise. I also show that an allied PM values confidentiality less than an enemy, and that a lobbyist has more influence on an allied PM than an enemy, but only if the PM’s expertise is sufficiently large. The discussion below focuses on the case of partial confidentiality, which provides richer results, but the main observations also apply to full confidentiality.²³

4.1 Value of Confidentiality

Expertise affects the equilibrium choice of investigation p and persuasion strategy π, as well as the probability of error given some strategies. For example, increasing expertise (B) can decrease the probability of error under transparency, but also make the lobbyist switch from a general persuasion strategy to a targeted one.

The first result is that the value of confidentiality varies nonmonotonically in expertise (B) for both enemies and allies as illustrated in Figure 8.

 

1. It is weakly increasing in expertise (B) at low levels and weakly decreasing at higher levels. For sufficiently high expertise, the PM is indifferent between transparency and confidentiality.

2. Holding the distance between μ₀ and $1/2$ constant, that value is higher for an enemy than an ally.

The nonmonotonicity in expertise is always strict for an enemy or an ally with sufficiently low prior μ₀. For an ally with prior close to 1, the value of confidentiality is always zero.

Intuitively, as the expertise of the PM increases, two opposite effects arise. The PM can extract more information from the lobbyist when keeping her information confidential: she can make her skeptical belief (⁠$μr0$⁠) more skeptical and force the lobbyist to produce more evidence. This increases the probability of choosing the right policy. On the other hand, expertise increases the value of information under transparency, which decreases the value of confidentiality. The second effect does not arise when expertise is low, so the value of confidentiality initially increases in expertise. When expertise is high, the probability of making the right policy choice increases under both transparency and confidentiality. However, the increase under confidentiality dampens as the PM needs to distort her information. As a result, the value of confidentiality decreases in expertise.

When expertise is very large, the PM prefers to make full use of her expertise (Proposition 3). The lobbyist uses a targeted strategy which induces the same probability of choosing the correct policy as in the transparency regime (Proposition 2). In this case, confidentiality does not improve policy making. In fact, the PM receives strictly less information under confidentiality than under transparency.²⁵

The second part of the proposition suggests that an ally would be more open to sharing information with lobbyists than an enemy. Intuitively, if expertise is low, an ally cannot extract additional information under confidentiality, while an enemy can, so an enemy benefits more. If expertise is high, both can extract information and the amount of information extracted is inversely proportional to the PM’s skeptical belief (⁠$m*(μ¯)$⁠), this belief is higher for an ally than an enemy, so the enemy benefits more.

Proposition 5 reveals that the effect of transparent institutions on the quality of policy making depends on both the policy environment and the political environment. Transparent institutions should be more prevalent in areas where the government is either not very competent, or on the contrary, very good at obtaining precise policy-relevant information. This depends on the policy’s complexity, on whether the government is composed of technocrats, or on the attractiveness of the civil service relative to the private sector for competent researchers. Transparency is also relatively more valuable when legislators are sufficiently aligned with lobbyists and cannot extract much information from them.

Even though the results presented here are for partial confidentiality, it is easy to see that partial confidentiality is weakly preferred to full confidentiality. When expertise is low, the incentive constraint does not bind and the equilibrium outcomes are the same in both confidentiality regimes. When expertise is intermediate, the PM can only extract additional information under partial confidentiality, when the lobbyist is aware of the type of investigation run by the PM. When expertise is high, the equilibrium outcomes are again the same in both regimes. If she can, the PM would therefore prefer to inform the lobbyist about the design of her investigation (but to withhold the results of that investigation).²⁶

4.2 Effect of Confidential Information on Influence

In this section, I analyze how the PM’s control over her investigation affects the lobbyist’s influence. In line with the existing literature (e.g., Cotton and Dellis 2016; Ellis and Groll 2020), I view influence as the change in policy choice that results from the presence of the lobbyist.²⁷ One might expect that the better informed a PM is, the less likely she is to be influenced by a lobbyist. I show that under certain conditions, an increase in government expertise can both increase the lobbyist’s influence and the probability of choosing the right policy. This result cautions against the popular view that external influence always has a negative impact on policy making.

Influence is a different metric than the lobbyist’s payoff because it accounts for the policy that the PM would choose in the absence of lobbying. Explicitly modeling the PM’s choice of information highlights that her equilibrium strategy in the counterfactual (in the absence of lobbyist) may be different.²⁸

In the absence of lobbying, the PM weakly prefers the most informative investigation. If her expertise is too low to ever change her choice (⁠$μ¯<1/2$ or $μ̱>1/2$⁠), the probability of choosing policy x = 1 is 0 for an enemy and 1 for an ally. Otherwise, it is equal to the probability of observing signal r₁.

In the presence of lobbying, the lobbyist chooses a general persuasion strategy π_G in equilibrium whenever the PM strictly prefers information to be confidential (⁠$B<B¯(μ0)$⁠). The probability of inducing policy x = 1 is therefore the probability of producing a signal s = s₁. When the PM prefers information to be public (⁠$B>B¯(μ0)$⁠), the probability that the policy is x = 1 is also the probability that s = s₁, unless the PM is already persuaded (r = r₁ and $μ¯>1/2$⁠).

As government expertise increases, the PM is better equipped to defend herself. She can make the lobbyist believe that she is very skeptical and force him to produce more evidence. Therefore, as expertise B increases, influence initially decreases.

However, influence can also increase in expertise. If the PM is an enemy (⁠$μ0<1/2$⁠), ω  =  0 is more likely than ω = 1. An increase in expertise makes the PM’s signal more precise, and her investigation is more likely to indicate that the state is ω = 0. In the absence of lobbying, the PM, therefore, becomes more likely to choose policy x = 0. In contrast, the presence of a lobbyist leads to a relatively high probability that the policy chosen is x = 1. As a result, influence increases in the PM’s expertise B. The co-movement of influence and the probability of making the correct decision is illustrated in Figure 9.²⁹

   

5. Discussion

5.1 Evolution of Institutions

The results from Section 4 provide a rationale for the development of information-gathering institutions and contribute to a more general literature on the role and development of legislative institutions (e.g., Krehbiel 1991, Bimber 1991, Krehbiel 2004). The diversity of sources of information within governments (agencies, legislative research services, public consultations, legislative hearings, etc.) raises a number of questions: why might legislators obtain redundant information from both internal and external sources? What explains transitions from confidentiality to transparency, such as congressional hearings in the USA in the 1970s and congressional research memos more recently?

The model shows that internal information is valuable to the PM even when that information would not impact policy in the absence of lobbying (Proposition 2). This can account for the puzzling observation that the government may choose to obtain information even if that information is redundant. Clark (2016) quotes an Appropriations Committee aide describing the CRS research as “quick and dirty analysis that is sometimes not perfect.” This suggests that the evidence obtained internally may not be precise enough to determine policy choices. The model shows that “quick and dirty analysis” can result in significant policy improvements in the presence of lobbyists, as the two sources can be strategic complements.

Proposition 5 also shows that, for a large enough level of expertise, confidentiality is no longer valuable to the PM. As a result, increases in expertise within the government can lead to more transparency of government information. This can arise independently of the role of transparency for accountability (as in Argenziano and Weeds 2019). In addition, since expertise varies across policy areas (Howlett 2015), it is possible for transparent institutions (such as hearings) to be used in certain domains, and confidential ones (such as agency memos) in others.

Finally, since the value of confidentiality decreases with expertise when expertise is high (Proposition 5), we should observe empirically a positive correlation between expertise and transparency. This is consistent with the findings of Islam (2006) that transparency (measured by the timeliness of governments in releasing economic information and the presence of freedom of information laws) is correlated with the quality of governance. However, it would be incorrect to conclude that transparency necessarily causes improvements in governance. In particular, Proposition 5 indicates that, at low levels of expertise, transparency would lead to worse policy making.

5.2 Risk of Leaks Under Partial Confidentiality

Under partial confidentiality, lobbyists are aware that the PM is obtaining some information. Publicizing the design of this investigation can expose the PM to risk of leaks, as lobbyists could request to obtain the results of that investigation under freedom of information requests. Under full confidentiality, this is less likely as lobbyists do not know for sure that the PM has investigated. In the Supplementary Appendix (Section B), I analyze a model where the outcome of the investigation r can leak with some positive probability under partial confidentiality but not under full confidentiality. In this case, the PM continues to prefer distorting her preliminary investigation when her expertise is intermediate, but only if the risk of leaks is not too high. Since the gains from distortions only arise when the investigation’s outcome remains confidential, distorting information becomes less valuable when the risk of leaks increase. In the presence of leaks, the PM would now strictly prefer the full confidentiality regime when expertise is low (⁠$B⩽Ḇ$⁠) since both regimes induce the same behavior from the lobbyist, but full confidentiality avoids the risk of damaging leaks. She would still prefer partial confidentiality otherwise. If the risk of leaks differs across policies areas, we should also expect differences in institutional design. For instance, exempting some policy areas from having to publish public notice or fulfilling freedom of information requests on national security grounds reduces the ability of legislators to share both the design and the results of their investigation. These exemptions can be valuable when the risks of leaks are high.³¹

5.3 Costly Investigations

The analysis conducted so far assumed that investigations were costless, to highlight that the PM might want to restrict her investigation even in the absence of costs. Introducing costs can lead to further distortions. Suppose, for example, that the players face a fixed cost to generate information, independent of the precision of the investigation, but that the PM’s investigation has limited precision as in the existing model.

If the fixed costs are sufficiently low for both players, then the results presented so far would not change.³² If the lobbyist’s costs are large enough, the PM might prefer to restrict her information in order to induce the lobbyist to generate information. By restricting her information, she makes persuasion easier and sufficiently valuable for the lobbyist to justify the fixed cost. As long as the distortions are not too important, this strictly benefits the PM. If not, the PM might give up on extracting information from the lobbyist and choose the most informative investigation. The PM only gains from these restrictions under partial confidentiality. Under transparency, she never strictly benefits from the lobbyist’s information and therefore does not want to incentivize the lobbyist to generate information. Under full confidentiality, she cannot credibly commit to these distortions. If introducing costs forces the PM to restrict her investigation beyond the restrictions already identified, then the more costly the lobbyist’s investigation is, the less valuable confidentiality is to the PM. In the Supplementary Appendix (Section F), I show that there always exists costs low enough that the results above are unchanged, and provide an example where costs force the PM to reduce the precision of her investigation.³³

6. Conclusion

This article examined the effect of a PM’s internal information on the provision of information by special interest groups. When PMs can control their preliminary investigations, they can extract additional evidence from special interest groups by distorting these investigations. Internal information becomes valuable even when it is limited, as long as it is confidential. This makes confidentiality valuable to PMs even in the absence of reputational concerns and explains why internal research is kept secret in many governments.

Governments can use confidential information to force special interest groups to adjust how they influence policy. This article, therefore, highlights that influence cannot be simply measured based on whether policy changed or not. The definition of influence should consider what policy would have been chosen in the absence of lobbying. Given this definition, it is possible that influence and welfare increase at the same time, when government expertise or ideological alignment change. When the CRS opens its research to the public, as is currently planned, it will increase the influence that interest groups exert on policy making. But if this move toward transparency is driven by an increase in expertise, then this increase in influence could be accompanied by an increase in welfare.

Understanding the control of governments over the production of internal information is therefore critical not only to the study of special interest groups but also to the relationship between transparency and the quality of government.

Footnotes

Appendix

Let $μsr(π)$ be the posterior induced by $s∈{s0,s1}$ from signal $π∈{πG,πT}$⁠, starting from interim $μr∈{μr0,μr1}$⁠, and $μs(π)$ the posterior belief induced by $s∈{s0,s1}$ from signal $π∈{πG,πT}$⁠, starting from the prior μ₀.

This defines a set of pairs of beliefs $(μr0,μr1)$⁠, denoted

The lobbyist, therefore, prefers π_G only if $μr0⩾m*(μr1)$⁠. It is easy to verify that for any $μ0∈[0,1]$ and $μr1∈[μ0,1]$⁠, we have $0⩽m*(μr1)⩽μ0$⁠.

Finally, the highest root $x¯$ is always greater than μ₀, so condition (Equation A2) is satisfied for all $μr0∈[m*(μr1),μ0]$⁠. As a result, for any $μr0∈[0,μ0], H(μr0)$ is positive if and only if $μr0≥x̱$⁠, so the lobbyist prefers π_G if and only if $μr0≥m*(μr1)$⁠.

The left-hand side of the first inequality in condition (Equation A2) is increasing in $μr0$ while the right-hand side is decreasing in $μr0$⁠. As $μr1$ increases, the right-hand side shifts down. The value of $μr0$ that makes the two sides equal is therefore decreasing in $μr1$⁠, so $m*(μr1)$ is decreasing in $μr1$⁠.

2. Lobbyist only needs to persuade lower type:

The left-hand side of this inequality is $H(μr0)$ from Equation (A3). The rest of the proof, therefore, follows the same logic as in the previous case. 

2. If the incentive constraint is not satisfied, confidentiality is not strictly preferred:

Therefore, for any $(μr0,μr1)$⁠, we have $UP(μr0,μr1)=UT(μr0,μr1)$

• When $B<Ḇ$⁠, the incentive constraint does not bind (⁠$m*(μ¯)<μ̱$⁠) and the optimal investigation induces $μr0=μ̱$⁠. The PM is indifferent between any $μr1$⁠, so any $(μr0,μr1)∈{μ̱}×[(m*)−1(μ̱),μ¯]$ can be in the support of an equilibrium investigation. Therefore, when $B<Ḇ$⁠, choosing $p¯$ is an equilibrium strategy for the PM.

• When $B>Ḇ$⁠, the constraint binds (⁠$m*(μ¯)>μ̱$⁠) and the optimal investigation induces $μr0=m*(μr1)$⁠. Since $m*(μr1)$ is decreasing in $μr1$ and $UG(μr0,μr1)$ is independent of $μr1$⁠, it is optimal to induce $μr1=μ¯$⁠. Therefore, when $B⩾Ḇ$⁠, the only equilibrium is for the PM to choose an investigation that induces $μr0 = m*(μ¯)$ and $μr1=μ¯$⁠.

2. Lobbyist only needs to persuade lower type:

Suppose now that $μ̱<1/2<μ¯$⁠. Note that $UG(μr0,μr1)$ is unchanged in this case. I show that the PM now sometimes prefers to induce π_T.

Claim 3: If the lobbyist were to choose π_T, the PM would choose $p¯$⁠.

Proof: Taking derivatives of $UT(μr0,μr1)$ (Equation (A5)) for the case $1/2<μr1$ shows that $UT(μr0,μr1)$ is increasing in $μr1$ and decreasing in $μr0$⁠. Therefore, the optimal investigation given π_T induces interim beliefs $(μ̱,μ¯)$⁠.

Claim 4: If $B<Ḇ$⁠, the PM strictly prefers $p¯$⁠.

Proof: As shown above, if $B<Ḇ$⁠, the PM induces the lobbyist to choose π_G even when using $p¯$⁠. From the proof of Proposition 2, we know that

Finally, from claim 2, we have $UG(μ̱,μ¯)⩾UG(μr0,μr1)$⁠. Therefore if the incentive constraint does not bind (⁠$B<Ḇ$⁠), the PM prefers $p¯$⁠.

Claim 5: If $B>Ḇ$⁠, there exists $B¯>Ḇ$ such that the PM prefers an investigation such that $μr0=m*(μ¯)$ and to induce the lobbyist to choose π_G if $B⩽B¯$⁠, and prefers an investigation such that $μr0=μ̱$ and to induce the lobbyist to choose π_T if $B⩾B¯$⁠.

Proof: If $B>Ḇ$⁠, the incentive constraint binds so the optimal investigation that induces π_G generates interim beliefs $μr0=m*(μ¯)$ and $μr1=μ¯$ (claim 2). Given claim 3, if the PM’s investigation induces π_T, it is optimal to generate interim beliefs $μr0=μ̱$ and $μr1=μ¯$⁠. To show the existence and uniqueness of $B¯$⁠, I proceed in three steps.

Step 1: At $B=Ḇ, m*(μ¯)=μ̱$ so $UG(m*(μ¯),μ¯)=UG(μ̱,μ¯)>UT(μ̱,μ¯)$ by claim 4.

Step 3: As $B→+∞, UG(m*(μ¯),μ̱)<UT(μ̱,μ¯)$⁠.

Therefore, $limB→+∞UT(μ̱,μ¯)>limB→+∞UG(m*(μ¯),μ¯)$ and the PM eventually prefers to give up trying to induce π_G.

Combining steps 1–3, and the intermediate value theorem implies that there exists a unique $B¯(μ0)>Ḇ$ that satisfies Claim 5.

Claim 6: If $μ0⩾(1+2/2+2)$⁠, then the PM always prefers an investigation such that $μr0=μ̱$ and to induce the lobbyist to choose π_T.

Proof: Let $B1(μ0)$ such that $m*(μ¯)=1/2$⁠. Note that $B1(μ0)<+∞$ if and only if $μ0<3/4$⁠, since $m*(μ¯)>1/2$ for any B whenever $μ0>3/4$⁠. Let $BH(μ0)$ such that $μ̱=1/2$⁠. If $μ0<(1+2)/(2+2)$⁠, then $B1(μ0)<BH(μ0)<Ḇ$⁠, so it is always possible to generate $m*(μ¯)<1/2$ whenever it is possible to generate some $μr0<1/2$⁠, and the logic of Claim 5 applies. If $μ0∈[1+22+2,34]$⁠, then $BH(μ0)<B1(μ0)<Ḇ$⁠.

For $B∈(BH(μ0),B1(μ0))$⁠, the PM receives no information if she induces $μr0⩾m*(μr1)>1/2$⁠. Therefore inducing the lobbyist to generate some information requires breaking the incentive constraint, and the optimal investigation is the most informative investigation, which induces the lobbyist to choose a targeted strategy.

And since by step 2 of Claim 5, we know that $UG(m*(μ¯),μ¯)−UT(μ̱,μ¯)$ is decreasing in B, we can conclude that it is never optimal for the PM to restrict her belief to $m*(μ¯)$ to induce π_G. We can define $B¯(μ0)=Ḇ$ to prove the statement in Proposition 3.

Finally, for any $μ0>34, m*(μ¯)>1/2$ for any B since $limB→∞m*(μ¯)=μ03−2μ0>1/2$ whenever $μ0>3/4$⁠. In this case, it is always optimal to choose $p¯$ since any belief that would satisfy the incentive constraint is above $1/2$ and would therefore induce the lobbyist to produce no information. Setting $B¯(μ0)=Ḇ$ again proves the statement in Proposition 3. Note that in these last two cases, the interval $[Ḇ,B¯]$ is empty and the PM always chooses $p¯$⁠.

1. Lobbyist does not need to persuade any type: Suppose that for any feasible pair of beliefs, $1/2<μr0,μr1$⁠: in this case, the lobbyist provides no information for any choice of investigation of the PM. The PM is therefore indifferent between any preliminary investigation, so choosing $p=p¯$ is an equilibrium. Setting $B¯(μ0)=Ḇ$ proves Proposition 3 for this case.

Claims 1–6 then imply that the PM makes full use of her expertise if either $B<Ḇ$ or $B>B¯(μ0)$ and otherwise distorts her investigation such that $μr0=m*(μ¯)$⁠. □ 

Lemma 3.

1. If $B⩽Ḇ$⁠, there exists a set of equilibria in which the lobbyist chooses $πG$ and the PM chooses a preliminary investigation which must induce $μr0=μ̱$ but can induce any $μr1∈(μ0,μ¯]$⁠. In particular, $p=p¯$ is an equilibrium strategy for the PM.

2. Such equilibria exist only if $B⩽Ḇ$⁠.

3. Every equilibrium in this set yields the same payoff to the PM.

Proof. The proof proceeds in three steps.

Claim 1: If an equilibrium exists where $πG$ is played, then $μr0=μ̱$⁠.

A deviation to $μr′0⩽μr0$ does not affect the strategy of the lobbyist, but changes (1) the relative likelihood of the two types of the PM and (2) the expected payoff conditional on r₀ and s₁, which now becomes $1−μs1r′0$ instead of $1/2$⁠. The derivative of this expected utility function with respect to $μr′0$ is negative and therefore it is always profitable to deviate to some $μr′0<μr0$⁠.

Claim 2: In any strategy profile such that the PM chooses an investigation p that induces interim beliefs $(μ̱,μr1)∈G$ and the lobbyist chooses a persuasion strategy π_G, the PM has no incentives to deviate to $p′$ such that $μr′0⩾μ̱$⁠.

Proof: Such a deviation yields $UG(μr′0,μr1)=PπG(s0)+μ0$⁠. Deviating to $μr′0⩾μ̱$ changes the expected payoff conditional on r₀ and s₁, which becomes $μs1r′0⩾1/2$ instead of $1/2$⁠. The indirect expected utility is linear in this region, so the deviation payoff is independent of $μr′0$ and there can be no gain.

Claim 3: In any strategy profile such that the PM chooses an investigation p that induces interim beliefs $(μ̱,μr1)∈G$ and the lobbyist chooses a persuasion strategy π_G, the PM has no incentives to deviate to $p′$ such that $μr′1≠μr1$⁠.

Proof: Such a deviation gives $UG(μr0,μr′1)=PπG(s0)+μ0$⁠. Deviating to $μr′1≠μr1$ changes the expected payoff conditional on r₁ and s₁, which now becomes $μs1r′1≠μs1r1$⁠, but such that $μs1r′1⩾1/2$ (since $μs1r′1>μs1r0⩾1/2$⁠). Because the indirect expected utility is linear in this region, the deviation payoff is independent of $μr1$ and there can be no gain.

Combining Claims 2 and 3 implies that the PM does not deviate from any investigation-inducing beliefs $(μr0,μr1)=(μ̱,μr1)∈G$ given that the lobbyist chooses π_G. Given Lemma 2, if $(μ̱,μr1)∈G$⁠, that is, if $B⩽Ḇ$ (and $μr1$ not too small), the lobbyist does want to play π_G so this is an equilibrium. This proves the “if” part of the existence statement.

In addition, since this holds for any $μr1$ such that $(μ̱,μr1)∈G$⁠, it holds in particular for $μr1=μ¯$⁠, so $p=p¯$ is an equilibrium. And by Claim 3, the expected utility of the PM is the same for any $μr1$⁠, so it is the same across all equilibria.

If $B>Ḇ$⁠, then $(μ̱,μr1)∉G$ for any $μr1∈[μ0,μ¯]$⁠, so Lemma 2 implies that the lobbyist chooses π_T if the PM’s strategy is such that $μr0=μ̱$⁠. Suppose that the PM chose an investigation such that $(μr0,μr1)∈G$⁠, then the lobbyist would choose strategy π_G, but Claim 1 implies that the PM would then prefer to deviate to some $μr′0<μr0$⁠. Therefore if $B>Ḇ$ there does not exist an equilibrium in which π_G is played. This proves the “only if” part of the existence statement. □

Lemma 4.

1. If $B>Ḇ$⁠, there exists a set of equilibria in which the lobbyist chooses $πT$ and the PM chooses an investigation which must induce $μr1=μ¯$ and some $μr0∈[μ̱,μ0)$⁠. In particular, $p=p¯$ is an equilibrium strategy for the PM.

2. Such equilibria exist only if $B>Ḇ$⁠.

3. Every equilibrium in this set yields the same payoff to the PM, which is the same utility as if information was public.

Proof.

If $B>Ḇ$⁠, Lemma 3 implies that the only possible equilibrium involves π_T.

Claim 1: If $μ¯⩽1/2$ (⁠$B⩽Bh(μ0)$⁠), $μr1=μ¯$ in any equilibrium where π_T is played.

Proof: By contradiction, suppose there was an equilibrium where $μr1<μ¯$⁠. In such an equilibrium, the PM would get utility $1−μ0$ because her posterior beliefs are always on the same linear portion of her expected utility function. If she deviates to $p′$ inducing $(μr0,μr′1)$ such that $μr1<μr′1<μ¯$⁠, then her posterior belief following s₁ will be such that $μs1r′1>1/2$⁠. This is therefore a profitable deviation.

Claim 2: If $μ¯⩽1/2$ (⁠$B⩽Bh(μ0)$⁠) the PM does not deviate from any investigation inducing $μr1=μ¯$⁠, and any $μr0$ such that $(μr0,μ¯)∉G$⁠.

Proof: The PM’s expected utility in equilibrium is $UT(μr0,μr1)$ as defined in Equation (A5). When $μr1=μ¯$⁠, this is equal to $UT(μr0,μ¯)=PπT(s0)+(1−μ0)μ¯1−μ¯=1−μ0$⁠. Deviating to $μr′1<μ¯$ gives the same expected utility $UT(μr0,μr′1)=PπT(s0)+(1−μ0)μ¯1−μ¯=1−μ0$⁠, independently of $μr′1$⁠, so the PM would not deviate. Deviating to $μr′0≠μr0$ gives expected utility of $UT(μr′0,μ¯)=PπT(s0)+(1−μ0)μ¯1−μ¯=1−μ0$⁠. This is also the same as the equilibrium utility, independently of $μr′0$⁠, so the PM would not deviate.

Claim 3: If $μ¯>1/2$ (⁠$B>Bh(μ0)$⁠) an investigation inducing $μr0=μ̱$ and $μr1=μ¯$⁠, and a persuasion strategy π_T is the only equilibrium strategy profile.

Proof: The PM’s expected utility in equilibrium is $UT(μ̱,μ¯)$ as defined in Equation (A5). Deviating to an investigation that induces a pair of interim beliefs $(μr′0,μr′1)$ such that $μr′0⩾μ̱$ and $μr′1⩽μ¯$ gives: $UT(μr′0,μr′1)=μ¯(1+μ0−2μ̱)−μ0(1−μ̱)μ¯−μ̱$⁠. Which is the same as when $μr′0=μ̱$ and $μr′1=μ¯$⁠, so the PM does not deviate to any $μr′0>μ̱$ and $μr′1<μ¯$⁠. This is the only equilibrium. Suppose there was an equilibrium such that interim beliefs were $(μr0,μr1)$ such that $μr0⩾μ̱$ and $μr1≤μ¯$ with at least one inequality strict. The derivatives of each payoff function when deviating show that: (1) If $μr0>μ̱$⁠, the PM would deviate to inducing $μr′0<μr0$⁠, and (2) If $μr1<μ¯$⁠, the PM would deviate to inducing $μr′1>μr1$⁠.

Claims 1 and 2 imply that there exists a set of equilibria in which π_T is played when $μ¯<1/2$⁠, that all these equilibria yield the same payoff to the PM and that in one of these equilibria, the PM chooses $p=p¯$⁠. Claim 3 implies that when $μ¯>1/2$⁠, there exists a unique equilibrium in which the lobbyist chooses π_T and the PM chooses $p=p¯$⁠.

Finally, since the PM’s expected utility is the same under confidentiality with π_T and transparency (Proposition 2), the two regimes yield the same utility whenever $B>Ḇ$⁠, so there are no gains from confidentiality in this case. □

Lemmas 3 and 4 imply Proposition 4: (a) If $B⩽Ḇ$⁠, then $p¯$ and π_G is an equilibrium, all equilibria yield the same payoff to the PM as this one, and the PM’s utility is strictly greater than under transparency. (b) If $Ḇ<B$⁠, then $p¯$ and π_T is an equilibrium, and when multiple equilibria exist, they all yield the same payoff to the PM as this one and the same payoff as under transparency. □

1. If $B<min{Ḇ,BH(μ0)}$ and $μ0⩾1/2$⁠, then $∂Wi(B)∂B=0$⁠.

2. If $B<min{Ḇ,Bh(μ0)}$ and $μ0<1/2 ∂Wi(B)∂B=μ0B2>0$⁠.

3. If $B∈(BH(μ0),Ḇ)$ and $μ0⩾1/2$ or if $B∈(Bh(μ0),Ḇ)$ and $μ0<1/2, ∂Wi(B)∂B=μ0(2B+1)−B2(1−μ0)(B(B+1))2⩾0$ if and only if $B≤μ0+μ0(1−μ0)$⁠.

4. If $B∈(Ḇ,Bh(μ0)), ∂Wi(B)∂B=−(1−μ0)1(1−m*(μ¯))2∂m*(μ¯)∂μ¯∂μ¯∂B>0$⁠, as $∂m*(μ¯)∂μ¯<0$ and $∂μ¯∂B>0$⁠.

5. If $B>max{Ḇ,Bh(μ0)}$⁠, $Wi(B)∂B=−1(B+1)2+10B−2+29B2−10B+1(3B+9B2−10B+1−1)29B2−10B+1μ0⩽0$⁠.

Finally, when $B⩾B¯(μ0)$ (or $μ0⩾1+22+2$⁠), the PM uses the most informative investigation, the lobbyist either chooses a targeted strategy or provides no information, and the probability of error is the same under transparency and confidentiality. Therefore, the value of confidentiality is $Wi(B)=0$ in all these cases.

To prove the second part of the proposition, let $μ0<1/2$ be the prior of an enemy, and $μ1=1−μ0>1/2$ be the symmetric prior for an ally. I show that in each case, $WE(B)⩾WA(B)$⁠.

1. If $μ1⩾1+22+2$⁠, then for an ally $WA(B)=0$⁠, while for an enemy $WE(B)⩾0$⁠.

2. If $μ1<1+22+2$ and $B⩾max{B¯(μ0),B¯(μ1)}$⁠, then $WA(B)=WE(0)=0$⁠.

3. If $μ1<1+22+2$ and $B¯(μ1)<B<B¯(μ0)$⁠, then $WE(B)>WA(B)=0$⁠. Note that $B¯(μ0)<B<B¯(μ1)$ is not possible as $B¯(μ0)$ is decreasing in μ₀ (see Claim 5 in proof of Proposition 3).

4. If $μ1<1+22+2$ and $Bh(μ0),Ḇ<B<min{B¯(μ1),B¯(μ0)}$ (note that $Bh(μ0)=BH(μ1)$ since $μ1=1−μ0$⁠), then $Wi(B)=UG(m*(μ̱),μ¯)−U¯P(μ̱,μ¯)$ which is strictly decreasing in μ₀ (see Claim 5 in proof of Proposition 3), so $WA(B)<WE(B)$⁠.

5. If $μ1<1+22+2$ and $Bh(μ0)<B<Ḇ$⁠, then $Wi(B)=UG(μ̱,μ¯)−U¯P(μ̱,μ¯)$ which is strictly decreasing in μ₀ (see Supplementary Appendix for details), so $WA(B)<WE(B)$⁠. Note that $Ḇ<Bh(μ0)=BH(μ1)$ is not possible when $μ1<1+22+2$⁠.

6. Finally, when $B<Ḇ, WE(B)=UG(μ̱,μ¯)−U¯P(μ̱,μ¯)$ and $WA(B)=0$⁠.

1. If $B∈(Bh(μ0),Ḇ), FE(B)=(3μ0−1)B+μ0B(B+1)<(3μ1−1)B+μ1B(B+1)=FA(B)$⁠.

2. If $B∈[Ḇ,B¯(μ1)), FE(B)=μ0+(1−μ0)m*(μ¯)(1−m*(μ¯))−μ0−μ̱μ¯−μ̱<μ1+(1−μ1)m*(μ¯)(1−m*(μ¯))−μ1−μ̱μ¯−μ̱+FA(B)$⁠, since $∂F(B)∂μ0=(2B2+8B+29B2−10B+1−2(B+1)(3B+9B2−10B+1−1))>0$ (see Supplementary Appendix for details).

3. If $B∈[B¯(μ1),B¯(μ0))$⁠, then we need to show that $FE(B)=μ0+(1−μ0)m*(μ¯)(1−m*(μ¯))−μ0−μ̱μ¯−μ̱<2μ1B+1=FA(B)$⁠. Since in this case, $FE(B)$ is increasing in μ₀ and $FA(B)$ is decreasing in μ₀ (increasing in $μ1=1−μ0$⁠), we just need to show that at $μ0=1/2, FE(B)>FA(B)$ for any $B∈[B¯(μ1),B¯(μ0))$⁠. At $μ0=1/2, FE(B)$ simplifies to: $FE(B)=(3B−1)−(B−1)(9B−1)4(B+1)$⁠, so $FE(B)>FA(B)$ if and only if $1B+1>(3B−1)−(B−1)(9B−1)4(B+1)$⁠. This is equivalent to $18+6B2−12B+8(B−1)(9B−1)>0$⁠, which always holds for any B.

4. If $B⩾B¯(μ0), FE(B)=2μ0B+1<2μ1B+1=FA(B)$⁠.

Suppose $μ1⩾1+22+2$⁠, since $B¯(μ1)=Ḇ<Bh(μ0)=BH(μ1)$⁠, there are two cases:

1. If $B∈(Bh(μ0),B¯(μ0)), FE(B)=μ0+(1−μ0)m*(μ¯)(1−m*(μ¯))−μ0−μ̱μ¯−μ̱<2μ1B+1=FA(B)$ as in case 3.

2. If $B⩾B¯(μ0), FE(B)=2μ0B+1<2μ1B+1=FA(B)$⁠.

Supplementary Material

Supplementary material is available at Journal of Law, Economics, & Organization online.

Funding

This work was supported by the Economic and Social Research Council grant number ES/J500070/1.

Conflict of interest statement. None declared.

I am deeply grateful to my supervisor Gilat Levy for invaluable guidance and to Ricardo Alonso and Stephane Wolton for precious advice. I would like to thank the editor and three anonymous referees for insightful suggestions, Simon Franklin, Andy Guess, Yingni Guo, Niall Hughes, Federica Izzo, Ronny Razin, Herschel Thomas, and Chris Tyson for helpful comments as well as seminar and conference participants at the LSE PSPE and Microeconomic Theory seminars, RES Junior Symposium 2018, MPSA Annual Conference 2018, LSE-Oxford Political Economy Conference, Queen Mary Economics and Finance Workshop, Max Plank Institute Political Economy workshop, and POLECON UK Conference 2019.

References