Abstract
This study examines the mechanism design problem for public goods provision in a large economy with n independent agents. We propose a class of dominantstrategy incentive compatible and expost individually rational mechanisms, which we call the adjusted meanthresholding (AMT) mechanisms. We show that when the cost of provision grows slower than the \(\sqrt{n}\)rate, the AMT mechanisms are both eventually exante budget balanced and asymptotically efficient. When the cost grows faster than the \(\sqrt{n}\)rate, in contrast, we show that any incentive compatible, individually rational, and eventually exante budget balanced mechanism must have provision probability converging to zero and hence cannot be asymptotically efficient. The AMT mechanisms have a simple form and are more informationally robust when compared to, for example, the secondbest mechanism. This is because the construction of an AMT mechanism depends only on the first moment of the valuation distribution.
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Notes
Myerson and Satterthwaite (1983) establish the impossibility result in the bilateral trade setting, but it extends directly to the public goods problem. We provide a more substantial overview of this problem below.
For any measurable set E, we use \({\mathbf {1}}E\) to denote the indicator function of E.
The central limit theorem states that the distribution of sum of iid random variables is approximately a normal distribution. The BerryEsseen theorem further specifies how close the normal approximation is.
This version can be found at: https://doi.org/10.48550/arXiv.2101.0242.
Hellwig (2003) also identifies the \(\sqrt{n}\)rate in the context of Bayesian incentive compatibility. Our results are focused on the dominantstrategy incentive compatibility. One takeaway here is that the weaker notion of Bayesian incentive compatibility does not relax this bound. This notion of equivalence between Bayesian incentive compatibility and dominantstrategy incentive compatibility represented by the requirement on the growth rate of cost can be seen as a complement to the theoretical results in Gershkov et al (2013).
This moment can come from any increasing and continuous transformation of the valuation. See Sect. 3.
In Sect. 3.3, we provide a more formal discussion of the qualitative difference between a moment and the distribution by using results in the information theory.
In Sect. 3, we discuss why the support is assumed to be in this form.
We can relax the assumptions that \({\bar{v}}\) is finite and that f is bounded away from zero. We only need to assume that the appropriate moments exist. However, that makes the exposition inconvenient.
In this paper, we use upper case letters to denote random variables and lower case letters to denote realizations or reported values.
This definition of social welfare is standard in the literature. From the social planner’s perspective, the production cost does not enter into welfare because it does not directly affect agents’ utility. The cost only plays a role through the budget constraint.
The valuations explored in Kunimoto and Zhang (2021) are discrete. However, this aspect does not impact the results qualitatively, as both discrete and continuous distributions can be addressed using the general measure theory. Specifically, discrete probability measures have density functions with respect to the counting measure, enabling the computation of virtual valuations accordingly.
They refer to the “regret ratio” in this paper as “competitive ratio.”
This definition is equivalent to Definition 3.8 in Börgers (2015).
Since F is supported on \([0,{\bar{v}}]\), we have \(\int _{0}^{{\bar{v}}} (1F(v)) dv = {\mathbb {E}}[V_i]\), \({\mathbb {E}}[\psi (V_i)] = {\mathbb {E}}[V_i]  \int _{0}^{{\bar{v}}} (1F(v)) dv\) = 0. See the end of Sect. 3.2 for a discussion of the support.
See Eq. (12) and the discussion before it.
A similar intuition can be found in the proof of Proposition 3 in Hellwig (2003). Here, we provide a more detailed discussion regarding the rate of the threshold \(\alpha _n\).
However, we can extend our analysis to allow for a point mass at 0 in the distribution F. In that case, the density f is the RadonNykodim derivative of the probability measure F with respect to the measure \(L + \delta _0\), where L is the Lebesgue measure on \({\mathbb {R}}_+\) and \(\delta _0\) is the Dirac measure at 0. This underlying measure is \(\sigma\)finite, and hence our analysis applies.
We are grateful to a referee for pointing this out.
For every random variable X and increasing functions \(g_1\) and \(g_2\), the random variables \(g_1(X)\) and \(g_2(X)\) are positively correlated (see, e.g., Thorisson 1995).
To see this, note that the definition of \(\psi\) implies \(d\log (1F(v)) / dv = 1/(v\psi (v))\).
See, for example, Stone (1980). If the density f is differentiable, then the optimal rate of convergence is \(m^{1/3}\). Please refer to that paper for the exact definition of optimal convergence rates.
The optimal uniform convergence rate of density estimators is derived in Stone (1983).
To obtain a crude estimate, we can use Lemma 7 in the Appendix, which indicates that the expected total payment from the pivot mechanism is at most \(O\left( n^{1/4}\right)\). Therefore, the pivot mechanism runs a growing budget deficit if the cost grows faster than that rate.
Notice that, however, the cost cannot grow exactly at the \(\sqrt{n}\)rate, because in that case we would need the adjustment term \(\alpha _n\) to also decrease at the \(\sqrt{n}\)rate to balance the budget, leading to an inefficient allocation decision.
The nature of the assumption on the growth rate of \(c_n\) is essentially about the production technology and how the marginal cost decreases with the number of agents. See Roberts (1976) for a discussion on the relationship between the cost of a public good and the number of consumers.
The first five notations are standard. The expression \(d_n = O_\varepsilon (e_n)\) means that \(d_n\) is asymptotically bounded above by \(e_n\) divided by some (sufficiently small) power of n. The expression \(d_n = \omega _{\varepsilon }(e_n)\) means that \(d_n\) asymptotically dominates \(e_n\) divided by any power of n.
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Acknowledgements
We appreciate the Editor Clemens Puppe, an Associate Editor, and two referees whose valuable comments and suggestions have significantly improved this paper. We are grateful to Joel Sobel for his constant support and helpful discussions on this paper. We also thank Songzi Du, Martin Hellwig, and Andrew Postlewaite.
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Appendices
Technical proofs
Appendix A.1 presents general convergence results dervied based on the BerryEsseen theorem. Appendix A.2 presents the proofs for theorems in the main text.
1.1 Preliminary results
This section provides some general results as useful lemmas. We denote \(\phi\) and \(\Phi\) respectively as the pdf and cdf of the standard normal distribution. We use C to denote a generic constant that does not depend on n, which may have different values at each appearance.
Lemma 2
Let X and Y be two random variables with \({\mathbb {E}}{X} < \infty\), then for any \(\alpha \in {\mathbb {R}}\),
In particular, if \(X=Y\), then
and
Proof of Lemma 2
Define \(X^+ = \max \{X,0\}\) and \(X^ = \max \{X,0\}\). For \(X^+\), notice that
Using Fubini theorem, we have
Similarly, for \(X^\), we have
The result then follows from the fact that \({\mathbb {E}}[X] = {\mathbb {E}}[X^+]  {\mathbb {E}}[X^]\). \(\square\)
Let \((X_i,Y_i), 1 \le i \le n\) be an independent sequence of random vectors in \({\mathbb {R}}^2\) with zero mean and \({\mathbb {E}}{X_i}^3,{\mathbb {E}}{Y_i}^3 < \infty\). We introduce the following set of notations for their marginal moments: \(\sigma ^2_{X} = {\mathbb {E}}X_i^2, \sigma _{Y} = {\mathbb {E}}Y_i^2,\rho _{X} = {\mathbb {E}}{X_i}^3, \rho _{Y} = {\mathbb {E}}{Y_i}^3,\) and correlation: \(\sigma _{XY} = {\mathbb {E}}[X_iY_i].\) All these moments are finite. Define \(S^X_n = \sum X_i\) and \(S^Y_n = \sum Y_i\). Let \((Z^X,Z^Y)\) be a joint normal random vector with zero mean and the same covariance structure as \((S^X_n,S^Y_n)\). We use \(\Vert {\cdot }\Vert\) to denote both the induced 2norm of matrices and the Euclidean norm of vectors.
We want to bound the difference between the distributions of \((S^X_n,S^Y_n)\) and \((Z^X,Z^Y)\) using the BerryEsseen theorem. The following lemma is the univariate BerryEsseen theorem.
Lemma 3
There is a constant \(C > 0\) such that
Then we prove the multivariate case.
Lemma 4
Assume X and Y are not perfectly correlated. The following bound holds between the joint distributions of \(\left( S^X_n,S^Y_n \right)\) and \(\left( Z^X,Z^Y \right)\): let \({\mathcal {B}}\) be the set of all measurable convex sets in \({\mathbb {R}}^2\), then there is a constant \(C>0\) such that
Proof of Lemma 4
We use \(\varSigma _{XY}\) to denote the normalized (by n) covariance matrix of \((S^X_n,S^Y_n)\),
The multivariate BerryEsseen theorem (Bentkus 2005) says there exists a universal constant C, such that
The induced 2norm of a positive semidefinite matrix equals to its largest eigenvalue. So we have
where \(\lambda _{\text {min}}\) is the smallest eigenvalue of \(\varSigma _{XY}\). We next compute \(\lambda _{\text {min}}\). The characteristic function of \(\varSigma _{XY}\) is
The smaller eigenvalue is
which is a positive constant when \(\sigma ^2_X\sigma ^2_Y  \sigma _{XY}^2 \ne 0\).
To deal with the remaining part of the bound, we employ the Minkowski inequality.
The last inequality follows from the fact that the function \(x \mapsto x^{2/3}\) is concave so that for any two positive numbers a and b, it holds that
\(\square\)
Lemma 5
The following expression of the normal truncated mean holds true:
In particular, for \(X = Y\), we have
Proof of Lemma 5
Define \(e = Z^X  \frac{{\sigma }_{XY}}{{\sigma }^2_{Y}} Z^Y\). It is straightforward to compute that \({\mathbb {E}}e=0\). Since e and \(Z^Y\) are jointly normal and uncorrelated, \(e \perp Z^Y\). Therefore,
\(\square\)
Lemma 6
For any sequence of numbers \(\alpha _n\), there is a constant \(C>0\) such that
Proof of Lemma 6
Using Chebyshev’s inequality, we have
Together with Lemma 3, we have
\(\square\)
Lemma 7
Suppose X and Y are not perfectly correlated. For any sequence of numbers \(\alpha _n\), there is a constant \(C>0\) such that
Proof of Lemma 7
By Lemma 2 and 5, we know that the absolute difference we want to bound is
We first deal with \(I_1\). Notice that
where the second inequality follows from the Chebyshev inequality. Similarly, we have
So putting these two tail bounds together, we have
Then, based on Lemma 4, we have
Following the same steps we can derive a same bound for \(I_2\). Then the result follows. \(\square\)
1.2 Proofs of results in the main text
We introduce some notations that are useful in the proofs. The moments of the valuation distribution are denoted by
They are all finite since we assume that f has a bounded support and is bounded away from zero on the support. To keep the asymptotic analysis concise, we use the notations in the following table.^{Footnote 36}
Notation  Definition  Short explanation 

\(d_n = o(e_n)\)  \(d_n/e_n \rightarrow 0\)  \(d_n \) dominated by \(e_n\) 
\(d_n = O(e_n)\)  \(\limsup {d_n}/e_n < \infty\)  \(d_n \) bounded above by \(e_n\) 
\(d_n = \omega (e_n)\)  \(d_n /e_n \rightarrow \infty\)  \(d_n \) dominates \(e_n\) 
\(d_n = \Omega (e_n)\)  \(\liminf d_n /e_n > 0\)  \(d_n \) bounded below by \(e_n\) 
\(d_n = \Theta (e_n)\)  \(d_n = O(e_n),d_n = \Omega (e_n)\)  \(d_n \) bounded below and above by \(e_n\) 
\(d_n = O_\varepsilon (e_n)\)  \(\exists \varepsilon >0, d_n = O(n^{\varepsilon }e_n)\)  \(d_n \) nearly bounded above by \(e_n\) 
\(d_n = \omega _{\varepsilon }(e_n)\)  \(\forall \varepsilon >0, d_n = \omega (n^{\varepsilon }e_n)\)  \(d_n \) nearly dominates \(e_n\) 
Proof of Theorem 1

(i)
We first obtain an asymptotic lower bound for the exante budget:
$$\begin{aligned} {\mathbb {E}}[b^{n}(\varvec{V})]&= {\mathbb {E}} \left[ \sum \psi (V_i) {\mathbf {1}}\big \{ \sum \psi (V_i) \ge \alpha _n \big \} \right]  c_n {\mathbb {P}}\left( \sum \psi (V_i) \ge \alpha _n \right) \\&\ge {\mathbb {E}} \left[ \sum \psi (V_i) {\mathbf {1}}\big \{ \sum \psi (V_i) \ge \alpha _n \big \} \right]  c_n \\&\ge \sqrt{n} {\sigma }_{\psi } \phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{\psi }} \right)  Cn^{1/4}  c_n, \end{aligned}$$where the last inequality follows from Lemma 6. Then we show that the first term on the RHS is the leading term. First, by the assumption \(\alpha _n = o\left( \sqrt{ n \log n } \right)\), we have
$$\begin{aligned} \log \left( \sqrt{n} \phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{\psi }} \right) \Big / n^{1/4} \right)&= \frac{1}{4}\log n \left( \frac{\alpha _n}{{\sigma }_{\psi }\sqrt{n}} \right) ^2 + C \\&= \log n \left( \frac{1}{4}  \left( \frac{\alpha _n}{{\sigma }_{\psi }\sqrt{n \log n}} \right) ^2 + o(1)\right) \\&= \log n (1/4 + o(1))\rightarrow \infty . \end{aligned}$$Next, by the assumption that \(c_n = O_{\varepsilon }\left( \sqrt{n}\right)\), there exist \(C,\varepsilon >0\) such that \(c_n \le C n^{1/2  \varepsilon }\) for large n. Then \(\log c_n / \log n \le 1/2  \varepsilon /2\) for large n. Therefore,
$$\begin{aligned}&\log \left( \sqrt{n} \phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{\psi }} \right) \Big / c_n \right) \\ =&\frac{1}{2} \log n  \left( \frac{\alpha _n}{{\sigma }_{\psi }\sqrt{n}} \right) ^2  \log c_n + C \\ =&\log n \left( \frac{1}{2}  \left( \frac{\alpha _n}{{\sigma }_{\psi }\sqrt{n \log n}} \right) ^2  \frac{\log c_n}{\log n} + o(1) \right) \\ \ge&\log n (\varepsilon /2 + o(1)) \rightarrow \infty . \end{aligned}$$The above asymptotic results show that the leading term in the lower bound is \(\sqrt{n} {\sigma }_{\psi } \phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{\psi }} \right)\). We then derive an upper bound for the exante budget:
$$\begin{aligned} {\mathbb {E}}[b^{n}(\varvec{V})]&\le {\mathbb {E}} \left[ \sum \psi (V_i) {\mathbf {1}}\big \{ \sum \psi (V_i) \ge \alpha _n \big \} \right] \\&\le \sqrt{n} {\sigma }_{\psi } \phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{\psi }} \right) + Cn^{1/4} , \end{aligned}$$where the last inequality again follows from Lemma 6. By the previous analysis, the leading term in the upper bound is also \(\sqrt{n}{\sigma }_{\psi } \phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{\psi }} \right)\). Therefore, we only need to derive the growth rate of that term. Take any \(\varepsilon >0\), we have
$$\begin{aligned} \log \left( \sqrt{n} \phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{\psi }} \right) \Big / n^{1/2\varepsilon } \right)&= \varepsilon \log n  \frac{1}{2} \left( \frac{\alpha _n}{{\sigma }_{\psi }\sqrt{n}} \right) ^2 +C \\&= \log n \left( \varepsilon  \frac{1}{2} \left( \frac{\alpha _n}{{\sigma }_{\psi }\sqrt{n \log n}} \right) ^2 + o(1) \right) \\&= \log n \left( \varepsilon + o(1) \right) \rightarrow \infty . \end{aligned}$$ 
(ii)
First notice that \({\mathbb {E}}[w^{n}(\varvec{V})] \le {\mathbb {E}}[w^*(\varvec{V})]\). We break the exante welfare into two parts
$$\begin{aligned} {\mathbb {E}}[w^{n}(\varvec{V})]&= {\mathbb {E}} \left[ \sum V_i q^{n}(\varvec{V})  \sum t_i^{n}(\varvec{V}) \right] \\&= {\mathbb {E}} \left[ \left( \sum V_i  c_n \right) q^{n}(\varvec{V})\right]  {\mathbb {E}}[b^{n}(\varvec{V})]. \end{aligned}$$We obtain a lower bound for the first term on the RHS:
$$\begin{aligned}&{\mathbb {E}} \left[ \left( \sum V_i  c_n \right) q^{n}(\varvec{V})\right] \\ =&{\mathbb {E}} \left[ {\mathbf {1}}\big \{ \sum \psi (V_i) \ge \alpha _n \big \} \sum V_i \right]  c_n {\mathbb {P}}\left( \sum \psi (V_i) \ge c_n \right) \\ \ge&{\mathbb {E}} \left[ {\mathbf {1}}\big \{ \sum \psi (V_i) \ge \alpha _n \big \} \right] {\mathbb {E}}\left[ \sum V_i \right]  c_n {\mathbb {P}}\left( \sum \psi (V_i) \ge c_n \right) \\ =&(n {\mu } c_n) {\mathbb {P}}\left( \sum \psi (V_i) \ge \alpha _n \right) \\ \ge&(n {\mu } c_n) \left( 1\Phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{\psi }} \right)  \frac{C}{\sqrt{n}} \right) \\ \ge&n {\mu } c_n  n {\mu }\left( \Phi \left( \frac{\alpha _n}{\sqrt{n}{\sigma }_{\psi }} \right) + \frac{C}{\sqrt{n}}\right) , \end{aligned}$$where the second inequality follows from the fact that \(\psi\) is nondecreasing and the second to last inequality follows from Lemma 3. By the result in part (i), we can bound the exante budget from above by
$$\begin{aligned} {\mathbb {E}}[b^{n}(\varvec{V})]&\le \sqrt{n}{\sigma }_{\psi } \phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{\psi }} \right) + Cn^{1/4}\\&\le \sqrt{n} {\sigma }_{\psi } / \sqrt{2 \pi } + Cn^{1/4} , \end{aligned}$$where the last inequality follows from the fact that \(\phi (\cdot ) \le 1/\sqrt{2 \pi }\). The exante efficient welfare is bounded above by \({\mathbb {E}}[w^*(\varvec{V})] \le {\mathbb {E}}\left[ \sum V_i \right] = n {\mu }\). Combining these results, we get
$$\begin{aligned} \frac{{\mathbb {E}}[w^*(\varvec{V})]  {\mathbb {E}}[w^{n}(\varvec{V})]}{{\mathbb {E}}[w^*(\varvec{V})]}&\le \frac{c_n}{n\mu } + \Phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{\psi }} \right) + \frac{C}{\sqrt{n}} \rightarrow 0. \\ \end{aligned}$$The remaining task is to show that the leading term in the above expression is \(\Phi \left( \alpha _n / \sqrt{n} {\sigma }_{\psi } \right)\). We use the wellknown lower bound on the tail of \(\Phi\):
$$\begin{aligned} \Phi (v) = 1\Phi (v) \ge {v}\exp \left( v^2/2 \right) /\big (\sqrt{2\pi }\left( 1+v^2 \right) \big ), v \le 0. \end{aligned}$$Then
$$\begin{aligned}&\log \left( \sqrt{n} \Phi \left( \frac{\alpha _n}{{\sigma }_{\psi }\sqrt{n}} \right) \right) \\ \ge&\frac{1}{2} \log n + \log \left( \frac{{\alpha _n}}{\sqrt{n}} \right)  \frac{1}{2} \left( \frac{\alpha _n}{{\sigma }_{\psi }\sqrt{n}} \right) ^2  \log \left( 1 + (\alpha _n/{\sigma }_{\psi }\sqrt{n})^2 \right) + C \\ =&\frac{1}{4} \log n + \log \left( \frac{{\alpha _n}}{\sqrt{n}} \right)  \frac{1}{2} \left( \frac{\alpha _n}{{\sigma }_{\psi }\sqrt{n}} \right) ^2 + \log \left( \frac{n^{1/4}}{1 + (\alpha _n/{\sigma }_{\psi }\sqrt{n})^2} \right) + C \\ =&\log n \left( \frac{1}{4}  \frac{1}{2} \left( \frac{\alpha _n}{{\sigma }_{\psi }\sqrt{n \log n}} \right) ^2 \right) + \log \left( \frac{{\alpha _n}}{\sqrt{n}} \right) \\ +&\log \left( \frac{n^{1/4}/\log n}{1/ \log n + (\alpha _n/{\sigma }_{\psi }\sqrt{n \log n})^2} \right) + C, \end{aligned}$$where the last line goes to \(\infty\). Therefore, \(\Phi \left( \alpha _n / \sqrt{n} {\sigma }_{\psi } \right)\) is \(\omega (1/\sqrt{n})\) and hence the leading term of the welfare ratio.

(iii)
The result comes directly from the previous two parts.
\(\square\)
Proof of Lemma 1
By the definition of \(\psi\), we have
By the Fubini theorem, the second term on the RHS is
The above quantity is finite since V has a bounded support and h is continuous. Then we perform integration by parts to the above LebesgueStieltjes integral:
Therefore,
which is positive if h is increasing. \(\square\)
For Theorem 2, we introduce the following notations for the moments of \(h(V_i)\):
They are finite when h is a continuous function on \([0,{\bar{v}}]\).
Proof of Theorem 2
If \(h(V_i)\) and \(\psi (V_i)\) are perfectly correlated (i.e., h and \(\psi\) are linearly dependent), then the result follows from Theorem 1. Therefore, we only need to study the case where \(h(V_i)\) and \(\psi (V_i)\) are not perfectly correlated.

(i)
We first obtain an asymptotic lower bound for the exante budget:
$$\begin{aligned} {\mathbb {E}}[b^{n}(\varvec{V})]&= {\mathbb {E}} \left[ \sum \psi (V_i) {\mathbf {1}}\big \{ \sum h(V_i) \ge \alpha _n \big \} \right]  c_n {\mathbb {P}}\left( \sum h(V_i) \ge \alpha _n \right) \\&\ge {\mathbb {E}} \left[ \sum \psi (V_i) {\mathbf {1}}\big \{ \sum h(V_i) \ge \alpha _n \big \} \right]  c_n \\&\ge \sqrt{n} \frac{{\sigma }_{\psi h}}{{\sigma }_{h}} \phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{h}} \right)  Cn^{1/4} c_n, \end{aligned}$$where the last inequality follows from Lemma 7. The first term is strictly positive by Lemma 1. Then following the same steps as in the proof of Theorem 1(i), we can show that the leading term in the above expression is \(\sqrt{n} \frac{{\sigma }_{\psi h}}{{\sigma }_{h}} \phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{h}} \right)\) under the assumptions \(\alpha _n = o(\sqrt{n \log n})\) and \(c_n = O_\varepsilon (\sqrt{n})\). We then derive an upper bound for the exante budget:
$$\begin{aligned} {\mathbb {E}}[b^{n}(\varvec{V})]&\le {\mathbb {E}} \left[ \sum \psi (V_i) {\mathbf {1}}\big \{ \sum h(V_i) \ge \alpha _n \big \} \right] \\&\le \sqrt{n} \frac{{\sigma }_{\psi h}}{{\sigma }_{h}} \phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{h}} \right) + Cn^{1/4} , \end{aligned}$$where the last inequality again follows from Lemma 7. By the previous analysis, the leading term in the upper bound is also \(\sqrt{n} \frac{{\sigma }_{\psi h}}{{\sigma }_{h}} \phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{h}} \right)\), and the result follows.

(ii)
We follow the proof of Theorem 1(ii). First notice that \({\mathbb {E}}[w^{n}(\varvec{V})] \le {\mathbb {E}}[w^*(\varvec{V})]\). We break the exante welfare into two parts
$$\begin{aligned} {\mathbb {E}}[w^{n}(\varvec{V})]&= {\mathbb {E}} \left[ \sum V_i q^{n}  \sum t^{n}_i(\varvec{V}) \right] \\&= {\mathbb {E}} \left[ \left( \sum V_i  c_n \right) q^{n}(\varvec{V})\right]  {\mathbb {E}}[b^{n}(\varvec{V})]. \end{aligned}$$We obtain a lower bound for the first term on the RHS:
$$\begin{aligned}{\mathbb {E}}& \left[ \left( \sum V_i  c_n \right) q^{n}(\varvec{V})\right] \\ =\,\,&{\mathbb {E}} \left[ {\mathbf {1}}\big \{ \sum h(V_i) \ge \alpha _n \big \} \sum V_i \right]  c_n {\mathbb {P}}\left( \sum h(V_i) \ge c_n \right) \\ \ge\,\,&{\mathbb {E}} \left[ {\mathbf {1}}\big \{ \sum h(V_i) \ge \alpha _n \big \} \right] {\mathbb {E}}\left[ \sum V_i \right]  c_n {\mathbb {P}}\left( \sum h(V_i) \ge c_n \right) \\ =\,\,&(n {\mu } c_n) {\mathbb {P}}\left( \sum h(V_i) \ge \alpha _n \right) \\ \ge\,\,&(n {\mu } c_n) \left( 1\Phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{h}} \right)  \frac{C}{\sqrt{n}} \right) \\ \ge\,\,&n {\mu } c_n  n \mu \left( \Phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{h}} \right) + \frac{C}{\sqrt{n}} \right) , \end{aligned}$$where the second line follows from the fact that h is nondecreasing and the second to last line follows from Lemma 3. Following the proof of part (i), we can upper bound the exante budget by
$$\begin{aligned} {\mathbb {E}}[b^{n}(\varvec{V})]&\le \sqrt{n} \frac{{\sigma }_{\psi h}}{{\sigma }_{h}} \phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{h}} \right) + C n^{1/4} \\&\le \sqrt{n} \frac{{\sigma }_{\psi h}}{{\sigma }_{h} \sqrt{2 \pi }} + C n^{1/4}. \end{aligned}$$We use the same upper bound on the efficient exante welfare \({\mathbb {E}}[w^*(\varvec{V})] \le n {\mu }\) as before. Combining these results together, we get a similar bound as in part (ii) of Theorem 1:
$$\begin{aligned} \frac{{\mathbb {E}}[w^*(\varvec{V})]  {\mathbb {E}}[w^{n}(\varvec{V})]}{{\mathbb {E}}[w^*(\varvec{V})]}&\le \frac{c_n}{n{\mu }} + \Phi \left( \frac{\alpha _n}{\sqrt{n} {\sigma }_{h}} \right) + \frac{C}{\sqrt{n}} \rightarrow 0. \end{aligned}$$Similar as in the proof of Theorem 1(ii), the leading term in the above expression is \(\Phi \left( \alpha _n / \sqrt{n} {\sigma }_{h} \right)\).

(iii)
The result comes directly from the previous two parts.
\(\square\)
Proof of Theorem 3
It is wellknown that the total expected payment of an incentive compatible and individually rational mechanism \((q^n,\{t_i^n\})\) is equal to
See, for example, Equation (3.6) in Hellwig (2003). Since the expected payment \({\mathbb {E}}[t^n_i(0,\varvec{V}_{i})]\) needs to be nonpositive for any i by the individually rational condition, the total expected payment is maximized by setting \(q^n(\varvec{v}) = {\mathbf {1}}\{ \sum \psi (v_i) \ge 0 \}\) and \(t^n_i(0,\varvec{v}_{i})=0\). Such a mechanism is in fact a special case of the mechanism in Example 3 with the adjustment term \(\alpha _n = 0\). From the proof of Theorem 1, we can see that the total expected payment is \(O(\sqrt{n})\). Notice that even though this particular mechanism may violate incentive compatibility when the distribution is not Myerson regular, the asymptotic order \(O(\sqrt{n})\) is nonetheless a valid upper bound on the growth rate of the maximum total expected payment.
Then for any sequence of incentive compatible and individually rational mechanisms \((q^n,\{t^n_i\})\) that satisfies eventually exante budget balanced, it must be true that
This implies that the provision probability of the public good is converging to zero:
\(\square\)
Discussion on Proposition 3 in Hellwig (2003)
In this section, we discuss the proof method of Proposition 3 in Hellwig (2003), hereafter H2003. We first translate the result and the proof with the terminology in our paper. Proposition 3 in H2003 shows that the secondbest mechanism is asymptotically efficient when the cost of the public good does not grow with n. The proof in H2003 essentially tries to show that the AMT mechanism in Example 3 is exante budget balanced and asymptotically efficient. Therefore, since the secondbest mechanism must have a higher welfare by definition, it is also asymptotically efficient.
To better explain the proof, we link the notations in H2003 to the ones in our paper. The mechanism \(Q^{nk}\) defined in (4.6) and (4.7) on p.597 of H2003 corresponds to the AMT mechanism with \(h=\varphi\) in our Example 3. The scalar k in the mechanism \(Q^{nk}\) corresponds to our adjustment term \(\alpha _n\). The discussion following Inequality (4.8) on p.598 of H2003 shows that when the adjustment term k is fixed (does not vary with n), the mechanism \(Q^{nk}\) is exante budget balanced. The discussion following Inequality (4.10) on p.598 of H2003 shows that when the adjustment term k decreases to \(\infty\), the mechanism \(Q^{nk}\) is asymptotically efficient.
We argue that this reasoning is incomplete because a discussion of the exante budget when k varies with n is lacking. More specifically, the proof in H2003 requires that Inequality (4.4) on p.597 to hold for any \(\varepsilon >0\) and n sufficiently large. In particular, we can take \(\varepsilon = 1/n\). Then the \(k(\varepsilon )\) on the second line after (4.10) depends explicitly on n and is decreasing as n increases. In this case, the discussion following Inequality (4.8) is no longer sufficient to show that the mechanism \(Q^{nk}\) is exante budget balanced. This is because a decreasing k would decrease the budget. The previous argument that \(Q^{nk}\) is exante budget balanced when k is fixed and \(n \rightarrow \infty\). However, this does not address the budget when k is decreasing. For example, in the extreme case where k decreases so fast that it is equal to \(\infty\), the revenue becomes zero in the limit, leading to a budget deficit.
This is where the BerryEsseen theorem becomes useful: we want to characterize the exante budget when n increases and the adjustment term k decreases. This is possible under the BerryEsseen theorem because it gives the convergence rate of the central limit theorem. For each n and k, we know not only that the budget can be approximated by a normal distribution but also how close this approximation is. Therefore, we can derive the rate of k under which the budget becomes balanced eventually. The details are described in Sect. 3.2 and in the proofs in Appendix A.
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Xi, J., Xie, H. Strength in numbers: robust mechanisms for public goods with many agents. Soc Choice Welf 61, 649–683 (2023). https://doi.org/10.1007/s00355023014662
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DOI: https://doi.org/10.1007/s00355023014662