| paper | d4-journals-geb-2011-0095 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | yes |
Theorem 2 and Theorem 3 characterize when cores are nonempty for all admissible preference profiles, but provide no algorithm, complexity bound, or effective construction. A continuous quota population gives a sensible theoretical analogue, yet no computational question is anchored by a named result of this paper. The computational-result requirement therefore fails.
fails bit a — no named computational result to mirror
The proposed analogue concerns Theorems 2 and 3's universal nonemptiness claims for quota-defined populations; it leaves the other structural results and computability examples aside.
The paper contains no named computational-complexity result that can serve as an anchor for the requested continuous computational mirror.
The nearest results are Theorem 2, proved here, and Theorem 3, also proved here. Theorem 2 says that \(\#B<\nu(W)\) is equivalent both to the core without majority dissatisfaction being nonempty for every measurable profile whose preferences have maxima on the agenda, and to the ordinary core being nonempty for every such profile. Theorem 3 gives the corresponding \(\kappa(W')\) characterization for the core without majority dissatisfaction, and says that this condition implies universal nonemptiness of the ordinary core, but is not implied by it. These are existence characterizations, not claims that a problem is in P, NP-hard, or in another complexity class. Theorem 1 is cited from Kumabe and Mihara (2008a); it is likewise a nonemptiness characterization.
There is a plausible *theoretical* population mirror: take a nonatomic electorate with a rational distribution \(\mu\) over finitely many complete preference types, and let winning coalitions be defined by a mass quota. Ask whether the core, or the core without majority dissatisfaction, is nonempty for every such profile on a given agenda. That is close to the paper’s subject, and its authors could recognize it as an analogue of their core-existence results. But it is not an eligible anchor for this task, and the paper already allows arbitrary, including uncountable, player sets. It supplies no computational question about a finite-support input, no objective over mass transfers, and no complexity result for such a formulation.
So my positive case stops at a credible theoretical continuization. I cannot state a continuous computational problem per anchor without inventing a new problem beyond the paper’s named computational results.
The proponent correctly identifies the main obstacle: the paper has no named computational result to mirror. Theorem 2 characterizes universal core nonemptiness by \(\#B<\nu(W)\); Theorem 3 gives a related characterization using \(\kappa(W')\). Neither gives an algorithm, a complexity bound, or a computational problem over an encoded society. Their subject is a mathematical existence property of a game across all admissible preference profiles.
The proposed finite-support mirror is sensible as a model, but a fixed-profile version has little computational content under the natural finite agenda and quota assumptions. For a quota game, each pairwise dominance test is just a sum of the masses of types preferring one alternative to another; core membership, and membership in the core without majority dissatisfaction, can then be checked by scanning the types and alternatives in \(O(\tau |B|^2)\) time. The universal version avoids that input distribution: it asks about *all* profiles and returns to the Nakamura-number characterization. With nonatomic mass-quota games, that parameter is determined by the quota geometry. Making the problem harder by giving \(W\) an arbitrary succinct encoding would put the computational difficulty in the representation of the voting game, not in continuizing the population.
Theorem 3 does not supply a stronger computational anchor. Its \(\kappa(W')\) ranges over covers in an abstract set system that may involve infinite cardinalities; the paper gives no finite input representation or effective procedure for computing it. Replacing that setting with a finitely encoded family of mass constraints could produce an algorithmic problem, but it would require choosing new computational structure the theorem does not provide.
There is one real limit to this negative case: the paper itself allows arbitrary player sets, and Example 4 explicitly uses a Lebesgue-measurable continuum of players with measure-defined winning sets. That makes a continuous *theoretical* analogue quite natural. It does not, however, answer a computational question. Within ChoCo’s computational remit, the strongest case against is therefore that a continuous population model can be attached to the paper, but no worthwhile computational mirror is anchored by its results. A new manipulation or optimization problem might be worthwhile in its own right; the paper does not establish it as the continuous version of one of these theorems.
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.