d4-journals-mss-2022-0039

The strongest mirror is the paper’s optimal-weight problem in the large-population regime. It is unusually plausible because the authors themselves take that limit and compute the optimum.

My anchor is Theorem 21, proved here. Under the paper’s sufficient-randomness and positive limiting group-share assumptions, with nontrivial local bias and a model that is not tightly correlated, the finite-population optimal weights \(w^{(N)}\) converge to the minimizers of the limiting democracy-deficit objective. Those limiting weights have the form \(w_\lambda=C_1\alpha_\lambda+C_2\), where \(C_1\) and \(C_2\) do not depend on the group shares, and
\[ C_1=\frac{\langle\overline m_1(\rho)\rangle-\langle m_1(\rho)d(\rho)\rangle}{1-a}, \qquad C_2=\frac{\langle m_1(\rho)d(\rho)\rangle-a\langle\overline m_1(\rho)\rangle}{(1-a)(1+(M-1)a)}. \]
Here \(a=\langle d(\rho)^2\rangle\), \(m_1(\rho)(z)=\int t\,\rho^z(dt)\), \(\overline m_1(\rho)(z)=\int|t|\,\rho^z(dt)\), and \(d(\rho)(z)=\rho^z((0,1])-\rho^z([-1,0))\). The theorem also gives the sum of the weights:
\[ \sum_\lambda w_\lambda= \frac{\langle\overline m_1(\rho)\rangle+(M-1)\langle m_1(\rho)d(\rho)\rangle}{1+(M-1)a}. \]
This is an exact result about the underlying optimization problem, not merely a formulation. The paper does not state a complexity classification, though, so I would not describe Theorem 21 as a theorem that the problem is “in P.”

Here is the continuous problem I would hand to a prover: CBM Continuous Democracy-Deficit Weighting. An instance consists of \(M\) constituencies with rational population shares \(\alpha_\lambda>0\) summing to \(1\), a symmetric finite-support rational distribution \(\mu\) for a global issue bias \(Z\), and, for each supported \(z\), a symmetric finite-support rational conditional distribution \(\rho^z\) for the constituencies’ local biases. Require the paper’s sufficient-randomness and non-tight-correlation conditions. For an issue, draw \(Z\sim\mu\), then draw \(T_1,\ldots,T_M\) independently from \(\rho^Z\). Constituency \(\lambda\) has mass \(\alpha_\lambda\), of which fraction \((1+T_\lambda)/2\) votes aye and the rest nay. Thus the realized society is a distribution over types \((\lambda,\text{aye})\) and \((\lambda,\text{nay})\), with masses \(\alpha_\lambda(1+T_\lambda)/2\) and \(\alpha_\lambda(1-T_\lambda)/2\).

The popular margin is \(R=\sum_\lambda\alpha_\lambda T_\lambda\). The council vote of constituency \(\lambda\) is \(\chi_\lambda=\operatorname{sgn}(T_\lambda)\); at \(T_\lambda=0\), use a fair sign, matching the finite-electorate limit in the paper. The task is to return weights \(w\in\mathbb R^M\) minimizing
\[ \mathbb E\left[\left(R-\sum_\lambda w_\lambda\chi_\lambda\right)^2\right]. \]
There is no nonnegativity constraint: the paper’s democracy-deficit optimization permits negative weights.

The natural regime is a large federation or union: each constituency contains many residents, while the number of constituencies and distinct constituency types is much smaller than the population. The type includes constituency affiliation because that determines how a voter’s stance contributes to the council. For a realized issue there are at most \(2M\) constituency-and-stance types. The shares \(\alpha_\lambda\) are masses, not rounded voter counts. The random bias draws describe the paper’s uncertainty over issues; they do not replace the population mass as the continuous object.

I expect this finite-support version to be tractable. The stated objective is a finite-dimensional quadratic optimization, and Theorem 21 supplies its unique optimizer when \(a<1\). With the rational support encoding above, the expectations in \(C_1,C_2\) are finite sums, so the formula yields an exact rational solution using polynomially many arithmetic operations in the input size. That algorithmic conclusion is an inference from the theorem and encoding, not a complexity claim made by the authors. Questions it opens include exact finite-\(N\) complexity and convergence rates, the version requiring nonnegative weights, and the extension to constituency-specific bias kernels. At \(a=1\), uniqueness fails; the paper’s Theorem 26 says the limiting objective depends only on the sum of the weights.

The authors should recognize this as their problem: the mirror preserves the two-tier majority rule, the random-issue CBM, and the expected squared discrepancy from a referendum. It replaces individual counts with fractional constituency populations. Theorem 15’s limit behavior for group margins and Theorem 21’s optimizer make that bridge explicit.

The scope is the optimal-weight and democracy-deficit analysis in the non-tightly-correlated, common-kernel CBM regime. The central weakness is also clear: this is a very close formalization of the paper’s own large-population analysis, and the paper has no hardness result whose dissolution the mirror could test. It is a strong, author-recognizable continuous analogue, but a modest target for the programme’s complexity landscape.