# TARGETING INTERVENTIONS IN NETWORKS

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

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## Overall Feedback

Here are some overall reactions to the document.

**Symmetric Network Assumption Excludes Directed Applications**

The main spectral characterization relies on Assumption 1 that the adjacency matrix G is symmetric, which guarantees real eigenvalues and orthogonal eigenvectors. However, many economic networks of interest—including supply chains, credit networks, and citation networks—are inherently directed and asymmetric. While Section OA3.2 discusses an extension using singular value decomposition, this changes the interpretation significantly by distinguishing left from right singular vectors and breaks the clean correspondence between strategic substitutes/complements and specific eigenvectors. It would be helpful to either elevate the asymmetric analysis to the main text or explicitly qualify that the primary results apply only to undirected networks, as the current framing suggests broader applicability than the mathematics supports.

**Full Network Observability Requirements for Implementation**

Theorem 1 characterizes optimal interventions using the principal components of G, which requires the planner to possess complete knowledge of the entire network structure. Section 5 relaxes the assumption that the planner knows standalone marginal returns (b), but maintains that G is common knowledge, without addressing robustness to measurement error in network links or the computational complexity of spectral decomposition. Readers might note that in practice, collecting full network data is often prohibitively expensive or impossible, and eigenvector-based targeting is sensitive to misspecification. It would be helpful to analyze the robustness of optimal interventions to network misspecification or to develop results for partial network observability (e.g., knowing only local neighborhoods or aggregate statistics).

**Restrictiveness of Property A Welfare Specification**

Theorem 1 and Corollary 1 assume Property A—that aggregate equilibrium utility is proportional to the sum of squares of equilibrium actions—which holds for specific investment and public goods examples but excludes many natural economic environments with linear externalities or other payoff structures. As shown in Corollary OA1, linear welfare objectives yield qualitatively different optimal interventions that target a single individual rather than using spectral decomposition across multiple principal components. While Section OA3.1 extends the analysis to more general externalities under additional restrictive assumptions, the main text presents the spectral characterization as applying to a "broad class" of games. It would be helpful to clarify upfront that Property A is a substantive restriction on payoff functions, not merely a technical convenience, and that the eigenvalue-based prioritization may not hold for alternative welfare specifications.

**Scope Limitations Under Equilibrium Multiplicity**

Assumption 2 requires that the spectral radius of βG be less than 1 to ensure a unique Nash equilibrium, excluding economically relevant settings with strategic complements and strong spillovers where multiple equilibria are common. The paper does not discuss what occurs when this condition fails—whether the spectral characterization remains valid under equilibrium selection mechanisms, or whether the planner's problem becomes ill-posed due to multiplicity. It would be helpful to clarify that the results apply only to games with strong contraction properties, and to address whether the targeting priorities (top vs. bottom eigenvectors) remain valid when the planner can select among multiple equilibria or when the spectral radius constraint binds.

**Neglect of Non-negativity and Corner Constraints**

The analysis assumes interior solutions where actions can take any real value, yet many economic applications involve non-negative actions (investment, effort, consumption). For large budgets, the optimal intervention characterized in Theorem 1 would require negative actions for some agents (since eigenvectors other than the first contain both positive and negative entries), violating feasibility constraints. The paper acknowledges in a footnote that non-negativity constraints will bind for large budgets, but does not characterize the optimal intervention when these constraints are active, meaning the welfare approximation in Proposition 2 may not hold for economically relevant parameter regions. It would be helpful to explicitly model non-negativity constraints in the main optimization problem and characterize when corner solutions emerge, as this affects the practical implementation of eigenvector-based targeting.

**Practical Constraints on Simple Interventions**

Proposition 2 establishes that optimal interventions become "simple" (concentrated on a single principal component) only when the budget exceeds a threshold proportional to ||b̂||²/(spectral gap)², which can be arbitrarily large for networks with small spectral gaps such as those with community structure. Furthermore, the large-budget characterization relies crucially on the quadratic cost structure, while alternative specifications such as linear costs lead to qualitatively different corner solutions targeting single individuals. The paper does not characterize optimal interventions for intermediate budget regimes where the asymptotic simplicity result does not apply, nor does it provide empirical guidance on typical spectral gaps in economic networks. It would be helpful to characterize optimal interventions for intermediate budgets or to assess the feasibility of the large-budget threshold in empirically relevant network architectures.

**Distinct Eigenvalues Assumption Violated by Circle Network Example**

It would be helpful to clarify how the optimal intervention is characterized when Assumption 2 (distinct eigenvalues) fails, as it does in Example 3. Assumption 2 requires that all eigenvalues of G are distinct to ensure a unique spectral decomposition. However, Example 3 and Figure 1 illustrate the results using a 14-node circle network (cycle graph C_14). For this network, the eigenvalues are λ_k = 2cos(2πk/14) for k=0,...,13, which yields λ_1 = λ_13 ≈ 1.802, λ_2 = λ_12 ≈ 1.247, and so on, with most eigenvalues having multiplicity 2. Readers might note that when eigenvalues repeat, the principal components (eigenvectors) are not uniquely defined—any rotation within the degenerate eigenspace yields a valid basis. Consequently, the similarity ratios r_ℓ* in Theorem 1 and the targeting weights on principal components become ill-defined or non-unique. It would be helpful to either (i) restrict illustrations to networks with distinct eigenvalues, (ii) explicitly handle the non-unique case by specifying a selection rule (e.g., targeting the centroid of the optimal set), or (iii) show that the welfare loss from arbitrary selection within the eigenspace is bounded.

**Exogenous Network Formation vs. Strategic Link Adjustment**

It would be helpful to reconcile the assumption of an exogenous, fixed network G with the economic applications described in Examples 1 and 2. The model treats G as a primitive that is invariant to the intervention (Section 2). However, Example 1 describes an investment game where "an individual's marginal returns are enhanced when his neighbors work harder," and Example 2 involves local public goods provision. Readers might note that in such settings, agents often have incentives to form or sever links based on the anticipated intervention. If the planner increases b_i for certain nodes, the value of connecting to those nodes changes, potentially altering the equilibrium network structure. It is not clear how the optimal intervention characterized in Theorem 1 performs when agents can reoptimize their links after the intervention is announced. It would be helpful to provide a robustness check or extension showing that the optimal intervention remains effective under small perturbations of G, or to explicitly model network formation as in Section OA3.2 but with endogenous link creation costs.

**Status**: [Pending]

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## Detailed Comments (22)

### 1. Asymptotic nature of simple interventions

**Status**: [Pending]

**Quote**:
> For large budgets, optimal interventions are simple — they involve a single principal component.²

**Feedback**:
The abstract states that for large budgets, optimal interventions involve a single principal component. However, Proposition 2 establishes that this is an asymptotic result requiring the budget to exceed a threshold proportional to $\|\hat{\bm{b}}\|^2/(\text{spectral gap})^2$, and the intervention only approaches simplicity as $C \to \infty$ with other components retaining non-zero weight. It would be helpful to revise the abstract to clarify that interventions become approximately simple in the limit of large budgets.

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### 2. Incorrect network multiplier in property (b)

**Status**: [Pending]

**Quote**:
> This basis has three special properties: (a) the effects of an intervention along a principal component is proportional to the intervention, scaled by a network multiplier; (b) the “network multiplier” is an eigenvalue of the network corresponding to that principal component; (c) the principal components are orthogonal, so that the effects along various principal components can be treated separately (in a suitable sense).

**Feedback**:
The text states that the network multiplier is an eigenvalue of $\bm{G}$, but the equilibrium condition $\underline{a}_{\ell}^{*} = \frac{1}{1-\beta\lambda_{\ell}}\underline{b}_{\ell}$ shows the multiplier is $(1-\beta\lambda_{\ell})^{-1}$. This is an eigenvalue of $(\bm{I}-\beta\bm{G})^{-1}$, not of $\bm{G}$ itself. It would be helpful to correct property (b) to state that the multiplier is $(1-\beta\lambda_{\ell})^{-1}$, derived from the eigenvalue $\lambda_{\ell}$.

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### 3. Typo in change of variables for local public good

**Status**: [Pending]

**Quote**:
> i}^{2}.$This is a game of strategic substitutes and positive externalities. Performing the change of variables $b_{i}=[\tau-b_{i}]/2$ and $\beta=-\tilde{\beta}/2$ (with the status quo equal to $\hat{b}_{i}=[\tau-\tilde{b}_{i}]/2$) yields a best-response structure exactly as in condition (2). The aggregate equilibrium uti

**Feedback**:
The change of variables is written as $b_{i}=[\tau-b_{i}]/2$, which is circular. Deriving from the first-order conditions of the local public good example yields $a_i = \frac{\tau - \tilde{b}_i}{2} - \frac{\tilde{\beta}}{2}\sum_j g_{ij}a_j$, confirming the correct definition is $b_i = [\tau - \tilde{b}_i]/2$. It would be helpful to replace $b_i$ with $\tilde{b}_i$ on the right-hand side.

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### 4. Inconsistency between principal components definition and spectral ordering

**Status**: [Pending]

**Quote**:
> ll}\|=1$.
> 
> For generic $\bm{G}$, the decomposition is uniquely determined, except that any column of $\bm{U}$ is determined only up to multiplication by $-1$.
> 
> An important interpretation of this diagonalization is as a decomposition into *principal components*. We can think of the columns of $\bm{G}$ as $n$ data points. The first principal component of $\bm{G}$ is defined as the $n$-dimensional vector that minimizes the sum of squares of the distances to the columns of $\bm{G}$. The first principal component can therefore be thought of as a fictitious column that “best summarizes” the dataset of all columns of $\bm{G}$. To characterize the next principal component, we orthogonally project all columns of $\bm{G}$ off this vector and repeat this procedure for the new columns. We continue in this way, projecting orthogonally off the (subspace generated by) vectors obtained to date, to find the next principal component. A well-known result is that the eigenvectors of $\bm{G}$ that diagonalize the matrix (i.e., the columns of $\bm{U}$) are indeed the principal components of $\bm{G}$ in this sense. Moreover, the eigenvalueTARGETING INTERVENTIONS IN NETWORKS

**Feedback**:
The geometric definition of principal components minimizes reconstruction error, which corresponds to eigenvectors of $\bm{G}^2$ ordered by $|\lambda_{\ell}|$, whereas the spectral decomposition orders by $\lambda_{\ell}$. For networks with negative eigenvalues (e.g., the circle network), the vector minimizing squared distances may be $\bm{u}^n$ rather than $\bm{u}^1$. Readers might note that the ordering by eigenvalue magnitude differs from the spectral ordering when $\bm{G}$ has negative eigenvalues.

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### 5. Assumption 3 missing square on norm

**Status**: [Pending]

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

**Feedback**:
Assumption 3 states $C < \|\hat{\bm{b}}\|$ when $w<0$, but achieving the bliss point $\bm{b}=\bm{0}$ requires $\|\bm{y}\|^2 = \|\hat{\bm{b}}\|^2 \leq C$. The condition should be $C < \|\hat{\bm{b}}\|^2$ to be dimensionally consistent with the budget constraint $\sum (b_i - \hat{b}_i)^2 \leq C$ and match the derivation in Appendix A. It would be helpful to correct the assumption to include the squared norm.

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### 6. Incorrect formula for alpha in text

**Status**: [Pending]

**Quote**:
> (via $\alpha_{\ell}=\frac{1}{1-\beta\lambda_{\ell}}$)

**Feedback**:
The text states $\alpha_{\ell}=\frac{1}{1-\beta\lambda_{\ell}}$, but the definition earlier in the section is $\alpha_{\ell}=\frac{1}{(1-\beta\lambda_{\ell})^{2}}$. The amplification factor is squared in the welfare expression $W = w\sum_\ell \alpha_\ell \underline{b}_\ell^2$, so the correct formula includes the square in the denominator. It would be helpful to correct the formula to include the exponent.

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### 7. Undefined subscript in Proposition 1

**Status**: [Pending]

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

**Feedback**:
Proposition 1 uses subscript $t$ in the numerator ($r_{t}^{*}$), but the limit expression uses $\ell$ ($\alpha_{\ell}$). Since the similarity ratio is defined as $r_{\ell}^{*}$ in Theorem 1, the subscript $t$ appears to be a typographical error for $\ell$. It would be helpful to replace $t$ with $\ell$ for consistency.

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### 8. Sign error in denominator of spectral gap expression

**Status**: [Pending]

**Quote**:
> ll. If $\beta<0$, then the term $\alpha_{n-1}/(\alpha_{n-1}-\alpha_{n})$ is small when the “bottom gap” of the graph, the difference $\lambda_{n-1}-\lambda_{n}$, is small.
> 
> We now examine what network features affect these gaps, and illustrate with e

**Feedback**:
For strategic substitutes ($\beta<0$), the amplification factor $\alpha_{\ell}=(1-\beta\lambda_{\ell})^{-2}$ is maximized at $\ell=n$, so $\alpha_{n} > \alpha_{n-1}$. Thus the denominator $\alpha_{n-1}-\alpha_{n}$ is negative, and the expression should be $\alpha_{n-1}/(\alpha_{n}-\alpha_{n-1})$ to yield a positive quantity that becomes large (not small) when the spectral gap is small. It would be helpful to correct the denominator.

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### 9. Incorrect proportionality claim for budget threshold

**Status**: [Pending]

**Quote**:
> Observe that the first term on the right-hand side of the inequality for $C$ is proportional to $\|\hat{\bm{b}}\|$.

**Feedback**:
The text claims the budget threshold is proportional to $\|\hat{\bm{b}}\|$, but Proposition 2 states the threshold as $C>\frac{2\|\hat{\bm{b}}\|^{2}}{\epsilon}(\frac{\alpha_{2}}{\alpha_{1}-\alpha_{2}})^{2}$. The dependence is on the squared norm $\|\hat{\bm{b}}\|^{2}$, not the norm itself. It would be helpful to correct the proportionality statement.

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### 10. Ambiguous principal components description for circle network

**Status**: [Pending]

**Quote**:
> A general feature that is worth noting is that the entries of the top principal components (smaller values of $\ell$) are clustered among neighboring nodes, while the bottom principal components (larger values of $\ell$) tend to be negatively correlated among neighboring nodes.

**Feedback**:
This description assumes a specific basis choice within degenerate eigenspaces. For the 14-node circle network, most eigenvalues have multiplicity 2, so the eigenvectors are not uniquely defined—any rotation within the eigenspace is valid. Readers might note that the clustering properties described apply to the standard sinusoidal basis but not to arbitrary rotations of degenerate eigenvectors.

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### 11. Missing square in Lagrangian expression

**Status**: [Pending]

**Quote**:
> pping the sign of $x_{\ell}^{*}$.
> 
> We now complete the proof. Observe that the Lagrangian corresponding to the maximization problem is
> 
> $\mathcal{L}=w\sum_{\ell}\alpha_{\ell}(1+x_{\ell})^{2}\hat{\underline{b}}_{\ell}+\mu\left[C-\sum_{\ell}\hat{\underline{b}}_{\ell}^{2}x_{\ell}^{2}\right].$
> 
> Taking our observation above that the constraint i

**Feedback**:
The objective function being maximized is $w \sum_{\ell} \alpha_{\ell} (1 + x_{\ell})^2 \hat{\underline{b}}_{\ell}^2$, where $\hat{\underline{b}}_{\ell}$ appears squared. The Lagrangian expression is missing this square on the first term. It would be helpful to correct the Lagrangian to include $\hat{\underline{b}}_{\ell}^{2}$ to match the subsequent first-order condition.

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### 12. Typo in norm equality in Lemma 1 proof

**Status**: [Pending]

**Quote**:
> The next equality follows by noticing that $\|\hat{\bm{b}}\|^{2}=\|\hat{\bm{b}}\|^{2}$.

**Feedback**:
The proof aims to use the invariance of the Euclidean norm under orthogonal transformation, where $\|\hat{\underline{\bm{b}}}\|^2 = \|\hat{\bm{b}}\|^2$ because $\underline{\hat{b}} = \bm{U}^\top \hat{\bm{b}}$. The statement $\|\hat{\bm{b}}\|^{2}=\|\hat{\bm{b}}\|^{2}$ is tautological. It would be helpful to replace the second $\hat{\bm{b}}$ with $\hat{\underline{\bm{b}}}$ to reflect the change of basis.

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### 13. Incorrect variance indices in Proposition 4 contradiction argument

**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 proof constructs an alternative intervention $\mathcal{B}^{**}$ by permuting components $\ell$ and $\ell'$, but incorrectly uses $\underline{b}^{*}$ instead of $\underline{b}^{**}$ for the post-permutation variances. The statement claims $\operatorname{Var}(\underline{b}_{\ell}^{*}) > \operatorname{Var}(\underline{b}_{\ell'}^{*})$, which contradicts the initial assumption. It would be helpful to correct the statement to $\operatorname{Var}(\underline{b}_{\ell}^{**}) = \operatorname{Var}(\underline{b}_{\ell'}^{*}) > \operatorname{Var}(\underline{b}_{\ell}^{*}) = \operatorname{Var}(\underline{b}_{\ell'}^{**})$.

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### 14. Intervention proportionality claim requires large budget qualification

**Status**: [Pending]

**Quote**:
> lso known as the vector of individuals’ eigenvector centralities. Thus, our results of Section 4 imply that, under strategic complementarities, interventions that aim to maximize the aggregate utility should change individuals’ incentives in proportion to their eigenvector centralities.
> 
> It is worth comparing this result with res

**Feedback**:
Theorem 1 shows the optimal intervention depends on $\frac{w\alpha_{\ell}}{\mu-w\alpha_{\ell}}\hat{\underline{b}}_{\ell}$ across all components, not just the first. Only as $C\to\infty$ does $\mu \to w\alpha_1$ and the intervention concentrate on the first principal component (eigenvector centrality). Readers might note that the claim holds asymptotically for large budgets but not for arbitrary budget levels.

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### 15. Incorrect coefficient in first-order condition

**Status**: [Pending]

**Quote**:
> It can be verified that the first-order condition for individual $i$ is given by
> 
> $$
> a_i = \frac{\tilde{b}_i}{1 + \gamma} + \frac{\tilde{b}_i + \gamma}{1 + \gamma} \sum g_{ij} a_j.
> $$

**Feedback**:
Deriving from the payoff function yields $\frac{\partial U_i}{\partial a_i} = \tilde{b}_i + \tilde{\beta}\sum_j g_{ij}a_j - a_i + \gamma\sum_j g_{ij}(a_j - a_i) = 0$, which implies $a_i = \frac{\tilde{b}_i}{1+\gamma} + \frac{\tilde{\beta}+\gamma}{1+\gamma}\sum_j g_{ij}a_j$. The coefficient should be $\frac{\tilde{\beta}+\gamma}{1+\gamma}$, not $\frac{\tilde{b}_i+\gamma}{1+\gamma}$. It would be helpful to correct the interaction coefficient.

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### 16. Incorrect threshold for positive externality

**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**:
The derivative of the payoff with respect to $a_j$ is $\frac{\partial U_i}{\partial a_j} = g_{ij}[\tilde{\beta}a_i - \gamma(a_j - a_i)] = g_{ij}[(\tilde{\beta}+\gamma)a_i - \gamma a_j]$. This is positive if and only if $a_j < (1 + \frac{\tilde{\beta}}{\gamma})a_i$, which is a weaker condition than $a_j < a_i$ since $\tilde{\beta}, \gamma > 0$. It would be helpful to correct the threshold.

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### 17. Incorrect coefficients in welfare weights

**Status**: [Pending]

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

**Feedback**:
Deriving the aggregate welfare from the definitions yields $w_1 = \frac{1}{2} + m_2 + m_4 + (n-1)m_5$ and $w_2 = nm_4(n-2)$. The constant term should be $1/2$ (from $\sum_i \hat{U}_i = \frac{1}{2}(\bm{a}^*)^\top\bm{a}^*$) and the coefficients $m_4$ and $m_5$ are interchanged in the text. It would be helpful to correct the expressions for $w_1$ and $w_2$.

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### 18. Inconsistency in Example OA1 welfare expression

**Status**: [Pending]

**Quote**:
> $W(\bm{b},\bm{G})=\frac{1}{2}\,(\bm{a}^{*})^{\mathsf{T}}\,\bm{a}^{*}-n\gamma\sum_{i}a_{i}^{*},$ (OA-1)

**Feedback**:
For Example OA1 with $m_3=-\gamma$ and other $m_k=0$, the general welfare formula yields a linear term coefficient of $\gamma(n-1)$, not $n\gamma$, because the externality applies to $j \neq i$ (giving $n-1$ neighbors). It would be helpful to correct the coefficient to $\gamma(n-1)$ for consistency with the general expression.

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### 19. Incorrect subscripts in Lemma OA2 derivative expression

**Status**: [Pending]

**Quote**:
> $\frac{d\mu}{dx_{1}}=\frac{\hat{b}_{1}^{2}x_{1}}{\sum_{\ell=2}\frac{w_{1}^{2}\hat{b}_{1}^{2}\alpha_{\ell}^{2}}{(\mu-w_{1}\alpha_{\ell})^{3}}}$

**Feedback**:
Implicit differentiation of the budget constraint yields $\frac{d\mu}{dx_1} = \frac{\hat{\underline{b}}_1^2 x_1}{\sum_{\ell=2} \hat{\underline{b}}_\ell^2 \frac{w_1^2 \alpha_\ell^2}{(\mu-w_1\alpha_\ell)^3}}$. The expression incorrectly uses $\hat{b}_1^2$ in the denominator sum instead of $\hat{\underline{b}}_\ell^2$. It would be helpful to correct the subscripts to reflect the variables from the budget constraint.

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### 20. Inconsistent notation for transformed status quo

**Status**: [Pending]

**Quote**:
> b. $x_{1}^* > 0$ if and only if $w_{1} + w_{2} + \frac{w_{3}}{2\sqrt{\alpha_{1}b_{1}}} > 0$.

**Feedback**:
The condition for $x_1^* > 0$ uses $b_1$ without decoration, but the derivation in Theorem OA1, part 1 uses $\hat{\underline{b}}_1$ (the projection of the status quo onto the first principal component). Since $\underline{b}_1 = \sum_i b_i/\sqrt{n}$ under Assumption OA1, the distinction between $b_1$ and $\hat{\underline{b}}_1$ is material. It would be helpful to use $\hat{\underline{b}}_1$ for consistency.

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### 21. Missing hat and underline in large-budget limit formulas

**Status**: [Pending]

**Quote**:
> ll^ {\prime}} ^ {*}} \rightarrow \frac {\alpha_ {\ell}}{\alpha_ {\ell^ {\prime}}} \text{ for all } \ell , \ell^ {\prime} \neq 1
> $$
> 
> $$
> \frac {x _ {1} ^ {*}}{x _ {\ell} ^ {*}} \rightarrow \frac {\alpha_ {1}}{\alpha_ {\ell} } \left[ w _ {1} + w _ {2} + \frac {w _ {3}}{2 \sqrt {\alpha_ {1} b _ {1}}} \right] \text{ for all } \ell \neq 1
> $$
> 
> 4. Suppose the game has strategic complements, $\beta > 0$. In the limit as $C \to \infty$, $\mu \to \max \{w_1\alpha_2, 

**Feedback**:
The limit expression as $C \to 0$ references $b_1$, but should reference $\hat{\underline{b}}_1$ to match the finite-$C$ formula in Theorem OA1, part 1 and indicate the status quo value. It would be helpful to include the hat and underline to maintain notational consistency.

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### 22. Incorrect interior set claim in Proposition OA3 proof

**Status**: [Pending]

**Quote**:
> 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**:
The inference that $\bm{b}^*$ in the interior of line segment $L$ implies it is in the interior of $E$ is incorrect; $\bm{b}^*$ lies on the boundary of $E = \{\bm{b} : W(\bm{b}) \leq W^*\}$ by definition of the optimum. The valid contradiction should instead use strict convexity of $W$ (since $\bm{M} = ([\bm{I}-\beta\bm{G}]^{-1})^\mathsf{T}[\bm{I}-\beta\bm{G}]^{-1}$ is positive definite) to show $W(\bm{b}^*) < \max\{W(\bm{b}_1), W(\bm{b}_2)\}$ for endpoints $\bm{b}_1, \bm{b}_2$ of $L$, contradicting optimality. Additionally, the symbol $U$ is undefined and should be $W$. It would be helpful to revise the proof to use the correct contradiction argument.

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