| paper | d4-journals-mss-2021-0018 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | yes |
Theorem 1 characterizes rationalizability by GARP, but states no algorithm, running-time bound, hardness result, or formulation used by the paper to compute. The proposed cohort question is concrete and could be a plausible application, but the analyst’s inferred finite-support test cannot supply the missing computational result.
fails bit a — no named computational result to mirror
The proposed mirror uses Theorem 1’s GARP rationalizability criterion; it leaves the game and consumption applications and the other corollaries outside.
The strongest honest positive case is Theorem 1, proved in this paper. It says that for a space \(X\) with a separable partial order, a data set \((\mathcal B,C)\) is rationalizable by a monotone utility function if and only if it satisfies GARP. In full, GARP requires that for every finite sequence of observed choices \(x_t\in C(B_t)\), if \(x_{t+1}\in B_t^\ge\) for each \(t<n\), then \(x_1\notin B_n^>\), where \(B^\ge=\{y:\exists z\in B,\ z\ge y\}\) and \(B^>=\{y:\exists z\in B,\ z>y\}\). The utility must satisfy \(U(x)\ge U(y)\) for every observed choice \(x\in C(B)\) and every \(y\in B\), while respecting the order. The proof is given here in Appendix A.3.
My lead mirror is Population GARP Rationalizability. Take \(X=\mathbb R_+^d\) with its coordinatewise order. An instance consists of \(\tau\) records \((B_r,x_r)\), each with a positive binary rational mass \(\mu_r\), where \(\sum_r\mu_r=1\), each \(B_r\) is a finite, explicitly listed menu of rational bundles, and \(x_r\in B_r\). Records with the same menu define the observed choice correspondence \(C_\mu(B)=\{x_r:B_r=B\}\). The question is whether there is a single monotone \(U:X\to\mathbb R\) such that \(U(x_r)\ge U(y)\) for every record \(r\) and every \(y\in B_r\). The solution is a yes/no answer: YES exactly when such a utility exists; a NO answer has a finite GARP-violating sequence as a witness. This is a feasibility question over the unknown utility, so it has no separate optimization objective.
The regime is a large, tightly defined consumer cohort: many households share the same taste class, while facing a small number of standardized budget or menu situations. Each type is a budget-and-observed-choice situation in that common class; its mass is the fraction of households in it. The intended case is \(N\gg\tau\), with \(\mu_r=n_r/N\) in a finite population representation. This keeps the paper’s common-utility hypothesis intact while making population mass, rather than one household at a time, the natural way to record prevalence.
The authors could recognize this as their rationalizability problem applied to a high-multiplicity cohort: the same menus, choices, order, common utility, and GARP condition remain in force. The society is the continuous object; the menu alternatives need not be made fractional. The mirror covers Theorem 1’s rationalizability result. It does not extend the paper’s applications to games or consumption streams.
For the finite encoding above, I expect this mirror to be Class A, tractable. A directed graph on the \(\tau\) records can encode the weak revealed-preference steps, and a GARP violation exists when a reachable pair also satisfies the strict terminal condition. Reachability and those finite-menu comparisons can be checked in time polynomial in \(\tau\), the menu-description length \(L\), and \(d\); masses matter for identifying positive support. This complexity conclusion is my inference from the theorem and the finite encoding. The paper itself states no named P or hardness result, and Theorem 1 is a characterization rather than an explicit complexity bound. If the anchor requirement is read to demand a theorem asserting a named complexity class, the paper supplies no qualifying anchor.
Natural follow-on questions include finding the least population mass of records that must be removed or corrected to restore GARP, and asking whether a rationalizing utility can also be required to be continuous or concave.
The weak point is that the mirror relies on a stringent shared-tastes assumption, and the mass in this decision problem affects which records are present but not how GARP weighs them. So the population interpretation is plausible for a homogeneous cohort with repeated standardized choice situations, but the tractability gain is mainly compression from \(N\) agents to \(\tau\) distinct records. It is a credible high-multiplicity mirror of the paper’s test, though a modest computational contribution by ChoCo’s standards.
Theorem 1 is a revealed-preference characterization, not a computational result. It says when a data set is rationalizable; it gives no complexity classification or algorithmic theorem for a population problem. Population GARP Rationalizability is a possible computational wrapper around that characterization, but it is not a complexity result this paper establishes. The proposed finite encoding and decision problem are supplied by the analyst.
The strongest version of that wrapper is a cohort with a common utility and different observed budget-choice records. That is a plausible economic question, but its population interpretation is fragile. The records in the paper are observations linked across situations to test the choices of one agent. Treating each record as a different household discards that link; treating each household’s full history as its type preserves it, but then rationalizability is assessed per history. Pooling records under a single utility instead adds a common-preferences assumption across households. The paper’s theorem can be applied to that pooled data, but the population-level restriction is an extra modeling choice.
Even granting the pooled version, the masses do no work in the rationalizability condition. For a finite support of records, the test depends on which records have positive mass, not their proportions. Reweighting the cohort while keeping its support fixed leaves the input to GARP unchanged. That is not an objection that the test might be easy; it is a reason the population’s continuous distribution is not the object being studied. The finite mirror is a support-based test of pooled observations.
The suggested repair question—minimize the mass of records to remove or correct so the remainder is rationalizable—does make mass relevant. But it adds an outlier or intervention objective absent from the paper. The theorem neither defines which households may be excluded nor gives a model of correction costs. That could be a separate project; it does not rescue a continuous mirror of Theorem 1. Corollary 1 keeps the focus on a specified player’s best responses in games, and Corollary 2 on one agent’s choices over time. Neither supplies a population-computational result.
The negative case is weakest on whether authors might recognize a common-utility cohort as an application of their theorem: that is quite plausible. But the anchor does not establish a worthwhile continuization of a computational result. Its most faithful population reading adds prevalence to observations while the theorem’s question remains mass-blind; the versions that make mass consequential change the problem.
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.