Lobbying, Inside and Out
How Special Interest Groups Influence Policy Choices∗
Stephane Wolton†
September 24, 2013
Abstract
Scholars have long recognized two classes of special interest group (SIG) spending: inside
lobbying, which is intended to influence the content of a bill, and outside lobbying, which is
intended to influence the probability a bill is enacted into law. Although a substantial body
of formal work investigates SIGs influence resulting from each category of lobbying activity,
none juxtaposes both within a single model. This paper does so. Considering a variety of SIG
environments, I demonstrate that the government bases his policy decisions on his assessment
of the risk of SIGs engaging in outside lobbying. Most importantly, the government’s expectations concerning the likelihood of outside lobbying activities are not measured adequately
by inside lobbying expenditures. Consequently, there is no clear correlation between an SIG’s
inside lobbying expenditures and her influence.
Keywords: special interest group, contributions, inside lobbying, policy-making, outside
lobbying
JEL Classification: D70, D72, D74, D78, D82
∗
I thank Scott Ashworth, Enghin Atalay, Christopher Berry, Ethan Bueno de Mesquita, Will Howell, Navin
Kartik, Antonio Nicolo, Joseph McMurray, Laura Pilossoph, Carlo Prato, Erik Snowberg, seminar participants at
the Harris School of Public Policy, and especially Richard Van Weelden for their insightful comments. All remaining
errors are the author’s responsibility.
†
University of Chicago, Department of Economics.
swolton@uchicago.edu.
1
1126 E. 59th St, Chicago, IL 60637.
E-mail:
1
Introduction
It is well known that special interest groups (SIGs) have more than one way to tilt political
decisions in their favor. SIGs can use inside lobbying expenditures (e.g., contributions, hiring a
lobbyist to draft a bill or to contact legislators) to influence the content of a bill. They can use
outside lobbying expenditures (e.g., running advertising campaigns, hiring petitioners to collect
signatures or to mobilize voters) to affect the likelihood a piece of legislation is enacted into law
(see Blaisdell, 1957; Wright, 1996; Kollman, 1998; Baumgartner et al., 2009). Scholars have
described SIGs’ efforts to employ both forms of mobilization in a variety of prominent legislative
battles, including Clinton’s 1993 health care reform (West et al., 1996; Goldstein, 1999), McCain’s
1998 Senate bill targeting tobacco companies (Jamieson, 2000; Derthick, 2012) and Obama’s 2010
Affordable Care Act (Hall and Anderson, 2012; LaPira, 2012).
In contrast, the formal literature focuses on either inside lobbying expenditures (e.g., Grossman
and Helpman, 1996) or outside mobilization (e.g., Kollman, 1998) on political decisions.1 This
paper is the first to combine both inside and outside lobbying activities in a single game-theoretic
model. Important insights are gained from the analysis of the complex interaction between both
classes of SIG spending. Results provide a rationale for why the most powerful groups in Congress
do not generally incur the highest inside lobbying expenditures.2 Contrary to common beliefs,
SIGs’s use of inside lobbying is not always meant to obtain a policy closer to their ideal point. In
some cases, the purpose of inside lobbying is to encourage the government to find a compromise
so as to avoid engaging in costly outside lobbying. Inside lobbying expenditures are a way to
“plead poverty” credibly. The theoretical findings explain why empirical tests of SIGs influence
that exclusively consider inside lobbying expenditures - as nearly all extant tests do - are likely to
produce inconsistent results.
I solve a model that consists of a three-stage game with a government, a pro-government
SIG (whose policy preferences align with the government), and an anti-government SIG (whose
preferences oppose the government).3 Before the government chooses the content of a bill, the proand anti-government SIGs can send a signal to reveal their capacity to bear the cost of outside
lobbying activities (i.e., their strength). These signals include a simple announcement regarding
1
The next section presents a more comprehensive review of the literature.
For example, the American Association of Retired Persons and the National Rifle–who topped the Fortune 25
Power List of interest groups in 1998 and 2001, respectively–are not among the top 100 contributors to candidates
or the top 30 biggest spenders when it comes to lobbying expenditures (source: Center for Responsive Politics).
3
Henceforth, I use pronouns ‘he’ and ‘she’ for the government and an SIG, respectively.
2
2
strengths (e.g., a phone call to a Congressman), and inside lobbying expenditures, part of them
could be direct financial contributions to the government. After observing the SIGs’ signals, the
government chooses the content of a bill. In the final stage, the pro- and anti-government SIGs
choose whether to engage in outside lobbying.4
An anti-government SIG can use inside lobbying expenditures to avoid paying the cost of outside
lobbying activities. I show that an anti-government SIG has sufficient incentives to incur inside
lobbying expenditures only for a certain range of parameter values; that is, only when the antigovernment SIG’s capacity to bear the cost of outside lobbying activities lies in an intermediate
range. As a consequence, for relatively strong and relatively weak SIGs, there is no correlation
between inside lobbying expenditures and political decisions.
A pro-government SIG can subsidize part of the government’s cost of defending his proposal
against the attacks of groups opposed to it. But she would rather have the government bear
this cost in full. The government and the pro-government SIG thus have a conflict of interests
concerning the non-policy dimension. A pro-government SIG incurs inside lobbying expenditures
to avoid bearing the cost of outside lobbying activities. Weak pro-government SIGs do not want
to engage in outside lobbying and thus use inside lobbying expenditures to “plead poverty”. I
demonstrate that for a range of parameter values, more inside lobbying expenditures lead to less
favorable political choices (the content of a bill) and outcomes (whether a bill is enacted) for a progovernment SIG. For some other parameter values, inside lobbying expenditures are uncorrelated
with political decisions.
By analyzing a formal model of inside and outside lobbying, this paper provides a new solution
to the old puzzle of how SIGs obtain favorable policies. As Leech (2010) argues, “The search for
a definitive statement about the power of lobbyists has become the Holy Grail of interest group
studies.” The theoretical findings explain why inside lobbying expenditures are not a sufficient
statistic to understand and measure SIGs influence. There is no clear correlation between inside
lobbying expenditures and an SIG’s capacity to obtain favorable policy. As a consequence, empirical tests of SIGs influence which exclusively consider this type of expenditures are bound to
produce inconsistent effect since the correlation between inside lobbying expenditures and influence
can be positive (as in Stratmann, 2002; Richter et al., 2009; Kang, 2013, e.g.), null (as in Grenzke,
1989; Bronars and Lott Jr., 1997, e.g.), or even negative (as in Wawro, 2001, e.g.) depending on
4
The paper considers only legal (and thus supposedly observable) means of influence; SIGs cannot bribe politicians.
3
the SIG’s strength. Finally, the model predicts that an anti-government SIG engages in outside
lobbying activities to complement her inside lobbying expenditures with positive probability when
a pro-government SIG is active.
The rest of the paper is organized as follows. Section 2 describes extant models of SIG influence.
In Section 3, I solve a model where the government faces only an anti-government SIG. In Section
4, I analyze a setting where an anti-government and a pro-government SIGs are present. Section
5 concludes. All proofs appear in the appendix, and in a supplemental appendix, I use the War
of Information developed by Gül and Pesendorfer to micro-found the influence of outside lobbying
on public opinion.
2
Existing models of SIG influence
It is well known in the literature on interest groups that SIGs use both inside and outside lobbying to influence political decisions. Inside lobbying is an attempt to affect the content of a bill
through transmission of information (Wright, 1996; Baumgartner and Leech, 1998; Ainsworth,
2002; Baumgartner et al., 2009). Congressional representatives use congressional hearings largely
to receive information regarding SIGs’ positions on a legislative proposal (Wright, 1996; Burstein
and Hirsh, 2007; Bertrand et al., 2011). Blaisdell (1957) describes testimonies in legislative committee hearings as “a common method of transmitting pressure to Congress.” SIGs often incur
costs (e.g., transporting witnesses to congressional hearings, contributions, hiring a lobbyist to
draft a legislative proposal, etc.) to bolster their credibility (Kollman, 1998; Lord, 2000).
Outside lobbying is an attempt to mobilize public opinion to alter the probability a bill is
enacted into law (Wright, 1996; Kollman, 1998; Hojnacki and Kimball, 1999; Baumgartner et
al., 2009; Nownes, 2013).5 In the words of Blaisdell (1957), it is a way to exert “maximum
persuasive power” on members of Congress “at the proper psychological moment”.6 Outside
lobbying activities can take the form of a media campaign such as the Harry and Louise ad
campaigns against Clinton’s Health Care Reform (West et al., 1996; Goldstein, 1999) or the tobacco
companies’ media efforts against the 1998 McCain Bill (Jamieson, 2000; Derthick, 2012). An SIG
can also hire petitioners to collect signatures or canvass to mobilize voters. Absent threat of outside
5
Political actors seem to make a similar distinction between the effect of both forms of SIGs’ mobilizations
(Lord, 2000).
6
In the literature, outside lobbying sometimes includes group members’ mobilization to get an issue on Congress’
agenda or electoral activities (Kollman, 1998). These behaviors are beyond the scope of this paper.
4
lobbying, the government is unconstrained because the public is inattentive. SIG mobilization
forces the government to justify his proposed policy change.7 Congressional representatives behave
strategically to avoid SIGs mobilizing against them in their district (Kingdon, 1981; Wolpe, 1990).
Nonetheless, the impact of outside lobbying on policy outcomes is uncertain since public reactions
are often unpredictable (Key, 1961; Anderson and Loomis, 1998).
In contrast, formal works generally consider only one form of SIG mobilization at a time,
and many theoretical papers assume inside lobbying expenditures buy favor from the government
directly (i.e., a quid pro quo relationship as in Grossman and Helpman, 1996 and 2001; Besley and
Coate, 2001) or indirectly (i.e., contributions buy access as in Lohmann, 1995a; Austen-Smith,
1998; Cotton, 2007, 2009 and 2011). In all of these papers, there is a direct link between inside
lobbying expenditures and policy choices. In comparison to empirical evidence (Ansolabehere et
al., 2003), these theoretical models overestimate the impact of this type of expenditures on political
decisions.
Two papers in the “contributions buy votes” literature assume SIGs have more than one way
to influence policy choices. Yu (2005) studies a model in which an environmentalist group and a
polluting industry can raise the salience of an issue (through outside lobbying) before engaging in
vote-buying contributions. Outside lobbying always complements inside lobbying, and the latter
is sufficient to measure SIG influence (contrary to empirical evidence). Bombardini and Trebbi
(2011) assume that SIGs can promise votes in addition to quid pro quo contributions. The authors
present empirical evidence that considering both types of mobilization better accounts for SIG
influence. Bombardini and Trebbi focus on the case where outside lobbying and inside lobbying
activities are pure substitute, unlike this paper where outside and inside lobbying activities do not
perform the same function. Bombardini and Trebbie also restrict attention to porks and subsidies;
they do not consider redistributive issues where one SIG’s gain is another’s loss.
Other studies assume SIGs use inside lobbying expenditures to credibly disclose information
about some of their characteristics as in Austen-Smith (1995), Ball (1995), Grossman and Helpman
(2001), and Gordon and Hafer (2005 and 2007).8 Gordon and Hafer (2005, 2007) consider a
7
Commenting on the success of the Harry and Louise campaign in the fight against Clinton’s Health Care
Reform in 1993, Bill McInturff, who helped in the campaign, explains, “In terms of the questions raised about
the “public policy process,” if the White House cannot build majority support faced with “soft” advertising that
raised simple and fundamental questions, it suggests to our firm that we have materially made a contribution to
the process by not allowing such a substantial piece of legislation to pass without a full airing of its consequences.”
(cited in Brodie, 2001).
8
Potters and Van Widen (1992), Lohmann (1995b), and Kollman (1998) consider the case in which SIGs use
inside lobbying to credibly reveal some information about the impact of a reform policy on voters’ welfare.
5
signaling model in which a firm signals to regulators her “willingness to fight” the decisions of a
regulatory agency, but do not explicitly model an SIG’s decision to contest the agency’s decisions.
Unlike the present paper, Gordon and Hafer cannot explain the interaction between inside and
outside lobbying, and how the latter limits the effect on political decisions of the former.
Several papers examine how threats and outside lobbying influence political decisions. Ellman
and Wantchekon (2000) study how the threat of conflict affects the electoral platforms of competing
parties. Dal Bó and Di Tella (2003) and Dal Bó et al. (2006) analyze how (physical) threats bias
policy in favor of powerful interest groups. These papers apply better to new democracies than
established ones like the United States. Kollman (1998) supposes that SIGs can raise the salience
of an issue to change a policy-maker’s decisions. Fox and Rothenberg (2011) show that politicians
cater to SIGs through fear of the SIG funding an opponent in a subsequent election. In all of these
papers, inside lobbying has limited use, unlike the present study.9
Finally, this paper relates to formal models in the international relations literature in which a
country can signal its resolve to its adversary in a territorial dispute and starts a conflict (Fearon,
1997; Arena, 2012). A country gains all or nothing in the conflict, and the cost of signaling is
constant across types. I relax these assumptions, which is significant since there is no separating
equilibrium when they hold. Furthermore, I also add signaling by allies.
3
Policy choices with a single SIG
I consider a two-player game with a government and an anti-government SIG who is opposed to
the government. The government chooses the content of a bill, b ∈ [0, 1]. After observing the
government’s proposal, the anti-government SIG decides whether to engage in outside lobbying to
mobilize public opinion and influence the probability that the government’s bill is enacted.10 The
anti-government SIG’s cost of outside lobbying activities–her strength–is her private information
(i.e., type). She is either strong (S) or weak (W), c̃a ∈ {caS , caW }, with 0 < caS < caW .11 It is
common knowledge that P rob(c̃ = caS ) = q a ∈ (0, 1). At the beginning of the game, the SIG can
decide whether to reveal her type to the government by sending a signal. The anti-government
9
In Dal Bó and Di Tella (2003) and Dal Bó et al. (2006), an SIG can bribe a decision-maker, but corruption
simply complements physical threats and so has little relevance by itself.
10
In the supplemental appendix, I show that the anti-government’s outside lobbying activities and the government’s response to it can be thought as a war of information to convince the public that the latter will benefit/be
hurt by the government’s proposed policy change (Gül and Pesendorfer, 2012).
11
All models in this paper can be solved with N types instead of two. This complicates analysis without providing
additional insights.
6
SIG’s signal takes the form of an announcement regarding the SIG’s type (t̂ ∈ {S, W }) and inside
lobbying expenditures, which play the role of signaling expenditures (lia ≥ 0): ζ a = (t̂, lia ).
When the anti-government SIG decides to engage in outside lobbying, the outcome depends
on the action the government chooses. If the government does not counter the anti-government
SIG’s attacks against his proposal, he must abandon his proposal, and the outcome is y = 0 with
probability 1 (0 can thus be assimilated to the status quo). When the government decides to
defend his proposal, he incurs costs, and the outcome of the game is stochastic. With probability
p, the SIG is successful at mobilizing public opinion against the government’s bill and the outcome
is: y = 0. Otherwise, the government is successful and his proposal is enacted: y = b. When the
anti-government SIG does not engage in outside lobbying, the outcome is y = b with probability
1.
The government’s utility increases with y. Therefore, everything else equal, the government’s
preferred outcome is y = 1. However, if the government decides to defend his proposal (g = 1),
he has to pay a cost k > 0. The government’s utility can depend on lia (if ν a > 0). Part of
the anti-government SIG’s inside lobbying (signaling) expenditures can thus be understood as a
transfer (i.e., contribution) to the government.12 The government’s utility function is:
ug (y, g) = y − kg + ν a lia , g ∈ {1, 0}, ν a ∈ [0, 1]
(1)
The SIG’s utility is decreasing with y, and her preferred outcome is y = 0. There thus is a
full conflict of interest on the policy dimension between the government and the anti-government
SIG. As for the government, outside lobbying is costly for the SIG and is parameterized by c̃a ,
which is the SIG’s private information. The SIG’s cost of outside lobbying activities depend
on her capacity to collect funds from her members, which is presumably unobservable by the
government (Ainsworth, 2002). There is thus some commonality of interest on the non-policy
dimension between the players. The cost of inside lobbying activities also depends on the SIG’s
fund-raising capacity, and thus depends on the SIG’s type, with the marginal cost of inside lobbying
expenditures equal to the cost of outside lobbying activities.13 The anti-government SIG’s utility
function is:
uasi (y, lia , loa ) = −αy − c̃a loa − c̃a lia , loa ∈ {1, 0}, α > 0
12
(2)
Results of this paper are unaffected if one assumes inside lobbying expenditures are contributions to an (unmodeled) opposition party and hurt the government (i.e., ν a < 0).
13
This assumption and the assumptions that the government and the SIG’s utility functions are linear in y are
for exposition purposes only; they do not drive the results of this paper.
7
The timing of the game is:
0. Nature draws the SIG’s cost of outside lobbying: c̃a ∈ {caS , caW }, with P rob(c̃a = caS ) = q a
1. After observing her type, the SIG sends a signal ζ a = (t̂, lia ) ∈ {S, W } × R+
2. The government chooses a bill: b ∈ [0, 1]
3. The anti-government SIG decides whether to engage in outside lobbying: loa ∈ {1, 0}
4. The government decides whether to defend his bill: g ∈ {1, 0}
Outcome:
1. If the anti-government SIG does not engage in outside lobbying (loa = 0): y = b


y = b with probability g(1 − p)
a
2. If the anti-government SIG engages in outside lobbying (lo = 1):

y = 0 with probability 1 − g(1 − p)
This game shares similarities with traditional signaling games, but includes an important variation.
In traditional signaling games, the Sender sends a signal, the Receiver decides what action to take
after observing the signal, and the game ends. Here, the Sender (the SIG) can react to the
Receiver’s action by engaging in outsside lobbying to mobilize public opinion (action loa = 1).
Inside lobbying expenditures increase the government’s utility, whereas outside lobbying expenditures impose a cost on the government.14 Moreover, inside lobbying expenditures occur before
the government drafts the details of the bill (i.e., chooses the value of b in this model) and can
influence its content. SIGs use these expenditures to credibly disclose information about some
of their characteristics as in Ball (1995), Austen-Smith (1995), Grossman and Helpman (2001),
Gordon and Hafer (2005, 2007).15 In contrast, outside lobbying expenditures are made after the
bill is drafted, and are intended to defeat the government’s legislative project.
The equilibrium concept used in this paper is Perfect Bayesian Equilibrium (PBE) in pure
strategies,16 a formal definition of which can be found in the Appendix (see Definition 2). In this
14
Campaign contributions reveal some information about the SIG’s strength, so there is no commitment problem
as in quid pro quo models (Grossman and Helpman,1996 and 2001).
15
One marginal difference is that in the studies cited, SIGs use inside lobbying expenditures to reveal information
about their preferences. The conclusions of this paper still hold as long as the government is uncertain about some
parameters of the SIG’s utility function.
16
Focusing on pure strategies is without loss of generality, except for knife-edge cases. A strong anti-government
SIG always pays the cost of outside lobbying activities if b > bS , and a weak SIG never does so if b ≤ bW (bJ is a
compromise bill with a type J SIG, J ∈ {S, W }, see below). The government thus never chooses a bill b such that
bS < b < bW since it gives him a lower expected utility than b = bW . In separating and semi-separating equilibria,
the government chooses between bS and bW and so allowing for mixed strategies, does not change the main results.
8
section and the remainder of the paper, out-of-equilibrium beliefs are refined using the Intuitive
Criterion (Cho and Kreps, 1987). In what follows, the term ‘equilibrium’ refers to this class of
equilibria.
Let’s denote by bS ≡
ca
S
αp
(bW ≡
ca
W
)
αp
the government’s legislative proposal, which makes strong
(weak) SIGs indifferent between engaging and not engaging in outside lobbying (given she expects
the government to defends his proposal). The bill bS (bW ) is the bill proposed by the government
when he wants to compromise with a strong (weak) SIG (see Lemma 1 in Appendix).
For the remainder of the paper, I assume the following inequality holds:
Assumption 1. The weak SIG is not too strong and not too weak:
max 1 − p − k,
k
1−p
≤ bW ≤ min
1−p−k
,1
p
The lower bound is a necessary condition for inside lobbying to have an effect on policy choices.
Inside lobbying cannot impact policy choice when the government never compromises with the antigovernment SIG (1 − p − k > bW ) or when he prefers his worst outcome to defending a compromise
bill ((1 − p)bW − k < 0). The upper bound is for exposition purposes only, and all results hold
when it is relaxed.
I study when a separating equilibrium exists. In such equilibrium, inside lobbying expenditures
influence political decisions strongly.
When a strong SIG (credibly) reveals her type, the government may choose a more moderate bill
(e.g., one closer to the status quo, her ideal point). This is traditional ‘benefit from differentiation’.
A strong SIG must pay a ‘signaling cost’ to disclose her private information credibly, as with all
signaling games. In addition, an SIG can mobilize public opinion with outside lobbying if she faces
a bill she dislikes. It is then less costly for a strong SIG to pretend to be weak; outside lobbying
is a strong SIG’s ‘outside option.’ A strong anti-government SIG is willing to separate from a
weak SIG only if her benefit from differentiation minus the cost of signaling (i.e., net benefit from
differentiation) is higher than her outside option; this inequality defines the strong SIG’s incentive
compatibility constraint. As Proposition 1 shows, this condition is satisfied when a strong SIG is
sufficiently different from a weak SIG, and when a strong SIG is not too strong.
Proposition 1. A separating equilibrium exists if and only if:
k
max 1 − p − k,
1−p
9
≤ bS ≤ (1 − p)bW
When a strong anti-government SIG is relatively similar to a weak anti-government SIG (bS >
(1 − p)bW ), she has little to gain from revealing her type, and it is very costly to do so since
the cost of signaling is type dependent. The net benefit from differentiation is very low and it is
impossible to satisfy the strong anti-government SIG’s incentive compatibility constraint because
of the SIG’s outside option, even though her cost of outside lobbying activities is large. When
a strong anti-government SIG is relatively strong (bS < 1 − p − k), the government prefers to
choose her preferred policy and defend it against the SIG’s attacks than compromise. The strong
anti-government SIG then has no incentive to reveal her type. When the strong SIG is very strong
((1 − p)bS − k < 0), the government prefers his worst outcome to defending the compromise bill.
In this latter case, the government wants to compromise, but is not credible doing so and knowing
the anti-government SIG always engages in outside lobbying, he proposes his preferred policy.17
When a separating equilibrium exists, there is a strong negative correlation between inside
lobbying expenditures and the content of a bill; a strong SIG incurs inside lobbying expenditures
in exchange for a bill closer to her preferred outcome. Inside lobbying expenditures are a sufficient
and necessary statistic to measure an SIG’s influence on political decisions since an anti-government
SIG does not engage in outside lobbying on the equilibrium path.
Corollary 1. In a separating equilibrium, an anti-government SIG does not engage in outside
lobbying on the equilibrium path.
A strong SIG’s inside lobbying expenditures depend on her cost of fighting (caS ). This cost
may reflect salience, import of an issue for the public. As a strong SIG becomes stronger (caS
decreases), her signaling expenditures increase (assuming a separating equilibrium exists). The
following corollary studies how inside lobbying expenditures influence the content of a law (how lia
affects b). I find that in line with intuition, as her inside lobbying expenditures increase, an SIG
obtains a more favorable piece of legislation (a lower b) from the government.18
Corollary 2. Denote by ζ a (caS ) and lia (caS ) a strong SIG’s signal and her inside lobbying expenditures in a separating equilibrium, respectively. The government’s proposal after he observes ζ a (caS ),
b(ζ a (caS )), is strictly decreasing with lia (caS ).
17
This game, like most signaling games, has many equilibria. However, Proposition 1 does not depend on
equilibrium selection.
18
To interpret this result, suppose that a strong SIG’s cost of fighting is drawn at the beginning of the game and
the weak SIG’s cost of fighting remains fixed. Both players observe the two costs (caW and caS ). Then Nature draws
the SIG’s type.
10
When a separating equilibrium exists, inside lobbying expenditures are a sufficient and necessary statistic to measure an SIG’s capacity to tilt political decisions in her favor because there is
one-to-one mapping between these expenditures and the risk of SIGs engaging in outside lobbying.
However, existence of a separating equilibrium is not always guaranteed in this model. When a
separating equilibrium does not exist (and we need to focus on pooling equilibria), the government
makes a policy choice based on the perceived threat of outside lobbying (i.e., his prior regarding
the probability he faces a strong type).
The threat of outside lobbying has an ambivalent effect depending on the strong SIG’s strength.
When a strong type is relatively strong (bS < max{1−p−k, k/(1−p)}), the government never wants
to (or cannot) compromise with a strong type; the government knows a strong SIG always engages
in outside lobbying. A high threat of outside lobbying (high q a ) reduces gains from compromising
with a weak type, and the government prefers b = 1. Inversely, when a strong SIG is relatively
weak, the government is always willing to compromise with at least one type (choosing b = 1 and
facing a group mobilizing public opinion is a strictly dominated strategy). High threat of outside
lobbying (high q a ) pushes the government to compromise with both types by choosing bS . This
result is summarized in Proposition 2.
Proposition 2. Suppose (1 − p)bW > 1 − p − k. In a pooling equilibrium:
n
o
k
i. When bS < max 1 − p − k, 1−p , there exists a unique q a ∈ (0, 1) such that the government
chooses b = bW if and only if q a ≤ q a . Otherwise, the government chooses b = 1.
n
o
k
ii. When bS > max (1 − p)bW , 1−p
, there exists a unique q a (bS ) ∈ (0, 1) such that the government chooses b = bS if and only if q a ≥ q a (bS ). Otherwise, the government chooses
b = bW .19
As the next corollary shows, we can observe inside lobbying expenditures in a pooling equilibrium, but there is no correlation between inside lobbying expenditures and the content of a bill.
These expenditures are only noise and cannot predict policy choices and outcomes.
Corollary 3. In a pooling equilibrium, SIGs can incur strictly positive signaling expenditures.
When a separating equilibrium does not exist, an anti-government SIG sometimes engages in
outside lobbying. These latter expenditures are also important to predict policy outcomes.
19
n
o
k
A pooling equilibrium also exists when max 1 − p − k, 1−p
≤ bS ≤ (1 − p)bW if and only if q a ≥ q a (bS ). A
pooling equilibrium when q a < q a (bS ) is ruled out by the Intuitive Criterion. When (1 − p)bW < 1 − p − k, one only
needs to replace (1 − p)bW with 1 − p − k above.
11
Corollary 4. In a pooling equilibrium, the probability of the anti-government SIG engaging in
o
n
k
outside lobbying is not null on the equilibrium path when bS < max 1 − p − k, 1−p or q a < q a (bS ).
Results of this section demonstrate that when the anti-government SIG can engage in outside
lobbying, a separating equilibrium exists only for certain parameter values, despite increasing differences in signaling costs. Inside lobbying expenditures are positively correlated with influence (a
group’s capacity to obtain more favorable policies) only when an SIG’s strength is in an intermediate range. For groups at both extremes (i.e., strong anti-government SIGs whose strength is a
relatively high or relatively low), the correlation between inside lobbying expenditures and influence is null. The unobservable government’s expectation regarding threat of outside lobbying is a
better predictor of policy choices (i.e., the content of a bill), and the observable outside lobbying
expenditures become a better predictor of policy outcomes (i.e., which bill is passed).
In the next section, I study a game when the government faces an anti-government SIG and a
pro-government SIG. When there is competition between SIGs, an anti-government SIG’s inside
lobbying expenditures are still not a sufficient statistic to measure her influence. Furthermore,
inside lobbying expenditures can be correlated negatively with a pro-government SIG’s influence.
4
Policy choices with competing SIGs
I now study a 3-player game with a government, a pro-government SIG, and an anti-government
SIG. As before, the government chooses a bill b ∈ [0, 1]. The anti-government SIG can decide
whether to engage in outside lobbying to mobilize public opinion against the government’s legislative proposal. If the anti-government SIG does not engage in outside lobbying, the game ends and
the outcome is the government’s proposal (y = b). When the anti-government SIG engages in outside lobbying, the government can decide to back down, ask for help, or defend his proposal alone.
If the government backs down, he must abandon his proposal (y = 0). If he fights alone, he incurs
a cost, and the outcome is stochastic as in Section 3. If he asks for help, the pro-government SIG
is pivotal. If the pro-government SIG pays the cost of outside lobbying activities, she subsidizes
part of the cost of defending the government’s bill, and the outcome is probabilistic as above. If
the pro-government SIG does not engage in outside lobbying (does not help), the anti-government
SIG is successful at mobilizing public opinion, with probability 1 the outcome is y = 0.
As in the previous section, I assume that an anti-government SIG’s strength (i.e., outside
lobbying capacity) is her private information. She is either strong (S) or weak (W): c̃a ∈ {caS , caW },
12
with 0 < caS < caW and P rob(c̃a = caS ) = q a ∈ (0, 1). I also assume that the pro-government
SIG’s cost of undertaking outside lobbying activities–her strength–is her private information. She
is either strong (S) or weak (W): c̃p ∈ {cpS , cpW }, with 0 < cpS < cpW and P rob(c̃p = cpS ) = q p ∈ (0, 1).
The pro-government and anti-government SIGs’ types are drawn independently, and both can
decide simultaneously to send a signal to the government at the beginning of the game. As above,
the anti-government (pro-government) SIG’s signal takes the form of an announcement regarding
her type and inside lobbying expenditures: ζ a = (t̂, lia ) (ζ p = (t̂, lip )) ∈ {S, W } × R+ .
As before and everything else equal, the government’s preferred outcome is y = 1. When the
government defend his proposal, he incurs a cost when he decides to defends it alone (g = 2) or ask
for support (g = 1). However, he pays a higher cost when he counters the anti-government SIG’s
attacks alone. Consequently, a pro-government SIG’s help is another form of legislative subsidy
(Hall and Deardoff, 2006). To simplify the exposition, I assume that the government’s cost doubles
when he defends his proposal alone.20 Part of the SIGs’ inside lobbying expenditures (lia and lip )
are contributions to the government. The government’s utility function is:
ug (y, g) = y −
kg
+ ν a lia + ν p lip , g ∈ {2, 1, 0}, (ν a , ν p ) ∈ [0, 1)2
2
(3)
The pro-government SIG’s ideal point is y = 1. She and the government have congruent
preferences on the policy dimension. The SIG pays the cost of outside lobbying activities if she helps
the government (lop = 1). As stated above, the pro-government SIG’s cost of outside lobbying (c̃p )
is her private information. This depends on an SIG’s capacity to collect funds from her members,
which are assumed unobservable by the government. Her capacity to collect influences the cost of
incurring inside lobbying expenditures. So the cost of the latter depends on the pro-government
SIG’s strength. To simplify the exposition, the marginal cost of inside lobbying expenditures is
the SIG’s type.21 The pro-government SIG’s utility function is:
upsi (y, lip , lop ) = γy − c̃p lop − c̃p lip , lop ∈ {1, 0}, γ > 0
(4)
The anti-government SIG’s utility function is the same as in Section 3:
uasi (y, lia , loa ) = −αy − c̃a loa − c̃a lia , loa ∈ {1, 0}
(5)
20
All results hold as long as his cost of fighting alone is strictly higher than his cost of fighting when he asks for
support.
21
This assumption, like the assumption of linear utility functions, is unessential to the results.
13
The timing and outcome of the game are:
0. Nature draws the anti-government SIG’s (c̃a ∈ {caS , caW }) and pro-government SIG’s (c̃p ∈
{cpS , cpW }) costs of outside lobbying independently.
1. After observing their type, both SIGs send simultaneously a signal: ζ a = (t̂, lia ) ∈ {S, W } ×
R+ , ζ p = (t̂, lip ) ∈ {S, W } × R+
2. The government chooses a proposal: b ∈ [0, 1]
3. The anti-government SIG decides whether to engage in outside lobbying: loa ∈ {1, 0}
4. The government decides to defend his proposal alone, ask for support, or back down: g ∈
{2, 1, 0}
5. The pro-government SIG decides to engage in outside lobbying: lop ∈ {1, 0}
Outcome:
1. If the anti-government SIG does not engage in outside lobbying (loa = 0): y = b
2. 
If the anti-government SIG engages in outside lobbying (loa = 1):

y = b with probability I{g+lp ≥2} (1 − p)
o

y = 0 with probability 1 − I p (1 − p)
{g+lo ≥2}
p
p
where I{g+lo ≥2} = 1 if g + lo ≥ 2, and 0 otherwise.
Since types are drawn independently, the pro-government SIG’s signal reveals no information
about the anti-government SIG’s strength, and vice versa. As before, inside lobbying expenditures
(e.g., contributions and hiring a lobbyist to draft a bill) can influence the content of a bill, and
outside lobbying expenditures (e.g., media expenditures, hiring canvassers) influence the fate of
the government’s proposal.
To make the problem interesting, I assume the following inequalities hold:
Assumption 2. The following inequalities hold:
1. The pro-government SIG’s costs of fighting satisfy: cpS < γ(1 − p) < cpW
2. The probability a pro-government SIG is strong satisfies: q p ≤ 1/2
14
If all types of pro-government SIG are willing to engage in outside lobbying, inside lobbying
expenditures have no influence on political decisions. Point 1 guarantees inside lobbying expenditures have some informational value. If the government’s default option is to ask for support, a
strong pro-government SIG has no interest in incurring inside lobbying expenditures to reveal her
strength. Point 2 guarantees the government always prefers to compromise if he does not know
whether he works with a strong pro-government SIG.
As before, the equilibrium concept is PBE in pure strategies, with out-of-equilibrium beliefs
refined using the Intuitive Criterion. I also impose the following equilibrium restriction; when
indifferent, the government prefers to compromise with the anti-government SIG and thereby
avoid paying the cost to counter her attacks.22
I first study how a pro-government SIG influences political decisions, and then examine how
competition between SIGs affects the correlation between the anti-government SIG’s inside lobbying expenditures and her influence.
Pro-government SIG and policy choices
I study how a pro-government SIG influences the government’s policy choice. To simplify the
exposition, I assume that a pro-government SIG reveals her type only if her signal influences the
government’s policy choice or strategy when an anti-government SIG mobilizes public opinion.23
Definition 1. The pro-government SIG plays a separating strategy only if her signal influences the
government’s actions: b(ζ a , ζ p (cpS )) 6= b(ζ a , ζ p (cpW )) and/or g(ζ a , ζ p (cpS )) 6= g(ζ a , ζ p (cpW )) for some
ζa
From the players’ utility functions (see (3) and (4)), the government and the pro-government
SIG have a conflict of interest regarding the non-policy dimension. The government wants the SIG
to subsidize the cost of defending the government’s proposal, and the SIG wants the government
to pay this cost in full. When the pro-government SIG plays a separating strategy,24 it must be
22
This restriction does not drive any result except in one knife-edge case: when the anti-government SIG does
not separate and the government is indifferent between a) compromising with a weak anti-government SIG and
defending his bill alone when the anti-government SIG engages in outside lobbying or b) compromising with both
types (see Lemma 5 for more details). Without this equilibrium criterion, Proposition 3 is generically true (i.e.,
everywhere, but for a set of measure 0 of parameter values).
23
There exist parameter values exist such that the government’s actions (i.e., policy choices b and strategy to
defend his proposal g) are independent of the pro-government SIG’s signal. I assume she plays a pooling strategy
since inside lobbying expenditures have no impact on political decisions in this case.
24
In Appendix, I show that there exist parameter values such that a separating strategy is the pro-government
SIG’s best response. As for the anti-government SIG, a separating strategy is not guaranteed for all parameter
values. See Lemma 8 for more details.
15
that an SIG is more prone to signal that she will not engage in outside lobbying to defend the
government’s proposal. A weak SIG is the least willing to pay the cost of outside lobbying and
should be the one incurring inside lobbying (signaling) expenditures. As the next proposition
shows, the weak pro-government SIG incurs inside lobbying expenditures to incentivize a strong
pro-government SIG to reveal her type.
Proposition 3. When the pro-government SIG plays a separating strategy on the equilibrium path,
a strong pro-government SIG are never incur any strictly positive inside lobbying expenditures.
Thus, only a weak pro-government SIG incurs positive inside lobbying expenditures.
Note that a weak pro-government SIG’s inside lobbying expenditures do not necessarily take
the form of contributions; a representative from a weak SIG could visit members of Congress and
“plead poverty.” This type of expenses is counted as lobbying expenditures according to the legal
definition of these activities (US Code Title 2 Chapter 26). A weak pro-government SIG does
not always incur inside lobbying expenditures. If a strong SIG’s cost of fighting is low enough, a
pro-government SIG’s type announcement is sufficient to reveal her type credibly. This is a major
difference with the anti-government SIG who must always incur inside lobbying expenditures to
reveal her type credibly.25
Although a strong SIG never incurs inside lobbying expenditures, she still obtains more favorable policy outcomes than a weak type. This result implies inside lobbying expenditures correlate
negatively with a pro-government SIG’s influence.
Proposition 4. When the pro-government SIG plays a separating strategy on the equilibrium path,
a strong pro-government SIG always obtains a more favorable policy outcome (in expectation) than
a weak pro-government SIG.
Interestingly, a weak pro-government SIG’s inside lobbying expenditures increase with the (expected) proposal the government chooses. However, it is incorrect to conclude from this result
that a pro-government SIG obtains a more favorable law because she increases her inside lobbying expenditures. Rather, the weak pro-government SIG needs to incur higher inside lobbying
expenditures to provide sufficient incentives for a strong pro-government SIG to reveal her type.
Corollary 5. When the pro-government SIG plays a separating strategy on the equilibrium path,
a weak pro-government SIG’s inside lobbying expenditures increases with the expected proposal the
government chooses when he knows he works with a weak pro-government SIG.
25
See the proof of Lemma 8 in the Appendix.
16
Results of this model suggest that by focusing solely on inside lobbying expenditures, researchers risk mismeasuring a pro-government SIG’s influence A better measure of influence is to
consider an SIG’s total expenditures: outside lobbying and inside lobbying expenditures combined.
In fact, a strong pro-government SIG’s observed total lobbying expenditures are always higher than
a weak SIG’s observed total lobbying expenditures.
Proposition 5. When the pro-government SIG plays a separating strategy on the equilibrium
path, a strong pro-government SIG’s observed total expenditures are always higher than a weak
pro-government SIG’s.
Inside lobbying expenditures have some influence on the content of a legislative proposal related
to the empirical findings of Hall and Wayman (1990).26 Outside lobbying expenditures influence
the fate of more dramatic policy changes. In some sense, they are a better predictor of how
legislators vote.27 Contributions go to allies, media campaigns target pivotal legislators, and total
expenditures measure influence.
To the best of my knowledge, no paper studies the impact of an SIG’s total expenditures
systemically. Ansolabehere et al. (2003) study contribution patterns, and many papers examine
the impact of contributions on members of Congress’ votes, finding statistically non-significant or
inconsistent results (Wright, 1985; Grenzke, 1989; Bronars and Lott Jr., 1997; Wawro, 2001). A few
papers study how lobbying affects policy choices (Richter et al., 2009; Kang, 2013). Ansolabehere
et al. (2002) link contributions and lobbying expenditures, but do not consider outside lobbing
expenditures.
Little is known about the effect of outside lobbying. Bergan (2009) finds that a grassroots
campaign had a significant effect on legislators’ votes in the New Hampshire state legislature
(for a more pessimistic assessment, see Fowler and Shaiko, 1987). A recent paper by Hall and
Reynold (2012) examines issue advocacy advertising expenditures. Hall and Reynold show that
SIGs’ advertising expenditures target swing legislators before important votes, but do not study
the impact of these expenditures on voting decisions.
Outside lobbying expenditures are not trivial. Falk et al. (2006) estimate that issue advocacy
advertising amounted to more than $400 million in the Washington DC media market alone dur26
Hall and Wayman (1990) do not study the effect of contributions on a bill’s content, but on committee members’
effort at the committee stage in Congress. However, if we assume that committee members’ effort affects the content
of a bill, Hall and Wayman’s conclusion that contributions buy time (to find a compromise, for example) can be
related to the results of this paper.
27
Since I consider a single decision-maker, how outside lobbying expenditures affect legislators’ vote cannot be
characterized fully.
17
ing the 108th Congress. In comparison, SIGs contributed approximately $570m to Members of
Congress, and spent $4bn on lobbying in the 2004 electoral cycle.28 Outside lobbying expenditures
can help understand how SIGs influence political decisions. For example, the Club for Growth
spent $1.5m in 2001 in ad campaigns to help pass Bush’s tax cut proposal. During the same period,
it contributed (through its members) less than $600,000 to various congressional representatives in
the 2000 electoral cycle (approximately $650,000 in the 2002 electoral cycle) and made no lobbying
expenditures.29 The influence of outside lobbying and SIGs’ total expenditures on policy choices
and outcomes remains untested.
Finally, this model stresses the importance of reverse lobbying, which has received no attention
in the formal literature, to understand pro-government SIGs’ influence. Baumgartner et al. (2009)
argue that lobbying is not a unidirectional process; legislators (and high-ranking civil servants) are
active lobbying actors who mobilize groups to facilitate or block policy change. Democratic and
Republican leaders in Congress occasionally ask for support from SIGs to implement their agendas
(Blaisdell, 1957; Shaiko, 1998; Ainsworth, 2002; Andres, 2009). The President also engages in
reverse lobbying. In 1993, President Clinton asked business groups to defend NAFTA against
trade union attacks (Kollman, 1998), and then enlisted trade unions to defend his Health Care
reform against attacks from business groups (Goldstein, 1999). More recently, President Obama
also secured help from various SIGs in the legislative fight to get the Affordable Care Act enacted
(Hall and Anderson, 2012; LaPira, 2012).
Anti-government SIG and policy choices
I now study an anti-government SIG’s incentives to play a separating strategy amidst competition
between SIGs. Her incentives are affected by a pro-government SIG’s presence only if the latter
separates. When the government knows he works with a strong pro-government SIG, he has fewer
incentives to compromise with a strong anti-government SIG since his ally subsidizes the cost of
defending his proposal partially. This naturally reduces the anti-government SIG’s benefits from
differentiation and incentives to separate.
Proposition 6.
i. When the pro-government SIG does not separate, an anti-government SIG’s
separating strategy is the best response to other players’ actions if and only if:
28
Source: Center for Responsive Politics–contributions to presidential candidate and Senator John Kerry were
excluded from the total.
29
Data on advertising expenditures were taken from Hamburger and VandeHei (2001). Data on campaign
contributions were taken from the Center for Responsive Politics.
18
n
max 1 − p − k,
k
1−p
o
≤ bS ≤ (1 − p)bW .
ii. When the pro-government SIG separates, there exists a unique b̄se ≤ (1 − p)bW such that
an anti-government SIG’s separating strategy is the best response to other players’ actions if
and only if:
n
o
k
max 1 − p − k, 1−p
≤ bS ≤ b̄se
I can now demonstrate that there exist parameter values such that both SIGs play a separating
strategy on the equilibrium path (using Proposition 6 and Lemma 8 in Appendix). In Proposition
7, I demonstrate that a strong anti-government SIG who reveals her type with inside lobbying
expenditures is uncertain to find a compromise with the government, and sometimes must engage
in outside lobbying.30 Despite that both forms of mobilization are strategic substitutes, outside
lobbying complements inside lobbying on some issues as documented in a variety of prominent
legislative battles, including Clinton’s 1993 healthcare reform (West et al., 1996; Goldstein, 1999),
McCain’s 1998 Senate bill that targeted tobacco companies (Jamieson, 2000; Derthick, 2012), and
Obama’s 2010 Affordable Care Act (Hall and Anderson, 2012; LaPira, 2012).
Proposition 7. There exists an open non-empty set of parameter values such that on the equilibrium path, an anti-government SIG makes inside lobbying expenditures and engages in outside
lobbying.
When a pro-government SIG is active (i.e., separates), the anti-government SIG’s inside lobbying expenditures do not necessarily influence a bill’s content. The correlation between these
expenditures and an anti-government SIG’s influence is weak because a strong pro-government
SIG has greater influence on policy choice than a strong anti-government SIG. In addition to inside lobbying expenditures, any measure of influence must include whether a pro-government SIG
is active and considers whether the anti-government SIG engages in outside lobbying.31
5
Conclusion
I analyze a game-theoretic model in which SIGs can use both inside lobbying (e.g., contributions
and hiring lobbyists) to influence the content of a bill, and outside lobbying (e.g., paid media
30
Contrary to a pooling equilibrium (see Corollaries 3 and 4), the anti-government SIG uses inside lobbying
expenditures to influence the content of a bill.
31
When an anti-government SIG does not separate, the choice of the government depends on the threat of outside
lobbying and the pro-government SIG’s strategy. An anti-government SIG’s inside lobbying expenditures have no
impact on policy choices, and the logic of Proposition 2 applies.
19
and hiring canvassers) to influence the fate of a government’s legislative proposal. When SIGs
can engage in outside lobbying, inside lobbying expenditures is not an adequate measure of SIGs
influence (the capacity to tilt political decision in their favor). Inside lobbying expenditures are
uncorrelated with influence for relatively weak and relatively strong anti-government SIGs. Inside
lobbying expenditures correlate negatively with a pro-government SIG’s influence. Empirical tests
of SIGs influence which consider only inside lobbying expenditures will produce inconsistent results.
The theoretical findings do not imply that SIG influence on policy choices and outcomes is marginal.
SIGs have a strong impact on political decisions through threat of or participation in outside
lobbying activities.
20
Appendix: Proofs
Definition 2. A PBE in pure strategies consists of: 1) the government’s decision to defend his
proposal (henceforth fighting decision): g ∗ (b, loa ) ∈ {1, 0}, 2) the anti-government SIG’s decision to
engage in outside lobbying (henceforth fighting decision): loa∗ (b) ∈ {1, 0}, 3)the choice of a proposal
by the government: b(ζ a ) ∈ [0, 1], 4) the anti-government SIG’s signal: ζ a (ca ) = (t̂(ca ), lia (ca )) ∈
{S, W } × R+ , 5) and beliefs P r(c̃a ≤ ca |ζ a ) satisfying the following conditions:
C1: g ∗ (b, loa ) = 1 if and only if (iff ): I{loa =1} (1 − p)b − k ≥ 0, ∀b ∈ [0, 1] (I{loa =1} = 1 iff loa = 1)
C2: loa∗ (b) = 1 iff: I{g=1} (−(1 − p)αb) − ca > −αb, ∀b ∈ [0, 1], ca ∈ {caS , caW }
C3: b(ζ a ) ∈ argmaxb∈[0,1] P rob(c̃a < α[1 − I{g=1} (1 − p)]b|ζ a )[I{g=1} (1 − p)b − k] + (1 − P rob(c̃a <
α[1 − I{g=1} (1 − p)]b|ζ a ))b
C4: Given the beliefs of the government: ζ a (ca ) ∈ argmaxt̂,
ca ] + 1 − I{loa =1} (−αb(ζ a )) − ca lia
lia ≥0
I{loa =1} [I{g=1} (−(1 − p)αb(ζ a )) −
C5: Beliefs, P r(c̃a = ca |ζ a ), ca ∈ {caS , caW }, satisfy Bayes’ rule whenever possible.
I assimilate participation in outside lobbying activities as a political conflict in what follows. I
first state three preliminary lemmata.
Lemma 1. In a separating assessment, when the government chooses to avoid a conflict with a type
J anti-government SIG (J ∈ {S, W }), he chooses a proposal: bJ =
ca
J
αp
whenever (1 − p)bJ − k ≥ 0
Proof. The government makes a take-it-or-leave-it offer to the anti-government SIG. He thus
chooses a proposal that makes a type J anti-government SIG indifferent between fighting or not
fighting (J ∈ {S, W }). Therefore, we have bJ such that: −αbJ = −(1 − p)αbJ − caJ ⇔ bJ =
ca
J
.
αp
Note that since (1 − p)bJ − k ≥ 0, the anti-government SIG expects the government to fight (g = 1)
if she starts a conflict (loa = 1).
Note that Lemma 1 and Assumption 1 imply that the government always chooses b = bW when
he knows he faces a weak SIG.
Lemma 2. In a separating equilibrium (supposing it exists), we must have: b(ζ a (caS )) < b(ζ a (caW )).
Proof. The proof is by contradiction. Suppose b(ζ a (caS )) > b(ζ a (caW )) = bW . The weak and strong
anti-government SIGs’ incentive compatibility constraints (IC) are, respectively:
21
−αbW − caW lia (caW ) ≥ −(1 − p)αb(ζ a (caS )) − caW − caW lia (caS )
−(1 − p)αb(ζ a (caS )) − caS − caS lia (caS ) ≥ −(1 − p)αbW − caS − caS lia (caW )
When a strong anti-government SIG imitates a weak anti-government SIG, she starts a conflict
with the government. And the government fights back by Assumption 1. Using the definition of
bW (see Lemma 1), this equivalent to:
caW (lia (caW ) − lia (caS )) ≤ (1 − p)α(b(ζ a (caS )) − bW )
caS (lia (caW ) − lia (caS )) ≥ (1 − p)α(b(ζ a (caS )) − bW )
Suppose lia (caS ) ≥ lia (caW ), then the left hand side of the second inequality is weakly negative,
whereas the right hand side is strictly positive since b(ζ a (caS )) > bW . This is impossible. Suppose
lia (caW ) > lia (caS ), we cannot satisfy the two inequalities simultaneously (since caS < caW ). Hence we
have reached a contradiction.
Lemma 3. In a separating equilibrium (supposing it exists), lia (caW ) = 0 and lia (caS ) > 0.
Proof. The proof of lia (caW ) = 0 is by contradiction and follows directly from Lemma 2. Using
Lemma 2 (i.e., b(ζ a (caS )) < bW ), a weak type’s incentive compatibility constraint is never satisfied
when lia (caS ) = 0, which proves point ii.
Proof of Proposition 1. Consider the following separating assessment. A strong type sends a signal ζ a (caS ) = (S, lia (caS )) and a weak type sends a signal ζ a (caW ) = (W, 0). The posterior of the
government is: P rob(c̃a = caS |ζ a ) = 1 if ζ a = (S, lia ) with lia ≥ lia (caS ) and 0 otherwise.
Suppose that the government compromises in a separating assessment. The government chooses
b = bS after observing ζ a = ζ a (caS ) and b = bW , otherwise. This assessment is a separating
equilibrium only if the Incentive Compatibility constraints (IC) of a strong and weak types are
satisfied. The weak type’s (IC) is: −αbW ≥ −αbS − cW lia (caS ). By the Intuitive Criterion, we have:
lia (caS ) =
α(bW −bS )
.
ca
W
The strong type’s (IC) is: −αbS − caS lia (caS ) ≥ −(1 − p)αbW − caS (she starts a
conflict when she mimics a weak type). Therefore, we must have:
−αbS − caS lia (caS ) ≥ −(1 − p)αbW − αpbS
⇔ (1 − p)(bW − bS ) ≥
caS
(bW − bS )
caW
⇔ (1 − p)bW ≥ bS
Where the first and last lines use the definition of bJ (J ∈ {S, W }, see Lemma 1). This is one of
the necessary condition for a separating equilibrium to exist.
22
It must also be that the government’s best response is b(ζ a (caS )) = bS (by Lemma 2). When
bS < 1 − p − k, the government chooses b(ζ a (caS )) = 1. Therefore, the separating assessment
is not an equilibrium in this case. When (1 − p)bS − k < 0, the government does not fight
in a political conflict. The anti-government SIG then always starts a conflict if the government
chooses bS . To compromise with the strong anti-government SIG, the government must then choose
b(ζ a (caS )) = caS /α. By Assumption 1, we have:
ca
S
α
= pyS < pyW ≤ 1 − p − k. So the government’s
best response is b(ζ a (caS )) = 1 and a separating assessment cannot be an equilibrium.
o
n
k
.
We thus have a second necessary condition bS ≥ max 1 − p − k, 1−p
n
o
k
To see that max 1 − p − k, 1−p
≤ bS ≤ (1−p)bW is a sufficient condition, we can consider the
following assessment. Define the belief of the government as P rob(c̃a = caS |ζ a ) = 1, ∀ζ a = (S, lia )
with lia ≥
α(bW −bS )
ca
W
and P rob(c̃a = caS |ζ a ) = 0, otherwise. Suppose the anti-government SIG and the
government’s actions are: ζ a (caS ) = (S, α(bWca−bS ) ), ζ a (caW ) = (W, 0), b(ζ a (caS )) = bS , and b(ζ a (caW )) =
W
bW , a type J SIG does not fight if b ≤ bJ (and (1 − p)bJ − k ≥ 0) and fight otherwise (J ∈ {S, W }),
the government fights if and only if the anti-government SIG fights and (1 − p)b − k ≥ 0. Using the
reasoning above, it is easy to check that 1) the belief correspondence satisfies Bayes’ Rule and the
Intuitive Criterion, 2) that b(ζ a (caS )) and b(ζ a (caW )) are the government’s best response to ζ a (caS )
and ζ a (caW ), 3) that the SIG’s decision whether to fight is a best response, and 4) that the (IC)
constraints are satisfied. Hence, this separating assessment is an equilibrium.
Proof of Corollary 1. From the proof of Proposition 1, in a separating equilibrium, we have b(ζ a (caS )) =
bS and b(ζ a (caW )) = bW . A SIG never starts a conflict then.
Proof of Corollary 2. From the proof of Proposition 1, lia (caS ) =
α(bW −bS )
ca
W
which decreases with
bS .
Proof of Proposition 2. In a pooling equilibrium, we have ζ a (caS ) = ζ a (caW ) ≡ ζ a = (t̂, lia ) and the
government’s belief is P r(c̃a = caS |ζ a ) = q a .
The government has three possible choices: i) choose b(ζ a ) = 1, fight both types, and get
EuFg (1) = 1 − p − k; ii) choose b(ζ a ) = bW , fight a strong SIG (with probability q a ), and get:
EuPg C (1) = q a [(1 − p)bW − k] + (1 − q a )bW ; iii) choose b(ζ a ) = bS , never fight, and get: uC
g = bS .
When bS < max{1 − p − k, k/(1 − p)}, b(ζ a ) = bS is strictly dominated by b(ζ a ) = 1. Simple
computations show the government chooses b(ζ a ) = bW if and only if q a ≤ q a ≡
23
bW −(1−p−k)
.
pyW +k
When bS > max{(1 − p)bW , k/(1 − p)}, b(ζ a ) = 1 is strictly dominated by b(ζ a ) = bS . Simple
computations show the government chooses b(ζ a ) = bw if and only if : qa ≥ q a (bS ) ≡
bW −bS
.
pyW +k
Proof of Corollary 3. For simplicity, I ignore the type announcement in the signal sent by the SIG
focus only on the SIG’s signaling expenditures: lia . This is without loss of generality by Lemma 3.
Suppose first that bS < max{1−p−k, k/(1−p)}. Consider the following government’s posterior:
P rob(c̃a = caS |lia < lbia ) = 1 and P rob(c̃a = caS |lia ≥ lbia ) = q a .
From the proof of Proposition 1, we know that the government chooses b(lia ) = 1, ∀lia < lbia .
After observing lia ≥ lbia , the government chooses: b(lia ) = 1 if q a > q a and b(lia ) = bW otherwise by
Proposition 2.
Now lbia must be such that the anti-government SIGs’ incentive compatibility constraints are
satisfied. Denote by lia the supremum signaling cost such that an anti-government SIG’s incentive
compatibility constraint is satisfied. When q a > q a , we have lia = 0. When q a ≤ q a , lia must be
such that:
−α(1 − p) − caS ≤ −α(1 − p)bW − caS − caS lia
−α(1 − p) − caW ≤ −α(1 − p)bW − caW − caW lia
where the second line uses the definition of bW . Simple computations lead to lia =
α(1−p)(1−bW )
.
ca
W
Therefore ∀lia ∈ [0, lia ], the anti-government SIG’s incentive compatible constraints are satisfied
and a pooling equilibrium with strictly positive signaling expenditures exists.
For the case when bS > max{(1 − p)bW , k/(1 − p)}. Consider the following government’s
posterior: P rob(c̃a = caS |lia < lbia ) = 0 and P rob(c̃a = caS |lia ≥ lbia ) = q a .
From the proof of Proposition 1, we know that the government chooses b(lia ) = bW , ∀lia < lbia .
After observing lia ≥ lbia , the government chooses: b(lia ) = bS if q a ≥ q a (bS ) and b(lia ) = bW otherwise
(see the proof of Proposition 2).
When q a < q a (bS ), then lia = 0. When q a ≥ q a (bS ), then ∀lia ∈ [0, 1 − α(bS − (1 − p)bW )/caS ],
the anti-government SIG’s incentive compatible constraints are satisfied and a pooling equilibrium
with strictly positive signaling expenditures exists.
Proof of Corollary 4. In a pooling equilibrium, when the government chooses b(ζ a ) > bS , then a
strong anti-government SIG starts a conflict on the equilibrium path. From Proposition 2, this
occurs when bS < max{1 − p − k, k/(1 − p)} or q a > q a (bS ).
24
I now prove the results of the model with competing SIGs. I introduce the following notation.
Let’s denote (b(ζ a (caJ ), ζ p (cpL )), g(b, loa )) the government’s best response: respectively policy choice
after observing signal ζ a (caJ ) = (t̂, lia (caJ )) and ζ p (cpL ) = (t̂, lip (cpL )), (J, L) ∈ {S, W }2 and fighting
decision after choosing policy b and observing the anti-government SIG’s choice whether to fight
(loa ). I also denote E(ug (b, g ∗ (b, loa ))|ζ a , ζ p ) the government’s expected utility from choosing b given
the government’s optimal fighting decision and the signals ζ a and ζ p from the anti-government and
pro-government SIGs, respectively.
I first study when a pro-government SIG’s best response is to play a separating strategy (as
defined in Definition 1). I suppose a strong (weak) pro-government SIG sends signal ζ p (cpS )
(ζ p (cpW ) 6= ζ p (cpS )). After observing ζ p (cpS ) (ζ p (cpW )), the government believes he works with a
strong (weak) pro-government SIG.
I describe below the players’ best response down the game tree (after the signaling and policy
choice stages). We have:
i. The anti-government SIG chooses loa = 1 if and only if (1 − p)b − caJ > 0, J ∈ {S, W }.
ii. The government chooses g(b, 0) = 0, ∀b ∈ [0, 1].
After observing ζ p = ζ p (cpS ) (6= ζ p (cpW )), he chooses:
a) g(b, 1) = 1 if (1 − p)b − k/2 ≥ 0 and (1 − p)b − cpS ≥ 0,
b) g(b, 1) = 2 if (1 − p)b − k ≥ 0 and (1 − p)b − cpS < 0,
c) g(b, 1) = 0 otherwise.
After observing ζ p = ζ p (cpW ), he chooses:
a) g(b, 1) = 2 if (1 − p)b − k ≥ 0,
b) g(b, 1) = 0 otherwise.
iii. A weak pro-government SIG never helps (lop (b, g; cpW ) = 0, ∀b ∈ [0, 1]);
a strong government helps lop (b, g; cpS ) = 1 if and only if (1 − p)b − cpS ≥ 0.
The following lemmata provide the key elements to prove Propositions 3 and 4.
Lemma 4. When an anti-government SIG and a pro-government SIG separate, we must have:
b(ζ a (caS ), ζ p (cpS )) = 1 and b(ζ a (caS ), ζ p (cpW )) = bS .
25
Proof. The proof is by contradiction. Suppose we have: b(ζ a (caS ), ζ p (cpS )) = 1 and b(ζ a (caS ), ζ p (cpW )) =
1. This implies that we have: b(ζ a (caW ), ζ p (cpS )) = 1 and b(ζ a (caW ), ζ p (cpW )) = bW or else either
the anti-government SIG or the pro-government SIG does not separate. But this implies that
b(ζ a (caS ), ζ p (cpW )) > b(ζ a (caW ), ζ p (cpW )). Using a similar logic as in Lemma 2, we can see that this
cannot be an equilibrium.
Suppose we have: b(ζ a (caS ), ζ p (cpS )) = bS and b(ζ a (caS ), ζ p (cpW )) = bS . Then it must be that
b(ζ a (caW ), ζ p (cpS )) = bW and b(ζ a (caW ), ζ p (cpW )) = bW (since bW > bS and since Assumption 1 holds).
But, in this case, a pro-government SIG does not play a separating strategy by definition. Hence
we have reached a contradiction.
Note that Lemma 4 implies: b(ζ a (caW ), ζ p (cpW )) = bW .
When a separating strategy is the pro-government SIG’s best response, denote p(lop = 1|ζ p (cpL ))
the probability that a type L pro-government SIG helps after sending signal ζ p (cpL ), L ∈ {S, W }
on the equilibrium path.
Lemma 5. A separating strategy is a best response for the pro-government SIG only if: i) p(lop =
1|ζ p (cpS )) > 0 and ii) p(lop = 1|ζ p (cpW )) = 0.
Proof. Point ii.
follows directly from Assumption 2.
Point i.
is always satisfied when the
anti-government SIG separates by Lemma 4. When the anti-government SIG does not separate
(ζ a (caS ) = ζ a (caW ) = ζ a ), the proof is by contradiction. Suppose p(lop = 1|ζ p (cpS )) = 0. This implies:
E(ug (b, g ∗ (b, loa ))|ζ a , ζ p (cpS )) = E(ug (b, g ∗ (b, loa ))|ζ a , ζ p (cpW )) since the government receives no help.
So it must be that b(ζ a , ζ p (cpS )) = b(ζ a , ζ p (cpW )) (since we assume that the government prefers to
avoid conflict when indifferent) and g(ζ a , ζ p (cpS )) = g(ζ a , ζ p (cpW )). But then a pro-government SIG
does not separate by Definition 1.32
Denote Y e (ζ p (cpL )), L ∈ {S, W } a type L pro-government SIG’s expected policy payoff from
revealing her type and Ŷ e (ζ p (cp−L ), cpL ) her expected policy payoff from pretending to be type -L,
L 6= −L, L ∈ {S, W }.
32
The assumption that the government avoids conflict when indifferent excludes the following case. Suppose that i) cpS > γ(1 − p)bW , ii) E(ug (bW , g ∗ (bW , loa ))|ζ a , ζ p ) = (1 − q a )bW + q a ((1 − p)bW − k) = bS =
E(ug (bS , g ∗ (bW , loa ))|ζ a , ζ p ), and iii) E(ug (bS , g ∗ (bS , loa ))|ζ a , ζ p ) ≥ E(ug (1, g ∗ (1, loa ))|ζ a , ζ p (cpS )) = 1 − p − k/2, ζ p ∈
{ζ p (cpS ), ζ p (cpW )}. For these parameter values, we can have the following equilibrium: ζ p (cpS ) 6= ζ p (cpW ),
b(ζ a , ζ p (cpS )) = bW and b(ζ a , ζ p (cpW )) = bS and p(lop = 1|ζ p (cpS )) = p(lop = 1|ζ p (cpW )) = 0. This equilibrium
relies strongly on the assumption that the government chooses bW (bS ) after observing ζ p (cpS ) (ζ p (cpW )) when indifferent between both policy choices. Note that, in this particular case, a strong pro-government SIG makes strictly
positive signaling expenditures (lip (cpS ) > 0 and lip (cpW ) = 0).
26
Lemma 6. The pro-government SIG plays a separating strategy on the equilibrium path only if:
Y e (ζ p (cpW )) < Y e (ζ p (cpS ))
Proof. Since p(lop = 1|ζ p (cpW )) = 0 (Lemma 5), we have Ŷ e (ζ p (cpW ), cpS ) = Y e (ζ p (cpW )). Since
p(lop = 1|ζ p (cpS )) > 0, we have: γ Ŷ e (ζ p (cpS ), cpW ) > γY e (ζ p (cpS )) − p(lop = 1|ζ p (cpS ))cpW (a weak type
never helps as it is a strictly dominated strategy by Assumption 2).
The proof is by contradiction. Suppose Y e (ζ p (cpS )) ≤ Y e (ζ p (cpW )). In a separating assessment,
the incentive compatibility constraint (IC) must be satisfied. The strong and weak types’ (IC) are,
respectively:
γY e (ζ p (cpS )) − p(lop = 1|ζ p (cpS ))cpS − cpS lip (cpS ) ≥ γY e (ζ p (cpW )) − cpS lip (cpW )
γY e (ζ p (cpW )) − cpW lip (cpW ) ≥ γ Ŷ e (ζ p (cpS ), cpW ) − cpW lip (cpS )
(6)
(7)
The first inequality is never satisfied when lip (cpS ) > 0 since Y e (ζ p (cpS )) ≤ Y e (ζ p (cpW )). So we
must have lip (cpS ) = 0 and lip (cpW ) > 0. This implies: lip (cpW ) =
γ(Y e (ζ p (cpW ))−Y e (ζ p (cpS )))+p(lop =1|ζ p (cpS ))cpS
.
cpS
Plugging this in the weak type’s (IC) we get the following necessary condition:
[γ(Y e (ζ p (cpW )) − Ŷ e (ζ p (cpS ), cpW )) − p(lop = 1|ζ p (cpS ))cpW ]cpS ≥ γ(Y e (ζ p (cpW )) − Y e (ζ p (cpS )))cpW
(8)
From the reasoning above, we have: γ(Y e (ζ p (cpW )) − Ŷ e (ζ p (cpS ), cpW )) − p(lop = 1|ζ p (cpS ))cpW <
γ(Y e (ζ p (cpW )) − Y e (ζ p (cpS ))). Since 0 < caS < caW , (8) can never be satisfied. We have thus reached
a contradiction.
Lemma 7. When a pro-government SIG plays a separating strategy, we must have: lip (cpS ) = 0.
Proof. We just need to show that Y e (ζ p (cpW )) > Ŷ e (ζ p (cpS ), cpW ), ∀ζ a . The Intuitive Criterion then
implies lip (cpS ) = 0.
Suppose the anti-government SIG separates (ζ a (caS ) 6= ζ a (caW )) so the government believes he
faces a strong (weak) anti-government SIG after observing ζ a (caS ) (ζ a (caW )). Then by Lemma 4,
we have: b(ζ a (caS ), ζ p (cpW )) = bS and b(ζ a (caW ), ζ p (cpW )) = bW . So Y e (ζ p (cpW )) = q a bS + (1 − q a )bW .
We also have b(ζ a (caS ), ζ p (cpS )) = 1 and b(ζ a (caW ), ζ p (cpS )) ∈ {bW , 1}. The government chooses
g ∗ (1, loa = 1) = 1 after observing ζ p (cpS ) and choosing b(ζ a , ζ p (cpS )) = 1, ζ a ∈ {ζ a (caS ), ζ a (caW )}.
Therefore, we have: Ŷ e (ζ p (cpS ), cpW ) ≤ (1 − q a )bW < Y e (ζ p (cpW )).
Suppose the anti-government SIG does not separate (ζ a (caS ) = ζ a (caW ) = ζ a ). From Lemma
6, we know that if b(ζ a , ζ p (cpW )) = bW , then we must have b(ζ a , ζ p (cpS )) = 1. This implies that
27
Ŷ e (ζ p (cpS ), cpW ) = 0 < (1 − q a p)bW = Y e (ζ p (cpW )). The left hand side follows from the fact that
the government asks for help in a conflict (which occurs with probability 1) and the weak progovernment SIG never helps so the status quo is upheld. The right hand side comes from the fact
that the anti-government SIG fights if she is strong (i.e., with probability q a ) when b(ζ a , ζ p (cpW )) =
bW , the government fights back alone (g ∗ (bW , 1) = 2 after observing ζ p (cpW ) by Assumptions 1 and
2) and wins the conflict with probability 1 − p.
Suppose b(ζ a , ζ p (cpW )) = bS . If b(ζ a , ζ p (cpS )) = 1, then the claim holds. Suppose b(ζ a , ζ p (cpS )) =
bW . The government’s best response after observing ζ p (cpS ) and loa = 1 is g ∗ (bW , 1) = 1 (ask for
help) and the strong pro-government SIG chooses lop (bW , loa ; cpS ) = 1 (see Lemma 5). So we have:
Ŷ e (ζ p (cpS ), cpW ) = (1 − q a )bW (a weak type never fights). But b(ζ a , ζ p (cpW )) = bS is a government’s
best response only if: bS ≥ E(ug (bW , g ∗ (bW , loa ))|ζ a , ζ p (cpW )) = (1 − q a )bW + q a ((1 − p)bW − k). The
government knows he must fight alone after observing ζ p (cpW ). By choosing bW , he has to fight
with probability q a (see Proposition 2 for more details). Since (1 − p)bW − k > 0 by Assumption
1, it must be that Y e (ζ p (cpW )) = bS > (1 − q a )bW .
We now need to prove that there exists parameter values such that a separating strategy is a
pro-government SIG’s best response to other players’ actions. Lemma 8 shows that it is the case.
Lemma 8.
i. When the anti-government SIG does not separate, ∃! Ξ ⊂ [0, 1]2 × (0, γ(1 − p))
such that a pro-governemnt SIG’s separating strategy is a best response to other players’
actions if and only if (bS , bW , cpS ) ∈ Ξ;
ii. When the anti-government SIG separates, ∃! c̄pse : [0, 1]2 → (0, γ(1 − p)) such that a progovernment SIG’s separating strategy is a best response to other players’ actions if and only
if: max{k/(1 − p), 1 − p − k} ≤ bS ≤ 1 − p − k/2
33
and 0 ≤ cpS ≤ c̄pse (bS , bW ).
Proof. Suppose the anti-government SIG does not separate.
33
Note this condition might not be binding, see Proposition 6.
28
We first prove necessity. Consider the following sets:



(b , b , cp ) ∈ [0, 1]2 × (0, γ(1 − p)) s.t.

 S W S
Υ1 = i. E[ug (bS , g ∗ (bS , loa ))|ζ a , ζ p (cpW )] ≥ max{E[ug (bW , g ∗ (bW , loa ))|ζ a , ζ p (cpW )], E[ug (1, g ∗ (1, loa ))|ζ a , ζ p (cpW )]



 ii. E[u (b , g ∗ (b , la ))|ζ a , ζ p (cp )] ≥ max{E[u (b , g ∗ (b , la ))|ζ a , ζ p (cp )], E[u (1, g ∗ (1, la ))|ζ a , ζ p (cp )]}
g
g S
S o
g W
W o
o
S
S
S



(b , b , cp ) ∈ [0, 1]2 × (0, γ(1 − p)) s.t.

 S W S
Υ2 = i. E[ug (bS , g ∗ (bS , loa ))|ζ a , ζ p (cpW )] ≥ max{E[ug (bW , g ∗ (bW , loa ))|ζ a , ζ p (cpW )], E[ug (1, g ∗ (1, loa ))|ζ a , ζ p (cpW )]



 ii. E[u (1, g ∗ (1, la ))|ζ a , ζ p (cp )] ≥ max{E[u (b , g ∗ (b , la ))|ζ a , ζ p (cp )], E[u (b , g ∗ (b , la ))|ζ a , ζ p (cp )]}
g
g S
S o
g W
W o
o
S
S
S


 (bS , bW , cpS ) ∈ [0, 1]2 × (0, γ(1 − p)) s.t.


Υ1 = i. E[ug (bW , g ∗ (bW , loa ))|ζ a , ζ p (cpW )] ≥ max{E[ug (bS , g ∗ (bS , loa ))|ζ a , ζ p (cpW )], E[ug (1, g ∗ (1, loa ))|ζ a , ζ p (cpW )]



 ii. E[u (1, g ∗ (1, la ))|ζ a , ζ p (cp )] ≥ max{E[u (b , g ∗ (b , la ))|ζ a , ζ p (cp )], E[u (b , g ∗ (b , la ))|ζ a , ζ p (cp )]}
g
o
g
S
S
S
o
S
g
W
W
o
We assume that the government avoids conflict when indifferent. Then ∀(bS , bW , cpS ) ∈ Υ1 , we
have: b(ζ a , ζ p (cpW )) = bS (and g ∗ (bS , 1) = 2) and b(ζ a , ζ p (cpS )) = bW (and g ∗ (bW , 1) = 1 by Lemma
5). The other sets are defined accordingly.
Using Lemma 7 and the strong type’s incentive compatibility constraints defined in Lemma 6
(see (6)), we have (by the Intuitive Criterion):
lip (cpW )
= max
γ(Y e (ζ p (cpW )) − Y e (ζ p (cpS ))) + p(lop = 1|ζ p (cpS ))cpS
,0
cpS
The weak type’s incentive compatibility (7) is always satisfied when lip (cpW ) = 0. When lip (cpW ) > 0,
then it is satisfied if and only if (after some simple algebra):
cpS ≤ cpW
γ(Y e (ζ p (cpS )) − Y e (ζ p (cpW )))
cpW p(lop = 1|ζ p (cpS )) + γ(Ŷ e (ζ p (cpS ), cpW ) − Y e (ζ p (cpW )))
The right-hand side is strictly positive since: i) Y e (ζ p (cpS )) > Y e (ζ p (cpW )) and ii) γ Ŷ e (ζ p (cpS ), cpW ) >
γY e (ζ p (cpS )) − cpW p(lop = 1|ζ p (cpS )) (see Lemma 6).
γ(Y e (ζ p (cpS ))−Y e (ζ p (cpW )))
p
p
p
p
e p p
e p p
W p(lo =1|ζ (cS ))+γ(Ŷ (ζ (cS ),cW )−Y (ζ (cW )))
We also have: ϑ(cpW ) = cpW cp
< γ(1 − p). Using Lemma 6,
we can check that ϑ0 (cpW ) < 0. By Assumption 2, we have: ϑ(cpW ) < ϑ(γ(1 − p)) < γ(1 − p) since
cpW p(lop = 1|ζ p (cpS )) + γ(Ŷ e (ζ p (cpS ), cpW ) − Y e (ζ p (cpW ))) > γ(Y e (ζ p (cpS )) − Y e (ζ p (cpW ))).
Furthermore, ∀(bS , bW , cpW ) ∈ Υ1 , we have ϑ(cpW ) < γ(1 − p)bW . To see that, note that
∀(bS , bW , cpW ) ∈ Υ1 , we have Y e (ζ p (cpS )) = (1 − q a p)bW , Y e (ζ p (cpW )) = bS , Ŷ e (ζ p (cpS ), cpW ) =
(1 − q a )bW , and p(lop = 1|ζ p (cpS )) = q a . Simple computations show: ϑ(γ(1 − p)bW ) = γ(1 − p)bW .
29
S
So ϑ(cpW ) < γ(1 − p)bW .
h
2
Denote Λ = [0, 1] × 0, cpW cp
γ(Y e (ζ p (cpS ))−Y e (ζ p (cpW )))
p
e
p p
e p p
W p(lo =1|ζ (cS ))+γ(ŶW (bS ,bW )−Y (ζ (cW )))
i
. Given that the anti-government
SIG does not separate and using Lemma 5, we have:
Y e (ζ p (cpS )) =b(ζ a , ζ p (cpS ))
Y e (ζ p (cpW )) =b(ζ a , ζ p (cpW ))



b if b(ζ a , ζ p (cpS )) = bS

 S
Ŷ e (ζ p (cpS ), cpW ) = (1 − q a )bW if b(ζ a , ζ p (cpS )) = bW



 0 if b(ζ a , ζ p (cp )) = 1
S


 0 if b(ζ a , ζ p (cpS )) = bS


p
p p
p(lo = 1|ζ (cS )) = q a if b(ζ a , ζ p (cpS )) = bW



 1 if b(ζ a , ζ p (cp )) = 1
S
Denote Ξ ≡ Υ1 ∪ Υ2 ∪ Υ3 ∩ Λ The pro-government SIG plays a separating strategy only if
(bS , bW , cpS ) ∈ Ξ
The sets Υk , k ∈ {1, 2, 3} defines the government’s best policy response after observing ζ p (cpW ) and
ζ p (cpS ). The government’s policy choice satisfies the necessary conditions described in Lemmata
4-7. By definition of the set Ξ, we know that a separating strategy is incentive compatible for a
pro-government SIG.
We now show sufficiency. Suppose (bS , bW , cpS ) ∈ Υ1 ∩ Λ and consider the following assessment.
i) A weak (strong) pro-government SIG sends signal ζ p (cpW ) = (W, lip (cpW )) (ζ p (cpS ) = (S, 0)), where
lip (cpW ) is defined above. ii) The government’s belief is: P rob(c̃p = cpW |ζ p (cpW )) = 1, and 0 otherwise.
iii) The government’s policy choice is: b(ζ a , ζ p ) = bW , ∀ζ p 6= ζ p (cpW ) and b(ζ a , ζ p (cpW )) = bS . The
remaining best responses down the game tree are defined above.
We can easily check that the belief satisfies Bayes’ rule, the government’s policy choice is a
best response given his belief. The pro-government SIG’s incentive compatibility constraints hold.
Hence, we can see that (bS , bW , cpS ) ∈ Υ1 ∩ Λ is a sufficient condition for a pro-government SIG to
play a separating strategy. Note that we have not proven that it is sufficient for the pro-government
SIG to separate in a PBE since we have simply assumed that the anti-government SIG does not
30
separate.
We can apply the same reasoning for (bS , bW , cpS ) ∈ Υk ∩ Λ, k ∈ {2, 3}.
Suppose the anti-government SIG separates. We just prove necessity. Sufficiency follows from
a similar argument as above.
From Lemma 4, we know that a necessary condition is: b(ζ a (caS ), ζ p (cpS )) = 1 and b(ζ a (caS ), ζ p (cpW )) =
bS . This is equivalent to: max{1 − p − k, k/(1 − p)} ≤ bS ≤ 1 − p − k/2. The first inequality
follows from Proposition 1. The second inequality just states that the government prefers to fight
with the pro-government SIG’s help than compromise with a strong anti-government SIG.
Using a similar reasoning as above, we can check that the pro-government SIG incentive compatγ(Y e (ζ p (cpS ))−Y e (ζ p (cpW )))
p
p
p
p
e p p
e p p
W p(lo =1|ζ (cS ))+γ(Ŷ (ζ (cS ),cW )−Y (ζ (cW )))
ibility constraints are satisfied only if: 0 < cpS ≤ c̄pse , where c̄pse = cpW cp
,
where we have:

 q a + (1 − q a )b if b ≥ 1 − p − k/2
W
W
e p p
a
a a
p p
a
a a
p p
Y (ζ (cS )) =q b(ζ (cS ), ζ (cS )) + (1 − q )b(ζ (cW ), ζ (cS )) =
 1 if b < 1 − p − k/2
W
Y e (ζ p (cpW )) =q a b(ζ a (caS ), ζ p (cpW )) + (1 − q a )b(ζ a (caW ), ζ p (cpW )) = q a bS + (1 − q a )bW

 (1 − q a )b if b ≥ 1 − p − k/2
W
W
p
e p p
Ŷ (ζ (cS ), cW ) =
 0 if b < 1 − p − k/2
W

 q a if b ≥ 1 − p − k/2
W
p
p p
p(lo = 1|ζ (cS )) =
 1 if b < 1 − p − k/2
W
When both SIGs separate, the government’s best response after observing ζ a (caW ) and ζ p (cpS ) is:
b(ζ a (caW ), ζ p (cpS )) = bW when bW ≥ 1 − p − k/2, and b(ζ a (caW ), ζ p (cpS )) = 1, otherwise (in this latter
case, the government chooses g ∗ (1, 1) = 1).
i
o
h
n
k
k
Hence bS ∈ max 1−p , 1 − p − k , 1 − p − 2 and cpS ∈ (0, c̄pse ] are necessary conditions for a
pro-government SIG to separate when the anti-government SIG separates as claimed.
The proof of Propositions 3 and 4 follow directly from Lemmas 6, 7, and 8.
Proof of Corollary 5. Using Lemmata 4-7 and the strong pro-government SIG’s Incentive Compatibility constraint (6), we have: lip (cpW ) =
p(lop =1|ζ p (cpS ))cpS +γ(Y e (ζ p (cpW ))−Y e (ζ p (cpS )))
.
cpS
increasing with Y e (ζ p (cpW )).
31
This is obviously
Proof of Proposition 5. The econometrician only observes inside lobbying expenditures (lip (cpW ))
and outside lobbying expenditures (normalized to 1). The observed expenditures are then (in
expectation): lip (cpW ) and p(lop = 1|ζ p (cpS )). Using (6), (7), and Lemma 7, we have that lip (cpW ) =
p(lop = 1|ζ p (cpS )) − γ
(Y e (ζ p (cpS ))−Y e (ζ p (cpW )))
cpS
< p(lop = 1|ζ p (cpS )), where Y e (ζ p (cpS )) > Y e (ζ p (cpW )) by
Lemma 6.
In what follows, we study under which conditions a separating strategy is a best response for
the anti-government SIG when a pro-government SIG is present. When the anti-government SIG
separates, a strong type sends signal ζ a (caS ) and a weak type sends signal ζ a (caW ) 6= ζ a (caS ). After
observing ζ a (caS ) (ζ a (caW )), the government believes he faces a strong (weak) type. We also assume
that all players play their best response down the game tree.
Proof of Proposition 6. Point i. of Proposition 6 follows directly from the proof of Proposition 1.
We thus focus on point ii. of the proposition.
We know from the proof of Proposition 1 that a necessary condition for a separating equilibrium
to exist is that the government wants to compromise with a strong type and is credible doing so by
o
n
k
.
Assumption 1 (see Proposition 1). This is also true here. So we need bS > max 1 − p − k, 1−p
By Lemma 4, we know that b(ζ a (caS ), ζ p (cpS )) = 1 and b(ζ a (caS ), ζ p (cpW )) = bS when both SIGs
separate. We also have: b(ζ a (caW ), ζ p (cpW )) = bW . So we only need to consider two cases: case a):
b(ζ a (caW ), ζ p (cpS )) = 1; case b): b(ζ a (caW ), ζ p (cpS )) = bW . Using a similar logic as in Lemma 3, we
can show that lia (caW ) = 0 and lia (caS ) > 0.
We first consider case a). The weak anti-government SIG’s (IC) is:
q p (−(1 − p)α − caW ) + (1 − q p )(−αbW ) ≥ q p (−(1 − p)α − caW ) + (1 − q p )(−αbS ) − caW lia (caS )
When the weak anti-government SIG faces a strong pro-government SIG working with the government, she has to fight (b(ζ a (caJ ), ζ p (cpS )) = 1, J ∈ {S, W }). When the weak anti-government
SIG faces a weak pro-government SIG working with the government, she gets either bW when she
reveals her type, or bS when she deviates (b(ζ a (caJ ), ζ p (cpW )) = bJ , J ∈ {S, W }). By the Intuitive
Criterion, we have: lia (caS ) = (1 − q p ) α(bWca−bS ) .
W
Now consider the strong type’s (IC):
q p (−(1 − p)α − caS ) + (1 − q p )(−αbS ) − caS lia (caS ) ≥ q p (−(1 − p)α − caS ) + (1 − q p )(−(1 − p)αbW − caS )
32
Substituting for lia (caS ) and rearranging, we get: (1 − p)(bW − bS ) ≥ caS (bWca−bS ) . This is exactly the
W
same condition as in the proof of Proposition 1. Therefore, we know that a necessary condition is:
bS ≤ (1 − p)bW .
We now consider case b). The weak anti-government SIG’s (IC) is:
−αbW ≥ q p (−(1 − p)α − caW ) + (1 − q p )(−αbS ) − caW lia (caS )
When a weak anti-government SIG reveals her type, the government chooses bW (b(ζ a (caW ), ζ p (cpL )) =
bW , L ∈ {S, W }). When she pretends to be strong, she has to fight when the government works
with a strong pro-government SIG (b(ζ a (caS ), ζ p (cpS )) = 1) and gets b = bS otherwise. By the Intuitive Criterion, we have: lia (caS ) =
p
αbW −q p [(1−p)α−ca
W ]−(1−q )αbS
.
ca
W
We claim this last term is positive
and verify the claim afterwards.
A strong type knows she gets the compromise bill only if the government works with a weak
pro-government SIG. We thus have the following (IC):
−q p ((1 − p)α + caS ) − (1 − q p )αbS − caS lia (caS ) ≥ −(1 − p)αbW − caS
α(bW − q p [(1 − p) − caW ] − (1 − q p )bS
≤ (1 − p)αbW + caS
a
c
W
ca
ca
caS
p
p
⇔ [q (1 − p) + (1 − q )αbS ] 1 − a + aS αbW ≤ (1 − p)αbW + αp aS bW
cW
cW
cW
p
(1 − p)bW − q (1 − p)
⇔ bS ≤
1 − qp
⇔ q p ((1 − p)α + caS ) + (1 − q p )αbS + caS
In the third line we rearrange terms and use the fact that bW =
−q p (1−p)
(1−p)bW
1−q p
ca
W
bS .
ca
S
Note that when bS =
, then lia (caS ) = 1 − q p > 0. Since lia (caS ) is decreasing with bS , we have lia (caS ) > 0 as
claimed.
Note also that we have:
(1−p)bW −q p (1−p)
1−q p
(1−p)bW −q p (1−p)
1−q p
< (1 − p)bW since bW < 1. Denote bse = (1 − p)bW or
depending on whether we are (respectively) in case a) or b). We thus get that an
anti-government SIG separates only if max{1 − p − k, k/(1 − p)} ≤ bS ≤ bse and bse ≤ (1 − p)bW
as claimed.
To see that the conditions are sufficient, we can construct a separating assessment in the
same way as in the proof of Proposition 1. The only difference is that we do not construct a
separating equilibrium, but verify that the anti-government SIG’s strategy is a best response to
33
the pro-government SIG’s and the government’s strategies.
Proof of Proposition 7. Suppose there exists a separating equilibrium when both SIGs separate.
By Lemma 4 and Proposition 6, a strong anti-goverment SIG makes inside lobbying expenditures
and starts a conflict whenever the government works with a strong pro-government SIG.
We thus just need to show that such a separating equilibrium exists. Consider the following
a
a
bS −(1−q )bW ]
case: bS ≤ (1 − p)bW , bW ≤ 1 − p − k/2, and cpS ≤ cpW γ[(1−p)−q
(this is not the only
cp −γ[q a bS +(1−q a )bW ]
W
possible case). Using Lemma 8 and Proposition 6, we know that, under these parameter values
there exist a separating equilibrium where both SIGs separate.
34
Supplemental Appendix
In this section, I micro-found the influence of outside lobbying on political decisions. To do so,
I use a simplified version of the War of Information (Gül and Pesendorfer, 2012). To simplify
the exposition of this subsection, I do not include a pro-government SIG. We know that an antigovernment SIG’s type announcement is not credible without inside lobbying expenditures (see
Lemma 3), so I assume that the anti-government SIG’s signal takes the form of inside lobbying
expenditures (i.e., ζ a ≡ lia ).
I consider a three-player game with the government, an anti-government SIG, and a representative voter. The government and the anti-government SIG play the same game as in the game
described in Section 3. However, there are now two states of the world: ω ∈ {I, L}. In state
I, the government’s legislative proposal increases the voter’s utility compared to the status quo.
In state L, the government’s proposal lowers the voter’s utility. No player knows the state of
the world at the beginning of the game. However, it is common knowledge that players’ (and,
in particular, the representative voter’s) prior is biased in favor of the government’s proposal (see
Assumption 3 below). This reflects the fact that the government has been elected by a majority
of the population.
As in Gül and Pesendorfer’s War of Information, the anti-government SIG’s outside lobbying
activities and the government’s response to them reveal information to the representative voter.
The voter receives no, one or two signals of the state of the world depending on other players’
actions. The voter then sides with the government or the SIG according to her belief regarding ω.
To provide information to voters, the government and anti-government SIG must pay a cost.34
To summarize, the timing of the augmented game is:
0. Nature draws the SIG’s type (c̃a ∈ {caS , caW })
1. After observing her type, the SIG sends a signal: lia ≥ 0
2. The government chooses a proposal: b ∈ [0, 1]
3. The SIG decides whether to start a war of information: loa ∈ {1, 0}
If there is no war, the representative voter sides with the government or SIG according to
her prior
34
Compared to Gül and Pesendorfer (2012), I assume that the anti-government SIG and the government can
provide information only once to the voter.
35
4. If there is a conflict, the voter receives a signal of the state of the world: w1 ∈ {i, l}. The
government observes the signal
5. The government decides to pursue or not the war of information: g ∈ {1, 0}
i. If he stops the war of information (g = 0), the voter sides with the government or SIG
according to her posterior after observing w1 ;
ii. If he continues the war of information (g = 1), the representative voter receives a second
signal w2 ∈ {i, l}
the voter sides with the government or the SIG according to her posterior after observing
w1 and w2 .
When there is a conflict, the representative voter receives a signal wt = i with probability ρ > 1/2
when the state of the world is I and with probability 1 − ρ when the state of the world is L,
t ∈ {1, 2}. She receives a signal wt = l with probability 1 − ρ when the state of the world is I and
with probability ρ when the state of the world is L, t ∈ {1, 2}. I also suppose that the two signals
received by the voter are independent once we condition on the state of the world. Denote by π0
the prior of the players that the state of the world is L. Denote by π1 (w1 ) the voter’s (and the
government’s) posterior that the state is L after observing signal w1 ∈ {i, l}. Denote by π2 (w1 , w2 )
the voter’s posterior that the state is L after observing two signals (if this occurs on the equilibrium
path). Since the equilibrium concept is PBE, the voter’s posterior must satisfy Bayes’ Rule.
The utility functions of the government and the anti-government SIG are respectively:
ug (y, g) = y − cg g, g ∈ {1, 0}
uasi (y, lia , loa ) = −αy − c̃a loa − c̃a lia , loa ∈ {1, 0}
(9)
(10)
Note that the cost of providing information is cg > 0 for the government.
The utility function of the representative voter is:
uv (y; ω) = h(ω)v(y)
36
(11)
with v(.) continuous and strictly increasing, and h(ω) a function with the following properties:
h(I) > 0 and h(L) < 0. For example, we can have:
h(ω) =


1 if ω = I

−1 if ω = L
To make the problem interesting, I assume that:
Assumption 3. ρ and π0 are such that:
π1 (l) ≡
ρπ0
−h(L)
>
> π0
ρπ0 + (1 − ρ)(1 − π0 )
h(I) − h(L)
The first inequality of Assumption 3 states that, after receiving a signal w1 = l, the voter
prefers the government to abandon his legislative proposal. This assumption is satisfied when the
prior is not too biased in the government’s direction (π0 is not too low) and the signal is sufficiently
informative (ρ is sufficiently high). When this assumption does not hold, the anti-government SIG
never engages in outside lobbying and the government always chooses b = 1. The second inequality
just implies that the voter’s prior is favorable to the government’s proposed bill.
It is obvious that the government does not pay the cost of sending a signal to the voter when
w1 = i. We have: π1 (i) < π0 <
−h(L)
h(I)−h(L)
and the voter always sides with the government in this
case. Providing information to the voter is costly to the government and does not change the
voter’s decision.35
I introduce some last pieces of notation. First, denote the probability that the voter receives
a signal w1 = l by p0 (N F ) = π0 ρ + (1 − π0 )(1 − ρ). This is also the ex-ante probability that
the anti-government SIG wins the war of information if the government does not stops the war of
information after signal w1 . The ex ante expected cost of a war of information for the government is
denoted: k = p0 (N F )cg . Denote the ex ante probability (before the start of the war of information)
that the anti-government SIG wins the war of attrition when the government pursues the war of
information: p0 (F ) = π0 ρ2 + (1 − π0 )(1 − ρ)2 . Lastly, denote the probability that the antigovernment SIG wins the war of information given that the voter has received a signal w1 = l by:
p1 (l) = π1 (l)ρ + (1 − π1 (l))(1 − ρ). I suppose that the following assumption holds:
Assumption 4. Denote bW =
35
ca
W
.
αp0 (F )
We have π2 (i, i) < π2 (i, l) = π0 <
We have: (1 − p1 (l))bW − k > 0 and bW > 1 − p0 (F ) − k
−h(L)
h(I)−h(L)
so the voter always side with the government.
37
This assumption is similar to Assumption 1. The inequality is no longer a necessary assumption for a separating equilibrium to exist. However, assuming this inequality holds simplifies the
exposition.
Using p0 (N F ), p0 (F ) and p1 (l), we can see that the primary result of section 1 holds when the
conflict between the government and the SIG takes the form of a war of information.
Proposition 8. Denote: bFS =
ca
S
αp0 (F )
F
and bN
=
S
ca
S
.
αp0 (N F )
A separating equilibrium exists if and only if:
1. 1 − p0 (F ) − k ≤ bFS ≤ (1 − p0 (F ))bW if bFS ≥
cg
1−p1 (l)
F
2. 1 − p0 (F ) − k ≤ bN
≤ (1 − p0 (F ))bW if bFS <
S
cg
1−p1 (l)
Proof of Proposition 8. I first show that the conditions are necessary.
First, by assumption 4, the government can credibly compromise with a weak SIG by proposing
bW =
ca
W
.
αp0 (F )
The government can also choose a radical bill and gets in expectation 1 − p0 (F ) − k.
Regarding the strong SIG, we need to distinguish between two cases depending on whether the
government will pursue the war of information when he compromises with a strong SIG.
When the government pursues the war of information, we have bFS =
ca
S
αp0 (F )
and following
Proposition 1, we need 1 − p0 (F ) − k ≤ bFS ≤ (1 − p0 (F ))bW . The government must be credible
when he chooses to compromise with a strong anti-government SIG (given that we consider a
PBE). The government pursues the war of information after the voter receives a signal w1 = l if
and only if (1 − p1 (l))bFS − cg > 0. Hence, we get that the conditions expressed in point 1. of the
proposition are necessary whenever bFS ≥ cg /(1 − p1 (l)).
If bFS < cg /(1 − p1 (l)), to avoid a war of information with the anti-government SIG, the govF
ernment must choose a proposal bN
such that a strong type is indifferent between starting a war
S
of information or not given that his probability of winning the conflict is p0 (N F ) (he wins when
the voter receives the signal w1 = l). Using this, we can apply the reasoning of Proposition 1. We
F
thus get that necessary conditions are: 1 − p0 (F ) − k ≤ bN
≤ (1 − p0 (F ))bW . This is point 2. of
S
the proposition.
To see that these conditions are sufficient, we can construct a separating assessment along the
same line as in the proof of Proposition 1 and verify that it is a PBE.
The main result described in the body of the paper still holds when outside lobbying activities
take the form of a war of information. A separating equilibrium exists only under some parameter
38
values (for strong anti-government SIG with intermediate strength). The analysis when SIGs
compete is more burdensome, but since modeling conflict as a war of information does not change
players’ incentives, the results are unaffected.
The war of information presented in this section fits relatively well with several important
political concepts. First, it assumes that most voters are inattentive to the government’s actions.
However, when there is a war of information between the government and the anti-government SIG,
voters become attentive, get new information, and are pivotal in the outcome of the conflict. This
version of the war of information corresponds to the idea of ‘latent opinion’ as expressed by V.O.
Key in his seminal book on public opinion (1961). Also, because of the voter’s lack of attention, the
government is able to pass any reform he desires when there is no opposition. The anti-government
SIG acts as a constraint on the government’s actions. This corresponds to Tocqueville (1840)’s
idea that associations have a key role to play in informing the public and providing a check on
the government in democracy. But note that a war of information is more informative when both
players pay the cost of providing information to the voters. In some sense, the truth (or some
signal of the true state of the world) is helped by the competition of ideas as stated by Mill (1859).
Mill (1859) also argued that the competition of ideas is welfare-improving. We can observe that
the war of information improves the voters’ welfare. They obtain better information on the impact
of the reform policy on their utility, but do not have to bear the cost of getting this information.
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- Mill, John Stuart. 1998 [1859]. On Liberty. Oxford: Oxford University Press;
- de Tocqueville, Alexis. 2005 [1840]. De la démocracy en Amérique, Volume II. Paris: folio
histoire;
39
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