| paper | d4-journals-mss-2014-0045 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | medium |
| authors would recognise it | yes |
P→P discrete result → its continuous mirror
Answer: The exact optimization problem is solvable in deterministic polynomial time, with a unique optimum given by the primitive integer boundary normal and its corresponding rational threshold. · discrete side: polynomial · adjudication: proved
full proof & adversarial review →
Theorem 4.9
Given positive rational type masses μ_1,μ_2 with μ_1+μ_2=1 and an explicitly described rational winning region \(W\subseteq[0,μ_1]\times[0,μ_2]\), promised to be exactly representable by ordered positive integer type weights and a positive rational quota, find \(w_1>w_2>0\) and θ∈ℚ_{>0} minimizing \(c_1w_1+c_2w_2+c_3θ\), for given rational \(c_i≥0\) not all zero, subject to \(x∈ W\iff w_1x_1+w_2x_2≥θ\) for every coalition mass vector \(x\) in the box.
A two-type society with masses \(\mu_1,\mu_2\); each coalition is any vector of type masses \(x\) in the corresponding box. Each type has an integer per-voter weight, while the threshold is a rational quota for the normalized society. The input is the continuous winning region \(W\); the objective minimizes the stated weighted sum of the two weights and quota.
The theorem combines per-voter weights with an integer quota, while the proposed mirror uses a normalized mass quota θ; the proponent motivates that quota change but does not specify a corresponding normalization for the objective coefficients.
fatal: False
Covers minimum integer representations for two-type continuous threshold games whose boundary crosses the interior of the mass box. It leaves the paper's finite-lattice results, the one-vector case, examples with more types, minimum-sum representations, and enumeration bounds outside the mirror.
The strongest anchor is Theorem 4.9, proved in this paper. It concerns a weighted game with two voter types, no null voters, and more than one shift-minimal winning vector. For every nonzero objective \(c_1w_1+c_2w_2+c_3q\), with \(c_i\ge 0\), minimizing over the paper’s inequalities for shift-minimal winning and shift-maximal losing vectors yields a unique integer solution \((q,w_1,w_2)\). It also bounds that solution by \(1\le w_1\le n_2\), \(1\le w_2\le n_1\), and \(w_2+1\le q\le 2n_1n_2\). The paper’s Remark 4.11 gives an algorithm for finding the representation. The paper has no named NP-hardness or complexity-class theorem to anchor on; its general ILP hardness observation is not a named result.
A continuous mirror is a Continuous Minimum Integer Representation problem. Take a large society split into two fixed, interchangeable blocs—for example, members of a federation with two classes of voting rights. The blocs have masses \(\mu_1,\mu_2>0\), with \(\mu_1+\mu_2=1\). A coalition is any mass vector \(x\in[0,\mu_1]\times[0,\mu_2]\), and a target game is a nontrivial rational polygon \(W\) of winning mass vectors, whose threshold boundary cuts through the interior of the box. Given \(W\) in an explicit rational polyhedral description and nonnegative rational coefficients \(c_1,c_2,c_3\), not all zero, find positive integer per-unit weights \(w_1,w_2\) and a positive rational normalized quota \(\theta\) such that, for every coalition mass vector,
\[
x\in W\quad\Longleftrightarrow\quad w_1x_1+w_2x_2\ge\theta,
\]
while minimizing \(c_1w_1+c_2w_2+c_3\theta\). A solution is an optimal triple implementing exactly the target region. The real quota is the natural co-modification when finite-population quota is normalized to population mass.
I expect this mirror to be tractable. The interior boundary segment determines the ratio of the two weights and the quota; clear denominators to get integer weights, then divide by their common factor to obtain the least integer scale. In the nondegenerate case, all exact representations lie on the same scaling ray, so the least scale uniquely optimizes any such nonnegative objective. The natural next questions are whether this extends cleanly to more types and what happens when the target boundary does not cut the interior.
The regime is \(N\) large with two fixed types: \(N\) grows while the number of distinct types stays at two. Members of a bloc are interchangeable under the target rule, and coalition size is honestly represented by mass. That is a credible high-multiplicity version of the paper’s voting game, and the authors would recognize the task as finding a minimum integer-weight representation of their game.
The weak point is also the source of the easy answer: the paper’s difficulty comes from classifying coalitions on a finite integer lattice, whereas this mirror requires exact agreement at every point in a continuous box. The full boundary then identifies the separating weights directly, potentially removing the combinatorics that made Theorem 4.9 interesting. I still think the question is a legitimate mirror of the underlying representation problem: it keeps the population continuous, retains integer per-unit weights, and asks whether the minimum-representation phenomenon survives when every fractional coalition is part of the game.
Theorem 4.9 is a real anchor: it proves a strong structural fact about minimizing representations, and Remark 4.11 gives an algorithm. So the negative case cannot dismiss the paper as lacking a computational result.
The vulnerability is the bridge from the paper’s object to the proposed continuous one. The paper gives a finite game: a Boolean label for each coalition vector on the lattice \(\{0,\ldots,n_1\}\times\{0,\ldots,n_2\}\). Its least representation is defined by agreement on that finite set. The proposed mirror instead supplies a winning region \(W\) over a continuous box and asks for a threshold that agrees with it everywhere. The finite game does not determine such a region: many different continuous extensions can agree on all its lattice points. If \(W\) is supplied as a rational threshold region, its boundary already gives the weight ratio, so the proposed task is chiefly about representing a continuous game specified independently of the paper’s finite one.
Taking a limit of growing games does not resolve that gap cleanly. The unit separation between winning and losing coalitions is one voter in the paper, and after normalizing population mass it shrinks to zero. Integer weight ratios can converge to an irrational boundary slope, even though every finite game in the sequence has integer weights. Thus exact integer representability is not preserved by the limit. Keeping the integer lattice preserves the paper’s representation question but keeps the population finite; replacing integer weights with real per-capita rates gives a natural continuous model but changes what “minimum integer representation” means. That is the more fundamental obstacle than any literal tie-breaking or quota convention.
There is a further mismatch in the proposed objective. The theorem minimizes combinations of \(w_1,w_2,q\), all defined in units of individual votes. In the mirror, the quota becomes a normalized mass threshold while the weights remain integers. Unless the model gives those units a population-independent interpretation, comparing them with the same coefficients has no canonical meaning. Normalizing the threshold can make a continuous game well-defined, but it does not recover the finite game’s least-representation order.
This is not a knockout argument against every possible continuous weighted-voting project. Two large blocs with interchangeable members are a credible high-multiplicity regime, and the authors might well recognize minimum representations of continuum voting games as an interesting extension. The negative case is narrower: the stated mirror either adds a continuous game not determined by the paper, or takes a limit that loses the lattice-scale information Theorem 4.9 uses. On the evidence here, that makes the anchor weak as a *continuization of this result*, though not as a basis for a separate continuous voting-game problem.
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.