Firms, Consumers, and Policies: Competing Through
Information Production∗
Jean-Philippe Bonardi†
Olivier Cadot‡
Lionel Cottier§
November 26, 2014
Draft - Please do not circulate or cite without authors’ permission
Abstract
We set up a model of costly information production between two lobbies, a firm and a
consumer group, competing for influence over an imperfectly-informed but benevolent
government. The government is endowed with a parametric amount of information and
chooses the best policy from a finite, countable feasible set given the information available (its own and that forwarded by lobbies). Lobbies have asymmetric preferences, the
firm being a “high-stakes” player with relatively extreme preferences and the consumer
a “low-stakes” player with preferences more aligned with the government’s. We show
that lobbies spend too much on information production in any Nash equilibium, but
the “over-research” is mitigated by a timing-game structure, in which the firm chooses
to play first at a low-search intensity. We also show that in some parameter configurations, the firm insures against a consumer win by forwarding unbiased information to
the government, in spite of its own extreme preferences and high stakes. The resulting
informational rent enables the government to adopt moderate policies aligned with its
own preferences, suggesting a new way in which lobby competition can produce good
policies even when the government is imperfectly informed.
Keywords: Game theory, lobbying model, imperfect information, timing game
JEL codes: H4, K0, P1, D72, F13
∗
This paper draws on previous work by José Anson as part of his Ph.D dissertation at the University
of Lausanne. Without implicating him, we would like to thank him for very useful conversations and for
motivating our work. We are also grateful to seminar participants at the DEEP seminar series, HEC Montreal, Lausanne’s Law and Economics seminar, and the XX conference in Chicago for useful comments.
Support from Switzerland’s NCCR under WP6 (Impact assessment) and from France’s Agence Nationale de
la Recherche under Investissement d’Avenir grant ANR-10-LABX-14-01 is gratefully acknowledged.
†
University of Lausanne; [email protected].
‡
University of Lausanne, CEPR, and FERDI; [email protected].
§
University of Lausanne; [email protected].
1
1
Introduction
Whether in environmental or public-health issues, firms find themselves frequently pitched
against organized consumer groups in battles for influence over key regulatory decisions, using
media coverage, professional lobbyists, and sometimes even academics, to sway the public and
decision-makers.1 In the 1990s, for instance, consumer groups in the US initiated a campaign
against monosodium glutamate (MSG), arguing that it was unsafe, generated obesity and behavioral disorders among children, and was even related to neuro-degenerative diseases such
as Alzheimer’s. Discovered and patented by Japan’s Ajinomoto in 1909 and manufactured
through bacterial fermentation, MSG is widely used in Asian cuisine, but also in packaged
savory foods in the West. Consumer groups and NGOs lobbied for MSG to be banned, producing studies that highlighted its alleged dangers for human health. In response, Ajinomoto
waged a relatively low-profile campaign, commissioning independent studies that suggested
there was no health hazard, although the studies failed to quell public anxieties. While
exposed to increasingly one-sided vibrations, governments in Western countries adopted a
middle-of-the road approach, imposing MSG labelling in prepared foods instead of the ban
demanded by consumer groups. Two features of the battle are of interest. First, while having
a high stake in the outcome, Ajinomoto refrained from waging a high-profile campaign on
its own. Second, the regulatory outcome was relatively balanced, in contrast with the conventional wisdom in the political-economy literature which suggests that the concentrated
producer interests typically win, enabling distortionary policies to generate private rents.
The fight over MSG does not fit with that narrative; and the rising stiffness of standards
around the world can be read as reflecting a growing power of relatively diffuse, low-stakes
consumer interests against high-stakes, concentrated producer lobbies.
What makes reasonable, middle-of-the road policy choices emerge when an imperfectlyinformed government relies, at least partly, on self-interested information from lobbies? How
does the government “aggregate” conflicting messages? Do balanced outcomes emerge from
symmetry of stakes between lobbies, from symmetry of access to the government, or from a
lack of credibility of extremist lobbies? The model we propose in this paper suggests that
the answer is “none of the above”; in our setup, a government extracts informational rents
from defensive lobbying by a high-stakes lobby, enabling it to make its own, informed choices.
We develop an information-production model à la Aghion-Tirole (1997, henceforth AT)
and adapt it to an informational lobbying set-up. While AT consider an organizational issue,
namely an agent’s effort allocation in a delegation context, their approach is well suited to
model interest-group competition to capture influence over an incompletely informed government having the final decision power — what AT call “formal authority”. Given the level of
independent information that the government can access, both the firm and the consumer
group try to capture decision power — what AT call “real authority” — in an environment
1
The rise in “consumer power” has been documented in a number of recent papers including King and
Soule (2007) or Spar and La Mur (2003). Consumer groups can challenge firms directly through boycotts or
protests (Baron and Diermeier, 2005; Feddersen and Gilligan, 2001; Lenox and Eesley, 2009) —what Baron
(2003) calls “private politics”. They can also confront corporations indirectly through lobbying for policies
and regulations (Bonardi and Keim, 2005; Lyon and Maxwell, 2004).
2
where their interests are conflicting and their stakes unequal. In order to capture decision
power, both interest groups produce efforts in information production. The model provides
a stylized representation of a situation where a high-stake producer group competes with a
low-stake consumer group whose interests are closer to society’s and therefore to those of
a benevolent government. Note that considering stake differences allows us to differentiate
firms and consumers in the lobbying game, and could help differentiate consumers from other
interest groups with more direct interests in the final policy decision.
A two-stage game is then solved; in stage one, both the firm and the consumers’ play an
information-production game and decide on a timing and disclosure strategy; in the second
stage, lobbies strategically forward some of the information produced in stage one to the
government who chooses the best policy among several alternatives given the information
available (its own and that forwarded by lobbies). For tractability, the government does not
choose its level of information production but is endowed with a given level of information
drawn from nature, which is, in our game, a comparative-statics parameter. Our approach
accommodates different levels of conflict between lobbies, different degrees of asymmetry in
stakes and preferences, and different levels of government information.
Our model relates to the existing literature on informational lobbying in several ways.
One strand of the literature depicts situations of rivalry between firms and interest groups
in the policy arena as cheap-talk games, i.e. games in which firms and interest groups know
the real state of the world but can’t convey it credibly to an uninformed policymaker because information is soft and messages are unverifiable (Crawford and Sobel, 1982). In that
context, the credibility of the source is key and firms are generally at a disadvantage. In
effect, companies cannot convey unverifiable information credibly to the government because
their payoffs are based on the policy imposed rather than on the underlying state of the
world (Lyon and Maxwell, 2004). By contrast, consumers’ positions are often seen as more
credible and closer to the public interest because they do not have much directly and individually at stake in the policy decision. In spite of this “credibility gap”, existing work
suggests that a firm can still influence the policy-making process, either because the two
opponents lobbying against each other reduce uncertainty for the policy-maker (Grossman
and Helpman, 2001; Krishna and Morgan, 2001), or because firms can use lobbying costs and
campaign contributions as a way to signal the truthfulness of unverifiable information and
gain credibility (Banerjee and Somanathan, 2001); or, finally, because firms can influence the
behavior of other interest groups with less biased objectives (Lyon and Maxwell, 2004). Our
results also relate to a distinct strand of the lobbying literature where information plays no
role but lobby rivalry prevents the emergence of extremist policies. For instance, in Grossman and Helpman’s common-agency game (Grossman and Helpman, 1994), rivalry between
principals (lobbies) with opposite preferences generates low-power incentives for the common
agent (the government), leading to middle-of-the-road policies, and even, when the incentive schedules of the lobbies perfectly cancel out, to socially optimal policies. In that strand
of the literature, governments are swayed not by information, but by campaign contributions.
We depart from the bulk of the literature by assuming that none of the players is fully
informed. Many situations such as those involving the impact of new technologies on public
3
health or on the natural environment are of this kind, and imply that both firms and consumer groups have to invest resources and efforts in finding out the true state of the world if
they want to effectively impact the lobbying game. They can do this in many different ways,
including producing research studies and reports of their own, or commissioning research
from external parties. The questions we ask are: (i) what determines whether and how much
the firm and consumer group will invest in information production? (ii) what is the impact
of the degree of conflict between the two types of actors? (iii) what are the implications for
the policy ultimately adopted?
In a paper related to ours, Henry (2009) uses a “persuasion game” with endogenous information production to explore the effect of mandatory disclosure of research results. In
his model, a “sender” (researcher) incurs a cost to produce a number of stochastic research
results (positive or negative) about a state of nature to influence the policy of a “receiver”
(say, a regulatory agency). The sender can withhold information but cannot misreport, information being verifiable. Suppose that the sender wants to induce the receiver to choose a
high level of the policy by convincing him that the state of nature is better than it actually
is. One way to do this is to withhold the negative results. Anticipating this, the receiver
calculates the unobserved total number of signals actually produced by the sender and interprets all of the unreported signals as negative, in accordance with Milgrom’s “unraveling
principle” (Milgrom 1981). If research effort is not observed by the receiver, Henry shows
that the sender will end up over-researching in equilibrium, wasting resources, so to speak,
to prove his honesty. Otherwise, the receiver would assume more negative signals than there
actually were.
“Over-researching” is also a central feature of our setup, but in an indirect way and in a
different setting with multiple, competing senders. A key driver of the model’s results is that
the lobbies’ research intensities are neither strategic complements nor strategic substitutes.
The consumers increase their information production in response to an increase in information
production by the firm, whereas the firm decreases its information production in response to
a higher effort in information production by the consumers. This unusual type of strategic
interaction generates two surprising results. First, our lobbyists mitigate over-researching
through a timing game in which the player with the strongest stake in keeping the research
effort at a low level plays first, producing a “better” equilibrium (for both players) than
the simultaneous game. Second, the lobby with the highest stakes, whether defensive (lots
to lose from losing the battle) or offensive (lots to gain from winning it) insures against
defeat by adopting a full-disclosure strategy that consists of providing the government with
unbiased information (the identity of the policy that is truly the best for the government),
thus neutralizing the information provided by the other lobby. This insurance strategy is
an informational rent for the government who gets access to unbiased information and can
choose its preferred policy, which generates, in our model (by assumption), a middle-of-theroad outcome.
4
2
The model
Consider a two-stage game between three players: two lobbies, labelled f (for a firm) and c
(for a consumer group), and a government, labelled g.
There are five feasible policies indexed by i = 1, ..., 5. Four of them are “reform” policies in
the sense that they depart from the status quo; the fifth
quo. Each policy i maps
is the status
f
g
c
into an outcome in the form of a payoff triplet ui = ui , ui , ui whose elements are payoffs
to the consumer group, the firm, and the government, in that order. Policy s, the status
quo,
has payoff us = (0, 0, 0) for all players; Policy w, the worst, has payoff uw = `c , `f , `g where
`j < 0 for j = c, f, g. Each of the remaining three policies is the best alternative for one of
the three players.2 Policy c, which delivers the highest payoff to the consumers,
has payoffs
f
g
c
f
g
c
uc = uc , uc , uc ; policy f , best for the firm, has payoffs uf = uf , uf , uf ; and policy g,
best for the government, has payoffs ug = ucg , ufg , ugg .
All five policies and outcomes are common knowledge. However, the mapping from policies to outcomes is unknown. That is, players (including the government) know what can
be done and what can happen, but they do not know what leads to what. That information
can be gathered only through costly search. Search intensities are denoted by ej , j = c, f for
the consumers and the firm respectively, with 0 ≤ ej ≤ 1, and the cost of search is (ej )2 /2.
As a simplification, the government does not search for information on its own but has a
parametric “information endowment” e, which is the probability that it is independently
informed. The information is indivisible in the sense that successful search reveals the entire
mapping from all policies to all payoffs. The probability of a successful search is just ej .
Once lobbies have spent ressources searching for information, they can forward part or all of
it to the government; whatever information they forward is verifiable. That is, lobbies can
withhold information, but they cannot misrepresent it. By contrast, a claim by a lobby that
its search was not successful is not verifiable.
The game’s timing is partly fixed, partly endogenous. The fixed part is the two-stage
structure. In stage one, lobbies search for information and strategically forward some of it
to the government. In stage two, the government chooses the policy it prefers given its information and that forwarded by the lobbies. The endogenous part is within stage one, where
the firm and the consumer group simultaneously decide on the timing of information search
and disclosure. That is, the stage-one subgame is itself a two-period timing game. If both
lobbies prefer searching in period one or both in period two, the subgame is simultaneous. If
one of them prefers period one and the other period two, it is sequential. Lobbies also choose
the timing of disclosure. If the search is simultaneous, so is the disclosure. If the search is
sequential, the leader (and only the leader) chooses to disclose either in period one (before
the follower searches) or in period two (after the follower has searched and simultaneously
with the follower’s own disclosure).
2
The set of policies can be enlarged to more than five policies without altering the results. What matters
is that there is one and only one best policy for each player.
5
Given the game’s structure, the lobbies’ strategy space has four dimensions: search intensity (a continuum between zero and one), search timing (a binary choice between period
one and period two within stage one), disclosure timing for the leader if the search game is
sequential (again, period one or period two), and disclosure itself (partial or full, in a sense
that we will make precise later on).
The following assumptions give more structure to the payoff matrix. To recall, a subscript
designates a policy and a superscript a player; so uij designates the payoff from policy j (i.e.
the one preferred by player j) to player i.
A1
uii = 1 ∀ i; uij < 1 ∀ i 6= j;
A2
ucf = −1; ufc = −2;
A3
0 < uig for i = f, c;
A4
0 < ugf < ugc ;
A5
1 + ugc + ugf + `g < 0;
A6
0 < e < 1.
A1 assigns a unitary payoff to each player’s preferred policy and less than unitary payoff
to all other ones; this is a normalization. A2 assigns negative cross payoffs to the firm’s and
the consumer’s policies, with a more negative payoff for the firm, making it a “high-stakes
player” because it has more to lose from the consumers’ policy than the consumers have
to lose from the firm’s. The normalization to -1 and -2 is inconsequential provided that
the inequality holds but facilitates the calculation of expected payoffs. A3 states that both
firm and consumers prefer policy g to the status quo, making reform socially beneficial. A4
states that the government prefers the consumers’ policy to the firm’s. A5 states that the
government’s expected utility from a random draw among all policies (including the worst)
is worse than the status quo. This generates a “conservative bias”: when the government
is completely uninformed, it prefers sticking to the status quo rather than firing a shot
in the dark.3 Finally, A6 states that the government is neither completely informed nor
completely uninformed. The resulting payoff structure is summarized in Table 1. While
these relationships are common knowledge, the identity of each policy (which payoff column
in Table 1 is under which column head) is revealed only through successful search.
Note that the payoff structure in Table 1 makes policy g a “middle-of-the-road” one, as its
payoffs for the firm and consumer can be expressed as convex combinations of the payoffs
from policies c and f .
3
A more general formulation preserving the conservative bias for any probability distribution over unknown
policies would have infinitely negative payoffs for the worst policy. Results are unaffected by this choice.
6
Table 1: Policy outcomes
Policy
c
f
g
s
w
3
Payoff to
c
f
g
1 −2 ugc
−1
1 ugf
ucg ufg
1
0
0
0
`c
`f `g
Equilibrium
The game is solved backwards, starting with the government’s policy decision at the end of
stage two. This decision is conditional on the government’s aggregate information, including
both its own and that forwarded by lobbies.
3.1
Stage two
Let the “government’s known set” be the set of policies whose outcomes have been revealed
to the government, either through its own information endowment or forwarded by lobbies;
we will call these policies “known policies”. Table 2 shows the government’s optimal policy
choice as a function of its known set. The first five columns code each of the policies by a one
if it is known, a zero otherwise, and a dot if it does not matter to the government’s choice.
The sixth column gives the government’s choice, and the seventh gives the corresponding
payoff vector.
The status quo, s, is coded “one” throughout because it is known by construction, being
the policy in force at the beginning of the game. The worst policy, w, is always coded with
a dot because its relevance is indirect; it is never the policy choice. In the first line, the
government knows its best policy, g. In that case, whether it knows other policies or not, it
chooses g; so the other policies do not matter and are marked by dots. In the second line,
the government knows the consumers’ best policy, c, but not its own. As c is its second-best
policy, it chooses c whether or not it knows policies f and w. In the third line, the government
knows f but neither c nor g. As f is its third-best, it chooses f . In the fourth line, it knows
only s and so, by A5, sticks to it. This exhausts the policy-relevant information partition.
3.2
Stage one
In stage one, the lobbies decide on the game’s timing, their search intensity, the timing of
disclosure, and the disclosure itself. Consider first the disclosure strategy.
If a lobby’s search is successful, the full mapping from policies to payoffs is revealed to it;
that is, all policies become “known” to the lobby, but the information is private. The choice
at this stage is how much to disclose. For the consumers, the choice is trivial because policy
7
Table 2: Information, policy decisions, and payoffs
Gov. known set
c f g s w
. . 1 1 .
1 . 0 1 .
0 1 0 1 .
0 0 0 1 .
Policy
choice
g
c
f
s
Payoffs
(uc , uf , ug )
(ucg , ufg , 1)
(1, −2, ugc )
(−1, 1, ugf )
(0, 0, 0)
c is the government’s second best; therefore, disclosing c and only c is always optimal. For
the firm, however, it is non-trivial. Suppose that the firm’s search is successful, and either
that (i) the game is simultaneous, or (ii) it is sequential with the firm playing first, or (iii) it
is sequential, but with the consumers playing first and delaying disclosure. Suppose further
that the firm discloses only f ; we will call this ‘partial disclosure’. If the consumers fail in
their search, the government will choose f , with a payoff equal to one for the firm. But if
the consumers also succeed, the government will know both f and c and will choose c, with
a payoff of minus two for the firm. Suppose now that the firm discloses both f and g; we will
call this ‘full disclosure’ (here, whether or not the firm also discloses policy c is irrelevant).
Then, whatever the outcome of the consumers’ search, the government will pick its first best,
g, with a payoff between zero and one for the firm. Thus, full disclosure is safe whereas
partial is risky, with the risk growing with the consumers’ search intensity. We will now solve
two versions of the game, one under partial disclosure, one under full, and calculate which
one yields the highest expected payoff to the firm given equilibrium search intensities.
3.2.1
Partial disclosure
Under partial disclosure, the firm reveals only its best policy. The government’s choices in
stage two (given by Table 2) can be used to derive expected
payoffs as
functions of search
intensities given partial disclosure. Let v j ec , ef , e = E uj (ec , ef , e) , where the expectation uses equilibrium probabilities of success, ec and ef , and the government’s parametric
probability of success e. Given A1-A5, the consumers’ expected payoff is given by
(ec )2
,
v c ec , ef , e = eucg + (1 − e) ec − (1 − ec )ef −
2
(1)
the firm’s by
2
ef
f
c f
c
v e , e , e = eug + (1 − e) (1 − e )e − 2e −
,
2
and the government’s by
v g ec , ef , e = e + (1 − e) ec ugc + (1 − ec )ef ugf .
f
c
f
(2)
(3)
The payoff functions’ cross-partial derivatives are
∂v c
= −(1 − e)(1 − ec ) ≤ 0
∂ef
8
(4)
for the consumers and
∂v f
= −(1 − e)(ef + 2) < 0
(5)
∂ec
for the firm. Thus, the consumers’ search exerts a negative externality on the firm and vice
versa except in a corner solution with ec = 1, where the information gathered by the firm
has no influence on the government (because the consumers’ information, which is always
available when ec = 1, dominates it). By contrast, for the government,
∂v g
= (1 − e)(ugc − ef ugf ) > 0
c
∂e
(6)
which is positive since ef ≤ 1 and ugc > ugf by A4; and
∂v g
= (1 − e)(1 − ec )ugf > 0.
∂ef
(7)
As (4)-(7) hold globally except at corner solutions, we can state without proof a first result:
Proposition 1 In any interior Nash equilibrium with partial disclosure, a decrease in the
search intensity of the firm and consumer group would make both lobbies better-off but the
government worse-off.
Lobby j’s maximization problem is
max
v j s.t. 0 ≤ ej ≤ 1, j = {c, f }.
j
(8)
e
Let λj and µj be two Lagrange multipliers. Kuhn-Tucker conditions are
(1 − e)(1 + ef ) − ec − λc ec − µc (1 − ec ) = 0,
λc ≥ 0, ec ≥ 0, λc ec = 0,
µc ≥ 0, ec ≤ 1, µc (1 − ec ) = 0.
for the consumers and
(1 − e)(1 − ec ) − ef − λf ef − µf 1 − ef
f
f
= 0,
f f
λ ≥ 0, e ≥ 0, λ e = 0,
µf ≥ 0, ef ≤ 1, µf 1 − ef = 0,
for the firm. Reaction functions in (ec , ef ) space are4
Rc (ef , e) = min (1 − e)(1 + ef ); 1 ,
Rf (ec , e) = (1 − e)(1 − ec ).
(9)
(10)
4
The formal definition of the reactions function is given by Rc (ef , e) = max 0; min (1 − e)(1 + ef ); 1 ,
and Rf (ec , e) = max {0; min {(1 − e)(1 − ec ); 1}} but since some of the inequality constraints are never binding given A1-A6, we only write the possibly binding ones to facilitate reading.
9
In an interior solution, the slopes of the consumers’ and firm’s reaction functions are respectively positive and negative:
∂Rc
= 1 − e > 0,
∂ef
and
∂Rf
= e − 1 < 0.
∂ec
Their intercepts are respectively
Rc (0, e)
Rc (1, e)
Rf (0, e)
Rf (1, e)
=
=
=
=
(1 − e),
min {2(1 − e); 1} ,
(1 − e),
0.
(11)
(12)
(13)
(14)
These reaction functions are shown in Figure 1 for e = 0.6.
Figure 1: Reaction functions under partial disclosure
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We now consider the timing game, in which each of the lobbies chooses whether it wants
to play in period one or in period two. Kempf and Rota-Graziosi (2010) showed that when
one player has a downward-sloping reaction function whereas the other has an upward-sloping
one (so the game is neither one of strategic substitutes nor one of strategic complements),
the player with the downward-sloping reaction function chooses period one whereas the other
chooses period two, so the equilibrium play is a Stackelberg game where the player with the
downward-sloping reaction function is the leader. In our case, this means that the firm is
10
the leader and the consumers the follower. As Stackelberg leader, the firm’s maximization
problem is
max v f (ec , ef , e) s.t. ec = Rc (ef , e),
(15)
ef
where v f (.) is given by (2) and Rc (.) by (9). This gives
(5 − 3e)e − 2
f?
e = max 0;
2(e − 2)e + 3
with the kink at e = 2/3, and
1−e
c?
e =
[1 + e2 (e − 2)] / [3 + 2e(e − 2)]
(16)
if e ≤ 2/3,
if ef > 0.
(17)
Lastly, we consider the timing of disclosure. The issue for the firm is whether it can trigger
a favorable update of the consumers’ beliefs or avoid an unfavorable one by either withholding
or revealing information. Suppose first that the firm’s search is unsuccessful. It has nothing
to disclose and cannot, by assumption, credibly claim that its search was unsuccessful. Thus,
consumers cannot update their beliefs and play (17). Suppose now that it is successful. If it
discloses its information in period one, before the consumers choose their search intensity, the
consumers’ optimal search intensity becomes Rc (1, e) = min{1; 2(1 − e)} using (12), which
is the worst outcome for the firm given (5). Thus, disclosure in case of success cannot be
optimal, and the firm always withholds information if its search is successful, waiting until
period two to disclose (disclosing in period two is of course optimal in order to influence the
government). It follows that the search subgame is sequential, but the disclosure subgame is
simultaneous.
3.2.2
Full disclosure
Under full disclosure, when successful, the firm discloses not only its preferred policy, but also
that preferred by the government, in order to “insure” against a consumer win. Omitting
subscripts to differentiate between partial and full disclosure for ease of notation (we will
revert to using them when comparing outcomes), expected payoffs become
(ec )2
v c ec , ef , e = eucg + (1 − e) ec 1 − ef + ef ucg −
2
(18)
for the consumers,
v
f
ef
e , e , e = eufg + (1 − e) ef ufg − 2ec 1 − ef −
2
c
f
2
(19)
for the firm, and
v g ec , ef , e = e + (1 − e) ec 1 − ef ugc + ef
(20)
for the government. The payoff functions’ cross-partial derivatives are
∂v c
= (1 − e)(ucg − ec ) ≷ 0
f
∂e
11
(21)
for the consumers and
∂v f
= −2(1 − e)(1 − ef ) ≤ 0
(22)
∂ec
for the firm. Thus again, consumer search exerts a negative externality on the firm except
in a corner solution; by contrast, the externality exerted by the firm’s search on consumers
is now ambiguous, depending on the value of a payoff, ucg , which we do not parameterize.
However, we do know that at ec = 1, it is negative because by assumption ucg < 1. We will
use this result later on. Again, for the government,
∂v g
= (1 − e)(1 − ef ) ≥ 0
∂ec
(23)
since ef ≤ 1; and
∂v g
= (1 − e)(1 − ec ugc ) > 0.
(24)
f
∂e
which is positive since ec ≤ 1 and ugc < 1 by A1. Omitting Kuhn-Tucker conditions for
brevity, the reaction functions are
Rc (ef , e) = (1 − e)(1 − ef ),
Rf (ec , e) = min (1 − e)(ufg + 2ec ); 1 .
(25)
(26)
In an interior solution, their slopes are again of opposite signs, but now the consumers’ is
negative whereas the firm’s is positive:
∂Rc
= −(1 − e) < 0,
∂ef
and
∂Rf
= 2(1 − e) > 0.
∂ec
Their intercepts are respectively
Rc (0, e) = (1 − e),
Rc (1, e) = 0,
Rf (0, e) = (1 − e)ufg ,
Rf (1, e) = min (1 − e)(ufg + 2); 1 .
(27)
(28)
(29)
(30)
These reaction functions are shown in Figure 2 for e = 0.6 (as before) and ufg = 0.5 (which
we did not need to parameterize before).
Using again Kempf and Rota Graziosi (2010), the timing game’s equilibrium has now
the consumers playing first and the firm second. The consumers’ maximization problem as
Stackelberg leader is
max
v c (ec , ef , e) s.t. ef = Rf (ec , e),
c
e
where v c (.) is given by (18) and Rf (.) by (26). This gives
12
(31)
Figure 2: Reaction functions under full disclosure
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f
c
(1
−
e)
1
−
u
(1
−
e)
+
2u
(1
−
e)
g
g
ec? =
4(e − 2)e + 5
(32)
and
(
ef ?
! )
c
f
(1
−
e)
(1
−
e)
+
2u
(1
−
e)
1
−
u
g
g
= min (1 − e) ufg + 2
;1 .
4(e − 2)e + 5
(33)
Again, we now consider the timing of disclosure. We know that it is optimal for consumers
to always disclose their policy only, but the question here is whether they can influence
the firm’s search intensity by withholding the result of their search. Suppose the search is
successful. Given that the firm now has an upward-sloping reaction function, wrongfully
claiming an unsuccessful search and withholding the information would be optimal if it could
convince the firm to set ef = Rf (0, e) = (1 − e)ufg , a low search intensity. However, by
assumption, such a claim is not credible, so withholding the search result does not lead to
any update of the firm’s belief. Thus, the firm will set its effort level at (33) if the consumer
group claims no success. If the consumer group
by contrast, the
discloses its information,
firm updates its belief and sets Rf (1, e) = min (1 − e)(ufg + 2); 1 using (28), which is bad
for the consumers given that (21) is negative at ec = 1. Thus, disclosure at the end of
period one in case of success is not optimal, and the consumers withhold information if their
search is successful. Suppose now that the consumers’ search is not successful. Given that
withholding when successful is optimal , a claim of “unsuccessful” is not credible and triggers
no updating of beliefs by the firm, which sets again ef = (1−e)ufg . Thus, consumers withhold
13
when successful, and the firm always plays (33). In period two, the consumers disclose their
preferred policy. Thus again, the search subgame is sequential, but the disclosure subgame
is simultaneous.
3.3
Equilibrium outcomes
We now combine the equilibrium outcomes of sections 3.2.1 and 3.2.2 to characterize fully
the game’s equilibrium outcomes under endogenous disclosure. So far, we have two sets of
results: Under partial disclosure, the firm, which has a downward-sloping reaction function
(upward-sloping for the consumers), searches first and withholds the results; under full disclosure, the opposite happens, with consumers having a downward-sloping reaction function
(upward-sloping for the firm), searching first, and again withholding. We now put the pieces
together, substituting optimal search intensities under each regime into value functions and
determining which disclosure strategy dominates for the firm (the only player making a disclosure decision). Then, we characterize equilibrium outcomes in terms of search intensities
and government policy choice, as functions of the government’s parametric information level,
e. The interest of characterizing equilibrium outcomes as functions of the government’s information endowment is that the latter can be taken as a proxy for an electoral cycle, with
an uninformed government (e = 0) taking power at the beginning of the cycle and gathering
information, or analytical capabilities, over the cycle until e = 1 at the end.
3.3.1
Equilibrium payoffs
By substituting (16) and (17) into (2), we can rewrite the firm’s value function under partial
disclosure as:
2
h
i
efP?
f?
f
?
f
?
c?
,
(34)
vP ec?
= eufg + (1 − e) (1 − ec?
P , eP , e
P )eP − 2eP −
2
where the subscript P stands for partial disclosure. The firm’s value function under full
disclosure is obtained by substituting (32) and (33) into (19):
vFf ?
2
i
h
efF?
f
?
f
?
f
?
f
f
c?
ec?
1 − eF
−
F , eF , e = eug + (1 − e) eF ug − 2eF
2
(35)
where the subscript F stands for full disclosure. The firm maximizes its expected payoff by
disclosure choice, which gives:
n
o
f?
f?
f ? c? f ?
,
e
,
e
,
e
=
max
v
(.)
;
v
(.)
.
(36)
v f ?? ec?
,
e
F
P
F
P
F
P
We can now state a second result, namely
Proposition 2 There is a critical value of e, ê, such that the firm chooses full disclosure
whenever e < ê and partial disclosure otherwise.
14
Figure 3: Firm’s payoff against government information
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The proposition is proved in the appendix. In order to fix ideas, Figure (3) plots (34) and
(35) against the government’s parametric information, e, for ufg = ucg = 0.6:
Given the graph’s parameter values, partial disclosure is the dominant strategy for the
firm when the government’s effort level is approximately given by e ∈ [0.91; 1[.
For the consumers, there is no disclosure choice but payoff functions switch between
regimes. Substituting (16) and (17) into (1), the consumers’ value function under partial
disclosure is:
vPc?
f?
ec?
P , eP , e
=
eucg
+ (1 − e)
h
ec?
P
− (1 −
f?
ec?
P )eP
i
2
(ec?
P)
.
−
2
(37)
Similarly substituting (32) and (33) into (18), the consumers’ value function under full disclosure is:
i (ec? )2
h f?
f? c
f?
F
c
c?
.
vFc? ec?
,
e
,
e
=
eu
+
e
u
+
(1
−
e)
e
1
−
e
F
g
F
F
F
F g −
2
(38)
Combining,
v
c??
f ? c? f ?
ec?
P , eP , eF , eF , e
=
vPc? (.)
vFc? (.)
if vPf ? (.) ≥ vFf ? (.) ,
otherwise.
(39)
Full payoff functions with the regime switch are illustrated in Figure (4) for the usual parameter values.
Interestingly, even though the firm is, by assumption, the player with the most extreme
preferences (the government’s preferences are assumed closer to those of the consumers),
its payoff rises monotonically with the government’s information. This is because a more
informed government is less likely to be swayed by the consumers, and here the firm has, by
assumption, much to lose from implementation of the consumers’ policy. When e is low, the
risk of a consumer win is high, so the firm’s defensive strategy is to disclose fully what it
finds, including the government’s preferred policy (what we call in the introduction “unbiased
information”). It is also, as we will now see, to search at a high intensity.
15
Figure 4: Players’ payoffs
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Search intensities
The lobbies’s search intensities are also subject to a regime switch based on the firm’s disclosure decision. They can be expressed as
c?
eP (.) if vPf ? (.) ≥ vFf ? (.),
c??
c? c?
e (eP , eF ) =
(40)
otherwise,
ec?
F (.)
for the consumers, and
e
f ??
ef ? (.)
f? f?
P
eP , eF =
efF? (.)
if vPf ? (.) ≥ vFf ? (.),
otherwise,
(41)
for the firm. Equilibrium search intensities, with the regime switch, are plotted against e in
Figure (5) for the usual parameter values.
Patterns are striking. The firm’s search intensity is monotone decreasing in the government’s
information. Under full disclosure (e < ê), this is intuitive, as the firm’s and the government’s information are perfect substitutes since the firm forwards unbiased information to
the government. Under partial disclosure, the result carries over although the firm’s and the
government’s information are now imperfect substitutes. The consumer’s search intensity is
not monotone anymore but has an inverse U-shape. Because the consumers are assumed to
be a low-stakes player, the convex cost of search leads to moderate search intensities at all
levels of the government’s information, and for high values of e, the consumers’ effort dips
to zero, albeit with a jump up when the firm switches from full to partial disclosure, which
makes the outcome of the game potentially more damaging for the consumers since past that
point there is a positive probability that the firm wins.
3.3.3
Policy outcomes
The probability of a given policy P being implemented can be computed in a similar fashion,
conditional on the firm’s disclosure decision. The probability of having the consumers’ policy
P c implemented is
(1 − e)ec?
if vPf ? (.) ≥ vFf ? (.),
c
P (.)
Pr(P = P ) =
(42)
f?
(1 − e)ec?
otherwise.
F (.)(1 − eF (.))
16
Figure 5: Effort levels
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The probability of having the firm’s policy P f implemented is
f?
(1 − e) (1 − ec?
if vPf ? (.) ≥ vFf ? (.),
f
P (.)) eP (.)
Pr(P = P ) =
0
otherwise.
The probability of having the government’s policy P g implemented is
e
if vPf ? (.) ≥ vFf ? (.),
g
Pr(P = P ) =
e + efF? (.)(1 − e) otherwise.
(43)
(44)
Finally, the probability of the status-quo P s is the probability that all three players fail in
their information search, i.e.
f?
if vPf ? (.) ≥ vFf ? (.),
(1 − e)(1 − ec?
s
P (.))(1 − eP (.))
(45)
Pr(P = P ) =
f?
(1 − e)(1 − ec?
otherwise.
F (.))(1 − eF (.))
Figure (6) plots these probabilities for usual parameter values and ufg = ucg = 0.2, which we
did not need to parameterize before.
The probability of implementation of the government’s policy (the “middle-of-the-road” one)
is strikingly high even at very low levels of the government’s information; in a neighborhood
of e = 0, it is over 0.65. Again, the reason is the firm’s defensive search/disclosure strategy
which consists of searching at a very high intensity and forwarding unbiased information to
the government. It is worth noting also that the firm’s very high search intensity is obtained
in a sequential equilibrium (full disclosure) where the consumers play first at a lower intensity
than in the simultaneous game in order to “contain” the firm (which has an upward-sloping
reaction function). However, at low levels of e, even the moderation generated by the timing
game fails to contain the firm’s defensive lobbying.
17
Figure 6: Probability of policy implementation
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Extension: Offensive Lobbying
Up till now, we assumed that the firm had more to lose from the consumers’ policy (ufc = −2)
than the consumers had to lose from the firm’s (ucf = −1); it was in this sense that the firm
was a “high-stakes” player. This particular payoff structure was meant to capture a situation
such as that discussed in the case of glutamate monosodium, where consumers were campaigning over relatively diffuse health hazards while the firm was struggling to keep one of
its cash cows on the market, a high-stakes issue for its shareholders. In other circumstances,
firms seek regulatory-induced rents (from trade protection or other distortions to competitions) at the expense of the public at large. In such cases, the appropriate payoff structure is
one in which the firm gains disproportionately from implementation of its preferred policy.
Intuitively, we would expect it to search at a high intensity to influence the government, not
so much to avoid the policy preferred by consumers, but to have its own implemented. The
question is whether this drastically affects the logic of our results in the sense of leading to
a more likely adoption of the firm’s extreme policy, as predicted by the political-economy
literature. In order to explore this, we now turn to a variant of our game with the payoff
structure shown in Table 3.
While payoffs change, the rules of the game remain the same and Table 4 shows that the
information partition is only marginally affected by the changes.
18
Table 3: Policy outcomes
Policy
c
f
g
s
w
Payoff to
c
f
g
1 −1 ugc
−1
2 ugf
ucg ufg
1
0
0
0
`c
`f `g
Table 4: Information, policy decisions, and payoffs
Gov. known set
c f g s w
. . 1 1 .
1 . 0 1 .
0 1 0 1 .
0 0 0 1 .
4.1
Policy
choice
g
c
f
s
Payoffs
(uc , uf , ug )
(ucg , ufg , 1)
(1, −1, ugc )
(−1, 2, ugf )
(0, 0, 0)
Partial disclosure
Under partial disclosure, payoff functions are
(ec )2
v c ec , ef , e = eucg + (1 − e) ec − (1 − ec )ef −
,
2
v
f
2
ef
f
c f
c
,
e , e , e = eug + (1 − e) 2(1 − e )e − e −
2
c
f
(46)
(47)
and
v g ec , ef , e = e + (1 − e) ec ugc + (1 − ec )ef ugf .
(48)
respectively. Their cross-partial derivatives are
∂v c
= −(1 − e)(1 − ec ) ≤ 0
∂ef
and
(49)
∂v f
= −(1 − e)(2ef + 1) < 0
(50)
∂ec
meaning that the direction of externalities remains the same. Omitting the maximization
problem and the Kuhn-Tucker conditions, which are unchanged, the new reaction functions
in (ec , ef ) space are
Rc (ef , e) = min (1 − e)(1 + ef ); 1 ,
(51)
19
Rf (ec , e) = min {2(1 − e)(1 − ec ); 1} .
(52)
In an interior solution, the consumers have an upward-sloping reaction function and the firm
has a downward-sloping one, as before:
∂Rc
= 1 − e > 0,
∂ef
∂Rf
= 2(e − 1) < 0.
∂ec
They are plotted in Figure 7 for e = 0.6.
Figure 7: Reaction functions under partial disclosure
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Following again Kempf and Rota-Graziosi (2010), the equilibrium play of the timing game
is a Stackelberg game where the firm is the leader, giving
(4 − 3e)e − 1
f?
e = max 0;
(53)
4(e − 2)e + 5
with the kink at e = 1/3, and
1−e
c?
e =
[(e − 2)2 (1 − e)] / [4(e − 2)e + 5]
if e ≤ 1/3,
if ef > 0.
(54)
The timing of disclosure remains unchanged, with the firm whitholding and disclosing in
period two.
20
4.2
Full disclosure
Payoff functions are the same as in section 3.2.2 except for the firm, with
2
f f
ef
f
c f
f
c
f
−
v e , e , e = eug + (1 − e) e ug − e 1 − e
.
2
(55)
with cross-partial derivatives
∂v c
= (1 − e)(ucg − ec ) ≷ 0
∂ef
(56)
∂v f
= −(1 − e)(1 − ef ) ≤ 0
∂ec
(57)
Reaction functions are
Rc (ef , e) = (1 − e)(1 − ef ),
Rf (ec , e) = min (1 − e)(ufg + ec ); 1 .
(58)
(59)
with opposite slopes compared to partial disclosure:
∂Rc
= −(1 − e) < 0,
∂ef
∂Rf
= (1 − e) > 0.
∂ec
The timing game’s equilibrium has now the consumers playing first and the firm second, with
(1 − e)(ucg (1 − e) + ufg e − ufg + 1)
e =
2(e − 2)e + 3
c?
and
(
ef ? = min
)
(1 − e) 2ucg (e − 1)2 + ufg − 2e + 2
;1 .
2(e − 2)e + 3
(60)
(61)
The timing of disclosure in unaffected, with simultaneous disclosure in period two.
4.3
Equilibrium outcomes
Following the same procedure as in Section 3.3, we obtain the firm’s payoff functions under
partial and full disclosure. Somewhat surprisingly, the firm still chooses full disclosure below
a critical value of e and partial above (see appendix), so the logic of the solution is intact.
Figure 8 illustrates this with the firm’s payoff functions under each regime as functions of e
with ufg = ucg = 0.6 as before.
Search intensities are plotted in Figure 9 while Figure 10 shows policy implementation probabilities for the usual parameter values (ufg = ucg = 0.2).
While discontinuities at the regime-switch point appear larger for the same parameter values,
the overall logic remains unaffected, with the firm putting in very high search intensities at
low values of e while the consumers have non-monotone search intensities.
21
Figure 8: Firm’s value functions
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The equilibrium implementation probability of the government’s policy remains disproportionately large in this new setting, even at very low levels of e; in fact, it is now even higher,
at more than 0.85 in a neighborhood of e = 0; while the probability that the firm’s extreme
policy is implemented is now the lowest, just above the status quo (Figure 10). Resulting
payoffs for the two players, still holding other parameters constant and equal to 0.6, are
reported in Figure 11.
Again, the firm’s payoff rises monotonically with the government’s information, because it
gets essentially the same outcome as under full disclosure but without having to incur the
search cost. By contrast, the consumers’ payoff is non-monotone, weakly decreasing up to
the regime switch point where it jumps down; this is because the firm then switches to a
partial disclosure strategy, with a positive probability of implementation of P f .
5
Concluding remarks
The objective of our model was to illustrate how a somewhat counter-intuitive result, namely
that mainstream policies can emerge even in the presence of extremist lobbies with persuasion power, can emerge in an informational-lobbying framework.
The result emerges in two distinct settings. Both settings share a set of common features,
including an asymmetry of preferences and stakes (in a cardinal-utility sense), an informational structure where both lobbies need to spend real resources to search for information,
and where strategic information takes place at two broad levels, information production and
information disclosure. The difference between the two settings lies in their payoff structures.
In one, the firm (the high-stakes player) loses disproportionately from implementation of the
consumers’ policy; we call this configuration one of “defensive lobbying” and use it to portray situations where consumer groups lobby against a particular firm, say by demanding a
ban on one of its core products. In the alternative, the firm gains disproportionately from
implementation of its preferred policy; we call this configuration one of “offensive lobbying”
and use it to portray situations of rent-seeking.
22
Figure 9: Effort levels
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Our thought experiment consists of deriving comparative-statics results on the government’s parametric level of information, which we use as a proxy for its experience or analytical
capabilities. The most intuitive interpretation of the comparative statics on government information is to think of it as an electoral cycle, with a fresh, inexperienced government
arriving in power and accumulating experience over the cycle. In both settings, when the
government is relatively uninformed (at the beginning of the cycle), the firm spends a large
amount of resources searching for information, but it adopts a conservative disclosure policy
where it feeds the government with unbiased information in order to neutralize the information conveyed by the consumer group. Thus, surprisingly, the most extremist lobby ends
up not sending extremely biased signals, and the probability that it gets its first-best policy
implemented is, in equilibrium one of the lowest if not the lowest. As the government gathers
experience, it relies less and less on the information conveyed by lobbies, substituting its own.
The incentive for costly information search for the lobbies shrinks, and equilibrium outcomes
increasingly reflect the government’s preferences based on its own information.
Thus, as in common-agency models, in our model lobby rivalry serves as a substitute for
government information to pull policies toward the center.
23
Figure 10: Probability of policy implementation
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