| paper | d4-journals-geb-2023-0079 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | no |
The named results characterize interim stability; none states an algorithm, complexity bound, hardness result, or solver formulation. Rescaling voter counts into population shares does not preserve the individual pivotal event that drives the equilibrium, while blocs or a limiting belief construction would add or change the strategic model. There is no paper-anchored continuous computational question to mirror.
fails bit a — no named computational result to mirror
No results are mirrored; the paper's stability characterizations, including the weighted-majority results and supporting lemmas, remain about its finite-agent game.
I don’t find a qualifying computational anchor in this paper. Its named results characterize interim stability; they do not assert that a computational problem is in \(P\), NP-hard, or in another complexity class.
For example, Proposition 1, proved here, says that a constitution \((f,F)\) of qualified majority rules is interim stable against all qualified majority alternatives if and only if \(q\ge q_f\) and \(q+1\ge n-q_f\). Theorem 1, also proved here, says that such a constitution is stable against all weighted majority alternatives if and only if it is stable against qualified majority alternatives. Corollary 1.1 says a qualified majority rule is interim self-stable against all weighted majority alternatives if and only if it is a standard or supermajority rule. These are substantive characterizations, but none is a named complexity result about a computational problem.
A continuous population model is easy to sketch: take two voter types, \(r\) and \(s\), with masses \(\mu_r\) and \(\mu_s=1-\mu_r\), and ask whether a pair of thresholds for outcome choice and rule change is stable. But that does not give me an honest continuous mirror of one of the paper’s computational anchors. The paper’s stability notion relies on individual voters being pivotal and on beliefs about their pivotality; in a nonatomic population, an individual has zero mass and cannot be pivotal. Restoring pivotality with blocs or atoms would change the model’s central mechanism.
So the strongest honest positive case is limited: the paper’s binary preference types make a continuous population representation tempting, but its results are equilibrium characterizations in a finite-agent model, not computational results for which this programme’s mirror test applies. There is no qualifying anchor here from which to state the required continuous problem.
The decisive point is that the paper has no named computational result to continuize. Proposition 1, Theorem 1, and Corollary 1.1 are substantive characterizations of interim stability, but they give no complexity bound or algorithm for a computational problem. A decision or search problem about recognizing stable constitutions could be added; that would be a new computational question, not a mirror of a result this paper establishes.
The most natural candidate mirror is Proposition 1, together with Corollary 1.1. One can replace the numbers of \(r\)- and \(s\)-voters by their population shares and scale the quotas by \(n\). The paper’s defining mechanism then becomes the obstacle: stability depends on an individual’s incentive conditional on being pivotal, using the posterior in (C.3). In a nonatomic population, an individual has zero voting mass, so that pivotal event has probability zero. Conditioning on it has no canonical meaning. Treating a positive-mass type group as the pivotal actor would give the group a coordinated bloc vote and change the game’s players and equilibrium concept.
Theorem 1 has the same problem in its most generous version. A continuum distribution over voting weights and preference types is plausible, especially for a large shareholder body. Yet each individual’s normalized weight vanishes, so individual pivotality again disappears. Keeping positive-mass blocs preserves pivotality by making groups the strategic actors; retaining finite voters preserves the original game but keeps the population discrete. Neither choice gives the paper’s interim-stability theorem a continuous-population mirror.
There is a serious possible rescue: take a sequence of large finite electorates and derive limiting beliefs conditional on pivotality, perhaps through a large-deviation or exponential-tilting calculation. That could be a worthwhile follow-on. It would require specifying how quotas, ties, and trembles scale, since those determine the limiting conditional beliefs. The paper’s propositions do not pose or solve that asymptotic computational problem. Moving to ex-ante stability offers another route around individual pivotality, but it changes the timing that distinguishes this paper’s contribution and still supplies no computational anchor.
So the negative case is strongest on fit: the paper offers equilibrium characterizations whose core object is finite-agent pivotality, while the programme asks for computational questions over continuous societies. The rescaled quota formulas are suggestive, and an asymptotic pivotality project may be interesting; neither makes a worthwhile continuous computational mirror of the paper’s named results.
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.