| paper | d4-journals-jet-2012-0107 |
| 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 solvable: \(\mathrm{P}\). · discrete side: other · adjudication: proved
full proof & adversarial review →
Proposition 4.2
Given two finite-support societies μ and ν on \(D × \{1,\ldots,H\}\), with rational incomes and binary-rational masses, decide whether ν is at least as good as μ for every \(U \in U_{22}\): equivalently, whether all three inequalities in Proposition 4.2 hold for every income threshold and ability prefix.
Represent each society by masses over household types \((x,h)\), where \(x \in D\) is income and \(h \in \{1,\ldots,H\}\) is ordinal ability. Define \(F_μ(y,h)\) as the mass with income at most \(y\) and ability at most \(h\), and likewise for ν; decide the Proposition 4.2 comparison using its prefix and integrated-CDF inequalities.
The mirror covers the unanimous dominance comparisons in Propositions 3.1, 3.2, 4.1, and 4.2; it leaves their normative interpretations, transformation characterizations, and the Jenkins–Lambert critique outside the computational question.
I find no qualifying anchor in this paper. Propositions 3.1, 3.2, 4.1 and 4.2 characterize when utilitarian unanimity agrees with particular stochastic-dominance conditions. They are normative characterizations, not named results about a problem being in P, NP-hard, FPT, or otherwise computationally classified. The paper gives no named computational result to mirror.
A plausible population model would assign mass to household types described by income and ability, with a type’s mass representing its share of a large society. But the paper’s questions concern welfare rankings and normative criteria, which the programme places out of scope; it does not formulate a computational task over those masses. Treating its dominance propositions as computational anchors would overstate what they establish, so I cannot give a valid continuous problem per anchor.
The attached case is right that the paper contains no theorem classifying a problem as polynomial-time solvable or hard. But that does not leave it without a computational question. Propositions 3.1, 3.2, 4.1, and 4.2 give exact tests for deciding whether one society is at least as good as another under *every* utilitarian welfare function in a specified class. The paper itself describes these as procedures for deciding whether a situation is preferable. A continuous mirror can ask how to decide those same comparisons when each society is given compactly as rational masses over household income and ability types.
For example, Proposition 3.1 gives the query “does \(\nu\) dominate \(\mu\) for all \(U\in U_{11}\)?” Its answer is determined by whether \(F_\nu(y,h)\le F_\mu(y,h)\) for every income threshold \(y\) and ability level \(h\). Propositions 3.2 and 4.1 give related queries with integrated distribution functions and ability prefixes; Proposition 4.2 combines those tests. These are named results supplying the predicates for a family of computational problems, even though the paper does not analyze their running time.
The population model is credible here: the paper’s welfare function is symmetric across households and invariant under population replication. Many households with the same income and ability are therefore represented exactly by one type and its mass. For finite support of size \(\tau\), the distribution functions and their integrals are piecewise constant or linear between income support points, so the tests reduce to finitely many exact comparisons, with running time polynomial in the number of ability types, \(\tau\), and the input bit length. This is a modest Class A mirror rather than a deep complexity challenge, but the programme explicitly treats tractability as a worthwhile outcome. The case for a mirror is strongest for the robust comparison in Proposition 4.2; the paper’s normative focus makes it less central to computational social choice than the programme’s usual intervention problems, but it does not make the question disappear.
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.