# TARGETING INTERVENTIONS IN NETWORKS

**Date**: 03/09/2026
**Domain**: social_sciences/economics
**Taxonomy**: academic/research_paper
**Filter**: Active comments

---

## Overall Feedback

Here are some overall reactions to the document.

**Property A Is a Load-Bearing Restriction Whose Scope and Limitations Deserve Prominent Treatment in the Main Text**

Theorem 1, Corollary 1, and Propositions 1–2 all rest on Property A, which requires aggregate equilibrium welfare to be proportional to the sum of squared equilibrium actions (W = w·(a*)ᵀa*). All three reviewers flag this as a substantive functional-form restriction, not merely a technical convenience. The paper's own Online Appendix OA3.1 demonstrates that natural environments—such as the peer-effects model with a global externality—violate Property A and require the substantially more complex Theorem OA1, whose large-budget characterization features two competing binding constraints rather than one, breaking the clean 'complements → top eigenvector, substitutes → bottom eigenvector' dichotomy of Corollary 1. The general extension in OA3.1 additionally requires Assumption OA1 (constant row sums), which itself rules out heterogeneous-degree networks common in empirical applications. It would be helpful to state explicitly in Section 4 which economic environments satisfy Property A, which qualitative conclusions survive its relaxation, and whether the ordering results of Corollary 1 can be reversed for plausible parameter values under Theorem OA1, rather than leaving this assessment entirely to the appendix.

**The Non-Degeneracy Condition on the Status-Quo Vector Is Implicit but Materially Affects Theorem 1 and Proposition 1**

The proof of Theorem 1 employs the change-of-variables xₗ = yₗ / b̂ₗ, which is undefined whenever b̂ₗ = 0, i.e., whenever the status-quo vector b̂ is orthogonal to some eigenvector of G. The paper acknowledges this only in passing ('We take a generic b̂ such that b̂ₗ ≠ 0 for all ℓ'), without incorporating it as a formal assumption alongside Assumptions 1–3 or discussing its economic content. A symmetric issue arises in Proposition 1, Part 2: the large-budget limit μ → wα₁ requires b̂₁ ≠ 0; if b̂₁ = 0, the first term in the binding constraint sum vanishes and the multiplier converges to wα₂ instead, reversing the conclusion that optimal interventions align with u¹(G). In economically natural cases—uniform status-quo vectors b̂ = c·1 on bipartite or regular graphs, where many eigenvectors are orthogonal to the all-ones vector—zero projections are the rule rather than the exception. It would be helpful to add the non-degeneracy condition as a formal assumption in both Theorem 1 and Proposition 1, to characterize the optimal intervention when some b̂ₗ = 0 (since the planner's problem remains well-posed in those components), and to verify that the numerical examples in Figures 2–4 all satisfy the condition.

**The Budget Threshold for Simple Interventions in Proposition 2 May Be Vacuous for Realistic Networks, and the Proof Contains Steps That Warrant Explicit Justification**

Proposition 2 gives a sufficient budget threshold C* = (2‖b̂‖²/ε)(α₂/(α₁−α₂))² above which the simple intervention achieves near-optimal welfare and near-unit cosine similarity with the leading eigenvector. The term α₂/(α₁−α₂) diverges as the spectral gap λ₁−λ₂ → 0, which occurs for Erdős–Rényi graphs near the connectivity threshold, stochastic block models with weak community structure, and scale-free networks where the gap is O(1/n), making C* grow as O(n²) or faster. The paper illustrates Proposition 2 only on small hand-crafted examples and does not analyze how the threshold scales with network size or topology for any canonical family, leaving open whether the 'simplicity' prescription is empirically relevant. Additionally, the proof's welfare-bound step—bounding ∑_{ℓ≠1} b̂ₗ² αₗ xₗ*(xₗ*+2) ≤ α₂ x₂*(x₂*+2) ∑_{ℓ≠1} b̂ₗ²—implicitly requires xₗ* ≥ −1 for the function t(t+2) to be increasing in |t|, and the final step substituting μ ≥ wα₁ to bound x₂* uses a one-sided bound that is stated but not derived at that point. It would be helpful to make these intermediate steps explicit, to provide a lower bound on C* or a tightness example, and to compute the threshold for at least one canonical network family such as a two-community stochastic block model.

**The Incomplete-Information Extension Does Not Fully Specify the Feasibility Set, the Planner's Commitment Technology, or the Equilibrium Concept**

Section 5 introduces the incomplete-information problem (IT-G) in which the planner chooses a random variable B to maximize E[W(b; G)] subject to K(B) ≤ C, but leaves several foundational elements underspecified. First, the paper does not clarify whether the budget constraint holds almost surely (ex-post feasibility) or only in expectation (ex-ante feasibility); for the variance-control case of Proposition 4, where K(B) is evaluated at the distribution rather than at each realization, this distinction materially affects the feasibility set and the permutation argument in the proof. Second, the mechanism by which a planner 'chooses a random variable B' over standalone returns is not microfounded: it is unclear whether this represents a randomized policy rule, a signal-contingent subsidy schedule, or a menu of contracts, and whether the planner's commitment to a distribution is credible before agents observe the realization. Third, the equilibrium concept is left implicit—agents play a complete-information game given each realization of b, while the planner optimizes over the distribution, creating an asymmetry that is not formally resolved as a well-defined extensive form. It would be helpful to write out the feasibility set explicitly as a set of distributions, to provide a concrete institutional interpretation of the planner's instrument, and to verify that the timing described in Section 2 is consistent with the incomplete-information setup.

**The SVD Extension to Non-Symmetric Networks Conflates Left and Right Singular Vectors and Does Not Verify the Required Singular-Value Ordering**

Online Appendix OA3.2 extends the analysis to non-symmetric G via the SVD M = USVᵀ of M = I − βG, asserting that 'Theorem 1 applies with αₗ = 1/sₗ².' Two distinct issues arise. First, the equilibrium gives Vᵀa* = S⁻¹Uᵀb, so welfare W = w(a*)ᵀa* = w‖Vᵀa*‖² is expressed in the V-basis while the cost ‖y‖² = ‖b − b̂‖² is expressed in the original (equivalently U-) basis; when U ≠ V—the generic case for non-symmetric G—the cost of changing b̄ₗ (U-basis) does not translate diagonally into a change in ā*ₗ (V-basis), so the claim that Theorem 1 applies requires either that M is normal (U = V) or a corrected derivation tracking the basis change explicitly. Second, the strategic complements/substitutes dichotomy requires the singular values of M to be ordered in the same way as the eigenvalue-based amplification factors 1/(1−βλₗ) in the symmetric case; for non-symmetric G, the singular values of M = I − βG are square roots of eigenvalues of MᵀM = (I−βGᵀ)(I−βG), whose ordering depends on both eigenvalues and eigenvectors of G in a complex way that is not verified. It would be helpful to either restrict OA3.2 to normal matrices, provide conditions on G ensuring the required singular-value ordering, or restate the large-budget result in terms of the smallest singular value of M without invoking the complements/substitutes framing.

**The Incremental Contribution Relative to Demange (2017) and Ballester et al. (2006) Is Not Precisely Delineated, and Policy Prescriptions Are Illustrated Only on Small Stylized Networks**

The paper claims that its principal-components approach yields eigenvector centrality as the optimal targeting rule, in contrast to the Bonacich centrality of Ballester et al. (2006), and notes that the two converge as the spectral radius of βG approaches 1. However, the discussion in OA2.1 does not formally delineate what is new relative to Demange (2017), which also studies a utilitarian planner in a network game of strategic complements under a budget constraint. A formal comparison—either a proposition or a worked example—showing precisely when and by how much the optimal intervention derived here differs from the Bonacich-centrality rule would clarify the incremental contribution. Separately, all policy illustrations use small hand-crafted networks (an 11-node two-hub graph in Figure 2, and the graphs in Figures 3–4). For scale-free networks, where the first eigenvector is dominated by high-degree hubs, the prescription under complements reduces to 'target hubs,' which is already well-known. It would be helpful to include at least one canonical large-network class—such as an Erdős–Rényi or Barabási–Albert graph—to demonstrate that the spectral decomposition approach yields genuinely new targeting insights beyond degree- or Bonacich-centrality-based rules, and to assess whether the simplicity thresholds from Proposition 2 are empirically relevant at realistic network scales.

**Assumption 1 (Symmetry of G) is load-bearing but its scope relative to directed/asymmetric networks deserves clearer framing**

Assumption 1 requires G to be symmetric, which is what licenses the real spectral decomposition G = UΛUᵀ used throughout (Fact 1, equation (4), Theorem 1, Corollary 1, Propositions 1–2). The paper does extend to non-symmetric G in Online Appendix OA3.2 via the singular value decomposition of M = I − βG, and it notes there that when G is non-symmetric the singular values of M are not equal to 1 − βλₗ and the left and right singular vectors need not coincide. However, the main-text results (Theorem 1 and all corollaries) are stated and interpreted exclusively under symmetry, and the abstract/introduction do not flag this restriction. In many empirically relevant settings—directed social networks, input-output matrices, citation networks—G is not symmetric. Readers might note that the welfare-maximizing targeting rules derived in the main text (e.g., 'target in proportion to eigenvector centrality under strategic complements') do not carry over directly to directed networks, because eigenvector centrality is no longer well-defined in the same sense and the amplification factors α_ℓ = (1 − βλₗ)⁻² rely on real eigenvalues. It would be helpful to add a brief remark in the main text acknowledging that the principal-component characterization is specific to symmetric interaction matrices and directing readers to OA3.2 for the directed case.

**Property A is assumed rather than derived for the main welfare analysis, and its failure mode is not fully integrated into the headline results**

Property A—that aggregate equilibrium utility is proportional to (a*)ᵀa*—is the linchpin of Theorem 1 and all main-text results. The paper verifies it for Examples 1 and 2 and the beauty-contest extension, and relaxes it in OA3.1 under the additional Assumption OA1 (constant row sums, ∑ⱼ gᵢⱼ = 1 for all i). However, Assumption OA1 is itself a strong restriction: it rules out heterogeneous degree distributions, which are the norm in empirical networks (e.g., scale-free or core-periphery structures). When Assumption OA1 fails and Property A does not hold, the welfare function acquires a linear term w₃∑ᵢaᵢ* and a squared-mean term w₂(∑ᵢaᵢ*)², as shown in Theorem OA1. The large-budget limit then depends on the comparison between w₁α₂ and (w₁+w₂)α₁ (parts 4a/4b of Theorem OA1), so the clean 'target the top/bottom eigenvector' prescription of Proposition 1 can break down. It would be helpful to note in the main text that the eigenvector-centrality targeting rule for large budgets is contingent on Property A (or Assumption OA1), and to give readers intuition for when the correction terms w₂ and w₃ are quantitatively important.

**Status**: [Pending]

---

## Detailed Comments (20)

### 1. Assumption 3 Uses Norm Instead of Squared Norm

**Status**: [Pending]

**Quote**:
> Either $w<0$ and $C<\|\hat{\bm{b}}\|$, or $w>0$.

**Feedback**:
The motivating argument states that when w < 0 the planner can achieve the first-best by setting bᵢ = 0 for all i at cost ‖0 − b̂‖² = ‖b̂‖², so the interesting case requires C < ‖b̂‖². The Appendix proof of Theorem 1 confirms this, writing the condition as ∑_ℓ b̂_ℓ² > C, i.e., ‖b̂‖² > C. The statement as written uses ‖b̂‖ (the norm, not its square), which is dimensionally inconsistent with the quadratic budget constraint ∑ᵢ(bᵢ − b̂ᵢ)² ≤ C. It would be helpful to rewrite 'C < ‖b̂‖' as 'C < ‖b̂‖²'.

---

### 2. Inline Reminder of α_ℓ Definition Drops the Square in the Denominator

**Status**: [Pending]

**Quote**:
> The second factor, $\frac{w\alpha_{\ell}}{\mu-w\alpha_{\ell}}$, is determined by two quantities: the eigenvalue corresponding to $\bm{u}^{\ell}(\bm{G})$ (via $\alpha_{\ell}=\frac{1}{1-\beta\lambda_{\ell}}$), and the budget $C$ (via the shadow price $\mu$).

**Feedback**:
Earlier in the same section α_ℓ is correctly defined as 1/(1 − βλ_ℓ)², with the square in the denominator, consistent with the Appendix definition and the equilibrium condition. The inline reminder here writes α_ℓ = 1/(1 − βλ_ℓ), missing the square. A reader relying on this reminder would compute incorrect amplification factors and misinterpret the ordering in Corollary 1. It would be helpful to rewrite 'α_ℓ = 1/(1 − βλ_ℓ)' as 'α_ℓ = 1/(1 − βλ_ℓ)²'.

---

### 3. Index Typo in Proposition 1 Part 1: 'r_t' Should Be 'r_ℓ'

**Status**: [Pending]

**Quote**:
> 1. As $C\to 0$, in the optimal intervention, $\frac{r_{t}^{*}}{r_{\ell^{\prime}}^{*}}\to\frac{\alpha_{\ell}}{\alpha_{\ell^{\prime}}}$.

**Feedback**:
The numerator on the left uses index 't' while the right-hand side uses α_ℓ and the denominator uses ℓ′, mixing three distinct indices and making the statement ill-formed. The Appendix proof of Proposition 1 confirms the intended claim is r_ℓ*/r_{ℓ′}* → α_ℓ/α_{ℓ′}. It would be helpful to rewrite 'r_{t}^{*}' as 'r_{\ell}^{*}'.

---

### 4. Proposition 2 Welfare Bound States Strict Inequality but Proof Derives Weak Inequality

**Status**: [Pending]

**Quote**:
> 1. If the game has the strategic complements property, $\beta>0$, then for any $\epsilon>0$, if $C>\frac{2\|\hat{\bm{b}}\|^{2}}{\epsilon}\left(\frac{\alpha_{2}}{\alpha_{1}-\alpha_{2}}\right)^{2}$, then $W^{*}/W^{s}<1+\epsilon$ and $\rho(\bm{y}^{*},\sqrt{C}\bm{u}^{1})>\sqrt{1-\epsilon}$.

**Feedback**:
The Appendix proof of Proposition 2 derives W* − Wˢ ≤ εWˢ, yielding W*/Wˢ ≤ 1 + ε (weak inequality). The proposition states a strict inequality W*/Wˢ < 1 + ε, but no additional argument is given for why equality is excluded when the budget threshold is met exactly. It would be helpful to rewrite 'W*/Wˢ < 1 + ε' as 'W*/Wˢ ≤ 1 + ε' in both Parts 1 and 2 of Proposition 2 to match the weak inequality established in the proof.

---

### 5. Lagrangian Missing Squared Term on Status-Quo Projection

**Status**: [Pending]

**Quote**:
> $\mathcal{L}=w\sum_{\ell}\alpha_{\ell}(1+x_{\ell})^{2}\hat{\underline{b}}_{\ell}+\mu\left[C-\sum_{\ell}\hat{\underline{b}}_{\ell}^{2}x_{\ell}^{2}\right].$

**Feedback**:
The transformed objective after the change of variables b̲_ℓ = b̂_ℓ(1+x_ℓ) is w∑_ℓ α_ℓ(1+x_ℓ)²b̂_ℓ², where each term is weighted by b̂_ℓ² (squared). Substituting b̲_ℓ = b̂_ℓ(1+x_ℓ) into w∑_ℓ α_ℓ b̲_ℓ² gives w∑_ℓ α_ℓ(1+x_ℓ)²b̂_ℓ². The first-order condition (10) is derived correctly with b̂_ℓ², confirming the Lagrangian as written contains a typographical omission of the square. It would be helpful to rewrite 'w∑_ℓ α_ℓ(1+x_ℓ)²b̂_ℓ' as 'w∑_ℓ α_ℓ(1+x_ℓ)²b̂_ℓ²'.

---

### 6. Bounding Step Cites Corollary 1 but Ordering Follows from Theorem 1

**Status**: [Pending]

**Quote**:
> $\leq\alpha_{2}x_{2}^{*}(x_{2}^{*}+2)\sum_{\ell\neq 1}\hat{\underline{b}}_{\ell}^{2}$ Corollary 1

**Feedback**:
The step claims ∑_{ℓ≠1} b̂_ℓ² x_ℓ*(x_ℓ*+2) ≤ x₂*(x₂*+2) ∑_{ℓ≠1} b̂_ℓ². For this to hold one needs x_ℓ*(x_ℓ*+2) ≤ x₂*(x₂*+2) for all ℓ > 1. Since x_ℓ* = wα_ℓ/(μ−wα_ℓ) > 0 (for w > 0) and t(t+2) is increasing for t > 0, this follows from x₂* ≥ x_ℓ* for ℓ ≥ 2, which in turn follows from α₂ ≥ α_ℓ (Theorem 1), not from Corollary 1 (which concerns similarity ratios r_ℓ*). It would be helpful to rewrite the citation as 'Theorem 1 and monotonicity of t(t+2) for t ≥ 0'.

---

### 7. Cosine Similarity Formula in Equation (11) Has Inconsistent Argument on Left-Hand Side

**Status**: [Pending]

**Quote**:
> $\rho(\Delta\bm{b}^{*},\sqrt{C}\bm{u}^{1})=\frac{(\bm{b}^{*}-\hat{\bm{b}})\cdot(\sqrt{C}\bm{u}^{1})}{\|\bm{b}^{*}-\hat{\bm{b}}\|\|\sqrt{C}\bm{u}^{1}\|}=\frac{(\bm{b}^{*}-\hat{\bm{b}})\cdot(\bm{u}^{1})}{\sqrt{C}},$ (11)

**Feedback**:
The algebra is correct: since ‖b*−b̂‖ = √C at the optimum and ‖√C u¹‖ = √C, the ratio simplifies to (b*−b̂)·u¹/√C. However, the left-hand side writes ρ(Δb*, √C u¹) while equation (12) and the lemma statement write ρ(Δb*, u¹). Cosine similarity is scale-invariant so the two are equal, but the inconsistency may confuse readers. It would be helpful to rewrite the left-hand side of (11) as ρ(Δb*, u¹) to match equation (12) and the lemma statement.

---

### 8. Entry-Wise Positivity of u¹ Invoked Without Non-Negativity or Connectivity Assumption on G

**Status**: [Pending]

**Quote**:
> Eigenvector $\bm{u}^{1}=\arg\max_{\bm{u}:\|\bm{u}\|=1}\sum_{ij}g_{ij}u_{i}u_{j}$ (corresponding to $\lambda_{1}$) assigns the same sign – say, positive to all nodes in the network. Clearly, eigenvector $\bm{u}^{2}$ must assign negative values to some of the nodes (as it is orthogonal to $\bm{u}^{1}$).

**Feedback**:
The claim that u¹ is entry-wise positive follows from the Perron–Frobenius theorem, which requires G to be non-negative and irreducible (connected). Assumption 1 requires only that G is symmetric; it does not require non-negative entries or connectivity. For a symmetric matrix with negative entries or a disconnected graph, the leading eigenvector need not be entry-wise positive, and the orthogonality argument for u² would also fail. It would be helpful to add a parenthetical noting that this conclusion uses the fact that G is non-negative and connected, so the Perron–Frobenius theorem guarantees u¹ is entry-wise positive, and to verify that this connectivity condition is either already assumed or should be added as a formal assumption.

---

### 9. Typographical Error in Beauty-Contest First-Order Condition: b̃ᵢ Instead of β̃

**Status**: [Pending]

**Quote**:
> $$
> a_i = \frac{\tilde{b}_i}{1 + \gamma} + \frac{\tilde{b}_i + \gamma}{1 + \gamma} \sum g_{ij} a_j.
> $$

**Feedback**:
Differentiating Uᵢ with respect to aᵢ and using ∑_j gᵢⱼ = 1 gives aᵢ(1+γ) = b̃ᵢ + (β̃+γ)∑_j gᵢⱼ aⱼ, so the coefficient on ∑_j gᵢⱼ aⱼ in the numerator of the second term should be (β̃ + γ), not (b̃ᵢ + γ). The subsequent reparametrization β = (β̃+γ)/(1+γ) is stated correctly and is consistent with the corrected FOC, but the displayed equation as written is internally inconsistent with that reparametrization. It would be helpful to rewrite the second term's numerator as '\tilde{\beta} + \gamma' rather than '\tilde{b}_i + \gamma'.

---

### 10. Sign of Cross-Effect ∂Uᵢ/∂aⱼ Stated with Incorrect Threshold

**Status**: [Pending]

**Quote**:
> This is a game of strategic complements; moreover, an increase in $j$'s action has a positive effect on individual $i$'s utility if and only if $a_j < a_i$.

**Feedback**:
Computing ∂Uᵢ/∂aⱼ = gᵢⱼ[β̃ aᵢ − γ(aⱼ − aᵢ)] = gᵢⱼ[(β̃+γ)aᵢ − γaⱼ], this is positive when aⱼ < (β̃+γ)/γ · aᵢ, not simply when aⱼ < aᵢ (which would require β̃ = 0, contradicting the maintained assumption β̃ > 0). For β̃ > 0 the threshold (1 + β̃/γ)aᵢ > aᵢ, so the region where the cross-effect is positive is strictly larger than {aⱼ < aᵢ}. It would be helpful to rewrite 'if and only if aⱼ < aᵢ' as 'if and only if aⱼ < (β̃+γ)/γ · aᵢ'.

---

### 11. Typo in SVD Equilibrium Condition: b_ℓ² Instead of b̲_ℓ

**Status**: [Pending]

**Quote**:
> Let $\underline{\bm{a}}=\bm{V}^{\mathsf{T}}\bm{a}$ and $\underline{\bm{b}}=\bm{U}^{\mathsf{T}}\bm{b}$; then the equilibrium condition implies that:
> 
> $\underline{a}_{\ell}^{*}=\frac{1}{s_{\ell}}b_{\ell}^{2},$

**Feedback**:
From the equilibrium condition a* = M⁻¹b = VS⁻¹Uᵀb, we have Vᵀa* = S⁻¹Uᵀb, giving ā*_ℓ = (1/s_ℓ)b̲_ℓ. The text writes b_ℓ² (b-sub-ℓ squared) instead of b̲_ℓ (b-underline-sub-ℓ), which is dimensionally inconsistent: the left side has units of action while the right side would have units of (standalone return)². The subsequent welfare calculation W = w∑_ℓ b̲_ℓ²/s_ℓ² would not follow from the squared expression. It would be helpful to rewrite '$\underline{a}_{\ell}^{*}=\frac{1}{s_{\ell}}b_{\ell}^{2}$' as '$\underline{a}_{\ell}^{*}=\frac{1}{s_{\ell}}\underline{b}_{\ell}$'.

---

### 12. Domain of x₁ Stated Inconsistently: √C/b̂₁ vs C/b̂₁

**Status**: [Pending]

**Quote**:
> Lemma OA2 implies that $\mu$ as a function of $x_{1}\in\left[-C/\hat{b}_{1},C\hat{b}_{1}\right]$ is U-shaped; the slope is $-\infty$ at $x_{1}=-C/\hat{b}_{1}$ and $+\infty$ at $x_{1}=C/\hat{b}_{1}$; and it reaches a minimum at $x_{1}=0$.

**Feedback**:
The feasibility condition C − b̂₁² x₁² ≥ 0 requires |x₁| ≤ √C/b̂₁. Lemma OA2 itself states limits as x₁ → ±√C/b̂₁, and the First Step of the proof writes x₁ ∈ [−√C/b̂₁, √C/b̂₁]. The passage quoted writes the domain as [−C/b̂₁, C b̂₁], which has two errors: the left endpoint is missing the square root, and the right endpoint is C·b̂₁ (a product) rather than √C/b̂₁ (a quotient). It would be helpful to rewrite 'x_{1}\in\left[-C/\hat{b}_{1},C\hat{b}_{1}\right]' as 'x_{1}\in\left[-\sqrt{C}/\hat{\underline{b}}_{1},\sqrt{C}/\hat{\underline{b}}_{1}\right]'.

---

### 13. Variance Swap Equation in Proof of Proposition 4 Is Self-Contradictory

**Status**: [Pending]

**Quote**:
> and so $\operatorname{Var}(\underline{b}_k^{**}) = \operatorname{Var}(\underline{b}_k^*)$ for all $k \notin \{\ell, \ell'\}$ and $\operatorname{Var}(\underline{b}_{\ell}^{*}) = \operatorname{Var}(\underline{b}_{\ell'}^{*}) > \operatorname{Var}(\underline{b}_{\ell'}^{*}) = \operatorname{Var}(\underline{b}_{\ell}^{*})$.

**Feedback**:
The permutation P swaps indices ℓ and ℓ′, so in the principal-component basis Var(b̲_ℓ^{**}) = Var(b̲_{ℓ′}^*) and Var(b̲_{ℓ′}^{**}) = Var(b̲_ℓ^*). The intended statement is Var(b̲_ℓ^{**}) = Var(b̲_{ℓ′}^*) > Var(b̲_ℓ^*) = Var(b̲_{ℓ′}^{**}), using the hypothesis Var(b̲_ℓ^*) < Var(b̲_{ℓ′}^*). As written, the chain 'A = B > B = A' is self-contradictory. It would be helpful to rewrite as 'Var(b̲_ℓ^{**}) = Var(b̲_{ℓ′}^*) > Var(b̲_ℓ^*) = Var(b̲_{ℓ′}^{**})'.

---

### 14. Variance-Covariance Transformation Formula for B** Omits U Factors

**Status**: [Pending]

**Quote**:
> Furthermore, the matrix
> 
> $$
> \boldsymbol {\Sigma} _ {\mathcal {B} ^ {* *}} = \boldsymbol {P} \boldsymbol {\Sigma} _ {\mathcal {B} ^ {*}} \boldsymbol {P} ^ {\top}
> $$

**Feedback**:
With B** = O B* and O = U P Uᵀ, the variance-covariance matrix in the original basis is Σ_{B**} = O Σ_{B*} Oᵀ = U P (Uᵀ Σ_{B*} U) Pᵀ Uᵀ = U P Σ_{b̲*} Pᵀ Uᵀ. The formula as written, Σ_{B**} = P Σ_{B*} Pᵀ, omits the U and Uᵀ factors and is incorrect in the original basis. What is correct is the principal-component-basis formula Σ_{b̲**} = P Σ_{b̲*} Pᵀ. It would be helpful to rewrite the displayed equation as Σ_{b̲**} = P Σ_{b̲*} Pᵀ (adding underlines to both sides) to indicate this holds in the principal-component basis.

---

### 15. Cost Computation in IT-P Section Has Asymmetric Factor-of-Two

**Status**: [Pending]

**Quote**:
> $K(\bm{y})$ $=$ $\frac{1}{2}\sum_{i}\bm{1}_{y_{i}>0}\int_{a_{i}(\bm{y})-y_{i}}^{a_{i}(\bm{y})}s_{i}^{1}(\tau_{i})d\tau_{i}+\sum_{i}(1-\bm{1}_{y_{i}>0})\int_{a_{i}(\bm{y})}^{a_{i}(\bm{y})+|y_{i}|}s_{i}^{0}(\tau_{i})d\tau_{i}$
> $=$ $\frac{1}{2}\sum_{i}y_{i}^{2}$

**Feedback**:
Computing each integral: for yᵢ > 0, ∫_{aᵢ−yᵢ}^{aᵢ} (τᵢ − (aᵢ−yᵢ)) dτᵢ = yᵢ²/2; with the prefactor 1/2 this contributes yᵢ²/4. For yᵢ < 0, ∫_{aᵢ}^{aᵢ+|yᵢ|} ([aᵢ+|yᵢ|] − τᵢ) dτᵢ = yᵢ²/2; without a prefactor this contributes yᵢ²/2. The two cases give different contributions (yᵢ²/4 vs yᵢ²/2), so neither case alone yields the claimed total (1/2)∑ᵢ yᵢ². For the result to hold, both integrals should appear without the asymmetric 1/2 factor (each already evaluates to yᵢ²/2). It would be helpful to remove the spurious 1/2 from the first term.

---

### 16. Proof of Lemma OA3 Does Not Establish Uniform Convergence

**Status**: [Pending]

**Quote**:
> Consider the Taylor expansion of $\kappa$ around $\bm{0}$ ($\kappa$ is defined by part (1) of the assumption). We will now study its properties under parts (2) to (5) of Assumption OA2. (5) ensures that the Taylor expansion exists. Local separability (4) says that there are no terms of the form $y_{i}y_{j}$. Non-negativity (3) ($\kappa$ is nonnegative and $\kappa(\bm{0})=0$) implies that all first-order terms are zero. Also, (5) says that terms of the form $y_{i}^{2}$ must have positive coefficients, and symmetry (2) says that their coefficients must all be the same. ∎

**Feedback**:
The proof identifies the leading term of the Taylor expansion as k‖y‖² but does not establish the uniform convergence claim that is the actual content of Lemma OA3. Writing κ(y) = k‖y‖² + R(y) with R(y) = O(‖y‖³), we get C⁻¹κ(C^{1/2}z) = k‖z‖² + C⁻¹R(C^{1/2}z). On a compact set ‖z‖ ≤ B, |C⁻¹R(C^{1/2}z)| ≤ M·C^{1/2}·‖z‖³ → 0 uniformly. This argument—substituting the scaling, bounding the remainder using the compact set—is entirely absent from the proof. It would be helpful to rewrite the proof to include the remainder bound showing C⁻¹R(C^{1/2}z) = O(C^{1/2}) uniformly on compact sets.

---

### 17. Proposition OA2 Conclusion Requires Full Rotational Invariance, Not Just Permutation Symmetry

**Status**: [Pending]

**Quote**:
> Consider a cost function that satisfies Assumptions OA2 and OA3. There is a function $f:\mathbb{R}_{+}\to \mathbb{R}_{+}$ such that
> 
> $\kappa(\bm{y})=f(\|\bm{y}\|).$

**Feedback**:
The conclusion κ(y) = f(‖y‖) requires κ to be invariant under all orthogonal transformations O(n), but Assumption OA2(2) gives only invariance under permutations of coordinates—a much smaller subgroup. For example, κ(y) = (∑ᵢ yᵢ⁴)^{1/2} satisfies OA2 and OA3 yet is not of the form f(‖y‖) since it distinguishes (1,0,…,0) from (1/√2, 1/√2, 0,…,0). The proof of Proposition OA2 is not provided, so it is unclear how the gap from permutation-symmetry to full rotational invariance is bridged. It would be helpful to either add the missing proof or strengthen Assumption OA2(2) to 'For any orthogonal matrix O, κ(Oy) = κ(y)'.

---

### 18. Proof of Proposition OA3 Contradiction Argument Is Logically Flawed

**Status**: [Pending]

**Quote**:
> Note that $L$ is contained in a convex set
> 
> $E=\{\bm{b}:W(\bm{b})\leq W^{*}\}.$
> 
> The point $\bm{b^{*}}$ is contained in the interior of $L$; thus $\bm{b^{*}}$ is in the interior of $E$. On the other hand, $\bm{b^{*}}$ must be on the (elliptical) boundary of $E$ because $U$ is strictly increasing in each component (by irreducibility of the network) and continuous. This is a contradiction.

**Feedback**:
Being in the interior of a line segment L does not imply being in the interior of the set E that contains L, since L could lie entirely on the boundary of E. The correct argument is: W(b) = ‖[I−βG]⁻¹b‖² is strictly convex in b, so its level sets are strictly convex and no line segment can lie in the level set {W(b) = W*}. But if b* is in the interior of L ⊂ F (the feasible set) and b* maximizes W on F, then all points of L satisfy W ≤ W*, and b* ∈ interior(L) with W(b*) = W* forces W to equal W* at interior points of L, contradicting strict convexity of W. It would be helpful to rewrite the contradiction argument using strict convexity of W rather than the interior-of-E claim.

---

### 19. Coefficient w₂ Formula Appears Inconsistent with Payoff Specification

**Status**: [Pending]

**Quote**:
> $w_{1}$ $=$ $1+m_{2}+m_{5}+(n-1)m_{4}$
> $w_{2}$ $=$ $nm_{5}(n-2)$
> $w_{3}$ $=$ $\sqrt{n}[m_{1}+(n-1)m_{3}].$

**Feedback**:
It is not immediately clear how w₂ = nm₅(n−2) arises. The term m₄(∑_{j≠i} aⱼ)² contributes m₄[(n−2)(∑_j aⱼ)² + ∑ᵢ aᵢ²] to aggregate welfare (after summing over i and expanding), so the coefficient of (∑ᵢ aᵢ*)²/n should receive a contribution from m₄, not m₅. The term m₅∑_{j≠i} aⱼ² contributes m₅(n−1)∑ᵢ aᵢ² to aggregate welfare, which enters w₁, not w₂. The presence of m₅ in w₂ rather than m₄ warrants explicit derivation. It would be helpful to add a brief derivation of w₁, w₂, w₃ from the payoff specification, or to recheck whether w₂ should be n·m₄·(n−2) rather than n·m₅·(n−2).

---

### 20. Large-Budget Limit for μ in Theorem OA1 Does Not Handle w₁ < 0 Case Clearly

**Status**: [Pending]

**Quote**:
> Concerning the first one, this is true if either $w_{1}\alpha_{2}$ or $w_{1}\alpha_{n}$ (depending on which one is positive) approaches $\mu$. The second denominator tends to $0$ if $(w_{1}+w_{2})\alpha_{1}$ tends to $\mu$. Both denominators are positive by definition of the Lagrange multiplier, so it will be the greater of $w_{1}\max\{\alpha_{2},\alpha_{n}\}$ and $(w_{1}+w_{2})\alpha_{1}$ which tends to $\mu$.

**Feedback**:
When w₁ < 0 (case (ii) of the theorem), w₁α_ℓ < 0 for all ℓ, so the first set of denominators (μ − w₁α_ℓ) = μ + |w₁|α_ℓ > 0 and bounded away from zero; they cannot tend to zero. The phrase 'depending on which one is positive' is ambiguous when w₁ < 0 (neither w₁α₂ nor w₁αₙ is positive). It would be helpful to restate the argument separately for w₁ > 0 and w₁ < 0: when w₁ < 0 only the second denominator can drive the limit, so μ → (w₁+w₂)α₁ provided (w₁+w₂)α₁ > 0.

---
