Keywords: local differential privacy; mechanism design; public-good provision; Myerson virtual values; large deviations.

JEL classification: D82, D44, C72, H41.

Credit authorship contribution statement:

Behrooz Moosavi: Review & editing, Writing original draft, Methodology, Investigation, Conceptualization.

Jordan Awan: Review & editing, Writing original draft, Methodology, Investigation, Conceptualization.

Conflict of Interest: The authors declare that they have no conflict of interest.

Funding: This work was supported in part by NSF award number SES-2610910.

1 Introduction↩︎

Classical mechanism design relies on the sensitivity of outcomes to individual reports. Vickrey–Clarke–Groves payments [1]–[3] charge each agent the externality it imposes on the others; the Myerson envelope theorem [4] recovers information rents from the interim allocation rule, since an agent’s equilibrium utility has derivative equal to its allocation probability and rents are the integral of that allocation over types. Differential privacy demands the opposite: the distribution of a released output must change only slightly when any single input changes [5], [6]. Local differential privacy enforces this by randomizing each input before it reaches the collector.

This architecture is already deployed at the scale of the world’s largest data collectors. Apple has used local differential privacy in iOS to gather usage statistics from hundreds of millions of devices, with randomization performed on the user’s device before any data reaches its servers [7], [8], and Google’s RAPPOR system applied the same local model in Chrome [9]. In both, the collector is untrusted and observes only a randomized signal, never the true input. These systems perform statistical estimation, with no transfers, incentive constraints, or collective decision at stake; the conflict with mechanism design arises only when such a channel is placed in front of a planner who must elicit preferences, make a public decision, and raise revenue from privatized signals alone. There, neither the standard allocation rule nor the standard payment identity survives unchanged.

This paper characterizes the optimal reduced-form mechanism and the asymptotic behavior of maximum reduced-form revenue in such an environment. A planner must decide whether to implement a binary public project—build or not build—when \(n\) agents hold privately known valuations \(X_1,\ldots,X_n\). The planner does not observe these valuations. Each agent submits a value to a fixed trusted mediator (privacy channel), which randomizes the submission before transmitting it to the planner; the agent controls the value it submits but neither observes nor controls the realized noise, and cannot bypass the protocol. The planner observes only the privatized signal profile \((Y_1,\ldots,Y_n)\), so both the project decision and the transfer schedule must be measurable functions of these signals alone. The mechanism must be Bayesian incentive compatible, so that truthful submission is optimal for every type; we analyze the resulting truthful equilibrium, along which the submitted value equals the valuation and the privatized signal is distributed as \[Y_i\sim K(\,\cdot\mid X_i).\] It must also be individually rational and meet an ex-ante revenue target.

This formulation fits settings in which preference-relevant information is generated or measured locally and passed through a prescribed privacy device before reaching an untrusted aggregator: federated rollout decisions based on privatized quality or demand reports, consortium or standard-adoption decisions in which members privatize their preferences from the aggregator, and more generally any binary collective decision mediated by a fixed local randomization protocol. Within this class we ask three questions. How strong can the privacy guarantee become before the project can no longer be financed, and how does this relate to the classical difficulty of public-good provision with many privately informed agents [10]? What form does the optimal decision rule take when the planner observes only privatized signals? And, once channels are calibrated to a common privacy benchmark, does the choice of noise distribution—Gaussian, Laplace, or logistic—affect welfare and revenue, or only the statistical precision of the planner’s information?

1.0.0.1 Our Contributions.

This paper makes four contributions.

• Privacy reshapes how the good is provided. The planner never sees agents’ true values, only reports passed through a fixed, privacy-protecting noise channel. We show that the planner should nonetheless act on a simple per-agent index that blends how much each agent is expected to value the good with how much revenue can be collected from them, and should provide the good when these indices sum to a positive total. The amount of noise the channel adds controls how strongly provision responds to people’s underlying preferences; coarser channels can leave the planner unable to tell apart agents whose reports are extreme.

• Knowing what to provide is not the same as being able to pay for it. We distinguish the allocation the planner would choose from the payments needed to make truth-telling optimal. Recovering payments that depend only on the noisy reports is a separate and harder problem—formally, solving an integral (Fredholm) equation—that may have no exact solution. When an exact solution exists it is unique, but existence is not automatic. The Online Supplement therefore also constructs payments that work approximately, with explicit bounds on how far incentives, participation, and revenue can be off.

• Privacy makes public goods harder to finance as the market grows. The revenue a privacy-respecting mechanism can raise is governed by the lower endpoint of the valuation distribution—the smallest value an agent could have. As the number of agents grows, revenue grows in proportion to the market when that lower endpoint is positive, grows only at a slow square-root rate when it is exactly zero, and shrinks to almost nothing when it is negative. The change between a fundable and an unfundable good is sharp, so a small tightening of the revenue requirement can switch provision off.

• The better privacy technology depends on how privacy is measured. Comparing two natural noise designs, Laplace and logistic, we find no design is better in every case. Holding the noise scale fixed—a common pure-\(\epsilon\) calibration—the Laplace design gives the planner more usable information; holding fixed instead a stricter hypothesis-testing privacy standard (Gaussian differential privacy), the ranking reverses in the hardest case, that of telling apart the highest- and lowest-value agents. Which design a planner prefers thus depends on the privacy standard used to compare them, and this reversal does not extend to a blanket ranking across all agents.

1.0.0.2 Related literature.

Our work lies at the intersection of differential privacy, mechanism design, and public-good provision. The privacy-aware mechanism-design literature begins with [11], who use the stability created by differential privacy to obtain approximate truthfulness. We study a different problem: the local privacy channel is fixed exogenously, privacy does not enter agents’ payoffs, and the planner must achieve exact Bayesian incentive compatibility using allocation and transfer rules that depend only on privatized signals. The classical transfer identity [4] therefore becomes an inverse problem: recovering a signal-contingent transfer requires solving a Fredholm equation of the first kind, where completeness of the channel kernel gives uniqueness when a solution exists and existence is a separate range condition. Recent work also studies privacy-constrained mechanism and information design: [12] incorporate belief-based privacy costs into mechanism design and show that privacy concerns can rationalize coarse mechanisms, while [13] characterize signals that conceal a protected attribute through garbling and mean-preserving contraction and [14] study signals that reveal information about a common state without revealing other agents’ signals. We share their use of garbling and the Blackwell order, but our planner does not design the information structure: it faces a fixed local differential-privacy channel and optimizes a public-good mechanism conditional on the information that channel produces. Two further strands bear on our setting and are developed in the Online Supplement: privacy as a payoff argument or a good to be compensated [15]–[17], and the statistical limits of local privacy [18], [19], where our concern is the privacy–revenue rather than the privacy–estimation frontier.

The paper most closely related to ours is [20], who study binary public-good provision with \(\pm1\) types transmitted through a randomized-response flip channel. They characterize Boolean decision rules trading off revenue, surplus, and noise sensitivity. We allow continuous valuations and a broader class of channels, so monotone likelihood ratios govern implementability and generate features without a Boolean analogue, including the distinction between strict and weak MLRP and Laplace tail pooling. We also characterize transfers through a Fredholm equation, allow a hierarchical prior, and derive three large-population revenue regimes—linear, square-root, and exponentially small—according to the sign of the lower endpoint of the valuation distribution.

Finally, our asymptotic analysis relates to the literature on public-good provision in large economies [10], [21], [22], where private information obstructs efficient provision and, under budget balance, the obstruction intensifies as the population grows. [23] identify a square-root boundary in the growth rate of project costs for asymptotically efficient provision. Their boundary concerns cost scaling and is therefore analogous to, but distinct from, our knife-edge regime, in which maximal reduced-form revenue is itself of square-root order. In our model the local privacy channel further determines the speed at which revenue feasibility deteriorates. The hierarchical extension is also related to [24]: a common latent parameter generates dependence across types, while privacy weakens the information each signal conveys about that parameter without eliminating the induced correlation.

1.0.0.3 Outline

The paper is organized as follows. Section 2 reviews the public-good problem and the privacy channels; Section 3 sets up the mediated reporting model and the posterior-score objects it relies on. Section 4 gives the main results: the optimal threshold rule, the Fredholm implementation problem, the linear/square-root/exponential revenue regimes, the hierarchical extension, and the channel welfare reversal. Section 5 illustrates some of the theoretical results derived in section 4. The appendices collect the proofs and additional numerical results, and the online supplement develops exact and approximate signal-measurable implementation, the revelation and completeness arguments, and the full numerical methodology.

2 Background↩︎

This section sets out the economic environment, the reporting and privacy architecture, and the privacy criteria. The planner’s mechanism-design problem is formulated in Section 3. Trade-off functions, privacy calibrations, and channel-specific likelihood-ratio calculations are collected in Appendix 7.

2.1 Agents and Preferences↩︎

A planner faces \(n\) agents indexed by \(i\in\{1,\ldots,n\}\). Agent \(i\) privately observes a valuation \[X_i\in\mathcal{X}=[\underline{x},\bar{x}]\subset\mathbb{R}.\] In the known-prior benchmark, the valuations are independently and identically distributed according to a commonly known distribution \(F\) with density \(f>0\) on \(\mathcal{X}\).

The planner decides whether to implement a binary public project, \(q\in\{0,1\}\), and assigns transfers \(t_i\in\mathbb{R}\). Preferences are quasilinear: \[\label{eq:utility} u_i(q,t_i\mid x_i) = q\,x_i-t_i.\tag{1}\] The valuation \(x_i\) is the net value of implementation, inclusive of any exogenously assigned cost share. If the planner observed the valuation profile \(x=(x_1,\ldots,x_n)\), the efficient allocation would be \[\label{eq:first95best} q^{\mathrm{FB}}(x) = \mathbf{1}\!\left\{ \sum_{i=1}^{n}x_i\ge0 \right\}.\tag{2}\]

2.2 Strategic Reports and the Privacy Channel↩︎

After observing \(X_i=x_i\), agent \(i\) submits a report \(R_i=r_i\in\mathcal{X}\) to a trusted local mediator. Conditional on the report, the mediator generates a privatized signal through a fixed Markov kernel: \[\label{eq:channel} Y_i\mid R_i=r_i \sim K(\cdot\mid r_i).\tag{3}\] When the kernel admits a density, it is denoted by \(k(y_i\mid r_i)\). Signals take values in a measurable space \(\mathcal{Y}\subseteq\mathbb{R}\), with \(\mathcal{Y}=\mathbb{R}\) for the additive channels considered below.

The agent controls the report entering the channel but not the realized channel noise. The planner observes only the signal profile \[Y=(Y_1,\ldots,Y_n)\] and observes neither \(X\) nor \(R\).

A randomized signal-measurable mechanism is a measurable map \[\label{eq:mechanism95def} (q,t): \mathcal{Y}^n \longrightarrow [0,1]\times\mathbb{R}^n, \qquad y \longmapsto \bigl(q(y),t_1(y),\ldots,t_n(y)\bigr),\tag{4}\] where \(q(y)\) is the probability of implementation and \(t_i(y)\) is agent \(i\)’s payment. Transfers are assumed measurable and integrable under the induced signal laws. The realized project decision is binary; \(q(y)\in[0,1]\) denotes its implementation probability conditional on the signal profile.

We study direct reporting mechanisms, in which the value an agent submits is itself the input to the fixed privacy channel. Bayesian incentive compatibility, imposed in Section 3.1, requires truthful reporting \(R_i=X_i\) to be optimal within this reporting game. Along the resulting truthful equilibrium path, \[\label{eq:truthful95signal95law} Y_i\mid X_i=x_i \sim K(\cdot\mid x_i).\tag{5}\] The report variable is retained to formulate deviations, whereas posterior beliefs, welfare, revenue, and allocation are evaluated under the truthful law 5 .

Conditional independence of the mediator’s randomizations gives \[\label{eq:truthful95joint95channel} Y\mid X=x \sim \prod_{i=1}^{n}K(\cdot\mid x_i).\tag{6}\] The truthful-path marginal density of one signal is³ \[\label{eq:marginal95signal95density} m(y) = \int_{\mathcal{X}}k(y\mid x)f(x)\,dx.\tag{7}\] Under the independent known-prior benchmark, \[\label{eq:joint95signal95density} f_Y(y) = \prod_{i=1}^{n}m(y_i), \qquad f_{Y_{-i}}(y_{-i}) = \prod_{j\ne i}m(y_j).\tag{8}\]

2.3 Privacy Criteria↩︎

Privacy is measured through hypothesis-testing trade-off functions, with Gaussian differential privacy providing the common cross-family benchmark [25]. For probability measures \(P\) and \(Q\) on a common measurable signal space, define \[\label{eq:tradeoff95function} \mathcal{T}(P,Q)(\alpha) = \inf_{\psi} \left\{ 1-\mathbb{E}_Q[\psi]: \mathbb{E}_P[\psi]\le\alpha \right\}, \qquad \alpha\in[0,1],\tag{9}\] where the infimum is over measurable randomized tests \(\psi:\mathcal{Y}\to[0,1]\). Thus, \(\mathcal{T}(P,Q)(\alpha)\) is the smallest attainable type-II error among tests whose type-I error is at most \(\alpha\). A larger trade-off function corresponds to a less informative binary experiment and therefore stronger hypothesis-testing privacy.

Smaller values of \(\mu\) correspond to stronger privacy. The Gaussian shift experiment \[\mathcal{N}(0,1) \quad\text{versus}\quad \mathcal{N}(\mu,1)\] has trade-off function \(G_\mu\).

The parameter \(\epsilon\) bounds multiplicative distinguishability, while \(\delta\) allows an additive exceptional probability [5], [6]. A \(\mu\)-GDP channel satisfies \((\epsilon,\delta_\mu(\epsilon))\)-LDP for every \(\epsilon\ge0\), where \[\label{eq:gdp95to95approx} \delta_\mu(\epsilon) = \Phi\!\left( -\frac{\epsilon}{\mu}+\frac{\mu}{2} \right) - e^\epsilon \Phi\!\left( -\frac{\epsilon}{\mu}-\frac{\mu}{2} \right)\tag{13}\] [25].

2.4 Channel Families↩︎

We study additive location channels of the form \[\label{eq:additive95channel} Y_i = R_i+\eta_i,\tag{16}\] where the noise variables are independent across agents and independent of \((X,R)\). Since reports lie in \([\underline{x},\bar{x}]\), the largest separation between any two reports is the type-space width \[\label{eq:type95sensitivity} \Delta_{\mathcal{X}} = \bar{x}-\underline{x},\tag{17}\] which plays the role of the sensitivity in the privacy calibrations below.

6pt

Along the truthful path, \[\label{eq:truthful95channel95summary} Y_i^{G} = X_i+\eta_i^{G}, \qquad Y_i^{L} = X_i+\eta_i^{L}, \qquad Y_i^{\mathrm{Log}} = X_i+\eta_i^{\mathrm{Log}}.\tag{18}\]

Privacy limits the distinguishability of reports. The monotone likelihood ratio property introduced in Section 3.2 is a distinct property: it governs how signals order posterior beliefs and therefore how the mechanism responds to signal realizations.

3 Setup↩︎

This section formulates the planner’s problem, states the maintained assumptions, and records the reduced-form identities used in Section 4. General incentive, integrability, posterior, and monotonicity arguments are collected in Appendix 8; mechanism-specific proofs are collected in Appendix 9.

3.1 The Planner’s Problem↩︎

Figure 1 summarizes the timing. Agent \(i\) observes \(X_i\), submits \(R_i\), the mediator draws \(Y_i\sim K(\cdot\mid R_i)\), and the planner chooses \(q(Y)\) and \(t(Y)\).

For a signal-measurable mechanism, define agent \(i\)’s interim allocation and interim payment after report \(r_i\), holding the other agents to truthful reporting, by \[\begin{align} Q_i(r_i) &= \int_{\mathcal{Y}^n} q(y)\, k(y_i\mid r_i)\, f_{Y_{-i}}(y_{-i})\,dy, \tag{19} \\ T_i(r_i) &= \int_{\mathcal{Y}^n} t_i(y)\, k(y_i\mid r_i)\, f_{Y_{-i}}(y_{-i})\,dy. \tag{20} \end{align}\] A type \(x_i\) reporting \(r_i\) obtains \[\label{eq:deviation95utility} U_i(x_i,r_i) = x_iQ_i(r_i)-T_i(r_i),\tag{21}\] and truthful interim utility is \[\label{eq:truthful95interim95utility} U_i(x_i) = U_i(x_i,x_i) = x_iQ_i(x_i)-T_i(x_i).\tag{22}\]

The planner maximizes expected surplus subject to incentive, participation, and revenue constraints: \[\tag{23} \begin{align} \sup_{q,t}\quad & \mathbb{E}\!\left[ q(Y)\sum_{i=1}^{n}X_i \right] \notag \\ \text{subject to}\quad & x_iQ_i(x_i)-T_i(x_i) \ge x_iQ_i(r_i)-T_i(r_i), && \forall x_i,r_i\in\mathcal{X}, \tag{24} \\ & U_i(x_i)\ge0, && \forall x_i\in\mathcal{X}, \tag{25} \\ & \mathbb{E}\!\left[ \sum_{i=1}^{n}t_i(Y) \right] = \sum_{i=1}^{n}\mathbb{E}[T_i(X_i)] \ge R, && \tag{26} \\ & q:\mathcal{Y}^n\to[0,1], \qquad t_i:\mathcal{Y}^n\to\mathbb{R} \text{ measurable and integrable.} && \notag \end{align}\] The constraints are Bayesian incentive compatibility, interim individual rationality, and an ex-ante revenue requirement. The case \(R=0\) imposes expected non-deficit [4], [27].

3.2 Structural Assumptions↩︎

This is the standard one-dimensional regularity condition [4]. Log-concavity of \(1-F\) is sufficient [28].

The domination condition 29 makes the conditional-expectation operator bounded on the maintained transfer space, while completeness 30 makes that operator injective. These are distinct properties. Boundedness guarantees that the operator is well defined and continuous; completeness identifies the opponent-averaged transfer whenever an implementation exists. Neither property implies that the desired interim payment belongs to the range of the operator, so existence remains a separate range condition.

Assumption 3 is used only for signal-measurable transfer implementation. It is not required for the reduced-form allocation or revenue characterizations. Completeness is maintained here as a high-level condition on the chosen transfer space \(\mathcal{H}\); verifying it for particular channel families and function spaces is a separate operator-specific question.

The signal space may be unbounded. The continuity and Fubini–Tonelli consequences of this assumption are collected in Appendix 8. Additional boundedness and strict-feasibility conditions used only for existence and multiplier arguments are stated in Appendix 9.

3.3 Interim Reduction and Posterior Objects↩︎

Although the privacy channel changes the mapping from reports to the planner’s observations, each agent’s deviation problem remains one-dimensional. By 21 , a type \(x_i\) that reports \(r_i\) obtains \[x_iQ_i(r_i)-T_i(r_i),\] which is the standard one-dimensional screening payoff associated with interim allocation \(Q_i\) and interim payment \(T_i\).

Bayesian incentive compatibility is therefore equivalent to weak monotonicity of \(Q_i\) together with the envelope identity \[\label{eq:envelope} U_i(x_i) = U_i(\underline{x}) + \int_{\underline{x}}^{x_i}Q_i(z)\,dz,\tag{32}\] or, equivalently, \[\label{eq:transfer95identity} T_i(x_i) = x_iQ_i(x_i) - \int_{\underline{x}}^{x_i}Q_i(z)\,dz - U_i(\underline{x}).\tag{33}\] Since \(Q_i\ge0\), truthful utility is weakly increasing, and interim IR is equivalent to \(U_i(\underline{x})\ge0\). For a fixed allocation, expected payments are maximized by imposing the zero-rent normalization \[\label{eq:lowest95type95zero95rent} U_i(\underline{x}) = 0.\tag{34}\] These claims are established in Proposition 13 of Appendix 8.

Under the zero-rent normalization, expected payment equals expected allocated virtual surplus: \[\label{eq:revenue95virtual} \mathbb{E}[T_i(X_i)] = \mathbb{E}\!\left[ Q_i(X_i)J(X_i) \right].\tag{35}\] The proof is given in Lemma 7 of Appendix 8.

The planner conditions on privatized signals rather than valuations. Define the posterior mean and posterior virtual value by \[\label{eq:posterior95objects95setup} \widehat x_i(y_i) = \mathbb{E}[X_i\mid Y_i=y_i], \qquad \widehat J_i(y_i) = \mathbb{E}[J(X_i)\mid Y_i=y_i].\tag{36}\] These functions are common across agents under symmetry. Independence of types and channel draws implies \[\mathbb{E}[X_i\mid Y] = \mathbb{E}[X_i\mid Y_i], \qquad \mathbb{E}[J(X_i)\mid Y] = \mathbb{E}[J(X_i)\mid Y_i].\] Consequently, \[\begin{align} \mathbb{E}\!\left[ q(Y)\sum_{i=1}^{n}X_i \right] &= \mathbb{E}\!\left[ q(Y)\sum_{i=1}^{n}\widehat x_i(Y_i) \right], \tag{37} \\ \mathbb{E}\!\left[ \sum_{i=1}^{n}t_i(Y) \right] &= \mathbb{E}\!\left[ q(Y)\sum_{i=1}^{n}\widehat J_i(Y_i) \right] \tag{38} \end{align}\] under the zero-rent normalization. For arbitrary lowest-type rents, the right-hand side of 38 is reduced by \(\sum_iU_i(\underline{x})\). These identities are proved in Lemma 9 of Appendix 8.

Privacy therefore enters the reduced-form problem through the posterior expectations induced by the channel. The planner evaluates each signal using the posterior mean \(\widehat x_i(Y_i)\) and posterior virtual value \(\widehat J_i(Y_i)\).

For \(\lambda\ge0\), define the generalized virtual value \[\label{eq:generalized95virtual95value} \Psi(x,\lambda) = x+\lambda J(x)\tag{39}\] and the posterior score \[\label{eq:posterior95score95setup} S_i(y_i,\lambda) = \widehat x_i(y_i) + \lambda\widehat J_i(y_i) = \mathbb{E}\!\left[ \Psi(X_i,\lambda) \mid Y_i=y_i \right].\tag{40}\] The score is the per-agent contribution to the planner’s weighted welfare–revenue objective. Regularity makes \(\Psi(\cdot,\lambda)\) weakly increasing, and MLRP implies that \(S_i(\cdot,\lambda)\) is weakly increasing. Under strict MLRP, if \(\Psi(\cdot,\lambda)\) is nonconstant on a set of positive prior probability, then \(S_i(\cdot,\lambda)\) is strictly increasing. The general argument is given in Lemma 10 of Appendix 8.

A useful identity for the large-population analysis is \[\label{eq:lower95endpoint} \mathbb{E}[J(X)] = \mathbb{E}[\widehat J_i(Y_i)] = \underline{x}.\tag{41}\] The proof is given in Lemma 11 of Appendix 8. Thus, \(\underline{x}\) is the mean posterior virtual value. When \(\underline{x}>0\), maximum reduced-form revenue has positive drift. When \(\underline{x}<0\), positive aggregate virtual surplus is a large-deviation event. The case \(\underline{x}=0\) is the central-limit boundary between these two regimes.

Define the reduced-form monotone allocation class by \[\label{eq:monotone95allocation95class} \mathcal{Q}^{\mathrm{mon}} = \left\{ q:\mathcal{Y}^n\to[0,1]: q\text{ is measurable and } Q_i^q\text{ is weakly increasing on }\mathcal{X} \text{ for every }i \right\}.\tag{42}\] When the dependence on the channel matters, the same class is denoted by \(\mathcal{Q}^{\mathrm{mon}}(K)\). For \(\lambda\ge0\), define the aggregate posterior score \[\label{eq:aggregate95score95def} G_\lambda(y) = \sum_{i=1}^{n}S_i(y_i,\lambda).\tag{43}\] The threshold characterization in Section 4 selects the project when \(G_\lambda>0\). If the aggregate score places positive probability on zero, the allocation on the zero-score set creates a separate selection problem: a tie-breaking rule must satisfy both the revenue requirement and the monotonicity restrictions. To obtain an unambiguous almost-sure threshold characterization, we impose the following condition.

A sufficient condition is that, at each relevant multiplier, at least one coordinate score \(S_i(Y_i,\lambda)\) have an atomless distribution and be independent of the sum of the remaining coordinate scores.

Continuity of the underlying signals alone is not sufficient. In particular, Laplace tail pooling creates atoms in the distribution of each individual posterior score (Remark 2). Although these atoms need not generate positive mass at zero in the aggregate score, they prevent aggregate atomlessness from being inferred automatically.

Assumption 5 must therefore either be verified for the channel and prior under study or maintained as an additional regularity condition.

The preceding identities characterize the reduced-form allocation and interim-payment requirements. Whether the required interim payment schedule can be generated by an actual signal-contingent transfer is a separate inverse problem, formulated as a Fredholm equation in Proposition 1.

4 Main Results↩︎

This section characterizes the optimal reduced-form allocation under a known prior, the implementation of its interim payment schedule through signal-measurable transfers, the asymptotic behavior of maximum reduced-form revenue, the hierarchical-prior extension, and welfare comparisons across privacy channels. The supporting functional-analytic and probabilistic arguments are collected in Appendix 9.

4.1 Optimal Allocation under a Known Prior↩︎

Under the zero-rent normalization \(U_i(\underline{x})=0\), the posterior welfare and revenue representations give \[\begin{align} \mathsf W(q) &= \mathbb{E}\!\left[ q(Y)\sum_{i=1}^{n}\widehat x_i(Y_i) \right], \tag{45} \\ \mathsf V(q) &= \mathbb{E}\!\left[ q(Y)\sum_{i=1}^{n}\widehat J_i(Y_i) \right]. \tag{46} \end{align}\] The reduced-form problem is therefore \[\label{eq:main95reduced95problem} \sup_{q\in\mathcal{Q}^{\mathrm{mon}}} \left\{ \mathsf W(q): \mathsf V(q)\ge R \right\}.\tag{47}\] The Lagrangian associated with the revenue constraint is \[\label{eq:main95known95prior95lagrangian} \mathcal{L}(q,\lambda) = \mathbb{E}[q(Y)G_\lambda(Y)] - \lambda R,\tag{48}\] with \(G_\lambda\) the aggregate posterior score 43 . Regularity makes \(x\mapsto x+\lambda J(x)\) weakly increasing, and MLRP makes each \(S_i(\cdot,\lambda)\) weakly increasing, so \(G_\lambda\) is coordinatewise weakly increasing.

Proof. Appendix 9.4.

The theorem has a simple interpretation. The planner selects the project whenever the sum of the agents’ posterior welfare–revenue scores is positive. If the revenue constraint is slack, complementary slackness implies \(\lambda^{*}=0\), and the rule reduces to the posterior-surplus-maximizing allocation. When \(\lambda^{*}>0\), the revenue constraint binds and posterior virtual values receive positive weight. Assumption 5 ensures that the zero-score set is null, so no separate tie-breaking rule is needed. A binding constraint may, in a degenerate case, still be supported by a zero multiplier, so the converse implication need not hold.

4.2 Implementation through Signal-Measurable Transfers↩︎

Theorem 1 characterizes the optimal allocation and its interim envelope payments. The existence of an actual transfer \(t_i:\mathcal{Y}^n\to\mathbb{R}\) generating those interim payments is a distinct inverse problem. Let \(\mathcal{H}\) be the Banach space of admissible opponent-averaged transfers specified in Appendix 9.5, and define the channel operator \[\label{eq:main95channel95operator} \mathcal{K}:\mathcal{H}\to C(\mathcal{X}), \qquad (\mathcal{K}g)(x) = \int_{\mathcal{Y}}g(y)k(y\mid x)\,dy.\tag{54}\]

Proof. Appendix 9.5.

4.3 Maximum Reduced-Form Revenue↩︎

Define aggregate posterior virtual surplus by \[\label{eq:aggregate95posterior95virtual95surplus} V_n(Y) = \sum_{i=1}^{n}\widehat J_i(Y_i).\tag{55}\] Since each \(\widehat J_i\) is weakly increasing under regularity and MLRP, the allocation \[\label{eq:revenue95maximizing95rule} q^{\mathrm{rev}}(y) = \mathbf{1}\!\left\{ V_n(y)>0 \right\}\tag{56}\] is coordinatewise weakly increasing and therefore belongs to \(\mathcal{Q}^{\mathrm{mon}}\). Because the pointwise maximizer of \(aV_n(y)\) over \(a\in[0,1]\) equals one when \(V_n(y)>0\), zero when \(V_n(y)<0\), and is immaterial to the objective when \(V_n(y)=0\), it follows that \[\label{eq:max95reduced95revenue} R_n^{*,\mathrm{red}}(K) = \sup_{q\in\mathcal{Q}^{\mathrm{mon}}} \mathbb{E}[q(Y)V_n(Y)] = \mathbb{E}[(V_n(Y))_+].\tag{57}\] By the lower-endpoint identity \(\mathbb{E}[\widehat J_i(Y_i)]=\underline{x}\), the asymptotic behavior of maximum reduced-form revenue is governed by the sign of \(\underline{x}\).

Proof. Appendix 9.6.

The theorem concerns maximum reduced-form revenue. Equality with revenue in the original mechanism problem requires that the envelope payment generated by \(q^{\mathrm{rev}}\) satisfy the range condition in Proposition 1.

4.4 Revenue of a Fixed Posterior-Score Rule↩︎

For \(\lambda\ge0\), let \(W_i(\lambda)=\widehat x_i(Y_i)+\lambda\widehat J_i(Y_i)=S_i(Y_i,\lambda)\) and let \[\label{eq:mean95score95def} \mu_S(\lambda) = \mathbb{E}[S_i(Y_i,\lambda)] = \mathbb{E}[X]+\lambda\,\underline{x}\tag{59}\] denote the per-agent mean posterior score; consider \(q_\lambda(Y)=\mathbf{1}\{\sum_iW_i(\lambda)\ge0\}\).

Proof. Appendix 9.7.

Proof. Appendix 9.8.

4.5 Hierarchical Prior↩︎

Let \(\theta\sim\pi\). Conditional on \(\theta\), suppose \(X_1,\ldots,X_n\overset{\mathrm{iid}}{\sim}F_\theta\) on the common support \(\mathcal{X}=[\underline{x},\bar{x}]\). Agents observe \(\theta\), while the planner observes only \(Y\); incentive compatibility and individual rationality are imposed conditional on \(\theta\). For every \(\theta\), define \(\widehat x^\theta(y_i)=\mathbb{E}[X_i\mid\theta,Y_i=y_i]\), \(\widehat J^\theta(y_i)=\mathbb{E}[J_\theta(X_i)\mid\theta,Y_i=y_i]\), and the hierarchical per-agent and aggregate scores \[\label{eq:hierarchical95score} \Psi_i(y,\lambda) = \mathbb{E}_{\theta\mid y} \left[ \widehat x^\theta(y_i) + \lambda\widehat J^\theta(y_i) \right], \qquad G_\lambda^{\mathrm H}(y) = \sum_{i=1}^{n}\Psi_i(y,\lambda).\tag{60}\] Unlike the known-prior score, \(\Psi_i(y,\lambda)\) generally depends on the entire signal profile through the posterior of \(\theta\). We first study a conditional reduced-form relaxation. In this relaxation, for each value of \(\theta\), the allocation must induce a conditionally monotone interim allocation and therefore admits a \(\theta\)-indexed envelope-payment schedule. Because the planner does not observe \(\theta\), these conditional payment schedules need not be jointly implementable by a single signal-measurable ex-post transfer. Full implementation therefore requires the additional simultaneous Fredholm compatibility condition described in Remark 10.

Proof. Appendix 9.9.

4.6 Welfare Comparisons across Channels↩︎

Let \(\mathcal{Q}^{\mathrm{mon}}(K)\) denote the reduced-form allocation rules that induce weakly increasing interim allocations under channel \(K\), and define \[\label{eq:reduced95value95channel} W^{\mathrm{red}}(R;K) = \sup_{q\in\mathcal{Q}^{\mathrm{mon}}(K)} \left\{ \mathbb{E}\!\left[ q(Y)\sum_{i=1}^{n}\widehat x_i(Y_i) \right]: \mathbb{E}\!\left[ q(Y)\sum_{i=1}^{n}\widehat J_i(Y_i) \right] \ge R \right\}.\tag{62}\]

Proof. Appendix 9.10.

4.6.1 Equal-Scale Laplace–Logistic Comparison↩︎

Proof. Appendix 7.7.

4.6.2 Common Tight-\(\mu\) Calibration↩︎

Let \(\mu_K(s)\) denote the smallest GDP parameter for which the channel at scale \(s\) satisfies \(\mu_K(s)\)-GDP.

Proof. Appendix 7.4.

Because \(\beta^*(\mu)<b_L^*(\mu)\), the common-\(\mu\) comparison is not an equal-scale comparison, so Proposition 5 does not determine the ordering under this calibration.

4.6.3 Endpoint Dominance under Common Tight \(\mu\)↩︎

Restrict the report space to the binary endpoint state space \(\{\underline{x},\bar{x}\}\), and let \(K_{\mathrm{Lap},\mu}^{\mathrm{end}}\) and \(K_{\mathrm{Log},\mu}^{\mathrm{end}}\) denote the tightly calibrated endpoint experiments.

Proof. Appendix 7.8.

Outside the exact comparisons above, no universal ranking of the Gaussian, Laplace, and logistic channels is asserted. The numerical comparisons in Section 5.4 are therefore specification dependent unless supported by Proposition 5 or Theorem 4.

5 Numerical Illustrations↩︎

This section ⁴ connects the main results to simulation. The four experiments trace, in turn, the price the planner pays for privacy in the responsiveness of provision, the knife-edge at which budget pressure switches a public good from always funded to never funded, the difficulty of financing provision when the marginal agent is break-even, and the channel-choice problem a regulator faces once privacy is budgeted in an economically meaningful unit. Throughout we use the three additive channels of Table 1—Gaussian \(Y=X+\eta^{G}\), Laplace \(Y=X+\eta^{L}\), and logistic \(Y=X+\eta^{\mathrm{Log}}\)—and a uniform type distribution, with the support chosen in each experiment to probe the relevant regime. Posterior moments \(\widehat x(y)=\mathbb{E}[X\mid Y=y]\) and \(\widehat J(y)=\mathbb{E}[J(X)\mid Y=y]\) are evaluated by quadrature against the prior; implementation frequencies and revenue are estimated by Monte Carlo with common random numbers across channels, so that channel comparisons are paired. The priors and privacy calibrations used in each experiment are recorded in Appendix 10.

5.1 The Price of Privacy in the Provision Rule↩︎

The optimal rule of Theorem 1 provides the public good when the aggregate posterior score \(G_\lambda=\sum_i S_i\) is positive (under the no-mass condition the threshold set is null, so the value assigned on \(\{G_\lambda=0\}\) is immaterial). Economically, each per-agent score \[S_i(y_i,\lambda) = \underbrace{\widehat x_i(y_i)}_{\text{expected value}} + \lambda\, \underbrace{\widehat J_i(y_i)}_{\text{revenue net of rents}}, \qquad J(x)=x-\frac{1-F(x)}{f(x)},\] is the planner’s case for building attributable to agent \(i\): the agent’s expected value for the good plus the revenue that can be extracted from that agent net of the information rent the agent commands, the two priced against each other by the shadow value \(\lambda\ge0\) of the revenue requirement. The privacy channel enters only through how sharply this case responds to the agent’s noisy report.

Figure 2 plots \(S(\cdot,\lambda)\) for each channel on a uniform prior. The Gaussian and logistic scores rise strictly with the signal: every report, however extreme, still moves the provision decision, so the good remains responsive to the private intensity of an agent’s preference. Under Laplace noise, the score varies over the interior region \(y\in(\underline{x},\bar{x})\) and is exactly flat on the global tails \(y\le\underline{x}\) and \(y\ge\bar{x}\): the privacy mechanism has pooled all sufficiently enthusiastic (or sufficiently reluctant) reporters into a single indistinguishable group, because the bounded prior makes the Laplace likelihood ratio constant in the tails (Remark 2). For those agents, expressing more enthusiasm buys no additional weight in the decision. The economic content of the channel is thus an elasticity: privacy noise governs how strongly public provision can track private values, and a coarser channel caps that responsiveness precisely for the types whose preferences are most extreme.

5.2 When Budget Pressure Switches the Good Off↩︎

The multiplier \(\lambda\) is the weight the planner places on revenue relative to welfare, and Proposition 2 shows that whether the good is funded is governed by the per-agent mean score \[\mu_S(\lambda)=\mathbb{E}[X]+\lambda\,\underline{x},\] the average net social value the planner perceives once revenue is priced at \(\lambda\). The decisive primitive is \(\underline{x}=\mathbb{E}[\widehat J_i(Y_i)]\) (Lemma 11), the lower endpoint of the type distribution, equal to the mean posterior virtual value, i.e.the average virtual surplus per agent. The mean score changes sign at the critical price \[\lambda_{\mathrm{crit}} = -\,\mathbb{E}[X]/\underline{x},\] where \(\mu_S(\lambda_{\mathrm{crit}})=0\). In the numerical design \(X\sim\mathop{\mathrm{Unif}}[-0.5,1.5]\), so \(\underline{x}<0\) and \(\mu_S'(\lambda)=\underline{x}<0\); hence \(\mu_S(\lambda)>0\) for \(\lambda<\lambda_{\mathrm{crit}}\) and \(\mu_S(\lambda)<0\) for \(\lambda>\lambda_{\mathrm{crit}}\), and by the law of large numbers the good is funded with probability tending to one for \(\lambda<\lambda_{\mathrm{crit}}\) and to zero for \(\lambda>\lambda_{\mathrm{crit}}\) (the two directions exchange when \(\underline{x}>0\)). The economic reading is stark: as the population grows the public good is essentially always provided while the budget is loose and essentially never provided once it tightens past \(\lambda_{\mathrm{crit}}\), with no smooth interior trade-off. A small increase in fiscal pressure can flip a project from “always built” to “never built.”

How privacy bears on this cliff is the content of the central-limit refinement. A separate local calculation around \(\lambda_{\mathrm{crit}}\), applied to sequences satisfying \(\sqrt n\,(\lambda_n-\lambda_{\mathrm{crit}})=O(1)\), gives a funding probability of approximately \(\Phi\!\big(\sqrt n\,\mu_S(\lambda)/\sigma_S(\lambda_{\mathrm{crit}})\big)\), where \(\sigma_S^2(\lambda_{\mathrm{crit}})=\mathop{\mathrm{Var}}\!\big(S_i(Y_i,\lambda_{\mathrm{crit}})\big)\). Writing the standardized budget pressure \[u = -\,\frac{\sqrt n\,\mu_S(\lambda)}{\sigma_S(\lambda_{\mathrm{crit}})} = -\,\frac{\sqrt n\,\underline{x}\,(\lambda-\lambda_{\mathrm{crit}})}{\sigma_S(\lambda_{\mathrm{crit}})},\] which absorbs the constant slope \(\mu_S'(\lambda_{\mathrm{crit}})=\underline{x}\), the probability converges to \(\Phi(-u)\) for every channel. Figure 3 confirms this: the curves for \(n\in\{25,\dots,500\}\) collapse onto the common Gaussian limit. For nondegenerate channels satisfying \(\sigma_S^2(\lambda_{\mathrm{crit}})>0\), the privacy channel enters the local transition through the scale \(\sigma_S(\lambda_{\mathrm{crit}})\). It stretches or compresses the transition, sharpening or blurring how precisely the planner can locate the funding margin, but it does not move the mean-score boundary \(\lambda_{\mathrm{crit}}\). The boundary is channel invariant because posterior expectations preserve the unconditional means \(\mathbb{E}[X]\) and \(\mathbb{E}[\widehat J(Y)]=\underline{x}\). Whether a public good is fundable is set by the mean posterior virtual value \(\underline{x}\), not by the privacy technology.

5.3 Financing a Good at the Break-Even Margin↩︎

Theorem 2 classifies optimal reduced-form revenue \(R_n^{*,\mathrm{red}}\) by the sign of the mean posterior virtual value \(\underline{x}\) of the previous subsection: revenue grows linearly with the market when \(\underline{x}>0\), decays when \(\underline{x}<0\), and—in the boundary case \(\underline{x}=0\)—grows only at square-root order \[R_n^{*,\mathrm{red}} \sim \frac{\sigma_J}{\sqrt{2\pi}}\,\sqrt n, \qquad \sigma_J^2=\mathop{\mathrm{Var}}\!\big(\widehat J_i(Y_i)\big).\] This boundary is the economically critical case: the lower endpoint equals zero, so average posterior virtual value has zero drift and the mechanism collects no systematic revenue. Per-capita revenue \(R_n^{*,\mathrm{red}}/n\to0\), so a privacy-respecting public-good mechanism cannot raise funds that scale with the market; what it collects comes entirely from the statistical luck of drawing a realized virtual surplus above its mean. The magnitude of that luck—and hence the financing the planner can hope for—is set by \(\sigma_J\), the informativeness of the privacy channel about types: a sharper channel raises the constant \(\sigma_J/\sqrt{2\pi}\), but no channel changes the \(\sqrt n\) rate. Figure 4 verifies both the rate and the constant, with the normalized revenue \(R_n^{*,\mathrm{red}}/\sqrt n\) flattening onto each channel’s level \(\sigma_J/\sqrt{2\pi}\). The message is that financing a public good from a population whose marginal member breaks even is fundamentally hard: privacy technology can buy a better constant, but not a better order of magnitude. We display \(\underline{x}=0\) because it gives the sharpest quantitative check; the funded (\(\underline{x}>0\)) regime is governed by its leading term \(n\underline{x}\), and the starved (\(\underline{x}<0\)) regime follows from the same theorem and is illustrated in Appendix 10.

5.4 Choosing a Privacy Technology: the Welfare Reversal↩︎

Which privacy technology should a planner adopt to maximize welfare? Figure 5 shows the answer is not a property of the noise distribution alone—it depends on the privacy-accounting standard used to hold privacy fixed. Raw noise scale, pure \(\epsilon\)-LDP, and \(\mu\)-GDP are distinct standards. For the Laplace–logistic comparison, a common noise scale is equivalent to a common tight pure-\(\epsilon\)-LDP calibration. Under this calibration, the Laplace channel Blackwell-dominates the logistic channel (Proposition 5), so \[W^{\mathrm{red}}(R;K_{\mathrm{Lap},s}) \ge W^{\mathrm{red}}(R;K_{\mathrm{Log},s}).\] Recalibrating both channels under a common tight \(\mu\)-GDP guarantee changes their relative scales and reverses the information ordering on the maximally separated endpoint experiment \(\{\underline{x},\bar{x}\}\): there the logistic channel takes the smaller scale and is the more informative experiment, so the welfare ranking flips (Theorem 4).

The left panel of Figure 5 plots the equal-scale welfare gap \((W_{\mathrm{Lap}}-W_{\mathrm{Log}})/W_{\mathrm{FB}}\ge0\) against the pure-\(\epsilon\) budget; the right panel plots the common-\(\mu\) endpoint welfare gap \((W_{\mathrm{Log}}-W_{\mathrm{Lap}})/W_{\mathrm{FB}}\ge0\) against the high-type probability, for three privacy levels \(\mu\). In each calibration the dominant channel’s welfare advantage is nonnegative, and the dominant channel flips from Laplace to logistic as the accounting standard changes from scale to GDP. The comparative reading is that no privacy channel is universally welfare-best; the welfare ranking depends on the privacy-accounting standard under which the channels are compared. We stress the scope. The reversal is established for the maximally separated endpoint experiment \(\{\underline{x},\bar{x}\}\) and does not lift to a Blackwell ranking over the full continuum of types (Remark 12); a planner therefore cannot conclude that logistic dominates Laplace under GDP for an arbitrary preference distribution. The sharp ROC-level form of the endpoint comparison, and the near-zero continuous-type gap that delimits its scope, are reported in Appendix 10.

6 Conclusion↩︎

We have studied binary public-good provision when a planner observes agents only through a fixed local-privacy channel. The privacy channel reshapes the information on which optimal provision is based: the optimal reduced-form rule thresholds an aggregate posterior score combining, for each agent, the posterior expected valuation and posterior virtual value. The channel enters through the responsiveness of this score to the agent’s privatized report.

Three economic conclusions emerge. First, the allocation the planner would like to choose and the payments needed to implement it are distinct objects. Implementing the envelope payments through transfers measurable in the privatized signals is an inverse problem governed by a Fredholm range condition: completeness gives uniqueness when a solution exists but does not guarantee existence. The Online Supplement complements this exact characterization with approximate implementations carrying explicit incentive, participation, and revenue bounds.

Second, maximum reduced-form revenue follows three asymptotic regimes governed by the lower endpoint of the valuation distribution. It is asymptotically linear, of square-root order, or exponentially small according as that endpoint is positive, zero, or negative. The privacy channel affects the quantitative difficulty within these regimes through the square-root constant and the large-deviation rate. For a fixed posterior-score rule, provision also changes sharply from asymptotically certain to asymptotically negligible at a mean-score boundary.

Third, welfare comparisons between privacy channels depend on how privacy is calibrated. For Laplace and logistic noise, a common scale is equivalent to the common tight pure-\(\epsilon\) calibration considered here, and Laplace Blackwell-dominates logistic noise on the full report space. Under a common tight \(\mu\)-GDP calibration, the ordering reverses for the maximally separated binary endpoint experiment. Thus the preferred channel may depend on the privacy standard used for comparison, while the endpoint reversal does not imply a general ordering over the full continuous type space.

The limits of these results also point to several directions for future work. The allocation and revenue results are reduced-form statements; connecting them to fully implemented mechanisms requires verifying the relevant transfer range condition. Characterizing which economically important envelope-payment schedules belong to the range of smoothing channel operators remains open. The hierarchical extension relies on a high-level coordinatewise-monotonicity condition that need not follow automatically once a latent common component links agents’ values. Identifying primitive conditions for this monotonicity, and studying the scope for surplus extraction under privatized correlated information, are natural next steps.

The channel comparisons leave a further boundary. Equal-scale Laplace-over-logistic dominance holds on the full report space, but the common-\(\mu\) reversal is established only for the endpoint experiment. Whether comparable welfare orderings hold on continuous type spaces, and how the conclusions change for nonadditive or nonsmoothing channels, remain open. More broadly, the framework suggests treating the privacy standard itself as part of the design problem: when welfare comparisons depend on the calibration used to hold privacy fixed, privacy accounting is not merely a technical convention.

7 Privacy and Channel Derivations↩︎

This appendix develops the privacy, likelihood-ratio, posterior, and comparison-of-experiments calculations used in the main text. We first derive the complete Neyman–Pearson trade-off functions for the Gaussian, Laplace, and logistic location channels. We then obtain their pure-LDP and tight GDP calibrations, establish the ordering of the common-\(\mu\) scales, describe posterior pooling under Laplace noise, derive the Gaussian posterior formula used in Example 2, and prove the two Blackwell comparisons stated in Section 4.6.

Throughout, \[\Delta_{\mathcal{X}} = \bar{x}-\underline{x}.\] For two reports \(r<r'\), write \[d=r'-r\in[0,\Delta_{\mathcal{X}}].\] For an additive location family, translation invariance reduces the binary experiment generated by \((r,r')\) to the experiment generated by locations \((0,d)\).

For probability measures \(P\) and \(Q\) on a common measurable space, recall the testing trade-off function \[\label{eq:app95tradeoff95definition} \mathcal{T}(P,Q)(\alpha) = \inf_{\psi} \left\{ 1-\mathbb{E}_Q[\psi]: \mathbb{E}_P[\psi]\le\alpha \right\}, \qquad \alpha\in[0,1],\tag{65}\] where the infimum is taken over measurable randomized tests \(\psi:\mathcal{Y}\to[0,1]\). The Neyman–Pearson lemma implies that, for each pair of simple hypotheses, the lower envelope is generated by likelihood-ratio tests.

7.1 Gaussian Location Channel↩︎

Let \[P_{0,\sigma} = \mathcal{N}(0,\sigma^2), \qquad P_{d,\sigma} = \mathcal{N}(d,\sigma^2).\] The log-likelihood ratio is \[\label{eq:app95gaussian95log95lr} \log \frac{dP_{d,\sigma}}{dP_{0,\sigma}}(y) = \frac{d}{\sigma^2}y - \frac{d^2}{2\sigma^2},\tag{66}\] which is strictly increasing in \(y\). Hence the most powerful test of \(P_{0,\sigma}\) against \(P_{d,\sigma}\) rejects for sufficiently large values of \(Y\).

For a level-\(\alpha\) test, choose \(c_\alpha\) such that \[\alpha = P_{0,\sigma}(Y\ge c_\alpha) = 1-\Phi(c_\alpha/\sigma).\] Thus \[c_\alpha = \sigma\Phi^{-1}(1-\alpha).\] The corresponding type-II error is \[P_{d,\sigma}(Y<c_\alpha) = \Phi\!\left( \frac{c_\alpha-d}{\sigma} \right).\] Therefore, \[\label{eq:app95gaussian95tradeoff} \mathcal{T}_{G,d,\sigma}(\alpha) = \Phi\!\left( \Phi^{-1}(1-\alpha)-\frac{d}{\sigma} \right) = G_{d/\sigma}(\alpha).\tag{67}\]

7.2 Laplace Location Channel↩︎

Let \[P_{0,b} = \mathop{\mathrm{Lap}}(0,b), \qquad P_{d,b} = \mathop{\mathrm{Lap}}(d,b), \qquad \rho = \frac{d}{b}.\] The likelihood ratio is \[\label{eq:app95laplace95lr} \frac{k_L(y\mid d)}{k_L(y\mid0)} = \exp\!\left\{ \frac{|y|-|y-d|}{b} \right\}.\tag{69}\] Equivalently, \[\label{eq:app95laplace95lr95piecewise} \frac{k_L(y\mid d)}{k_L(y\mid0)} = \begin{cases} e^{-\rho}, & y\le0, \\[3pt] e^{(2y-d)/b}, & 0<y<d, \\[3pt] e^\rho, & y\ge d. \end{cases}\tag{70}\] The ratio is weakly increasing but constant in each outer tail. Thus the Laplace channel satisfies weak, but not strict, MLRP.

7.3 Logistic Location Channel↩︎

The centered logistic distribution with scale \(\beta>0\) has cdf \[F_\beta(y) = \frac{1}{1+e^{-y/\beta}}\] and density \[\label{eq:app95logistic95density} g_\beta(y) = \frac{e^{-y/\beta}}{\beta(1+e^{-y/\beta})^2}.\tag{72}\] Let \[P_{0,\beta} = \mathop{\mathrm{Logistic}}(0,\beta), \qquad P_{d,\beta} = \mathop{\mathrm{Logistic}}(d,\beta), \qquad \rho = \frac{d}{\beta}.\]

7.4 Tight GDP Indices and Common-\(\mu\) Scales↩︎

For a trade-off function \(T\), define its Gaussian separation profile by \[\label{eq:app95gaussian95separation} D_T(\alpha) = -\Phi^{-1}(\alpha) - \Phi^{-1}(T(\alpha)), \qquad \alpha\in(0,1).\tag{73}\] Because \[G_\mu(\alpha) = \Phi\!\left( -\Phi^{-1}(\alpha)-\mu \right),\] the inequality \[T(\alpha)\ge G_\mu(\alpha)\] is equivalent to \[D_T(\alpha)\le\mu.\] Consequently, the tight GDP index of \(T\) is \[\label{eq:app95tight95mu95functional} \mu(T) = \sup_{\alpha\in(0,1)}D_T(\alpha).\tag{74}\]

Solving \(\mu_L(\rho)=\mu\) gives \[\label{eq:app95laplace95rho95mu} \rho_L^*(\mu) = -2\log\!\bigl(2\Phi(-\mu/2)\bigr),\tag{81}\] while solving \(\mu_{\mathrm{Log}}(\rho)=\mu\) gives \[\label{eq:app95logistic95rho95mu} \rho_{\mathrm{Log}}^*(\mu) = 2\log\!\left( \frac{\Phi(\mu/2)}{\Phi(-\mu/2)} \right).\tag{82}\]

7.5 Laplace Posterior Pooling↩︎

Suppose \[X\in[\underline{x},\bar{x}], \qquad Y=X+\eta, \qquad \eta\sim\mathop{\mathrm{Lap}}(0,b).\]

7.6 Uniform Prior with Gaussian Noise↩︎

Suppose \[X\sim\mathop{\mathrm{Unif}}[-1,1], \qquad Y=X+\eta, \qquad \eta\sim\mathcal{N}(0,\sigma^2).\] Conditional on \(Y=y\), \(X\) has a \(\mathcal{N}(y,\sigma^2)\) law truncated to \([-1,1]\). Define \[a(y) = \frac{-1-y}{\sigma}, \qquad b(y) = \frac{1-y}{\sigma}.\]

For \(X\sim\mathop{\mathrm{Unif}}[-1,1]\), \[F(x)=\frac{x+1}{2}, \qquad f(x)=\frac{1}{2},\] so \[J(x) = x-\frac{1-F(x)}{f(x)} = 2x-1.\] Therefore, \[\label{eq:uniform95gaussian95posterior95virtual} \widehat J(y) = 2h(y,\sigma)-1,\tag{83}\] and \[\label{eq:uniform95gaussian95posterior95score95appendix} S(y,\lambda) = (1+2\lambda)h(y,\sigma)-\lambda.\tag{84}\]

7.7 Equal-Scale Laplace–Logistic Ordering↩︎

The characteristic functions of centered Laplace and logistic random variables with scale \(s\) are \[\phi_{\mathrm{Lap},s}(t) = \frac{1}{1+s^2t^2}\] and \[\phi_{\mathrm{Log},s}(t) = \frac{\pi st}{\sinh(\pi st)}.\] Euler’s product formula gives \[\frac{\sinh(\pi z)}{\pi z} = \prod_{k=1}^{\infty} \left( 1+\frac{z^2}{k^2} \right).\] Hence \[\label{eq:app95logistic95cf95factorization} \phi_{\mathrm{Log},s}(t) = \frac{1}{1+s^2t^2} \prod_{k=2}^{\infty} \left( 1+\frac{s^2t^2}{k^2} \right)^{-1}.\tag{85}\]

7.8 Endpoint Ordering under Common Tight \(\mu\)↩︎

Fix \(\mu>0\), and define \[p = \Phi(-\mu/2) \in(0,1/2).\] Under common tight-\(\mu\) calibration, \[e^{-\rho_L^*} = 4p^2, \qquad e^{\rho_{\mathrm{Log}}^*} = \left( \frac{1-p}{p} \right)^2.\] Thus the calibrated endpoint trade-off functions are \[\label{eq:app95calibrated95laplace95tradeoff} T_L(\alpha) = \begin{cases} 1-\dfrac{\alpha}{4p^2}, & 0\le\alpha\le2p^2, \\[8pt] \dfrac{p^2}{\alpha}, & 2p^2\le\alpha\le\frac{1}{2}, \\[8pt] 4p^2(1-\alpha), & \frac{1}{2}\le\alpha\le1, \end{cases}\tag{86}\] and \[\label{eq:app95calibrated95logistic95tradeoff} T_{\mathrm{Log}}(\alpha) = \frac{1-\alpha}{ 1-\alpha+ \left( \frac{1-p}{p} \right)^2\alpha }.\tag{87}\]

8 Incentive, Posterior, and Monotonicity Results↩︎

This appendix collects the general one-dimensional incentive arguments and posterior identities used throughout the paper. These results depend on the reporting structure and the truthful signal law, but not on the particular form of the optimal allocation.

8.1 Interim Incentive Compatibility↩︎

8.2 Expected Payments↩︎

8.3 Posterior Factorization and Reduced-Form Representations↩︎

8.4 Posterior Monotonicity under MLRP↩︎

8.5 Lower-Endpoint Identity↩︎

8.6 Continuity of Interim Allocations↩︎

9 Functional Analysis and Proofs of the Main Results↩︎

This appendix establishes existence and strong duality for the reduced-form problem, proves the posterior-score characterization, develops the Fredholm implementation operator, proves the three revenue regimes and the fixed-score asymptotics, provides the hierarchical extension, and establishes Blackwell monotonicity.

9.1 Weak-\(^*\) Compactness↩︎

Let \[\nu(dy) = f_Y(y)\,dy\] denote the truthful signal law on \(\mathcal{Y}^n\). We identify allocations that agree \(\nu\)-almost surely and equip \(L^\infty(\nu)\) with the weak-\(^*\) topology \[\sigma(L^\infty(\nu),L^1(\nu)).\]

Define the allocation cube \[\label{eq:app95allocation95cube} \mathcal{A} = \left\{ q\in L^\infty(\nu): 0\le q\le1 \quad \nu\text{-a.e.} \right\}.\tag{91}\]

For each \(i\) and \(x\in\mathcal{X}\), define \[\ell_{i,x}(y) = \frac{k(y_i\mid x)}{m(y_i)}\] on the support of \(m\). Then \[Q_i^q(x) = \int_{\mathcal{Y}^n} q(y)\ell_{i,x}(y)\,\nu(dy),\] and \[\|\ell_{i,x}\|_{L^1(\nu)} = 1.\]

Define \[\mathcal{Q}^{\mathrm{mon}} = \left\{ q\in\mathcal{A}: Q_i^q(x')\ge Q_i^q(x) \text{ whenever }x'\ge x, \;\forall i \right\}.\]

Define \[w(y) = \sum_{i=1}^{n}\widehat x_i(y_i), \qquad v(y) = \sum_{i=1}^{n}\widehat J_i(y_i).\] Compactness of \(\mathcal{X}\) and boundedness of \(J\) imply \[w,v\in L^1(\nu).\]

9.2 Supporting Multipliers and Strong Duality↩︎

Define the downward-comprehensive attainable set \[\mathcal{C} = \left\{ (r,z)\in\mathbb{R}^2: \exists q\in\mathcal{Q}^{\mathrm{mon}} \text{ with } r\le\mathsf V(q), \quad z\le\mathsf W(q) \right\}.\]

9.3 Pointwise Lagrangian Maximization↩︎

9.4 Proof of the Optimal-Allocation Theorem↩︎

9.5 The Channel Operator and Fredholm Implementation↩︎

Let \(\mathcal{H}\) be the Banach space of admissible opponent-averaged transfers specified in Assumption 3. Define \[\mathcal{K}:\mathcal{H}\to C(\mathcal{X}), \qquad (\mathcal{K}g)(x) = \int_{\mathcal{Y}}g(y)k(y\mid x)\,dy.\]

9.6 Proof of the Revenue Regimes↩︎

Set \[Z_i = \widehat J_i(Y_i), \qquad S_n = \sum_{i=1}^{n}Z_i.\] The variables \(Z_i\) are i.i.d. and bounded, with \[\mathbb{E}[Z_i] = \underline{x}.\] Moreover, \[R_n^{*,\mathrm{red}}(K) = \mathbb{E}[(S_n)_+].\]

9.7 Proof for a Fixed Posterior-Score Rule↩︎

9.8 Local Transition at the Mean-Score Boundary↩︎

9.9 Hierarchical Prior↩︎

Define the hierarchical monotone allocation class by \[\mathcal{Q}_{\mathrm H}^{\mathrm{mon}} = \left\{ q: Q_{i,\theta}^q \text{ is weakly increasing in }x \text{ for every }i,\theta \right\}.\]

Conditional on \(\theta\), the envelope theorem gives \[U_i^\theta(x) = U_i^\theta(\underline{x}) + \int_{\underline{x}}^{x} Q_i^\theta(z)\,dz.\] Under conditional zero rents, \[\mathbb{E}[T_i(X_i)\mid\theta] = \mathbb{E}[ Q_i^\theta(X_i)J_\theta(X_i) \mid\theta ].\] Moreover, \[\mathbb{E}[X_i\mid\theta,Y] = \widehat x^\theta(Y_i), \qquad \mathbb{E}[J_\theta(X_i)\mid\theta,Y] = \widehat J^\theta(Y_i).\]

A common ex-post transfer must satisfy, for every \(\theta\) and \(x_i\), \[\begin{align} & \int_{\mathcal{Y}^n} t_i(y) k(y_i\mid x_i) \prod_{j\ne i} m_\theta(y_j)\,dy \\ &\qquad = \int_{\mathcal{Y}^n} q_{\mathrm H}^*(y) \left[ x_i k(y_i\mid x_i) - \int_{\underline{x}}^{x_i} k(y_i\mid z)\,dz \right] \prod_{j\ne i} m_\theta(y_j)\,dy. \end{align}\] Conditional completeness identifies the conditional opponent average if a solution exists. It does not imply that a single ex-post transfer belongs to the intersection of the ranges of all conditional channel operators.

9.10 Blackwell Monotonicity↩︎

10 Additional Numerical Results↩︎

This appendix records the numerical illustrations not shown in Section 5. Posterior quantities are computed by quadrature and Monte Carlo comparisons use common random numbers across channels, so that all channel comparisons are paired.

10.1 Simulation Design and Calibration↩︎

10.1.0.1 Priors.

The posterior-score panel uses \[X\sim\mathop{\mathrm{Unif}}[-1,1].\] The phase-transition, budget-tradeoff, and approximate-LDP experiments use \[X\sim\mathop{\mathrm{Unif}}[-0.5,1.5],\] for which \[\mathbb{E}[X]=0.5, \qquad \underline{x}=-0.5, \qquad \lambda_{\mathrm{crit}}=-\,\mathbb{E}[X]/\underline{x}=1.\] The square-root revenue experiment uses \[X\sim\mathop{\mathrm{Unif}}[0,1],\] so that \(\underline{x}=0\). The exponential-regime experiment uses a prior with \(\underline{x}<0\).

10.1.0.2 Common tight-\(\mu\) calibration.

For support width \(\Delta_{\mathcal{X}}\), the frontier scales are \[\sigma = \frac{\Delta_{\mathcal{X}}}{\mu}, \qquad b_L = \frac{\Delta_{\mathcal{X}}}{-2\log(2\Phi(-\mu/2))}, \qquad \beta = \frac{\Delta_{\mathcal{X}}}{2\log\!\big(\Phi(\mu/2)/\Phi(-\mu/2)\big)},\] as in Proposition 6.

10.1.0.3 Equal-scale calibration.

For a common pure-\(\epsilon\) frontier, \[b_L=\beta=\frac{\Delta_{\mathcal{X}}}{\epsilon}.\]

10.2 The Exponential Revenue Regime↩︎

Of the three regimes in Theorem 2, the linear case \(\underline{x}>0\) is governed by its leading term \(n\underline{x}\) and needs no separate plot; the boundary case \(\underline{x}=0\) is shown in Section 5. Here we illustrate the remaining case. When \(\underline{x}<0\), Theorem 2 predicts \[R_n^{*,\mathrm{red}} = \exp\{-nI_K(0)+o(n)\},\] and the probability of positive aggregate posterior virtual surplus obeys the same exponential decay exponent \(I_K(0)\) (Remark 7).

10.3 Hierarchical Unknown-Endpoint Diagnostic↩︎

The hierarchical theorem assumes a common conditional support. The numerical specification \[X_i\mid\theta\sim\mathop{\mathrm{Unif}}[\underline{x},\theta]\] instead has a parameter-dependent support and is nonregular, so it falls outside Theorem 3 (Remark 9). It is included only as a descriptive diagnostic.

10.4 Equal-Scale Welfare–Revenue Frontiers↩︎

At equal scale, Laplace Blackwell-dominates logistic (Proposition 5). Both the maximal reduced-form revenue and the full reduced-form welfare–revenue frontier are therefore weakly higher under Laplace.

10.5 Common-\(\mu\) Endpoint and Continuous-Type Comparisons↩︎

The left panel of Figure 10 displays the exact endpoint trade-off difference \[\mathcal{T}_{\mathrm{Lap}}(\alpha) - \mathcal{T}_{\mathrm{Log}}(\alpha),\] which is nonnegative by Theorem 4 and vanishes at \[\alpha\in\{0,\Phi(-\mu/2),1\},\] the three testing levels at which both endpoint experiments touch the Gaussian benchmark (Remark 13).

The right panel reports the corresponding continuous-type welfare difference. That difference is a numerical quantity, not a consequence of the endpoint Blackwell theorem. Its confidence band overlaps zero over a substantial part of the displayed range, so the simulation does not support extending the endpoint Blackwell order to the full continuum (Remark 12).

10.6 Welfare–Revenue Trade-off under Approximate LDP↩︎

This experiment uses a common \((\epsilon,\delta)\)-LDP frontier with \[\delta=10^{-5},\] under the prior \(X\sim\mathop{\mathrm{Unif}}[-0.5,1.5]\) of the design section. The first panel reports welfare efficiency against the privacy budget. The second reports welfare and signed reduced-form revenue as the score multiplier varies, with the vertical reference at the mean-score boundary \(\lambda_{\mathrm{crit}}=1\).

References↩︎

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1. Department of Economics, University of Pittsburgh. Email: bem159@pitt.edu (Corresponding Author)↩︎

2. Department of Statistics, University of Pittsburgh. Email: jaa557@pitt.edu↩︎

3. Throughout, \(m(\cdot)\) denotes the one-signal marginal density. The mean of the posterior score introduced later is denoted by \(\mu_S(\lambda)\).↩︎

4. Note: The code for all numerical results, together with the additional material, is available at https://github.com/BehroozMoosavi/Codes/tree/main/private_mechanism_design.↩︎