# Mechanism Design by a Politician††thanks: I thank Laura Doval, Fikri
Pitsuwan, Marek Pycia, Florian Scheuer, Yves Sprumont, Julian Teichgräber.

Giovanni Valvassori Bolgè
Center of Economic Research
at ETH Zürich
Leonhardstrasse 21
8092 Zürich
Switzerland
<EMAIL_ADDRESS>
A set of agents has to make a decision about the provision of a public good
and its financing. Agents have heterogeneous values for the public good and
each agent’s value is private information. An agenda-setter has the right to
make a proposal about a public-good level and a vector of contributions. For
the proposal to be approved, only the favourable votes of a subset of agents
are needed. If the proposal is not approved, a type-dependent outside option
is implemented. I characterize the optimal public-good provision and the
coalition-formation for any outside option in dominant strategies. Optimal
public-good provision might be a non-monotonic function of the outside option
public-good level. Moreover, the optimal coalition might be a non-convex set
of types.

Keywords: legislative coalitions, public goods, mechanism design, reservation
utility

JEL Classification: D02, D82, H41

##  1 Introduction

This paper revisits a classic mechanism design problem, the provision of a
public good, by considering a political economy environment. To this end, I
aim at unifying two different views on political economy.

From a macroeconomics perspective, political economy developed at the
intersection of Economics and Political Science as an attempt to incorporate
political dynamics into the analysis of economic policy. In the words of
Alesina and Perotti, (1994):

> “Political-economy models begin with the assertion that economic policy
> choices are not made by social planners, who live only in academic papers.
> Rather, economic policy is the result of political struggle within an
> institutional structure.”

On the other hand, drawing on the developments in the theory of
screening111For a recent and thorough overview, see Börgers et al., (2015).,
an alternative view of political economy has been put forth by the work of
Jean-Jacques Laffont and coauthors.222For an extensive treatment, see Laffont,
(2001). The Constitution-politician agency problem is reminiscent of the
principal-agent problem in the theory of contracts. As it was made clear in
Laffont, (2000),

> “[p]olitical economy is the recognition of the incompleteness of the
> Constitution […]. Because the Constitution is an incomplete contract,
> politicians have a lot of discretion, that they use to further interest
> groups."

In particular, the incompleteness of the Constitution entails the absence of
adequate institutional tools to cope with asymmetric information. In fact,
according to Laffont, (2001):

> “… a major inefficiency of political conflicts follows from the inefficiency
> of redistributive instruments due to asymmetric information. […] Politics is
> then a game of redistribution of information rents within the realm of
> discretion left by the Constitution."

In this paper, I consider a public-good provision model with privately-
informed, heterogeneous agents where the mechanism designer is one of the
agents. In particular, this agent, whom I refer to as the ‘politician’, is the
agenda-setter in a collective choice problem with three main features.

First, the collective choice entails the provision of a nonrival,
nonexcludable public good and its financing. Second, the contributions borne
by different agents are allowed to be different. Third, in order for the
proposal to be implemented, the agenda-setter only needs the support of a
subset of the agents (e.g., simple or supermajority).

If the proposal is not approved, a type-dependent outside option is
implemented. Drawing from insights in the theory of screening, I characterize
optimal public-good provision and coalition formation for any outside option.
I focus on dominant-strategy implementation.

The main idea is a fundamental connection between monopolistic screening and
agenda-setting. When there is a single agent, the two problems are identical.
Like a monopolist, the agenda-setter has to elicit from the agent his/her
value for the public-good and offer him/her a contract involving a quantity
(in this case, a public-good level) in exchange for a transfer. When there are
multiple agents, the two problems are still mathematically equivalent ex ante.
However, the ex post constraints are fundamentally different.

In fact, whereas in monopolistic screening the quantity of the good traded is
consumed privately by each agent, the public good is nonrival and
nonexcludable; hence, once a public-good level has been chosen, all agents
will consume it, regardless of the individual contributions to its financing.

Building on key contributions in monopolistic screening with type-dependent
outside options (Lewis and Sappington, (1989), Maggi and Rodriguez-Clare A.,
(1995), Jullien, (2000)), I show that the shape of the reservation utility
profile, namely the utility profile yielded by the outside option, crucially
determines the optimal contract, and hence public-good provision and coalition
formation.

Intuitively, for a given outside option, once a public-good level has been
provided, some agents will have the ex post incentive to understate their
preference for the public good, whereas others will have the ex post incentive
to overstate it. If the agenda-setter can provide a public-good level which is
consistent with all these constraints, then the public-good provision will
differ from the outside option. Otherwise, the outside option will be
implemented.

When the agenda-setter only needs a subset of favourable votes for the
proposal to be implemented, some agents can be excluded from the winning
coalition and therefore their participation constraints can be violated. If
the incentive compatibility constraints are required to hold only for the
members of the coalition, then it is optimal to randomly choose the agents and
propose the mechanism to $q-1$ of them, while taxing the others to some upper
bound $\bar{\tau}$.

If the incentive compatibility constraints have to hold for all agents,
including those left out of the winning coalition, then also this set of
excluded agents will affect the optimal public-good provision, since those
agents still have to be given the incentive to truthfully reveal their value
for the public good.

Interestingly, the winning coalition varies significantly with the outside
option and the shape of the reservation utility profile. When the outside
option public-good level is sufficiently low (high), it is optimal to choose
the agents with the ‘highest’ (‘lowest’) value for the public good, regardless
of the shape of the reservation utility profile.

On the other hand, for intermediate values of the outside option public-good
level, the winning coalition crucially depends on the concavity or convexity
of the reservation utility profile. When it is concave, the winning coalition
can be a non-convex set of agents, comprising agents with high and low values
for the public good, while excluding other agents with intermediate values.
When the reservation utility profile is convex, the winning coalition is a
convex set of agents, excluding either the agents with the highest, or lowest,
value for the public good.

Optimal public-good provision is indeed the result of a ‘game of
redistribution of information rents’, as pointed out by Laffont, (2001).
The rest of the paper is structured as follows. Section 2 reviews the
literature. Section 3 introduces the model. Section 4 and Section 5 lay out
the analysis. Section 6 foreshadows the main results, which are then presented
in Section 7. Section 8 characterizes the implications of the main results.
Section 9 discusses the relaxations of the assumptions and Section 10
concludes. All proofs are in Appendix A.

##  2 Related Literature

This paper contributes to the literature on public-good provision in political
economy and mechanism design. In political economy, an established literature
develops a positive theory of public-good provision in a legislative
bargaining framework, as in Baron and Ferejohn, (1989), Battaglini and Coate,
(2007), Battaglini and Coate, (2008) and Bowen et al., (2014). These
contributions study the bargaining between heterogeneous players in a
complete-information setting. In contrast to them, I consider a setting in
which agents are privately informed about their valuations for the public
good.

Within mechanism design, the contribution of this paper is twofold. First,
following the pioneering contribution of Green and Laffont, (1977), a branch
of the literature has studied the design of optimal mechanisms for the
provision of a public good in dominant strategies.333For a comprehensive
treatment, see Green and Laffont, (1979). For recent contributions, see
Kuzmics and Steg, (2017) and the references therein. The optimality of the
mechanisms in these models hinges on the standpoint of a benevolent social
planner seeking to maximize aggregate utilitarian welfare. Moreover, these
models concern the provision of a fixed-size public project.

In this paper, the mechanism designer is one of the agents; hence, s/he is
privately informed and seeks to maximize his/her utility, subject to the
institutional constraints. In addition to this, the size of the project (the
public-good level) is endogenous to the coalition-formation problem, and not
fixed a priori.

There is also a relatively small literature on monopolistic screening with
agents having type-dependent outside options (Lewis and Sappington, (1989),
Baron and Myerson, (1982), Maggi and Rodriguez-Clare A., (1995), Jullien,
(2000), Figueroa and Skreta, (2009)). These models consider the optimal
contract of a monopolist to a privately-informed agent who has an outside
option, which also depends on his/her type.

As an interesting application of the same framework, Teichgräber et al.,
(2022) consider the optimal design of short-time work schemes when firms have
private information about their productivity and type-dependent outside
options.

The present paper adopts a similar viewpoint, whereby the agenda-setter needs
to propose a mechanism to the agents in order to maximize his/her utility.
However, the mechanism designer is providing the agents with a nonrival, non-
excludable public good, and not with a private good. Therefore, the ex post
constraints differ from the ones in those models.

At the intersection of political economy and mechanism design, Bierbrauer and
Hellwig, (2016) consider coalition-incentive compatible mechanisms and show
that they are voting mechanisms. Scheuer and Wolitzky, (2016) study the
optimal dynamic capital tax policy for a government with limited commitment
power due to political coalition formation. In order for a policy to be
credible, the government needs the support of a large enough share of the
electorate, hence leading to an endogenous coalition formation. To the best of
my knowledge, this is the first paper studying a coalition-formation problem
in a mechanism design setting.

The closest contributions to the present paper are Gersbach et al., (2023) and
Dragu and Laver, (2017).

The former studies the coalition-formation problem of an agenda-setter in an
environment where agents have heterogeneous preferences for the public good
and their utilities are logarithmic. Importantly, however, the authors
consider a complete information setting. Instead the focus of the present
paper is on a private information setting; moreover, a more general utility
function is considered. A thorough comparison between the two contributions is
deferred to Section 9.6.

Dragu and Laver, (2017) adopts a mechanism design approach to study which
legislative majority coalitions will be formed in an environment where
politicians are privately informed about their policy preferences. It is shown
that the equilibrium policy coalitions will be either center left or center
right, depending on whether the median party has a policy position to the left
or right of the outside option.

The present paper departs from this analysis in two fundamental regards.
First, the agenda-setter is one of the agents and s/he is able to control the
resource constraint; i.e., in addition to a public-good level (or a specific
policy), s/he can propose a vector of net contributions which can differ
across agents, reflecting the information rents that have to be paid for them
to report their type truthfully.

Second, whereas Dragu and Laver, (2017) normalize the outside option to zero,
I characterize the solution for any outside option. I show that, depending on
the shape of the reservation utility profile and the outside option public-
good level, very different coalitions can arise in equilibrium.

##  3 Model

There is a set $N=\\{1,\dots,n\\}$ of agents, with $n\geq 2$. They have to
make a collective choice about the provision of a nonrival, nonexcludable
public good and its financing. Agents are heterogeneous with respect to their
value for the public good. For each agent $i\in N$, the value for the public
good (or the ‘type’) $\theta_{i}$ is private information. In particular,
$\theta_{i}\in\Theta_{i}\equiv[\underline{\theta},\overline{\theta}]$ has a
cumulative distribution function $F_{i}$ with strictly positive density
$f_{i}(\theta_{i})>0$. Denote by $\theta$ the generic vector
$(\theta_{1},\theta_{2},...,\theta_{n})$, where
$\theta\in\Theta\equiv[\underline{\theta},\overline{\theta}]^{N}$. Moreover,
denote $f(\theta)=\prod_{i\in N}f_{i}(\theta_{i})$.
The agenda-setter, $a\in N$, makes a proposal consisting of a public-good
level and a vector of contributions. Formally, the proposal is a $n+1$-tuple
$(g,t_{i})_{i\in N}$, where $g\in\mathbb{R}_{+}$ is the public-good level and
$t_{i}\in\mathbb{R}$ is the contribution of agent $i$. The cost of producing
the public good is linear: $c(g)=g$. The proposal has to satisfy a resource
constraint

$g\leq\sum_{i\in N}t_{i}\ .\\\ $

In other words, the total contributions by the agents need to be at least as
high as the cost of providing the public good. Given a policy proposal, agent
$i$’s utility is given by

$u_{i}(\theta_{i},g,t_{i})=\theta_{i}\phi(g)-t_{i}$

with $\phi:\mathbb{R}_{+}\longrightarrow\mathbb{R}_{+}$ being a twice-
differentiable, non-decreasing and concave function. Moreover, it satisfies
$\phi(0)=0$.
The agenda-setter needs the support of $q\leq n$ agents to have his/her policy
implemented. If the proposal is not approved, then an outside option public-
good level is implemented. This yields utility
$\bar{v}_{i}(\theta_{i},g^{\circ})$, where $\bar{v}_{i}$ is non-decreasing in
$g^{\circ}$, twice-differentiable and it satisfies $\bar{v}_{i}(\cdot,0)=0$.
Importantly, $\bar{v}_{i}(\theta_{i},g^{\circ})$ is type-dependent. Moreover,
the utility levels from the agenda-setter’s proposal may differ from those
under the outside option. This generality is meant to capture some potential
irreversibilities entrenched in the institutional framework.

For example, a set of local governments might want to provide a nation-wide
public-good provision. In this case, the provision of the public good at the
national level generates utility levels which are different under the outside
option. This might be due to the presence of fixed costs or the impossibility
of providing the same public-good level at the same cost with the local
technology.444Caveat: different technologies might imply different utility
function $\bar{v}_{i}$.

Alternatively, the outside option might be linked to the marginal benefit of
the policy proposal in a nonlinear way. For example, some agents might benefit
from the policy proposal the most because they would be the worst-off under
the outside option.

Therefore, no restriction is placed on the shape of the reservation utility
profile.

##  4 First-Best Analysis

What is the best the agenda-setter can do? By the Revelation Principle
(Myerson, (1979)), we can restrict our attention to direct mechanisms without
loss of generality.

######  Definition 1.

A direct mechanism is a set of functions $g$ and $t_{i}$, $\forall i\in N$

$\displaystyle g$ $\displaystyle:\Theta\longrightarrow\mathbb{R}_{+},$
$\displaystyle t_{i}$ $\displaystyle:\Theta\longrightarrow\mathbb{R}\ .$

###  4.1 VCG Mechanisms

We now consider the class of VCG mechanisms and see whether they can implement
the ‘efficient’ (utilitarian first-best) public-good level.555The discussion
in this section follows Börgers et al., (2015). Is this allocation
implementable? We first need two definitions.

######  Definition 2.

The public-good level $g$ is efficient if it solves

$\sum_{N}\theta_{i}\phi^{\prime}(g)=1\ .$ (1)

######  Definition 3.

A direct mechanism $(g,t_{i})_{i\in N}$ is called a ‘Vickrey-Clarke-
Groves’(VCG) mechanism if $g$ is efficient and if for every $i\in N$ there is
a function

$\tau_{i}:\Theta_{i}\longrightarrow\mathbb{R}$

such that

$t_{i}(\theta)=-\sum_{j\neq i}u_{j}(\theta,g(\theta))+\tau_{i}(\theta_{-i})\
.$

######  Proposition 1 (Krishna and Maenner, (2001)).

Suppose that for every $i\in N$ the set $\Theta_{i}$ is a convex subset of a
finite-dimensional Euclidean space. Moreover, assume that for every $i\in N$
the function $u_{i}(\theta_{i},g(\theta))$ is a convex function of
$\theta_{i}$. Suppose that $(g,t_{i})_{i\in N}$ is a dominant strategy
incentive-compatible mechanism. Then, a direct mechanism with the same
$(g(\theta),t^{\prime}_{i})_{i\in N}$ is dominant strategy incentive-
compatible if and only if for every $i\in N$ and every $\theta_{-i}$ there is
a number $\tau_{i}(\theta_{-i})\in\mathbb{R}$ such that

$t^{\prime}_{i}(\theta)=t_{i}(\theta)+\tau_{i}(\theta_{-i})\quad\forall\theta\in\Theta\
.$

######  Corollary 1 (Börgers et al., (2015)).

Suppose that for every $i\in N$ the set $\Theta_{i}$ is a convex subset of a
finite-dimensional Euclidean space. Moreover, assume that for every $i\in N$
the function $u_{i}(\theta,g(\theta))$ is a convex function of $\theta_{i}$.
Suppose that $(g,t_{i})_{i\in N}$ is a dominant strategy incentive-compatible
mechanism, and suppose that $g$ is efficient. Then, $(g,t_{i})_{i\in N}$ is a
VCG mechanism.

In our setting, the two convexity conditions are met. Hence, VCG mechanisms
are the only mechanisms that make efficient public-good level decision rules
dominant strategy incentive-compatible.
However, we have the following

######  Theorem 1.

No VCG mechanism is implementable.

Proof. For simplicity, assume $\phi(g)=\ln(1+g).$ Consider the public good
$g^{*}$ and the transfers $t^{*}_{i}(\theta,g^{*})_{i\in N\setminus\\{a\\}}$
of a VCG mechanism. By standard arguments (Green and Laffont, (1979)),
transfers do not cover the cost of the public good and external funds are
needed to balance the budget. Now, pick the agent with the lowest type, and
denote it $\theta_{j}$. Consider agent $\theta_{j}$’s variation in utility
brought about by a slight decrease in $g^{*}$ offset by a decrease in
$t^{*}_{j}$. Formally

$\displaystyle\theta_{j}\ln(1+g^{*}-\delta)-\theta_{j}\ln(1+g^{*})$
$\displaystyle=\underbrace{t^{*}_{j}(\theta,g^{*}-\delta)-t^{*}_{j}(\theta,g^{*})}_{\epsilon}$
$\displaystyle\theta_{j}\ln(\sum_{i\in
N}\theta_{i}-\delta)-\theta_{j}\ln(\sum_{i\in N}\theta_{i})$
$\displaystyle=\epsilon$ $\displaystyle\delta$ $\displaystyle=\sum_{i\in
N}\theta_{i}(e^{\frac{\epsilon}{\theta_{j}}}-1)$

By construction, agent $j$’s utility is unchanged. All other agents
$\theta_{k}$, with $k\in N\setminus\\{a,j\\}$, have a utility loss equal to
$\epsilon-\frac{\theta_{k}}{\theta_{j}}\epsilon$. Therefore, for the transfers
$t^{*}_{i}(\theta,g^{*}-\delta)_{i\in N\setminus\\{a\\}}$ to be still
incentive-compatible, the proposer needs to compensate them. Now, consider the
variation in the proposer’s utility.

$\displaystyle\theta_{a}\ln(1+g^{*}-\delta)-\theta_{a}\ln(1+g^{*})-\sum_{i\in
N\setminus\\{a\\}}\left(\frac{\theta_{i}}{\theta_{j}}-1\right)\epsilon-[(g^{*}-\delta)-g^{*}]$
$\displaystyle=\sum_{i\in
N}\theta_{i}(e^{\frac{\epsilon}{\theta_{j}}}-1)-\sum_{i\in
N}\frac{\theta_{i}}{\theta_{j}}\epsilon$ $\displaystyle=O(\epsilon)>0\ ,$

where the last equality follows from the Taylor expansion of $e^{x}$ around
$x=0$. This argument generalizes to arbitrary functions $\phi(g)$. Hence,
$g^{*}$ is not implementable. ∎
Theorem 1 has a simple interpretation. Suppose that the efficient public-good
level is known and it is set in the Constitution. Moreover, suppose that there
is availability of external funds for the set of agents to implement the
efficient public-good provision. Even in that case, it would not be possible
to find a mechanism to implement that public-good level in dominant
strategies.

##  5 Second-best Mechanisms

We now consider second-best mechanisms. Formally, there is the following
sequence of events.

  1. 1.

An agenda-setter $a\in N$ has the right to make a proposal.

  2. 2.

Each agent $i\in N$ is privately informed about his/her own value for the
public-good $\theta_{i}$.

  3. 3.

The agenda-setter asks each agent $i\in N$ to reveal his/her value for the
public good and simultaneously makes a take-it-or-leave-it policy proposal
consisting of a public-good level $g(\theta)$ and a vector of transfers to
finance it, $t_{i}(\theta)$.

  4. 4.

The voting stage occurs secretly. If at least $q\leq n$ agents vote in favor
of the policy proposal, then it is implemented. If not, an outside option
public-good level is implemented, yielding utilities
$\bar{v}_{i}(\theta_{i},g^{\circ})$ to each agent $i\in N$.

Denote

$\mathcal{Q}\equiv\left\\{Q\subseteq N:a\in Q\;\text{and}\;|Q|=q\right\\}$

as the set of all minimal winning coalitions that include $a$. The agenda-
setter solves the following problem

$\begin{gathered}\max_{g(\theta),t(\theta),Q}\qquad\int\theta_{a}\phi(g(\theta))-t_{a}(\theta)\
dF(\theta)\\\ \text{subject to}\qquad\;g(\theta)\leq t_{a}(\theta)+\sum_{i\in
N\setminus\\{a\\}}t_{i}(\theta),\\\ \text{and for some $Q\in\mathcal{Q}$},\\\
\theta_{i}\phi(g(\theta))-t_{i}(\theta)\geq\theta_{i}\phi(g(\theta^{\prime}_{i},\theta_{-i}))-t_{i}(\theta^{\prime}_{i},\theta_{-i})\qquad\forall\theta_{i},\theta^{\prime}_{i}\in\Theta_{i},\theta_{-i}\in\Theta_{-i}\
\text{and}\ i\in N\setminus\\{a\\},\\\
\theta_{j}\phi(g(\theta))-t_{j}(\theta)\geq\bar{v}_{j}(\theta_{j},g^{\circ})\qquad\forall\;j\in
Q.\end{gathered}$

The first constraint is the resource constraint: the net contributions of the
agents have to be enough to cover the public-good expenditure.
The second is the incentive compatibility constraint: every agent $i\in
N\setminus\\{a\\}$ needs to have an incentive to reveal his/her true type
truthfully to the agenda-setter.
The last constraint is the participation constraint: the utility yielded by
the proposed policy to the agents in the coalition needs to be at least as
high as the one yielded by the outside option.

The participation constraint needs to be satisfied only for those $q-1$ agents
included in the coalition; the remaining agents can be forced to participate.
However, they have to be given the incentive to reveal their type truthfully;
hence, the incentive compatibility constraint has to be satisfied for all
agents.

This is consistent with alternative interpretations of the model. First, there
might be political economy constraints preventing the agenda-setter to extract
resources from agents who do not participate in the mechanisms.666By this, I
do not mean the agents who participate but whose participation constraints are
violated. Rather, they are agents who are not even asked by the agenda-setter
to participate in the mechanism; e.g., this would be the case if the agenda-
setter chose randomly $q-1$ agents and proposed them a direct mechanism.
Second, the agenda-setter might be thought of as a government placing any set
of welfare weights to the agents. For example, an egalitarian government might
place a unit weight on the agent(s) with the lowest type. In order to identify
the type(s) of interest, the types of all the agents have to be
elicited.777However, when the agenda-setter is indeed one of the agents
seeking to maximize his/her expected utility, it might be reasonable to assume
that the incentive compatibility constraint has to hold only for the members
of the winning coalition. In Section 9, it is shown that the agents in the
winning coalition $Q$ will be chosen randomly.
To begin with, a few assumptions are in order. They will be relaxed in Section
9.

######  Assumption 1.

The agenda-setter has full commitment power.

This assumption rules out the possibility that the agenda-setter might modify
the allocation rule after it has been announced and the preferences of the
other agents have been revealed.

######  Assumption 2.

The density functions $f_{i}(\theta_{i})$, for every $i\in N$, are log-
concave.

This assumption is standard in the literature and it ensures that the inverse
of the hazard rate is increasing in types.

######  Assumption 3.

The reservation utility profile is implementable.

This assumption is referred to as homogeneity in Jullien, (2000). It requires
that the reservation utility profile be implementable without excluding any
agent. It implies that $\bar{v_{i}}$ is non-decreasing in types.
Standard results in mechanism design theory (Maggi and Rodriguez-Clare A.,
(1995)) yield the following lemma.

######  Lemma 1.

A direct mechanism $(g,t_{i})_{i\in N}$ is dominant-strategy incentive
compatible if and only if for every $\theta_{i}\in\Theta_{i}$ and
$\theta_{-i}\in\Theta_{-i}$:

  * (i)

$g(\theta_{i},\theta_{-i})$ is increasing in $\theta_{i}$;

  * (ii)

For every $\theta_{i}\in[\underline{\theta},\overline{\theta}]$ we have

$dU_{i}/d\theta_{i}=\phi(g(\theta))-\bar{v}_{i}^{\prime}(\theta_{i},g^{\circ})\
,$ (2)

where
$U_{i}(\theta_{i})=\theta_{i}\phi(g(\theta))-\bar{v}_{i}(\theta_{i},g^{\circ})$.

The first condition is the standard monotonicity constraint, a well-known
necessary condition for implementability of the allocation rule.

Condition $(\ref{eq:envelope})$ is an envelope condition which shows the two
‘countervailing incentives’888To the best of my knowledge, the term was coined
by Lewis and Sappington, (1989) and later used by Maggi and Rodriguez-Clare
A., (1995). at play. On the one hand, agents have the incentive to understate
their value for the public-good: since it is nonrival and nonexcludable, all
agents will benefit from its consumption, independently of who contributes to
its financing.

On the other hand, agents have the incentive to overstate their types so as to
overstate their reservation utility, the utility yielded by the outside
option. Depending on which effects dominates, agents will have the overall
incentive to understate or overstate their value for the public good.

Condition $(\ref{eq:envelope})$ implies that, over the set of types for whom
the dominating incentive is to understate, information rents will be
increasing in types. Analogously, when the dominating incentive is to
overstate the types, information rents will be decreasing over this set of
types.

This implication of dominant-strategy incentive compatibility will play a
major role in the subsequent analysis.

##  6 Linear Reservation Utility Profile

In order to foreshadow the main results, we consider a particular functional
form for the reservation utility profiles, namely the case in which
$\bar{v}_{i}(\cdot)$ is linear in types:

$\bar{v}_{i}(\theta_{i},g^{\circ})=\theta_{i}\phi(g^{\circ})-\frac{g^{\circ}}{n},\qquad\forall
i\in N.$

The outside option is given by a outside option public-good level $g^{\circ}$,
and a uniform array of transfers to finance it.999This is a common assumption
in the political economy literature; see e.g., Battaglini and Coate, (2008)
and Gersbach et al., (2023). Importantly, by condition (2), the incentive to
understate or overstate the type is given by

$dU_{i}/d\theta_{i}=\phi(g(\theta))-\phi(g^{\circ})\ .$ (3)

Hence, the sign of (3) is type-independent. The analysis is thus greatly
simplified.

###  6.1 Unanimity

Suppose first that the institutional constraints require that the proposal be
approved by unanimity; in other words, $q=n$. The agenda-setter solves the
following problem

$\begin{gathered}\max_{g(\theta),\mathbb{t}(\theta)}\qquad\int\theta_{a}\phi(g(\theta))-t_{a}(\theta)\
dF(\theta)\\\ \text{subject to}\qquad\;g(\theta)\leq t_{a}(\theta)+\sum_{i\in
N\setminus\\{a\\}}t_{i}(\theta),\\\ \frac{dg(\theta)}{d\theta}\geq 0,\\\
dU_{i}/d\theta_{i}=\phi(g(\theta))-\phi(g^{\circ}),\\\
\theta_{i}\phi(g(\theta))-t_{i}(\theta)\geq\theta_{i}\phi(g^{\circ})-\frac{g^{\circ}}{n}\qquad\forall\;i\in
N\setminus\\{a\\}.\end{gathered}$ ($\mathcal{LU}$)

Denote $g^{*}(\theta)$ as the solution to $\mathcal{LU}$.

######  Lemma 2.

If $g^{*}(\theta)\neq g^{\circ}$, then the participation constraint can bind
at most at one type.

Proof. Suppose not. Then, there exist two agents, $\theta_{k}$ and
$\theta_{l}$, with $\theta_{k}>\theta_{l}$, whose participation constraints
are binding. However, by condition (3) the incentive to understate or
overstate the type is type-independent.

But then, if $g^{*}(\theta)>g^{\circ}$, all agents have ex post the incentive
to understate their type. Analogously, if $g^{*}(\theta)<g^{\circ}$, all
agents have ex post the incentive to overstate their type. Therefore, in the
first case $\theta_{k}$ could pay a lower contribution by reporting
$\theta_{l}$; in the second case, $\theta_{l}$ could pay a lower contribution
by reporting $\theta_{k}$. This contradicts the optimality of the
participation constraints of both agents being binding. ∎
Then, from Lemma 1 and Lemma 2 we immediately have the following result.

######  Proposition 2.

There exist two threshold values, $g_{U}^{L}$ and $g_{U}^{H}$, such that:

$g^{*}(\theta)=\begin{cases}g_{U}^{L}&\mbox{if $g^{\circ}<g_{U}^{L}$,}\\\
g^{\circ}&\mbox{if $g_{U}^{L}\leq g^{\circ}\leq g_{U}^{H}$,}\\\
g_{U}^{H}&\mbox{if $g^{\circ}>g_{U}^{H}$.}\end{cases}$

where $g_{U}^{L}$ and $g_{U}^{H}$ solve, respectively,

$\displaystyle\left[\theta_{a}+\sum_{i\in
N\setminus\\{a\\}}\left(\theta_{i}-\frac{1-F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\right]\phi^{\prime}(g(\theta))$
$\displaystyle=1$ $\displaystyle\left[\theta_{a}+\sum_{i\in
N\setminus\\{a\\}}\left(\theta_{i}+\frac{F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\right]\phi^{\prime}(g(\theta))$
$\displaystyle=1\ .$

Figure 1 illustrates the solution.

$g^{\circ}$$g^{*}(\theta)$$g_{U}^{L}$$g_{U}^{L}$$g_{U}^{H}$$g_{U}^{H}$$g^{*}(\theta)=g^{\circ}$
Figure 1: $g^{*}(\theta)$ in the linear case with unanimity.

The intuition behind the result is very simple. The incentive to understate or
overstate the type is ex post type-independent: once a public-good level
$g^{*}(\theta)$ has been provided, all agents will have the ex post incentive
to either understate or overstate their type, depending on the sign of (3).

If the outside option public-good level is sufficiently low $(g^{\circ}\leq
g_{U}^{L})$, all agents will have the incentive to understate their type. If
the outside option public-good level is sufficiently high $(g^{\circ}\geq
g_{U}^{H})$, all agents will have the incentive to overstate their type.

Finally, if the outside option public-good level assumes intermediate values
$(g_{U}^{L}\leq g^{\circ}\leq g_{U}^{H})$, then some agents will have the
incentive to understate, whereas some other agents will have the incentive to
overstate. The two conflicting incentives cannot be consistent ex post  with
(3). Hence, the agenda-setter can only propose the outside-option public-good
level.

###  6.2 Majority

Suppose now that $q\leq n$. The agenda-setter solves

$\begin{gathered}\max_{g(\theta),\mathbb{t}(\theta),Q}\qquad\int\theta_{a}\phi(g(\theta))-t_{a}(\theta)\
dF(\theta)\\\ \text{subject to}\qquad\;g(\theta)\leq t_{a}(\theta)+\sum_{i\in
N\setminus\\{a\\}}t_{i}(\theta),\\\ \frac{dg(\theta)}{d\theta}\geq 0,\\\
dU_{i}/d\theta_{i}=\phi(g(\theta))-\bar{v}_{i}^{\prime}(\theta_{i},g^{\circ})\qquad\forall\;i\in
N\setminus\\{a\\},\\\ \text{and for some $Q\in\mathcal{Q}$},\\\
\theta_{j}\phi(g(\theta))-t_{j}(\theta)\geq\theta_{j}\phi(g^{\circ})-\frac{g^{\circ}}{n}\qquad\forall\;j\in
Q.\end{gathered}$ ($\mathcal{LM}$)

The first three constraints are the same as in the unanimity problem. However,
in addition to $g(\theta)$ and $\mathbb{t}(\theta)$, the agenda-setter now
also needs to choose a coalition $Q\in\mathcal{Q}$.
For any $Q\in\mathcal{Q}$, let $\Xi(Q)=\sum_{i\in Q}\theta_{i}$ be $Q$’s value
for the public good. Define

$\underline{Q}\equiv\operatorname*{arg\,min}_{Q\in\mathcal{Q}}\Xi(Q)\quad\text{and}\quad\overline{Q}\equiv\operatorname*{arg\,max}_{Q\in\mathcal{Q}}\Xi(Q),$

to be, respectively, the ‘lowest’ and ‘highest’ coalitions as measured by
their value for the public good. Along the same lines as in the unanimity
case, we obtain the following

######  Proposition 3.

There exist a set of agents $I$, with $|I|\leq n-q$, two cutoff types,
$\tilde{\theta}$,
$\tilde{\tilde{\theta}}\in[\underline{\theta},\overline{\theta}]$ and two
threshold public-good values, $g_{M}^{L}$, $g_{M}^{H}\in\mathbb{R}_{+}$, such
that, if $g_{M}^{L}<g_{M}^{H}$,

$g^{*}(\theta)=\begin{cases}g_{M}^{L}&\mbox{if $g^{\circ}<g_{M}^{L}$,}\\\
g^{\circ}&\mbox{if $g_{M}^{L}\leq g^{\circ}\leq g_{M}^{H}$,}\\\
g_{M}^{H}&\mbox{if $g^{\circ}>g_{M}^{H}$,}\end{cases}$

where $g_{M}^{L}$ and $g_{M}^{H}$ solve, respectively,

$\displaystyle\left[\theta_{a}+|I|\cdot\tilde{\theta}+\sum_{i\in\overline{Q}\setminus\\{a\\}}\left(\theta_{i}-\frac{1-F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\right]\phi^{\prime}(g(\theta))$
$\displaystyle=1\ ,$
$\displaystyle\left[\theta_{a}+|I|\cdot\tilde{\tilde{\theta}}+\sum_{i\in\underline{Q}\setminus\\{a\\}}\left(\theta_{i}+\frac{F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\right]\phi^{\prime}(g(\theta))$
$\displaystyle=1\ .$

Moreover,

$Q^{*}=\begin{cases}\overline{Q}&\mbox{if $g^{\circ}<g_{M}^{L}$,}\\\
\textrm{any}\ Q\in\mathcal{Q}&\mbox{if $g_{M}^{L}\leq g^{\circ}\leq
g_{M}^{H}$,}\\\ \underline{Q}&\mbox{if $g^{\circ}>g_{M}^{H}$.}\end{cases}$

If $g_{M}^{L}>g_{M}^{H}$ for some $q\leq n$, then

$g^{*}(\theta)=\begin{cases}g_{M}^{L}&\mbox{if $g^{\circ}<g_{M}^{L}$,}\\\
g_{M}^{H}&\mbox{if $g^{\circ}>g_{M}^{L}$.}\end{cases}$

Moreover,

$Q^{*}=\begin{cases}\overline{Q}&\mbox{if $g^{\circ}<g_{M}^{L}$,}\\\
\underline{Q}&\mbox{if $g^{\circ}>g_{M}^{L}$}.\end{cases}$

Figure 2 and Figure 3 illustrate the solution.

$g^{\circ}$$g^{*}(\theta)$$g_{M}^{L}$$g_{M}^{L}$$g_{M}^{H}$$g_{M}^{H}$$g^{*}(\theta)=g^{\circ}$
Figure 2: $g^{*}(\theta)$ in the linear case with majority
($g_{M}^{L}<g_{M}^{H}$).

$g^{\circ}$$g^{*}(\theta)$$g_{M}^{H}$$g_{M}^{H}$$g_{M}^{L}$$g_{M}^{L}$$g^{*}(\theta)=g^{\circ}$
Figure 3: $g^{*}(\theta)$ in the linear case with majority
($g_{M}^{L}>g_{M}^{H}$).

The reasoning is similar to the unanimity case. Notice, however, that the
thresholds $g_{M}^{L}$ and $g_{M}^{H}$ are different.

In fact, unlike the unanimity case, the agenda-setter can now exclude some
agents from the winning coalition; i.e., some agents can be forced to
participate in the collective choice. However, they have to be given the
incentive to truthfully report their types. Therefore, they will also affect
public-good provision. The intuition behind Proposition 3 is again related to
(3).

If $g^{*}(\theta)>g^{\circ}$, then all agents have the ex post incentive to
understate their types. Moreover, information rents have to be increasing in
types.

Intuitively, when the outside option is low, incentive constraints bind
downwards and the participation constraint binds on some type
$\tilde{\theta}$. All agents with higher types must earn information rents;
however, the types below can be bunched at the same allocation of
$\tilde{\theta}$, thus violating their participation constraints. It is then
optimal to choose $\tilde{\theta}$ in such a way that the highest $q-1$ agents
earn information rents. An analogous reasoning goes through for the case where
$g^{\circ}$ is high.

Interestingly, when for some $q$ it is the case that $g_{M}^{L}>g_{M}^{H}$,
public-good provision becomes non-monotonic in the outside option public-good
level. This is the same result obtained in Gersbach et al., (2023).
The results in this section come as no surprise. When the reservation utility
profile is linear in types, condition (3) states that the agents’ incentive to
misreport their types is type-independent. Hence, the agenda-setter is able to
implement a collective choice which is different from the outside option only
when the incentives are aligned for all agents. This is the case when the
outside option public-good level is sufficiently low or sufficiently high.

On the other hand, when the outside option public-good level assumes
intermediate values, the only collective choice which can be implemented in
dominant strategies is proposing the outside option.

The same reasoning applies when the agenda-setter only needs the support of a
subset of agents. Dominant strategy incentive compatibility severely restricts
the ability of the agenda-setter to ‘exploit’ the agents excluded from the
winning coalition.

##  7 Main Results

In this section, I consider a general reservation utility profile
$\bar{v}_{i}$. Before stating the two main results of this paper, some
preliminary definitions and results are in order.

A key insight from Jullien, (2000) is that the shadow value on the
participation constraints at the optimal allocation can be represented by
$\gamma^{*}(\theta_{i})$, which is a cumulative distribution function over the
set of types. The agenda-setter surplus can be rewritten so as to account for
the information rents brought about by incentive compatibility.

######  Lemma 3.

The agenda-setter’s objective function can be rewritten as

$\sigma(\theta,\gamma(\theta),g)\equiv\int_{\Theta}\left[\sum_{i\in
N}\theta_{i}\phi(g(\theta))-\sum_{i\in
N\setminus\\{a\\}}\left(\frac{\gamma^{*}(\theta_{i})-F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\phi(g(\theta))-g(\theta)\right]f(\theta)d\theta\
.$ (4)

Since (4) is strictly quasi-concave in the public-good level, the objective
function will have a unique maximizer.

######  Definition 4.

$\xi(\gamma(\theta),\theta)=\operatorname*{arg\,max}_{g}\
\sigma(\theta,\gamma(\theta),g).$

One last piece of notation. It is useful to define the partition of
$N\setminus\\{a\\}$ depending on whether agents have the ex post incentive to
understate or overstate their types.

######  Definition 5.

$N\setminus\\{a\\}=\ K(g^{\circ})\ \cup\ L(g^{\circ})\ \cup\ M(g^{\circ})\ ,$

where

$\displaystyle K(g^{\circ})$ $\displaystyle=\left\\{i\in N\setminus\\{a\\}\ |\
dU_{i}/d\theta_{i}<0\right\\}$ $\displaystyle L(g^{\circ})$
$\displaystyle=\left\\{i\in N\setminus\\{a\\}\ |\
dU_{i}/d\theta_{i}=0\right\\}$ $\displaystyle M(g^{\circ})$
$\displaystyle=\left\\{i\in N\setminus\\{a\\}\ |\
dU_{i}/d\theta_{i}>0\right\\}\ .$

###  7.1 Unanimity

The agenda-setter solves

$\begin{gathered}\max_{g(\theta),\mathbb{t}(\theta)}\qquad\int\theta_{a}\phi(g(\theta))-t_{a}(\theta)\
dF(\theta)\\\ \text{subject to}\qquad\;g(\theta)\leq t_{a}(\theta)+\sum_{i\in
N\setminus\\{a\\}}t_{i}(\theta),\\\ \frac{dg(\theta)}{d\theta}\geq 0,\\\
dU_{i}/d\theta_{i}=\phi(g(\theta))-\bar{v}_{i}^{\prime}(\theta_{i},g^{\circ}),\\\
U_{i}(\theta_{i},g(\theta))\geq\bar{v}_{i}(\theta_{i},g^{\circ})\qquad\forall\;i\in
N\setminus\\{a\\}.\end{gathered}$ ($\mathcal{U}$)

######  Theorem 2.

Under Assumptions (1)-(3), there exists a unique optimal solution to
$\mathcal{U}$. An implementable allocation $g^{*}(\theta)$ is optimal if and
only if there exists a cumulative distribution function
$\gamma^{*}(\theta_{i})$ such that

$\begin{cases}&(\mathcal{A})\qquad
d\gamma^{*}(\theta_{i})>0\qquad\textrm{for}\ \ \theta_{i}\in L(g^{\circ}),\\\
&(\mathcal{B})\qquad
g^{*}(\theta)=\xi(\gamma^{*}(\theta_{i}),\theta).\end{cases}$

The optimal allocation partitions the set of agents $N\setminus\\{a\\}$ into
$K(g^{\circ})$, $L(g^{\circ})$ and $M(g^{\circ})$.
Otherwise, $g^{*}(\theta)=g^{\circ}$.

Theorem 2 is closely related to Theorem 1 in Jullien, (2000). Condition
$\mathcal{A}$ identifies which participation constraints bind and, hence,
which types earn no information rents. It states that the participation
constraint can bind on a single interior type and at the extremes: the lowest
and highest types.

Condition $\mathcal{B}$ characterizes the optimal public-good provision as the
public-good level which maximizes the agenda-setter’s utility adjusted to take
into account the information rents induced by incentive compatibility.
Depending on the outside option public-good level $g^{\circ}$, some agents
will have the incentive to understate their type, whereas others will have the
incentive to overstate it. In particular, if
$\gamma^{*}(\theta_{i})>F_{i}(\theta_{i})$, the dominating incentive is to
understate the type; if $\gamma^{*}(\theta_{i})<F_{i}(\theta_{i})$, the
dominating incentive is to overstate the type.

The main idea behind Theorem 2 is very simple: if there exists a cdf
$\gamma^{*}(\theta_{i})$ such that the collective choice, the public-good
provision $g^{*}(\theta)$, is consistent with the agents’ ex post incentives
to misreport their types, then such collective choice will be implemented. If
not, the agenda-setter can do no better than proposing the outside option.

###  7.2 Majority

We consider the following problem faced by the agenda-setter

$\begin{gathered}\max_{g(\theta),\mathbb{t}(\theta),Q}\qquad\int\theta_{a}\phi(g(\theta))-t_{a}(\theta)\
dF(\theta)\\\ \text{subject to}\qquad\;g(\theta)\leq t_{a}(\theta)+\sum_{i\in
N\setminus\\{a\\}}t_{i}(\theta),\\\ \frac{dg(\theta)}{d\theta}\geq 0\\\
\text{and for some $Q\in\mathcal{Q}$},\\\
dU_{i}/d\theta_{i}=\phi(g(\theta))-\bar{v}_{i}^{\prime}(\theta_{i},g^{\circ})\qquad\forall\;i\in
N\setminus\\{a\\},\\\
U_{j}(\theta_{j},g(\theta))\geq\bar{v}_{j}(\theta_{j},g^{\circ})\qquad\forall\;j\in
Q\ .\end{gathered}$ ($\mathcal{M}$)

We now state the second main result.

######  Theorem 3.

There exists a unique optimal solution to $\mathcal{M}$. An implementable
allocation $g^{*}(\theta)$ is optimal if and only if there exists a cumulative
distribution function $\gamma^{*}(\theta_{i})$ such that

$\begin{cases}&(\mathcal{A})\qquad
d\gamma^{*}(\theta_{i})>0\qquad\textrm{for}\ \ \theta_{i}\in L(g^{\circ}),\\\
&(\mathcal{B})\qquad
g^{*}(\theta)=\xi(\gamma^{*}(\theta_{i}),\theta).\end{cases}$

The optimal allocation partitions the set of agents $N\setminus\\{a\\}$ into
$K(g^{\circ})$, $L(g^{\circ})$ and $M(g^{\circ})$. Moreover, there exists a
set of agents

$I(g^{\circ})=\left\\{i\in N\ |\ U_{i}(\theta_{i})<0\right\\}\ ,$

with $|I(g^{\circ})|\leq n-q$. Otherwise, $g^{*}(\theta)=g^{\circ}$.

The interpretation of Theorem 3 is very similar. Conditions $\mathcal{A}$ and
$\mathcal{B}$ still apply. Moreover, the solution will induce a partition of
agents into $K(g^{\circ})$, $L(g^{\circ})$ and $M(g^{\circ})$. However, the
agenda-setter needs to choose a coalition $Q$, for whose members the
participation constraints need to be satisfied. There is a set $I(g^{\circ})$
for whom the participation constraints is violated.

Notice that this type of exclusion is different from the exclusion of some
agents in Jullien, (2000). The reason is that, in the present framework,
agents can be forced to participate in the mechanism, and they will still
consume the public good once it has been provided. However, they still need to
be given the incentives to reveal their types truthfully ex post.

The idea is again very simple. $K(g^{\circ})$ and $M(g^{\circ})$ are two
convex sets.101010Hence, for any three types
$\theta_{1}<\theta_{2}<\theta_{3}$, if $\theta_{1}$ and $\theta_{3}$ are
included in either set, then also $\theta_{2}$ is included in the same set.
This is because, by condition (2), information rents have to be increasing
(decreasing) over the set of agents who have the incentive to understate
(overstate) their types. The agenda-setter has to identify a set of agents
whose participation constraints can be violated, but whose incentives to
report truthfully are still required to hold. The choice of this set crucially
depends on the shape of the reservation utility profile.

Moreover, depending on how far public-good provision is from the efficient
level, a subset of $I(g^{\circ})$ will be bunched at the allocation of the
agent(s) whose participation constraints is (are) binding.

Therefore, Theorem 2 and Theorem 3 imply that the optimal public-good
provision is either equal to - or a step function of - the outside option
public-good level.

$g^{\circ}$$g^{*}(\theta)$$g^{L}$$g^{L}$$g^{H}$$g^{H}$$g^{*}(\theta)=g^{\circ}$
Figure 4: $g^{*}(\theta)$ is a step function of $g^{\circ}$.

##  8 Implications

Theorem 2 and Theorem 3 show that optimal public-good provision crucially
depends on the shape of the reservation utility profile $\bar{v}_{i}$.

When the outside option public-good level $g^{\circ}$ is low, all agents have
the incentive to understate their types. By familiar arguments, it is then
optimal to underprovide the public good by distorting the allocations
downwards. The participation constraint binds for the lowest type; all others
earn information rents. By Condition (2), rents are increasing in types.

Analogously, when the outside option public-good level is sufficiently high,
all agents have the incentive to overstate their types. The optimal public-
good provision now entails distorting the allocation upwards. The
participation constraints binds only for the highest type. All other agents
earn information rents, which are now decreasing in types by Condition (2).

When $g^{\circ}$ assumes intermediate values, however, agents have
countervailing incentives: some agents will have the incentive to understate
their types, whereas others will have the incentive to overstate their types.
Therefore, the optimal public-good provision depends on the concavity or
convexity of the reservation utility profile.

Section 6 considered the linear case. We now consider the concave and convex
cases.

###  8.1 $\bar{v}_{i}$ Concave

######  Lemma 4.

Suppose $\bar{v}^{\prime\prime}_{i}\leq 0$. Then, for some
$k,l\in\\{0,1,\dots,r\\}$ with $k\leq l$,

$\displaystyle K(g^{\circ})$
$\displaystyle=\\{\theta_{1},\dots,\theta_{k}\\},$ $\displaystyle
L(g^{\circ})$ $\displaystyle=\\{\theta_{k+1},\dots,\theta_{l}\\},$
$\displaystyle M(g^{\circ})$
$\displaystyle=\\{\theta_{l+1},\dots,\theta_{r}\\}.$

Moreover, the indices $k$ and $l$ are increasing in $g^{\circ}$. If values are
all distinct, then $L(g^{\circ})$ has at most one agent.

Lemma 4 orders the partition of the agents, depending on whether they have the
incentive to understate or overstate their types, or they are indifferent
between the two.

When values are all distinct, there will be a single type $\tilde{\theta}$
whose participation constraint binds. When $g^{\circ}$ is very low, then
$\tilde{\theta}=\theta_{1}$. As the outside option public-good level
increases, the cutoff type $\tilde{\theta}$ will be an interior type. By Lemma
4, all types below (above) $\tilde{\theta}$ will have the incentive to
overstate (understate) their types. Figure 5 displays the information rents as
a function of the types in this case.

$\underline{\theta}$$\overline{\theta}$$\tilde{\theta}$ Figure 5:
$\bar{v}_{i}$ concave: unanimity

In the majority case, the same argument applies. However, Theorem 3 states
that a subset of agents $I$ might be bunched at the allocation of the type
whose participation constraint binds. Depending on $g^{\circ}$, the
participation constraint binds on a different type and hence also the set of
agents excluded from the coalition differs. Figure 6 displays the solution
when the cutoff type is interior.

$\underline{\theta}$$\overline{\theta}$$\tilde{\theta}$$\theta_{p}$$\theta_{q}$$Q$$I$
Figure 6: $\bar{v}_{i}$ concave with exclusion

Interestingly, the winning coalition can be a non-convex set: there can be
three types $\theta_{i}<\theta_{k}<\theta_{j}$ such that types $\theta_{i}$
and $\theta_{j}$ might be included in $Q$, whereas $\theta_{k}$ is excluded.

###  8.2 $\bar{v}_{i}$ Convex

######  Lemma 5.

Suppose $\bar{v}^{\prime\prime}_{i}\geq 0$. Then, for some
$k,l\in\\{0,1,\dots,r\\}$ with $k\leq l$,

$\displaystyle K(g^{\circ})$
$\displaystyle=\\{\theta_{r-1},\dots,\theta_{l}\\},$ $\displaystyle
L(g^{\circ})$ $\displaystyle=\\{\theta_{1},\theta_{r}\\},$ $\displaystyle
M(g^{\circ})$ $\displaystyle=\\{\theta_{k},\dots,\theta_{2}\\},$

Moreover, the indices $k$ and $l$ are decreasing in $g^{\circ}$.

Analogously to Lemma 4, Lemma 5 orders the partition of the agents. However,
the transition between sets is more involved than in the concave case. The
reason is again Condition (2).

As $g^{\circ}$ increases, the first agent whose incentive switches from
understating to overstating is the agent with the highest type. However, the
information rents have to be decreasing for the set of agents having the
incentive to overstate. The only way to preserve incentive compatibility is
then to have the participation constraint binding at both extremes of the type
space. Interestingly, there will be a single interior type where the two
countervailing incentives cross out and the agent will be just indifferent
between understating or overstating.111111See Maggi and Rodriguez-Clare A.,
(1995). This cutoff type is decreasing in $g^{\circ}$: it coincides with
$\theta_{n}$ when $g^{\circ}$ is very low and it coincides with lower types as
$g^{\circ}$ increases. Figure 7 illustrates this property.

$\underline{\theta}$$\overline{\theta}$$\tilde{\theta}$ Figure 7: Information
rents when $\bar{v}_{i}$ is convex

In the majority case, there will be a set of agents $I$ whose participation
constraint is violated. These agents might be bunched at the allocation of the
type(s) with binding participation constraint(s). Contrary to the concave
case, the winning coalition is now a convex set. Figure 8 illustrates this
property.

$\underline{\theta}$$\overline{\theta}$$\tilde{\theta}$$Q$$I$$I$ Figure 8:
$\bar{v}_{i}$ convex with exclusion

##  9 Extensions

###  9.1 Limited Commitment

Suppose that the agenda-setter cannot fully commit to reporting her true
parameter $\theta_{a}$ after collecting the preferences of the other agents.
Recall the efficient public-good level in (1) and denote it $g^{FB}$. Then, we
have the following result.

######  Proposition 4.

The optimal report $\hat{\theta}_{a}$ is given by

$\hat{\theta}_{a}=\begin{cases}0&\mbox{if $g^{*}(\theta)\geq g^{FB}$},\\\
\min\\{g^{FB}-g^{*}(\theta),\overline{\theta}\\}&\mbox{if
$g^{*}(\theta)<g^{FB}$},\end{cases}$

if $g^{*}(\theta)\neq g^{\circ}$. Otherwise, $\hat{\theta}_{a}=\theta$, for
any $\theta\in[\underline{\theta},\overline{\theta}]$.

Proof. The agenda-setter’s objective function is concave and maximized at
$g^{FB}$. Therefore, any reduction in the public-good level by reporting a
lower type has a first-order effect on the objective function only when
public-good provision is higher than the efficient level. ∎

###  9.2 Bunching

####  9.2.1 Relaxing (2)-(3)

####  9.2.2 Negatively-sloped Reservation Utility Profile

Suppose $\bar{v}_{i}^{\prime}(\theta_{i},g^{\circ})<0$, for every $i\in N$.
Then, condition (2) yields

$dU_{i}/d\theta_{i}=\phi(g(\theta))-\bar{v}_{i}^{\prime}(\theta_{i},g^{\circ})\
>0$

for all $\theta_{i}\in\Theta_{i}$. The dominating incentive is then always to
understate the type. The equilibrium public-good provision is then the same as
in Proposition 2 and Proposition 3 when $\bar{v}_{i}(\theta,g^{\circ})\equiv
0$.

###  9.3 Common Values

###  9.4 Stochastic Mechanisms

Mechanisms can be stochastic in two ways: for each agent, the inclusion in the
winning coalition $Q$ (stochastic coalitions); or the public-good provision
$g$ (stochastic public-good provision).

####  9.4.1 Stochastic Coalitions

Suppose now that the incentive compatibility constraint has to be satisfied
only for the members of the coalition $Q\in\mathcal{Q}$. We consider the
following problem faced by the agenda-setter

$\begin{gathered}\max_{g(\theta),\mathbb{t}(\theta),Q}\qquad\int\theta_{a}\phi(g(\theta))-t(\theta_{a})\
dF(\theta)\\\ \text{subject to}\qquad\;g(\theta)\leq t(\theta_{a})+\sum_{i\in
N\setminus\\{a\\}}t(\theta_{i}),\\\ \frac{dg(\theta)}{d\theta}\geq 0,\\\
\text{and for some $Q\in\mathcal{Q}$},\\\
dU_{j}/d\theta_{j}=\phi(g(\theta))-\bar{v}_{j}^{\prime}(\theta_{j},g^{\circ}),\\\
U_{j}(\theta_{j},g(\theta))\geq\bar{v}_{j}(\theta_{j},g^{\circ})\qquad\forall\;j\in
Q.\end{gathered}$ ($\mathcal{S}$)

Problem $\mathcal{S}$ differs from Problem $\mathcal{M}$ only insofar as the
individual rationality constraints of the agents excluded from the coalition
are concerned. The resource constraint is relaxed, since the agents in
$N\setminus Q$ can now be taxed to some upper bound $\bar{\tau}$.

######  Proposition 5.

The coalition $Q$ is randomly chosen. Within this set of agents, the solution
from Problem $\mathcal{U}$ applies.

####  9.4.2 Stochastic Public-Good Provision

Since the utility function is linear in types and quasilinear in the
transfers, the agents are risk-neutral. Moreover, the agenda-setter ’s
objective function is concave in the public good. Therefore, it is
straightforward to show that there is no gain in proposing a stochastic
public-good provision.

######  Proposition 6.

Deterministic public-good provision is optimal.

###  9.5 Dynamics

Consider now the simplest possible dynamic setting. The agenda-setter is the
same in each period $\tau=1,\dots,T$ and the agents’ values for the public
good do not evolve over time. Denote $\delta\in[0,1)$ the common discount
factor of the agents. Moreover, let $g_{\tau}$ and $t_{\tau}$ be the public-
good provision and transfer in period $\tau$, respectively. Agent $i$’s total
utility is then given by

$U_{i}^{T}(\theta_{i})=\sum_{\tau=1}^{T}\delta^{\tau-1}(\theta_{i}\phi(g_{\tau})-t_{\tau})\
.$

Analogously to Definition 1, the notion of direct mechanism can be extended to
this simple dynamic setting.

######  Definition 6.

A dynamic direct mechanism is a set of functions $g=(g_{1},\dots,g_{T})$ and
$t_{i}=(t_{i,1},\dots,t_{i,T})$, $\forall i\in N$

$\displaystyle g$ $\displaystyle:\Theta\longrightarrow\mathbb{R}_{+}^{T},$
$\displaystyle t_{i}$ $\displaystyle:\Theta\longrightarrow\mathbb{R}^{T}\ .$

Moreover, as in Section 5, the notions of incentive compatible and
individually rational (static) mechanisms can be naturally extended to dynamic
mechanisms.

######  Definition 7.

A dynamic direct mechanism is incentive compatible if, $\forall i\in N$,

$U_{i}^{T}(\theta)=\sum_{\tau=1}^{T}\delta^{\tau-1}(\theta_{i}\phi(g_{\tau}(\theta))-t_{\tau}(\theta))\geq\sum_{\tau=1}^{T}\delta^{\tau-1}(\theta_{i}\phi(g_{\tau}(\theta_{i}^{\prime},\theta_{-i}))-t_{\tau}(\theta_{i}^{\prime}))\
\qquad\forall\theta_{i},\theta_{i}^{\prime}\in[\underline{\theta},\overline{\theta}]\
.$

######  Definition 8.

A dynamic direct mechanism is individually rational if, $\forall i\in N$,

$U_{i}^{T}(\theta)\geq\bar{v}_{i}(\theta_{i},g^{\circ})\beta\
\qquad\forall\theta_{i}\in[\underline{\theta},\overline{\theta}]\ ,$

where $\beta=\frac{1-\delta^{T}}{1-\delta}$.

In this simple dynamic setting, the agenda-setter has full commitment power
and s/he proposes an intertemporal profile of public-good levels. If the
agents vote in favour, then the public-good provision $g_{\tau}$ will be
implemented in each period $\tau$. If not, the outside option
$\bar{v}_{i}(\theta_{i},g^{\circ})$ will be implemented in the current and
future periods, without the possibility to renegotiate the public-good
provision.

In the screening literature, the seller cannot benefit from ‘deviating’ from
the (static) optimal mechanism: posting the same price $p^{*}$ is the best
s/he can do.121212See Chapter 11 in Börgers et al., (2015). Then, a natural
conjecture in the present setting is that optimal dynamic public-good
provision will be the same as if agents were ‘myopic’ in the sense of Gersbach
et al., (2023); i.e., at each period $\tau$, they take into account only the
payoff in period $\tau$. If the participation constraint in Definition 8 has
to hold only for a coalition $Q\in\mathcal{Q}$, then public-good provision
might assume more complicated intertemporal dynamics. A more thorough analysis
is then left for future research.

###  9.6 Comparison with Gersbach et al., (2023)

As noted in Section 2, Gersbach et al., (2023) consider a very similar model
in a complete information setting. The presence of private information implies
several significantly different results.

First, the optimal choice of the winning coalition depends on the shape of the
reservation utility profile and the outside option public-good level. In
Gersbach et al., (2023), it is optimal to choose the highest or lowest
coalition, depending on whether the outside option public-good level is lower
or higher than a cutoff value.

As shown in Section 8, extreme coalitions are optimal when the outside option
public-good level is sufficiently high or low, regardless of the shape of
$\bar{v}_{i}$. When it assumes intermediate values, however, optimal
coalitions might be a non-convex set of types and exclude the highest or
lowest types.

Second, in Gersbach et al., (2023) public-good provision is a non-monotonic
function of the outside option public-good level; moreover, it is never
optimal to propose the outside option. As shown Section 6, however, even when
the reservation utility profile is the same as in Gersbach et al., (2023), the
latter result is reversed.

Third, lowering $q$ never decreases public-good provision. This is the exact
opposite of Gersbach et al., (2023), whereby increasing $q$ can never decrease
public-good provision.

##  10 Conclusions

In this paper, I embedded a mechanism design problem about the provision of a
public good into a political economy framework. In line with the previous
contributions from the screening literature, the shape of the outside option
crucially determines the optimal public-good provision and coalition
formation. Several directions for future research remain to be pursued.

First, dominant-strategy implementation circumvents the issue of the ‘informed
principal’ in the sense of Myerson, (1983). An analysis in Bayesian
implementation would raise significant complications, as there is no guarantee
that the revelation principle holds.131313See Mylovanov and Tröger, (2014) and
the references therein. However, relaxing the requirement of dominant-strategy
implementation constitutes a promising direction for future research.

Second, the present paper only considered a static problem. Exploring public-
good provision and coalition formation in dynamic setting and limited
commitment, along the lines of Doval and Skreta, (2022), would also constitute
an interesting exercise.

Finally, the paper considered a single dimension of private information.
Multidimensional screening problems are well-known to pose severe difficulties
(e.g., Rochet and Choné, (1998)). Nonetheless, members of organizations are
likely to be heterogeneous along different dimensions of private information.
Making progress on extending the model to multiple dimensions would provide
additional insights into the coalition formation problem.

##  Appendix A Proofs

###  A.1 Proof of Lemma 1

Proof. Consider $\theta_{i}>\theta_{j}$. By incentive compatibility, we have
that

$U_{i}(\theta_{i})\geq\theta_{i}\phi(g(\theta_{j},\theta_{-i}))-t(\theta_{j})-\bar{v}_{i}(\theta_{i},g^{\circ})=U_{j}(\theta_{j})+(\theta_{i}-\theta_{j})\phi(g(\theta_{j},\theta_{-i}))-(\bar{v}_{i}(\theta_{i},g^{\circ})-\bar{v}_{j}(\theta_{j},g^{\circ}))$
(5)

and

$U_{j}(\theta_{j})\geq\theta_{j}\phi(g(\theta_{i},\theta_{-j}))-t(\theta_{i})-\bar{v}_{j}(\theta_{j},g^{\circ})=U_{i}(\theta_{i})+(\theta_{j}-\theta_{i})\phi(g(\theta_{i},\theta_{-j}))-(\bar{v}_{j}(\theta_{j},g^{\circ})-\bar{v}_{i}(\theta_{i},g^{\circ}))$
(6)

which imply

$\phi(g(\theta_{i},\theta_{-i}))-\frac{\bar{v}_{i}(\theta_{i},g^{\circ})-\bar{v}_{j}(\theta_{j},g^{\circ})}{\theta_{i}-\theta_{j}}\geq\frac{U_{i}(\theta_{i})-U_{j}(\theta_{j})}{\theta_{i}-\theta_{j}}\geq\phi(g(\theta_{j},\theta_{-j}))-\frac{\bar{v}_{j}(\theta_{j},g^{\circ})-\bar{v}_{i}(\theta_{i},g^{\circ})}{\theta_{i}-\theta_{j}}.$

As $\theta_{j}\longrightarrow\theta_{i}$, the conditions in Lemma 1 are
implied.
To establish the converse, assume the above mentioned conditions hold. Then,

$\displaystyle U_{i}(\theta_{i})-U_{j}(\theta_{j})$
$\displaystyle=\int_{\theta_{j}}^{\theta_{i}}\phi(g(x,\theta_{-j}))-\bar{v}^{\prime}_{j}(x,g^{\circ})\
dx$
$\displaystyle\geq\int_{\theta_{j}}^{\theta_{i}}\phi(g(\theta_{j},\theta_{-j}))-\bar{v}^{\prime}_{j}(\theta_{j},g^{\circ})\
dx$
$\displaystyle=(\theta_{i}-\theta_{j})\phi(g(\theta_{j},\theta_{-i})-(\bar{v}_{i}(\theta_{i},g^{\circ})-\bar{v}_{j}(\theta_{j},g^{\circ}))\
.$

This implies (5). An analogous argument proves (6). ∎

###  A.2 Proof of Proposition 2

Proof.

##### Case I

$dU_{i}/d\theta_{i}>0$, $\forall i\in N$.
The proposer’s problem Equation $\mathcal{LU}$ becomes

$\begin{gathered}\max_{g(\theta),\mathbb{t}(\theta)}\qquad\int\theta_{a}\phi(g(\theta))-t(\theta_{a})\
dF(\theta)\\\ \text{subject to}\;\;g(\theta)\leq t(\theta_{a})+\sum_{i\in
N\setminus\\{a\\}}t(\theta_{i}),\\\ \frac{dg(\theta)}{d\theta}\geq 0,\\\
t(\theta_{i})=\theta_{i}\phi(g(\theta_{i},\theta_{-i}))-\int_{\underline{\theta}}^{\theta_{i}}\phi(g(x,\theta_{-i}))dx,\\\
t(\theta_{i})\leq\theta_{i}\phi(g(\theta))\qquad\forall\;i\in
N\setminus\\{a\\}\ .\end{gathered}$

Consider the proposer’s expected utility. Since all constraints are binding,
it is equal to

$\displaystyle\int_{\Theta}\left[\theta_{a}\phi(g(\theta))-\sum_{i\in
N\setminus\\{a\\}}t(\theta_{i})-g(\theta)\right]f(\theta)d\theta$
$\displaystyle=\int_{\Theta}\left[\theta_{a}\phi(g(\theta))+\sum_{i\in
N\setminus\\{a\\}}\left[\theta_{i}\phi(g(\theta))-\int_{\underline{\theta}}^{\theta_{i}}\phi(g(x,\theta_{-i}))dx\right]-g(\theta)\right]f(\theta)d\theta$
$\displaystyle=\int_{\Theta}\left[\sum_{i\in
N}\theta_{i}\phi(g(\theta))-\sum_{i\in
N\setminus\\{a\\}}\left(\int_{\underline{\theta}}^{\theta_{i}}\phi(g(x,\theta_{-i}))dx\right)-g(\theta)\right]f(\theta)d\theta$
$\displaystyle=\int_{\Theta}\left[\sum_{i\in
N}\theta_{i}\phi(g(\theta))-\sum_{i\in
N\setminus\\{a\\}}\left(\frac{1-F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\phi(g(\theta))-g(\theta)\right]f(\theta)d\theta$

where the last equality follows from Fubini’s theorem. Pointwise maximization
with respect to $g(\theta)$ yields

$\left[\theta_{a}+\sum_{i\in
N\setminus\\{a\\}}\left(\theta_{i}-\frac{1-F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\right]\phi^{\prime}(g(\theta))=1\
.$ (7)

2 ensures the monotonicity constraint is satisfied.
The same goes through for any $g^{\circ}\leq g^{L}$.

##### Case II

$dU_{i}/d\theta_{i}<0$, $\forall i\in N$.
The proposer’s problem Equation $\mathcal{LU}$ becomes

$\begin{gathered}\max_{g(\theta),\mathbb{t}(\theta)}\qquad\int\theta_{a}\phi(g(\theta))-t(\theta_{a})\
dF(\theta)\\\ \text{subject to}\;\;g(\theta)\leq t(\theta_{a})+\sum_{i\in
N\setminus\\{a\\}}t(\theta_{i}),\\\ \frac{dg(\theta)}{d\theta}\geq 0,\\\
t(\theta_{i})=\theta_{i}\phi(g(\theta_{i},\theta_{-i}))-\int_{\theta_{i}}^{\overline{\theta}}\bar{v}_{i}(x,g^{\circ})-\phi(g(x,\theta_{-i}))\
dx,\\\
\theta_{i}\phi(g(\theta))-t(\theta_{i})\geq\theta_{i}\phi(g^{\circ})-\frac{g^{\circ}}{n}\qquad\forall\;i\in
N\setminus\\{a\\}\ .\end{gathered}$

Consider the proposer’s expected utility. Since all constraints are binding,
it is equal to

$\displaystyle\int_{\Theta}\left[\theta_{a}\phi(g(\theta))-\sum_{i\in
N\setminus\\{a\\}}t(\theta_{i})-g(\theta)\right]f(\theta)d\theta$
$\displaystyle=\int_{\Theta}\left[\theta_{a}\phi(g(\theta))+\sum_{i\in
N\setminus\\{a\\}}\left[\theta_{i}\phi(g(\theta))-\int_{\theta_{i}}^{\overline{\theta}}\bar{v}_{i}(x,g^{\circ})-\phi(g(x,\theta_{-i}))\
dx\right]-g(\theta)\right]f(\theta)d\theta$
$\displaystyle=\int_{\Theta}\left[\sum_{i\in
N}\theta_{i}\phi(g(\theta))-\sum_{i\in
N\setminus\\{a\\}}\left(\int_{\theta_{i}}^{\overline{\theta}}\bar{v}_{i}(x,g^{\circ})-\phi(g(x,\theta_{-i}))\
dx\right)-g(\theta)\right]f(\theta)d\theta$
$\displaystyle=\int_{\Theta}\left[\sum_{i\in
N}\theta_{i}\phi(g(\theta))+\sum_{i\in
N\setminus\\{a\\}}\left(\frac{F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\phi(g(\theta))-\sum_{i\in
N\setminus\\{a\\}}\left(\int_{\theta_{i}}^{\overline{\theta}}\bar{v}_{i}(x,g^{\circ})\right)-g(\theta)\right]f(\theta)d\theta\
.$

Pointwise maximization with respect to $g(\theta)$ yields

$\left[\theta_{a}+\sum_{i\in
N\setminus\\{a\\}}\left(\theta_{i}+\frac{F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\right]\phi^{\prime}(g(\theta))=1\
.$ (8)

Again, by 2, the monotonicity constraint is satisfied. Moreover, the same goes
through for any $g^{\circ}\geq g^{H}$.

##### Case III

$g^{L}<g^{\circ}<g^{H}$.
Consider a simple three-agent example, with
$N=\\{\theta_{a},\theta_{k},\theta_{l}\\}$. Suppose $\theta_{a}=0$ and
$\theta_{k}>\theta_{l}\neq 0$. This is a classic screening problem as in Maggi
and Rodriguez-Clare A., (1995). Figure 9 illustrates the problem faced by the
agenda-setter.

$\underline{\theta}$$\overline{\theta}$$g^{H}$$g^{L}$$\bar{v}^{\prime}(\theta_{i},g^{\circ})$$\tilde{\theta}$$\tilde{\tilde{\theta}}$$\theta_{k}$$\theta_{l}$
Figure 9: Two-agent screening example

Suppose $\theta_{k}\in(\underline{\theta},\tilde{\theta})$ and
$\theta_{l}\in(\tilde{\tilde{\theta}},\overline{\theta})$ and recall the IC
condition (3). For both,

$dU_{k,l}/d\theta_{k,l}=\phi(g(\theta))-\phi(g^{\circ})\ .$

On the one hand, if $g(\theta)>g^{\circ}$, then $\theta_{k}$ has ex post the
incentive to overstate. On the other hand, if $g(\theta)<g^{\circ}$, then
$\theta_{l}$ has ex post the incentive to understate. Hence, $a$ can do no
better than choosing the outside option. ∎

###  A.3 Proof of Proposition 3

Proof. We begin with the following two lemmata.

######  Lemma 6.

The optimal choice of coalition is

$Q^{*}=\begin{cases}\overline{Q}&\mbox{if $g^{\circ}<g_{M}^{L}$,}\\\
\textrm{any}\ Q\in\mathcal{Q}&\mbox{if $g_{M}^{L}\leq g^{\circ}\leq
g_{M}^{H}$,}\\\ \underline{Q}&\mbox{if $g^{\circ}>g_{M}^{H}$,}\end{cases}$

if $g_{M}^{L}<g_{M}^{H}$. Otherwise

$Q^{*}=\begin{cases}\overline{Q}&\mbox{if $g^{\circ}<g_{M}^{L}$,}\\\
\underline{Q}&\mbox{if $g^{\circ}>g_{M}^{L}$}.\end{cases}$

Proof. Consider first the case $g^{\circ}<g_{M}^{L}$. Obviously, the solution
from the analogous case in Proposition 2 is still feasible. However, it is
possible for the solution to the majority case to violate some individual
rationality constraints, since they are required to hold only for $q-1$
agents. Denote this set of agents $I$, with $|I|\leq n-q$. Condition (3)
requires that information rents be increasing over the whole set of types.
Moreover, incentive constraints are pointing in the direction of the lowest
type. Also, if the participation constraint is satisfied for type
$\theta_{1}$, then it is also satisfied for $\theta_{2}>\theta_{1}$.
Therefore, utility maximization requires that the cutoff agent
$\tilde{\theta}$ be the agent with the lowest type still included in the
coalition:

$\theta_{1},\theta_{2},\dots,\underbrace{\tilde{\theta},\dots,\theta_{n-1},\theta_{n}}_{\text{at
most}\ q-1}\ .$

An analogous reasoning applies to the case $g^{\circ}>g_{M}^{L}$, with the
corresponding cutoff type $\tilde{\tilde{\theta}}$:

$\underbrace{\theta_{1},\theta_{2},\dots,\tilde{\tilde{\theta}}}_{\text{at
most}\ q-1},\dots,\theta_{n-1},\theta_{n}\ .$

∎
The number of agents whose participation constraint is violated depends on the
value of $q$.

######  Lemma 7.

The cardinality of the set $I$, $|I|$, is inversely related to $q$.

Proof. Recall (7). If agents with the lowest types are bunched at the same
allocation of a cutoff type $\tilde{\theta}$, the analogous condition is

$\left[\theta_{a}+|I|\cdot\tilde{\theta}+\sum_{i\in\overline{Q}\setminus\\{a\\}}\left(\theta_{i}-\frac{1-F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\right]\phi^{\prime}(g(\theta))=1\
.$ (9)

Thus, the optimal public-good provision $g^{*}(\theta)$ is higher when
$I\neq\varnothing$. However, as $q$ decreases further, by concavity of the
utility functions the efficient (´first-best’) level (1) is reached for some
$\tilde{q}$. For any $q<\tilde{q}$, it is optimal to maintain the same cutoff
type $\tilde{\theta}$. ∎
With the help of Lemma 6, the analogous cases as in Proposition 2 now follow.

##### Case I

$dU_{i}/d\theta_{i}>0$, $\forall i\in N$.
The agenda-setter’s problem Equation $\mathcal{LM}$ becomes

$\begin{gathered}\max_{g(\theta),\mathbb{t}(\theta)}\qquad\int\theta_{a}\phi(g(\theta))-t(\theta_{a})\
dF(\theta)\\\ \text{subject to}\;\;g(\theta)\leq t(\theta_{a})+\sum_{i\in
N\setminus\\{a\\}}t(\theta_{i}),\\\ \frac{dg(\theta)}{d\theta}\geq 0,\\\
t(\theta_{i})=\theta_{i}\phi(g(\theta_{i},\theta_{-i}))-\int_{\tilde{\theta}}^{\theta}\phi(g(x,\theta_{-i}))dx\qquad\forall\;i\in
N\setminus\\{a\\},\\\
t(\theta_{i})\leq\theta_{i}\phi(g(\theta))\qquad\forall\;i\in Q\
.\end{gathered}$

By Lemma 6, it is then optimal to make the participation constraint of
$\tilde{\theta}$ binding.

######  Claim 1.

For all $\theta_{i}\leq\tilde{\theta}$, the participation constraint is
violated.

Proof. We have that

$\displaystyle t(\theta_{i})$
$\displaystyle=\theta_{i}\phi(g(\theta))-\int_{\tilde{\theta}}^{\theta_{i}}\phi(g(x,\theta_{-i}))dx$
$\displaystyle=\theta_{i}\phi(g(\theta))+\int_{\theta_{i}}^{\tilde{\theta}}\phi(g(x,\theta_{-i}))dx$
$\displaystyle\geq\theta_{i}\phi(g(\theta))\ ,$

where the last inequality follows from the monotonicity of the allocation
rule. Notice that, in order to preserve incentive compatibility, it must be
the case that $t(\theta_{i})=t(\tilde{\theta})$, for all
$\theta_{i}\leq\tilde{\theta}$. ∎
Consider now the expected transfer of types $\theta_{i}\geq\tilde{\theta}$. It
is equal to

$\int_{\underline{\theta}}^{\overline{\theta}}t(\theta_{i})\
f_{i}(\theta_{i})d(\theta_{i})=\int_{\underline{\theta}}^{\overline{\theta}}\theta_{i}\phi(g(\theta_{i},\theta_{-i}))-\int_{\underline{\theta}}^{\overline{\theta}}\int_{\tilde{\theta}}^{\theta}\phi(g(x,\theta_{-i}))dx\
f_{i}(\theta_{i})d(\theta_{i})\ .$

Consider the second term.

$\displaystyle\int_{\underline{\theta}}^{\overline{\theta}}\int_{\tilde{\theta}}^{\theta}\phi(g(x,\theta_{-i}))dx\
f_{i}(\theta_{i})d(\theta_{i})$
$\displaystyle=\int_{\underline{\theta}}^{\overline{\theta}}\int_{\underline{\theta}}^{\theta}\phi(g(x,\theta_{-i}))dx\
f_{i}(\theta_{i})d(\theta_{i})-\int_{\underline{\theta}}^{\overline{\theta}}\int_{\underline{\theta}}^{\tilde{\theta}}\phi(g(x,\theta_{-i}))dx\
f_{i}(\theta_{i})d(\theta_{i})\ .$

Simplifying the first term is a standard procedure in the literature (Börgers
et al., (2015)). A very similar procedure can be applied to simplify also the
second term. A graphical representation helps to illustrate how the standard
procedure has to be modified in order to take into account the excluded
types.141414The steps follow closely Chapter 2 in Börgers et al., (2015).For
completeness, I include the derivations for both terms.

$\theta_{i}$$x$$\underline{\theta}$$\underline{\theta}$$\underline{\theta}$$\underline{\theta}$$\tilde{\theta}$$\tilde{\theta}$
Figure 10: Change in the order of integration.

The first term simplifies as follows:

$\displaystyle\int_{\underline{\theta}}^{\overline{\theta}}\int_{\underline{\theta}}^{\theta}\phi(g(x,\theta_{-i}))dx\
f_{i}(\theta_{i})d(\theta_{i})$
$\displaystyle=\int_{\underline{\theta}}^{\overline{\theta}}\int_{\underline{\theta}}^{\theta}\phi(g(x,\theta_{-i}))\
f_{i}(\theta_{i})dxd(\theta_{i})$
$\displaystyle=\int_{\underline{\theta}}^{\overline{\theta}}\int_{\underline{\theta}}^{\theta}\phi(g(x,\theta_{-i}))\
f_{i}(\theta_{i})\mathbb{1}_{x\leq\theta}\ d(\theta_{i})dx$
$\displaystyle=\int_{\underline{\theta}}^{\overline{\theta}}\phi(g(x,\theta_{-i}))\int_{\underline{\theta}}^{\theta}f_{i}(\theta_{i})\mathbb{1}_{x\leq\theta}\
d(\theta_{i})dx$
$\displaystyle=\int_{\underline{\theta}}^{\overline{\theta}}\phi(g(x,\theta_{-i}))\int_{x}^{\overline{\theta}}f_{i}(\theta_{i})\
d(\theta_{i})dx$
$\displaystyle=\int_{\underline{\theta}}^{\overline{\theta}}\phi(g(x,\theta_{-i}))\
\left[1-F(x)\right]dx$
$\displaystyle=\int_{\underline{\theta}}^{\overline{\theta}}\phi(g(\theta_{i}))\
\left[1-F_{i}(\theta_{i})\right]d\theta_{i}\ .$

When applying Fubini’s theorem in the second equality, the whole black
triangle is considered. On the other hand, for the second term, only the red
polygon has to be taken into account.

$\displaystyle\int_{\underline{\theta}}^{\overline{\theta}}\int_{\underline{\theta}}^{\tilde{\theta}}\phi(g(x,\theta_{-i}))dx\
f_{i}(\theta_{i})d(\theta_{i})=\int_{\underline{\theta}}^{\tilde{\theta}}\int_{\underline{\theta}}^{\overline{\theta}}\phi(g(x,\theta_{-i}))\
f_{i}(\theta_{i})\mathbb{1}_{\theta\leq\tilde{\theta}}\ d(\theta_{i})dx$
$\displaystyle=\int_{\underline{\theta}}^{\overline{\theta}}\phi(g(x,\theta_{-i}))\int_{x}^{\overline{\theta}}f_{i}(\theta_{i})\
d(\theta_{i})dx$
$\displaystyle=\int_{\underline{\theta}}^{\overline{\theta}}\phi(g(x,\theta_{-i}))\int_{\underline{\theta}}^{\overline{\theta}}f_{i}(\theta_{i})\
d(\theta_{i})dx-\int_{\underline{\theta}}^{\overline{\theta}}\phi(g(x,\theta_{-i}))\int_{\underline{\theta}}^{x}f_{i}(\theta_{i})\
d(\theta_{i})dx$
$\displaystyle=\int_{\underline{\theta}}^{\tilde{\theta}}\phi(g(x,\theta_{-i}))F(\tilde{\theta})dx-\int_{\underline{\theta}}^{\tilde{\theta}}\phi(g(x,\theta_{-i}))F(x)dx$
$\displaystyle=\int_{\underline{\theta}}^{\overline{\theta}}\phi(g(x,\theta_{-i}))\
\left[F(\tilde{\theta})-F(x)\right]\mathbb{1}_{x\leq\tilde{\theta}}dx$
$\displaystyle=\int_{\underline{\theta}}^{\overline{\theta}}\phi(g(\theta))\
\left[F(\tilde{\theta})-F_{i}(\theta_{i})\right]\mathbb{1}_{\theta_{i}\leq\tilde{\theta}}d\theta_{i}\
.$

Then, the expected transfer of the excluded types are given by

$\displaystyle\int_{\underline{\theta}}^{\overline{\theta}}t(\theta_{i})\
f_{i}(\theta_{i})d(\theta_{i})=\int_{\underline{\theta}}^{\overline{\theta}}\theta_{i}\phi(g(\theta_{i},\theta_{-i}))-\int_{\tilde{\theta}}^{\theta}\phi(g(x,\theta_{-i}))dx\
f_{i}(\theta_{i})d(\theta_{i})$
$\displaystyle=\int_{\underline{\theta}}^{\overline{\theta}}\tilde{\theta}\phi(g(\theta))d\theta_{i}-\left[\int_{\underline{\theta}}^{\overline{\theta}}\phi(g(\theta))\
\left[1-F_{i}(\theta_{i})\right]d\theta_{i}-\int_{\underline{\theta}}^{\overline{\theta}}\phi(g(\theta))\
\left[F(\tilde{\theta})-F_{i}(\theta_{i})\right]\mathbb{1}_{\theta_{i}\leq\tilde{\theta}}d\theta_{i}\right]$
$\displaystyle=\int_{\underline{\theta}}^{\overline{\theta}}\left[\tilde{\theta}-\frac{1-F(\tilde{\theta})}{f_{i}(\theta_{i})}\right]\phi(g(\theta))d\theta_{i}$

In other words, types $\theta_{i}\leq\tilde{\theta}$ are bunched at the same
allocation of the cutoff type $\tilde{\theta}$, whose participation constraint
is binding. Putting everything together, and going through the same
derivations as for Proposition 2, yields the following result.

######  Lemma 8.

Optimal public-good provision solves

$\left[\theta_{a}+|I|\cdot\tilde{\theta}+\sum_{i\in\overline{Q}\setminus\\{a\\}}\left(\theta_{i}-\frac{1-F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\right]\phi^{\prime}(g(\theta))=1\
.$ (10)

##### Case II

$dU_{i}/d\theta_{i}<0$, $\forall i\in N$.
Along the same lines as in the previous case, the following result holds.

######  Lemma 9.

Optimal public-good provision solves

$\left[\theta_{a}+|I|\cdot\tilde{\tilde{\theta}}+\sum_{i\in\underline{Q}\setminus\\{a\\}}\left(\theta_{i}+\frac{F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\right]\phi^{\prime}(g(\theta))=1\
.$ (11)

The same goes through for any $g^{\circ}\geq g^{H}$.

##### Case III

$g^{L}<g^{\circ}<g^{H}$.
The argument is the same as in Case III in Proposition 2. ∎

###  A.4 Proof of Theorem 2

####  A.4.1 Proof of Lemma 3

Proof. Consider the expected transfer $\int t(\theta_{i})f_{i}(\theta_{i})\
d\theta_{i}$ from agent $i\in N$. As standard in the literature, we can
rewrite it as

$\int_{\underline{\theta}}^{\overline{\theta}}s_{i}(\theta_{i},g(\theta))-u_{i}(\theta_{i})\
f_{i}(\theta_{i})\ d\theta_{i}$

where $s_{i}(\theta_{i},g(\theta))=\theta_{i}\phi(g(\theta))$. As in Jullien,
(2000), define $\gamma^{*}(\theta_{i})$ to be the shadow value of a uniform
reduction in the participation constraint for types between
$\underline{\theta}$ and $\theta_{i}$. Then, $d\gamma^{*}(\theta_{i})$ is the
shadow value on the individual rationality constraint for agent with type
$\theta_{i}$.

##### Step 1.

$\gamma^{*}(\theta_{i})$ is a cumulative distribution function.
Increasing $\theta_{i}$ extends the set of agents for whom the constraint is
relaxed; hence, $\gamma^{*}(\theta_{i})$ is positive and nondecreasing.
Furthermore, relaxing the participation constraint for the whole set of agents
is equivalent to a uniform increase in $u_{i}(\theta_{i})$. Therefore,
$\gamma^{*}(\overline{\theta})=1$.

##### Step 2.

From a single to many agents.
The expected transfer is then

$\int_{\underline{\theta}}^{\overline{\theta}}s_{i}(\theta_{i},g(\theta))-u_{i}(\theta_{i})\
f_{i}(\theta_{i})\
d\theta_{i}+\int_{\underline{\theta}}^{\overline{\theta}}u_{i}(\theta_{i})d\gamma^{*}(\theta_{i}).$
(12)

Integrating by parts (12) twice yields

$\displaystyle
s_{i}(\theta_{i},g(\theta))-u_{i}(\theta_{i})F_{i}(\theta_{i})\Big{|}_{\underline{\theta}}^{\overline{\theta}}-\int_{\underline{\theta}}^{\overline{\theta}}\big{[}s^{\prime}_{i}(\theta_{i},g(\theta))-u^{\prime}_{i}(\theta_{i})\big{]}F_{i}(\theta_{i})\
d\theta_{i}+u_{i}(\theta_{i})\gamma(\theta_{i})\Big{|}_{\underline{\theta}}^{\overline{\theta}}-\int_{\underline{\theta}}^{\overline{\theta}}u^{\prime}_{i}(\theta_{i})\gamma^{*}(\theta_{i})$
$\displaystyle=s_{i}(\overline{\theta},g(\theta))-\int_{\underline{\theta}}^{\overline{\theta}}\big{[}s^{\prime}_{i}(\theta_{i},g(\theta))-u^{\prime}_{i}(\theta_{i})\big{]}F_{i}(\theta_{i})\
d\theta_{i}-\int_{\underline{\theta}}^{\overline{\theta}}u^{\prime}_{i}(\theta_{i})\gamma^{*}(\theta_{i})$
$\displaystyle=s_{i}(\overline{\theta},g(\theta))-\Bigg{[}s_{i}(\theta_{i},g(\theta))F_{i}(\theta_{i})\Big{|}_{\underline{\theta}}^{\overline{\theta}}-\int_{\underline{\theta}}^{\overline{\theta}}s_{i}(\theta_{i},g(\theta))f_{i}(\theta_{i})\
d\theta_{i}-\int_{\underline{\theta}}^{\overline{\theta}}u^{\prime}_{i}(\theta_{i})F_{i}(\theta_{i})\Bigg{]}-\int_{\underline{\theta}}^{\overline{\theta}}u^{\prime}_{i}(\theta_{i})\gamma^{*}(\theta_{i})$
$\displaystyle=\int_{\underline{\theta}}^{\overline{\theta}}s_{i}(\theta_{i},g(\theta))f_{i}(\theta_{i})\
d\theta_{i}+\int_{\underline{\theta}}^{\overline{\theta}}u^{\prime}_{i}(\theta_{i})F_{i}(\theta_{i})-\int_{\underline{\theta}}^{\overline{\theta}}u^{\prime}_{i}(\theta_{i})\gamma^{*}(\theta_{i})$
$\displaystyle=\int_{\underline{\theta}}^{\overline{\theta}}\Bigg{[}s_{i}(\theta_{i},g(\theta))-\frac{\gamma^{*}(\theta_{i})-F_{i}(\theta_{i})}{f_{i}(\theta_{i})}u^{\prime}_{i}(\theta_{i})\Bigg{]}f_{i}(\theta_{i})\
d\theta_{i}$

Summing over $i\in N\setminus\\{a\\}$ and substituting back the original
utility functions into the budget constraint yield (4). ∎

####  A.4.2 Defining the thresholds

Consider now the implicit function

$G(\theta_{i};g^{\circ})\equiv\phi(g(\theta))-\bar{v}^{\prime}_{i}(\theta_{i},g^{\circ})=0\
.$ (13)

The following result ensures that, for given public-good provision $g(\theta)$
and outside option public-good level $g^{\circ}$, agents are ordered according
to their incentive to misreport their types. A crucial role is played by the
shape of the reservation utility profile.

######  Lemma 10.

For $\theta_{i}\in[\underline{\theta},\overline{\theta}]$ and
$x\in\mathbb{R}_{+}$, there exists a unique $\bar{g}(\theta_{i},x)$ such that

$G(\theta_{i};\bar{g}(\theta_{i},x))=x.$

In addition, if $\bar{v}_{i}$ is concave (convex), then
$\bar{g}(\theta_{i},x)$ is strictly increasing in $x$ and strictly increasing
(decreasing) in $\theta_{i}$.

Proof of Lemma 10. It can be readily checked that $G(\theta_{i};g^{\circ})$
has the following properties:

  * -

Suppose $\bar{v}$ is concave. Then, for $\theta_{i}<\theta_{j}$,
$G(\theta_{i};g^{\circ})\leq G(\theta_{j};g^{\circ})$ on its domain. Equality
holds if and only if $g^{\circ}=0$.

  * -

Suppose $\bar{v}$ is convex. Then, for $\theta_{i}<\theta_{j}$,
$G(\theta_{j};g^{\circ})\leq G(\theta_{i};g^{\circ})$ on its domain. Equality
holds if and only if $g^{\circ}=0$.

Moreover, $G(\theta_{i};g^{\circ})$ is monotone increasing in $g^{\circ}$, for
all $g^{\circ}\in\mathbb{R}_{+}$. ∎
As Lemma 10 shows, the ordering of agents according to their types hinges on
the concavity or convexity of the reservation utility profile. Therefore, each
case is considered separately.

###  A.5 Concavity

I begin with the proof of Lemma 4.
Proof. Consider $\theta_{k}\in K(g^{\circ})$, $\theta_{l}\in L(g^{\circ})$ and
$\theta_{m}\in M(g^{\circ})$, with $\theta_{k}\leq\theta_{l}\leq\theta_{m}$.
We have that

$\displaystyle
dU_{m}/\theta_{m}=\phi(g(\theta))-\bar{v}^{\prime}_{m}(\theta_{m},g^{\circ})$
$\displaystyle\geq\phi(g(\theta))-\bar{v}^{\prime}_{l}(\theta_{l},g^{\circ})$
$\displaystyle=dU_{l}/d\theta_{l}$ $\displaystyle=0$
$\displaystyle\geq\phi(g(\theta))-\bar{v}^{\prime}_{k}(\theta_{k},g^{\circ})$
$\displaystyle=dU_{k}/d\theta_{k}\ .$

∎
I now define a partition of the support of $g^{\circ}$ according to the
agents’ incentive to understate or overstate their types. In particular,
define $\bar{g}^{|K|,|M|}$ as the outside option public-good level solving
(13) when $G(\cdot)$ solves the following condition

$\left[\theta_{a}+\sum_{k\in
K}\left(\theta_{k}+\frac{1-F_{k}(\theta_{k})}{f_{k}(\theta_{k})}\right)+\sum_{m\in
M}\left(\theta_{m}-\frac{F_{m}(\theta_{m})}{f_{m}(\theta_{m})}\right)\right]\phi^{\prime}(g(\theta))=1.$
(14)

In order to keep notation compact and consistent with Section 6, define the
following thresholds as

$\displaystyle\bar{g}^{L}$ $\displaystyle\equiv\left[\theta_{a}+\sum_{i\in
L,M}\left(\theta_{i}-\frac{1-F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\right]\phi^{\prime}(g(\theta))=1,$
$\displaystyle\bar{g}^{L-1}$
$\displaystyle\equiv\left[\theta_{a}+\left(\theta_{1}+\frac{F_{1}(\theta_{1})}{f_{1}(\theta_{1})}\right)+\sum_{i\in
L,M}\left(\theta_{i}-\frac{1-F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\right]\phi^{\prime}(g(\theta))=1,$
$\displaystyle\bar{g}^{H-1}$ $\displaystyle\equiv\left[\theta_{a}+\sum_{i\in
K}\left(\theta_{i}+\frac{F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)+\left(\theta_{r}-\frac{1-F_{r}(\theta_{r})}{f_{r}(\theta_{r})}\right)\right]\phi^{\prime}(g(\theta))=1,$
$\displaystyle\bar{g}^{H}$ $\displaystyle\equiv\left[\theta_{a}+\sum_{i\in
L,K}\left(\theta_{i}+\frac{F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\right]\phi^{\prime}(g(\theta))=1\
.$

Then, Lemma 4 and Lemma 10 imply the following proposition.

######  Proposition 7.

Suppose $\bar{v}_{i}^{\prime\prime}(\theta_{i},g^{\circ})\leq 0$. The indices
$k=k(g^{\circ})$ and $l=l(g^{\circ})$ are given by:

$\displaystyle k(g^{\circ})$
$\displaystyle=\left\\{\begin{array}[]{lcl}0&\mbox{if}&0\leq
g^{\circ}<\bar{g}^{L}\\\ 1&\mbox{if}&\bar{g}^{L}\leq
g^{\circ}<\bar{g}^{L-1}\\\ \;\vdots&\mbox{if}&\qquad\vdots\qquad\vdots\\\
r-2&\mbox{if}&\bar{g}^{H-1}\leq g^{\circ}<\bar{g}^{H}\\\
r-1&\mbox{if}&\bar{g}^{H}\leq g^{\circ}\ ,\end{array}\right.$ $\displaystyle
l(g^{\circ})$ $\displaystyle=\left\\{\begin{array}[]{lcl}1&\mbox{if}&0\leq
g^{\circ}<\bar{g}^{L}\\\ 2&\mbox{if}&\bar{g}^{L}\leq
g^{\circ}<\bar{g}^{L-1}\\\ \;\vdots&\mbox{if}&\qquad\vdots\qquad\vdots\\\
r-1&\mbox{if}&\bar{g}^{H-1}\leq g^{\circ}<\bar{g}^{H}\\\
r&\mbox{if}&\bar{g}^{H}\leq g^{\circ}\ .\end{array}\right.$

Proof.

  * •

$k=0$, $l=1$. Same as in Proposition 2.

  * •

$0<k<l<r$. By Lemma 4, the set of types is ordered. But then, rents are
minimized if the participation constraint binds on a single interior type.
Denote this type by $\theta_{l}$. Formally, the proposer’s problem becomes

$\begin{gathered}\max_{g(\theta),t(\theta)}\qquad\int\theta_{a}\phi(g(\theta))-t(\theta_{a})\
dF(\theta)\\\ \text{subject to}\;\;g(\theta)\leq t(\theta_{a})+\sum_{i\in
N\setminus\\{a\\}}t(\theta_{i}),\\\ \frac{dg(\theta)}{d\theta}\geq 0,\\\
t(\theta_{i})=\left\\{\begin{array}[]{lcl}\theta_{i}\phi(g(\theta_{i},\theta_{-i}))-\int_{\underline{\theta}}^{\theta_{i}}\phi(g(x,\theta_{-i}))dx&\mbox{for}&\theta_{i}\geq\theta_{l},\\\
\theta_{i}\phi(g(\theta_{i},\theta_{-i}))-\int_{\theta_{i}}^{\overline{\theta}}\bar{v}_{i}(x,g^{\circ})-\phi(g(x,\theta_{-i}))\
dx&\mbox{for}&\theta_{i}<\theta_{l},\end{array}\right.\\\
u_{i}(\theta_{i},g(\theta))\geq\bar{v}_{i}(\theta_{i},g^{\circ})\qquad\forall\;i\in
N\setminus\\{a\\}\ .\end{gathered}$

Then, going through the same derivations as in Proposition 2 yields the
following condition for the optimal $g(\theta)$

$\left[\theta_{a}+\sum_{i<l}\left(\theta_{i}+\frac{F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)+\sum_{i\geq
l}\left(\theta_{i}-\frac{1-F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\right]\phi^{\prime}(g(\theta))=1\
.$

  * •

$k=r-1$, $l=r$. Same as in Proposition 2.

∎

###  A.6 Convexity

I begin with the proof of Lemma 5.
Proof. Consider $\theta_{k}\in K(g^{\circ})$, $\theta_{l}\in L(g^{\circ})$ and
$\theta_{m}\in M(g^{\circ})$, with $\theta_{m}\leq\theta_{l}\leq\theta_{k}$.
We have that

$\displaystyle
dU_{m}/\theta_{m}=\phi(g(\theta))-\bar{v}^{\prime}_{m}(\theta_{m},g^{\circ})$
$\displaystyle\geq\phi(g(\theta))-\bar{v}^{\prime}_{l}(\theta_{l},g^{\circ})$
$\displaystyle=dU_{l}/d\theta_{l}$ $\displaystyle=0$
$\displaystyle\geq\phi(g(\theta))-\bar{v}^{\prime}_{k}(\theta_{k},g^{\circ})$
$\displaystyle=dU_{k}/d\theta_{k}\ .$

However, in contrast to Lemma 4, there cannot be a single interior type in
$L(g^{\circ})$.

Intuitively, as $g^{\circ}$ increases, the agents with the highest types
transition into $K(g^{\circ})$. By Condition (2), information rents have to be
decreasing in $K(g^{\circ})$ and increasing in $M(g^{\circ})$. Since
$\gamma^{*}(\theta_{i})$ is nondecreasing, it must be constant for all types;
i.e., $\gamma^{*}(\theta_{i})=\gamma^{*}$, for all $i\in N\setminus\\{a\\}$.
But then, information rents are minimized if the IR constraints of the agents
with the highest and lowest types bind.

Formally, define $R(\gamma^{*})$ to be the utility differential between the
high type and the low type

$R(\gamma^{*})=U(\overline{\theta})-U(\underline{\theta})=\int_{\underline{\theta}}^{\overline{\theta}}\phi(\hat{g}(\theta,\gamma^{*}))-\bar{v}^{\prime}_{i}(\theta_{i},g^{\circ})\
d\theta\ ,$

where $\hat{g}$ solves (4). Clearly, $R(\gamma^{*})$ is decreasing. To
determine the optimal choice of $\gamma^{*}$, we have the following lemma.

######  Lemma 11 (Maggi and Rodriguez-Clare A., (1995)).

The optimal $\gamma^{*}$ satisfies

$\gamma^{*}\equiv\begin{cases}1&\mbox{if $R(1)\geq 0$}\\\ R^{-1}(0)&\mbox{if
$R(1)<0<R(0)$}\\\ 0&\mbox{if $R(0)\leq 0$}\ .\end{cases}$

Proof. When $R(1)\geq 0$ or $R(0)\leq 0$, the same reasoning as in Proposition
2 applies. Consider now the case $R(1)<0<R(0)$. Choosing $\gamma^{*}$ ensures
that $R(\gamma^{*})=0$ and incentive compatibility is preserved. Moreover,
denote

$J(\theta;g^{\circ})\equiv\int_{\underline{\theta}}^{\theta}\phi(\hat{g}(x;\gamma^{*}))-\bar{v}^{\prime}_{i}(x,g^{\circ})\
dx$ (15)

where $\hat{g}$ solves (4). We have the following lemma.

######  Lemma 12.

$J(\theta;g^{\circ})$ is quasiconcave in $\theta$.

Proof. $J^{\prime}(\theta;g^{\circ})=0$ when $G(\theta_{i};g^{\circ})=0$,
which occurs for a single $\theta_{i}$. By convexity of $\bar{v}_{i}$,
$J^{\prime}(\theta;g^{\circ})$ is decreasing for $\theta^{\prime}>\theta$. ∎
Then, $U(\theta)=J(\theta;g^{\circ})+U(\underline{\theta})$ and, by
quasiconcavity,
$U(\theta)\geq\min\left\\{U(\underline{\theta}),U(\overline{\theta})\right\\}$
for all $\theta_{i}\in\Theta_{i}$. Hence, if the individual rationality
constraint is satisfied for the highest and lowest types, it is satisfied for
all types. ∎
This concludes the proof of Lemma 5.

As in Section A.5, it is useful to define the following thresholds of the
support of $g^{\circ}$ recursively as follows:

$\displaystyle\bar{g}^{L}$ $\displaystyle\equiv\left[\theta_{a}+\sum_{i\in
L,M}\left(\theta_{i}-\frac{1-F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\right]\phi^{\prime}(g(\theta))=1$
$\displaystyle\bar{g}^{L-1}$
$\displaystyle\equiv\left[\theta_{a}+\left(\theta_{r}-\frac{\gamma^{*}_{L-1}-F_{r}(\theta_{r})}{f_{r}(\theta_{r})}\right)+\sum_{i\in
L,M}\left(\theta_{i}-\frac{\gamma^{*}_{L-1}-F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\right]\phi^{\prime}(g(\theta))=1$
$\displaystyle\bar{g}^{H-1}$ $\displaystyle\equiv\left[\theta_{a}+\sum_{i\in
K}\left(\theta_{i}-\frac{\gamma^{*}_{H-1}-F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)+\left(\theta_{1}-\frac{\gamma^{*}_{H-1}-F_{1}(\theta_{1})}{f_{r}(\theta_{1})}\right)\right]\phi^{\prime}(g(\theta))=1$
$\displaystyle\bar{g}^{H}$ $\displaystyle\equiv\left[\theta_{a}+\sum_{i\in
L,K}\left(\theta_{i}+\frac{F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\right]\phi^{\prime}(g(\theta))=1\
.$

Notice that $\gamma^{*}$ changes with the outside option public-good level, as
shown in Lemma 11.
Lemma 5 and Lemma 10 imply the following proposition.

######  Proposition 8.

Suppose $\bar{v}_{i}^{\prime\prime}(\theta_{i},g^{\circ})\geq 0$. The indices
$k=k(g^{\circ})$ and $l=l(g^{\circ})$ are given by:

$\displaystyle k(g^{\circ})$
$\displaystyle=\left\\{\begin{array}[]{lcl}0&\mbox{if}&0\leq
g^{\circ}<\bar{g}^{L}\\\ 1&\mbox{if}&\bar{g}^{L}\leq
g^{\circ}<\bar{g}^{L-1}\\\ \;\vdots&\mbox{if}&\qquad\vdots\qquad\vdots\\\
r-2&\mbox{if}&\bar{g}^{H-1}\leq g^{\circ}<\bar{g}^{H}\\\
r-1&\mbox{if}&\bar{g}^{H}\leq g^{\circ}\ ,\end{array}\right.$ $\displaystyle
l(g^{\circ})$ $\displaystyle=\left\\{\begin{array}[]{lcl}r&\mbox{if}&0\leq
g^{\circ}<\bar{g}^{L}\\\ r-1&\mbox{if}&\bar{g}^{L}\leq
g^{\circ}<\bar{g}^{L-1}\\\ \;\vdots&\mbox{if}&\qquad\vdots\qquad\vdots\\\
2&\mbox{if}&\bar{g}^{H-1}\leq g^{\circ}<\bar{g}^{H}\\\
1&\mbox{if}&\bar{g}^{H}\leq g^{\circ}\ .\end{array}\right.$

Proof.

  * •

$k=0$, $l=1$. Same as in Proposition 2.

  * •

$0<k<l<r$. In order to preserve incentive compatibility, according to
condition (2) information rents have to be increasing (decreasing) over the
set of types who have the incentive to understate (overstate) their type. By
Lemma 5, the set of types is ordered. But then, rents are minimized if the
participation constraints of the lowest and highest types bind. Formally, the
proposer’s problem becomes

$\begin{gathered}\max_{g(\theta),t(\theta)}\qquad\int\theta_{a}\phi(g(\theta))-t(\theta_{a})\
dF(\theta)\\\ \text{subject to}\;\;g(\theta)\leq t(\theta_{a})+\sum_{i\in
N\setminus\\{a\\}}t(\theta_{i}),\\\ \frac{dg(\theta)}{d\theta}\geq 0,\\\
t(\theta_{i})=\left\\{\begin{array}[]{lcl}\theta_{i}\phi(g(\theta_{i},\theta_{-i}))-\int_{\underline{\theta}}^{\theta_{i}}\phi(g(x,\theta_{-i}))dx&\mbox{for}&\theta_{i}\leq\theta_{l},\\\
\theta_{i}\phi(g(\theta_{i},\theta_{-i}))-\int_{\theta_{i}}^{\overline{\theta}}\bar{v}_{i}(x,g^{\circ})-\phi(g(x,\theta_{-i}))\
dx&\mbox{for}&\theta_{i}>\theta_{l},\end{array}\right.\\\
u_{i}(\theta_{i},g(\theta))\geq\bar{v}_{i}(\theta_{i},g^{\circ})\qquad\forall\;i\in
N\setminus\\{a\\}\ .\end{gathered}$

Then, going through the same derivations as in Proposition 2 yields the
following condition for the optimal $g(\theta)$

$\left[\theta_{a}+\sum_{i>l}\left(\theta_{i}-\frac{\gamma^{*}-F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)+\sum_{i\leq
l}\left(\theta_{i}-\frac{\gamma^{*}-F_{i}(\theta_{i})}{f_{i}(\theta_{i})}\right)\right]\phi^{\prime}(g(\theta))=1\
.$

  * •

$k=r-1$, $l=r$. Same as in Proposition 2.

∎
This concludes the proof.
∎

###  A.7 Proof of Theorem 3

Proof. From Theorem 2, we immediately have the following corollaries.

######  Corollary 2.

For a given coalition $Q\in\mathcal{Q}$, the solution to the unanimity problem
has to hold.

######  Corollary 3.

The participation constraint holds for at least $q$ agents.

######  Corollary 4.

$K(g^{\circ})$ and $M(g^{\circ})$ are convex sets.

What remains to be done is identifying the set of agents whose participation
constraint is not satisfied.

######  Proposition 9.

Suppose $dU_{i}/d\theta_{i}$ has the same sign for all agents. Then, the
participation constraint binds on a single type $\theta_{l}$. There exists a
set $I(g^{\circ})=\left\\{i\in N\ |\ U(\theta_{i})<0\right\\}$, with
$|I(g^{\circ})|=n-q$, such that

  * (i)

If $dU_{i}/d\theta_{i}<0\ \forall i\in N$, then

$I=\left\\{i\in N\ |\ \theta_{i}\leq\theta_{l}\right\\}\ ,$

  * (ii)

If $dU_{i}/d\theta_{i}>0\ \forall i\in N$, then

$I=\left\\{i\in N\ |\ \theta_{i}\geq\theta_{l}\right\\}\ .$

Moreover, a subset of agents in $I$ is bunched at the allocation of
$\theta_{l}$.

Proposition 9 applies when $g^{\circ}$ is sufficiently high or low. When
agents have heterogeneous incentives to misreport their types -
$dU_{i}/d\theta_{i}$ does not have the same sign for all agents - then, the
choice of the set $I(g^{\circ})$ hinges on the distribution of information
rents induced by incentive compatibility; hence, it crucially depends on the
shape of the reservation utility profile. The concave and convex cases are
therefore dealt with separately.

####  A.7.1 Concavity

The construction needed is shown in Figure 6 and can be derived with a very
similar reasoning as in Lemma 11. Intuitively, it is efficient for the agenda-
setter to violate $n-q$ participation constraints. To preserve incentive
compatibility, we need the following lemma.
Formally, define $R(\gamma^{*})$ to be the utility differential between
$\theta_{p}$ and $\theta_{q}$

$R(\gamma^{*})=U(\theta_{q})-U(\theta_{p})=\int_{\theta_{p}}^{\theta_{q}}\phi(\hat{g}(\theta,\gamma^{*}))-\bar{v}^{\prime}_{i}(\theta_{i},g^{\circ})\
d\theta\ ,$

where $\hat{g}$ solves (4). Clearly, $R(\gamma^{*})$ is decreasing.

######  Lemma 13.

The optimal $\gamma^{*}$ satisfies

$\gamma^{*}\equiv\begin{cases}F_{q}(\theta_{q})&\mbox{if
$R(F_{q}(\theta_{q}))\geq 0$}\\\ R^{-1}(0)&\mbox{if
$R(F_{q}(\theta_{q}))<0<R(F_{p}(\theta_{p}))$}\\\ F_{p}(\theta_{p})&\mbox{if
$R(F_{p}(\theta_{p}))\leq 0$}\ .\end{cases}$

Proof. When $R(F_{q}(\theta_{q}))\geq 0$ or $R(F_{p}(\theta_{p}))\leq 0$,
Proposition 9 applies. Consider now the case
$R(F_{q}(\theta_{q}))<0<R(F_{p}(\theta_{p}))$. Choosing $\gamma^{*}$ such that
$R(\gamma^{*})=0$ ensures incentive compatibility is preserved.
As above, denote

$J(\theta;g^{\circ})\equiv\int_{\theta_{p}}^{\theta}\phi(\hat{g}(x;\gamma^{*}))-\bar{v}^{\prime}_{i}(x,g^{\circ})\
dx$ (16)

where $\hat{g}$ solves (4). We have the following lemma.

######  Lemma 14.

$J(\theta;g^{\circ})$ is quasiconvex in $\theta$.

Proof. $J^{\prime}(\theta;g^{\circ})=0$ when $G(\theta_{i};g^{\circ})=0$,
which occurs for a single $\theta_{i}$. By concavity of $\bar{v}_{i}$,
$J^{\prime}(\theta;g^{\circ})$ is increasing for $\theta^{\prime}>\theta$. ∎
Then, $U(\theta)=J(\theta;g^{\circ})+U(\theta_{p})$ and, by quasiconvexity,
$U(\theta)\leq\max\left\\{U(\theta_{p}),U(\theta_{q})\right\\}$ for all
$\theta\in[\theta_{p},\theta_{q}]$. Hence, if the participation constraint is
satisfied for $\theta_{p}$ and $\theta_{q}$, it is violated for all types in
between. ∎
Finally, the choice of the interval $(\theta_{p},\theta_{q})$ can only the
largest interval comprising $n-q$ agents.

######  Lemma 15.

The choice of the interval $(\theta_{p},\theta_{q})$ is such that:

$\theta_{1},\theta_{2},\dots,\underbrace{\theta_{p},\theta_{p+1}\dots,\theta_{q-1},\theta_{q}}_{|I|\leq
n-q},\dots,\theta_{n-1},\theta_{n}\ .$

Proof. Suppose not. If $(\theta_{p},\theta_{q})$ comprises more than $n-q$
agents, then the participation constraint is satisfied for fewer than $q-1$
agents, and the proposal cannot be implemented. On the other hand, if
$(\theta_{p},\theta_{q})$ comprises fewer than $n-q$ agents, then extending it
to include just $n-q$ agents does not affect incentive compatibility and
decreases the rents paid out by the agenda-setter, a contradiction. ∎

####  A.7.2 Convexity

When the reservation utility profile is convex, it was shown in Section A.6
that the participation constraints binds on the highest and lowest types.
Along a very similar reasoning as in Proposition 9, we have the following
proposition.

######  Proposition 10.

There exist two types $\theta_{p}$ and $\theta_{q}$ such that

$I(g^{\circ})=\left\\{i\in N\ |\ \theta_{i}\leq\theta_{p}\ \cup\
\theta_{i}\geq\theta_{q}\right\\}\ .$

Moreover, a subset of agents might be bunched at the same allocation of
$\theta_{p}$ and $\theta_{q}$.

Proof. When $g^{\circ}$ is sufficiently low or high, Proposition 9 applies.
For intermediate values of $g^{\circ}$, in the unanimity case the
participation constraints are binding for the lowest and highest types, as
shown in Section A.6. Incentive compatibility is preserved if transfers are
increased uniformly so as to induce the participation constraints of types
$\theta_{p}>\underline{\theta}$ and $\theta_{q}<\overline{\theta}$ bind.
Moreover, whenever $g(\theta)$ is lower than (1), agents
$\theta_{i}<\theta_{p}$ can be bunched at the allocation of $\theta_{p}$ so as
to increase public-good provision. Analogously, when $g(\theta)$ is higher
than (1), agents $\theta_{i}>\theta_{q}$ can be bunched at the allocation of
$\theta_{q}$in order to decrease public-good provision. ∎

∎

###  A.8 Proof of Proposition 6

As in Jullien, (2000), define $g(\theta)$ as the public-good level such that

$\phi(g(\theta))=\mathbb{E}_{\mu}\left[\phi(g(\theta))\ |\ \theta\right]\ ,$

where $\mathbb{E}_{\mu}\left[\cdot\ |\ \theta\right]$ is the conditional
expectation operator for the distribution $\mu(\theta,dg)$.

Proof. Recall condition (2) and consider two types, $\theta_{1}$ and
$\theta_{2}$. We have that

$\displaystyle\int_{\theta_{1}}^{\theta_{2}}\phi(g(\theta))-\bar{v}^{\prime}_{i}(\theta_{i},g^{\circ})\
d\theta_{i}$
$\displaystyle\geq\int_{\theta_{1}}^{\theta_{2}}\int_{g}\phi(g(\theta))\
\mu(\theta_{1},dg))\
d\theta_{i}-\int_{\theta_{1}}^{\theta_{2}}\bar{v}^{\prime}_{i}(\theta_{i},g^{\circ})\
d\theta_{i}$
$\displaystyle=\int_{\theta_{1}}^{\theta_{2}}\phi(g(\theta_{1},\theta_{-i}))\
d\theta_{i}-\bar{v}^{\prime}_{i}(\theta_{i},g^{\circ})\ d\theta_{i}\ ,$

where the first inequality follows from Jensen’s inequality and the last
equality by the linearity in types of the utility function. Therefore,
$g(\theta)$ is incentive compatible.
Moreover, by concavity of the objective function, we have that

$\sigma(\theta,\gamma(\theta),g)\geq\sigma\left(\theta,\gamma(\theta),\mathbb{E}_{\mu}[g]\right).$

∎

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