Voting with Limited Energy: A Study of Plurality and Borda

The paper has no named theorem, lemma, proposition, or corollary asserting complexity or an algorithmic guarantee. The proponent gives a plausible continuous optimization direction for platform-scale electorates, but it is introduced from the conclusion rather than mirroring a qualifying named computational result. Thus the paper fails the required computational-anchor condition even though its proposed continuous model is worthwhile.

The objection that survived

Neither Proposition 3 nor Proposition 4 is a named computational result, so neither can anchor the newly formulated display-design optimization problem.

What the mirror covers

The proposed model extends the paper's concluding question about optimal show orders and is conceptually related to Propositions 3 and 4; it does not cover a named computational result because the paper has none.

Open questions for a prover

The case FOR (proponent)

Strictly, this paper contains no named computational-complexity result: no theorem states NP-hardness, polynomial-time solvability, FPT, or an approximation guarantee. Its named results are welfare and manipulability statements. So there is no qualifying computational anchor in the paper itself. Still, I think there is a genuinely good continuization here, centred on the paper’s explicitly posed open problem of finding an optimal show order from a profile distribution; Propositions 3 and 4 are the strongest conceptual anchors for it.

My lead problem would be Continuous Energy-Aware Display Design for Plurality. An instance has candidates \(A\), and a finite collection of voter types \(t=(\pi,e)\), where \(\pi\) is a complete intrinsic ranking and \(e\in\{2,\ldots,m\}\) is the attention/energy cap. The society is a rational distribution \(\mu_t\). A designer selects, for every type \(t\), nonnegative masses \(q_{t,S}\) over \(e\)-element displayed sets \(S\subseteq A\), with \(\sum_S q_{t,S}=\mu_t\). Thus \(q_{t,S}\) is the fraction of this type shown precisely \(S\); its reported ranking is \(\pi\!\restriction_S\). The plurality scores, and hence the lexicographically tie-broken winner, are induced by these masses.

Given a welfare threshold \(W\), the question is whether there is such a display policy whose plurality winner \(c\) has intrinsic per-capita welfare

\[ \sum_t\mu_t\,(m-\operatorname{rank}_{\pi_t}(c))\geq W. \]

A solution is the display policy \(q\), together with its resulting winner. Equivalently, one can maximize this welfare. This is the continuous version of selecting show orders to improve collective outcomes, rather than merely estimating their effect.

The natural regime is a large platform-mediated electorate: a city-wide participatory-budgeting vote, a large member association, or a consumer/cooperative platform choosing one option from a common slate. The platform has a preference model and attention estimates, not named voters: it knows that, say, 18% are of one ranking/attention segment and 7% of another. Many people share each segment, while the relevant number of occupied types is far smaller than the population. A segment-level display policy is also more defensible than individually engineered persuasion: it may be based on declared interest class, locale, or a privacy-preserving model class.

This is recognisably the authors’ problem. The paper already treats the show order as a designer-controlled function and ends by asking for “the computational complexity of discovering the optimal show order given a profile distribution.” Replacing integer counts by type masses is exactly the high-multiplicity version of that question, not a different outcome-space or probabilistic-social-choice problem.

The best named anchor is Proposition 3, proved in this paper: for every profile and energy function, some show order preserves the full-information plurality winner. Its continuous analogue is immediate and substantive: for every type, display a set containing its intrinsic top alternative. That preserves each type’s plurality contribution, so the full-energy plurality winner is preserved. The mass formulation makes the statement a population-level design guarantee rather than a voter-by-voter construction.

I expect the optimization version to be Class A. For a fixed intended winner, winner constraints are linear in \(q\). Although there are exponentially many displayed sets, the pricing problem is simple for plurality: for a type, one selects an \(e\)-set whose top-ranked member yields the best dual contribution. This is precisely the sort of column-generation LP that continuization exposes.

A useful second, narrower mirror is Continuous Borda Full-Winner Preservation: with the same input \((A,\mu)\), ask whether a display policy makes the Borda winner equal the full-energy Borda winner. Here the solution is again a feasible mass policy \(q\). This mirrors Proposition 4, proved in this paper, which gives a discrete instance where no show order can preserve the full-energy Borda winner. Its example embeds as a rational society, so the negative phenomenon survives continuization. More generally, Borda’s pricing subproblem appears tractable by dynamic programming along a type’s intrinsic ranking: selecting an \(e\)-set determines selected candidates’ Borda positions, and one can optimize the dual-weighted contribution by number selected so far. I would therefore also expect this feasibility/design problem to be Class A, while retaining the paper’s substantive impossibility boundary.

The main further questions are: what privacy or fairness constraints should restrict segment-specific displays; what happens when \(\mu\) is estimated rather than known; and how much welfare is lost if every segment must receive the same randomized display policy?

The weak point is real: the paper’s motivating examples include very small groups of friends, where mass is plainly the wrong abstraction, and its individual show-order function permits more granular personalization than a realistic segment policy may. The case survives because the paper itself points toward profile distributions and large voting contexts, and because the continuous problem should be claimed only for those high-multiplicity platform regimes—not as a replacement for the small-group model.

The case AGAINST (opponent, writing after the proponent)

The formal negative case is strong only on scope: this paper proves no named computational-complexity result. Propositions 3 and 4 are structural welfare statements, not algorithms or hardness claims. Thus neither supplies a computational result for ChoCo to continuize. The proponent’s “anchors” really use the paper’s concluding open direction to introduce a new design/optimization problem; that may be worthwhile research, but it is not a continuous counterpart of either proposition.

Proposition 3 particularly cannot carry the claimed anchor. Its content is an existential preservation observation: show every voter her intrinsic top option. Recasting voters as masses changes the quantifier from voters to types but adds no computational question. The LP for welfare-maximising displays is a distinct problem with a new objective, new input assumption (the designer knows a ranking/energy distribution), and—crucially—a new policy space that permits splitting a single type across displays. Calling that the continuous analogue of Proposition 3 overstates the connection.

The Proposition 4 anchor has the same issue in reverse. The proposition is one three-voter counterexample to exact Borda-winner preservation. Embedding its rational counts into a distribution merely repeats that counterexample in different units. The proposed general feasibility problem is again a fresh design problem, not a result whose complexity is being transferred or relaxed. Nothing in the proposition identifies a population-multiplicity bottleneck for continuization to resolve.

That said, I do not think the stronger universal negative claim is honestly defensible. The proponent’s large platform regime is a sensible high-multiplicity regime: ranking-plus-energy segments can recur at scale, and a platform may choose displays by segment. Once one accepts that regime, continuous display design is a well-posed computational problem over a distribution of types. Its randomized/segment-split policy can be read either as randomized presentation or as deterministic allocation across a large cohort, so it does not collapse for lack of a mass interpretation.

My conclusion is therefore narrow but important: this paper should not be greenlit *on the advertised named-result criterion*, because it contains no computational anchor and Propositions 3 and 4 do not become such anchors by relabelling them. But it cannot support the broader verdict that no worthwhile continuous mirror exists in any scenario. The paper’s own final question about optimal show orders, combined with platform-scale segmented electorates, leaves a credible continuous research direction.

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.