Abstract: A mechanism design approach is used to characterize communication patterns between lobbying groups and a decision-maker when those groups ...
Communication by Multiple Interest Groups1
David Martimort2 and Aggey Semenov3
This Version: February 10, 2005
Abstract: A mechanism design approach is used to characterize communication patterns between lobbying groups and a decision-maker when those groups have private information on their ideal points in a one-dimensional policy space. First, we derive the optimal mechanism for a single group and show that it depends on the conflict of interests with the policy-maker but also on how informative the distribution of the interest group’s ideal point is. The analysis is then extended to the case of multiple interest groups. Although dealing with a coalition of those groups allows the policy-maker to benefit from a more precise information (an informativeness effect), the optimal organization may nevertheless require a decentralized mechanism where groups compete because this is the only way to transmit information on the relative strength of their preferences (a screening effect). From the decision-maker’s viewpoint, a coalition dominates for small conflicts of interests whereas competing interest groups are preferred otherwise. Interest groups also win from merging when their conflicts of interest vis-`a-vis the principal are small enough.
Keywords: Communication mechanisms, Lobbying, Coalition. JEL Classification : D72; D82.
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We thank Jean-Jacques Laffont, Michel Lebreton, Eric Maskin, Shlomo Weber, Fran¸cois Salani´e, Wilfried Sand-Zantman and seminar participants in Toulouse, Stockholm and Madrid for helpful discussions. 2 University of Toulouse, (IDEI, GREMAQ) and Institut Universitaire de France. 3 University of Toulouse (GREMAQ).
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1
Introduction
This paper present a theoretical framework to understand the structure of communication between interest groups and a decision-maker in a multigroup environment. This framework is based on the by-now well admitted fact, both among political scientists and economists, that lobbying has an important informational role.1 Lobbyists spend time and resources to convey information on their preferences to uninformed political decision-makers to influence their choices. Informational asymmetries are thus crucial to understand the relationships between lobbying groups and policy-makers. That lobbying groups have private information puts them in a unique position to influence policy-making at least on the very issues on which their private interests are at stake. Because there may exist a conflict between the preferences of policy-makers and those of interest groups, the latter have incentives to manipulate information. The implemented policies result thus from a trade-off between choosing a rigid policy independent on the interest group preferences and communicating with the interest group to implement a more flexible policy but at the cost of letting the preferences of this group drive policy choice. Clearly, informational asymmetries not only explain much of the policies implemented in practice but also generate the transaction costs that plague the relationships between policy-makers and interest groups. The overall organization of lobbying should then certainly be driven by the desire to minimize these transaction costs. In this paper, we will give a particular attention to the different communication patterns and their consequences on payoffs when interest groups may either merge or remain split apart in the political arena. By remaining split apart, interest groups make it more likely that their own preferences will be represented in the political process; a screening effect. However, this may be contradict by the fact that, by merging, two otherwise distinct interest groups may be better able to communicate with a decision-maker; an informativeness effect. To understand first how informational asymmetries shape policy design, we start with a simple relationship involving a single interest group and a policy-maker both having different ideal points in a one-dimensional space of policy. The interest group has private information on his ideal point which is relevant to determine the optimal policy from the decision-maker’s viewpoint. In that simple context, we analyze the determinants of the policy choice and provide simple comparative statics. As the conflict of interests between the interest group and the policy-maker is more pronounced, a rigid policy involving pooling across a larger set of preference profiles becomes more attractive. Communication is less valuable when the interest group has stronger incentives to manipulate information 1
See Milbrath (1963), Wright (1990), Hansen (1991) and more recently Grossman and Helpman (2001).
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to influence decision-making. The same is true when the decision-maker has already a relatively precise idea of the interest group’s preferences. Indeed, as the distribution of these preferences puts more weight around its mean, the decision-maker relies less on active communication. The optimal policy rule comes closer to the ex ante optimal rigid policy that the policy-maker chooses without relying on the interest group’s information. This informativeness effect is particularly crucial to understand one first force which could justify why separate interest groups may want to merge. Consider indeed two interest groups with independently and identically distributed ideal points who credibly share information and merge as one active player. That coalition only needs to communicate the average preferences of the two groups to the decision-maker. This signal is less noisy than the type of each individual interest group taken separately. Dealing with such a coalition, the policy-maker gets a more precise idea of the relevant information needed to implement a policy and, by the same token, does not need to rely so much on communication. Scope economies in communication emerge. Those scope economies favor large coalitional groups compared with smaller ones holding more diffuse private information which is harder to communicate. Although intuitive, this insight turns out to be incomplete. When all interest groups merge as one, the statistics used by this coalition to communicate with the decision-maker may indeed be too rough to describe with enough precision the exact preferences profile. The information aggregation which already takes place within such a large group smooths out large discrepancies in the preferences of his individual members and cannot account for the diversity of opinions which may be quite relevant when the ideal points of those interest groups are far apart. Indeed, because less communication is needed with a coalition, a rigid policy is chosen for a larger set of preference profiles and, most noticeably, when the preferences of the groups are sufficiently far away from each other. A better communication pattern can then be obtained with a decentralized mechanism such that either one or the other interest group’s ideal point is ultimately chosen by the policy-maker. Either group becomes then residual claimant for the policy implemented depending on how close he is to the decision-maker. When the conflict of interests between the principal and the interest groups is sufficiently pronounced, it is more efficient to deal separately with each of them than with a coalition.The corresponding decentralized mechanism improves both the decision-maker and the interest groups’ expected payoffs. This screening effect justifies that it can be preferable to keep interest groups apart from each other even though this decentralized organization may not enjoy the scope economies of merging highlighted above. For lower levels of conflict instead, the informativeness effect dominates. Both the decisions-maker and the interest groups prefer to merge. To derive those insights on the organizational pattern of lobbying activity, we depart significantly from the existing literature on the informational role of interest groups. 3
Starting with the seminal paper of Crawford and Sobel (1982), this literature has indeed viewed lobbying groups as informed Stackelberg leaders in the communication game played with an uninformed policy-maker.2 Although attractive on positive grounds, that approach faces some difficulties when it comes to discuss the overall organization of lobbying. Signaling games are generally plagued with a multiplicity of inefficient partition equilibria3 and in the absence of a convincing equilibrium refinement, the comparison of outcomes between alternative scenarios and organizational forms is only indicative of the forces shaping the overall organization of lobbying groups. Instead, we adopt in this paper a mechanism design perspective.4 We revert the timing of standard lobbying games and assume that the policy-maker commits ex ante to a mechanism describing policy choices as a function of the various messages that might be sent by lobbyists on their preference profiles at the communication stage. This approach is in lines with Melumad and Shibano (1991) who earlier adopted that perspective although in the context of a single informed agent, Baron and Meirowitz (2001) who more recently offered some further links between the signaling and the mechanism design approach,5 and Baron (2000) who introduced mechanism design to model legislative organization.6 The mechanism design approach is useful on normative grounds because the Revelation Principle gives us a complete characterization of the set of implementable allocations achieved as equilibria7 of the communication game played by lobbies. The comparison between the costs and benefits of different organizational forms of lobbying and the corresponding optimal mechanisms is thus facilitated. When analyzing communication in a decentralized organization with multiple interest groups reporting their ideal points non-cooperatively to the decision-maker, we rely on dominant strategy mechanisms. The characterization of those for the large domain of single-peaked preferences is already available from Moulin (1980). Using the smaller class of quadratic single-peaked preferences, and restricting to continuous and symmetric mechanisms, we recover Moulin’s result that decision rule should be a generalized median with phantom bids.8 More importantly, this restriction on preferences allows us to 2
See Austen-Smith (1997) and Grossman and Helpman (2001) for synthetic views of this approach. A noticeable exception is the multidimensional world described in Battaglini (2002). 4 Laffont (2000) offers a more general defense of this perspective to explain constitutional choices. 5 More specifically, these authors argue that with a convenient modeling of out-of equilibrium beliefs capturing the lack of commitment assumption included in the signaling perspective, the two approaches are equivalent. 6 Other contributions to that mechanism design literature involves Green and Stokey (1981), Holmstr¨om (1984) and Armstrong (1994). On the relationship between models with or without commitment, see also Mylovanov (2004) who shows that any incentive compatible mechanism without transfers can be implemented with very little commitment using a veto-based delegation. 7 Modulo an implementation concept and here we will focus on dominant strategy for simplicity. 8 Our proof based on a differentiable approach is more direct than Moulin (1980)’s one. Restricting to the domain of quadratic single-peaked preferences also introduces non-continuous schemes which are not our main focus below. 3
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compare the expected payoffs that both the interest groups and the policy-maker obtain under various organizational forms. This is an important step of the analysis to get some normative insights on why groups coalesce or not. Baron (2000) has analyzed a model where legislators in a Committee have similar preferences (driven by the same common shock) and monetary resources can be exchanged between the uninformed floor and the informed Committee to help screening. We depart from this work by focusing essentially on the case where such monetary transfers are not feasible. Also our interest groups may have different pieces of private information and the issue of finding the optimal communication pattern and organization of lobbying is not tackled by Baron (2000) since he a priori assumes that the only differences between the legislators’ ideal points is common knowledge. Wolinsky (2002) considers a setting where a decision-maker has to elicit information from informed experts who have similar preferences but have conditionally independent signals on the desirability of an action. Glazer and Rubinstein (1998) consider also the problem of designing a mechanism without transfers to achieve a given target when agents are driven by different motives. Even though our mechanism design approach can be viewed as an alternative to the signaling literature, this literature has already provided valuable insights on organizational issues by comparing equilibria outcomes. Austen-Smith (1990, 1993a, 1993b) has analyzed communication patterns when groups report either sequentially or simultaneously. Krishna and Morgan (2000) have studied a lobbying game with two informed lobbyists who share the same information on the state of nature but may have conflicting or congruent views on what should be the optimal policy. They showed that conflicting views help the policy-maker to extract information. A mechanism design perspective would predict in such a context the existence of a costless fully communicative equilibria irrespectively of the bias.9 This leads us to focus on the case where each interest group has idiosyncratic private information on their ideal points. Grossman and Helpman (2001, Chapter 4) have also compared the informativeness of various equilibria depending on whether the lobbyists messages are public or private, an issue which is also put aside by the mechanism design approach.10 Finally, Battaglini and Benabou (2003) have recently analyzed the incentives of interest groups to coalesce or not using the signaling approach. They study a common value environment where the extend of one group’s lobbying depends on whether other groups show up in the lobbying process and on whether the decision9
This would be obtained by using a revelation mechanism `a la Maskin. The issue may nevertheless be the existence of other non-truthful equilibria. 10 Indeed, the fact that communication with each interest group is publicly observable is part of the mechanism. Relaxing that assumption would move us towards a third-best world where bilateral communication between the decision-maker and each interest group may create some scope for vertical collusion. On this issue in a related model, see Gromb and Martimort (2004).
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maker finds those activities trustworthy or not. Even though their methodology and the informational structure they use both differ from ours, we share with them results which are qualitatively similar. For small levels of conflict, they also show that a coalition of interest groups is preferred whereas a non-coordinated behavior dominates otherwise. Section 2 presents the model. In Section 3, we derive the optimal mechanism which maximizes the expected payoff of the policy-maker when there is a single interest group. We give sufficient conditions for that mechanism to be continuous and highlight then its simple structure. Continuity is important to get a simple characterization of optimal mechanisms which is more easily amenable to comparisons between different organizational forms. We provide there also some comparative statics analyzing the impact of the level of conflict between the policy-maker and the interest groups and the role of the informativeness of the distribution of the latter’s ideal point. This section is thus of independent value for readers interested in the mechanism design approach for lobbying games. In Section 4, we analyze the scope economies which arise when two groups coalesce to influence a policy-maker. Finally, in Section 5, we compare two organizations of the playing field for lobbying: one with a coalition of two groups, the other with two competing groups. We highlight there the better possibilities for screening which arise in the decentralized organization where groups remain split apart, at least when the conflict of interests is large enough. Section 6 concludes. Proofs are relegated to an Appendix.
2
The Model
In the economy, there are n special interest groups (agents) indexed by 1, . . . , n and a single policy-maker (principal). The policy q chosen by the principal is one-dimensional.11 Interest groups have single-peaked quadratic preferences with ideal points θi : 1 Ui (q, θi ) = − (q − θi )2 .12 2 The principal aggregates the preferences of those interest groups but may also take into account the preferences of the rest of society. We capture this effect in assuming that the principal has also a quadratic utility function given by: 2 n 1 1 q− V (q, θ1 , . . . , θn ) = − θi − δ . 2 n i=1 The parameter δ > 0 represents the bias in the preferences of the policy-maker compared to the benchmark policy which would consist in averaging the interest groups’ ideal points. 11
Depending on the application, this can be a tax, an import tariff for instance, a regulated price, a number of allowed permits, etc. 12 Symmetry in the preferences is assumed for simplicity.
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That the policy-maker cares also on the interests of the general public even though it is not organized as an active lobby might be justified if, for instance, he has reelection concerns. Importantly, the principal and the agents have non-monotonic preferences, a condition which is necessary for communication between the principal and the agents to take place in a mechanism design framework without monetary transfers. Interest group i has private information on his ideal point θi . The preference parameters θ1 , . . . , θn are drawn identically and independently on the same bounded interval ¯ according to the cumulative distribution F (·) which has an atomless and everyΘ = [θ, θ] where density f = F . We will sometimes assume that F (·) is log-concave (i.e. positive F (θ) d > 0). We denote by Ef (θ) the expectation of θ according to f (·). Even though, dθ f (θ) we want to make some of the analysis below as general as possible, we will mostly deal with the case of a uniform distribution in the case of a single interest groups. However, a general formulation is useful when we will study a merged group whose ideal point is the average of two types which are uniformly distributed. Note that our formulation is such that all groups have a priori the same expected conflict with the principal even though ex post their ideal points may differ. They both tend to prefer on average a more conservative policy than the principal’s ideal point.13 We follow a mechanism design approach where the principal commits to a policy rule stipulating which decision should be made as a function of the lobbyists’ reports on their preferences. From the Revelation Principle, there is indeed no loss of generality in restricting the principal to offer a direct revelation mechanism {q(θˆ1 , . . . , θˆn )}θˆi ∈Θ which is truthful. When we deal with multiple interest groups, we use dominant strategies as the implementation concept. The motivation for the commitment assumption is two fold. From a practical viewpoint first, this assumption seems reasonable. The principal may have already promised to the interest groups which have earlier contributed to his electoral campaign a particular policy platform to respond to their needs. He must thus stick to this scheme when in office to maintain his reputation in view of raising future campaign contributions. More importantly maybe, the commitment assumption is also attractive because it solves the equilibrium indeterminacy that arises in the signaling environment where lobbyists would move first. The mechanism design approach allows also a full characterization of communication patterns achievable at any equilibrium of a communication game among the agents. This property seems also quite attractive as far as one is concerned with the normative comparison between various organizational forms of lobbying. 13
This assumption is mainly done to simplify. We could account for differences among interest groups in the directions in which they want the policy to be pushed by having preferences being drawn from different supports.
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The timing of the lobbying game is as follows. First, each lobbyist observes only his own preferences. Second, the policy-maker offers a mechanism {q(θˆ1 , . . . , θˆn )}θˆi ∈Θ . Third, the lobbyists report their preferences to the policy-maker. Fourth, the corresponding policy is implemented. For further references, let us determine the policy chosen when the policy-maker knows the whole vector of preferences (θ1 , . . . , θn ). The first-best policy chosen by the principal is then: n 1 q ∗ (θ1 , . . . , θn ) = θi + δ. (1) n i=1 Instead, the principal might remain uninformed on the preferences profile and commit ex ante to a policy without communicating with interest groups. He would then choose a θ¯ rigid policy q p to maximize an ex ante criterion, namely − 12 θ (q − θ − δ)2 f (θ)dθ, i.e., q p = δ + Ef (θ). Remark: By the Law of Large Numbers, the complete information policy converges towards the rigid policy when the number of agents is large. This effect is of a particular importance in view of what follows. It already suggests that communication is in fact less needed as the principal faces more interest groups.
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One Interest Group
Let us suppose that the principal faces a single privately informed interest group. Incentive compatibility constraints for that group write as: 1 ˆ 1 − θ)2 − (q(θ) − θ)2 ≥ − (q(θ) 2 2
ˆ ∈ Θ2 . ∀(θ, θ)
(2)
The principal’s mechanism design problem can thus be expressed as: ¯ 1 θ (q(θ) − θ − δ)2 f (θ)dθ max − {q(·)} 2 θ subject to (2).
3.1
Incentive Compatibility
The general structure of the set of incentive compatible mechanisms with quadratic preferences can be easily derived following the work of Melumad and Shibano (1991). 8
Lemma 1 : (Melumad and Shibano (1991)14 ) An incentive compatible scheme q(·) must satisfy the following conditions: • q(θ) is weakly increasing and thus almost everywhere differentiable; • if q(θ) is strictly increasing, q(θ) = θ; • if q(θ) is discontinuous at a point θ1 then: −
q(θ1+ ) + q(θ1− ) = 2θ1 ,
−
q(θ) is flat on the right and the left of θ1 ,
−
q(θ1 ) belongs to the pair {q(θ1− ), q(θ1+ )}.
(3)
The proof of this lemma is instructive because it yields some insights on the nature of incentive compatible schemes. At any differentiability point of q(·), we must indeed have: (q(θ) − θ)q(θ) ˙ =0
(4)
This incentive constraint is satisfied by two interesting classes of schemes: the pooling ones where q(θ) = q on all Θ, and the fully separating one corresponding to the interest group’s ideal point, q(θ) = θ on all Θ. The optimal mechanism will in fact be a compromise between such schemes. Continuous mechanisms have a simple form in this environment. It can be easily seen that they have at most one strictly increasing part. Typically, let us denote θˆ1 and θˆ2 the boundary of the segment where q(θ) = θ, a continuous scheme writes as:
q(θ) = min{θˆ2 , max{θ, θˆ1 }}.
(5)
The reader accustomed with the social choice literature will have recognized the minmax rule (in the case of one agent only) due to Moulin (1980). Moulin’s characterization of dominant strategy incentive mechanisms was obtained by assuming that the domain of single-peaked preferences is much larger than the specific quadratic preferences used throughout. The cost of such an enlargement is that the differentiable approach we rely on is lost. The benefit of focusing on continuous mechanisms is a gain in tractability, an important step in view of proceeding to a normative analysis comparing communication patterns and payoffs across organizational modes. 14
Melumad and Shibano (1991) have proved this lemma for general single-peaked utility functions satisfying the single-crossing property. We get a shorter proof by specializing to the quadratic case.
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Incidentally, the comparison of Lemma 1 and Moulin’s result shows also that enlarging the preferences domain beyond quadratic to all single-picked preferences rules out only discontinuous schemes. This robustness of continuous schemes when preferences go beyond the quadratic case might be viewed as another justification to focus on schemes given in (5). Although keeping with the quadratic case allows us to make explicit comparisons of expected payoffs whether groups merge or not; something that would be out of reach otherwise. Lastly even if one is not ready to focus a priori on continuous schemes, next Lemma provides a sufficient condition on the types distribution to ensure that the optimal mechanism is actually continuous with quadratic preferences. Lemma 2 : The optimal mechanism is continuous at point x ∈ Θ if for any ∆ ≥ 0:15 x x+∆ δ (2F (x) − F (x − ∆) − F (x + ∆)) − F (y)dy + F (y)dy ≥ 0. (6) x−∆
x
Condition (6) is somewhat weaker than requiring the concavity of the cumulative distribution F (·). However, for any decreasing density f (·), the optimal mechanism is in fact continuous.16 A special case of interest is that of a uniform distribution that will be analyzed below.17 Still keeping in mind our concern for tractability, Lemma 2 is useful since it will allow us to directly check on distributions that continuity of the optimal mechanism is ensured. This will be particularly interesting when we will move to the case where two groups merge and change accordingly their distribution of ideal points. Of course, the continuity of the optimal mechanism is generally not ensured for any distribution. For instance, in the case of a normal distribution, discontinuities appear when δ is not too large.18 Condition (6) is indeed harder to satisfy when distributions put enough mass on a hump. The loss of tractability associated to the treatment of discontinuities forces us to limit the scope of our comparison between a coalition and separated groups in some of the Propositions below by restricting to the case of a uniform distribution and/or conditions on δ ensuring continuity. 15
If x+∆ > θ¯ or x−∆ < θ then in (6) x+∆ is replaced by θ¯ and x−∆ is replaced by θ correspondingly. Lemma 2 shows that, when (6) holds, any scheme with a discontinuity at a point x is dominated by a continuous scheme which follows the most preferred policy of the interest group on an interval around x. 17 When Lemma 2 holds, discontinuous schemes which would look like the partition equilibria found in the signaling model Crawford and Sobel (1982) can thus never be optimal. 18 A proof is available upon request. 16
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3.2
Optimal Mechanism
We are now ready to give a simple characterization of the optimal mechanism. Proposition 1 : Assume that condition (6) holds, the distribution F (·) is strictly logconcave, and the degree of conflict between the policy-maker and the interest group is not θ¯ too severe, i.e., δ < θ F (y)dy = θ¯ − Ef (θ). The optimal mechanism is then unique it is continuous and of the form: ˆ q(θ) = max{θ, θ} where the cut-off θˆ is uniquely defined by the condition 1 δ= ˆ F (θ)
θˆ
ˆ F (θ)dθ = θˆ − Ef (θ|θ ≤ θ).
(7)
θ
That proposition highlights the trade-off faced by the policy-maker in designing an optimal communication mechanism. On the one-hand, the principal would like to learn the true ideal point θ of the interest group to implement his own most preferred policy q ∗ (θ) = θ + δ. However, this choice is not incentive compatible from (4). If the principal wants to follow a flexible type-dependent policy, it has to necessarily be the interest group’s ideal point and thus q(θ) = θ. This policy differs from the policy-maker’s own ideal point by an amount δ and is thus “too” low compared to q ∗ (θ). Instead, the principal might want to avoid any communication and implement a more rigid policy. This policy entails much pooling but is closer on average of his own ex ante optimal policy. The key insight provided by Proposition 1 is that the principal always benefits from some communication on the upper tail of the types distribution whereas pooling is instead preferred on the lower tail. The intuition behind this result is the following. First, we know that a continuous mechanism has at most one interval where effective communication takes place between the principal and the agent (see (5)). Clearly, having a pooling mechanism on an upper tail θ ≥ θˆ2 cannot be optimal. Indeed, the principal can offer an incentive scheme with effective communication inducing a policy q(θ) = θ for θ ≥ θˆ2 which is closer to his own ideal point q ∗ (θ) = θ + δ than any pooling scheme q(θ) = θˆ2 when θ is large enough. Hence, communication is valuable on the upper tail. Second, consider now the lower tail of the type distribution. The policy-maker prefers to offer a pooling policy on such an interval. The cost of doing so is of course that the policy is not responsive to the preference parameter θ. The benefit is that, for the highest values of θ on that interval, this pooling policy is closer on average to the principal’s ideal point. The benefits of having a near-by policy for these intermediate values may exceed 11
the cost borne on the very lowest values with a pooling scheme even though this pooling may be quite further away from the principal’s ideal policy. This is so when relatively little weight is left to that lower tail. Log-concavity ensures this latter property. Proposition 1 highlights a fundamental trade-off between rigidity on the lower tail and flexibility on the upper tail of the types distribution. The cut-off value θˆ summarizes how those two forces compensate each other. The alternative expression in (7) reinforces our understanding of this trade-off. On the lower tail, the principal chooses a policy ˆ whereas on the upper tail, the interest group is residual claimant for the δ + Ef (θ|θ ≤ θ) policy which fits his own preferences. The cut-off is determined by making the principal just indifferent between these two options. Corollary 1 : Assume that the conditions of Proposition 1 hold and that the conflict of θ¯ interests between the principal and the agent is large enough, i.e., δ ≥ θ F (θ)dθ then θ¯ there is no communication at the optimal mechanism. When δ < θ F (θ)dθ, there is always some partial communication. ¯ we have Example: For a uniform distribution on [θ, θ], δ=
1 θˆ − θ
θˆ
(θ − θ)dθ = θ
so that communication never takes place when δ >
θˆ − θ , 2
∆θ . 2
It is striking to see that a similar upper bound on the level of conflict is needed to ensure no-communication in the signaling environment where the interest group moves first.19 Large conflicts of interest destroy any possibility for communicating preferences in an incentive compatible way even when the principal recovers some commitment power in the mechanism design environment. In the sequel, we will focus on levels of the conflict of interests small enough to still justify some (even partial) communication. This is only in that case that the comparison between organizational forms is really meaningful.
3.3 3.3.1
Comparative Statics Conflict of Interests
In this subsection, we analyze how the policy-maker and the interest group’s expected utility vary with the degree of conflict δ. 19
See Grossman and Helpman (2001, Section 4.1.4) for instance.
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First, note that the expected utilities of the policy-maker and the interest group can be written respectively as ˆ θ(δ) 1 ˆ − θ − δ)2 f (θ)dθ + δ 2 (1 − F (θ(δ))) ˆ V (δ) = − (θ(δ) 2 θ and 1 U (δ) = − 2
ˆ θ(δ)
ˆ − θ)2 f (θ)dθ (θ(δ)
θ
ˆ is defined implicitly from (7). where θ(δ) Proposition 2 : Assume that the optimal mechanism is continuous, then the following properties hold: ˆ increases with • There is less communication as the degree of conflict increases (θ(δ) δ) when F (·) is log-concave. • Both the principal and the agent’s expected utilities decrease with the degree of conflict (V˙ (δ) < 0, U˙ (δ) < 0). The intuition behind these results is straightforward. As the conflict of interests between the policy-maker and the interest group is more pronounced, communication is less attractive to the principal who prefers a rigid policy for a greater set of values of the preference parameter. Of course, enlarging that pooling region hurts the interest group who prefers a flexible policy since the latter fits his own ideal point. Moreover, more conflict hurts also the principal because it increases the wedge between his ideal point and that of the interest group over the range of values of θ where communication indeed takes place. 3.3.2
Informativeness of the Distribution
Let us now keep the bias δ as fixed and consider some comparative statics with respect to the informativeness of the distribution. For simplicity, we will now assume that f (·) is distributed on the whole real line, +∞ +∞ centered at zero so that Ef (θ) = −∞ θf (θ)dθ = 0 and Varf (θ) = −∞ θ2 f (θ)dθ < +∞. Consider now the distribution G(θ, σ) with density g(θ, σ) = σf (σθ) where σ is a positive parameter. It is clear that +∞ Eg (θ) = σθf (σθ)dθ = 0 −∞
13
and Varg (θ) =
+∞
−∞
1 σθ f (σθ)dθ = 2 σ 2
+∞
u2 f (u)du =
−∞
Varf (θ) . σ2
An increase in σ means an increase in the informativeness of the distribution since G(·) second-order stochastically dominates F (·) by sweeping more weight around a common mean Ef (θ) = Eg (θ) = 0. As a benchmark, note that a policy-maker who never communicates with the interest group would choose a rigid policy q p = δ but his expected payoff would always obviously be greater when the distribution of types is more centered around its mean. Let us turn to the case where communication is possible. We define now the principal’s expected welfare V (σ) as a function of the parameter σ: ˆ θ(σ) 1 ˆ − θ − δ)2 g(θ, σ)dθ + δ 2 (1 − G(θ(σ), ˆ V (σ) = − (θ(σ) σ)) 2 −∞ ˆ where the optimal cut-off θ(σ) is defined implicitly by: δ=
1 ˆ G(θ(σ), σ)
ˆ θ(σ)
−∞
1 G(θ, σ)dθ = ˆ σF (σ θ(σ))
ˆ σ θ(σ)
F (θ)dθ.
(8)
−∞
˜ δσ). ˜ δ) the solution to (8) when σ = 1, we observe that θ(σ) ˆ If we denote θ(1, = σ1 θ(1, From (8), we obtain after integrating by parts 1 ˜ δσ) − δσ = θ(1, ˜ δσ)) F (θ(1,
˜ θ(1,δσ)
θf (θ)dθ −∞
and thus ˜ δσ) δσ ∼ θ(1, σ→∞
so that
ˆ θ(σ) −→ δ. σ→∞
This first result shows that, as the distribution on the interest group’s ideal point becomes more informative, the principal chooses a policy which comes closer to what he would do with a rigid policy based on no communication at all. Indeed, such policy is almost first-best when the distribution puts increasingly more weight around the mean. Proposition 3 : The principal’s expected payoff increases with the informativeness of the distribution: V˙ (σ) > 0. The interest group’s expected payoff decreases with the informativeness of the distribution: U˙ (σ) < 0.
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This result is of a particular importance since it gives us some insights on why a policymaker might want to face “larger” groups to facilitate communication. As the distribution of the ideal point θ becomes more centered around the mean, the principal does not really care about the policy out of this particular value and communication has less value for him than in the case of a less informative distribution. Relying on the interest group’s report is of a lesser importance.
4
Coalition
The previous section has shown the nature of the transaction costs associated to asymmetric information between an interest group and the policy-maker. Communication is needed for distributions which are not too informative on the preferences of the interest group but the cost of this communication is that policies depart from the principal’s ideal point. We now focus on the properties of those transaction costs when groups coalesce. Let us now consider two groups with preference parameters θ1 and θ2 respectively. Those parameters are identically and independently distributed in the same uniform distribution on [0, 1]. Those two groups merge as one by credibly sharing their information on preferences and have an equal bargaining weight of one half in the coalition. To motivate information sharing, we will informally invoke the repeated relationship of those interest groups. Whenever the discounted benefit from such sharing information on preferences exceeds the benefit of a short-run deviation consisting in hiding the true value of his preferences to the other group, such credible disclosure is efficient. That interest groups share credibly information in repeated relationships has been documented in the political science literature. Laumann and Knoke (1987) and Heinz and al. (1990) examine information exchanges between group dyads as a key to intergroup coordination. Hula (1999, Chapter 4) reported that interest groups are linked by the career path of their staff members and that a phenomenon akin to the “revolving door” takes place between groups. This phenomenon certainly facilitates information flows between distinct organizations. For the purpose of this paper, we will keep this process of coalition formation as given and draw its implications for communication patterns and expected payoffs. By assuming credible information sharing, we thus characterize an upper bound on the benefits of such a coalitional merger between the groups. Henceforth, whenever a decentralized organization where groups are kept split apart increases the groups’ payoffs, it will be also so if the coalitional behavior of those groups was plagued with some frictions and was not as perfect as assumed.
15
Given our assumptions, the utility of the merged group can be written as:
2 1 1 1 1 (θ2 − θ1 )2 θ1 + θ 2 2 2 UM (q, θ1 , θ2 ) = − − (q − θ1 ) + (q − θ2 ) = − q− . 2 2 2 2 2 8
(9)
Up to some terms which are independent from the policy chosen and thus cannot be screened by the policy-maker, the utility of that coalition only depends on the average ideal point between group members. The game of communication between that merged group and the uninformed policymaker is thus similar to that we solved in Section 3. The only difference comes now from the fact that the initial distribution of types F (·) is replaced by the distribution G(·) of the “average” between two independent draws of the initial distribution. Of course, G(·) is more informative than F (·) since it shifts more weight around the same mean. Although G(·) is not obtained with a direct transformation of F (·) as assumed in 3.3.2., the logic behind Proposition 3 should again apply and that increase in the informativeness of the distribution should definitively improve the principal’s expected payoff. Below, we show this result formally in the case of a uniform distribution on [0, 1]. From Lemma 2, we then know that the optimal mechanism for a single group is necessarily continuous and that the cut-off θˆf between the pooling and separating regions satisfies θˆf = 2δ.20 Consider now the average between two independent draws given by: 4θ if 0 ≤ θ ≤ 12 g(θ) = 4 − 4θ if 1 ≥ θ ≥ 12 .
θ1 +θ2 . 2
It has a density g(·)
We show in the Appendix (Lemma 3) that the optimal mechanism for such a distribution is in fact continuous. It is characterized by a cut-off satisfying (7) and solving: 0
θˆg
(θˆg − θ − δ)g(θ)dθ = 0.
For δ ≤ 16 , the solution is θˆg = 3δ > θˆf meaning that the pooling region increases for small conflicts of interests. For δ ≥ 16 , the cut-off θˆg is determined by21 (θˆg − δ)(12θˆg2 − 24θˆg + 6) + 12θˆg2 − 8θˆg3 − 1 = 0. 20
(10)
Remember that partial communication takes only place if 2δ ≤ 1. The equation (10) has three parametric roots: θ1 (δ) , θ2 (δ) , θ3 (δ) . We have to select one for which 1 θi 2 = 1. Then put θg (δ) = θi (δ) . 21
16
The solution θˆg (δ) of (10) is increasing in the size of the conflict of interests δ and it is also concave in δ. Moreover, θˆf (δ) = 2δ is such that the left-hand side of (10) is negative so that θˆg (δ) > θˆf (δ) = 2δ. We summarize these findings as: Proposition 4 : Assume that the ideal points θi (i = 1, 2) of the interest groups are uniformly and independently distributed on [0, 1]. The policy-maker uses an optimal mechanism which is continuous if he deals with a coalition. It entails less communication than what occurs with each individual interest group taken separately. When dealing with a coalition, the principal benefits from a more precise information. As in Proposition 3, a more precise signal makes it less necessary to rely on communication and a rigid policy over a greater pooling region performs relatively well. Proposition 4 suggests a basic tension that drives the formation of a coalition. On the one hand, by dealing with a coalition, the policy-maker gets a more precise information on preferences and communication becomes less necessary. On the other hand, individual interest groups may be reluctant to form such a coalition because their respective ideal points are never chosen as communication is less necessary and, when it takes place, this is only an aggregate of the preferences of both groups which ends up being used by the principal. To sharpen intuition, let consider the case of n groups merging as one. Each group i has his ideal point θi normally distributed with zero mean and variance σ12 . The normal n θi distribution is interesting since the average ideal point i=1 is itself normally distributed n 1 with the same mean and variance nσ2 . Using (7) in the case of a normal distribution we ˆ get, after integrating by parts, the simple expression of the cut-off θ(n) ˆ ˆ + 1 f (θ(n), n) , δ = θ(n) ˆ nσ 2 F (θ(n), n)
(11)
2 2 2 θ − nσ2 x where F (θ, n) = nσ e dx is the cumulative distribution function for the law 2π −∞ n 1 θ i ˆ N 0, nσ2 of the average i=1 . It is easy to check on (11) that θ(n) converges towards n p δ as n gets large. The rigid policy q = δ is thus almost the optimal mechanism offered by the policy-maker.
An alternative interpretation of our result is worth to be stressed. As n increases, the ideal point of each individual group becomes less relevant for the chosen policy. The individual incentives to lie of each individual interest group get pooled altogether in a coalition and the collective incentives to lie are weak. Indeed, by the Law of Large n θi Numbers, uncertainty on the average type i=1 is washed out in the aggregate. Because n 17
a larger group better communicates a piece of information which is known for sure by the policy-maker, the incentive problem is easier to solve. Having a grand-coalition of interest groups creates some economies of scope in the transaction costs associated to asymmetric information. Those economies of scope take an extreme form as the number of interest groups gets large.22
5
Coalition versus Multiple Interest Groups
In the previous section, we limited ourselves to comparing what happens if the principal deals with a coalition of interest groups and what happens if he deals with each of them separately. In this section, we undertake a less biased comparison by considering the case where the policy-maker deals simultaneously with both groups. To model the groups’ non-cooperative behavior, we use dominant strategy as the relevant implementation concept. Although Bayesian implementation would relax incentive constraints, dominant strategy implementation is amenable to a more direct comparison with the case on a single interest group.23 Another motivation for this implementation concept is that it is less sensitive to the beliefs that interest groups may have on each other. Finally, focusing on dominant strategy is already enough to obtain the result that dealing separately with the different groups may sometimes improve on the outcome achieved with a coalition. With dominant strategy, incentive compatibility for interest group i can be written as 1 1 − (q(θi , θ−i ) − θi )2 ≥ − (q(θˆi , θ−i ) − θi )2 , ∀i = 1, 2, (θi , θˆi , θ−i ) ∈ Θ3 . 2 2 Proceeding with the same differentiable approach as in Section 3,24 we can easily prove that q(·) is monotonically increasing in each of its arguments and thus almost everywhere differentiable in (θ1 , θ2 ). At any point of differentiability, incentive constraints can be written as: ∂q (θi , θ−i )(q(θi , θ−i ) − θi ) = 0, for i = 1, 2, (θi , θ−i ) ∈ Θ2 . ∂θi 22
(12)
In a signaling models, Lohman (1993, 1994) showed that agents may still have an incentives to signal their types even though they are informationally small. The mechanism design approach that we use in this paper shows that dealing with a single large coalition weakens indeed the cost of credible communication. 23 It is well known from Mookherjee and Reichelstein (1992) that dominant strategy entails no loss compared to Bayesian implementation in some models with monetary transfers and quasi-linearity. This equivalence of course fails in our context without monetary transfer. 24 See Laffont and Maskin (1980) for the differentiable approach of dominant strategy mechanisms in settings with monetary transfers.
18
Hence, q(·) is either constant along θi or equal to the ideal point of group i and thus independent on θ−i . This leads to the following characterization of dominant strategy continuous schemes. Lemma 3 : For any symmetric dominant strategy and continuous mechanism q(·), there exist cut-offs θ∗ , θ∗∗ and θ∗∗∗ such that θ∗ ≤ θ∗∗ ≤ θ∗∗∗ and : q(θ) = min{θ∗∗∗ , max{θ1 , θ∗∗ }, max{θ2 , θ∗∗ }, max{θ1 , θ2 , θ∗ }}.
(13)
These continuous mechanisms generalize those found with a single interest groups. Indeed, along each argument θi , the trace of the mechanism q(θi , ·) looks like a onedimensional continuous mechanism. Again, we recover with our differentiable approach the minmax mechanisms of Moulin (1980), now expressed in a two-agent framework. Those dominant strategy mechanisms can be given an interesting interpretation. For the region where the policy is flexible, the ideal point of either of the interest groups is implemented by the policy process. Everything happens as if this group becomes then residual claimant for the policy. Of particular importance in the analysis below are the following mechanisms depending only on the two parameters (θ∗ , θ∗∗ ):25 ∗ if max(θ1 , θ2 ) ≤ θ∗ θ q(θ1 , θ2 ) = max(θ1 , θ2 ) if θ∗ ≤ max(θ1 , θ2 ) ≤ θ∗∗ (14) ∗∗ ∗∗ 26 θ if θ ≤ max (θ1 , θ2 ) . Those schemes implement thus a policy which fits with the highest ideal point of the interest groups when there is screening in the intermediate region and with the smallest ideal point when there is screening in the upper-right corner of the square [0, 1] × [0, 1]. Denoting then by V (θ∗ , θ∗∗ ) the principal’s expected payoff with such a dominant strategy continuous scheme, we have, using symmetry and the uniform distribution:
2 θ∗ θ1 θ + θ 1 2 θ∗ − (15) V (θ∗ , θ∗∗ ) = − − δ dθ2 dθ1 2 0 0
2 θ∗∗ θ1 θ1 + θ 2 θ1 − − δ dθ2 dθ1 − 2 θ∗ 0
2 1 θ1 + θ θ 1 2 − δ dθ2 dθ1 θ∗∗ − − 2 ∗∗ θ 0 25
It will be shown in Appendix that these mechanisms are in fact optimal ones. Note that those mechanisms are such that θ∗∗∗ = θ∗∗ . We show in the Appendix that it is indeed optimal to do so. 26
19
Optimizing with respect to θ∗ and θ∗∗ we obtain θ∗ (δ) = 2δ
1 + 2δ, 2
and θ∗∗ (δ) =
and θ∗∗ (δ) < 1 if and only if δ ≤ 14 . For θ∗ (δ) = 2δ.
1 2
(16)
≥ δ ≥ 14 , the scheme has only one cut-off
We are now ready to state the main result of this section. Proposition 5 : Assume that θ1 and θ2 are independently and uniformly distributed on [0, 1]. Then, the optimal dominant strategy incentive compatible mechanism is continuous. Moreover, there exist two critical levels of the conflict of interests δ ∗ and δ ∗∗ such that δ ∗∗ > δ ∗ and the following ranking holds: • When δ < δ ∗ (resp. δ ≥ δ ∗ ), the policy-maker is better (resp. worse) off with a coalition than with separated groups. • When δ < δ ∗∗ (resp. δ ≥ δ ∗∗ ), interest groups are better (resp. worse) off if they merge than if they remain separated.
The main idea of the proof is to compare the optimal mechanism with a merged group with a dominant strategy mechanism of type (14) dealing with both groups. To do that, the following figure is useful. θ2 1
@ @
6
@ @
C
@
B
@
@• C @ @ @
@
A
B
@ @
θˆg (δ) > 2δ
A
1
- θ 1
Figure 1: Comparing dominant strategy mechanisms and a coalition. The case δ > 16 . 20
2 On Figure 1, we have drawn the line θ1 +θ = θˆg > 2δ which separates the areas A + B 2 where a coalition of both groups receives a rigid policy from the area C where the ideal policy of this coalition is chosen. On C, isopolicy lines are 450 downward slopping lines which thus correspond to none of the ideal points of either group except on the measure zero set of events where both groups have the same preferences.
Consider now the dominant strategy mechanism of type (14) having a cut-off θ∗ = θˆg (δ). On area A, this mechanism is pooling and yields to the principal the same expected payoff as the mechanism given to a coalition since the rigid policy chosen is the same in both cases. On area B, dealing separately with each group allows to make a policy contingent on the preferences of the group who has the highest ideal point. Screening although imperfect and biased in one direction is possible whereas dealing with a coalition still entails a rigid policy. This screening effect captures the benefits of keeping groups split apart. When the interest group’s preferences are sufficiently apart one from the other, the dominant strategy mechanism elicits the preferences of the group with the highest ideal point whereas communicating with a coalition does not allow this. To understand this effect, it is useful to come back on the comparative statics made in Section 3.3.1. We showed there that the optimal policy with a single group requires more communication as the conflict of interests between this group and the policy-maker is less pronounced. With multiple interest groups, the same logic applies. The “virtual conflict of interests” between group 1 and the policy-maker depends now on the realized preferences of group 2 and will typically be equal to δ = δ +
θ1 + θ2 (θ1 − θ2 ) − θ1 = δ − . 2 2
This virtual conflict is a decreasing function of the “distance” between the ideal points of both groups. As the distance between the groups’ ideal points increases, the principal’s preferences are more aligned with the group having the highest ideal point given that δ is large enough to ensure that the virtual conflict of interests remain positive. Everything happens as if an endogenous bias towards that group appears. Instead, the mechanism dealing with a coalition smooths out this discrepancy between preferences and is thus unable to account for this endogenous bias. On area C, dealing with a coalition allows a more efficient communication with the 2 principal since only the statistics θ1 +θ is relevant for decision-making and isopolicy lines 2 cannot depend on this statistics with dominant strategy. However, since the density θ1 +θ2 of 2 decreases over θˆg (δ), 1 , this potential benefit of a coalition is not big enough to offset the cost of an excessive bunching when the interest groups’ preferences are sufficiently far apart. 21
To give further intuition on the benefits of a decentralized mechanism it is useful to think about the case of a sufficiently large conflict of interests. Typically, we will set δ = 12 −ε for ε small enough so that, with a coalition, θˆg (δ) is close to 1 and there is almost no communication with a coalition. The screening area C is then a small triangle having an area of order ε2 . The gain of communicating with a coalition rather than imposing pooling everywhere is thus of order ε2 . Instead, by dealing separately with each interesting group with a dominant strategy mechanism, the principal can screen preferences on areas B + C and obtains a gain of communicating of order ε everywhere. This dimensionality argument underscores the benefits of a dominant strategy mechanism at least for large conflicts of interests. 2 For small values of the conflict of interests, the 450 degree line θ1 +θ = θˆg lies on the 2 very south-west of the [0, 1] × [0, 1] square. There is communication almost everywhere with a coalition and the communicated statistics perfectly fits what is needed by the principal for deciding which policy to implement. Clearly, dealing with a merger dominates in that case. In the extreme case where δ is close to zero and thus there is almost no pooling, using a decentralized mechanism is clearly suboptimal whatever the values of the cut-offs θ∗ and θ∗∗ .Indeed, such mechanisms have isopolicy lines which are never aligned with those of the principal contrary to what happens with a coalition.
Proposition 5 shows that interest groups also gain from being split apart for sufficiently large δ. Indeed, when they remain separated, there is a positive likelihood that the principal’s policy is exactly the ideal points of each of those groups. This benefit is strong enough to dominate the cost of having the preferences of the other group being exactly implemented sometimes. Instead, the more average policy that would be chosen with a coalition never fits the ideal points of either group except, with probability zero, when these ideal points are the same and above θˆg (δ). It is striking that, as δ decreases, both the principals and the interest groups gain from a merger. One should thus expect less (respectively more) interest groups in environment with little (respectively significant) conflicts.
6
Conclusion
In this paper,we have proposed a mechanism design approach useful to analyze the determinants of group formation in an environment where interest groups communicate information which is relevant for optimal policy-making. We have first characterized the optimal mechanism and communication pattern in a framework with only one interest group. More communication takes place as the con-
22
flict between the interest group and the decision-maker is less pronounced and as the distribution of the interest group’s preference parameter becomes less informative. This last point shows that a policy-maker could have strong incentives to facilitate the formation of large coalition of interest groups. Indeed, the “average” ideal points of such coalitions are better known and communication becomes less of an issue. This information effect is however not enough in itself to explain alone the organization of lobbies. A coalition of interest groups, although it transmits a more precise information to the decision-maker, is unable to account for the possible large discrepancies in the preferences of the colluding interest groups. The policy-maker and the interest groups as well may sometimes prefer a more decentralized way of communication information where each interest group is approached separately. Communication is then easier, more specifically when the interest groups preferences are sufficiently far apart and the conflict of interests with the principal significant. A playing field where several competing interest groups independently communicate with the decision-maker is optimal from the decisionmaker’s viewpoint and may also be preferred by the interest groups themselves. Our model based on the precise analysis of the transaction costs due to asymmetric information in lobbying games offers thus a justification for the coexistence of multiple interest groups on the policy arena, especially on very conflicting issues. Of course, those results are partial and mostly obtained by making restrictive assumptions on the distribution of idea points. A more thorough analysis which would generalize our findings to other distributions is called for but is likely to face strong technical difficulties coming from the fact that optimal mechanisms cannot be easily characterized and compared without specifying those distributions. An alternative path would be to relax incentive constraints and the constraints on communication by allowing monetary transfers between the policy-maker and the interest groups. Although, those transfers can be motivated in practice,27 such an extension of the model might lead to qualitatively different insights.28 It would also be worth to generalize our analysis to the cases where interest groups have different biases and to other information structures involving for instance correlation between the various interest groups’ ideal points. Finally, the organization of lobbying which emerges in our model depends of course of the decision-maker’s objective function. Here, we have assumed that this principal is somewhat a social welfare maximizer, ideally choosing an average between the interest groups’ ideal points and that of the general public. That decision-maker could follow 27
They may for instance stand for bribes, campaign contributions, or even official redistributive policies. For instance, in the I.O. literature, Baron and Besanko (1992), Gilbert and Riordan (1995), Dana (1993), and others have found conditions under which dealing with a coalition of suppliers Pareto dominates decentralized mechanisms in a context where agents have quasi-linear utility functions. 28
23
another objective. For instance, in a political contest environment, he could certainly be incline to choose a policy corresponding to the median of the public and the interest groups’ ideal points. The whole pattern of communication and transaction costs would be modified accordingly leading potentially to an interesting feed-back on the incentives of groups to coalesce or not. We hope to investigate some of these issues in future research.
References Armstrong, M., 1994, “Delegation and Discretion”, Discussion Papers in Economics and Econometrics, n0 9421, University of Southampton. Austen-Smith, D., 1990, “Information Transmission in Debate,” American Journal of Political Science, 34: 124-152. Austen-Smith, D., 1993a, “Information and Influence: Lobbying for Agendas and Votes”, American Journal of Political Science, 37: 799-833. Austen-Smith, D., 1993b, “Interested Expects and Policy Advice: Multiple Referrals under Open Rule”, Games and Economic Behavior, 5: 3-44. Austen-Smith, D., 1997, “Interest Groups: Money, Information and Influence” in Perspectives on Public Choice, D. Mueller, ed. Cambridge University Press. Baron, D., 2000, “Legislative Organization with Informational Committees”, American Journal of Political Sciences, 44: 485-505. Baron, D. and D. Besanko, 1992, “Information, Control and Organizational Structures”, Journal of Economics and Management Strategy, 1: 237-275. Baron, D. and A. Meirowitz, 2001, “Relations between Fully-Revealing Equilibria of Multiple-Sender Signaling and Screening Models”, Research Paper, Stanford. Battaglini, M., 2002, “Multiple Referrals and Multidimensional Cheap-Talk”, Econometrica, 70: 1379-1401. Battaglini, M. and R. Benabou, 2003, “Trust, Coordination and the Industrial Organization of Political Activism”, Journal of European Economic Association, 1: 851-889. Crawford, V. and J. Sobel, 1982, “Strategic Information Transmission”, Econometrica, 50: 1431-1451.
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Dana, J., 1993, “The Organization and Scope of Agents: Regulating Multiproduct Industries”, Journal of Economic Theory, 59: 288-310. Gilbert, R. and M. Riordan, 1995, “Regulating Complementary Products: A Comparative Institutional Analysis”, Rand Journal of Economics, 26: 243-256. Glazer, J. and A. Rubinstein, 1998, “Motives and Implementation: On the Design of Mechanisms to Elicit Opinions”, Journal of Economic Theory, 19: 157-173 Green, J. and N. Stokey, 1981, “The Value of Information in the Delegation Problem”, Discussion Paper Harvard University. Gromb, D. and D. Martimort, 2004, “The Organization of Delegated Expertise”, Mimeo IDEI Toulouse. Grossman, G. and E. Helpman, 2001, Special Interest Politics, MIT Press, Cambridge. Hansen, J., 1991, Gaining Access: Congress and the Farm Lobby, 1919-81, University of Chicago Press. Heinz, J., E. Laumann, R. Salisbury and R. Nelson, 1990, “Inner Circles or Hollow Cores? Elite Networks in National Policy Systems”, Journal of Politics, 52: 356-390. Holmstr¨om, B. 1984, “On the Theory of Delegation”, in M. Boyer and R. Kihlstrom; (eds.), Bayesian Models in Economic Theory, Elsevier Science Publishers B.V. Hula, K., 1999, Lobbying Together: Interest Group Coalitions in Legislative Politics, Georgetown University Press. Krishna, V. and J. Morgan, 2000, “A Model of Expertise”, Quarterly Journal of Economics, 116: 747-775. Laffont, J.J., 2000, Incentives and Political Economy, Oxford University Press. Laffont, J.J. and E. Maskin, 1980, “A Differentiable Approach to Dominant Strategy Mechanisms”, Econometrica, 48: 1507-1520. Laumann, E. and D. Knoke, 1987, The Organizational State, Madison: University of Wisconsin Press. Lohmann, S., 1993, “A Signalling Model of Informative and Manipulative Political Action”, American Political Science Review, 87: 319-333.
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Lohmann, S., 1994, “A Welfare Analysis of Political Action”, Chapter 17 in Political Economy: Institutions, Information, Competition and Representation, W. Barnett and al. ed., 437-461, Cambridge University Press. Melumad, N. and T. Shibano, 1991, “Communication in Settings with no Transfers”, Rand Journal of Economics, 22: 173-198. Milbrath, L., 1963, The Washington Lobbyists, Rand McNally. Mookherjee, D. and S. Reichelstein, 1992, “Dominant Strategy Implementation of Bayesian Incentive Compatible Allocation Rules”, Journal of Economic Theory, 56: 378-399. Moulin, H., 1980, “On Strategy-Proofness and Single-Peakedness”, Public Choice, 35: 437-455. Mylovanov, T., 2004, “Veto-Based Delegation”, Working Paper, University of Bonn. Wolinsky, A., 2002, “Eliciting Information from Multiple Experts”, Games and Economic Behavior, 41: 141-160. Wright, J., 1990, “Contributions, Lobbying and Committee Voting in the U.S. House of Representatives”, American Political Sciences Review, 84: 417438.
Appendix • Proof of Lemma 1: From standard revealed preferences arguments, we have indeed ˆ ∈ Θ2 such that θ > θ: ˆ for any (θ, θ) 1 1 ˆ − (q(θ) − θ)2 ≥ − (q(θ) − θ)2 2 2 and 1 ˆ ˆ 2 ≥ − 1 (q(θ) − θ) ˆ 2. − (q(θ) − θ) 2 2 Summing those two constraints yields ˆ ˆ ≥ 0. (θ − θ)(q(θ) − q(θ)) Therefore, q(·) is weakly increasing and thus almost everywhere differentiable. At any point of differentiability, we must have (4) and thus q(θ) is either flat or q(θ) = θ, i.e., corresponds to the most-preferred choice of the lobbying group. 26
If q(·) is discontinuous at θ1 , q(θ1+ ) = q(θ1− ), but type θ1 must be indifferent between choosing the policies which are respectively on the left and on the right of θ1 1 1 1 − (q(θ1− ) − θ1 )2 = − (q(θ1 ) − θ1 )2 = − (q(θ1+ ) − θ1 )2 . 2 2 2
(A1)
Using the right- and left-hand sides, we then get (3). Because q(θ1+ ) = q(θ1− ) at a discontinuity and q(·) is differentiable on both sides, it cannot be that q(θ1 ) is not flat on those sides. Finally, (A1) shows that either q(θ1 ) = q(θ1+ ) or q(θ1 ) = q(θ1− ). • Proof of Lemma 2: Suppose to the contrary that the optimal mechanism q(θ) is discontinuous at x ∈ [θ, θ]. Then using the structure of the incentive compatible mechanisms given in Lemma 1, there exists ∆ ≥ 0 such that on the interval [x − ∆, x + ∆] the contract q(θ) has the form:29 x + ∆ if θ ∈ (x, x + ∆] q(θ) = x − ∆ if θ ∈ [x − ∆, x) We can consider the new mechanism q(θ, ε) which is parameterized by ε ∈ [0, ∆], and which differs from q(θ) only in [x − ∆, x + ∆] where it is defined by : θ if θ ∈ [x + ∆ − ε, x + ∆) x + ∆ − ε if θ ∈ (x, x + ∆ − ε] q(θ, ε) = x − ∆ + ε if θ ∈ [x − ∆ + ε, x) θ if θ ∈ [x − ∆, x − ∆ + ε]. This new mechanism preserves incentive compatibility. The derivative of the utility of the principal on this interval with respect to ε evaluated at ε = 0 under condition (6) is positive, which contradicts the optimality of q(θ). • Proof of Proposition 1: Note that the principal’s expected payoff with a continuous scheme characterized by the cut-offs θˆ1 and θˆ2 can be written as ˆ θˆ2 θ¯ θ1 1 2 2 2 V (θˆ1 , θˆ2 ) = − (θˆ1 − θ − δ) f (θ)dθ + δ f (θ)dθ + (θˆ2 − θ − δ) f (θ)dθ . 2 θˆ2 θˆ1 θ
Optimizing with respect to θˆ2 with the constraint θˆ2 ≥ θˆ1 yields θ¯ ∂V ˆ ˆ (θ1 , θ2 ) = − (θˆ2 − θ − δ)f (θ)dθ < 0 ∂ θˆ2 θˆ2 29
We take the interval [x − ∆, x + ∆] ⊂ [θ, θ]. The cases when x − ∆ < θ or x + ∆ > θ are treated similarly. Also we ignore the value of the contract at x.
27
so that θˆ2 = θˆ1 and we can rewrite V (·) as a function of θˆ1 only. Abusing slightly notations, we have: ˆ θ1 1 V (θˆ1 ) = − (θˆ1 − θ − δ)2 f (θ)dθ + δ 2 (1 − F (θˆ1 )) . 2 θ The first-order condition
d V dθˆ1
(θˆ1 )
θˆ1 =θˆ
= 0 yields the expression of θˆ given by (7).
Moreover, we have d2 V (θˆ1 ) = δf (θˆ1 ) − F (θˆ1 ) dθˆ1 and thus the second-order condition
d2 ˆ − F (θ) ˆ ≤0 V (θˆ1 ) = δf (θ) ˆ ˆ dθˆ1 θ1 =θ
is satisfied if and only if ˆ θˆ f (θ) F (θ)dθ ≤ 1. ˆ θ F 2 (θ) θ However, consider the function ϕ(θ) = F 1(θ) θ F (x)dx. We have θ d f (θ) f (θ) F (x)dx = 1 − (ϕ(θ)) = 1 − 2 ϕ(θ). dθ F (θ) θ F (θ) F (θ) d But when F (·) is log-concave, dθ f (θ) > 0 and 1 ϕ(θ) = F (θ) so that
d (ϕ(θ)) dθ
θ
θ
(A2)
F (θ) F (x) f (x)dx ≤ , f (x) f (θ)
≥ 0. Finally, (A2) holds.
• Proof of Corollary 1: Since ϕ(θ) is increasing, θˆ ≤ θ¯ if and only if δ ≤
θ¯ θ
F (θ)dθ.
ˆ • Proof of Proposition 2: Note first that δ = ϕ(θ(δ)). Because ϕ(·) is increasing when ˆ F (·) is log-concave, θ(δ) increases with δ. Using the Envelope Theorem: θ(δ) ˆ ˆ − θ − δ)f (θ)dθ − δ(1 − F (θ(δ))). ˆ ˙ V (δ) = (θ(δ) θ
ˆ Because (7) holds, the first term on the r.h.s. is zero and V˙ (δ) = −δ(1 − F (θ(δ))) < 0. Also, we have: ˆ θ(δ) d ˆ ˆ = −δF (θ(δ)) ˆ ˆ − θ)f (θ)dθ d θ(δ) U˙ (δ) = − θ(δ) < 0. (θ(δ) dδ dδ θ 28
• Proof of Proposition 3: Using the Envelope Theorem, we get: ∂G ˆ 2V˙ (σ) = δ 2 (θ(σ), σ) − ∂σ
ˆ θ(σ)
−∞
ˆ − θ − δ)2 (θ(σ)
∂g (θ, σ)dθ. ∂σ
Integrating by parts, V˙ (σ) = − = −
ˆ θ(σ)
ˆ − θ − δ) (θ(σ)
−∞ ˆ ∂G θ(σ) (θ, σ) ∂σ −∞
∂G (θ, σ)dθ ∂σ
g(θ, σ)
ˆ − θ − δ) dθ. g(θ, σ)(θ(σ)
Integrating by parts again, θ
θ(σ) ˆ ∂G (θ, σ) ∂σ ˆ − θ˜ − δ)g(θ, ˜ σ)dθ˜ ∂ V˙ (σ) = (θ(σ) dθ. ∂θ g(θ, σ) −∞ −∞ Note that
∂G (θ,σ) ∂σ
g(θ,σ)
=
θ σ
θ
−∞
ˆ and that, by definition of θ(σ), ˆ − θ˜ − δ)g(θ, ˜ σ)dθ˜ ≥ 0 for θ ≤ θ(σ). ˆ (θ(σ)
Therefore, V˙ (σ) ≥ 0. For the interest group, we have: ˆ θ(σ) 1 ˆ − θ)2 g(θ, σ)dθ . U (σ) = − (θ(σ) 2 −∞ Thus, ˆ θ(σ) ˆ θ(σ) 1 ˆ + ˆ − θ)g(θ, σ)dθ d θ(σ) ˆ − θ)2 ∂ g(θ, σ)dθ U˙ (σ) = − (θ(σ) (θ(σ) 2 2 dσ ∂σ −∞ −∞ and 1 U˙ (σ) = − 2
ˆ θ(σ)
−∞
Finally, since
ˆ − θ)2 f (θ, σ)dθ + 2 (θ(σ)
d ˆ θ(σ) dσ
ˆ θ(σ)
−∞
ˆ − θ)f (θ, σ)dθ (θ(σ)
> 0 we get that U˙ (σ) < 0.
• Proof of Proposition 4: Now, we prove the following Lemma: 29
d ˆ 1 + σ θ(σ) . dσ
Lemma 4 : The optimal mechanism for the distribution G(·) corresponding to the coalition is necessarily continuous. Proof: First, consider the following property of the optimal mechanism q(θ) which is a direct consequence of Lemma 2. If at a point x for ∆ such that both x + ∆ and x − ∆ belong to Θ,30 the following is true: for all t ∈ (0, ∆) g(x − t) − δg (x − t) + g(x + t) − δg (x + t) > 0
(A3)
then there is no discontinuity of the optimal mechanism q (θ) on [x − ∆, x + ∆] of the type x − ∆ if x − ∆ ≤ θ < x, (A4) q(θ) = x + ∆ if x < θ ≤ x + ∆. Indeed, consider the function ϕ(t), defined on [0, ∆] by ϕ (t) = δ (2G (x) − G (x − t) − G (x + t)) − x x+t − G (y) dy + G (y) dy. x−t
x
Then, if condition (A3) is satisfied we have ϕ (t) > 0. Since ϕ(0) = 0, therefore ϕ(∆) > 0 and acting as in the proof of Lemma 2 we get that the optimal mechanism cannot have any discontinuity of type (A4). Suppose, we have an optimal mechanism q(θ) which has on the interval [0, x + ∆] a discontinuity at x of the form x − ∆ if 0 ≤ θ < x, (A5) q(θ) = x + ∆ if x < θ ≤ x + ∆. Six cases should be considered depending on the locations of x − ∆, x and x + ∆.31 1.
1 2
< x + ∆ < 1 and x − ∆ > 12 . Then, since the whole interval [x − ∆, x + ∆] lies in the area where the density g(θ) is decreasing, it is clear that the condition (A3) is satisfied.32
30
If x ± ∆ are out of Θ we have to use the footnote in Lemma 2. It is easy to see that the first discontinuity of q(θ) from the left should have the form (A5). Particularly, the optimal contract cannot start with an interval of separation or have such an interval before the point of discontinuity. Also, because we consider bounded support of the distribution, the possibility of an optimal contract with an infinite number of discontinuities is ruled out. 32 The problem may arise because of discontinuity of g (θ) . However, this does not affect the proof. 31
30
2.
1 2
< x + ∆ < 1 and 12 > x − ∆ > 0.Taking into account the expression of g(·), (A3) can be rewritten as: x − ∆ − δ + 1 − x − ∆ + δ = 1 − 2∆ > 0, which holds for ∆ < 12 .
3.
1 2
< x + ∆ < 1 and 0 > x − ∆. Then (A3) becomes
g(x + ∆) − δg (x + ∆) > 0.
(A6)
This amounts to 1 − x − ∆ + δ > 0 which is always true. 4. x + ∆
δ. If x < δ, then it is easy to see that q(θ) is dominated by q(θ) if x + ∆ < θ, q˜(θ) = x + ∆ if θ < x + ∆, 5. x + ∆
0
which is true if x + ∆ > δ. If x + ∆ < δ, then as in the previous case x < δ and the contract q(θ) is dominated. 6. x + ∆ > 1. In this case, it is easy to see that: if x > 12 , then the contract derived by replacing q(θ) on [x − ∆, x + ∆] by q(θ) = θ dominates q(θ). If x < 12 , then since x + ∆ > 1, we have x − ∆ < 0 and the mechanism is dominated by the rigid mechanism q (θ) = x + ∆. Putting everything together, we can conclude that the optimal mechanism for the coalition with uniform priors is continuous. • Proof of Lemma 3: Consider an arbitrary continuous dominant strategy mechanism q(θ1 , θ2 ). By Lemma 1 there exist cut-offs θ∗ and θ∗∗ such that the uni-dimensional scheme q(θ) = q(θ, θ) has the form: ∗∗ if θ∗∗ ≤ θ ≤ θ θ q (θ) = θ if θ∗ ≤ θ ≤ θ∗∗ ∗ θ if θ ≤ θ ≤ θ∗ 31
Consider the most interesting case θ < θ∗ < θ.33 Step 1. Take ε > 0 such that θ + ε ≤ θ∗ qε (θ) = q(θ, θ + ε). By Lemma 1 there exist θε∗ ∗∗ θε if qε (θ) = θ if ∗ θε if
and consider the uni-dimensional scheme and θε∗∗ such that: θε∗∗ ≤ θ ≤ θ θε∗ ≤ θ ≤ θε∗∗ θ ≤ θ ≤ θε∗
Suppose first that θε∗ < θ∗ . Then take µ > 0 such that θε∗ +µ < θ∗ and q(θε∗ +µ, θ+ε) = θε∗ + µ. Then q (θε∗ + µ, θ + ε) = θε∗ + µ < θ∗ = q (θε∗ + µ, θ) which contradicts the monotonicity of the uni-dimensional scheme q (θ) = q(θε∗ + µ, θ). If θε∗ > θ∗ then q (θ∗ , θ + ε) = θε∗ > θ∗ = q (θ∗ , θ) Therefore, since the uni-dimensional contract q (θ) = q(θ∗ , θ) is continuous and it = θ < θ + ε, on the increases from θ to θ + ε there exists at least one θ such that q(θ∗ , θ) other hand θ > θ∗ . This yields θ + ε > θ∗ . But by construction, θ + ε ≤ θ∗ . Contradiction, therefore θε∗ = θ∗ . Repeatedly applying this argument we conclude that ∀(θ1 , θ2 ) such that (θ1 , θ2 ) ∈ Ω∗ = {(θ1 , θ2 ) : max{θ1 , θ2 } ≤ θ∗ } we have q(θ1 , θ2 ) = θ∗ . Geometrically Ω∗ is the square in the down-left corner of Θ × Θ where the continuous dominant strategy contract is constant and equal to θ∗ . ∗∗ ≥ θ−i , θi ≥ θ−i }. For the set Step 2. Consider the sets Ω∗∗ i = {(θ1 , θ2 ) : θi ≥ θ ∗∗ Ω1 , similar arguments as in Step 1 show that ∀ε such that θ + ε ≤ θ∗∗ and ∀θ ≥ θ∗∗ , q(θ, θ + ε) = θ∗∗ . For the set Ω∗∗ (θ) = q(θ, θ) 2 we consider the uni-dimensional contract q and using the symmetric argument of Step 1, we can see that necessarily for the cut-offs θ ∗ and θ ∗∗ of the contract q (θ) : θ ∗ = θ∗ , and θ ∗∗ = θ∗∗ . Therefore for all (θ1 , θ2 ) ∈ Ω∗∗ = ∗∗ ∗∗ Ω∗∗ 1 ∪ Ω2 we have q(θ1 , θ2 ) = θ .
Step 3. Consider the set Ωi = {(θ1 , θ2 ) : θ∗ ≤ θi ≤ θ∗∗ , θi ≥ θ−i }. The sets Ωi are ∗ ∗∗ located between the sets Ω∗ and Ω∗∗ i . By construction q(θ) = q(θ, θ) = θ for θ ≤ θ ≤ θ . For the set Ω1 we take θ ∈ [θ∗ , θ∗∗ ]. We show that if ε > 0 is such that θ + ε ≤ θ then q(θ , θ + ε) = θ . Again, if q(θ , θ + ε) < θ we have a contradiction with the monotonicity. If q(θ , θ + ε) > θ = q(θ , θ) then again as in Step 1, there exists at least one θ such that 33
The other cases are treated similarly.
32
= θ < θ + ε, on the other hand θ > θ > θ + ε, contradiction. Using the symmetric q(θ , θ) argument for the set Ω2 , we conclude that ∀(θ1 , θ2 ) ∈ Ω1 ∪ Ω2 , q(θ1 , θ2 ) = max{θ1 , θ2 }. Proceeding as in Steps 1,2,3 with the set Ω = {(θ1 , θ2 ) : min{θ1 , θ2 } ≥ θ∗∗ } which is located in the upper-right corner of Θ × Θ, we obtain a contract of the form that is stated in the Lemma 3.34
• Proof of Proposition 5: Let us provide a Lemma which shows the conditions under which continuity of that dominant strategy mechanism arises. Lemma 5 : For distributions satisfying condition (6), the optimal dominant strategy incentive compatible mechanism q(·) is necessarily continuous. Proof: Suppose that the optimal mechanism q(θ1 , θ2 ) has a discontinuity at point a = (a1 , a2 ). Assume, without loss of generality, that a1 ≤ a2 . Then it has to be that at least for one of the uni-dimensional mechanisms determined by q(θ1 , θ2 ) and θ1 = a1 or θ2 = a2 there is a discontinuity at either point a2 or a1 correspondingly. Now suppose, without loss of generality, that for the uni-dimensional mechanism q1 (θ1 ) = q(θ1 , a2 ), there is a discontinuity at point a1 . Denote q(a− 1 , a2 ) =
+ + + − lim q(a1 , a2 ) by a− 1 , and q(a1 , a2 ) by a2 (note that a2 > a1 ).
θ1 →a1 −0
Then from Lemma 1 the uni-dimensional mechanism q1 (θ1 ) = q(θ1 , a2 ) on the interval + [a− 1 , a2 ] has a form if a1 < θ1 ≤ a+ a+ 1 1 + q1 (a1 ) ∈ a− , a q1 (θ1 ) = 1 1 a− if a− 1 1 ≤ θ1 < a1 + − − Particularly, q1 (a+ 1 ) = a1 and q1 (a1 ) = a1 . − − Consider two uni-dimensional contracts: q2+ (θ2 ) = q(a+ 1 , θ2 ) and q2 (θ2 ) = q(a1 , θ2 ). Since by dominant strategy incentive compatibility they are both incentive compatible in uni-dimensional case we can consider the following cases in the neighborhood of a2 :
Case 1: q2+ (θ2 ) is locally strictly increasing at a2 , i.e., there exists ε > 0 such that − q2+ (θ2 ) = θ2 for all θ2 ∈ [a2 − ε, a2 + ε]. Then a2 = q2+ (a2 ) = a+ 1 . But then, q2 (θ2 ) also has 35 to be locally strictly increasing at a2 and therefore a2 = q2− (a2 ) = a− Contradiction. 1. 34
We have to take the uni-dimensional contract q(θ) = q(θ, θ) and then Lemma 1 and considerations as in Steps 1 and 2 yield the required form of the optimal contract on the set Ω. 35 − q2 (θ2 ) cannot be locally constant at a2 since it will violate the monotonicity of the uni-dimensional contract q(θ2 ) = lim q(θ1 , θ2 ) at point a2 . By the same reason q2− (θ2 ) may not have a jump at a2 . θ1 →a1 +0
33
− + + + Case 2: q2+ (θ2 ) has a jump at a2 . Then it must be that q2+ (a− 2 ) = a1 and q2 (a2 ) = a1 . + + − + + Therefore by Lemma 1: 2a1 = a− 1 + a1 = q2 (a2 ) + q2 (a2 ) = 2a2 . So it must be that a1 = a 2 .
Case 3: q2+ (θ2 ) and q2− (θ2 ) are constant in the neighborhood of a2 . From these cases we can determine locally the structure of the contract q(θ1 , θ2 ) in the connected area. + If q2+ (θ2 ) has no jump, then in the area S = {(θ1 , θ2 ) : a− 1 ≤ θ1 ≤ a1 } the contract q(θ1 , θ2 ) has a form:36 + a1 if a1 < θ1 ≤ a+ 1 q (θ1 , θ2 ) = a− if a− 1 1 ≤ θ1 < a1 − + + Since q(a− 1 , θ2 ) = a1 and q(a1 , θ2 ) = a1 we can replace q(θ1 , θ2 ), without violating dominant strategy incentive compatibility, by the new contract q (θ1 , θ2 ) which is identical to q(θ1 , θ2 ), except on the area S where it is equal to θ1 . The argument of the Lemma 2 shows that the new contract strictly dominates q(θ1 , θ2 ). Contradiction with the optimality of q(θ1 , θ2 ).
If q2+ (θ2 ) has a jump, it may happen only for θ2 = a1 . Then in the area S = {(θ1 , θ2 ) : + a− 1 ≤ min{θ1 , θ2 } ≤ a1 } the contract has a form: + a1 if a1 < min {θ1 , θ2 } ≤ a+ 1 q (θ1 , θ2 ) = a− if a− 1 1 ≤ min {θ1 , θ2 } < a1 Replacing q(θ1 , θ2 ) by min{θ1 , θ2 } on S leads to the new contract which strictly dominates q(θ1 , θ2 ). Remark 1: First we prove that the optimal mechanism has indeed the form (14). Denote by V (θ∗ , θ∗∗ , θ∗∗∗ ) the payoff of the principal for dominant strategy incentive compatible 36
We ignore the value of the contract q(θ1 , θ2 ) on the line θ1 = a1 which has a measure zero.
34
scheme for uniform distribution:
2 θ + θ 1 2 θ∗ − V (θ∗ , θ∗∗ , θ∗∗∗ ) = − − δ dθ2 dθ1 2 0 0
2 θ∗∗ θ1 θ1 + θ2 θ1 − − − δ dθ2 dθ1 2 θ∗ 0
2 1 θ∗∗ + θ θ 1 2 − δ dθ2 dθ1 θ∗∗ − − 2 ∗∗ θ 0
2 θ∗∗∗ 1 θ1 + θ 2 θ1 − − − δ dθ2 dθ1 2 θ1 θ∗∗
2 1 θ1 + θ θ 1 2 θ∗∗∗ − − − δ dθ2 dθ1 2 θ∗∗∗ θ∗∗∗
θ∗
θ1
Then it is easy to see, that the derivative of V with respect to θ∗∗∗ is always negative: ∂V 1 (1 − θ∗∗∗ )2 (1 + 2δ − θ∗∗∗ ) < 0 = − ∗∗∗ ∂θ 2 Therefore θ∗∗∗ = θ∗∗ . That means that indeed the schemes with two cut-offs are optimal.37 ! Optimizing (15) with respect to θ∗ and θ∗∗ , we can conclude that for δ ∈ 14 , 12 the optimal dominant strategy mechanism has a form with one cut-off value θ∗ = 2δ and ! for δ ∈ 0, 14 there are two cut-off values θ∗ = 2δ and θ∗∗ = 12 + 2δ. From Proposition 4 we know the structure of the optimal contract with a coalition. In order to prove Proposition 5 thus we have to consider three cases depending on the value of the conflict ! ! ! δ : δ ∈ 0, 16 , δ ∈ 16 , 14 , δ ∈ 14 , 12 . ! Consider the case δ ∈ 14 , 12 .38 For a coalition between groups, the optimal mechanism yields to the decision-maker:
2 + θ θ 1 2 θˆg − − δ dθ2 dθ1 2 0 0
2 1 2θˆg −θ1 θ + θ 1 2 θˆg − − δ dθ2 dθ1 2 θˆ 0 1 θ1 − δ 2 dθ2 dθ1 ,
1 VM (θˆg ) = − 2 −
θˆg
θˆg
θˆg
2θˆg −θ1
37
It is routine to show that the dominant incentive compatible schemes which are dual to schemes 14 are not optimal. ! ! 38 The cases δ ∈ 16 , 14 and δ ∈ 0, 16 are treated similarly.
35
where θˆg is the cut-off in the case of a coalition determined by (10). Let us consider now the dominant strategy incentive compatible mechanism such that θ∗ = θˆg . It yields to the decision-maker an expected payoff
2 θ1 + θ 2 ˆ θg − − δ dθ2 dθ1 2 0 0
2 1 θ1 θ1 + θ 2 θ1 − − δ dθ2 dθ1 . 2 θˆg 0
1 VD (θˆg ) = − 2 −
θˆg
θˆg
Considering the difference VD (θˆg ) − VM (θˆg ), we have: 1 2 2 ˆ ˆ ˆ ˆ ˆ VD (θg ) − VM (θg ) = (1 − θg ) 2θg (1 + 12δ) − 1 − 7θg 24
(A7)
Using the expression (10) for δ, the sign of (A7) is the same as the sign of the expression: 2θˆg4 − 16θˆg3 + 7θˆg2 + 2θˆg − 1 2θˆ2 − 4θˆg + 1 g
which is positive for θˆg ∈
! 39 .
1 ,1 2
Let us turn to the interest groups. We consider the optimal mechanisms in both cases (coalition and multiple interest groups) for arbitrary δ ∈ [ 14 , 12 ]. The expected utility of such a group in a coalition is: UM
1
ˆ 2 1 2θg (δ)−1 ˆ ˆ θg (δ) = − θg − θ1 dθ2 dθ1 − 2 0 0 ˆ
2 1 2 2θg (δ)−θ1 1 1 θ + θ 1 2 − θˆg (δ) − θ1 dθ2 + − θ1 dθ2 dθ1 = 2 2θˆg (δ)−1 2 2θˆg (δ)−θ1 0 1ˆ 2 3 ˆ ˆ ˆ θg (δ) −1 + 3θg (δ) − 4θg (δ) + θg (δ) = 6
where θˆg (δ) again is determined by (10). In the case of the decentralized mechanism, we have instead:
θˆf (δ) θˆf (δ) 1 θ2 2 1 1 2 UD θˆf (δ) = − θˆf (δ) − θ1 dθ2 dθ1 − (θ2 − θ1 ) dθ1 dθ2 = 2 0 2 θˆf (δ) 0 0 1 4 ˆ 1 + 3θf (δ) = − 24 here θˆf (δ) = 2δ. 39
Computations for δ ≤
1 4
show that VD > VM for δ > δ ∗ ≈ 0.11 and for δ < δ ∗ , VM > VD .
36
It can be shown that the difference between the expected payoffs of an interest group: UD (θˆf (δ)) − UM (θˆg (δ)) is positive for δ ∈ [ 14 , 12 ]. Therefore, we have UM (θˆg (δ)) < UD (θˆf (δ)).40
40
We can show that for δ > δ ∗∗ ∈ [ 16 , 14 ], UD > UM and for δ < δ ∗∗ , we have the reverse.
37
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