| paper | d4-journals-geb-2015-0052 |
| 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 decidable for \(m=\tau=3\); a stable partition can also be constructed in polynomial time as an exact algebraic mass table whenever one exists. · discrete side: other · adjudication: proved_with_minor_gaps
full proof & adversarial review →
Proposition 5
Given three productivity types with rational masses \(\mu_h,\mu_m,\mu_l\) summing to \(1\), their productivity levels \(\lambda_h>\lambda_m>\lambda_l\), and a minimum productive organization mass \(\alpha\), decide whether a measurable partition into organizations exists in which every productive organization has mass at least \(\alpha\), its mass median selects meritocracy or egalitarianism under the paper’s voting rule, and no positive-mass group can form a blocking organization; construct such a partition when one exists.
The society is an atomless population with finite support over complete productivity types. Decision variables assign each type’s mass across organization compositions, conserving \(\mu_t\); productive organizations have mass at least \(\alpha\), output is the integral of member productivities, and the mass median determines the reward rule. The task is a feasibility and construction problem, with no separate optimization objective.
Proposition 5 gives no stated runtime or complexity bound, so the complexity of deciding existence or constructing a stable partition must be established anew.
fatal: False
The mirror covers Proposition 5’s three-type core-existence and stable-structure result. It leaves the broader clustered-society results, the segregation and reward-regime characterizations, comparative statics, and endogenous effort without computational mirrors.
The strongest positive case is that the paper’s population has a natural high-multiplicity regime. Imagine a large workforce with many employees at each of a few productivity levels, forming organizations that need at least a fraction \(\alpha\) of the workforce to operate. A type is a productivity level, and mass is the fraction of workers at that level. Organizations are measurable groups of mass at least \(\alpha\); their output is the integral of member productivity. A mass-weighted median chooses between meritocratic pay and equal division, and a core-stable structure is one with no group of agents who would all strictly prefer to form a new organization. Scaling the paper’s minimum size \(v\) with population size as \(v\approx \alpha n\) preserves its coalition-size mechanism. This is a plausible continuous population mirror, and the authors say in Section 7 that further work tested a continuum formulation retaining the discrete model’s characteristics.
There is, however, no qualifying computational anchor in this paper. Its numbered propositions characterize existence and form of core-stable structures; they do not establish that a computational problem is in P, hard, fixed-parameter tractable, or otherwise classify its complexity. Proposition 5 is the closest candidate: for three-type societies, core-stable structures exist if and only if the society meets one of the three “structured” conditions in Definition 6. That is a mathematical characterization, not a named complexity result. I would not present it as an anchor.
An unanchored follow-on question could be: given rational productivity levels \(\lambda_1>\cdots>\lambda_k\), rational masses \(\mu_1,\ldots,\mu_k\), and a rational minimum organization mass \(\alpha\), does the resulting atomless society have a core-stable partition under the paper’s voting and preference rules? A solution would be such a partition, with no measurable subpopulation able to block it. This is a plausible continuation of the authors’ model, but it is an analytic existence question as stated. The paper supplies no computational anchor from which to predict its complexity class, and it reports continuum work without giving its details here. Those are the main limits on a ChoCo case.
The strongest case against is that this paper has no computational result to mirror. Propositions 2, 5 and 7 characterize when core-stable structures exist and what they look like; none classifies the complexity of a computational problem. Asking for an algorithm that decides core existence or constructs a stable structure from rational masses would be a new computational question, not a continuous mirror of a result the paper proves. If ChoCo is selecting papers for direct computational mirrors, this is a poor fit.
That case does not dispose of the proposed workforce scenario. Productivity levels can be complete types, the input can be a finite-support distribution, and a mass-weighted median and blocking coalition have natural interpretations. Allowing a minimum organization mass \(\alpha\) is a recognizable extension of the paper’s size threshold. The resulting problem seems well-posed; I see no sound objection based on identity, lack of multiplicity, or a degenerate limit.
Section 7 also says the authors tested a continuum formulation that retained the model’s characteristics. That makes a new structural existence question less compelling, but the paper gives no details and reports no continuous computational analysis, so it does not establish that the proposed algorithmic question is already answered.
So the negative case is strong only under a narrow standard: this is not a mirror of a named computational result. I cannot honestly defend the broader claim that no worthwhile continuous computational project exists here. The better mass-based version survives the modelling objections; rejecting it would require showing that turning this structural theory into a computational problem falls outside ChoCo’s remit.
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.