| paper | d4-journals-scw-2002-0033 |
| 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: Both feasibility and expediency are in \(\mathrm{P}\). Every valid instance is feasible. Expediency holds exactly when \(\sigma_iw_i+\sigma_jw_j>0\) and \(\sum_t(\mu_t-\sigma_t)w_t^2>0\), with \(\sigma_t=0\) outside the bloc types. For \(m=2\), both decisions and rational witness construction take \(O(\tau L^2)\) bit operations. · discrete side: other · adjudication: proved
full proof & adversarial review →
Theorem 4.2
statement extracted from the paper’s text layer
Given rational masses \(\mu_t\) of voting-mandate types, per-member weights \(w_t\), and a bloc comprising positive masses \(\sigma_t\) from two types, replicate the types in regular finite societies with exact counts and weighted-majority quotas at half the total weight. For type-symmetric internal weighted-majority rules with rational per-type weights \(v_t\), decide whether some \(v\) makes every bloc type’s limiting absolute Banzhaf influence \(\lim_{N\to\infty}\sqrt{N}\,\beta'_a[W_N\parallel W_{S,N}]\) at least its separate baseline \(\lim_{N\to\infty}\sqrt{N}\,\beta'_a[W_N]\), and whether strict gains are possible, restricting to sequences for which these limits exist.
A society is a distribution \(\mu\) over complete voting-mandate types. A mass \(\sigma_t\) of two types joins a bloc; the decision variables are the bloc’s internal type weights \(v_t\), and the objective is to maximize the minimum gain in \(\sqrt{N}\)-scaled absolute Banzhaf influence over its represented types.
The \(\sqrt{N}\) influence density is regime-dependent: under supermajority quotas it may vanish or require a different scale, so it does not cover the paper’s full class of simple voting games.
fatal: False
The mirror covers alliance feasibility and expediency in Theorems 4.2–4.3; it leaves aside P-power and index comparisons, annexation, and the finite CMEC calculations.
I can’t make a positive case that meets the required anchor standard. The paper has numbered theorems, but none asserts a complexity result or presents an algorithm for a named computational problem. The closest results are Theorem 4.2, which says a two-voter bloc is never expedient and is feasible exactly when the voters have equal Banzhaf power or one is a dummy, and Theorem 4.3, which gives a feasibility and expediency condition for a particular three-voter bloc. They are structural voting-power results, not tractability or hardness results.
A tempting mirror would take voter-weight classes as types, their population shares as mass, and ask whether some internal rule lets every type gain Banzhaf influence through a bloc. But in a nonatomic population, an individual’s probability of being pivotal collapses; defining a meaningful per-person limit requires a normalization or a new influence measure. The paper supplies neither, and its theorems do not establish results for that model.
So there is no honest continuous problem per qualifying anchor to hand to a prover. The strongest positive observation is that the paper studies alliance feasibility, a property one could investigate in a high-multiplicity voting regime. But turning that observation into an anchored computational mirror would overstate what the paper establishes.
The absence of a numbered complexity theorem is a real limitation, but it does not mean the paper has no computational question to mirror. Its central notion of feasibility asks whether there exists an internal voting game \(W_S\) that lets every bloc member retain or increase their influence. Searching over such rules is an existence problem already implicit in the paper. A continuous version can ask how to choose the bloc’s composition and internal rule in a high-multiplicity society.
A natural setting is a large federation or member organization with many interchangeable voters in each mandate or weight class. Let \(\mu_t\) be the share of members of type \(t\), with type specifying everything relevant to voting power. This is a sensible high-multiplicity regime: members are anonymous within a class, and the object being compared is their per-member influence. Existing high-multiplicity voting work would support the regime’s relevance.
The proponent is right that raw individual Banzhaf power tends to zero as the population grows. That does not force the continuous question to be vacuous. In regular majority-voting sequences, use the asymptotic influence density \(\lim_{N\to\infty}\sqrt{N}\,\beta'_a\). For example, with \(N\) equal-weight voters under simple majority, a voter’s absolute Banzhaf power is \(\binom{N-1}{(N-1)/2}/2^{N-1}\), which is asymptotic to \(\sqrt{2/(\pi N)}\). The scaled quantity is finite. One can define it through finite societies whose type proportions approach \(\mu\), then use Theorem 4.1’s exact factorization to study the limit for a bloc. The quota regime and scaling need to be specified; that is a modelling task, not a universal obstruction.
This gives concrete continuations of both results the proponent raised. For Theorem 4.2, ask whether a bloc drawn from two positive-mass type classes can be feasible or expedient, and find the mass compositions and internal rules that allow it. For Theorem 4.3, ask whether three type classes with equal baseline influence densities admit an expedient internal rule when the third class has positive external pivotal influence. These are not limits of two or three named individuals. They are questions about many members per type, with influence compared per member.
A computational version is: given rational type masses, external voting weights and quota, does there exist a type-symmetric internal weighted rule that gives every represented type at least its baseline asymptotic influence density? One can also seek the smallest bloc mass or the rule maximizing the least gain. The paper’s results provide the structural starting point; classifying these decision or optimization problems would be new work. The continuous answer might be easy in some regimes and hard in others. Either way, it is a well-posed question about population mass and alliance design. The case for a mirror is therefore limited by the need to choose a defensible influence limit and rule family, but it is stronger than the claim that no worthwhile mirror exists.
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.