| paper | d4-conf-aaai-2017-0813 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | medium |
| authors would recognise it | yes |
other→P discrete result → its continuous mirror
Answer: Polynomial-time solvable in m, τ, and L. Every promised instance has finite dimension, and attaining rational rules can be output in polynomial time. · discrete side: other · adjudication: proved
full proof & adversarial review →
Theorem 1
statement extracted from the paper’s text layer
Given rational masses \(\mu_t\) for \(\tau\) membership types and an explicitly encoded monotone, closed rational polyhedral winning region \(W\subseteq\prod_t[0,\mu_t]\), find the least \(d\) for which rational vectors \(w^1,\ldots,w^d\ge 0\) and positive rational quotas \(q_1,\ldots,q_d\) satisfy \(x\in W\) exactly when \(w^j\cdot x\ge q_j\) for every \(j\); output the rules or report that no finite representation exists.
A society consists of membership types with rational masses \(\mu_t\); a coalition is a vector \(x\) of participating mass by type. The monotone winning region \(W\) is represented by a finite union of rational polytopes. Each rule is a weighted threshold on type masses, and the objective is to minimize the number of rules whose intersection exactly represents \(W\).
There is no refinement rule connecting discrete games and their one-player Hamming covers to the continuous winning region, so this extends the dimension objective without transferring Theorem 1’s quantitative bound.
fatal: False
The mirror covers the constructive dimension objective in Theorem 1. It leaves aside Theorem 2’s SECDED pairing bound, the extremal bounds on \(d_n\), and the coding-theoretic estimates.
The strongest honest anchor is Theorem 1, proved in this paper. It says that for a simple game \(\Gamma(N,LM)\), if \(C\subseteq 2^N\) has the property that every maximal losing coalition is within Hamming distance one of some member of \(C\), then the dimension of \(\Gamma\) is at most \(|C|\). The proof constructs the representation: assign each maximal losing coalition to a nearby member of \(C\), and represent the game defined by each resulting group as a weighted game; intersect those weighted games to recover \(\Gamma\). This is a constructive bound on the number of weighted rules needed, though it is not a conventional claim such as “dimension is in P” or “dimension is NP-hard.” Its significance is the underlying object it computes: the minimum number of weighted games whose intersection implements a monotone coalition rule.
A plausible continuous population is a large federation or membership organization with a few statutory voting classes—say, thousands of members in each class, with members of a class sharing the same voting rights and eligibility. Let \(T\) be those classes, with population masses \(\mu_t\) summing to one. A coalition is represented by \(x\in\prod_{t\in T}[0,\mu_t]\), where \(x_t\) is the mass of class \(t\) that joins. The number of members can be very large while the number of distinct types \(\tau=|T|\) remains small. The rule is anonymous within each class, but can still be an arbitrary monotone rule over class participation.
I would call the mirror Continuous Coalition Dimension. An instance consists of the rational class masses \(\mu\) and a monotone, closed, rational polyhedral winning region \(W\subseteq\prod_t[0,\mu_t]\), specified by a finite union of rational polytopes, with the empty coalition losing and the full society winning. The question is to find the least \(d\), or report that no finite \(d\) exists, for which there are nonnegative rational weight vectors \(w^1,\ldots,w^d\) and positive rational quotas \(q_1,\ldots,q_d\) satisfying, for every coalition mass vector \(x\),
\[
x\in W
\quad\Longleftrightarrow\quad
w^j\cdot x\ge q_j\ \text{for every }j\in\{1,\ldots,d\}.
\]
A solution is the list of those weighted rules, or a correct report that no such finite intersection represents \(W\). The objective—minimize the number of weighted rules—is exactly the dimension objective in the paper, with the population represented by type masses.
I think the authors would recognize that as a continuous analogue of their underlying dimension problem: it asks how many weighted voting rules are needed to implement a monotone coalition outcome. It also fits the paper’s institutional-voting motivation. My strongest claim is for this problem, not for a continuous version of Theorem 1’s covering-code bound. I would expect continuum-specific geometric difficulty when \(\tau\) varies: intersections of weighted rules are convex, whereas a general monotone winning region need not be. For fixed \(\tau\), polyhedral methods may make the problem tractable; the paper itself proves neither complexity claim.
The mirror covers the paper’s constructive upper-bound result about dimension. It does not cover its cited lower bounds, the full extremal analysis of \(d_n\), or its specific coding-theoretic estimates. Nor does it assert that the paper proves a complexity classification.
The soft spot is substantial: Theorem 1 relies on maximal losing coalitions and one-player Hamming neighborhoods. In a continuum, maximal losing points may disappear, and the normalized size of one player tends to zero. There is no evident way to carry that covering argument over unchanged. Many continuous monotone rules also cannot be represented by any intersection of weighted rules because they are nonconvex. So the defensible case is that the dimension question has a meaningful population-continuous form, not that the paper’s particular bound survives. That still seems worth asking: it preserves the paper’s central implementation objective, and the gap between finite coalition geometry and continuous mass geometry is itself a concrete research question.
Theorem 1 has a constructive proof, so it would be unfair to say the paper has no computational content. But it states no complexity result: its claim is a combinatorial upper bound on dimension, obtained by covering maximal losing coalitions with radius-one Hamming neighborhoods. The proposed “Continuous Coalition Dimension” changes that into a new minimization problem over real-valued coalition masses. That may be interesting, but it does not yet mirror the result the paper proves.
The class-based federation is a plausible high-multiplicity setting, and fractional participation is a reasonable model there. The difficulty is the theorem’s unit of analysis: one player joining or leaving a coalition. If a type has \(k\) members, a one-member change is \(1/k\) of its normalized mass, which vanishes as \(k\) grows. Keeping the exact one-player notion gives a zero-radius cover in the limit; replacing it with an \(\varepsilon\)-radius cover introduces a scale the paper does not supply and changes an exact representation theorem into an approximation question. A geometric covering result might be worth developing, but it would need its own motivation and dictionary to high-multiplicity games.
The stronger proposed mirror—minimizing the number of weighted rules that represent a monotone region \(W\) on the whole mass box—does preserve the paper’s dimension objective. But it imposes a sharp geometric restriction: an intersection of weighted rules is convex. Nonconvex monotone regions therefore have infinite dimension, even though every finite simple game has finite dimension. If \(W\) is restricted to a convex rational polyhedron, the finite answer is essentially its number of necessary supporting facets with admissible monotone normals. That is a legitimate polyhedral representation problem, but it is no longer the paper’s covering-code question or its bound on \(d_n\). Without a rule for how discrete games on increasingly fine grids determine \(W\), the continuous model also adds a substantive interpolation assumption.
So the strongest case against is that the proposed mirror is a new continuous voting-game problem, not a continuization of the paper’s named result. The paper offers no computational complexity theorem for that problem and no scaling principle that carries its radius-one construction to a population continuum. I would not call the negative case airtight: the federation scenario is credible, and authors could reasonably recognize continuous coalition dimension as a worthwhile extension. The case is strongest if the standard is “a mirror of this paper’s result,” and weaker if an independently motivated continuous representation problem is enough.
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.