| paper | d4-journals-jet-2020-0107 |
| 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: in \(\mathrm{P}\), with running time \((m+\tau+L)^{O(1)}\) bit operations, hence \(L^{O(1)}\) for the stated explicit encoding. · discrete side: other · adjudication: proved
full proof & adversarial review →
Proposition 5
Given a fixed finite set of welfare-response types \(T\), rational shares \(\mu_t>0\) summing to \(1\), and finite weak and strict comparisons between profiles \(x,y\in\mathbb{Q}^T\), decide whether there exist \(c\in\mathbb{R}\) and \(a_t\in\mathbb{R}\) such that \(U_\mu(x)=c\sum_{t\in T}\mu_t x_t^2+\sum_{t\in T}\mu_t a_t x_t\) respects every comparison, weakly or strictly as specified.
A society consists of fixed welfare-response cohorts \(T\) with rational masses \(\mu_t\); each observed policy gives a type-level welfare profile in \(\mathbb{Q}^T\). The decision variables \(c\) and \(a_t\) define a weighted spherical welfare representation, and the task is to test its feasibility against the observed comparisons.
The paper specifies no fixed cohort partition, so types inferred only from observed profiles may split as additional policy comparisons are added.
fatal: False
The mirror covers Proposition 5’s finite-data rationalizability test for spherical preferences over welfare profiles; it leaves Theorem 1’s full axiomatic characterization, Theorem 4’s welfare-function decomposition, and Theorem 6’s topology result unaddressed.
The strongest honest anchor is Proposition 5, proved in this paper using a duality result cited from elsewhere. The paper does not give a named complexity classification such as “in P” or “NP-hard.” Proposition 5 is nevertheless a computationally relevant result: it gives a finite-data test for rationalizability and a dual way to falsify the model.
In full, let \(R\) and \(P\) be finite collections of weak and strict comparisons between vectors in \(\mathbb R^n\). They are rationalizable by a spherical preference if and only if, for every \(\lambda\in\Delta(R\cup P)\) assigning positive total weight to comparisons in \(P\), at least one of these holds:
\[
\sum_{(x,y)\in R\cup P}\lambda(x,y)\|x\|^2
\ne
\sum_{(x,y)\in R\cup P}\lambda(x,y)\|y\|^2,
\]
or
\[
\sum_{(x,y)\in R\cup P}\lambda(x,y)x
\ne
\sum_{(x,y)\in R\cup P}\lambda(x,y)y.
\]
Equivalently, the comparisons are rationalizable exactly when there exist \(c\in\mathbb R\) and \(u\in\mathbb R^n\) such that \(c\|x\|^2+u\cdot x\) respects every weak comparison and strictly respects every strict one. The proposition’s point is that finite observations can be tested through linear inequalities, and a suitable \(\lambda\) can witness failure.
A continuous mirror I would put to a prover is Continuum Spherical-Welfare Rationalizability. Its instance has a finite set of resident types \(T\), with positive rational masses \(\mu_t\) summing to \(1\), and a finite dataset of weak and strict comparisons between welfare profiles \(x,y\in\mathbb Q^T\). Here a type is a cohort with the same welfare level under each policy in the dataset and the same attributes relevant to the model; \(\mu_t\) is that cohort’s share of the population. The intended regime is a large population—perhaps millions of residents—in a much smaller number \(\tau=|T|\) of welfare-response cohorts.
The question is whether there are \(c\in\mathbb R\) and type-level coefficients \(a_t\in\mathbb R\) such that the functional
\[
U_\mu(x)=c\sum_{t\in T}\mu_t x_t^2+\sum_{t\in T}\mu_t a_t x_t
\]
satisfies all the observed comparisons, weakly where the data say weakly and strictly where they say strictly. A yes-solution is such a pair \((c,a)\); a no-certificate can be a dual witness of the kind in Proposition 5. This keeps the alternatives as policy-induced welfare profiles and makes the population continuous through its type masses. It does not make outcomes or lotteries continuous.
This is a plausible mirror of the paper’s social-choice application: the coordinates of its welfare vectors are individuals, and its spherical preferences aggregate those welfare vectors. The continuous model replaces a long list of repeated coordinates with type shares and uses the corresponding weighted inner product. For a homogeneous finite electorate, the weighted representation recovers the repeated-coordinate one up to a positive rescaling of utility. The authors could reasonably recognize the question as testing whether observed social rankings fit their spherical welfare-aggregation model.
For rational masses and rational profile data, the mirror reduces directly to feasibility of finitely many linear inequalities in \(c\) and the \(a_t\). Thus I expect this particular problem to be tractable, in time polynomial in the number of comparisons, \(\tau\), and the input bit length. That is a conclusion about the proposed mirror, not a complexity result claimed by Proposition 5. It also supplies the continuous counterpart of the proposition’s dual falsification test: a convex mixture of observations that cancels both the weighted squared-norm differences and the profile differences rules out a rationalizer.
The scope is deliberately limited to Proposition 5’s finite-data rationalizability problem. Theorem 1 is the paper’s axiomatic characterization of spherical preferences, but it is not itself a computational result, and I would not claim this mirror covers the theorem’s full characterization or the paper’s topology results. Further questions include whether the same test works for a genuinely nonatomic type distribution under useful input representations, and whether the paper’s full OIOI characterization extends to an infinite-dimensional welfare-profile space.
My weakest point is the type-level restriction: the mirror assumes a meaningful regime in which many residents share the same policy-relevant welfare profile and coefficient class. If welfare responses or relevant attributes are nearly unique to each person, then \(\tau\) approaches the population size and the high-multiplicity motivation fades. That limits the regime, but it does not erase the mirror for cohort-based welfare analysis.
The strongest objection is that Proposition 5 is a finite-data rationalizability criterion, not a complexity result about a society or an intervention. The paper’s main contributions characterize preferences and welfare functions; none asks how to change a population. Proposition 5 does give a computationally relevant test, but it tests whether observed comparisons fit a model. Treating it as an anchor therefore broadens the programme’s target from computational problems over societies to any computational task attached to a social-choice model.
The proposed types also come from the data table: two people count as one type if their welfare coordinates agree across every comparison currently observed. Add a policy comparison and that partition may change. This is a useful compression of the dataset, but it does not establish that the society has a fixed, low-support type structure. For those observed profiles, arbitrary coordinate weights \(u_i\) can be summed within each block, so the weighted type model preserves rationalizability. That makes the proposed mirror faithful to the finite observations, while leaving open whether its types describe the population beyond them.
A stronger mirror could define a type by its entire policy-to-welfare function and give types a fixed mass distribution. That is coherent, and a finite set of such types with rational masses fits the programme’s high-multiplicity model. But it adds a population structure and a policy domain absent from the paper. An atomless version also needs an input representation for the welfare functions and the measure. If those functions reduce to finitely many features, the question returns to finite-data rationalizability; otherwise the computational problem depends on choices the paper never makes.
That is a scope objection, not a decisive defeat. The paper explicitly interprets coordinates as individuals’ welfare, and cohort-level welfare aggregation is plausible. If ChoCo accepts finite-data rationalizability as a computational anchor and fixed welfare cohorts as a natural population regime, Proposition 5 supports a continuous mirror. I cannot honestly argue that no worthwhile mirror exists in any scenario; the strongest negative case is that this mirror is adjacent to the paper’s contribution and requires a new, fixed-type population model.
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.