| paper | d4-journals-jet-2011-0095 |
| 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 theorem asserts that an object can be computed or that computing it is hard; both characterize properties of choice functions. The suggested task of computing a self-stable choice set from a distribution requires selecting a rule and is a new computational problem, so no anchor satisfies bit (a).
fails bit a — no named computational result to mirror
Theorems 2 and 3 provide axiomatic characterizations; the paper's examples and discussion of social choice functions do not supply a computational result to mirror.
This paper has no named computational result to anchor a continuous mirror. Theorem 2 characterizes set-rationalizable choice functions by \(\hat{\alpha}\); Theorem 3 characterizes self-stable choice functions by \(\hat{\alpha}\) and \(\hat{\gamma}\). Both are axiomatic equivalences. Neither gives an algorithm, a complexity classification, or a computational formulation the paper works with.
A distribution over preference rankings could supply a continuous population model for majority-based social choice. One might then ask how to compute a self-stable choice set from that distribution. But that would be a new computational question, not a continuous version of a named computational result in this paper. It cannot serve as the required anchor, so I can’t honestly give the requested per-anchor problems or predict their complexity.
The strongest case against a mirror is that the paper’s results have no computational problem to lift. Theorem 2 characterizes which choice functions are set-rationalizable; Theorem 3 characterizes which are self-stable. Both take a choice function \(S\) as given and state properties of its behavior across feasible sets. Neither takes a voter population as input or asks for an output, an algorithm, or a complexity bound. Replacing a finite electorate with a distribution over rankings does not turn either characterization into a computational result.
The suggested rescue—given a distribution \(\mu\), compute a self-stable choice set—needs extra choices before it becomes a problem. Theorem 3 characterizes a class of functions; it does not select one to compute. One could choose a named example such as the minimal covering set and ask for its output on an agenda. That is a plausible high-multiplicity winner-determination problem, but it is a computational companion to the paper’s example, not a mirror of Theorem 3. Similarly, testing \(b\alpha\) or \(b\gamma\) for a profile-induced rule would require specifying the rule and asking a new verification question; minimizing mass changes to alter its output would add a campaign or robustness objective the paper never studies. These may be worthwhile projects, but the paper does not supply the computational question they would continuize.
The case is strongest against calling any of these a *mirror*, and weaker against calling them worthwhile follow-up work. A reader could reasonably think that computing the output of a self-stable correspondence is close enough to the paper’s subject. But that possibility depends on broadening “mirror” to include computational questions newly attached to an axiomatic result. On the paper’s own results, there is no named computational anchor to continuize.
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.