A Brief Critical Overview of Common Agency Models and Politics

Following the seminal work of Reference Bernheim and WhinstonBernheim and Whinston (1986a, Reference Bernheim and Whinston1986b), those situations have been modeled as common agency games. In a nutshell, a common agency model works as follows. Several principals (interest groups, public bodies) noncooperatively design contribution schedules to influence a single policy-maker. This common agent chooses which offers to accept and which decision should be taken. The decision may be a vector of regulated prices on domestic and international markets or whether a particular reform should be implemented or not. The schedule offered by each principal stipulates how much that principal is ready to pay for a given decision. Reference Bernheim and WhinstonBernheim and Whinston (1986b) demonstrated that the set of equilibria of this three-stage game may be quite large. To refine within this equilibrium set, they observed that each principal has always in his best-response correspondence a truthful contribution schedule. A truthful contribution schedule perfectly reflects the principal’s preferences over alternative actions.Footnote ¹ Truthful contributions make de facto the policy decision-maker residual claimant for the decision made. As it is so for all principals, the agent fully internalizes the whole welfare of principals at any truthful equilibrium. The public decision maximizes the aggregate payoff of the grand coalition formed by the contributing principals and the common agent and thus ends up being always efficient. Although different truthful equilibria may entail different distributions of the overall surplus among the principals and their common agent, the policy chosen is welfare maximizing and thus essentially unique under weak conditions of strict concavity.

This model of the political process yields a striking and, in a sense, quite surprising result. According to this so-called Pluralistic View of Politics, the political process is frictionless. This Chicago-like approach of politics thus suggests that free entry in the political arena together with some form of perfect competition among interest groups would definitively ensure social-welfare maximization. And this even if policy decision-makers are not subject to any formal constraint on behavior and conduct.

This conclusion is clearly at odds with reality for at least three reasons. First, because it does not generate transaction costs of any sort, the standard common agency approach cannot explain why some groups form and are active players in the political arena, whereas others remain inactive.

Second, and for the same reason, the paradigm is silent on the boundaries of interest groups. Indeed, the common agency approach leaves no room for active groups to build coalition or even merge to leverage their influence (unless such merger allows to better extract the agent’s rent). Indeed, in any case, the decision implemented is always efficient and thus remains unchanged. By the same token, the paradigm fails to explain why individual players should ever join groups instead of acting as separate principals on their own; a point forcefully emphasized by Reference MallardMallard (2014) and Reference MartimortMartimort (2018). In other words, the paradigm has been developed without any consideration for the free-riding problem that Reference OlsonOlson (1965) stressed as a key determinant of collective action problems. As a model of political economy, this weakness is actually a serious blow.

Third, the common agency model also treats separately redistributive and allocative issues. Again, because the decision chosen by the common agent is always efficient, the distribution of welfare among interest groups has no impact whatsoever on this decision and vice versa. As a model of political economy, this feature is at best unpleasant. If anything, political economy should precisely be concerned by the link between the policies chosen on the one hand and the payoff distribution they induce among stakeholders on the other hand. Indeed, the common agency model under complete information beautifully derives the feasible redistributions of the surplus among the interest groups which are assumed to actively influence the policy-maker, but, again, this redistribution has no impact on the decision which ends up being taken.

Moral Hazard as a Source of Contractual Frictions

In this Element, we take stock of the lessons of the existing common agency literature but modify this basic framework in a crucial direction. We explicitly introduce an agency problem between political principals and the policy-maker under the form of moral hazard. Moral hazard is a quite natural assumption in the framework of political delegation as it has been argued forcefully by several political scientists.Footnote ²

Our running example throughout the Element has a decision-maker exerting a nonobservable effort that affects the probability that a reform be passed. Interest groups can favor or oppose the reform. Interest groups aim at influencing the decision-maker by means of contribution schedules that stipulate a payment to the decision-maker for each possible outcome: typically, whether the reform passes or fails. Although the principals’ contributions can be made contingent on outcomes, they cannot depend explicitly on the action taken by the decision-maker. For example, contribution schedules cannot depend on whether the policy-maker in charge has worked hard or not to convince other politicians or the rest of citizens of the benefits of his decision.

This agency problem potentially may generate contractual frictions. Those frictions shape the interest groups’ incentives to become active or not and, if active, to engage in coalition building with other groups if it helps leveraging collective influence. In other words, contractual frictions are key not only to explain the landscape of active interest groups but also to provide the missing link between the redistributive and the allocative sides of the political process that the previous complete information literature unfortunately failed to recognize.

Moral Hazard Might Not Always Suffice to Generate Contractual Frictions

Our analysis of moral hazard in common agency lobbying games starts by considering the most general set of feasible contributions given the informational constraint faced by interest groups. In this most unrestricted contracting scenario, each of those principals is able to offer payments contingent for all possible outcomes of the political process. For instance, payments to the decision-maker only depend on whether a reform passes or fails. In this unrestricted contracting environment, making a risk-neutral decision-maker residual claimant for the interest group’s benefits of the chosen policy is always a best response for this principal whatever contributions other competing groups are actually offering. This well-known result from the principal-agent literature carries thus over in a noncooperative context where groups compete for influence. At equilibrium, contribution schedules align each group’s preferences over alternatives with those of the common agent. Contributions are again truthful in this moral hazard scenario.

However, the meaning of truthfulness slightly differs from what is commonly understood in the complete information scenario. In contrast with Reference Bernheim and WhinstonBernheim and Whinston (1986b)’s original work, contributions now perfectly reflect the principals’ relative preferences among outcomes and not among the actions that the agent could entertain. In a moral hazard environment with a standard full support assumption, all outcomes are feasible with some probability whatever the common agent’s action. To illustrate, a reform can pass or fail with some probability and although this probability may change with the policy-maker’s effort, there always remains uncertainty on final outcomes. From a technical viewpoint, the issue of extending the contribution schedules for outcomes which are not reached on the equilibrium path thus disappears in our framework. It is not necessary to impose any equilibrium refinement to justify truthfulness. Equilibrium schedules must satisfy this property. Compared with the case of complete information, the introduction of moral hazard significantly increases the predictive power of the common agency model.

Yet, even in a moral hazard environment, the main lesson of the common agency literature pertains. The risk-neutral decision-maker again ends up being made residual claimant by all principals for all consequences of his or her action. Accordingly, all interest groups are active at equilibrium, and the agent chooses an action which maximizes the payoff of the grand coalition he or she forms with those active principals. The political process remains frictionless; thereby confirming the main takeaway of the Pluralistic Approach of Politics. Moral hazard alone is not enough to challenge this efficiency view of competition in politics.

Nonnegative Contributions and Frictions

To get some meaningful contractual frictions, we have to consider a more restricted contracting environment where interest groups can no longer make the agent residual claimant for their own objectives. In fact, the residual claimancy argument relies on the possibility for the policy-maker to compensate the interest group in case it is hurt by the decision taken: an unpalatable conclusion. The policy-maker could indeed always have the option to renege on an earlier agreement with an interest group and he will certainly do so whenever asked to compensate the group if some political outcome that hurts this group realizes.

A contrario, consider thus the scenario where interest groups only pay for the outcome that pleases them the most. Imposing this additional condition that transfers should be nonnegative generates meaningful agency frictions that render the picture of the game of influence much more realistic. To study in fine details these frictions, we distinguish two cases. In the first one, all principals have congruent preferences and are pleased when the reform succeeds. To illustrate, two interest groups are congruent if they want to push toward the adoption of the reform. In the second scenario, the groups have conflicting preferences and compete head-to-head for the policy-maker’s services.

With congruent preferences, all interest groups prefer that the policy-maker exerts more effort toward implementing the reform. This effort becomes thus a public good and interest groups contribute to its provision by independently rewarding the common agent. Now, each principal enjoys only form a fraction of the aggregate benefit of a reform but, under a noncooperative behavior, has to pay for the full agency cost (as it resorts from the analysis) of implementing it. Because principals are not cooperating in designing contributions, free riding in the provision of incentives follows. This leads to a familiar under-provision of the agent’s effort and the likelihood of a reform decreases accordingly.

This negative externality provides the key ingredient to explain entry into the political process. When an interest group’s valuation for the reform is less than the corresponding agency cost it bears from its relationship with the decision-maker, its equilibrium contribution is null. Those principals do not contribute and remain out of the political process. The policy chosen by the common agent in equilibrium reflects thus only the preferences of the principals who are the most willing to intervene. This effect is crucial. It gives us a link between the allocative efficiency of the policy choice, the endogenous structure of active groups, and the distribution of surplus across those groups.

Importantly, heterogeneity is key to explain that some interest groups do not act. If all congruent interest groups are alike, they all intervene even though the political outcome still reflects the existing free riding among them and the likelihood of a reform is inefficiently low. This shows that redistribution of the benefits of the reform among interest groups has a nonneutral impact on the policy chosen.

The extent of the free-riding problem and its determinants, namely, the size of a group and the heterogeneity among its members, is now a fundamental driver of the game of influence. While adding new members to a group, by increasing the aggregate valuation, strengthens this group’s influence, size in itself decreases a group’s ability to exert influence. That is, for a given aggregate valuation, smaller groups will be more efficient than larger ones. In addition, groups with more homogeneous members tend to be more effective than groups whose members are more diversely affected by the policy.

Those frictions in the interaction among principals also modify the scope for coalition building among groups. Under unrestricted contracting, all the collective gains are exhausted via groups’ competition. Coalition building can only be valuable when it helps in reducing the rent the agent can extract from groups’ competition, while it has no impact on the policy decision. By contrast, free riding provides groups with an incentive to merge in order to cooperatively design their contributions. Such a joint design of contribution not only affects the distribution of the collective gains, but also the eventual decision and therefore the collective gain achieved from the policy process.

In the case of conflicting principals, most of the aforementioned features carry over although details change. In particular, conflicting principals may face different agency costs and evaluate in quite different ways the cost of entering the political arena. The result is that head-to-head competition between conflicting groups is more likely to induce a very asymmetric political landscape; the weaker group remaining outside the political arena.

Organization of the Element

Section 2 reviews the relevant literature. Section 3 presents our model of a political process leading to adopting or not a reform. This problem is viewed as a delegated common agency game under moral hazard. In Section 4, we derive the properties of equilibria when principals are unrestricted in the contribution schedules they may use. In Section 5, we introduce the nonnegativity constraints on payments. We analyze two important benchmarks for the rest of the analysis: the cooperative outcome and the scenario of intrinsic common agency game. We deduce from there the moral hazard frictions that affect the political process. Section 6 develops the case of congruent principals, whereas Section 7 deals with the scenario of conflicting interests. Section 8 analyzes the incentives of interest groups to merge to leverage their common influence. Section 9 briefly concludes. Proofs are relegated to Appendix A. In addition, Appendix B develops etc. an alternative scenario where the source of the agency frictions is the agent’s risk aversion.

On the Political Economy Front

Over the past few decades, applications of the common agency model to various political economy contexts have flourished as exemplified by the work of Reference AidtAidt (1998), Reference Dixit, Grossman and HelpmanDixit, Grossman, and Helpman (1997), Reference Rama and TabelliniRama and Tabellini (1998), Reference Besley and CoateBesley and Coate (2001), Reference Grossman and HelpmanGrossman and Helpman (1994, Reference Grossman and Helpman2001), Reference Helpman and PerssonHelpman and Persson (2001), and Reference YuYu (2005), among many others. Reference MallardMallard (2014) and Reference MartimortMartimort (2018) have written extensive surveys of this literature but have also stressed some of the limits of this approach. As the common agency model became extensively used as a description of political influence, getting a more realistic picture of the political process has been high on the research agenda.

In an attempt to reconcile redistributive and efficiency concerns, Reference Dixit, Grossman and HelpmanDixit, Grossman, and Helpman (1997) have departed from the quasi-linear world inherited from Reference Bernheim and WhinstonBernheim and Whinston (1986b) by introducing income effects. Unfortunately, the efficiency property of the truthful equilibria of the corresponding common agency games is preserved. In other words, redistributive concerns alone are unable to generate any frictions in the political process.

A key aspect of realism is certainly asymmetric information. Assuming asymmetric information on the agent’s side and ex post contracting, Reference Laussel and Le BretonLaussel and Le Breton (1998b), Reference Le Breton and SalaniéLe Breton and Salanié (2003), Reference Martimort and SemenovMartimort (2007b, Reference Martimort and Semenov2008), and Reference MartimortMartimort (2018) have also analyzed, as we do hereafter, the magnitude of the free-riding problem among interest groups. Putting an agency problem at the core of the analysis, be it induced by hidden information or hidden action, gives a fresh look at the Olsonian program of finding the determinants of groups’ collective action.Footnote ³ Olson identified free riding among actors with similar preferences as a major impediment to collective action. We will argue in Section 6 that this free-riding problem can be endogenized as coming from a contractual externality among contributors. Moreover, free riding is not enough alone to justify that an interest group fails to intervene. This failure is also linked to the existence of other competing groups. The whole landscape of competitors of an interest group matters to determine whether the latter intervenes or not.

In this respect, Reference Lefebvre and MartimortLefebvre and Martimort (2020) have offered a model that explicitly integrates a group-formation stage in the analysis of the lobbying process. Forming an active group is actually a collective action problem that is made difficult by the fact that individual members have private information on their own marginal benefit of the group’s influence on policy. Such private information is the source of a free-riding problem which now bites within groups. This free riding might undermine their influence in the political process, sometimes up to the point of eschewing any influence at all.Footnote ⁴ Asymmetric information on the principals’ side in common agency model has also been studied by Reference Martimort and MoreiraMartimort and Moreira (2010) and Reference Lima and MoreiraLima and Moreira (2014) who argue that it is a major source of inefficiency.

Still assuming that decision-makers hold private information, Reference Martimort and StoleMartimort and Stole (2018) have also shown that, if the principals’ preferences are close to each other, all principals contribute in equilibrium; confirming thereby one of our results in corollary 1 in Section 6. This latter paper can indeed be viewed as the adverse selection version of the present work, with many results mirroring those found in this Element, especially in Section 6, under moral hazard. There also, an interest group intervenes if its benefit of doing so exceeds the corresponding agency costs. Of course, with asymmetric information, agency costs are of a different nature. As a result, marginal contributions are no longer truthful. Contributions offered by a group have to be discounted below its (marginal) willingness to pay to reflect the group’s difficulty in solving the asymmetric information problem vis-à-vis the policy-maker. This point was made in various contexts by Reference Martimort and SemenovMartimort and Semenov (2007b, Reference Martimort and Semenov2008) and Reference Martimort and StoleMartimort and Stole (2018).

In the standard Chicago view of the political process pushed forward by Reference PeltzmanPeltzman (1976) and later on by Reference BeckerBecker (1983, Reference Becker1985), the influence function which describes how interest groups exert pressure on a policy-maker is ad hoc and given at the outset. In Reference BeckerBecker (1983, Reference Becker1985), for instance, this function may not only depend on whether a group favors or not more spending, but also on the group size. The common agency approach makes progress on that front insofar as actual competition among interest groups is now explicitly modeled as a menu auction where interest groups design contribution schedules to influence the decision-maker. Of course, because this approach also implies that the equilibrium is efficient, it fails to offer a link between the size of the group and its political pressure. We will argue in Section 6 that the common agency approach, when conveniently amended to account for agency costs, is able to endogenize pressure functions and link them to group size.

Finally, this Element also significantly differs from most of the applied common agency literature because the subset of interest groups which intervene is not specified a priori but instead derived from the equilibrium analysis. To fill this gap of the earlier literature, Reference MitraMitra (1999) was the first to introduce exogenous fixed costs of entering the political process, a route which was then also taken by Reference Martimort and SemenovMartimort and Semenov (2007a) and Reference BombardiniBombardini (2008) among others. Entry costs somewhat endogenize the set of active principals and thus offer a direct link between the policy chosen in response to those active groups and the distribution of surplus. In this Element, the same link arises but this exogenous fixed cost is now replaced by agency costs which are endogenous to the problem.

On the Methodological Front

Several contributions dealing with moral hazard, most noticeably those of Reference Bernheim and WhinstonBernheim and Whinston (1986a) in a general agency framework and Reference DixitDixit (1996) in a political economy environment, have focused on intrinsic common agency games. Under intrinsic common agency, the agent accepts or refuses all contracts at once.Footnote ⁵ Although possibly attractive to model political settings like bureaucratic oversight by multiple committees or split control of a regulated firm between different regulatory bodies, those models seem less relevant to capture issues related to the competition among various interest groups willing to influence a common policy-maker and attract his favors. In sharp contrast, this Element analyzes the issue of delegated common agency in a moral hazard environment.

In this respect, this Element is tangentially related to a broader literature on competitive equilibrium in insurance markets plagued with moral hazard (Reference PaulyPauly, 1974; Reference Helpman and LaffontHelpman and Laffont, 1975; Reference HellwigHellwig, 1983; Reference Arnott and StiglitzArnott and Stiglitz, 1991; Reference Bisin and GuaitoliBisin and Guaitoli, 2004; Reference Attar and ChassagnonAttar and Chassagnon, 2009; Reference Attar, Casamatta, Chassagnon and DécampsAttar et al., Reference Attar, Casamatta, Chassagnon and Décamps2019). There also multiple principals post contracts to attract an agent. This agent exerts a 0–1 effort; a feature which introduces a fundamental nonconvexity into the model. The issues that are investigated by this literature are mostly the constrained-efficiency properties of competitive equilibria, whether exclusive contracting is feasible and its consequences and, finally, the role of off-path contracts to sustain equilibrium allocations. Those concerns are not really relevant in our political economy context that has a finite number of strategic principals acting without any of those being able to enforce any exclusivity clause.

As mentioned in the introduction, the truthfulness criterion used in the complete information literature is actually a refinement of the equilibrium set. Contributions stipulating threats for unexpected changes in the agent’s decision make it easy to sustain many other equilibrium outcomes. Some of these outcomes are possibly not efficient. This point was made in Reference Martimort, Blundell, Newey and PerssonMartimort (2007) and Reference Chiesa and DenicolòChiesa and Denicolò (2009) among others. Although found attractive for applications, the truthfulness criterion has thus been questioned in terms of its foundations. Under complete information, it seems hard to a priori preclude the use of simple forcing contracts, in which case, a plethora of (inefficient) equilibria emerges. In this respect, Reference Kirchsteiger and PratKirchsteiger and Prat (2001) defined natural equilibria as associated with forcing contracts. By running an experiment, those authors showed that, under complete information, players tend to play natural rather than truthful equilibria. Note also that forcing contracts may significantly save on menu costs. In a complete information private common agency context where each principal is only affected by a subset of the agent’s actions, Reference Chiesa and DenicolòChiesa and Denicolò (2009) have also shown that truthful contracts might even be Pareto-dominated from the principals’ viewpoint.

Our analysis of the case of unrestricted contracting demonstrates how truthfulness arises there as a best-response property at all equilibria. It is thus no longer a refinement criterion. Reference Laussel and Le BretonLaussel and Le Breton (1998a) have also obtained a similar result in a model with hidden information on a parameter of the agent’s utility function and ex ante contracting. Incentive constraints then pin down the slope of the contribution schedule at any equilibrium decision. With ex ante contracting, efficiency is still achieved for any realization of the underlying uncertainty. However, truthful contributions might also be used off equilibrium to compute the agent’s reservation payoffs if he deviates and refuses one of the proposed contracts. Those extensions have thus a nontrivial redistributive impact since they determine how much each principal can extract from the common agent and thus how much he gets at equilibrium. This issue is also investigated in Reference Martimort and StoleMartimort and Stole (2009) under the assumption of ex post contracting, that is, when the agent already knows his type before contracting. Working in a delegated common agency model with moral hazard avoids those arbitrary extensions since every political outcome has some nonzero probability on the equilibrium path when a standard assumption of full support is made.

Lastly, Reference Bernheim and WhinstonBernheim and Whinston (1986b) have demonstrated that truthful equilibria have also the additional property that they are immune to deviations by coalitions of principals which are themselves robust to further deviations by sub-coalitions. Reference Furusawa and KonishiFurusawa and Konishi (2011) have extended the notion of coalition-proofness when entry is also a strategic decision for principals. Yet, we will show in Section 8 that this quite demanding criterion does not exhaust incentives for coalition building with moral hazard frictions.

Preferences

There are $n$ interest groups (sometimes referred to as the principals) in the economy. Those groups are indexed by $i\in N=\{0,1,\dots ,n\}$ .

A policy-maker (the agent) exerts an action $e$ which affects the probability that a reform is enacted. One may think of this reform as, for instance, opening trade barriers and adopting free trade or tightening existing regulatory standards. For most of this Element, we thus focus on a 0–1 political decision. This assumption simplifies the analysis and allows to get clear results which have nevertheless a broader generality.Footnote ⁶ To fix ideas, one can think of $e$ as the effort and resource that the policy-maker devotes to convince other policy-makers but also voters that the reform should be adopted. To simplify notations, we identify $e$ with the probability that the reform passes. With complementary probability $1-e$ , the status quo instead prevails.

Interest group $i$ thus gets an expected payoff given by

$V_i=E\left({\tilde{S}}_i-{\tilde{t}}_i|e\right),$(3.1)

where $E(\cdot |e)$ denotes the expectation operator with respect to the distribution of political outcomes induced by effort $e$ , and ${\tilde{S}}_i$ and ${\tilde{t}}_i$ are random variables that stand for the benefit that accrues to group $i$ and the payment it makes to the decision-maker. To be more precise, let ${\bar{S}}_i$ (resp. ${\underset{\_}{S}}_i$ ) be interest group $i$ ’s gross payoff when the reform passes (resp. fails). Similarly, let ${\bar{t}}_i$ (resp. ${\underset{\_}{t}}_i$ ) represent the interest group’s monetary contribution to the policy-maker in case the reform passes (resp. fails).

Two polar cases will be of a particular interest for what follows:

• Congruent interest groups: When ${\bar{S}}_i>{\underset{\_}{S}}_i$ for all $i\in \{1,..,n\}$ , all principals find it worth to implement the reform. Passing the reform can thus be viewed as a public good to which all groups may want to contribute.

• Conflicting interest groups: When ${\bar{S}}_i>{\underset{\_}{S}}_i$ for $i\in \{1,..,k\}$ and ${\bar{S}}_i<{\underset{\_}{S}}_i$ for $i\in \{k+1,n\}$ , the first $k$ principals favor the reform, whereas the $n-k$ others are opposed to it. This corresponds to a fierce head-to-head competition between two opposite sets of interest groups.

It should be clear that we can normalize payoffs for each principal so that the worst political outcome from its own viewpoint yields zero payoff. In the case of groups who favor the reform, it boils down to adopting the normalization ${\underset{\_}{S}}_i=0$ , while ${\bar{S}}_i=0$ prevails for groups that are opposed to this reform.Footnote ⁷

The risk-neutral policy-maker has an opportunity cost $\psi \left(e\right)$ of increasing the probability that the reform is implemented. This disutility may stand for the cost of not allocating resources to private ends or, alternatively, the cost of not defending other causes in the policy arena. It is assumed that $\psi \left(e\right)$ is increasing, convex with a positive third derivative $({\psi }^{\prime }>0,{\psi }^{\prime \prime }>0,{\psi }^{\prime \prime \prime }\ge 0)$ and such that $\psi \left(0\right)=0$ . The Inada conditions ${\psi }^{\prime }\left(0\right)=0$ and ${\psi }^{\prime }\left(1\right)=+\infty$ hold to insure interior solutions. Observe that the policy-maker has thus a bias toward the status quo since, in the absence of any influence, he would not exert any effort and the status quo would prevail.

The agent’s expected payoff can thus be written as:

$U=E\left(\sum _{i=1}^n{\tilde{t}}_i|e\right)-\psi \left(e\right).$(3.2)

For further references, we, respectively, denote by ${\bar{T}}_I=\sum _{i\in I}{\bar{t}}_i$ and ${\underset{\_}{T}}_I=\sum _{i\in I}{\underset{\_}{t}}_i$ the aggregate contribution of a coalition made of principals belonging to any subset $I\subseteq N$ while ${\bar{T}}_{-i}$ and ${\underset{\_}{T}}_{-i}$ denote the sum of the contributions of all principals except $i$ in each state of nature. Similar notations are used throughout for the aggregate benefits ${\bar{S}}_I$ and ${\underset{\_}{S}}_I$ of such a coalition and for other variables of interest as well.Timing: The lobbying game unfolds as follows.

1. 1. Principals offer noncooperatively their state-contingent contribution schedules $\left\{\right({\bar{t}}_i,{\underset{\_}{t}}_i){\}}_{1\le i\le n}$ .

2. 2. The policy-maker decides which subset of contributions he should accept. If he refuses all such schedules, he obtains an exogenous payoff that is normalized at zero.

3. 3. The policy-maker chooses the nonverifiable action $e$ .

4. 4. Finally, the political outcome, that is, whether the reform passes or fails, realizes. The payments stipulated by the contracts are made accordingly.

We are interested in studying the subgame-perfect equilibria of this delegated common agency game (Reference Bernheim and WhinstonBernheim and Whinston, 1986b). The equilibrium set of bilateral relationships between the principals and the agent emerges endogenously at those equilibria.Benchmarks. For further references, let us compute the socially optimal action $e_N$ as

$e_N=\arg \underset{e\in [0,1]}{\max }{\rm E}\left({\tilde{S}}_N\right|e)-\psi (e).$

The necessary and sufficient first-order condition for an interior optimum takes the form of a familiar Lindahl–Samuelson condition:

${\bar{S}}_N-{\underset{\_}{S}}_N={\psi }^{\prime }\left(e_N\right).$(3.3)

The marginal cost of effort is equal to the sum of the (algebric) marginal benefits.

As long as ${\bar{S}}_N>{\underset{\_}{S}}_N$ , an assumption that is made throughout, $e_N$ remains positive. When all interest groups have congruent preferences, this condition of course holds. More generally, it must be that principals in favor of the reform have a greater aggregate valuation for than principals against to get a positive effort, that is, $\sum _{i=1}^k{\bar{S}}_i>\sum _{i=k+1}^n{\underset{\_}{S}}_i.$ When instead $\sum _{i=k+1}^n{\underset{\_}{S}}_i>\sum _{i=1}^k{\bar{S}}_i,$ the optimal effort is at a corner, namely, $e_N=0$ . The socially optimal action is then to never implement the reform.

Consider now a coalition of principals $I\subset N$ and define as well the optimal action $e_I$ for such a coalition as:

$e_I=arg\underset{e\in [0,1]}{max}E\left({\tilde{S}}_I|e\right)-\psi \left(e\right).$

Using a first-order condition yields

${\bar{S}}_I-{\underset{\_}{S}}_I={\psi }^{\prime }\left(e_I\right)$(3.4)

for an interior solution, while the corner solution $e_I=0$ prevails if ${\bar{S}}_I<{\underset{\_}{S}}_I$ .

For any subset of groups $I\subset N$ , the aggregate payoff of the grand coalition made of the principals belonging to $I$ together with the agent is defined as:

$W_I=E\left({\tilde{S}}_I|e_I\right)-\psi \left(e_I\right).$(3.5)

4.1 Preliminaries

We start with a simple definition.

It is slightly simpler for expositional purposes to start by writing down the conditions for a putative equilibrium where all principals would be active and have each their offer being accepted by the agent. We will then show in Appendix A that there cannot exist any other equilibrium with some principals not being active in the scenario of unrestricted contracting – that is, here under scrutiny.

Since all principals’ offers are accepted at such an equilibrium, the following participation constraint of the agent has to be satisfied:

$\underset{e\in [0,1]}{max}E\left({\tilde{t}}_i+{\tilde{T}}_{-i}^D|e\right)-\psi \left(e\right)\ge max\left(0,E\left({\tilde{T}}_S^D|e_S\right)-\psi \left(e_S\right)\right),\text{for any}S\subset N.$(4.1)

The r.h.s of (4.1) describes what the agent can get by either refusing all offers or only accepting a subset of those offers. In particular, let $e_S$ denote the optimal effort supply made by the agent when he accepts the aggregate contribution of principals in a coalition $S$ . In fact, for an interior solution, the following first-order condition holds:

${\bar{T}}_S^D-{\underset{\_}{T}}_S^D={\psi }^{\prime }\left(e_S\right).$

In turn, maximizing the l.h.s. of (4.1) immediately yields the following incentive constraint for an interior solution:

${\bar{t}}_i+{\bar{T}}_{-i}^D-\left({\underset{\_}{t}}_i+{\underset{\_}{T}}_{-i}^D\right)={\psi }^{\prime }\left(e\right).$(4.2)

Consider now principal $i$ ’s optimal offer when all other principals are active. It solves the following best-response problem:

$\left(P_i^D\right):\underset{\{e,({\bar{t}}_i,{\underset{\_}{t}}_i\left)\right\}}{max}E\left({\tilde{S}}_i-{\tilde{t}}_i|e\right)\text{subject to (4.1) and (4.2).}$

Let us omit for the moment the incentive constraint (4.2). As it will be seen in Section 4.2, the equilibrium payments $({\bar{t}}_i^D,{\underset{\_}{t}}_i^D)$ found will satisfy this constraint. Because principal $i$ can decrease the expected payment to the common agent up to the point where the participation constraint (4.1) is binding, $\left(P_i^D\right)$ can be written in a more compact way as:

$\underset{e\in [0,1]}{max}\left(E\left({\tilde{S}}_i+{\tilde{T}}_{-i}^D|e\right)-\psi \left(e\right)-max\left(0,\underset{S\subseteq N}{max}\left(E\left({\tilde{T}}_S^D|e_S\right)-\psi \left(e_S\right)\right)\right)\right).$

The solution to this problem is to induce an effort $e^D$ such that:

$e^D\in arg\underset{e\in [0,1]}{max}E\left({\tilde{S}}_i+{\tilde{T}}_{-i}^D|e\right)-\psi \left(e\right).$(4.3)

This condition simply means that principal $i$ does not want to induce another action than $e^D$ given that the offers of all other principals are accepted. Each principal offers a contract which induces an action that maximizes the bilateral surplus of the coalition he forms with the common agent.

The following Lemma provides necessary and sufficient conditions for the $n+1$ -uple $\left(({\bar{t}}_i^D,{\underset{\_}{t}}_i^D{)}_{1\le i\le n},e^D\right)$ to be such an equilibrium where all principals are active.

Lemma 1 The $n+1$ -uple $\left(({\bar{t}}_i^D,{\underset{\_}{t}}_i^D{)}_{1\le i\le n},e^D\right)$ is an equilibrium of the delegated common agency game where all principals are active if and only if:

$\begin{array}{cc}& e^D\in arg\underset{e\in [0,1]}{max}\left(E\left({\tilde{T}}_N^D|e\right)-\psi \left(e\right)\right),\end{array}$(4.4)

$\begin{array}{cc}& e^D\in arg\underset{e\in [0,1]}{max}\left(E\left({\tilde{S}}_i+{\tilde{T}}_{-i}^D|e\right)-\psi \left(e\right)\right),\mathrm{for\; all}i\in N,\end{array}$(4.5)

$\begin{array}{cc}& E\left({\tilde{S}}_i+{\tilde{T}}_{-i}^D|e^D\right)-\psi \left(e^D\right)\ge \underset{\left\{i\right\}\subseteq S\subset N}{max}\left(E\left({\tilde{S}}_i+{\tilde{T}}_{S-\left\{i\right\}}|e_S\right)-\psi \left(e_S\right)\right),\end{array}$(4.6)

$\begin{array}{cc}& \text{for all}i\in N,\\ & E\left({\tilde{T}}_N^D|e^D\right)-\psi \left(e^D\right)=\underset{S\subset N}{max}\left(0,E\left({\tilde{T}}_S^D|e_S\right)-\psi \left(e_S\right)\right).\end{array}$(4.7)

This Lemma extends the analysis of Reference Bernheim and WhinstonBernheim and Whinston (1986b) (complete information) and Reference Laussel and Le BretonLaussel and Le Breton (1998a) (hidden information with ex ante contracting) to the case of moral hazard.

To build intuition behind conditions (4.4)–(4.7), note that (4.4) is simply the agent’s incentive constraint when he faces an aggregate contribution $({\bar{T}}_N,{\underset{\_}{T}}_N)$ . Condition (4.5) has been discussed above (see (4.3)). Condition (4.6) indicates that no principal finds it worth to make an offer that would bring the agent to exclude some principals. The grand-coalition forms. If condition (4.6) were not to hold for one principal, this principal could make an offer accepted by the agent who would contract then with only a subset of the principals.

Finally, condition (4.7) shows that the common agent should be indifferent between accepting the contracts of the grand-coalition $N$ and accepting the next best option which is either to only accept contracts from a proper sub-coalition or to refuse all contracts at once. Indifference is definitively needed since, otherwise, one of the principals in the grand coalition could deviate, reduce payments in each state of nature ${\bar{t}}_i$ and ${\underset{\_}{t}}_i$ uniformly by the same small amount $\epsilon$ , while still inducing the same action $e^D$ and acceptance of the whole range of contracts offered by the grand-coalition $N$ .

4.2 Truthfulness and Efficiency

To characterize which (interior) efforts are equilibrium candidates, we use the necessary and sufficient first-order conditions for (4.4) and (4.5). This yields

${\psi }^{\prime }\left(e^D\right)={\bar{T}}_N^D-{\underset{\_}{T}}_N^D,$(4.8)

and

${\psi }^{\prime }\left(e^D\right)={\bar{S}}_i+{\bar{T}}_{-i}^D-\left({\underset{\_}{S}}_i+{\underset{\_}{T}}_{-i}^D\right).$(4.9)

A common agency equilibrium with a nonnegative level of effort is thus such that:

${\psi }^{\prime }\left(e^D\right)={\bar{T}}_N^D-{\underset{\_}{T}}_N^D={\bar{S}}_i+{\bar{T}}_{-i}^D-\left({\underset{\_}{S}}_i+{\underset{\_}{T}}_{-i}^D\right).$(4.10)

From this, we deduce that the marginal contribution of each principal necessarily reflects his relative preferences among alternative political outcomes:

${\bar{t}}_i^D-{\underset{\_}{t}}_i^D={\bar{S}}_i-{\underset{\_}{S}}_i,$(4.11)

or, with more compact notations,

${\tilde{t}}_i^D={\tilde{S}}_i-C_i^D,$(4.12)

where $C_i^D$ is a fixed fee independent of the realization of the reform or not.

Equation (4.12) simply states that principal $i$ ’s contribution is truthful. Payments perfectly reflect the benefits of the principal for each realized political outcome. Such a contribution schedule (4.12) thus aligns the objective of principal $i$ with that of the common agent. Henceforth, even with moral hazard, the agent’s incentive constraint is costless for the principals when contracts are unrestricted.

By offering a truthful schedule, a principal becomes indifferent at equilibrium among all realizations of the contractible outcomes. In our reform example, each principal can secure the same final payoff, namely $C_i^D$ , whether the reform fails or succeeds. Determining the value of the principals’ equilibrium payoff, and the sign of the transfers in each state of nature, is not yet possible at this stage. These issues are addressed in Section 4.3.

Since each principal offers a contribution which reflects his own relative valuation between alternative outcomes, the common agent ends up being residual claimant for the decision taken. He thus chooses the efficient level of effort from the point of view of the grand coalition he forms with all the principals. Indeed, using (4.8) and (4.12), an interior solution solves:

${\psi }^{\prime }\left(e^D\right)={\bar{S}}_N-{\underset{\_}{S}}_N.$(4.13)

From standard moral hazard theory when applied to a single principal–agent relationship,Footnote ⁹ we already know that this principal obtains the first-best outcome by making the risk-neutral agent residual claimant for whatever decision he takes when this action is nonverifiable. This is obtained by using a truthful contribution of the form (4.12). Such contract makes the principal indifferent between all possible outcomes and has the agent pay a fixed fee $C_i^D$ to the principal for the right of taking decisions. With multiple competing principals, the same result still holds. At a best response, each principal makes also the agent residual claimant and efficiency necessarily follows from the grand coalition’s viewpoint.

4.3 Distribution of the Surplus

The only two remaining issues are to find whether such equilibria where all principals are active exist and if yes what are the sets of fixed fees offered by the principals. These fees determine the possible distributions of the aggregate surplus among principals.

From the binding condition (4.7), which expresses the indifference of the agent between contracting with the grand coalition and with the next best option with a proper one, and the form of the equilibrium schedules, we deduce that the vector $(C_i^D{)}_{1\le i\le n}$ is a solution to the fundamental equations:

$W_N-C_N^D=max\left(0,\underset{S\subset N}{max}\left(W_S-C_S^D\right)\right).$(4.14)

In a public good model with hidden information and ex ante contracting, Reference Laussel and Le BretonLaussel and Le Breton (1998a) got a similar characterization of equilibrium payoffs by means of such equations. Our analysis borrows much from theirs.

Congruent Interest Groups. Here, we get:

Proposition 2 The only equilibria of the delegated common agency game when principals have congruent preferences entail all principals being active. In any such equilibrium, principals get positive payoffs $(C_i^D{)}_{1\le i\le n}$ and the common agent gets zero rent with:

$C_i^D\ge W_i>0,$(4.15)

and

$C_N^D=W_N.$(4.16)

With congruent preferences, the set of equilibrium payoffs of the delegated common agency game can be identified with the nonempty core of a cooperative game among principals having characteristic form $(W_S{)}_{S\subseteq N}$ . Existence and properties immediately follow.

Two features of the equilibrium are worth noticing. First, when principals have congruent preferences, they all have in common the desire to extract the common agent’s rent. Second, each of those principals has strong incentives to join the grand coalition of active principals to benefit from the increasing returns coming from the convexity of the cooperative game characterized with (3.5). Actually, the stand-alone payoffs $W_i$ are payoffs obtained when each principal pays all the cost of the agent’s effort. With congruent interests, this cost of effort, instead of being somehow duplicated for each principal when acting alone, is now jointly supplied by the coalition. Hence, there exists a surplus resulting from the joint contribution of multiple principals and the grand-coalition forms. This equilibrium aggregate surplus can be distributed in many different ways to give each principal more than the stand-alone payoff that he would get by being alone contracting with the decision-maker.

Conflicting Interest Groups. To simplify the analysis, we focus on the case of two principals who have opposite preferences on the benefit of the reform.

Proposition 3 Assume that $n=2$ and principals have conflicting preferences with principal 1 being dominant, that is, ${\bar{S}}_1>{\underset{\_}{S}}_2$ . Then, there exists a unique equilibrium of the delegated common agency game. In this equilibrium, the common agent gets a positive rent and both principals are active. Principal $i$ gets a payoff:

$C_i^D=W_{12}-W_{-i}>0.$(4.17)

The agent gets a residual payoff:

$U=W_1+W_2-W_{12}>0.$(4.18)

With conflicting preferences, the common agent can play one principal against the other and gets a positive rent out of this head-to-head competition. At the unique equilibrium of the game, each principal gets a payoff just equal to his incremental contribution to the aggregate coalition, as expressed in (4.17).

As noticed in Reference Bernheim and WhinstonBernheim and Whinston (1986b) under complete information, this feature of the equilibrium under head-to-head competition is reminiscent of payments in a Groves mechanism although the comparison is somewhat misleading since the preferences of the competing principals are here common knowledge whereas, in the standard framework of Reference GrovesGroves (1973), Reference ClarkeClarke (1971), and Reference Green and LaffontGreen and Laffont (1977), the preferences of contributing players are unknown to the mechanism designer.

Two common lessons can be withdrawn from Propositions 2 and 3. First, the equilibria of the delegated agency under moral hazard remain efficient, that is, there is neither free riding nor wasteful competition among principals. Second, all principals find it worth to intervene when they are unrestricted in the kind of contributions they may offer. This is the case whatever the nature of the competition between interest groups.

The agency problem that could arise under moral hazard between the common agent and his principals is not enough alone to explain why some groups may fail to operate in the policy arena. Under moral hazard, risk-neutrality, and unrestricted contracting, the political process efficiently aggregates the preferences of all principals and they are all active. If some groups are active in the lobbying process while others are not, the reasons must be found by modifying the lobbying model as it is; a route taken in Section 5.

5.1 A Useful Lemma

Next Lemma simply states that groups that favor the reform do not pay when it is not passed, while it is the reverse for those groups hurt by the reform. This result significantly simplifies exposition.

Lemma 2 There is no loss of generality in assuming that the agent accepts all contracts proposed when payments are nonnegative. Moreover, in any equilibrium, each principal offers a null contribution for an outcome that he does not favor:

${\tilde{S}}_i=0\Rightarrow {\tilde{t}}_i=0,\forall i\in N.$

Equipped with this result, we can now turn to the impact that various configurations of preferences and the types of behavior they involve have on equilibrium outcomes.

Imposing a nonnegativity constraint on payments simplifies the analysis of the acceptance stage of the game. Indeed, the agent is always (at least weakly) willing to accept any such contract. When principal $i$ is not active, that is, proposes a null contribution for all outcomes, the agent is indifferent between accepting or rejecting his offer.

5.2 The Cooperative Outcome

It it useful, both for getting a clue upfront for some of the frictions at play and for relevant comparisons, to start our analysis with the cooperative outcome. For future references, consider thus the hypothetical scenario where principals jointly design a nonnegative aggregate contribution. Possibly, this agreement may also stipulate ex ante lump sum payments to redistribute payoffs among principals. The policy-maker may either accept this joint offer or refuse all those offers at once.

The cooperative principals do not pay for their least-preferred outcome and thus

${\underset{\_}{T}}_N=0.$(5.1)

The relevant nonnegativity constraint on aggregate payments thus boils down to

${\bar{T}}_N\ge 0.$(5.2)

Given an aggregate incentive scheme collectively offered by the $n$ principals, the policy-maker thus chooses now an action $e$ which solves:

${\bar{T}}_N={\psi }^{\prime }\left(e\right).$(5.3)

So doing yields to the common agent a payoff worth

$U=e{\bar{T}}_N-\psi \left(e\right)=R\left(e\right),$(5.4)

where the moral hazard rentFootnote ¹³ $R\left(e\right)=e{\psi }^{\prime }\left(e\right)-\psi \left(e\right)$ is nonnegative, increasing and convex with the assumptions made on $\psi \left(e\right)$ (namely, $R^{\prime }\left(e\right)=e{\psi }^{\prime \prime }\left(e\right)>0$ , $R^{\prime \prime }\left(e\right)=e{\psi }^{\prime \prime \prime }\left(e\right)+{\psi }^{\prime \prime }\left(e\right)>0$ ). From (5.4), it then follows that the agent’s participation constraint necessarily holds. In other words, with nonnegative payments, the agent must necessarily be rewarded when the reform passes since he cannot be punished when it fails. He thus obtains a positive rent.

The cooperative outcome is then solution to the following problem:

$\left(P^C\right):\underset{e\in [0,1]}{max}e\left({\bar{S}}_N-{\underset{\_}{S}}_N\right)-\psi \left(e\right)-R\left(e\right).$

This expression showcases that the cooperative solution must balance efficiency considerations and the fact that inducing effort requires to leave a moral hazard rent $R\left(e\right)$ to the agent. This rent is costly from the cooperating principals’ viewpoint. Of course, a similar trade-off arises in the noncooperative scenarios that will be investigated in Section 6.

The solution to the cooperative problem $\left(P^C\right)$ is straightforward. Because the reform is viewed as being valuable from the principals’ grand-coalition viewpoint, namely ${\bar{S}}_N>{\underset{\_}{S}}_N$ , the optimal effort $e^C$ is interior and solves:

${\bar{S}}_N-{\underset{\_}{S}}_N={\psi }^{\prime }\left(e^C\right)+e^C{\psi }^{\prime \prime }\left(e^C\right).$(5.5)

This condition is again a familiar Lindahl–Samuelson condition for effort once it has been conveniently modified to account for the extra (marginal) cost of the agent’s moral hazard rent $R^{\prime }\left(e\right)=e{\psi }^{\prime \prime }\left(e\right).$

To reduce the agent’s moral hazard rent $R\left(e\right)$ , the cooperating principals jointly reduce the aggregate payment made when the reform succeeds, namely, using (5.11):

${\bar{T}}^C={\bar{S}}_N-{\underset{\_}{S}}_N-e^C{\psi }^{\prime \prime }\left(e^C\right).$(5.6)

As this condition shows, the existence of an agency cost $R\left(e\right)$ implies that, even when cooperating, principals as a whole shade their overall valuation for the reform. Because of such shading, their contribution is no longer truthful. We have

$0={\underset{\_}{T}}^C<{\bar{T}}^C<{\bar{S}}_N-{\underset{\_}{S}}_N.$

5.3 A Useful Benchmark: The Scenario of Intrinsic Common Agency

Consider now the hypothetical scenario where principals no longer cooperate in designing contracts but the game remains one of intrinsic common agency, an expression coined by Reference Bernheim and WhinstonBernheim and Whinston (1986a). In this scenario, the agent has only two options. Either, he accepts all the principals’ offers or none of those and thereby gets his reservation payoff which is normalized at zero. In this context, the only nonnegativity constraint on payments applies on the aggregate, namely:

${\bar{t}}_i+{\bar{T}}_{-i}={\bar{T}}_N\ge 0$(5.7)

while (5.1) still holds. Of course, the agent’s rent in this context is still given by (5.4).

To give some intuition on the nature of the distortion, consider the case where all principals cooperate and share the agency cost needed to induce an effort $e$ . Principal $i$ then offers a contribution

${\bar{t}}_i^C={\bar{S}}_i-{\underset{\_}{S}}_i-\frac{1}{n}{e}^C{\psi }^{\prime \prime }\left(e^C\right)$(5.8)

in case the reform passes which yields a payoff worth

$V_i^C={\underset{\_}{S}}_i+\frac{1}{n}\left(e^C{)}^2{\psi }^{\prime \prime }\right(e^C).$(5.9)

Starting from such a cooperative contract, principal $i$ , if he deviates by offering a lower transfer ${\bar{t}}_i$ , induces another effort $e_i$ which satisfies

${\psi }^{\prime }\left(e_i\right)={\bar{t}}_i+{\bar{S}}_{-i}-{\underset{\_}{S}}_{-i}-\frac{n-1}{n}{e}^C{\psi }^{\prime \prime }\left(e^C\right).$

Footnote ¹⁴Such a deviation thus yields an expected payoff to principal $i$ which is worth:

$V_i^d\left(e_i\right)={\underset{\_}{S}}_i+e_i\left({\bar{S}}_i-{\underset{\_}{S}}_i-{\bar{t}}_i\right)={\underset{\_}{S}}_i+e_i\left({\bar{S}}_N-{\underset{\_}{S}}_N-\frac{n-1}{n}{e}^C{\psi }^{\prime \prime }s\left(e^C\right)-{\psi }^{\prime }\left(e_i\right)\right).$

It immediately follows that

${\dot{V}}_i^d\left(e^C\right)=-\frac{n-1}{n}{e}^C{\psi }^{\prime \prime }\left(e^C\right)<0.$

Hence, starting from the cooperative solution, principal $i$ would like to induce a lower effort; an instance of negative externality among principals. By reducing his contribution and inducing a lower effort, principal $i$ benefits from the fact that the remaining principals offer higher contributions while he also saves on the associated agency cost. This is the essence of the free-riding problem among noncooperating principals.

It turns out that the characterization of the equilibrium under intrinsic common agency already contains much information about what will happen under delegated common agency. In a first step toward such characterization, we observe that Lemma 2 and the fact that ${\bar{S}}_N>{\underset{\_}{S}}_N$ altogether imply that the aggregate incentive scheme offered by $n$ principals satisfies

${\underset{\_}{T}}_N={\underset{\_}{t}}_i+{\underset{\_}{T}}_{-i}=0.$(5.10)

The policy-maker now chooses an action $e$ which solves:

${\psi }^{\prime }\left(e\right)={\bar{T}}_N={\bar{t}}_i+{\bar{T}}_{-i}.$(5.11)

So doing yields an expected payoff to the agent which is still given by (5.4).

Definition 2 A vector $\left(({\bar{t}}_i^I,{\underset{\_}{t}}_i^I{)}_{i\le n},e^I\right)$ is an equilibrium of the intrinsic common agency game if and only if it solves for each principal $i$ the following best-response optimization problem:

$\left(P_i^I\right):\underset{\{{\tilde{t}}_i,e\}}{max}E\left({\tilde{S}}_i-{\tilde{t}}_i|e\right)\text{subject to (5.10) and (5.11).}$

Inserting (5.10) into the above maximand and using the incentive constraint (5.11), we rewrite this program in terms of a sole maximization with respect to the effort level that principal $i$ can induce as

$(P_i^I{)}^{\prime }:\underset{e\in [0,1]}{max}e\left({\bar{S}}_i-{\underset{\_}{S}}_i+{\bar{T}}_{-i}^I-{\underset{\_}{T}}_{-i}^I\right)-\psi (e)-R(e).$

The intrinsic common agency game is known to be an aggregate game at least since Reference Bernheim and WhinstonBernheim and Whinston (1986a). In such a game, the payoff of each principal depends on his own payments and those of others only through the overall aggregate payment that induces a given effort level. Sometimes, those games are known to have nice properties in the sense that their equilibria can be found as solution to a self-generating problem as we will soon see below. This is the Principle of Aggregate Concurrence as coined by Reference Martimort and StoleMartimort and Stole (2012). This reduction captures the fact that, in such an aggregate game, the objectives of the different principals can be somehow aligned. Intuitively, because the non-negativity and the agent’s incentive constraints depend only on the aggregate payment, any given principal, since he is unrestricted in his own payment, can always undo whatever payments are offered by others without changing the feasibility set. This creates a common concern across principals and aligns their objectives.

To see how, observe that if the equilibrium effort $e^I$ is a solution for all problems $(P_i^I{)}^{\prime }$ , it is also a solution for the sum of those optimization problems, namely, for the so-called self-generating optimization problem:

$\left(P^{SE}\right):\underset{e\in [0,1]}{max}e\left({\bar{S}}_N-{\underset{\_}{S}}_N+(n-1)({\bar{T}}^I-{\underset{\_}{T}}^I)\right)-n\psi \left(e\right)-nR\left(e\right),$

where ${\underset{\_}{T}}^I=0$ from (5.10) and the equilibrium level of effort satisfies

${\psi }^{\prime }\left(e^I\right)={\bar{T}}^I.$(5.12)

This problem is self-generating because its solution $e^I$ is also an argument of the maximand.

It should be clear from the nonnegativity constraint (5.10) that there exists a continuum of possible equilibria of the intrinsic common agency game that are defined up to a redistribution of payments across principals even though the equilibrium effort remains the same across such equilibria. For the sake of comparison with the delegated common agency scenario where this constraint is taken principal by principal, we shall concentrate on one such redistribution where

${\underset{\_}{t}}_i^I=0\forall i.$(5.13)

We summarize the result of this optimization and the main features of this equilibrium in the next proposition.

In the intrinsic common agency game, each principal takes as given the contributions of others when choosing how much to pay himself for the agent’s services. Inducing a higher level of effort from the agent becomes a public good as it can be easily seen from the agent’s incentive constraint (5.11). In equilibrium, underprovision of this public good follows from the principals’ noncooperative behavior. The equilibrium effort is lower than if principals were jointly designing the agent’s contribution. Indeed, all principals care about extracting the agent’s moral hazard rent $R\left(e\right)$ and, doing so, they induce a lower level of effort than at the first best. From (5.15), everything happens as if each principal had to pay exactly for the marginal agency cost of this rent, namely: $R^{\prime }\left(e\right)=e{\psi }^{\prime \prime }\left(e\right)$ . As a result of this compounding of rent extraction across principals, there is now a $n$ -fold distortion of the effort level below the first best. The equilibrium effort level under intrinsic common agency is thus lower than if principals had cooperated and internalized the free-riding externality.

It should be clear from our analysis that the free-riding problem under intrinsic common agency comes from the fact that the agent withdraws some positive rent when aggregate payments are constrained to be nonnegative and, because actions are nonverifiable, this rent is linked to his effort level. It is in sharp contrast with the case of unrestricted contracting. There, the agent can still withdraw some positive rent; the case of conflicting interests analyzed in Section 4.3 is an example in order. Yet, this rent only depends on how principals compete to get a share of the overall surplus. With unrestricted contracting, redistributive and allocative issues are disentangled and the agent’s effort is set at the first best independently on how the surplus is shared among principals. Because the agent always exerts the first-best effort, no free riding arises.

6.1 Contributions and Allocative Inefficiency

To better understand the structure of the equilibrium, we now rank the different principals according to their increasing valuations for the reform in such a way that ${\bar{S}}_1<{\bar{S}}_2\cdots <{\bar{S}}_n$ . To simplify presentation, we assume for the time being that all principals have different preferences. A straightforward extension would consist in allowing for different principals with the same preferences, an exercice that we will undertake later.

The allocative properties of the equilibrium are investigated in the next proposition.

6.1.1 Active Set of Principals

To understand this equilibrium characterization, it is useful to start with what we already know from the equilibrium characterization under intrinsic common agency. From (5.15) and taking into account our assumption that ${\underset{\_}{S}}_i=0$ for principals who favor the reform, we observe that payments in any intrinsic common agency equilibrium are given as

${\bar{t}}_i^I={\bar{S}}_i-e^I{\psi }^{\prime \prime }\left(e^I\right).$

Clearly, those payments cannot be part of an equilibrium under delegated agency if ${\bar{S}}_i$ is small enough since it would violate the nonnegativity constraint (6.1). Those principals with the lowest valuations, that is, valuations which stand below the marginal agency cost $R^{\prime }\left(e^I\right)=e^I{\psi }^{\prime \prime }\left(e^I\right)$ , are pushed out of the political process. Under delegated agency, those principals being out, the magnitude of the free-riding problem diminishes among the remaining active principals. As a result the agent’s effort increases, which means that the (marginal) agency cost also increases and might push out of the process even principals with higher valuation. Because principals are ranked according to their increasing valuations for the reform, that subset of active principals is an upper tail of the distribution of principals. Equation (6.4) defines then the endogenous cut-off on valuations above which principals become active contributors. The size $k$ of the active coalition is precisely found when the bringing in of an extra principal with a “marginal valuation” just covers the marginal agency cost at the effort level that such coalition would induce. More formally, adding a $k+1$ th inframarginal principal with valuation ${\bar{S}}_{n-k}$ only increases the effort level if and only if

${\bar{S}}_{n-k}>e_{k+1}{\psi }^{\prime \prime }\left(e_{k+1}\right).$

6.1.2 Virtual Valuations

Under moral hazard and with the added constraint of having nonnegative contributions, everything happens as under complete information except that the valuation of each principal ${\bar{S}}_i$ is now replaced by a virtual valuation which is effort-dependent, namely

${\tilde{\bar{S}}}_i\left(e\right)\equiv max\{0,{\bar{S}}_i-e{\psi }^{\prime \prime }(e\left)\right\}.$

This virtual valuation now takes into account the marginal cost of the agent’s rent which is borne by this principal. Since this cost is the same for all principals, some of them, the weakest ones, fail to intervene.

This remark been made, it becomes useful to rewrite the equilibrium condition in a more compact way as the solution to the following Lindahl–Samuelson condition conveniently modified to account for informational frictions:

$\sum _{i=1}^{n}max\left(0,{\bar{S}}_i-e^D{\psi }^{\prime \prime }\left(e^D\right)\right)={\psi }^{\prime }\left(e^D\right).$(6.6)

This more compact expression first stresses the nature of the exact contribution of each principal. Second, it implicitly encompasses also the determination of the set of active principals. Lastly, it easily generalizes the previous approach to the case where principals may have the same preferences, a scenario that is covered in Corollary 1 below.

6.1.4 Homogenous Principals

Consider the case where all principals are alike and have congruent preferences in favor of the reform.

Corollary 1 Assume that all principals gets a fixed benefit from the reform, that is, ${\bar{S}}_i=\bar{S}>0$ for any $i\in \{1,\dots ,n\}$ .

1. 1. The equilibrium effort level entails a $n$ -fold distortion:

   $n\bar{S}={\psi }^{\prime }\left(e^D\right)+n{e}^D{\psi }^{\prime \prime }\left(e^D\right).$(6.7)

2. 2. All principals are active in equilibrium:

   ${\bar{t}}_n=\bar{S}-e^D{\psi }^{\prime \prime }\left(e^D\right)=\frac{{\psi }^{\prime }\left(e^D\right)}{n}>0.$(6.8)

When $n$ increases, two effects are at play. First, when the per capita benefit $\bar{S}$ of the reform remains unchanged, the overall benefits of the reform that accrue to the group grows linearly with $n$ . Second, as the size of the group increases, the free-riding problem also gets worse. It turns out that, as a result of those two countervailing forces, the equilibrium effort $e^D$ increases as it can be readily seen from (6.7). Such congruent principals actually display all features of what Reference OlsonOlson (1965) coined as being an inclusive group. An inclusive group has its members willing to accept newcomers and, in this case, the supply of the collective good increases with the size of the group.

As $n$ goes large, each individual contribution should have little impact on the probability of a reform. Minimizing his own contribution is then the sole objective of any given principal. When evaluating whether to contribute or not, this individual thus compares his benefit  $\bar{S}$ with the marginal agency cost at the (limiting) equilibrium effort $e_{\infty }$ . This effort is determined by the indifference condition:Footnote ¹⁵

$\underset{n\to +\infty }{lim}{\bar{t}}_n=0\iff \bar{S}=e_{\infty }{\psi }^{\prime \prime }\left(e_{\infty }\right).$

Although each individual contribution becomes arbitrarily small as the size of the group goes out of bound, the probability of the reform nevertheless remains bounded away from zero.

Consider instead the alternative scenario where the per capita benefit of the reform writes as $\bar{S}\left(n\right)=\frac{\Sigma }{n}$ ; in other words, each principal now gets an equal share of a fixed-size benefit of the reform. It is straightforward to adapt our previous reasoning to get that, in the limit, while $\underset{n\to +\infty }{lim}{\bar{t}}_n=0$ , we have also $e_{\infty }=0$ . In other words, there is now a strong form of free riding with no reform being implemented in the limit.

This discussion is reminiscent of some earlier findings in the mechanism design literature. There, free riding might arise due to private information on the valuations of agents who voluntarily participate to a mechanism for the provision of a public good. Whether the free-riding problem holds or not in the limit of a large economy depends on the comparison between the per capita cost of provision and the sum of the individuals’ information rent. In this respect, Reference Mailath and PostlewaiteMailath and Postlewaite (1990) and Reference HellwigHellwig (2003) reach different conclusions on the magnitude of the free-riding problem in the limit of a large economy when making different assumptions on how this per capita cost varies with the size of the population.

6.1.5 Heterogenous Principals

The striking lesson of Proposition 5 and Corollary 1 is that heterogeneity among the principals is a key factor to explain the failure of some of them to intervene and contribute. The virtual valuation of the weakest principals may be zero, and those principals end up having no influence on the political process.

Example 1 Denote $n=n_1+n_2$ where $n_i$ is the number of principals having valuations ${\bar{S}}_i(i=1,2)$ , with ${\bar{S}}_1<{\bar{S}}_2$ . Suppose also that $\psi \left(e\right)=\frac{e^2}{2\mu }$ for some positive parameter $\mu$ small enough.Footnote ¹⁶ From (6.6), the effort level is given by

$e_k=\mu \frac{(k-n_2){\bar{S}}_1+n_2{\bar{S}}_2}{1+k}$

if the number of active principals $k\ge n_2$ is such that $e_{k+1}>\mu {\bar{S}}_1>e_k$ . This gives:

$k=\left(\begin{array}{cc}n\text{if}& {\bar{S}}_2>{\bar{S}}_1\ge \frac{n_2}{n_2+1}{\bar{S}}_2,\\ n_2\text{if}& \frac{n_2}{n_2+1}{\bar{S}}_2>{\bar{S}}_1.\end{array}\right)$(6.9)

When the preferences of the two interest groups are sufficiently far apart, only those principals who value the good the most contribute. Instead, when there is less heterogeneity among groups, all principals find it worth to intervene. This result offers a striking illustration of Reference OlsonOlson (1965)’s well-known exploitation of the great by the small. Interestingly, condition (6.9) that determines whether principals with a lower valuation will participate to the political process depends on $n_2$ alone. When considering to become active, a principal with valuation ${\bar{S}}_1$ will only compare the cost of increasing the agent’s effort with his own benefit from the decision. As $n_2$ increases, the free-riding problem will become more and more severe, up to the point where all principals with a lower valuation remain inactive, even though the aggregate value for them, as a group, of an increase of the effort by the agent may become arbitrarily large.

As discussed in the introduction, this result is driven by the fact that the aggregate value of the decision increases for active principals. If we make $n_2$ vary while keeping $n_2{\bar{S}}_2\equiv {\bar{\Sigma }}_2$ constant, principals from the first group will remain inactive when ${\bar{S}}_1<\frac{1}{1+n_2}{\bar{\Sigma }}_2$ . This condition becomes more stringent when $n_2$ increases. As free riding gets stronger among high valuation principals, it becomes more difficult to rely only on their contributions, and principals with a lower valuation are more likely to enter the game of influence.

Using ${\bar{\Sigma }}_2$ as the measure of the aggregate valuation of the reform for high valuation principals, and $n_2$ as a measure of the size of this group ceteris paribus, in order to better separate the different effects, we can summarize this discussion:

Corollary 2 An interest group with a low valuation is less likely to contribute when it faces another group with a higher aggregate valuation, ${\bar{\Sigma }}_2$ , and a smaller size, $n_2$ .

Putting Corollaries 1 and 2 altogether suggests that the form that free riding among interest groups takes, that is whether it is by means of weak contributions or even no participation at all, indeed depends on the surrounding environment and the force that existing competing groups exert also on the political process. It is not enough to specify exogenously a pressure function which depends on group size as Reference BeckerBecker (1983, Reference Becker1985) repeatedly did. This function should also depend on the composition of other groups as well.Footnote ¹⁷

Redistributing benefits across contributors making them more symmetric while keeping the total benefit constant might thus have a nonneutral impact on whether the reform is implemented or not. Indeed, political participation by more principals is more likely as those principals are more symmetric. However, as more principals get involved, free riding is also exacerbated and the probability of a reform decreases. This tension is illustrated in the following example.

Example 2 Take $n=2$ and suppose also that, as before, $\psi \left(e\right)=\frac{e^2}{2\mu }$ . Consider first a setting with principal 1 having valuation $\bar{S}$ and principal 2 having zero valuation for the reform. The equilibrium level of effort is such that

$e^D=\frac{\mu \bar{S}}{2}.$

Consider now a redistribution of the benefits of the reform toward a symmetric setting where both principals get $\frac{\bar{S}}{2}$ out of the reform. The equilibrium level of effort is now such that

$e^D=\frac{\mu \bar{S}}{3}.$

The redistribution, although it increases political participation since now both principals intervene, decreases the likelihood of the reform. These results are in line with the Olson-Stigler’s proposition that states that collective action will be more successful as benefits are more heterogeneous among actors with congruent preferences.

This result should of course be contrasted with the well-known Neutrality Theorem in the public economics literature. This theorem establishes that the voluntary provision of public good is independent of the wealth distribution among contributors – a result found in Reference WarrWarr (1983), Reference Bergstrom, Blume and VarianBergstrom, Blume, and Varian (1986), and Reference BernheimBernheim (1986) among others. The second of these papers nevertheless argues that the neutrality result holds only if the set of contributors does not change as a result of the wealth redistribution among contributors. This is precisely a similar failure of neutrality which may happen here as well when some groups fail to intervene because of the moral hazard frictions.Footnote ¹⁸

6.2 Distribution of the Equilibrium Surplus

We conclude this section with some results on the distribution of the equilibrium surplus.

The fact that active principals all get the same payoffs is already an important feature of the intrinsic common agency scenario.Footnote ¹⁹ It carries over to delegated common agency. This feature captures the fact that those active principals end up having aligned objectives. In other words, the Principle of Aggregate Concurrence applies for this subset. For those active principals, any ex ante redistribution of the benefits of the reform by means of lump-sum payments has also no impact on their equilibrium payoff. This payoff only depends on the equilibrium effort and this variable remains unchanged through such redistribution. Instead, the equilibrium payoffs of inactive principals depend on their valuation and such ex ante redistribution makes winners and losers.