How bad is forming your own opinion?

My strongest case is built around Proposition 4.2.

Proposition 4.2, proved in this paper rather than cited, says: given an unweighted opinion network, a distinguished node \(w\), and internal opinions, finding the best set of edges to add from \(w\), so as to minimize the Nash-equilibrium social cost of all nodes except \(w\), is NP-hard. The reduction is from Subset Sum: \(w\) has internal opinion \(1\), its \(n\) leaves have opinion \(0\), and there are isolated nodes with opinions \(-a_i/t\). The equilibrium cost is zero exactly when the selected opinions sum to \(-1\), which is equivalent to selecting numbers summing to \(t\).

I would mirror it with the following problem.

Continuous Source-Exposure Opinion Control\(_\infty\). The population consists of a source cohort \(W\), a leaf cohort \(L\), and item cohorts \(1,\ldots,\tau\). Cohort \(i\) has mass \(\mu_i\), intrinsic opinion \(s_i\), and otherwise identical network position; each cohort contains many interchangeable agents. The source has intrinsic opinion \(s_w\), each leaf listens to the source with unit weight, and each item cohort is otherwise isolated. The source’s own cost is excluded, exactly as in Proposition 4.2.

The decision is a mass vector \(x=(x_1,\ldots,x_\tau)\), where \(0\le x_i\le \mu_i\) is the mass of cohort \(i\) to which the source adds an outgoing link. In the continuum, individual identities disappear: selecting half of a cohort is represented by \(x_i=\mu_i/2\). With unit aggregate influence per unit selected mass, the equilibrium opinions are \(z_i=s_i\), \(z_w(x)=\frac{s_w+\sum_i x_i s_i}{1+\sum_i x_i}\), and \(z_L(x)=\frac{s_L+z_w(x)}{2}\). The objective is the total equilibrium cost of the leaf mass, namely \(F(x)=\mu_L\big((z_L-s_L)^2+(z_L-z_w)^2\big)\). The problem asks for a feasible \(x\) minimizing \(F(x)\), together with the resulting equilibrium.

This is a genuine population mirror, not merely fractionalizing the opinion variable. Opinions are already numerical in the paper; what becomes continuous is the society and the intervention target. A natural regime is a large platform or media ecosystem with \(M\) nearly interchangeable users in each of \(\tau\) behavioural cohorts, where \(M\gg\tau\). A type records the intrinsic opinion, baseline network role, and eligibility for exposure. In an \(M\)-fold blow-up, microscopic edge weights can be scaled by \(1/M\), so the aggregate influence remains finite.

The original authors should recognize this as their problem: a media source still chooses whom to expose itself to, the Nash equilibrium is still recomputed from the same averaging rule, and the objective is still equilibrium disagreement cost. The only change is that a large cohort is controlled at mass resolution, which is precisely the high-multiplicity interpretation.

I would expect this mirror to be in Class A. Here \(F(x)\) is the square of a linear-fractional function. Its attainable equilibrium source opinions form an interval, and the optimum can be found by linear-fractional programming or by checking whether the target opinion is reachable through a linear equation in \(x\). Thus the Subset Sum hardness disappears because the continuum permits partial use of an item cohort. That is an informative result rather than a distortion: the hardness in Proposition 4.2 is driven by all-or-nothing selection of individually named nodes, while the high-multiplicity version asks how much mass of each type should be selected.

The natural follow-up questions are whether one can round \(x\) for a finite population with additive error, how the answer changes with an exposure budget, and whether hardness returns if whole cohorts must be selected indivisibly. The latter would be a discrete restriction, not evidence against the continuous mirror.

A second, weaker but useful anchor is Theorem 4.11, also proved here. It states that, given a graph, internal opinions, and an ordered edge \((i,j)\), the optimal nonnegative weight \(\rho\) to add to that edge—minimizing the social cost at the new Nash equilibrium—can be computed in polynomial time. The proof uses Sherman–Morrison to express the new equilibrium as a one-parameter perturbation and observes that the resulting cost is a quadratic over a computable interval.

Its type-level mirror is Continuous Type-Channel Weight Design\(_\infty\). The input is a distribution \(\mu\), intrinsic opinions \(s_t\), and a directed interaction kernel \(w_{tu}\) between types. The equilibrium \(z^\rho\) satisfies \(z_t^\rho=\frac{s_t+\sum_u\mu_u w_{tu}z_u^\rho+\mathbf 1_{t=p}\rho z_q^\rho}{1+\sum_u\mu_u w_{tu}+\mathbf 1_{t=p}\rho}\), where \(\rho\) is additional influence from type \(p\) toward type \(q\). The objective is the mass-weighted social cost, including the added term \(\mu_p\rho(z_p^\rho-z_q^\rho)^2\). The task is to return the optimal \(\rho\), allowing \(\rho=\infty\) to denote the limiting hard-equality channel.

This remains Class A. The type-level update is still a rank-one matrix modification, so the paper’s argument carries over with the number of agents replaced by the number of types \(\tau\). The result is less powerful as a mirror than Proposition 4.2 because edge-weight choice is already continuous in the original paper, but it confirms that the authors’ linear-algebraic method survives the population quotient.

My weakest point is that the Proposition 4.2 mirror is most persuasive when there are genuinely repeated cohorts and a distinguished media-source role. The original reduction itself gives every item a different intrinsic opinion, so \(\tau\) may still grow with the number of Subset Sum items; the claim is not that a fixed finite type set preserves NP-hardness. Nor does this mirror cover arbitrary network topology or the unrestricted edge-addition problems in Propositions 4.4 and 4.6. It covers one named result emphatically: the paper’s source-exposure problem has a sensible high-multiplicity form, and its discrete NP-hardness becomes a tractable continuous optimization problem for a concrete, recognizable reason.