| paper | d4-journals-geb-2011-0084 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | medium |
| authors would recognise it | yes |
Answer: Every element of \(M\) is a Condorcet winner, and computing the prescribed output \(M\) is polynomial-time solvable in \(m,\tau,L\). · discrete side: other · adjudication: proved
full proof & adversarial review →
Theorem 2
statement extracted from the paper’s text layer
Given a finite ordered agenda \(A\), a finite support \(T\) of complete weak preference orders with singleton peaks, and binary-rational masses \(\mu_t\) summing to \(1\), promised to be peak-monotonic relative to the order on \(A\), compute the weighted-median peak alternatives and return them as Condorcet winners under mass-weighted majority, counting indifferent mass on both sides.
Each type is a complete weak order over \(A\), and \(\mu_t\) is its share of the population. The output is the set of weighted-median peaks; pairwise majority uses the total mass on each side, with indifferent mass counted for both.
Theorem 2’s explicit median construction applies to peak-monotonic profiles, so the mirror should be restricted to that case rather than claiming the full top-monotonic winner problem is anchored by the same construction.
fatal: False
The mirror covers Theorem 2’s peak-monotonic median-winner claim with rational type masses; it leaves Theorem 1’s domain implications, Proposition 1’s necessary condition, and the broader top-monotonic result outside this subcase.
The strongest honest positive case is a plausible follow-up, but it does not meet your required anchor test. A large public-school district or municipality is a convincing high-multiplicity setting: many households vote over tax or service levels, and the paper’s Appendix II already models households with a continuous income distribution. For the majority-winner question, a type can be a complete preference order over a finite agenda; \(\mu_t\) is the share of households with that order, with many households per type.
A concrete follow-up would be Mass-Condorcet under top monotonicity. Given a finite ordered agenda \(A\), a support \(T\) of complete weak orders, and rational masses \(\mu_t\ge 0\) summing to \(1\), promised to be top monotonic, return every alternative \(x\) such that
\[
\sum_{t:\,x\succeq_t y}\mu_t
\;\ge\;
\sum_{t:\,y\succeq_t x}\mu_t
\qquad\text{for every }y\in A.
\]
The answer is the set of Condorcet winners under mass-weighted majority, counting indifferent voters on both sides as the paper does. This is a precise population-continuous version of the paper’s majority-winner question. It should be tractable: pairwise mass sums compute the answer in polynomial time in \(m\), \(\tau\), and the input bit length. It also suggests follow-up questions about recognizing top monotonicity and measuring how winner sets change as masses shift.
The problem is that this is not anchored in a named computational result of the paper. The nearest results are mathematical domain and existence results:
None asserts that a computational problem is in P, hard, or belongs to another complexity class. The paper’s remarks that some cases are easy to check do not supply a formal complexity result. Under your anchor rule, there are therefore no qualifying anchors.
That is the weak point, and it is substantial: the paper studies domain restrictions and majority-rule existence, not computational problems. Its closest computational mirror—finding the winners from rational type masses—is an easy pairwise calculation, while rational masses can be converted exactly to a finite electorate by taking a common denominator. So there is a sensible continuous-population follow-up, but I would not present it as an anchored positive case for this programme.
The strongest negative case is that this paper supplies no computational result to continuize. Theorem 1 relates preference domains; Theorem 2 proves existence and locates Condorcet winners; Proposition 1 gives a necessary condition for top monotonicity. None studies the complexity of a computational problem. That is a target problem, not a flaw in the proposed mass model: a follow-up could be well posed, but it would not be a continuous mirror of a named computational result here.
Theorem 1 does not gain a computational question when voters become mass. Top monotonicity depends on the preference profile and its order, not on how many agents hold each preference. With finitely many alternatives, a distribution over complete preference types has finite support, and changing the masses while preserving that support does not change whether the condition holds. Theorem 2 has a more plausible mirror: compute Condorcet winners from rational type masses. But for each pair \(x,y\), one sums the masses preferring each alternative; doing so for all pairs takes polynomial time in \(m\), \(\tau\), and the input bit length. Weighted median tops are likewise weighted order statistics. That is a sensible mass version of the existence question, but it offers no apparent complexity problem of the kind the programme is meant to map. Proposition 1 fares no better: with an order specified, checking its condition is a finite comparison task; searching for an order would be a separate recognition problem, and population mass does not create that problem.
The best richer version would use the paper’s economic setting: households drawn from a continuous income distribution, with preferences over tax or public-service levels induced by their utilities. One could ask for the mass of households preferring each alternative and compute the Condorcet set from those quantities. But the computation would then depend on how the density and utility functions are represented and on whether the required integrals can be evaluated. If those primitives are simple, the winner calculation is straightforward; if they are hard to process, the difficulty lies in the chosen function representation. The paper already uses continuous income distributions to discuss majority-rule existence analytically, but gives no computational result for this setting either.
So none of the three proposed anchors survives as an anchor for ChoCo’s complexity programme, even under a better population model. The honest limit to this case is that one could invent a new problem—bribery or robustness, for example—on a top-monotonic society. But that would be a new computational question attached to the paper’s model, not a continuous mirror of what the paper proves.
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.