Proportional Public Decisions

The strongest honest case is positive at the level of the model, but the paper has a serious evidentiary limitation: it contains no eligible computational-complexity anchor. The numbered results—Theorems 1–7, Propositions 1–2, and Corollaries 1–2—prove proportionality guarantees, not membership in \(P\), NP-hardness, W[1]-hardness, or FPT results. Thus Theorem 1 or Theorem 6 must not be misreported as an algorithmic-complexity theorem. There are zero valid anchors under the brief’s strict rule.

The best substantive mirror is nevertheless clear. I would lead with the paper’s Theorem 1, proved in this paper (with its proof omitted there and obtained from the later general argument), together with Corollary 1. It gives the cleanest continuous formulation.

Call the problem Continuous PAV Public-Decision Winner Determination. An instance consists of \(m\) binary issues, a finite set of complete voter types \(T\subseteq\{0,1\}^{m}\), and rational masses \(\mu_t\ge 0\) summing to \(1\). A type records the voter’s preferred decision on every issue; \(t_a=1\) means YES and \(t_a=0\) means NO. The intended regime is a large housing cooperative, municipality, or professional association with many residents but relatively few recurring preference profiles: perhaps \(n\) in the hundreds of thousands and \(\tau=|T|\) in the hundreds or low thousands. Agents sharing a type have identical preferences and are indistinguishable for the problem.

The decision variable remains a binary outcome \(W\subseteq [m]\). Continuization applies to the population, not to the decisions. Define

\[ u_t(W)=\sum_{a=1}^{m}\mathbf{1}\bigl[t_a=1\ \text{and}\ a\in W\bigr] +\mathbf{1}\bigl[t_a=0\ \text{and}\ a\notin W\bigr], \]

and let \(H(0)=0\), \(H(k)=\sum_{j=1}^{k}1/j\). The task is to output

\[ W^\star\in\arg\max_{W\subseteq [m]} \sum_{t\in T}\mu_t H\bigl(u_t(W)\bigr). \]

This is not a softened or welfare-only variant of the paper’s problem. If a finite electorate has \(n_t\) voters of type \(t\), setting \(\mu_t=n_t/n\) makes the continuous objective exactly \(1/n\) times the paper’s finite PAV objective, so the maximizers are identical. Conversely, rational masses are precisely high-multiplicity electorates under the usual encoding caveat.

The proportionality question attached to this problem is also exact. For a subpopulation represented by masses \(0\le\nu_t\le\mu_t\), let \(\alpha=\sum_t\nu_t\) and

\[ \overline u_\nu(W)= \frac{\sum_t\nu_tu_t(W)}{\sum_t\nu_t}. \]

The continuous analogue of Theorem 1 asks whether every PAV optimum satisfies

\[ \overline u_\nu(W^\star)>\frac{\alpha m}{2}-1 \]

for every \(0<\alpha\le 1\) and every subpopulation of mass at least \(\alpha\). The proof in the paper is a weighted argument, so its sums should extend directly to these type masses. Fractionally selecting part of a type is legitimate here: it is exactly what a continuum population means, and identical agents remain indistinguishable.

I would expect the winner-determination problem itself to fall in Class B rather than Class A, if the corresponding discrete variable-winner PAV problem is hard. Every discrete instance embeds by taking its empirical type distribution, so hardness driven by the agenda and the \(2^m\) possible outcomes transfers. Continuity does not remove that combinatorics. By contrast, once \(W\) is fixed, finding the worst-represented subpopulation of a given mass is a straightforward type-sorting problem, with a fractional final type, and is polynomial in \(\tau\) and \(m\).

This mirror covers the no-abstention PAV results, specifically Theorem 1 and Corollary 1. The abstention version would replace \(\{0,1\}^m\) by \(\{0,1,\bot\}^m\) and use the common relevant-issue count from Theorem 5. A second natural extension would continuize Theorem 6 by replacing the finite MES-style payments with integrals of the form

\[ \sum_t \mu_t\min\{\rho,b_t\}, \]

but I would not present that as a separate anchor without more precision.

The weakest point is decisive: this paper itself does not establish any computational landscape. A referee could therefore regard the mirror as merely the high-multiplicity encoding of a finite PAV instance, rather than a result already pointing toward a new complexity theorem. The positive case survives only in the narrower sense required by the programme: the paper’s population model is genuinely suitable for continuization, its proportionality statements are naturally mass statements, and its discrete objective has an exact high-multiplicity counterpart. What remains to be supplied is precisely the missing ChoCo contribution—winner-determination complexity, approximation or parameterized algorithms, and a proof of whether the population continuum changes anything computationally.