| paper | d4-conf-ijcai-2011-0484 |
| 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: Yes: mass-weighted CSS always certifies a necessary co-winner when \(\mathrm{MMR}=0\), and otherwise returns a positive-mass type and an incomparable pair. Computing the complete CSS step is in \(\mathrm{P}\), polynomial in \(m,\tau,L\). · discrete side: other · adjudication: proved
full proof & adversarial review →
Proposition 3
statement extracted from the paper’s text layer
Given alternatives \(A\), positive rational masses \(\mu_t\) on partial-ranking types \(P_t\), and the assumption that each type shares one hidden complete ranking, can a mass-weighted CSS procedure certify \(\mathrm{MMR}=0\), or, whenever \(\mathrm{MMR}>0\), return a positive-mass type \(t\) and a pair incomparable in \(P_t\) for its next comparison query?
The input is a support of partial rankings \(P_t\) with binary-rational masses \(\mu_t\) summing to one; each type has one hidden completion \(v_t\in C(P_t)\) shared by its mass. For a completion vector \(\mathbf v\), let \(S_{\mathbf v}(c)=\sum_t\mu_t s_B(c,v_t)\) and \(\mathrm{MMR}=\min_{c\in A}\max_{\mathbf v\in\prod_t C(P_t)}\max_{c'\in A}[S_{\mathbf v}(c')-S_{\mathbf v}(c)]\). A query selects \(t\) and an incomparable pair in \(P_t\); its answer refines the partial ranking for that type’s mass.
The mirror covers Proposition 3’s next-query guarantee for mass-weighted Borda elicitation. It leaves the paper’s other rule-specific regret results, experiments, and full query-sequence optimization outside the anchor.
The paper has no numbered Theorem, Lemma, Corollary, or Proposition that classifies minimax-regret computation as polynomial-time or hard. Its strongest named computational anchor for a positive case is Proposition 3, proved here: unless \( \mathrm{MMR}=0 \), CSS selects a voter \(k\) and a comparison query \(a_i\succ_k a_j\) such that \(a_i\) and \(a_j\) are incomparable in that voter’s current partial ranking. This guarantees a valid next query while regret remains positive; it does not bound the number of queries or prove a complexity class. If an anchor must literally assert a complexity classification, then the supplied text has no qualifying anchor.
The most credible continuous mirror of Proposition 3 is a type-compressed CSS step for Borda. An instance has alternatives \(A\), a list of \(\tau\) positive-mass partial-ranking types \(P_t\), and binary-rational masses \(\mu_t\) summing to one. Members of each type share the same latent complete ranking, which extends \(P_t\), so a query to one representative reveals that comparison for the whole cohort. For a completion \(v=(v_t)_t\), define \(S_v(c)=\sum_t\mu_t s_B(c,v_t)\), and define regret and \( \mathrm{MMR} \) by maximizing over completions and then minimizing over the proposed winner.
The question is: given this state, can a CSS-style procedure either certify \( \mathrm{MMR}=0 \), or return a positive-mass type \(t\) and a pair incomparable in \(P_t\) for its next comparison query? A solution is that certificate or query; the response refines \(P_t\) for its full mass. A fitting regime is a large consultation or election with standardized, coordinated ballot blocs: the number of participants \(n\) is much larger than the number of distinct ranking types \(\tau\). The population remains the object being continued; regret is measured over mass-weighted Borda scores, and uncertainty still comes from the possible completions of partial rankings.
I expect this step-selection problem to be tractable in \(m,\tau,L\). Section 4 describes polynomial Borda regret calculations by decomposing regret across partial votes; for identical members of a type, their contribution can be weighted by \(\mu_t\), and the query choices can be checked over the finite support. That is a plausible high-multiplicity extension, though the polynomial-time claim is not itself a numbered theorem in the supplied paper. The mirror covers CSS’s next-query guarantee, not the paper’s other rule-specific calculations, its cited complexity results, or the full optimal query-sequence problem. Further questions include minimizing total query cost to reach a regret threshold and what remains true when members of a cohort can give different answers.
The soft spot is the type assumption. It is convincing only where a cohort really shares one latent ranking and a representative can answer for it. Also, Proposition 3 is a progress guarantee rather than a complexity classification, so this is a modest anchor: the best positive case in the text, not a claim that the paper already proves a continuous algorithm.
The proponent’s anchor is weaker than its framing suggests. Proposition 3 is a numbered computational statement, but it is not a complexity result: it says that, while regret is positive, CSS can find an unresolved comparison to ask. It gives no bound on the value of that query, the number of queries, or the cost of reaching a regret threshold. The paper does discuss polynomial-time regret algorithms in Section 4, but those results are not stated as numbered complexity theorems.
There is also a mismatch between the continuous model and what an elicitation query learns. In ChoCo’s model, a type is a complete ranking. If the distribution over those types is known, the society’s scores are known and its minimax regret is zero. If the input instead groups voters by their *partial* rankings, members of a group may have different hidden completions; asking one person does not reveal the answer for the group’s mass. Letting one answer update the entire group requires an extra assumption: the group shares one latent ranking, or its representative speaks for it. That may fit a coordinated ballot bloc, but it is not a general high-multiplicity interpretation of the paper’s elicitation setting.
The stronger weighted-regret mirror is harder to dismiss. For societies of partial-ranking masses, worst-case regret can be defined over completions, and the paper’s additive score calculations appear to extend by weighting each type’s contribution. That is a plausible continuous computational question, and its bloc version has a sensible scenario. Calling the extension straightforward would not defeat it: an easy answer can still be a worthwhile mirror under the programme’s standard.
So the negative case can challenge Proposition 3 as a substantial complexity anchor and expose the assumption needed to make type-level queries meaningful. It cannot honestly establish that no worthwhile mirror exists in any scenario. Coordinated blocs remain a plausible survivor, and I would not argue for a red verdict on the universal claim.
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.