| paper | d4-conf-aaai-2018-0028 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | yes |
The continuous Dodgson winner question is concrete and fits a credible population regime, but the article attributes the \(\Theta_2^p\)-completeness result to [HHR97a] and states no numbered computational result of its own. It therefore lacks the required result anchor for a green mirror.
fails bit a — no named computational result to mirror
The conditional mirror covers only the Dodgson winner-recognition discussion in §2.1; it leaves the article's other examples and broader thesis untouched.
The article contains computational-complexity claims, but no numbered Theorem, Lemma, Corollary, or Proposition. So it has no anchor that meets your stated rule. The strongest candidate is its §2.1 discussion of Dodgson elections: it reports that deciding whether a candidate wins is \(\Theta_2^p\)-complete. That result is cited from elsewhere, [HHR97a], not proved in this article, and it has no theorem number printed here. I can’t present it as a fully compliant anchor.
If that unnumbered claim is acceptable as a conditional anchor, the strongest mirror I see is Continuous Dodgson Winner Recognition. The anchor’s content is: given a finite electorate of tie-free rankings and a candidate, decide whether the candidate has minimum Dodgson score, where a candidate’s score is the minimum number of adjacent swaps needed to make that candidate a Condorcet winner. The article attributes the \(\Theta_2^p\)-completeness result to [HHR97a].
For the mirror, an instance is a candidate set \(C\), a rational distribution \(\mu\) over a listed support \(S\) of complete rankings, and a target candidate \(c\). A transfer plan \(y_{r,r'}\ge 0\) assigns mass initially of type \(r\) to final ranking \(r'\), with \(\sum_{r'}y_{r,r'}=\mu_r\). The resulting distribution is \(\mu'_{q}=\sum_r y_{r,q}\). Define \(D_\mu(c)\) as the minimum of \(\sum_{r,r'} d_K(r,r')y_{r,r'}\), where \(d_K\) is Kendall distance, subject to \(c\) receiving at least half the mass in every head-to-head comparison. This is the infimum cost of making \(c\) a strict Condorcet winner; the weak inequalities express that infimum in a closed optimization problem. The question is whether \(D_\mu(c)\le D_\mu(a)\) for every \(a\in C\). A solution gives that yes/no answer, with the exact scores or minimizing transfer plans as certificates of the computation.
A plausible regime is a large electorate divided among a modest number \(\tau\) of shared ballot types: many voters, but relatively few distinct rankings, each type’s mass representing a population fraction. The decision variable is mass moved between rankings; the objective is total per-capita adjacent-swap cost. This retains the underlying Dodgson question—who wins by minimum distance to Condorcet status—while making the population the continuous object.
I would expect this mirror to be tractable, in Class A, through a structural LP approach; the continuization programme specifically identifies continuous Dodgson as an earlier example of that pattern. The research questions would include whether exact winner recognition is polynomial in \(m,\tau,L\) despite the \(m!\) possible rankings, and how its answer and scores relate to finite elections when transfer masses must be multiples of \(1/n\). This covers only the article’s Dodgson example in §2.1, not its broader claims about complexity and computational social choice.
The weakest point is that the article uses Dodgson to illustrate a natural \(\Theta_2^p\)-complete problem, but does not itself study fractional electorates or mass transfers. The mirror is plausible because it preserves the rule and its score; its connection to the article is still indirect, and the anchor remains formally ineligible under your numbering requirement.
The strongest case against starts with the anchor: this is a perspective essay, not a paper proving a computational result. It has no numbered theorem, lemma, corollary, or proposition asserting a problem’s complexity. Its Dodgson \(\Theta_2^p\)-completeness claim is attributed to [HHR97a], so mirroring it would be a project about that cited result, not a continuous mirror of a result established in this paper.
Even granting Dodgson as a conditional anchor, the proposed model is sensible: a large electorate sharing a modest set of ballot types is a credible high-multiplicity regime, and mass transfers with Kendall cost preserve the relevant question. But the strongest version of that mirror—compute continuous Dodgson scores and recognize winners over a distribution—is already among ChoCo’s worked examples: continuous Dodgson is efficiently solvable by structural insights and LP. A more elaborate version with type-specific prices or constraints would pose a different problem, rather than uncover a missing computational question in this anchor.
A finite-to-continuous comparison could still yield a useful rounding or approximation theorem. That is the weak point in the negative case: ChoCo explicitly values a formal dictionary between discrete and continuous variants, so I can’t honestly dismiss every such project as worthless. But it would be new bridge work around a cited Dodgson result, not a fresh continuous computational treatment of a result in this article. The case against is therefore strong on paper-specific relevance and novelty, but weak as a universal claim that no worthwhile question can be extracted.
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.