| paper | d4-journals-mss-2013-0067 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | medium |
| authors would recognise it | yes |
Proposition 3.4
statement extracted from the paper’s text layer
Given a finite set of ballot types \(\mathcal T\), a rational population vector \(\mu\in\Delta(\mathcal T)\), a positional scoring rule with the paper’s tie-breaking and manipulation assumptions, and the induced region \(\mathcal W\subseteq\prod_t[0,\mu_t]\) of manipulation-winning coalition masses, compute \(\theta=\inf\{u\in[0,1]:u\mu\in\mathcal W\}\). This is the limiting fraction of the population that must be queried, with \(1-\theta\) giving the continuous limit of the query-based measure.
Treat complete ballots as types with population masses \(\mu\). A queried fraction \(u\) has limiting type-mass vector \(u\mu\); the region \(\mathcal W\) is induced by the paper’s manipulation condition, and the objective is to find the smallest \(u\) for which \(u\mu\in\mathcal W\).
The manipulation-winning region \(\mathcal W\) and the high-multiplicity election family are not specified in enough detail to determine the complexity of computing the threshold.
fatal: False
The mirror covers the collective query-based measure, including its specialization \(Q_0^*\) as used in the manipulation example; it leaves the paper’s individual semivalue characterizations, duality results, and axiomatic relations untouched.
The paper has no numbered computational-complexity result to anchor a compliant case. Its propositions characterize power measures and give formulas, but do not classify an underlying problem as tractable or hard, or analyze an algorithm or encoding for it. For example, Proposition 3.4, proved here, gives \(Q_F^*(G)=\sum_k f(n,k)|W_k|\); it is a formula for the measure, not a complexity result about computing it. So I can’t honestly name a lead anchor or predict its complexity class.
The strongest positive lead is nevertheless quite natural. The paper’s Example 6.2 motivates its query process with an election where the electorate is known but voter identities are not. In a large electorate, a type could be a complete ballot (and any other attributes relevant to manipulation), with \(\mu_t\) the fraction of voters of type \(t\). Let \(\mathcal W\subseteq\prod_t[0,\mu_t]\) be the upward-closed region of coalition-mass vectors whose members can manipulate the outcome under the paper’s assumptions. A candidate continuous query problem is: given rational \(\mu\) and a representation of \(\mathcal W\), compute
\[
\theta=\inf\{u\in[0,1]:u\mu\in\mathcal W\},
\]
the fraction of the population that must be queried before the sampled coalition becomes winning. Under the usual regularity assumptions on the finite populations’ winning conditions, \(1-\theta\) is the continuum limit of the paper’s \(Q_0^*=1-Q/(n+1)\).
This is a plausible high-multiplicity regime and closely follows the paper’s motivating story. But it is an unanchored research direction: the paper neither formulates nor analyzes the complexity of computing that limit.
The strongest objection is that this paper has no computational result to continuize. Its propositions characterize indices and give formulas; they do not analyze the complexity of computing an index or solving an algorithmic problem. The proposed anchor is Example 6.2, which motivates the query measure but does not formulate or classify such a problem. On the programme’s terms, that makes the proposal an adjacent research direction, not a mirror of a result in this paper.
The proposed limit is coherent: with finitely many voter types, a uniformly queried prefix containing fraction \(u\) of a large electorate has type-mass vector close to \(u\mu\). If \(\mathcal W\) is the region of winning coalition masses, the stopping fraction can converge to \(\theta=\inf\{u:u\mu\in\mathcal W\}\), and \(Q_0^*\) then converges to \(1-\theta\). But that also removes the query process’s randomness: every prefix has essentially the same type proportions. For the paper’s small positional-scoring example, once the manipulation condition is expressed in those masses, finding \(\theta\) may reduce to a one-dimensional threshold calculation. That could be a useful limit calculation, but it does not supply the missing complexity question from the paper.
A stronger repair would specify a high-multiplicity election and an encoding of its manipulation region, then study the complexity of computing \(\theta\) or a continuous power index. That is a sensible problem to formulate. Its difficulty and value, though, would come from choices of election model, manipulation condition, and input representation that this paper neither makes nor analyzes. The negative case is weakest here: such a follow-on might be worthwhile, and I cannot rule out a compelling computational formulation. What I can argue is that the proponent has not yet shown a worthwhile *continuous mirror of this paper’s results*; the paper offers motivation for one, not a computational result to anchor it.
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.