Information sharing in democratic mechanisms

Abstract

We examine how democratic mechanisms can yield socially desirable outcomes in the presence of uncertainty about an underlying state of nature. We depart from a conventional mechanism design approach because we aim for democratic mechanisms to reflect some basic properties of decision-making in democracies. In particular, actual decisions are made by majority voting. The proposals to be voted upon are made by a selfish agenda-setter. Moreover, communication is limited to a binary message space (that is, voting Yes or No). We show how suitable democratic mechanisms can resolve uncertainty, reveal the state of nature, and implement the Condorcet winner. We demonstrate that this implementation result requires (at most) two voting stages regardless of the number of states or the number of alternatives. We also show that implementation requires a conditional privilege for a small representative subset of the population.

Appendices

Appendix A

In Appendix A, we provide a detailed account of assumptions on the cost function, the probability distributions, the type space, and the Condorcet winners which generate the properties imposed in Sect. 2 and, in particular, in Assumption 1.

1.1 Cost function

Although we will eventually consider a discrete set of possible quantities, it is convenient to start by defining continuous cost and utility functions on ${\mathbb {R}}_{+}.$ In particular, we assume that the per capita cost of providing a quantity $q\in {\mathbb {R}}_+$ of the public good is given by a twice continuously differentiable, strictly increasing, and strictly convex function $c: {\mathbb {R}}_{+} \rightarrow {\mathbb {R}}_{+}$ with $c(0)=0.$ This implies in particular that average cost c(q)/q is strictly increasing. One typical interpretation of such a cost function is that each citizen is initially endowed with $w \ (w>0)$ units of a private consumption good which can either be consumed or transformed into the public good. The per capita costs c(q) then represent the utility losses due to foregone private consumption.

1.2 Probability distributions

We assume that the family of probability distributions from which the types are drawn in each state can be represented by continuously differentiable probability density functions which satisfy a property known as the monotonicity of likelihood ratios. More formally, let the first derivative of $f_{k}$ be $(f_{k}^{\prime })_{k=1,\ldots ,n}.$ Then, we assume that $f_k (z)>0$ for all $k\in N$ and all $z\in Z$ and, moreover, that

$$\begin{aligned} \frac{f'_{k+1} (z)}{f_{k+1} (z)}>\frac{f'_{k} (z)}{f_{k} (z)}, \quad \forall z\in Z, \ \forall k\in N\setminus \{n\}. \end{aligned}$$

(3)

The monotonicity of likelihood ratios property has three key implications. First, the probability distribution associated with $F_{k+1}$ (strictly) first-order stochastically dominates the one associated with $F_{k}$ for every $k\in N\setminus \{n\},$ that is, $F_{k+1}(z) < F_{k}(z)$ for every $z\in int(Z).$ In that sense, the benefits from the public good are higher in state $k+1$ than in state k. Second, the monotonicity of likelihood ratios implies a single-crossing property of the probability density functions, which will be crucial for our analysis. Finally, the monotonicity of likelihood ratios imposes monotone Bayesian updating (Milgrom 1981). More specifically, citizen z believes in state $k=1,\ldots ,n$ with probability

$$\begin{aligned} \beta _{k}(z)=\frac{f_{k}(z) p_k}{\sum _{j=1}^{n} f_{j}(z) p_j}, \end{aligned}$$

(4)

and the associated probability distributions for citizen $z_{2}$ stochastically dominates the one for citizen $z_{1}$ if $z_{1}< z_{2}.$

We note that the monotonicity of likelihood ratios implies, loosely speaking, that a higher type tends to believe in higher states of nature with higher probability.

1.3 Type space

We assume that, for every $z\in Z,$ the most desired public good level of citizen z belongs to Q. Formally, since the most preferred public good level of citizen z is given by $z=c^{\prime }(q),$ we assume $c^{\prime }(0) \le inf(Z)$ and $c^{\prime }(q^{max}) \ge sup (Z)$ where inf(Z) and sup(Z) are the infimum and supremum of the type space Z and $q^{max}$ the highest possible public good level in Q.¹²

Together with the strict convexity of the cost function, this property implies that the preferences of each type $z\in Z$ are single-peaked, and the most preferred quantity of citizen z is that q which solves $c^{\prime }(q)=z.$

Moreover, we assume $c(q_{1})/q_{1} \in int(Z).$

1.4 Condorcet winner

Citizen z is the median voter in state k if $F_{k} (z) = 1/2.$ We define for each state $k\in N$ a quantity $q_{k} \in {\mathbb {R}}_{+}$ as the unique solution to

$$\begin{aligned} F_{k} \left( c'(q_{k}) \right) = {1}/{2}, \end{aligned}$$

so that $q_{k}$ is the most preferred quantity of the median voter in state k. The definition and the assumptions regarding the cost function and the monotonicity of likelihood ratios imply that $0<q_{1}<\cdots <q_{n}.$ Due to the strict convexity of the cost function, we have

$$\begin{aligned} \frac{c(q_{k})-c(q)}{q_{k}-q}< & {} c^{\prime } (q_{k}), \quad \forall q < q_{k}, \ \forall k\in N,\\ \frac{c(q_{k})-c(q)}{q_{k}-q}> & {} c^{\prime } (q_{k}), \quad \forall q > q_{k}, \ \forall k\in N.\\ \end{aligned}$$

Due to the single-peaked preferences, these statements can be expressed in terms of the distribution functions as Inequalities (5) and (6) below. These expressions are well-defined since $c(q_{1})/q_{1} \in int(Z).$

$$\begin{aligned} F_{k} \left( \frac{c(q_{k}) - c(q) }{q_{k} - q} \right)< 1/2, \quad \forall q<q_{k}, \ \forall k\in N, \end{aligned}$$

(5)

$$\begin{aligned} F_{k} \left( \frac{c(q_{k}) - c(q) }{q_{k} - q} \right)> 1/2, \quad \forall q>q_{k}, \ \forall k\in N. \end{aligned}$$

(6)

Verbally, in state k,  a simple majority of citizens prefers $q_k$ over any other quantity $q\in {\mathbb {R}}_{+} \setminus \{q_{k} \}.$ Thus, $q_{k}$ is the Condorcet winner in state k.

1.5 Point of indifference

Moreover, for every $k\in N,$ we define the quantity $\widetilde{q}_{k} \in {\mathbb {R}}_{+}$ as the unique¹³ solution to

$$\begin{aligned} F_{k}\left( c({\widetilde{q}}_{k}) / {\widetilde{q}}_{k} \right) = 1/2. \end{aligned}$$

(7)

In state k,  a majority of citizens prefers any quantity $q<{\widetilde{q}}_k$ to zero public good, but prefers zero public good to any quantity $q>{\widetilde{q}}_k.$ Given that average cost c(q)/q is strictly increasing and that $c(q)/q < c^{\prime }(q),$ we have ${\widetilde{q}}_{k} > q_{k}$ for every $k\in N.$

Appendix B

1.1 Proof of Proposition 1

Take any $P\in {\mathcal {P}},$ and suppose that citizens do not manipulate information sharing. That is, they use a communication strategy $\sigma ^{P}$ such that $\varphi (\delta _{i}(\sigma ^{P})) = q_{i}$ for every $i\in N.$ Due to sincere voting, if the true state is i,  the agenda-setter anticipates that any proposal $q\in Q$ such that $q < \widetilde{q}_{i}$ is going to win against the status quo. Suppose that the agenda-setter’s type z is such that $z=c^{\prime }(q)$ for some $q\in Q$ such that $q < {\widetilde{q}}_{i}$ and $q\ne q_{i}.$ In this case, it is in the agenda-setter’s interest to choose $\rho ^{P} (z, \delta _{i}(\sigma ^{P})) = q$ rather than $\rho ^{P} (z, \delta _{i}(\sigma ^{P})) = q_{i}.$ Indeed, he can exploit information sharing. $\square$

1.2 Proof of Proposition 2

Suppose by way of contradiction that a baseline democratic mechanism implements the Condorcet winner. Take some public good problem $P\in {\mathcal {P}}$ which has the distance property, and in which a state $i\in N\setminus \{n\}$ is concealable. Let $\sigma ^{P}$ be a communication strategy and $\rho ^{P}$ a proposal strategy such that

$$\begin{aligned} \varphi ^{P}(\delta _{k}(\sigma ^{P})) = \rho ^{P}(z,\delta _{k}(\sigma ^{P})) = q_{k} \end{aligned}$$

for every $k\in N$ and every $z\in Z.$ By construction of $\varphi ^{P},$ the communication strategy $\sigma ^{P}$ is such that citizen z votes Yes if and only if $z\ge z^{P}.$ We show that $\sigma ^{P}$ cannot be optimal.

Due to the premise that state $i\in N\setminus \{n\}$ is concealable, the function $F_{i}(z)-F_{i+1}(z)$ has a unique maximizer $z^{*}_{i}$ which belongs to $Z_{-}.$

Let ${\widetilde{z}} = \min \{z^{P}, z^{*}_{i}\}.$ We have

$$\begin{aligned} \delta _{i+1} (\sigma ^{P}) - \delta _{i} (\sigma ^{P}) \le F_{i} ({\widetilde{z}}) - F_{i+1}({\widetilde{z}}). \end{aligned}$$

In particular, this inequality implies

$$\begin{aligned} \delta _{i} (\sigma ^{P}) + F_{i}({\widetilde{z}}) \ge \delta _{i+1}(\sigma ^{P}). \end{aligned}$$

Due to continuity of the cumulative distribution function, there is a $\zeta$ such that $\zeta \le {\widetilde{z}} \le z^{*}_{i} \le {\widehat{z}}$ and

$$\begin{aligned} \delta _{i}(\sigma ^{P}) + F_{i}(\zeta ) = \delta _{i+1}(\sigma ^{P}). \end{aligned}$$

Consider a joint deviation from $\sigma ^{P},$ under which citizens $z\le \zeta$ vote Yes, and citizens $z> \zeta$ vote according to $\sigma ^{P}.$ Since $\zeta \le {\widehat{z}},$ this is a deviation by members of $Z_{-}$ only. If the true state is i,  then the resulting share of Yes-votes is $\delta _{i}(\sigma ^{P}) + F_{i}(\zeta ) = \delta _{i+1}(\sigma ^{P}).$ Thus, under this deviation, the proposal $q_{i+1}$ will be made at the voting stage with probability one if the true state is i. Since citizens vote sincerely, this proposal will be rejected since $q_{i+1} > {\widetilde{q}}_{i},$ and the resulting public good level will be zero. By construction, members of $Z_{-}$ prefer zero over $q_i.$ Now we have shown that the deviation by $Z_{-}$ is (strictly) profitable if the true state is i. Suppose next that the true state is some $j\in N\setminus \{i\}.$ Due to the distance property, no quantity $q\in Q$ with $q>q_{j}$ would be accepted at the voting stage in state j. Thus, under the deviation, the outcome must be some $q^{\prime }_{j}\le q_{j}$ if the state is j. All members of $Z_{-}$ weakly prefer $q^{\prime }_{j}$ to $q_{j}.$ Since $p_{i} > 0$ and thus $\beta _{i}(z)> 0$ for all $z\in Z,$ the deviation is (strictly) profitable in expectation. $\square$

Appendix C

1.1 Proof of Theorem 1

Now we turn to the proof of the implementation result. First, we are going to consider the final voting stage. The criterion for sincere voting differs depending on whether the final vote is between the proposal and the status quo or between the amendment and the status quo. We state the appropriate definition of sincere voting. Second, we are going to establish conditions under which decisions at the selection stage are also sincere, in a sense to be made precise. Third, we state a profile of strategies and beliefs which is a Bayesian equilibrium, involves sincere selection, and leads to the implementation of the Condorcet winner.

1.2 Sincere voting

The voting stage of the democratic mechanism with sampling and two-stage voting is a final and binary choice which leaves no room for profitable strategic behavior. As a result, citizens vote sincerely at the voting stage. However, the sample group receives a privilege conditional on final acceptance of the proposal. This conditional privilege must be taken into account in order to determine the appropriate meaning of “sincere voting” in the present context. Suppose that at the voting stage, the alternative $q\in Q$ is pitted against the status quo, which is zero public good provision. Consider first the case where q is not the original proposal, but the amendment. In that case, neither tax-exemptions nor transfers will be granted, so that sincere voting simply means that citizen z votes for q if and only if $z\ge c(q)/q.$

Now suppose instead that q is the original proposal made by the agenda-setter. Since $zq>0$ for all $q\in Q \setminus \{0\}$ and all $z\in Z,$ it is sincere for every sample group member to vote in favor of q. Citizen z,  who is not a sample group member, votes sincerely in favor of q if

$$\begin{aligned} z> \frac{ c(q) + \lambda \tau }{(1-\lambda )q}. \end{aligned}$$

The conditional privilege distorts sincere voting behavior for citizens outside the sample group. We argue that this distortion is negligible, however, when $\lambda >0$ is small enough.

For any given values of $\lambda$ and $\tau ,$ and for every state $i\in N,$ define ${\widetilde{q}}^{(\lambda , \tau )}_{i}$ as the solution to the equality

$$\begin{aligned} F_{i} \left( \frac{ c(\widetilde{q}^{(\lambda , \tau )}_{i} ) +\lambda \tau }{(1-\lambda ) \widetilde{q}^{(\lambda , \tau )}_{i} } \right) = 1/2. \end{aligned}$$

We note that ${\widetilde{q}}^{(\lambda , \tau )}_{i}<{\widetilde{q}}_{i}$ for $\lambda >0$ and $\tau \ge 0.$ Moreover, for any value of $\tau \ge 0,$ we find that ${\widetilde{q}}^{(\lambda , \tau )}_{i}$ converges to ${\widetilde{q}}_{i}$ as $\lambda \downarrow 0.$ Verbally, considering the limit as the sample group becomes arbitrarily small, the distortion of sincere voting behavior due to the conditional privilege vanishes.

1.3 Sincere selection

We turn to the selection stage which precedes the voting stage. In this selection stage, voters select either a proposal q or an amendment $\psi _{-}(q).$ In principle, a “selection strategy” should specify whether a citizen of a given type with a given belief about the state of nature selects any given proposal or the associated amendment. For the purpose of our argument, however, it will be enough to specify the optimal selection by citizens who believe with probability one that any alternative selected at the selection stage will win against the status quo at the voting stage. In that case, the decision to select the proposal or the amendment is a final and binary choice of the alternative to be implemented. Consequently, it is optimal to select sincerely, and we only have to spell out what sincere selection means: A citizen z outside the sample group prefers proposal q over amendment $\psi _{-}(q)$ if $zq - \frac{c(q) + \lambda \tau }{(1-\lambda )} > z\psi _{-}(q) -c(\psi _{-}(q)).$ A sample group member always prefers q over $\psi _{-}(q)$ because $\tau + zq > z\psi _{-}(q) -c(\psi _{-}(q)),$ that is, a sample group member sincerely selects the proposal.

1.4 Strategies and beliefs

We now proceed with the construction of a communication strategy, a proposal strategy, and some consistent beliefs. The communication strategy is as follows:

$$\begin{aligned} \sigma ^{P} (z) = {\left\{ \begin{array}{ll} Yes &{}\quad \text {if } \ z\ge z^{P},\\ No &{}\quad \text {otherwise.} \end{array}\right. } \end{aligned}$$

The proposal strategy is such that the agenda-setter does not exploit information sharing, that is, for every $z\in Z,$ we have $\rho ^{P}(z,\delta ) = \varphi ^{P}(\delta ),$ thus:

$$\begin{aligned} \rho ^{P}(z,\delta ) = {\left\{ \begin{array}{ll} q_{1} \ &{}\quad \text {if} \ 0\le \delta< 1-F_{2} (z^{P}),\\ q_{k} \ &{}\quad \text {if} \ 1-F_{k}(z^{P}) \le \delta < 1-F_{k+1}(z^{P}), \ k=2,\ldots , n-1,\\ q_{n} \ &{}\quad \text {if} \ 1-F_{n} (z^{P}) \le \delta \le 1. \end{array}\right. } \end{aligned}$$

The agenda-setter (regardless of his type) believes with probability one that information sharing has been “successful” so that the alternative identified by the information mapping is the Condorcet winner. Thus, using $e_{i}$ to denote the n-vector with component i equal to one and all other components equal to zero, we can write his beliefs as follows:

$$\begin{aligned} \pi ^{P}_{AS} (\delta ) = {\left\{ \begin{array}{ll} e_{1} \ &{}\quad \text {if} \ 0\le \delta< 1-F_{2} (z^{P}),\\ e_{k} \ &{}\quad \text {if} \ 1-F_{k}(z^{P}) \le \delta < 1-F_{k+1}(z^{P}), \ k=2,\ldots , n-1, \\ e_{n} \ &{}\quad \text {if} \ 1-F_{n} (z^{P})\le \delta \le 1. \end{array}\right. }\end{aligned}$$

Citizens other than the agenda-setter, regardless of their type, believe with probability one that the proposal $q\in Q$ made by the agenda-setter is the Condorcet winner, thus:

$$\begin{aligned} \pi ^{P} (q) = {\left\{ \begin{array}{ll} e_{1} \ &{}\quad \text {if} \ q<q_{2},\\ e_{k} \ &{}\quad \text {if} \ q_{k}\le q < q_{k+1}, \ k=2,\ldots , n-1,\\ e_{n} \ &{}\quad \text {if} \ q \ge q_{n}. \end{array}\right. } \end{aligned}$$

We note that citizens other than the agenda-setter base their beliefs only on the proposal q made by the agenda-setter, while the observed share $\delta$ of positive signals determines the agenda-setter’s beliefs. This construction of beliefs has allowed us to specify optimal selection only for the case where citizens believe that whichever alternative they select at the selection stage is going to become the final outcome of the mechanism.

In order to establish the theorem, we need to show two things: first, the communication strategy $\sigma ^{P}$ and the proposal strategy $\rho ^{P}$ as defined above actually lead to the implementation of the Condorcet winner in every state. Second, these strategies are optimal given the beliefs $\pi ^{P}_{AS} (\delta )$ and $\pi ^{P}(q).$

1.4.1 Implementation

We see that $\rho ^{P} (z, \delta _{k}(\sigma ^{P})) = \varphi ^{P} (\delta _{k} (\sigma ^{P})) = q_{k}$ for every $k\in N.$ Thus, if the communication strategy $\sigma ^{P}$ and the proposal strategy $\rho ^{P}$ are played, then the agenda-setter believes with probability one in the true state, and he proposes the Condorcet winner. The beliefs $\pi ^{P}_{AS}$ and $\pi ^{P}$ are such that citizens believe with probability one that the quantity proposed by the agenda-setter is the Condorcet winner. Hence, all citizens believe that both the proposal made by the agenda-setter and the amendment would prevail against the status quo at the voting stage. Citizens select sincerely at the selection stage. In particular, all sample group members select the proposal. Since the proposal is the Condorcet winner, also at least half of the citizens outside the sample group select it. Hence, in the entire population, a share of at least $(1+\lambda )/2$ select the proposal. Because of sincere voting at the voting stage and the fact that the proposal is the Condorcet winner on the path of play, the Condorcet winner does become the final outcome of the mechanism, as desired.

1.4.2 Optimal signaling and agenda-setting

Now we show that no deviation by the agenda-setter or by sample group members can be profitable. Notice that sample group members and agenda-setter receive the payoff $zq_{k}+\tau \ge zq_{k}$ on the path induced by the profile under consideration when the state is k. Suppose by way of contradiction that there is a profitable deviation for the agenda-setter or (some coalition within) the sample group. For every $k\in N,$ let ${\widehat{q}}_{k}$ be the outcome which follows after a profitable deviation. Due to the premise that the deviation is profitable, there must be a state $i\in N$ such that ${\widehat{q}}_{i}>q_{i}.$ This is only possible if the agenda-setter proposed either ${\widehat{q}}_{i}$ or $\psi _{+}({\widehat{q}}_{i}).$ Either way, by construction of $\pi ^{P},$ citizens believe with probability one that both the proposal and the amendment would win against the status quo at the voting stage. Hence, citizens outside the sample group select sincerely, while sample group members select the proposal. However, the majority of citizens outside the sample group prefer $q_{i}$ over ${\widehat{q}}_{i}$ simply because $q_{i}$ is the Condorcet winner in state i. Hence, ${\widehat{q}}_{i}>q_{i}$ can only be the outcome of the mechanism if it has been an amendment at the selection stage, and has then prevailed against the status quo $q=0$ at the voting stage. Hence, under the deviation, there is no tax exemption and no transfer. Since the deviation is profitable for an agenda-setter or sample group member of type z,  it must hold that $z{\widehat{q}}_{i} -c({\widehat{q}}_{i}) > zq_{i} + \tau .$ However, for any $z\in Z,$ this inequality is violated for sufficiently large $\tau .$ Indeed, we obtain a contradiction for large enough $\tau .$$\square$

Appendix D

In Appendix D, we show that the democratic mechanism with sampling and one-stage voting implements the Condorcet winner in those public good problems which satisfy the distance property.

1.1 Sincere voting

From the point of view of citizens outside the sample group, the provision of a public good quantity $q\in Q$ is no longer associated with a tax burden of c(q),  but of $\left( \frac{1}{1-\lambda }\right) c(q).$ Consequently, the utility of citizen z outside the sample group from the public good level $q\in Q$ is given by

$$\begin{aligned} {\widehat{u}} (z,q) = zq -\left( \frac{1}{1-\lambda }\right) c(q). \end{aligned}$$

The popular vote in the democratic mechanism with sampling is a binary and final choice and thus strategic voting can never be beneficial. The citizens vote sincerely in the democratic mechanism with sampling and one-stage voting. We now formally spell out the sincere voting rule in the democratic mechanism with sampling and one-stage voting. Of course, this rule is different for sample group members and for regular citizens. Indeed, suppose that at the voting stage of the democratic mechanism with sampling and one-stage voting the alternative $q\in Q$ is pitted against the status quo, which is zero public good provision. Since $Z\subset {\mathbb {R}}_{++},$ we have that $zq>0$ for every $q\in Q\setminus \{0\},$ and thus all sample group members vote sincerely in favor of the proposal. Citizen z,  who is not a sample group member, prefers q to zero public good provision if

$$\begin{aligned} z > \frac{c(q)}{(1-\lambda )q }.\end{aligned}$$

Analogously to the argument in Appendix C, the privileges for the sample group distort sincere voting behavior of citizens outside the sample group. Again, we argue that this distortion is negligible when $\lambda$ is small.

For every state $i\in N,$ define ${\widetilde{q}}^{\lambda }_{i}$ as the solution to the equality

$$\begin{aligned} F_{i} \left( \frac{c(\widetilde{q}^{\lambda }_{i})}{(1-\lambda ){\widetilde{q}}^{\lambda }_{i}} \right) = 1/2.\end{aligned}$$

In words, if the true state is i,  then a majority of citizens outside the sample group prefers any quantity $q\in Q$ with $q<{\widetilde{q}}^{\lambda }_{i}$ over zero public good provision, and prefers zero public good provision to any quantity $q\in Q$ such that $q>{\widetilde{q}}^{\lambda }_{i}.$ The assumptions introduced on the cumulative distribution functions $(F_{i})_{i\in N}$ guarantee that ${\widetilde{q}}^{\lambda }_{i}$ exists.

For every $\lambda > 0,$ we have ${\widetilde{q}}^{\lambda }_{i} < {\widetilde{q}}_{i},$ and we find that ${\widetilde{q}}^{\lambda }_{i}$ converges to ${\widetilde{q}}_{i}$ as $\lambda \downarrow 0.$ Suppose that $q\in Q$ is a proposal which is preferred by the majority to zero public good provision under uniform taxation. Then, q is also preferred to zero public good provision by the majority under the tax-exemption, provided that $\lambda >0$ is small enough.

1.2 Strategies and beliefs

Define the following communication strategy $\sigma ^{P}$:

$$\begin{aligned} \sigma ^{P} (z) = {\left\{ \begin{array}{ll} \text {Yes} \ &{}\quad \text {if } \quad z\ge z^{P},\\ \text {No} \ &{}\quad \text {otherwise.} \end{array}\right. } \end{aligned}$$

Moreover, consider the proposal strategy:

$$\begin{aligned} \rho ^{P}(z,\delta ) = {\left\{ \begin{array}{ll} q_{1} \ &{}\quad \text {if} \ 0\le \delta< 1-F_{2} (z^{P}),\\ q_{k} \ &{}\quad \text {if} \ 1-F_{k} (z^{P}) \le \delta < 1-F_{k+1} (z^{P}), \ k=2,\ldots , n-1,\\ q_{n} \ &{}\quad \text {if} \ 1-F_{n} (z^{P}) \le \delta \le 1 . \end{array}\right. } \end{aligned}$$

Define a belief for the agenda-setter as:

$$\begin{aligned} \pi ^{P}_{AS} (\delta ) = {\left\{ \begin{array}{ll} e_{1} \ &{}\quad \text {if} \ 0\le \delta< 1-F_{2} (z^{P}),\\ e_{k} \ &{}\quad \text {if} \ 1-F_{k} (z^{P}) \le \delta < 1-F_{k+1} (z^{P}), \ k=2,\ldots , n-1,\\ e_{n} \ &{}\quad \text {if} \ 1-F_{n} (z^{P}) \le \delta \le 1 . \end{array}\right. } \end{aligned}$$

It is straightforward that the belief $\pi ^{P}_{AS}$ is consistent with the strategies $\sigma ^{P}$ and $\rho ^{P}.$ Moreover, if these strategies are played, then the outcome is the Condorcet winner. In order to establish the implementation result, we need to show that $\sigma ^{P}$ and $\rho ^{P}$ are optimal given the beliefs $\pi ^{P}_{AS}.$ We show first that the proposal strategy $\rho ^{P}$ is optimal. Indeed, let i be the true state, so that $q_{i}$ is the Condorcet winner. Due to the distance property and sincere voting, any proposal $q\in Q$ such that $q>q_{i}$ will be rejected, and any proposal $q\in Q$ such that $q \le q_{i}$ will be accepted in the popular vote. Since the agenda-setter is tax-exempt, it follows immediately that it is optimal for him to propose the quantity $q_{i}$ whenever his belief is $e_{i}.$ Otherwise, if the agenda-setter makes any proposal $q\in Q\setminus \{q_{i}\},$ the outcome of the mechanism is some $q^{\prime }\in \{q, 0\},$ and $q^{\prime } < q_{i}.$ Next we show that the communication strategy is optimal. All members of the sample group as well as the agenda-setter are tax-exempt. A joint deviation by a subset of the sample group (or a deviation by the agenda-setter) could only be profitable if it led to some public good level $q > q_{i}.$ Due to sincere voting, however, such a public good level will not be implemented.

$\square$