Separating and Collapsing Electoral Control Types

The proposed mass-partition problem is a plausible direct high-multiplicity extension of approval voter partition, but the paper's anchor is not computational under the required criterion. Theorem 20 proves a semantic strict containment, not a complexity classification or algorithmic result, so there is no named computational result of this paper to mirror.

The objection that survived

Theorem 20 is rule-theoretic rather than a named computational complexity result, so its continuous analogue cannot satisfy the required computational-anchor condition.

What the mirror covers

The proposed mirror follows the voter-partition semantics underlying Theorem 20, but it does not mirror any named computational result in the paper.

Open questions for a prover

The case FOR (proponent)

The strongest positive case is narrow but real: voter-partition control in approval elections has a very natural high-multiplicity regime. I would anchor it on Theorem 20, proved in this paper:
\[ \text{Approval-DC-PV-TP-UW}\subsetneq \text{Approval-DC-PV-TE-NUW}. \]
This is not an NP-hardness or P theorem—the paper is explicitly about equivalences and containment among control types—but it is a named computational reachability result. Its two decision problems are exactly the sort of objects whose continuous versions should be studied.

Call the lead problem Continuous Approval Destructive Control by Voter Partition, TP–UW, or \( \mathrm{C\mbox{-}Approval\mbox{-}DC\mbox{-}PV\mbox{-}TP\mbox{-}UW}_\infty\). An instance consists of a finite candidate set \(C\), a target \(p\), and a rational society \(\mu\) over approval types \(T\subseteq\{0,1\}^{C}\). Here a type is a complete approval ballot: citizens of that type approve exactly the same candidates. A solution is a mass partition \(y_t\in[0,\mu_t]\), with the two preliminary electorates \(y\) and \(\mu-y\). Each preliminary election is run by approval; under TP every score-maximizer advances. The final is run on the full society \(\mu\), restricted to the candidates advancing from either preliminary. The question is whether a partition makes \(p\) fail to be the unique final winner.

The companion problem is \( \mathrm{C\mbox{-}Approval\mbox{-}DC\mbox{-}PV\mbox{-}TE\mbox{-}NUW}_\infty\). It has the same instance and mass-partition variable, but preliminary ties eliminate, and the question is whether \(p\) can be made not to be a final winner at all.

These are not merely “approval voting with real-valued scores.” The population, rather than the outcome, is continuous: \(y_t\) says what fraction of a homogeneous constituency is assigned to each preliminary. A rational discrete election is recovered by setting \(\mu_t=n_t/n\) and requiring \(y_t\) to be a multiple of \(1/n\). Dropping that integrality condition is precisely the high-multiplicity relaxation.

There is a credible regime for it. Think of a national membership organization, platform with a very large user base, or a public participatory-budgeting system that conducts two qualifying deliberations before a final vote. There may be \(10^5\) to \(10^7\) participants but only tens or hundreds of stable approval blocs: for example, people who approve the same subset of a fixed slate of projects or candidates. The organizer allocates participation quotas across two panels, districts, or online deliberation streams; the meaningful decision is then “how much of each bloc goes to each stream,” not which named citizen goes where. This is particularly plausible when the assignment is mediated by registration slots, regional capacities, or randomized invitations. The type count can of course grow as \(2^{|C|}\) in the worst case, but the relevant high-multiplicity regime is one with modest observed support \(T\), not an assertion that every approval election has few types.

The authors should recognize this as their problem. Their action is literally a partition of voters, their vote type for approval is already a bit-vector, and approval scores are additive. The continuous version retains the two-stage procedure, TP/TE distinction, final-round rule, and destructive objectives. It changes only the granularity of the voter collection.

I expect both continuous problems to be Class A. For every candidate \(c\), its score in either preliminary is linear in \(y\):
\[ s_y(c)=\sum_{t\in T}y_t t(c),\qquad s_{\mu-y}(c)=\sum_{t\in T}(\mu_t-y_t)t(c). \]
Whether a named candidate is promoted under TP or uniquely promoted under TE is therefore a finite system of linear weak or strict inequalities. To decide successful destructive control, one need only enumerate the few candidate witnesses relevant to \(p\)’s failure: competitors that tie or beat \(p\) in each preliminary, or a final-round competitor whose full-society score defeats \(p\). Each choice produces an LP in the \(|T|\) mass variables; strict inequalities can be handled by maximizing a common rational slack. Thus the decision problems admit a polynomial-time algorithm in \(|C|,|T|\), and the encoding length, without expanding the society into individual voters.

Theorem 20 then becomes a particularly good continuous research question:

Does the strict containment
\[ > \mathrm{C\mbox{-}Approval\mbox{-}DC\mbox{-}PV\mbox{-}TP\mbox{-}UW}_\infty > \subsetneq > \mathrm{C\mbox{-}Approval\mbox{-}DC\mbox{-}PV\mbox{-}TE\mbox{-}NUW}_\infty > \]
hold for every rational approval society, and can both sides be decided by the above LP formulation?

I expect yes. The distinction is substantive in the continuum just as in the discrete model: under TE, preventing \(p\) from uniquely winning a preliminary can eliminate \(p\) from the final altogether; TP instead promotes tied preliminary winners, so it offers a different and weaker destructive mechanism. Strictness should not depend on individual voter identities and so should survive rational high-multiplicity witnesses.

The most useful follow-up questions are the integrality gap between the continuous partition and a population of \(n\) people, rounding guarantees for a feasible mass partition, and what happens once the organizer must respect locality, indivisible precincts, or lower/upper quotas. Those constraints may reintroduce discrete hardness; that would identify exactly where continuization stops helping rather than undermine the basic mirror.

The weakest point is institutional, not mathematical. Ordinary election administrators normally cannot split an ideological type fractionally, and arbitrary voter partition is already an artificial control action. The case therefore should not be sold as a model of every election. It survives because there are genuine allocation-to-preliminary settings where people are interchangeable within broad approval blocs and quotas, rather than named targeting, are the actual decision variable. In that regime, the continuous model is the honest aggregate form of the paper’s voter-partition problem.

The case AGAINST (opponent, writing after the proponent)

The strongest objection is that Theorem 20 is not a complexity result to continuize. This paper explicitly says that collapse and separation of control types are “not directly about complexity.” Theorem 20 classifies the inclusion of two extensional yes-instance sets for approval; it supplies neither an algorithm nor a hardness result. The proposed LP analysis is therefore new research on a newly formulated mass-partition problem, not a continuous counterpart of a computational result established here. At most, the paper supplies two adjacent control definitions and a structural comparison between them.

That matters because the proposed continuous theorem would also largely be a comparison of control semantics, rather than the programme’s target: a complexity landscape created by making a society continuous. Whether TP–UW mass partitions are included in TE–NUW mass partitions is a rule-theoretic statement. The genuinely computational claim—polynomial-time feasibility via LP enumeration, plus integrality gaps and rounding—is not inherited from Theorem 20. It would need justification as an independently valuable control model.

There is also a serious institutional caveat. The mass variable \(y_t\) grants the controller ballot-type-aware authority to route arbitrary fractions of every approval type into either preliminary. In ordinary elections that is not merely an aggregate description of a feasible action: voters’ approval types are normally private until they vote, and assignment to districts, panels, or streams is constrained by location, eligibility, capacity, and individual participation. Once those attributes affect assignment, they belong in the complete type. The clean “tens of stable approval blocs” account then has to assume that, within each bloc, all such attributes are also identical and that the organizer may deliberately split that bloc by its political preferences. That is a much narrower and more artificial regime than the proponent’s national-election rhetoric suggests.

But this does not defeat the anchor universally. A deliberately designed two-panel process with predeclared approval ballots, interchangeable participants within each approval-and-eligibility type, and quota-based or randomized allocation really can make \(y_t\) the honest aggregate decision variable. Approval scores remain linear, no individual identity is needed by the objective, and exact ties do not disappear in a rational nonatomic model because the controller chooses the split. The proponent’s strongest constructed setting therefore survives the usual objections about identity, multiplicity, and degeneration.

So the negative case is procedurally strong but substantively insufficient. This paper itself offers no named complexity theorem for ChoCo to mirror, and Theorem 20 should not be presented as though it did. Yet if the programme permits an adjacent continuous control-feasibility problem to be motivated by a paper’s control definitions rather than by one of its complexity results, approval voter partition has a credible high-multiplicity scenario. I would not claim that there is *no* worthwhile continuous mirror in any scenario; that universal negative is not supportable against this anchor.

The adversarial triple: the proponent anchors on up to three named results; the opponent sees that case and must defeat every anchor; the judge decides which case convinced it. These are the pipeline’s own outputs, generated by tools/triple_run.py — no human edited them. The paper’s own text is not reproduced here beyond the quoted statement above.