| paper | d4-journals-geb-2016-0079 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | no |
Neither named result asserts an algorithm, hardness bound, or computable formulation: Proposition 1 is an invariance result, and Theorem 1 is a structural characterization. The opponent also explains why an atomless version would lose the finite coalition structure, but the decisive failure is bit (a).
fails bit a — no named computational result to mirror
No result is covered by a computational mirror. The paper’s characterization, constitution conditions, and heterogeneous-utility extension are theoretical results without computational guarantees or hardness claims.
This paper has no named computational result to anchor a continuous mirror. Its closest results are Proposition 1, which proves that self-stability is independent of the preference-probability vector \(p\), and Theorem 1, which characterizes self-stable weighted majority rules by their veto, normal, and null players, with an additional case when \(r\) is a reciprocal integer. Both are proved here, but they are structural characterization results: neither asserts a complexity classification nor gives an algorithm for a computational problem.
A continuous-population version of self-stability could be posed, but it would introduce a new problem whose computational status this paper does not establish. Presenting it as a mirror of a named computational result would overstate the case.
Neither result is a computational anchor. Proposition 1 proves that self-stability is invariant to \(p\); Theorem 1 characterizes which finite weighted majority rules are self-stable. Neither defines an algorithmic problem or reports a complexity result. One could invent a recognition task—given a rule, decide whether it is self-stable—or a design task—find a stable rule under added constraints. But those would be new computational questions, with no unique choice of input, output, or objective supplied by this paper.
Proposition 1 is especially poor support for a continuous mirror. Its point is that the preference-probability vector does not affect the classification. Replacing a list of \(p_i\)'s with a distribution over voter types would therefore add a continuous object that the paper’s main result deliberately makes irrelevant. Extending the invariance proof to distributions could be mathematically interesting, but it would still establish an identity, not a complexity landscape.
The strongest version of Theorem 1’s mirror would model an atomless electorate with a measurable voting weight \(w(x)\), and ask which weighted rules are self-stable. That changes the mechanism the theorem classifies. With atomless voting power, no individual is a veto player. Moreover, for a weighted quota rule, any winning coalition with weight strictly above quota can be trimmed to a smaller winning coalition, so inclusion-minimal winning coalitions disappear. The theorem’s finite partition into veto, normal, and null players—and its bounds on the sizes of minimal winning coalitions—has no direct population-level counterpart.
There are two natural repairs, and neither gives a strong mirror. If voters have independent preferences and the electorate is atomless, the realized mass of each preference type is fixed almost surely by its probability distribution. The original random coalition profile, over which rules are compared in expected utility, collapses to a fixed aggregate profile. If instead positive-mass groups share perfectly correlated preferences so the profile remains random, those groups act as finitely many bloc-voters. The result is again an ordinary weighted voting model; its real-valued bloc weights are already allowed by the paper’s finite model. Theorem 2’s heterogeneous utilities and the constitution results do not change this basic issue: they extend the finite characterization, but do not introduce a computational task.
This is a strong case against treating the paper as a ChoCo mirror, not proof that no related continuous model could be worth studying. A continuum-of-types stability or rule-design project could be legitimate. But it would need its own model and computational question; it would not continue the paper’s named results in a way that makes them useful anchors for the programme.
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.